# Completed Research and Evaluation

A large body of literature exists that focuses on or is related to the *Connected Mathematics Project*. Here, you will find articles on CMP that we have compiled over the past thirty years. These include research, evaluation and descriptions from books, book chapters, dissertations, research articles, reports, conference proceedings, and essays. Some of the topics are:

- student learning in CMP classrooms
- teacher's knowledge in CMP classrooms
- CMP classrooms as research sites
- implementation strategies of CMP
- longitudinal effects of CMP in high school math classes
- students algebraic understanding
- student proportional reasoning
- student achievement
- student conceptual and procedural reasoning and understanding
- professional development and teacher collaboration
- comparative studies on different aspects of mathematics curricula
- the CMP philosophy and design, development, field testing and evaluation process for CMP

This list is based on thorough reviews of the literature and updated periodically. Many of these readings are available online or through your local library system. A good start is to paste the title of the publication into your search engine. Please contact us if you have a suggestion for a reading that is not on the list, or if you need assistance locating a reading.

Adams, L. M., Tung, K. K., Warfield, V. M., Knaub, K., Mudavanhu, B., & Yong, D. (2002). *Middle school mathematics comparisons for Singapore Mathematics, Connected Mathematics Program, and Mathematics in Context*. Report submitted to the National Science Foundation by the Department of Applied Mathematics, University of Washington.

Adams, R. L. (2005).** ***Standards-based accountability: Improving achievement for all students through standards based mathematics instruction*. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 66(6). (ProQuest ID No. 932378841)

ABSTRACT: The purpose of this study was to conduct evaluation research on the professional development intervention implemented to address the effectiveness of standards-based instruction in improving the mathematic achievement of all student subgroups in Yolo County schools. The question addressed in this study was "Does standards-based instruction in mathematics, coupled with professional development on the standards-based content of California State Board of Education-approved text books, lead to increases in student achievement and high school graduation rates for all subgroups in Yolo County schools?"

The Yolo County Office of Education (YCOE) university partnership designed the professional development intervention for teachers delivering math at grade levels 5th through algebra I. Twelve teachers (treatment group) participated in 40-hour institutes; follow-up sessions, and data gathering to measure the effectiveness of the training and support. Ten teachers (control group) recruited as 2005 institute participants simultaneously gathered like data.

Twelve schools participated in the study. The teacher index ranges from 0.00 teachers trained in standards-based mathematics instruction to 0.50 with a mean of 0.19 indicating that the schools hadn't implemented school-wide professional development.

There was a significant difference between the treatment group scores on the post-survey and the control group scores (p = .011) (effect size > 1.0). The treatment group results indicate that the treatment group's beliefs on standards based instruction shifted significantly into the high-reform range after the intervention.

Curriculum calibration indicates that the use of the textbook as the main teaching resource did not ensure that the instruction was on grade-level over 75% of the time. The control group used the textbook as the main teaching resource 30% of the time compared to 55% by the treatment group, yet taught on grade-level more often then the treatment group.

The implications of this program evaluation point to continued organizational improvement through reducing gaps in: content knowledge, motivation, and organization support. Based on the research cited, and the practical implications from the intervention piloted in the Yolo County schools, the county partnership must continue to build systems of support that embrace standards-based mathematic instruction.

Aisling, L. M., Friel, S. N., & Mamer, J. D. (2009). It’s a fird!: Can you compute a median of categorical data? *Mathematics Teaching in the Middle School*, 14(6), 344-351.

Description: Students need time and experience to develop essential understandings when they explore data analysis. In this article, the reader gains insight into confusion that may result as students think about summarizing information about a categorical data set that is attempting to use, in particular, the median. The authors highlight points to consider in helping students unpack these essential understandings.

Alibali, M. W., Stephens, A. C., Brown, A. N., Yvonne, S., & Nathan, M. J. (2014). M*iddle school students’ conceptual understanding of equations: Evidence from writing story problems. *International Journal of Educational Psychology, 3(3), 235–264. doi:10.4471/ijep.2014.13

ABSTRACT: This study investigated middle school students’ conceptual understanding of algebraic equations. 257 sixth- and seventh-grade students solved algebraic equations and generated story problems to correspond with given equations. Aspects of the equations’ structures, including number of operations and position of the unknown, influenced students’ performance on both tasks. On the story-writing task, students’ performance on two-operator equations was poorer than would be expected on the basis of their performance on one-operator equations. Students made a wide variety of errors on the story-writing task, including (1) generating story contexts that reflect operations different from the operations in the given equations, (2) failing to provide a story context for some element of the given equations, (3) failing to include mathematical content from the given equations in their stories, and (4) including mathematical content in their stories that was not present in the given equations. The nature of students’ story-writing errors suggests two main gaps in students’ conceptual understanding. First, students lacked a robust understanding of the connection between the operation of multiplication and its symbolic representation. Second, students demonstrated difficulty combining multiple mathematical operations into coherent stories. The findings highlight the importance of fostering connections between symbols and their referents.

Read Middle School Students Conceptual Understanding of Equations

American Association for the Advancement of Science: Project 2061 (2000). *Middle grades mathematics textbooks: A benchmarks-based evaluation.* Evaluation report prepared by the American Association for the Advancement of Science.

Anderson, N. C. (2008). Walk the line: Making sense of y = mx + b. In C. E. Greenes & R. Rubenstein (Eds.), *Algebra and algebraic thinking in school mathematics, 70th yearbook* (pp. 233-246). Reston, VA: National Council of Teachers of Mathematics.

Anderson, V. J. (2010). Connected Mathematics Project, 2nd edition, implementation in Seattle: The experience of teachers and principals. Dissertation Abstracts International Section A: Humanities and Social Sciences, 71 (2-A), 432.

Arbaugh, F., Lannin, J., Jones, D. L., & Park Rogers, M. (2006). Examining instructional practices in Core-Plus lessons: Implications for professional development. *Journal of Mathematics Teacher Education*, 9(6), 517-550.

ABSTRACT: In the research reported in this article, we sought to understand the instructional practices of 26 secondary teachers from one district who use a problems-based mathematics textbook series (Core-Plus). Further, we wanted to examine beliefs that may be associated with their instructional practices. After analyzing data from classroom observations, our findings indicated that the teachers’ instructional practices fell along a wide continuum of lesson implementation. Analysis of interview data suggested that teachers’ beliefs with regard to students’ ability to do mathematics were associated with their level of lesson implementation. Teachers also differed, by level of instructional practices, in their beliefs about appropriateness of the textbook series for all students. Results strongly support the need for professional development for teachers implementing a problems-based, reform mathematics curriculum. Further, findings indicate that the professional development be designed to meet the diverse nature of teacher needs.

Artzt, A. F., & Curcio, F. R. with Sultan, A. & Wachter, T. (2003). Rethinking secondary mathematics teacher preparation. In D. Kaufman, D. M. Moss, & T. A. Osborn (Eds.), *Beyond the boundaries: A transdisciplinary approach to learning and teaching* (pp. 69-80). Westport, CT: Greenwood Publishing Group.

Asquith, P., Stephens, A.C., Knuth, E.J., Alibali, M.W. (2005). Middle school mathematics teachers' knowledge of students' understanding of core algebraic concepts: Equal sign and variable. *Mathematical Thinking and Learning,* 9(3), 249-272.

ABSTRACT: This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.

Ball, D. L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. *Phi Delta Kappan*, 77(7), 500-508.

ABSTRACT: In order to improve mathematics education, a close examination of assumptions about teacher learning and the teaching of mathematics must be made. Teachers and others participating in the reform process will have to learn many new ideas and unlearn many previous assumptions.

Banilower, E. R. (2010). *Connected Mathematics, 2nd Edition: A three-year study of student outcomes.* Chapel Hill, NC: Horizon Research, Inc.

Banilower, E. R., Smith, P. S., Weiss, I. R., Malzahn, K. A., Campbell, K. M., & Weis, A. M. (2013). *Report of the 2012 National Survey of Science and Mathematics Education.* Chapel Hill, NC: Horizon Research, Inc.

Bay, J. M. (1999). *Middle school mathematics curriculum implementation: The dynamics of change as teachers introduce and use standards-based curricula.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 60(12). (ProQuest ID No. 730586091)

ABSTRACT: Two case studies of school districts were developed to study the district-level constraints and considerations during adoption of standards-based middle school mathematics curricula. In addition, the nature of implementation within classrooms was described through six teacher case studies. The two school districts were in their third year of full implementation of a curricula, with one school district implementing the Connected Mathematics Project and the other MATH Thematics. Data collected included interviews, surveys, and classroom observations. Factors influencing teacher decision-making and district-level decision-making were analyzed.

Several themes emerged related to the district-level issues of implementation. First, teacher leadership and/or participation in the professional development and district decision-making throughout the implementation had an impact on the nature of the teachers' perceptions of the need for change. Those who were involved in professional development or provided leadership in the district had a stronger commitment to the implementation. Teacher turnover constrained the level of implementation in the classroom and the level of interaction among teachers. Perceptions of parents, expectations for students, and state/national assessments were important considerations as districts selected and implemented their curriculum.

Successful implementation of standards-based curriculum in the classroom appeared to be related to several factors. First, the extent to which teachers were involved in the process of implementation, including choosing the curriculum and participating in professional development, influenced the degree to which their classrooms were aligned with recommendations from the curricula and the NCTM Standards (1989, 1991, 1995). Collaborative relationships that were developed during the selection and first year of implementation continued to function productively in the third year of implementation, which happened to be the first year the districts were not participating in any externally-sponsored professional development. All teachers were concerned with the level of skill development that students needed beyond what was provided in the curriculum and made adjustments accordingly.

Bay, J. M., Beem, J. K., Reys, R. E., Papick, I., & Barnes, D. E. (1999). Student reactions to standards-based math-ematics curricula: The interplay between curriculum, teachers, and students. *School Science and Mathematics, 99*(4), 182–188.

ABSTRACT: As standards-based mathematics curricula are used to guide learning, it is important to capture not just data on achievement but data on the way in which students respond to and interact in a standards-based instructional setting. In this study, sixth and seventh graders reacted through letters to using one of two standards-based curriculum projects ("Connected Mathematics Project or Six Through Eight Mathematics. Letters were analyzed by class, by teacher, and by curriculum project. Findings suggest that across classrooms students were positive toward applications, hands-on activities, and working collaboratively. The level of students’ enthusiasm for the new curricula varied much from class to class, further documenting the critical role teachers play in influencing students’ perceptions of their mathematics learning experiences. The results illustrate that, while these curricula contain rich materials and hold much promise, especially in terms of their activities and applications, their success with students is dependent on the teacher.

Bay, J. M., Reys, B. J., & Reys, R. E. (1999). The top 10 elements that must be in place to implement standards-based mathematics curricula. *Phi Delta Kappan, 80*(7), 503 506.

ABSTRACT: Teachers' work with four National Science Foundation-funded curricula in the Missouri Middle-School Mathematics Project has disclosed 10 critical implementation elements: administrative support, opportunities for study, curriculum sampling, daily planning, interaction with experts, collaboration with colleagues, incorporation of new assessments, student adjustment time, and planning for transition.

Bay-Williams, J., Scott, M., & Hancock, M. (2007). Case of the mathematics team: Implementing a team model for simultaneous renewal. *Journal of Educational Research, 100*(4), 243-253.

ABSTRACT: Simultaneous renewal in teacher education is based on the notion that improvement at 1 level requires improvement at all levels and that all stakeholders are responsible for such improvement. The authors discuss the creation and impact of a mathematics team as a vehicle for simultaneous renewal by using the team model for simultaneous renewal for improved teacher-education courses, student achievement in an elementary school, and curriculum changes in K-16 mathematics. Participation in the mathematics team created awareness and respect for the teachers, mathematicians, and mathematics educators.

Beaudrie, B. P., & Boschmans, B. (2013). Transformations and handheld technology. *Mathematics Teaching in the Middle School*, 18(7), 444-450.

Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (1997a). *Development of Proportional Reasoning in a Problem-Based Middle School Curriculum.* Paper presented at the Annual Meeting of the American Educational Research Association. Chicago, IL.

ABSTRACT: Contemporary constructivist views of mathematical learning have encouraged curriculum developers to devise instructional materials that help students build their own understanding and procedures for doing rational number computations, solving proportions, and applying those skills to real and whimsical problems. The Connected Mathematics Project (CMP) curriculum supports construction of rational number knowledge by presenting students with a series of units based on contextual problems that require proportional reasoning and computation. The goal of this study was to describe the character and effectiveness of proportional reasoning by students with different curricular experiences as they face problems in which ratio and proportion ideas are appropriate and useful. Performance task papers and follow-up interviews with selected students from the study indicated that, in addition to a greater frequency of correct answers and reasoning compared with control group students, CMP students appeared to have developed greater ability to articulate their thinking. Students from CMP classes had a generally broader and more flexible repertoire of strategies available for problem solving. The results suggest that problem-based curriculum and instruction can be effective in helping students construct effective personal understanding and skill in one of the core strands of middle grade mathematics.

Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (1997b). *A study of proportional reasoning among seventh and eighth grade students.* Paper presented at the annual meeting of the American Educational Research Association. Chicago, IL

Ben-Chaim, D., Fey, J., Fitzgerald, W., Benedetto, C., & Miller, J. (1998). Proportional reasoning among 7th grade students with different curricular experiences. *Educational Studies in Mathematics, 36*(3), 247-273.

ABSTRACT: Contextual problems involving rational numbers and proportional reasoning were presented to seventh grade students with different curricular experiences. There is strong evidence that students in reform curricula, who are encouraged to construct their own conceptual and procedural knowledge of proportionality through collaborative problem-solving activities, perform better than students with more traditional, teacher-directed instructional experiences. Seventh grade students, especially those who study the new curricula, are capable of developing their own repertoire of sense-making tools to help them to produce creative solutions and explanations. This is demonstrated through analysis of solution strategies applied by students to a variety of rate problems.

Ben-Chaim, D., Keret, Y., & Ilany, B-S. (2012). *Ratio and proportion: Research and teaching in mathematics teachers’ education (pre- and in-service mathematics teachers of elementary and middle school classes)*. Rotterdam, The Netherlands: Sense Publishers.

Ben-Haim (1983). *Spatial Visualization: Sex Differences, Grade Level Differences and the Effect of Instruction on the Performance and Attitudes of Middle School Boys and Girls. *(Doctoral dissertation).

Bennett, C. L. (2007). *A curriculum project of vocabulary development in the Connected Math program Moving Straight Ahead. *Unpublished master’s thesis. State University of New York College at Brockport.

Ben-Zvi, D. (2004). Reasoning about data analysis. In D. Ben-Zvi & J. B. Garfield (Eds.), *The challenge of developing statistical reasoning, literacy and thinking *(pp. 121-146). Dordrecht, Netherlands: Kluwer.

Bieda, K. (2010a). Enacting proof in middle school mathematics: Challenges and opportunities. *Journal for Research in Mathematics Education, 41*(4), 351-382.

ABSTRACT: Discussions about school mathematics often address the importance of reasoning and proving for building students’ understanding of mathematics. However, there is little research examining how teachers enact tasks designed to engage students in justifying and proving in the classroom. This article presents results of a study investigating the processes and outcomes of implementing proof-related tasks in the classroom. Data collection consisted of observations of 7 middle school classrooms during implementation of proof-related tasks-tasks providing opportunities for students to produce generalizations, conjectures, or proofs-in the Connected Mathematics Project (CMP) curriculum by teachers experienced in using the materials. The findings suggest that students’ experiences with such tasks are insufficient for developing an understanding of what constitutes valid mathematical justification.

Bieda, K. (2010b). Stuck in the concrete: Students’ use of manipulatives when generating proof. *Mathematics Teaching in the Middle School, 15(*9), 540-546.

Bieda, K. N., Ji, X., Drwencke, J., & Picard, A. (2014). Reasoning-and-proving opportunities in elementary mathematics textbooks. *International Journal of Educational Research*, 64, 71–80. doi:10.1016/j.ijer.2013.06.005

ABSTRACT: Over the past two decades, standards documents have emphasized the importance of developing students’ abilities to generate and critique mathematical arguments across all grade levels. However, little is known about the opportunities elementary textbooks provide for students to learn mathematical argumentation. We analyzed seven upper elementary (ages 9–11) mathematics textbooks published in the U.S., focusing specifically on reasoning-and-proving opportunities in written tasks, and found that the average percentage of such tasks was 3.7%. Further, analyses of the task purpose and type of justification warranted revealed distinctions between the text materials in terms of the kinds of reasoning-and-proving activities prompted and the placement of tasks in the lesson sections. Specifically, textbooks developed based on research and written to align with curriculum and instruction standards were more likely to have reasoning-and-proving tasks within the narrative and student exercise sections than other texts. We discuss implications for the opportunities to learn reasoning-and-proving in elementary classrooms.

Bieda, Kristen N., Bowers, David, & Kuchle, Valentin A.B. (2019). The Genre(s) of Argumentation in School Mathematics. *Michigan Reading Journal. *(41)

Bledsoe, A. M. (2002). *Implementing the Connected Mathematics Project: The interaction between student rational number understanding and classroom mathematical practices. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 63(12). (ProQuest ID No. 765115471)

ABSTRACT: The Research Advisory Council (RAC, 1991) of the National Council of Teachers of Mathematics (NCTM) called for research on the effects of Standards -based (NCTM, 1989, 1991, 2000) curricula. Following a qualitative design, this dissertation study provides insight into what it means to know and do mathematics in one seventh-grade classroom in which one such curriculum was implemented. More specifically, this study provides a thick description of the teaching and learning of rational number concepts in a classroom where the Bits and Pieces I unit (Lappan, Fey, Fitzgerald, Friel, & Phillips, 1997) from the Connected Mathematics Project (CMP) was used.

Through the lens of the Emergent Perspective (Cobb & Yackel, 1996), this study investigates the relationship between students' initial and developing understandings and the evolving classroom mathematical practices. Results indicate that students' rational number understandings and the teacher's proactive role contributed to the establishment of the classroom mathematical practices. These mathematical practices serve to document the development of the collective understandings as the students engaged in activities from Bits and Pieces I (Lappan et al., 1997). Findings suggest that students did make significant growth in their rational number understandings as a consequence of engaging in these activities and participating in these mathematical practices. In particular, results indicate that participation in conceptually-based mathematical practices provided greater opportunities for students' to advance in their rational number understandings than participation in those that were procedurally-based. In fact, participation in procedurally-based mathematical practices actually constrained some students' advance in their rational number understandings.

Booth, J. L., & Koedinger, K. R. (2012). Are diagrams always helpful tools? Developmental and individual differences in the effect of presentation format on student problem solving. *British Journal of Educational Psychology, 82*(3), 492–511.

ABSTRACT: Background. High school and college students demonstrate a verbal, or textual,advantage whereby beginning algebra problems in story format are easier to solve than matched equations (Koedinger & Nathan, 2004). Adding diagrams to the stories may further facilitate solution (Hembree, 1992; Koedinger & Terao, 2002). However, diagrams may not be universally beneficial (Ainsworth, 2006; Larkin & Simon, 1987).

Aims. To identify developmental and individual differences in the use of diagrams, story, and equation representations in problem solving. When do diagrams begin to aid problem-solving performance? Does the verbal advantage replicate for younger students?

Sample. Three hundred and seventy-three students (121 sixth, 117 seventh, 135 eighth grade) from an ethnically diverse middle school in the American Midwest participated in Experiment 1. In Experiment 2, 84 sixth graders who had participated in Experiment 1 were followed up in seventh and eighth grades.

Method. In both experiments, students solved algebra problems in three matched presentation formats (equation, story, story + diagram).

Results. The textual advantage was replicated for all groups. While diagrams enhance performance of older and higher ability students, younger and lower-ability students do not benefit, and may even be hindered by a diagram’s presence.

Conclusions. The textual advantage is in place by sixth grade. Diagrams are not inherently helpful aids to student understanding and should be used cautiously in the middle school years, as students are developing competency for diagram comprehension during this time.

Boston, M. D., & Wilhelm, A. G. (2015). Middle school mathematics instruction in instructionally focused urban districts. *Urban Education*, 1-33.

ABSTRACT: Direct assessments of instructional practice (e.g., classroom observations) are necessary to identify and eliminate opportunity gaps in students’ learning of mathematics. This study examined 114 middle school mathematics classrooms in four instructionally focused urban districts. Results from the Instructional Quality Assessment identified high percentages of lessons featuring cognitively challenging tasks, but declines in cognitive challenge during implementation and discussions. Overall instructional quality exceeded results from studies with nationally representative samples and paralleled results of studies of instructionally focused urban middle schools. Significant differences existed between districts, favoring the district with veteran teachers, long-term use of Standards-based curricula, and professional development initiatives.

Bouck, E. C., & Kulkarni, G. (2009). *Middle-School Mathematics Curricula and Students with Learning Disabilities: Is One Curriculum Better?* Learning Disability Quarterly, 32(4), 228-244.

ABSTRACT: Little is known about how best to teach mathematics to students with learning disabilities. This study explored the performance and self-reported calculator use of 13 sixth-grade and 15 seventh-grade students with learning disabilities educated in either standards-based or traditional mathematics curricula on multiple-choice and open-ended assessments. Across both groups of students: (a) curriculum did not impact the number of problems students answered correctly, (b) students answered more problems correctly on the multiple-choice than on the open ended assessments, (c) students self-reported low percentages of calculator use, and (d) curriculum did not impact students' self-reported calculator use. Overall, the results suggest that students with learning disabilities are not advantaged or disadvantaged by receiving either a traditional or a standards-based mathematics curriculum.

Bouck, E. C., Joshi, G. S., & Johnson, L. (2013). Examining calculator use among students with and without disabilities educated with different mathematical curricula. Educational Studies in Mathematics, 83(3), 369-385.

ABSTRACT: This study assessed if students with and without disabilities used calculators (four function, scientific, or graphing) to solve mathematics assessment problems and whether using calculators improved their performance. Participants were sixth and seventh-grade students educated with either National Science Foundation (NSF)-funded or traditional mathematics curriculum materials. Students solved multiple choice and open-ended problems based on items from the State’s released previous assessments. A linear mixed model was conducted for each grade to analyze the factors impacting students’ self-reported calculator use. Chi Square tests were also performed on both grade’s data to determine the relationship between using a calculator and correctly solving problems. Results suggested only time as a main factor impacting calculator use and students who self-reported using a calculator were more likely to answer questions correctly. The results have implications for practice given the controversy over calculator use by students both with and without disabilities.

Bouck, E. C., Kulkarni, G., & Johnson, L. (2011). *Mathematical performance of students with disabilities in middle school: Standards-based and traditional curricula. *Remedial and Special Education, 32(5), 429–443.

ABSTRACT: This study investigated the impact of mathematics curriculum (standards based vs. traditional) on the performance of sixth and seventh grade students with disabilities on multiple-choice and open-ended assessments aligned to one state’s number and operations and algebra standards. It also sought to understand factors affecting student performance on assessments: ability status (students with and without disabilities), curriculum (standards based vs. traditional), and assessment type (multiple choice vs. open ended). In all, 146 sixth grade students and 149 seventh grade students participated in the study. A linear mixed model for each grade revealed students with disabilities did not perform better in either curriculum. Furthermore, curriculum type was not a significant factor affecting student performance; however, ability status, time, and assessment type were. The implications of these results are discussed.

Bouck, M., Keusch, T., & Fitzgerald, W. (1996). Developing as a teacher of mathematics. *The Mathematics Teacher, 89*(9), 769-73.

ABSTRACT: This study investigated the impact of mathematics curriculum (standards based vs. traditional) on the performance of sixth and seventh grade students with disabilities on multiple-choice and open-ended assessments aligned to one state’s number and operations and algebra standards. It also sought to understand factors affecting student performance on assessments: ability status (students with and without disabilities), curriculum (standards based vs. traditional), and assessment type (multiple choice vs. open ended). In all, 146 sixth grade students and 149 seventh grade students participated in the study. A linear mixed model for each grade revealed students with disabilities did not perform better in either curriculum. Furthermore, curriculum type was not a significant factor affecting student performance; however, ability status, time, and assessment type were. The implications of these results are discussed.

Bray, M. S. (2005). *Achievement of eighth grade students in mathematics after completing three years of the Connected Mathematics Project.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 66(11). (ProQuest ID No. 1031063341)

ABSTRACT: The purpose of this study was to examine the three-year effect of the Connected Mathematics Project (CMP) on the mathematics achievement of middle school students in a southeastern Tennessee public school district. This was accomplished by (1) comparing the mathematics achievement of eighth graders who have completed three years of CMP with their mathematics achievement after completing one and two years of CMP; (2) comparing the achievement of male and female students during the same period of time; and (3) comparing the mathematics achievement of historically underrepresented students after completing one, two, and three years of CMP.

In order to provide for a richer analysis of the CMP experience, the overall design employed quantitative and qualitative methodologies. The quantitative section of the study examined the mathematical achievement of almost 2,900 of the 2001-2002 eighth graders, over 3,000 of the 2000-2001 seventh graders, and over 3,100 1999- 2000 sixth graders as evidenced by their Tennessee Comprehensive Assessment Program (TCAP) test scores. The qualitative segment of the study explored the experiences of the textbook adoption committee members, teachers, administrators, and parents.

Using the Tennessee Comprehensive Assessment Program mathematics total battery test score as the dependent variable, there was no significant difference between the mathematics achievement of students completing one or two years of CMP. However, there was a significant difference in the mathematics achievement between students completing three years of CMP when compared to their mathematics scores after one and two years. There was also a significant difference between male and female students after completing one and two years of CMP but no significant difference was detected after the completion of three years. Though there was a significant difference revealed in the achievement between African Americans and Non African Americans after completing one, two, and three years of CMP the gap closed slightly after completing three years. Overall, CMP students performed better on the state achievement assessment the longer they were being instructed using the standards based curriculum.

Breyfogle, M. L., & Lynch, C. M. (2010). Van Hiele revisited. *Mathematics Teaching in the Middle School, 16*(4), 232-238.

ABSTRACT: Assessment is a tool used in the classroom as a way to deepen students' learning and to allow the educator to make informed decisions regarding instruction. In this article, the authors focus on the role of assessment, both in terms of teachers and students, while developing students' understanding of geometry. In particular, the authors are interested in using authentic assessment to develop students' geometric thought using the van Hiele model. The van Hiele model of the development of geometric thought was created in the 1980s by two Dutch middle school teachers and researchers, Dina van Hiele-Geldhof and Pierre van Hiele. The model described levels of understanding through which students progress in relation to geometry (Crowley 1987). The authors examine authentic assessment and its use in encouraging students to progress along the van Hiele levels. To analyze students' geometric thinking, the authors suggest using both formative and summative assessments to move students along the van Hiele model of thought. (Contains 4 figures and 2 tables.)

Breyfogle, M. L., McDuffie, A. R., & Wohlhute, K. A. (2010). Developing curricular reasoning for grades pre-K-12 mathematics instruction. In R. Reys & B. Reys (Eds.), *K-12 mathematic curriculum: Issues, trends, and future directions, 72nd yearbook* (pp. 307-320). Reston, VA: National Council of Teachers of Mathematics.

Brucker, E. L. (2008). Journey into a Standards-based mathematics classroom. *Mathematics Teaching in the Middle School, 14*(5), 300-303.

ABSTRACT: A standards-based approach to mathematics involves using story problems to allow students to investigate a solution. This approach emphasizes an understanding of concepts and processes and assumes mastery of basic computation skills. This article will encourage teachers to continue teaching standards-based mathematics and to take advantage of available training to produce students who are better prepared in mathematics and who enjoy the process.

Burdell, C., & Smith III, J. P. (2001). *“The math is different, but I can deal”: Studying students’ experiences in a reform-based mathematics curriculum.* Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA.

ABSTRACT: The research reported in this paper describes the mathematical experiences of 9 students who moved from a traditional mathematics program in junior high school to a high school mathematics program structured by current reforms in curriculum and teaching. We will refer to the high school site of this work as Logan High (though the name is fictitious). Logan has for some years implemented the Core-Plus Mathematics Project materials for most of its grade 9–12 students, including some (but not all) students who come out of the “advanced” mathematics track in the junior high school. We recruited 24 Logan student volunteers starting in January 2000 and have tracked these students in their mathematics work for 2.5 semesters.

We report on the experiences of 9 of these students, drawing on a maximum of 3 semesters of mathematics coursework (Spring 2000, Fall 200, and Spring 2001). We have analyzed their mathematical experiences along 4 dimensions: (1) performance in mathematics, (2) disposition towards the subject, (3) approach to learning the subject, and (4) differences students see between traditional and Core-Plus mathematics curricula and teaching. All of our 9 students reported differences between their past and present mathematics programs as they moved into Core-Plus, but in only 2 cases was there any significant change in performance across the curricular shift.

Cady, J. A., Hodges, T. E., & Collins, R. L. (2015). A comparison of textbooks’ presentation of fractions. *School Science & Mathematics, 115(*3), 105–116. doi:10.1111/ssm.12108.

ABSTRACT: In the United States, fractions are an important part of the middle school curriculum, yet many middle school students struggle with fraction concepts. Teachers also have difficulty with the conceptual understanding needed to teach fractions and rely on textbooks when making instructional decisions. This reliance on textbooks, the idea that teaching and learning of fractions is a complex process, and that fraction understanding is the foundation for later topics such as proportionality, algebra, and probability, makes it important to examine the variation in presentation of fraction concepts in U.S. textbooks, especially the difference between traditional and standards-based curricula. The purpose of this study is to determine if differences exist in the presentation of fractions in conventional and standards-based textbooks and how these differences align with the recommendations of National Council of Teachers of Mathematics, Common Core State Standards, and the research on the teaching and learning of fractions.

Cai, J. & Nie, B. (2007). Problem solving in Chinese mathematics education: Research and practice. *ZDM Mathematics Education. 39*, 459-473

ABSTRACT: This paper is an attempt to paint a picture of problem solving in Chinese mathematics education, where problem solving has been viewed both as an instructional goal and as an instructional approach. In discussing problem-solving research from four perspectives, it is found that the research in China has been much more content and experience-based than cognitive and empirical-based. We also describe several problem-solving activities in the Chinese classroom, including "one problem multiple solutions," "multiple problems one solution," and "one problem multiple changes." Unfortunately, there are no empirical investigations that document the actual effectiveness and reasons for the effectiveness of those problem solving activities. Nevertheless, these problem-solving activities should be useful references for helping students make sense of mathematics.

Cai, J. (2014). Searching for evidence of curricular effect on the teaching and learning of mathematics: Some insights from the LieCal project. *Mathematics Education Research Journal, 26*, 811-831.

ABSTRACT: Drawing on evidence from the Longitudinal Investigation of the Effect of Curriculum on Algebra Learning (LieCal) Project, issues related to mathematics curriculum reform and student learning are discussed. The LieCal Project was designed to longitudinally investigate the impact of a reform mathematics curriculum called the Connected Mathematics Project (CMP) in the USA on teachers' teaching and students' learning. Using a three-level conceptualization of curriculum (intended, implemented, and attained), a variety of evidence from the LieCal Project is presented to show the impact of mathematics curriculum reform on teachers' teaching and students' learning. This paper synthesizes findings from the two longitudinal studies spanning 7 years of the LieCAl Project both to show the kind of impact curriculum has on teachers' teaching and students' learning and to suggest powerful but feasible ways researchers can investigate curriculum effect on both teaching and learning.

Cai, J. (2015). Curriculum reform and mathematics learning: Evidence from two longitudinal studies. In S. J. Cho (Ed.), *Selected regular lectures from the 12th International Congress on Mathematical Education *(pp. 71–92). Gewerbestrasse, Switzerland: Springer International Publishing.

ABSTRACT: Drawing on longitudinal evidence from the LieCal Project, issues related to mathematics curriculum reform and student learning are discussed. The LieCal Project was designed to longitudinally investigate the impact of a reform mathematics curriculum called the Connected Mathematics Project (CMP) in the United States on teachers’ teaching and students’ learning. Using a three-level conceptualization of curriculum (intended, implemented and attained), a variety of evidence from the LieCal Project is presented to show the impact of mathematics curriculum reform on teachers’ teaching and students’ learning. The findings from the two longitudinal studies in the LieCal Project serve both to show the kind of impact curriculum has on teachers’ teaching and students’ learning and to suggest powerful ways researchers can investigate curriculum effect on both teaching and learning.

Cai, J., & Moyer, J. C. (2006). A conceptual framework for studying curricular effects on students’ learning: Conceptualization and design in the LieCal project. Poster presented at the 2006 Annual Meeting of the International Group of Psychology of Mathematics Education, Prague, Czech Republic.

Cai, J., Hwang, S., & Moyer, J.C. (2016) Mathematical problem posing as a measure of curricular effect on students’ learning: A response. *Educational Studies in Mathematics, 91*(1), 9–10.

Cai, J., Moyer, J. C., & Wang, N. (2013). Longitudinal investigation of the effect of middle school curriculum on learning in high school. In A. Lindmeier & A. Heinze (Eds.), *The proceedings of the 37th conference of the International Group for the Psychology of Mathematics Education* (pp. 137–144). Kiel, Germany: The International Group for the Psychology of Mathematics Education.

Cai, J., Moyer, J. C., Wang, N., & Nie, B. (2009). Curricular impact on the development of algebraic thinking: A longitudinal study. In M. Tzekaki, M. Kaldrimidou, & H. Sakonidis (Eds.), *Proceedings of the 33rd Conference of the International Group for the Psychology of Mathematics Education *(Vol. 2, pp. 241 – 248). Thessaloniki, Greece: PME.

Cai, J., Moyer, J. C., Wang, N., & Nie, B. (2011). Examining students’ algebraic thinking in a curricular context: A longitudinal study. In J. Cai & E. Knuth (Eds.), *Early algebraization: A global dialog from multiple perspectives* (pp. 161-186). New York: Springer.

ABSTRACT: This chapter highlights findings from the LieCal Project, a longitudinal project in which we investigated the effects of a Standards-based middle school mathematics curriculum (CMP) on students’ algebraic development and compared them to the effects of other middle school mathematics curricula (non-CMP). We found that the CMP curriculum takes a functional approach to the teaching of algebra while non-CMP curricula take a structural approach. The teachers who used the CMP curriculum emphasized conceptual understanding more than did those who used the non-CMP curricula. On the other hand, the teachers who used non-CMP curricula emphasized procedural knowledge more than did those who used the CMP curriculum. When we examined the development of students’ algebraic thinking related to representing situations, equation solving, and making generalizations, we found that CMP students had a significantly higher growth rate on representing-situations tasks than did non-CMP students, but both CMP and non-CMP students had an almost identical growth in their ability to solve equations. We also found that CMP students demonstrated greater generalization abilities than did non-CMP students over the three middle school years.

The research reported in this chapter is part of a large project, Longitudinal Investigation of the Effect of Curriculum on Algebra Learning (LieCal Project). The LieCal Project is supported by a grant from the National Science Foundation (ESI-0454739). Any opinions expressed herein are those of the authors and do not necessarily represent the views of the National Science Foundation.

Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garber, T. (2013). Mathematical problem posing as a measure of curricular effect on students’ learning. *Educational Studies in Mathematics, 83*(1), 57–69.

ABSTRACT: In this study, we used problem posing as a measure of the effect of middle-school curriculum on students' learning in high school. Students who had used a standards-based curriculum in middle school performed equally well or better in high school than students who had used more traditional curricula. The findings from this study not only show evidence of strengths one might expect of students who used the standards-based reform curriculum but also bolster the feasibility and validity of problem posing as a measure of curriculum effect on student learning. In addition, the findings of this study demonstrate the usefulness of employing a qualitative rubric to assess different characteristics of students' responses to the posing tasks. Instructional and methodological implications of this study, as well as future directions for research, are discussed.

Cai, J., Moyer, J. C., Wang, N., Hwang, S., Nie, B., & Garger, T. (2012). Mathematical problem posing as a measure of the curricular effects on students’ learning. *Educational Studies in Mathematics, 83*(1), 57-69.

ABSTRACT: In this study, we used problem posing as a measure of the effect of middle-school curriculum on students' learning in high school. Students who had used a standards-based curriculum in middle school performed equally well or better in high school than students who had used more traditional curricula. The findings from this study not only show evidence of strengths one might expect of students who used the standards-based reform curriculum but also bolster the feasibility and validity of problem posing as a measure of curriculum effect on student learning. In addition, the findings of this study demonstrate the usefulness of employing a qualitative rubric to assess different characteristics of students' responses to the posing tasks. Instructional and methodological implications of this study, as well as future directions for research, are discussed.

Cai, J., Moyer, J., Nie, B., & Wang, N. (2009). Learning mathematics from classroom instruction using Standards-based and traditional curricula: An analysis of instructional tasks. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), *Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (Vol. 5, pp. 692-699). Atlanta, GA: Georgia State University.

ABSTRACT: The LieCal Project longitudinally investigates the effects of the Connected Mathematics Program (CMP) and more traditional middle school curricula (non-CMP) on students’ learning of algebra. To ascertain the curricular effects, we must attend to aspects of teaching that influence students’ learning opportunities. In this paper, we particularly focused on the mathematical tasks to understand the instructional experiences provided when using CMP and Non-CMP curricula. We found that teachers in CMP classrooms implemented significantly more cognitively demanding tasks than teachers in Non-CMP classrooms. Also, teachers are much more likely to encourage multiple strategies in CMP classrooms than in Non-CMP classrooms.

Cai, J., Nie, B., & Moyer, J. (2010). The teaching of equation solving: Approaches in Standards-based and traditional curricula in the United States.* Pedagogies: An International Journal. 5*(3), 170-186.

ABSTRACT: This paper discusses the approaches to teaching linear equation solving that are embedded in a Standards-based mathematics curriculum (CMP) and in a traditional mathematics curriculum (Glencoe Mathematics) in the United States. Overall, the CMP curriculum takes a functional approach to teach equation solving, while Glencoe Mathematics takes a structural approach to teach equation solving. The functional approach emphasizes the important ideas of change and variation in situations and contexts. It also emphasizes the representation of relationships between variables. The structural approach, on the other hand, requires students to work abstractly with symbols, and follow procedures in a systematic way. The CMP curriculum may be regarded as a curriculum with a pedagogy that emphasizes predominantly the conceptual aspects of equation solving, while Glencoe Mathematics may be regarded as a curriculum with a pedagogy that emphasizes predominantly the procedural aspects of equation solving. The two curricula may serve as concrete examples of functional and structural approaches, respectively, to the teaching of algebra in general and equation solving in particular.

Cai, J., Nie, B., Moyer, J. C., & Wang, N. (2014). Teaching mathematics using standards-based and traditional curricula: A case of variable ideas. In Y. Li & G. Lappan (Eds.), *Mathematics curriculum in school education *(pp. 391–415). Dordrecht, Netherlands: Springer Netherlands.

ABSTRACT: This chapter discusses approaches to teaching algebraic concepts like variables that are embedded in a Standards-based mathematics curriculum (CMP) and in a traditional mathematics curriculum (Glencoe Mathematics). Neither the CMP curriculum nor Glencoe Mathematics clearly distinguishes among the various uses of variables. Overall, the CMP curriculum uses a functional approach to teach equation solving, while Glencoe Mathematics uses a structural approach to teach equation solving. The functional approach emphasizes the important ideas of change and variation in situations and contexts. The structural approach, on the other hand, avoids contextual problems in order to concentrate on developing the abilities to generalize, work abstractly with symbols, and follow procedures in a systematic way. This chapter reports part of the findings from the larger LieCal research project. The LieCal Project is designed to investigate longitudinally the impact of a Standards-based curriculum like CMP on teachers’ classroom instruction and student learning. This chapter tells part of the story by showing the value of a detailed curriculum analysis in characterizing curriculum as a pedagogical event.

Cai, J., Wang, N., Moyer, J. C., Wang, C., & Nie, B. (2011). Longitudinal investigation of the curricular effect: An analysis of student learning outcomes from the LieCal project in the United States. *International Journal of Educational Research, 50*(2), 117–136.

ABSTRACT: In this article, we present the results from a longitudinal examination of the impact of a Standards-based or reform mathematics curriculum (called CMP) and traditionalmathematics curricula (called non-CMP) on students’ learning of algebra using various outcome measures. Findings include the following: (1) students did not sacrifice basic mathematical skills if they are taught using a Standards-based or reform mathematics curriculum like CMP; (2) African American students experienced greater gain in symbol manipulation when they used a traditional curriculum; (3) the use of either the CMP or a non-CMP curriculum improved the mathematics achievement of all students, including students of color; (4) the use of CMP contributed to significantly higher problem-solving growth for all ethnic groups; and (5) a high level of conceptual emphasis in a classroom improved the students’ ability to represent problem situations. (However, the level of conceptual emphasis bears no relation to students’ problem solving or symbol manipulation skills).

Cai, J., Yujing N., & Hwang, S. (2015). Measuring change in mathematics learning with longitudinal studies: Conceptualization and methodological issues. In J. Middleton, J. Cai, & S. Hwang (Eds.), *Large-scale studies in mathematics education* (pp. 293–309). Gewerbestrasse, Switzerland: Springer International Publishing.

ABSTRACT : Learning is fundamentally about growth and change. Longitudinal studies of mathematics learning must therefore conceptualize, measure, analyze, and interpret changes in students’ mathematical thinking. This chapter provides a perspective on how researchers can deal with issues entailed in researching such change over time, drawing on the authors’ experiences with two longitudinal projects in the USA and China. Both the LieCal (Longitudinal Investigation of the Effect of Curriculum on Algebra Learning) project and the China project studied the effects of curriculum on student learning. Based on these projects, several challenges are discussed, including the complexity of conceptualizing and measuring change in mathematical thinking, the importance of appropriate analytic techniques, the need to consider long-term change, and critical concerns when interpreting the correlates or causes of observed change.

Cain, J. S. (2002). An evaluation of the Connected Mathematics Project. *Journal of Educational Research*, 95(4), 224-33.

ABSTRACT: Evaluated the Connected Mathematics Project (CMP), a middle school reform mathematics curriculum used in Louisiana's Lafayette parish. Analysis of Iowa Test of Basic Skills and Louisiana Education Assessment Program mathematics data indicated that CMP schools significantly outperformed non-CMP schools. Surveys of teachers and students showed that both groups believed the program was helping students become better problem solvers.

Camenga, K. A., & Johnson Yates, R. B. (2014). Connecting the dots: *Rediscovering continuity. Mathematics Teacher,* 108(3), 212–217.

Capraro, M. M., Kulm, G., & Capraro, R. M. (2005). Middle grades: Misconceptions in statistical thinking. *School Science and Mathematics, *105, 165-174.

ABSTRACT: A sample of 134 sixth-grade students who were using the Connected Mathematics Project (CMP) curriculum were administered an open-ended item entitled, Vet Club (Balanced Assessment, 200). This paper explores the role of misconceptions and naïve conceptions in the acquisition of statistical thinking for middle grades students. Students exhibited misconceptions and naïve conceptions regarding representing data graphically, interpreting the meaning of typicality, and plotting 0 above the x-axis.

Castro, A. M. (2006). *Planning for mathematics instruction: A study of the teacher guide as a resource.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 67(10). (ProQuest ID No. 1251814391)

ABSTRACT: Planning is an important, and often underappreciated, phase of teaching, during which teachers make decisions and draw upon a variety of resources, such as curriculum materials, that shape students' opportunities to learn. The teacher guide (TG) is a particularly important curricular resource be cause it is designed to assist teachers in making decisions that affect these opportunities. Prior research has established that teachers' use of curriculum materials is affected by a range of factors, such as state level policies, knowledge of mathematics, and the nature and extent of their teaching experience. What is less clear, and far less examined, in prior research is the role that the TG may play in mediating the influence of these and other factors on teachers' decisions and actions during planning and instruction. Accordingly, this study examines how four experienced 6th grade teachers use the TG from Connected Mathematics Project (CMP) as a resource in making planning and enactment decisions, and factors associated with patterns of TG use.

Using interpretive case study methodology, the author examined teachers' use of the CMP TG in planning for and implementing an entire unit. In addition to observing their implementation of the unit, teachers were interviewed prior to and immediately following each observation to understand how they used the TG to plan for and enact different mathematical tasks. The author then developed case studies of teachers' use of the TG in implementing the unit.

Through cross-case analysis, the author found that teachers seemed to draw largely from their personal resources when making planning and enactment decisions related to mathematical tasks, and not particularly from the TG. For example, when faced with certain planning and instructional challenges, such as anticipating how students would work on a task or students struggling with the content, teachers tended to rely on their particular conceptions of mathematics teaching to address these challenges. Despite the fact that the TG provided suggestions for teachers as to how address such challenges, it was not extensively used as a resource by the teachers in this study in their planning and enactment of classroom lessons. Based on these and other findings the author identifies important questions and potential implications for curriculum developers, teacher educators, and researchers.

Cavanagh, J. M. (2012). *An organizational case study: The impact of an initiation, implementation, and institutionalization of a curricular change* (Doctoral dissertation). Available from ProQuest Dissertations and Theses data-base. (UMI No. 1015379520)

ABSTRACT: Successful change in schools is planned, expected and managed with the objective focused on benefiting the students, not just converting the staff. This investigation is a case study of a public school district that opted to implement curricular change following an examination of the district's performance toward adequate yearly progress. This case study utilized a quantitative design to address: the process and impact of an initiation, implementation and institutionalization of a district level curricular change, the roles that emerged among participants in this process, the influence of stakeholders, the dynamics and processes of change, and the impact of the curricular change on student achievement. Surveys were distributed to 18 teachers, three middle level administrators and four central office personnel in order to analyze the organizational processes and the perceived roles of stakeholders in the curricular change process. The overall participation rate was 68%. Surveys were analyzed to examine three themes: if the curricular change process was triggered by external stakeholders that had legitimate claims on the operation of the organization, if the curricular change process was initiated and dictated by the high level district stakeholders, and if the curricular change process was implemented and carried through by high level internal stakeholders. Additionally, Pennsylvania System of School Assessment math scores for eighth grade were collected and analyzed comparing four groups based upon the amount of Connected Mathematics Project instruction the students received. Analysis of Pennsylvania System of School Assessment eighth grade math scores revealed that scores increased progressively with each additional year of Connect Mathematics Project completion. Further research involving the surveying of high school teachers, as well as review of eleventh grade Pennsylvania System of School Assessment math scores may be helpful. Review of additional performance indicators, such as classroom mathematics grades may also be beneficial.

Celedon, S. (1998). *An analysis of a teacher's and students' language use to negotiate meaning in an ESL/mathematics classroom. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 69(9). (ProQuest ID No.732855961)

ABSTRACT: The research reviewed indicates a paucity of studies addressing issues regarding language as used by linguistically diverse students and its role in mathematics problem solving, especially at the secondary level. The purpose of this qualitative study was threefold: (1) to describe how English as a second language (ESL) students and their teacher used language (Spanish and English) to negotiate mathematical meaning in an ESL/Mathematics classroom, (2) to explore problem-solving strategies used by ESL students and examine how these connect, or not, to those presented by their teacher, and (3) to generate a theory about the use of language to teach mathematics to ESL students. Research was conducted in a self-contained ESL/Mathematics classroom at the middle school level (6th-8th grade). The study included participant observations, in-depth interviews with a representative sample of nine students and the teacher, and written documents.

Analysis of the data collected throughout a nineteen-week period indicated that Spanish was the language used by most ESL students to express themselves when they needed to elaborate on their responses orally or in written form as they engaged in a curriculum, the Connected Mathematics Project(CMP), that promoted higher order thinking skills. From the teacher-student discourse samples, it was evident that using Spanish created more opportunities for students to participate in discussions where an explanation of their responses was needed. Furthermore, these students felt comfortable expressing themselves in their first language when explaining their problem-solving strategies during think-aloud protocols. Overall, the accuracy of these nine students improved by one or two word problems (out of five)in the Spanish version. These results indicate the importance of making both languages accessible to students during mathematics problem solving. While I am not advocating that Spanish be used as the only language of instruction, I am suggesting that students' sociocultural and linguistic experiences be used to make the mathematical connections between the everyday use of English and the language that is specific to mathematics.

Studying how ESL students used language when engaged in mathematical problem solving provides educators insight as to how they can help students make connections between their existing everyday language and the mathematical language necessary for problem solving. In addition, these findings provide both ESL and mathematics teachers with detailed information regarding the variety of problem-solving strategies used by ESL students.

Chapin, S. H. (2003). Crossing the bridge to formal proportional reasoning. *Mathematics Teaching in the Middle School, 8*(8), 420-425.

Chappelle, M. (2003). Keeping mathematics front and center: Reaction to middle-grades curriculum projects research. In S. Senk & D. Thompson (Eds.), *Standards-based school mathematics curricula: What are they? What do students learn?* (pp. 285- 298). Mahwah, NJ: Erlbaum

Charalambos, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship. *Journal of Curriculum Studies, 44*(4), 443- 466.

ABSTRACT: The set of papers presented in this issue comprise a multiple-case study which attends to instructional resources—teacher knowledge and curriculum materials—to understand how they individually and jointly contribute to instructional quality. We approach this inquiry by comparing lessons taught by teachers with differing mathematical knowledge for teaching who were using either the same or different editions of a US Standards-based curriculum. This introductory paper situates the work reported in the next four case-study papers by outlining the analytic framework guiding the exploration and detailing the methods for addressing the research questions.

Charalambos, C. Y., Hill, H. C., & Mitchell, R. N. (2012). Two negatives don't always make a positive: Exploring how limitations in teacher knowledge and the curriculum contribute to instructional quality. *Journal of Curriculum Studies, 44*(4), 489-513.

ABSTRACT: This paper examines the contribution of mathematical knowledge for teaching (MKT) and curriculum materials to the implementation of lessons on integer subtraction. In particular, it investigates the instruction of three teachers with differing MKT levels using two editions of the same set of curriculum materials that provided different levels of support. This variation in MKT level and curriculum support facilitated exploring the distinct and joint contribution of MKT and the curriculum materials to instructional quality. The analyses suggest that MKT relates positively to teachers' use of representations, provision of explanations, precision in language and notation, and ability to capitalize on student contributions and move the mathematics along in a goal-directed manner. Curriculum materials set the stage for attending to the meaning of integer subtraction and appeared to support teachers' use of representations, provision of explanations, and precision in language and notation. More critically, the findings suggest that less educative curriculum materials, coupled with low levels of MKT, can lead to problematic instruction. In contrast, educative materials can help low-MKT teachers provide adequate instruction, while higher MKT levels seem to enable teachers to compensate for curriculum limitations.

Chiappetta, C. (2015). *Impact of a Mathematics Intervention on Achievement of Urban Middle School Students.* (Unpublished doctoral dissertation). Nova Southeastern University, Fort Lauderdale, FL.

ABSTRACT: The problem addressed in this study was that it was necessary to assess the efficacy of the Connected Mathematics Project 2 that was implemented in five middle schools beginning in 2008 to improve the students’ mathematics skills. The purpose of this study was to determine the efficacy of the mathematics program at the sixth-, seventh-, and eighth-grade levels using an ex post facto approach with an interrupted time-series design.

To compare the mathematics academic achievement of students before and after implementation of the intervention, pretest and posttest archival data from the state mastery test were analyzed. A questionnaire completed by the middle school mathematics teachers was used to ascertain teachers’ perceptions about the new mathematics program, how the program impacted students during the first year of implementation, and perceptions of the professional development teachers received for this intervention.

The results show that implementation of the Connected Mathematics Project 2 improved the overall mathematics achievement of students in Grades 6, 7, and 8 on the state standardized assessments. However, the year-to-year growth of students’ performance on the assessment did not improve significantly. Most of the students in specific populations in Grades 6, 7, and 8 also had improved achievement. Furthermore, the achievement gaps between White students and both African American and Hispanic students, as well as the economic achievement gap between economically disadvantaged students and all students, although still significant, were reduced. However, special education students in Grade 7 and English-language learners in Grades 7 and 8 did not experience improvement. Teachers indicated that the professional development they received improved their practice, and they also believed that students benefited from the implementation of the intervention.

Choppin, J. (2006). *Studying a curriculum implementation using a communities of practice perspective. *Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mèrida, Mèxico.

ABSTRACT: The publication of the 1989 NCTM Standards (NCTM, 1989) marked the launch of extensive efforts to reform mathematics teaching and learning. These efforts have included the development and publication of curricula which implicate constructivist instructional practices. Implementing reform curricula in a way that changes core teaching practices has proven to be a difficult endeavor (Spillane & Zeuli, 1999), especially so in urban settings, which are typically stressed in terms of teacher turnover, lack of material resources, and funding for professional development. A number of researchers have noted the importance – if not necessity – of professional community in facilitating and sustaining teacher change towards constructivist-based pedagogy (Cobb, McClain, Lamberg, & Dean, 2003; Secada & Adajian, 1997; Stein, Silver, & Smith, 1998). In this study I use Wenger’s (1998) three dimensions of community of practice (CoP) to analyze the extent to which core learning principles exist within the professional communities in my study. I focus on the learning principles of collaboration, reflection, recognition, and autonomy, which have been identified as characteristics of effective learning in communities of practice (Gee, 2003; Schon, 1983; Secada & Adajian, 1997; Wenger, 1998). This study describes characteristics of CoP’s in an urban school system implementing the Connected Mathematics Project (CMP) (Lappan, Fey, Fitzgerald, Friel & Phillips, 1998) curriculum.

Choppin, J. (2006). *Design rationale: Role of curricula in providing opportunities for teachers to develop complex practices.* Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mèrida, Mèxico.

ABSTRACT: This study analyzes the potential of two similar tasks to generate dialogic classroom interactions. Although both tasks were similar in context and outcome, one affords teachers’ actions to elicit and build from diverse student explanations. This would require greater teacher expertise – both mathematically and pedagogically – and an articulation of conditions when more potentially dialogical tasks should be implemented.

Choppin, J. (2009). Curriculum-context knowledge: Teacher learning from successive enactments of a Standards-based mathematics curriculum. *Curriculum Inquiry, 39*(2), 287- 320.

ABSTRACT: This study characterizes the teacher learning that stems from successive enactments of innovative curriculum materials. This study conceptualizes and documents the formation of curriculum-context knowledge (CCK) in three experienced users of a Standards-based mathematics curriculum. I define CCK as the knowledge of how a particular set of curriculum materials functions to engage students in a particular context. The notion of CCK provides insight into the development of curricular knowledge and how it relates to other forms of knowledge that are relevant to the practice of teaching, such as content knowledge and pedagogical content knowledge. I used a combination of video-stimulated and semistructured interviews to examine the ways the teachers adapted the task representations in the units over time and what these adaptations signaled in terms of teacher learning. Each teacher made noticeable adaptations over the course of three or four enactments that demonstrated learning. Each of the teachers developed a greater understanding of the resources in the respective units as a result of repeated enactments, although there was some important variation between the teachers. The learning evidenced by the teachers in relation to the units demonstrated their intricate knowledge of the curriculum and the way it engaged their students. Furthermore, this learning informed their instructional practices and was intertwined with their discussion of content and how best to teach it. The results point to the larger need to account for the knowledge necessary to use Standards-based curricula and to relate the development and existence of well-elaborated knowledge components to evaluations of curricula.

Choppin, J. (2011). Learned adaptations: Teachers’ understanding and use of curriculum resources. *Journal of Mathematics Teacher Education, (published online: DOI: 10.1007/s10857-011-9170-3)*

ABSTRACT: This study focused on the use of curriculum materials for three teachers who had enacted instructional sequences from the materials on multiple occasions. The study investigated how the teachers drew on the materials, what they understood about the curriculum resources, and how they connected their use of the materials to their observations of student thinking. There were similarities across the teachers, particularly with respect to their goals and how they read and followed recommendations in the teacher resource materials. There were differences in how their task revisions were in response to what they observed about student thinking. The teacher who most intensively observed student thinking made connections between her interpretations of students’ strategies and her use of the curriculum resources, allowing her to design learned adaptations. Learned adaptations required both an understanding of the design rationale and empirically developed knowledge of how that rationale played out in practice. The empirically developed knowledge could not be totally anticipated by the designers, in part because it developed within a particular context by a teacher with particular characteristics. The case of the teacher who developed learned adaptations showed how these complementary forms of knowledge helped her to use the curriculum resources in ways that enhanced students’ opportunities for sense making. Furthermore, her adaptations were intended to facilitate success not only at the task level, but also across instructional sequences as well. This study also shows how professional vision is not limited to informing only in-the-moment instructional decisions, but also to the use of curriculum materials.

Choppin, J. (2011). The impact of professional noticing on teachers’ adaptations of challenging tasks.* Mathematical Thinking and Learning, 13*(3), 175-191.

ABSTRACT: This study investigates how teacher attention to student thinking informs adaptations of challenging tasks. Five teachers who had implemented challenging mathematics curriculum materials for three or more years were videotaped enacting instructional sequences and were subsequently interviewed about those enactments. The results indicate that the two teachers who attended closely to student thinking developed conjectures about how that thinking developed across instructional sequences and used those conjectures to inform their adaptations. These teachers connected their conjectures to the details of student strategies, leading to adaptations that enhanced task complexity and students' opportunity to engage with mathematical concepts. By contrast, the three teachers who evaluated students' thinking primarily as right or wrong regularly adapted tasks in ways that were poorly informed by their observations and that reduced the complexity of the tasks. The results suggest that forming communities of inquiry around the use of challenging curriculum materials is important for providing opportunities for students to learn with understanding.

Choppin, J. (2011). The role of local theories: Teacher knowledge and its impact on engaging students with challenging tasks.* Mathematics Education Research Journal, 23*(1), 5-25.

ABSTRACT: This study explores the extent to which a teacher elicited students’ mathematical reasoning through the use of challenging tasks and the role her knowledge played in doing so. I characterised the teacher’s knowledge in terms of a local theory of instruction, a form of pedagogical content knowledge that involves an empirically tested set of conjectures situated within a mathematical domain. Video data were collected and analysed and used to stimulate the teacher’s reflection on her enactments of an instructional sequence. The teacher, chosen for how she consistently elicited student reasoning, showed evidence of possessing a local theory in that she articulated the ways student thinking developed over time, the processes by which that thinking developed, and the resources that facilitated the development of student thinking. Her knowledge informed how she revised and enacted challenging tasks in ways that elicited and refined student thinking around integer addition and subtraction. Furthermore, her knowledge and practices emphasised the progressive formalisation of students’ ideas as a key learning process. A key implication of this study is that teachers are able to develop robust knowledge from enacting challenging tasks, knowledge that organizes how they elicit and refine student reasoning from those tasks.

Choppin, J. M., Callard, C. H., & Kruger, J. S. (2014). Interpreting Standards as Sense-Making Opportunities. Mathematics Teaching in the Middle School, 20(1), 24-29.

Description: “The authors are a team of two teachers and a researcher who for several years have studied the teachers’ enactment of Accentuate the Negative, a unit on rational numbers that is part of the Connected Mathematics Project (CMP) curriculum (Lappan et al. 2006). We show how allowing students to create algorithms provided opportunities for them to reason about rational number addition and subtraction.”

Choppin, J. M., Cancy, C. B., & Koch, S. J. (2012). Developing formal procedures through sense-making. *Mathematics Teaching in the Middle School, 17*(9), 552-557.

ABSTRACT: The eight mathematical practices explored in the Common Core Math Standards are the following: (1) Make sense of problems and persevere in solving them; (2) Reason abstractly and quantitatively; (3) Construct viable arguments and critique the reasoning of others; (4) Model with mathematics; (5) Use appropriate tools strategically; (6) Attend to precision; (7) Look for and make use of structure.; and (8) Look for and express regularity in repeated reasoning. If teachers are going to take the Common Core Math Standards seriously, they need to think of them as more than simply a reordering of content. That means focusing on the practices they associate with mathematical understanding. A major implication is that "developing practices" rather than "covering content" requires a focus on task sequences rather than singular lessons; these sequences provide repeated opportunities for students to reason about ideas before they are formalized. Most students can reason mathematically but few get the opportunity to publicly test ideas and conjectures as they are forming. Participation in such practices leads not only to increased understanding but also to the development of mathematical dispositions that are valuable as students move to more advanced mathematics.

Chval, K., Chávez, Ó., Reys, B., & Tarr, J. (2009). Considerations and limitations related to conceptualizing and measuring textbook integrity. In J.T Remillard, B. A Herbel-Eisenmann, & G.M Lloyd (Eds.), *Mathematics teachers at work: Connecting curriculum materials to classroom instruction* (Studies in Mathematical Thinking and Learning Series, A. Schoenfeld, Ed.) (pp. 70-84). New York: Routledge.

Chval, K., Chávez, Ó., Reys, B., & Tarr, J. (2009). Considerations and limitations related to conceptualizing and measuring textbook integrity. In J. T. Remillard, B. A. HerbelEisenmann, & G. M. Lloyd (Eds.), *Mathematics teachers at work: Connecting curriculum materials and classroom instruction* (Studies in Mathematical Thinking and Learning Series, A. Schoenfeld, Ed.) (pp. 70-84). New York: Routledge.

Clarkson, L. M. (2002). *The effects of the Connected Mathematics Project in a Midwestern urban middle school district.* Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.

Cobb, P., & Jackson, K. (2012). Analyzing educational policies: A learning design perspective. *The Journal of the Learning Sciences, 21*(4), 487-521.

ABSTRACT: In this article, we describe and illustrate an analytical perspective in which educational policies are viewed as designs for supporting learning. This learning design perspective is useful when designing policies, when adapting policies to particular school and district settings during implementation, and when revising policies after implementation to make them more effective. Analyzed from this perspective, a policy comprises the goals for the learning of members of the target group, the supports for their learning, and an often implicit rationale for why these supports might be effective. We clarify that this perspective on policies has broad generality. In addition, we illustrate that personnel at all levels of the US education system both formulate policies designed to influence others’ practices, and are practitioners targeted by others’ policies. The standard image of a single policy traveling down though an education system with more or less fidelity is therefore displaced by that of people at multiple levels of a system reorganizing their practices in school and district settings shaped by others’ policymaking efforts.

Collins, A. M. (2000). Yours is not to reason why. *Education Week, 20*(1), 60.

Collins, A. M. (2002). *What happens to student learning in mathematics when a multifaceted, long-term professional development model to support Standards-based curricula is implemented in an environment of high stakes testing?* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 65(2). (ProQuest ID No. 765336031)

ABSTRACT: Assessment and accountability have created a high-stakes environment for districts, schools, teachers, and students. Assessment is driving most educational decisions. In Massachusetts graduation is contingent upon passing the mathematics and English language arts subtests of the Massachusetts Comprehensive Assessment System (MCAS). Teachers in schools where 30% or more students fail MCASare required to take a mathematics proficiency test. Middle schools not exhibiting improvement in their mathematics scores are identified as under-performing and are subject to interventions by the Department of Education. Not surprisingly, students in urban districts score significantly lower than those in more affluent suburban districts. To date only urban schools have been declared under-performing. It is within this environment of high-stakes testing and As repercussions that this study was undertaken.

In an effort to change the unsuccessful experiences of many urban students, the Noyce Foundation and Raytheon Company made a commitment to funding a long-term professional development intervention. This study investigates the impact of that sustained professional development program in one urban district. The professional development was designed to support the implementation of The Connected Mathematics Project (CMP) and to assess its impact on student learning. This dissertation presents a quantitative comparison between student scores on two standardized tests in schools whose teachers availed themselves of all available professional development surrounding the implementation process for CMP with schools whose teachers chose only to participate in contractually mandated district professional development.

Results indicate that students in schools whose teachers received sustained professional development designed to meet the needs of the participating teachers performed significantly higher on both the MCAS and a nationally normed achievement test, Terra Nova, than did those students whose teachers had not participated in consistent professional development. Evidence is included to document the positive impact on student achievement as a result of changing teacher practice and beliefs through mentoring and coaching in teachers' own classrooms.

Confrey, J. (2006). Comparing and contrasting the National Research Council report on evaluating curricular effectiveness with the What Works Clearinghouse approach. *Educational Evaluation and Policy Analysis, 28*(3), 195-213.

ABSTRACT: This article summarizes the findings of the National Research Council (NRC) report On Evaluating Curricular Effectiveness and examines the reviews in middle grades mathematics undertaken by the What Works Clearinghouse (WWC). The NRC report reviewed and assessed 147 key evaluations of 13 National Science Foundation–supported K–12 mathematics curricula and six commercially generated curricula. The report found that the evaluations overall were not sufficiently robust to permit confident judgments on individual programs, so it instead focused on how to define effectiveness in conducting future evaluations. Effectiveness was defined as “an integrated judgment based on interpretation of a number of scientifically valid evaluations that combine social values, empirical evidence, and theoretical rationales” (NRC, 2004, p. 4). The report introduced a model for curricular evaluation that includes program theory, implementation, and outcome measures, and reviewed three major methodologies found in the literature: content analysis, comparative analysis, and case study. This article then examines the What Works Clearinghouse’s exclusive emphasis on experimental and quasi-experimental designs from the perspective of an author of the NRC report. The two reports agree in recognizing the need for significant improvement in evaluation quality; however, they differ in four areas: standards for individual studies, need for multiple methods, the way to accumulate information across a set of studies, and how to communicate results with the public. This article concludes with a call for focused efforts to address several shared targets needed to make further progress on how to establish curricular effectiveness.

Conklin, M., Grant, Y., Rickard, A., & Rivette, K. (2006). *Prentice Hall Connected Mathematics Project: Research and evaluation summary*. Upper Saddle River, NJ: Pearson Education, Inc.

Covington Clarkson, L. M. (2001). *The effects of the Connected Mathematics Project on middle school mathematics achievement*. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 61(12). (ProQuest ID No. 727079071)

ABSTRACT: The purpose of this study was to examine the three-year effect of the Connected Mathematics Project (CMP) on the mathematics achievement of middle school students in an urban school district. This was accomplished by (1) comparing the mathematics achievement of eighth graders who have completed three years of CMP with the achievement of eighth graders who have completed three-years of a traditional curriculum; (2) comparing the interaction and communication patterns in the two types of classrooms; and (3) comparing the mathematics achievement of historically underrepresented students in both curricula. In order to provide for a richer analysis of the CMP experience, the overall design employed quantitative and qualitative methodologies. The quantitative section of the study examined the mathematical achievement of 700 of the 1999- 2000 eighth graders as evidenced by their State Basic Standards Test(BST) scores. The qualitative segment of the study explored the experiences of the primary participants, the teacher and the students.

Using the State Basic Standards Test as the dependent variable, there was no significant difference between the mathematics achievement of CMP students and that of traditional students after three years of the respective curricula. The achievement gap between CMP Caucasian students and CMP African American students was smaller than the achievement gap between these groups in the traditional curricula. The classroom interaction and communication patterns were very different. CMP classrooms provided more opportunities to learn mathematics than traditional classes. Moreover, CMP students demonstrated algebraic reasoning skills at the same level as the traditional students and demonstrated conceptual understanding through the use of multiple strategies at a higher level than traditional students. Overall, CMP students had a higher level of satisfaction and more positive experiences in their mathematics classes than did traditional students.

Ben-Haim, G Lappan, RT Houang (1985). *Visualizing rectangular solids made of small cubes: Analyzing and effecting students' performance*. Educational studies in Mathematics

Danielson, C. (2005).* Walking a straight line: Introductory discourse on linearity in classrooms and curriculum.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 67(2). (ProQuest ID No. 1095417771)

ABSTRACT: The current curricular reform in US mathematics education has changed many aspects of classroom teaching. Commonly, discussions about this curricular reform presume an unproblematic relationship between textbooks and classroom instruction. This study contributes to the understanding of the relationship between one published reform curriculum, Connected Mathematics (CMP) (Lappan, Fey, Fitzgerald, Friel & Phillips, 2001) and classroom instruction. The study characterizes teaching and learning in terms of communication patterns---discourse ---and analyzes the discourse of CMP, of a traditional US curriculum, Mathematics, Structure and Method (Dolciam, Sorgenfrey & Graham, 1992), and of two teachers in urban classrooms---focusing on the introductory lessons on linear relationships in each case. Results include full descriptions of the introductory discourse on linearity in the textbooks and changes that the CMP textbook discourse undergoes as the curriculum is implemented in these two classrooms.

Danielson, C. (2015). *They'll Need it for Calculus. Mathematics Teaching in the Middle School, 20*(5), 260-265.

Description: “This article focuses on the big question of what it means to be ready for calculus; it also explores the role of the middle school curriculum in preparing students to study calculus later.” Specific to CMP, this article cites the bike shop problems from Variables and Patterns and finite difference problem(s) in Frogs, Fleas, and Painted Cubes as examples of middle school tasks that give students opportunities to think about rates of change, exposure to which may help students prepare for similar ideas in calculus.

Davenport, J., Kao, Y. S., & Schneider, S. A. (2013). Integrating cognitive science principles to redesign a middle school math curriculum. In M. Knauff, M. Pauen, N. Sebanz, & I. Wachsmuth (Eds.), *Proceedings of the 35th Annual Conference of the Cognitive Science Society.* Austin, TX: Cognitive Science Society.

ABSTRACT: Does a middle school mathematics curriculum that is redesigned using principles based in cognitive research improve student outcomes? To test whether research can be effectively translated into practice, the Connected Mathematics Project 2 (CMP2) curriculum was revised according to four principles 1) integrating visual with verbal information, 2) prompting for self-explanation of correct and incorrect worked examples, 3) spacing learning over time, and 4) using formative assessment. This study of 6th grade and 8th grade mathematics education addresses the research question: “Do students who are exposed to specific redesigned CMP2 curriculum modules (treatment) exhibit greater improvements in mathematics performance in the module-specific content area than their counterparts exposed to the regular CMP2 curriculum (control)?” Preliminary analyses show statistically significant effects of the redesigned CMP2 units in three of the four curricular units in this study.

De Groot, C. (2000). *Three female voices: The transition to high school mathematics from a reform middle school mathematics program.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 61(4). (ProQuest ID No. 731933601)

ABSTRACT: In this ethnographic study, the transition experiences and coping mechanisms of three female students are reported. These students were members of a cohort in grades 6, 7, and 8 (ages 12-14) that participated in the field testing of the Connected Mathematics Project (1990-1995), a middle school curriculum closely reflecting recommendations of the National Council of Teachers of Mathematics. The participants of the study were in the same mathematics class during their grade 8 experience, but went to different high schools.

Two interviews were conducted toward the end of their grade 9 experience and six interviews were conducted during their grade 10 experience. Middle school mathematics teachers and high school mathematics teachers were interviewed as well as one parent. One observation of each of their tenth grade mathematics classes was conducted. The reported characteristics of transition in this study focus mainly on changes or discontinuities in the learning of mathematics. Data were analyzed by coding processes and presented in narratives and Qualitative Schematics of Dimensions of Transition in Learning Mathematics Thematic interpretations are given with respect to coping mechanisms that were revealed.

One of the major findings of this study is that early in grade 9 these three students related their learning of mathematics in high school closer to their (traditional) elementary experience, which was termed as regular mathematics, than to their reform middle school experience, which was more constructivist in design. In grade 10 they seemed to connect more with their middle school experience, for example, while doing proofs and related this to "explaining your thinking." Another major finding was that these three students experienced a gradual individualization during this transition together with increased in-class competition among students, particularly for attention from the teacher. In high school, they appeared to cope with this lack of student-to-student discourse by forming out of-class support networks.

Suggestions for future research are made regarding the transition discontinuity from learning in a reform environment to learning in a traditional environment, as well as the need to investigate how transitional standards-based curricula, steeped in problem solving, supports students' development of mathematical proof.

DeBoer, G., Morris, K., Roseman, J. E., Wilson, L., Capraro, M. M., Capraro, R., & Manon, J. (2004). *Research Issues in the Improvement of Mathematics Teaching and Learning through Professional Development.* Paper presented at the American Educational Research Association, San Diego, CA.

ABSTRACT: The purpose of this paper is to describe a study we are conducting on the improvement of mathematics teaching and learning at the middle school level through professional development and to discuss some of the research issues that we have encountered in conducting the study. The paper will lay out the various rationales for our initial design and for the adjustments that we made along the way. We are nearing the end of year two of a five-year study, so this is very much a work in progress. The study is not large in terms of the number of teachers involved (approximately 50 teachers and 1,000 students per year in the early stages of the study), but it is a complex study involving many interconnected elements. In Part I we lay out the design of the study, and in Part II we discuss some of the issues that we are facing as we progress through our work.

Ding, M., & Li, X. (2014). Facilitating and direct guidance in student-centered classrooms: addressing “lines or pieces” difficulty. *Mathematics Education Research Journal, 26*(2), 353-376.

ABSTRACT: This study explores, from both constructivist and cognitive perspectives, teacher guidance in student-centered classrooms when addressing a common learning difficulty with equivalent fractions—lines or pieces—based on number line models. Findings from three contrasting cases reveal differences in teachers’ facilitating and direct guidance in terms of anticipating and responding to student difficulties, which leads to differences in students’ exploration opportunity and quality. These findings demonstrate the plausibility and benefit of integrating facilitating and direct guidance in student-centered classrooms. Findings also suggest two key components of effective teacher guidance including (a) using pre-training through worked examples and (b) focusing on the relevant information and explanations of concepts. Implementations are discussed.

Ding, M., Li, X., Piccolo, D., & Kulm, G. (2007). Teacher interventions in cooperative learning math classes. *The Journal of Educational Research, 100*(3), 162-175.

ABSTRACT: The authors examined the extent to which teacher interventions focused on students' mathematical thinking in naturalistic cooperative-learning mathematics classroom settings. The authors also observed 6 videotapes about the same teaching content using similar curriculum from 2 states. They created 2 instruments for coding the quality of teacher intervention length, choice and frequency, and intervention. The results show the differences of teacher interventions to improve students' cognitive performance. The authors explained how to balance peer resource and students' independent thinking and how to use peer resource to improve students' thinking. Finally, the authors suggest detailed techniques to address students' thinking, such as identify, diversify, and deepen their thinking.

Durkin, N. M. (2005). *Using Connected Math program: Its impact on the Delaware State Testing scores of 8th-grade students at Milford Middle School.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 66(4). (ProQuest ID No. 913516241)

ABSTRACT: This study was designed to investigate the impact of the Connected Math Project curriculum on the student achievement of eighth grade students participating in the Delaware State Testing Program from 1998-2004. The study included an investigation of overall student achievement of students participating in the Connected Math Project as well as specific subgroup populations such as the Black and Special Education students

The investigation revealed that overall student performance and subgroup population performance has increased since the first administration of the Delaware State Testing Program in 1998. A pair wise comparison probability for all test years indicates the increase in mean math scale scores was significant. However, additional pair wise comparison probabilities indicate the percentages of students meeting the state math standard are significant for comparison of test year 2000 with 2003 only. This indicates that although student mean math scale scores are increasing the percentage of students meeting the standard has not increased significantly. Student scores may be approaching the standard but not meeting or exceeding the standard. Pair wise comparison probabilities for the subgroup populations Black and Special Education also indicate a significant increase in the mean math scale scores but not a significant increase in the percentage of students meeting the standard.

Eddy, R. M., Berry, T., Aquirre, N., Wahlstrand, G., Ruitman, T., & Mahajan, N. (2008). The effects of Connected Mathematics Project 2 on student performance: Randomized control trial. Claremont, CA: Claremont Graduate University Institute of Organizational and Program Evaluation Research. Pearson's CMP2 Efficacy Study

Claremont Graduate University (CGU) conducted an efficacy trial of the Connected Mathematics Project 2 (CMP2) curriculum in sixth grade classrooms (across six schools in three states including more than 1,000 students), during the 2007-08 school year. This study was funded by Pearson Education. This report provides an overall description of the study as well as a summary of results based on the major outcome measures. The results are drawn from student performance on the Iowa Test of Basic Skills (ITBS), the Balanced Assessment in Mathematics (BAM), and responses on a student attitudes survey.

Edson, A.J., Phillips, E., Slanger-Grant, Y., & Stewart J. (2018). The Arc of Learning framework: An ergonomic resource for design and enactment of problem-based curriculum. *International Journal of Educational Research*.

Edson, A.J.,** **Phillips, E.D**.**, & Bieda, K. (2018). Transitioning a problem-based curriculum from print to digital: New considerations for task design. In H-G Weigand, A. Clark-Wilson, A. Donevska-Todorova, E. Faggiano, N. Gronbaek & A. Trgalova (Eds.), *Proceedings of the Fifth ERME Topic Study on Mathematics in the Digital Age *(p. 59-67). Copenhagen, Denmark: University of Copenhagen.

Edwards, C. M., & Townsend, B. E. (2012). Diary of change: Shifting mathematical philosophies. *Mathematics Teaching in the Middle School, 18*(3), 174-179.

Ellis, A. (2007a). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. *Journal of the Learning Sciences, 16(*2), 221-262.

ABSTRACT: This article presents a cohesive, empirically grounded categorization system differentiating the types of generalizations students constructed when reasoning mathematically. The generalization taxonomy developed out of an empirical study conducted during a 3-week teaching experiment and a series of individual interviews. Qualitative analysis of data from teaching sessions with 7 seventh-graders and individual interviews with 7 eighth-graders resulted in a taxonomy that distinguishes between students' activity as they generalize, or generalizing actions, and students' final statements of generalization, or reflection generalizations. The three major generalizing action categories that emerged from analysis are (a) relating, in which one forms an association between two or more problems or objects, (b) searching, in which one repeats an action to locate an element of similarity, and (c) extending, in which one expands a pattern or relation into a more general structure. Reflection generalizations took the form of identifications or statements, definitions, and the influence of prior ideas or strategies. By locating generalization within the learner's viewpoint, the taxonomy moves beyond casting it as an activity at which students either fail or Succeed to allow researchers to identify what students see as general, and how they engage in the act of generalizing.

Ellis, A. (2007b). The influence of reasoning with emergent quantities on students' generalizations. *Cognition and Instruction, 25(*4), 439-478.

ABSTRACT: This paper reports the mathematical generalizations of two groups of algebra students, one which focused primarily on quantitative relationships, and one which focused primarily on number patterns disconnected from quantities. Results indicate that instruction encouraging a focus on number patterns supported generalizations about patterns, procedures, and rules, while instruction encouraging a focus on quantities supported generalizations about relationships, connections between situations, and dynamic phenomena, such as the nature of constant speed. An examination of the similarities and differences in students' generalizations revealed that the type of quantitative reasoning in which students engaged ultimately proved more important in influencing their generalizing than a mere focus on quantities versus numbers. In order to develop powerful, global generalizations about relationships, students had to construct ratios as emergent quantities relating two initial quantities. The role of emergent-ratio quantities is discussed as it relates to pedagogical practices that can support students' abilities to correctly generalize.

Ellis, A. B. (2007). Connections between generalizing and justifying: Students reasoning with linear relationships. *Journal for Research in Mathematics Education, 38*(3), 194–229.

ABSTRACT: Research investigating algebra students’ abilities to generalize and justify suggests that they experience difficulty in creating and using appropriate generalizations and proofs. Although the field has documented students’ errors, less is known about what students do understand to be general and convincing. This study examines the ways in which seven middle school students generalized and justified while exploring linear functions. Students’ generalizations and proof schemes were identified and categorized in order to establish connections between types of generalizations and types of justifications. These connections led to the identification of four mechanisms for change that supported students’ engagement in increasingly sophisticated forms of algebraic reasoning: (a) iterative action/reflection cycles, (b) mathematical focus, (c), generalizations that promote deductive reasoning, and (d) influence of deductive reasoning on generalizing.

Ellis, A. B., Özgür, Z., Kulow, T., Williams, C. C., & Amidon, J. (2015). *Quantifying exponential growth: Three concep-tual shifts in coordinating multiplicative and additive growth. Journal of Mathematical Behavior, 39*, 135–155.

ABSTRACT: This article presents the results of a teaching experiment with middle school students who explored exponential growth by reasoning with the quantities height (y) and time (x) as they explored the growth of a plant. Three major conceptual shifts occurred during the course of the teaching experiment: (1) from repeated multiplication to initial coordination of multiplicative growth in y with additive growth in x; (2) from coordinating growth in y with growth in x to coordinated constant ratios (determining the ratio of f(x2) to f(x1) for corresponding intervals of time for (x2− x1) ≥ 1), and (3) from coordinated constant ratios to within-units coordination for corresponding intervals of time for (x2− x1) < 1. Each of the three shifts is explored along with a discussion of the ways in which students’ mathematical activity supported movement from one stage of understanding to the next. These findings suggest that emphasizing a coordination of multiplicative and additive growth for exponentiation may support students’ abilities to flexibly move between the covariation and correspondence views of function.

Ellis, J. D. (2011). Middle school mathematics: A study of three programs in south Texas. (Doctoral dissertation). *Available from ProQuest Dissertations and Theses database. *(UMI No. 3483008)

Ellis, J. D., Kupczynski, L., Mundy, A., & Jones, D. (2012). Middle school mathematics: A study of three programs in south Texas. *Journal of Modern Education Review, 2*, 9-17.

ABSTRACT: The purpose of the study was to determine if there was a significant difference in three math programs within one school district and their impact on student performance as measured by the Texas Assessment of Knowledge and Skills (TAKS). All campuses involved in this study were designated as middle schools containing grade levels six through eight. Mathematics teachers at each of these middle school campuses teach students the mathematics objectives outlined in the Texas Essential Knowledge and Skills (TEKS) through their school’s curriculum. Of the campuses in the study, one campus used the Texas MathWorks Program for every student in grades six and seven, two campuses used the Connected Mathematics Program in grades six through eight, and four campuses use the district approved state adopted textbook, Glencoe, in grades six through eight. The study determined if there were significant differences in test scores among these three math programs in grades six and seven for the academic school years 2008–2009 and 2009–2010. Campus scores on TAKS from the campuses involved in this study were reviewed using the Academic Excellence Indicator System (AEIS) provided by the Texas Education Agency for TAKS results as well data provided by the south Texas school district. From the analysis of data, it can be concluded that students enrolled in Connected Mathematics did better on the TAKS test than those in the two other instructional programs, Glencoe and Texas MathWorks.

Fauth, T. (2007). *Using the Connected Math Project to improve seventh grade math scores at Wapato Middle School.* (Unpublished master’s thesis). Heritage University, Toppenish, WA.

Fey, J. T., & Philips, E. D. (2005). A course called Algebra 1. In C. Greenes & C. Findell (Eds.),* Developing students’ algebraic reasoning abilities *(pp. 4-16). Lakewood, CO: National Council of Supervisors of Mathematics.

ABSTRACT: As suggested by the NCTM Principles and Standards 2000, an overarching focus for algebra is on developing student ability to represent and analyze relationships among quantitative variables. From this perspective, variables are not letters that stand for unknown numbers-they are quantitative attributes of objects (like measurements of size), patterns, or situations that change in response to other quantities or with the passage of time. Understanding and predicting patterns of change in variables emerges as the most important goal of algebra, with linear functions a cornerstone of beginning algebra. This paper provides a framework for ways to organize these ideas into a comprehensive and coherent curriculum and a set of dispositions that should be outcomes for students.

Folsom, M. L. (2002). *Empowering girls in math: The influence of curriculum on female beliefs about mathematics.* (Doctoral dissertation). Retrieved from Masters Abstracts International, 41(2). (ProQuest ID No. 766367131)

ABSTRACT: This qualitative inquiry examines the belief systems of female students in a sixth grade mathematics classroom and explores how a middle school math curriculum influences these beliefs. Specifically, this inquiry focuses on two of four internal beliefs posited by Gilah C. Leder: confidence and usefulness of mathematics. The design of this inquiry is loosely based on the research tradition of ethnography. Data collection consisted of classroom observations, teacher surveys, standardized test scores, and student questionnaires. The inquiry found that the math curriculum had some influence on the girls' overall attitude towards and enjoyment of math classes. Despite confusing explanations with overly complicated language and editing errors, the girls' enjoyed working through the math curriculum's small group activities and experiments. The inquiry found that the Connected Mathematics Project curriculum connected with the sixth grade girls.

Friel, S. N. & O’Connor, W. T. (1999). Sticks to the roof of your mouth?* Mathematics Teaching in the Middle School, 4*(6), 404–11.

ABSTRACT: Part of a special issue on teaching and learning the concepts of data and chance in the middle school. An activity that involves students comparing data sets by using data about 37 brands of peanut butter and their quality ratings is presented. The testing of the peanut butter, the graphing of the data, the determination of outliers, and the extension of the data analysis are discussed.

Friel, S. N. (1998). Teaching statistics: What’s average? In L. J. Morrow & M. J. Kenney (Eds.),* The Teaching and Learning of Algorithms in School Mathematics, 60th yearbook *(pp. 208- 217). Reston, VA: National Council of Teachers of Mathematics.

Garet, M., Wayne, A., Stancavage, F., Taylor, J., Eaton, M., Walters, K., Doolittle, F. (2011). *Middle school mathematics professional development impact study: Findings after the second year of implementation.* Washington, DC: U.S. Department of Education.

Garrison, A. L. (2013). *Understanding teacher and contextual factors that influence the enactment of cognitively demanding mathematics tasks.* (Unpublished doctoral dissertation). Vanderbilt University, Nashville, TN.

ABSTRACT: The level of challenge, or cognitive demand, of the tasks students solve is the foundation for their learning opportunities in mathematics classrooms. Unfortunately, it is difficult for teachers to effectively use cognitively demanding tasks (CDTs). I seek to understand how to support and improve mathematics teachers’ enactment of CDTs at scale. The purpose of this three-paper dissertation is to address some of the key unresolved questions and to set a direction for future research.

In paper 1, based on a comprehensive literature review, I identify 13 potentially relevant factors and elaborate a method for building on results from small-scale studies to better understand the enactment of CDTs across large samples of teachers.

Paper 2 investigates how teachers’ mathematical knowledge for teaching and their beliefs about teaching and learning mathematics are related to their enactment of CDTs. I found that aspects of teachers’ knowledge and beliefs are interconnected and are significantly related to their enactment of CDTs.

Paper 3 investigates changes in teachers’ enactment of CDTs over time and whether their interactions with colleagues (e.g., work with a math coach, advice-seeking interactions) are related to these changes. I found that the mere occurrence of interactions was generally not sufficient to support teachers’ development, and expertise available within interactions did not influence the productivity of those interactions. However, advice-seeking interactions were significantly related to teachers’ development. Further, the lack of expertise within interactions might have contributed to these findings.

These three studies suggest that there is much more to be understood about supporting teachers’ enactment of CDTs. There is, however, evidence that teachers’ mathematical knowledge for teaching and their beliefs about teaching and learning mathematics are integral to their enactment of CDTs, and that they are interrelated. In addition, it is clear that in designing supports for teachers’ enactment of CDTs, schools and districts should go beyond policies that provide only opportunities for interaction, and should specifically plan productive activities and enhance the available expertise within those interactions.

Gencturk, Y. C. (2012). *Teachers’ mathematical knowledge for teaching, instructional practices, and student outcomes.* (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign, Champaign, IL.

ABSTRACT: This dissertation examines the relationships among teachers’ mathematical knowledge, their teaching practices, and student achievement. Quantitative and qualitative data collection techniques (content knowledge assessments, surveys, interviews, and classroom observations) were used to collect data from 21 teachers and 873 students. Twenty-one in-service teachers who enrolled in a master’s program designed specifically for the needs of a partnership district were followed for 4 years to study how their mathematical knowledge as well as their teaching changed over time. Of the 21 teachers, 8 teachers were chosen for additional classroom observations and interviews. For the quantitative part of the study, two-level linear growth models were used to examine the effects of the mathematical knowledge of K-8 teachers on their instructional practices. After student-level data were added, three-level growth models were used to analyze the effects of teachers’ knowledge and instructional practices on students’ gain scores. Teachers’ beliefs about teaching and learning mathematics were also included in some analyses. The results indicated that, compared with the initial baseline data, teachers’ mathematical knowledge increased dramatically, and the teachers made statistically significant changes in their lesson design, mathematical agenda of the lessons, task choices, and classroom climate. The gains in teachers’ mathematical knowledge predicted changes in the quality of their lesson design, mathematical agenda, and classroom climate. Teachers’ beliefs were related to the quality of their lesson design, mathematical agenda, and the quality of the tasks chosen. However, only student engagement was significantly related to students’ gain scores. Neither teachers’ mathematical knowledge nor other aspects of instruction (inquiry-oriented teaching, the quality of task choices, and the classroom climate) were associated with students’ gain scores. The qualitative analyses revealed particular strands of the complex relationship between teachers’ mathematical knowledge and their instructional practices. Teachers’ beliefs played a mediating role in the relationship between teachers’ mathematical knowledge and instructional practices. Teachers favoring standards-based views of mathematics tended to teach in more inquiry-oriented ways and ask more questions of students; however, among teachers with limited mathematical knowledge, these practices seemed superficial. Additionally, the teachers’ task choices appeared to be confounded by teachers’ current level of mathematical knowledge and their textbook use.

Genz, R. (2006). *Determining high school students’ geometric understanding using Van Hiele Levels: is there a difference between Standards-based curriculum students and non-Standards-based curriculum students?* (Unpublished master’s thesis). Brigham Young University, Provo, UT.

ABSTRACT: Research has found that students are not adequately prepared to understand the concepts of geometry, as they are presented in a high school geometry course (e.g. Burger and Shaughnessy (1986), Usiskin (1982), van Hiele (1986)). Curricula based on the National Council of Teachers of Mathematics (NCTM) Standards (1989, 2000) have been developed and introduced into the middle grades to improve learning and concept development in mathematics. Research done by Rey, Reys, Lappan and Holliday (2003) showed that Standards-based curricula improve students’ mathematical understanding and performance on standardized math exams. Using van Hiele levels, this study examines 20 ninth-grade students’ levels of geometric understanding at the beginning of their high school geometry course. Ten of the students had been taught mathematics using a Standards-based curriculum, the Connected Mathematics Project (CMP), during grades 6, 7, and 8, and the remaining 10 students had been taught from a traditional curriculum in grades 6, 7, and 8. Students with a Connected Mathematics project background tended to show higher levels of geometric understanding than the students with a more traditional curriculum (NONcmp) background. Three distinctions of students’ geometric understanding were identified among students within a given van Hiele level, one of which was the students’ use of language. The use of precise versus imprecise language in students’ explanations and reasoning is a major distinguishing factor between different levels of geometric understanding among the students in this study. Another distinction among students’ geometric understanding is the ability to clearly verbalize an infinite variety of shapes versus not being able to verbalize an infinite variety of shapes. The third distinction identified among students’ geometric understanding is that of understanding the necessary properties of specific shapes versus understanding only a couple of necessary properties for specific shapes.

Goodell, J. E., & Parker, L. H. (2001). Creating a connected, equitable mathematics classroom: Facilitating gender equity. In B. Atweh, H. Forgasz, & B. Nebres (Eds.), *Sociocultural research on mathematics education: An international perspective* (pp. 411- 431). Hillsdale, NJ: Lawrence Erlbaum Associates.

Goodman, E. (2004). *Connected Mathematics Project: A constructivist view of mathematics education in the middle grades. *(Masters thesis). Retrieved from Masters Abstracts International, 43(2). (ProQuest ID No. 813809801)

ABSTRACT: For decades, education critics have been debating what and how mathematics should be taught. The following Master's thesis examines a new mathematics curriculum, Connected Math Project, geared to teach mathematics from a constructivist approach. It examines whether or not the students are able to reflect knowledge or understanding of mathematical concepts as well as their ability to learn from group motivated investigation. It also looks at the view and beliefs of mathematics teachers towards a constructivist program. This thesis is founded on the notion that public school educators must introduce a mathematics curriculum that enables all children to increase their problem solving skills and abilities with regards to mathematics.

Grandau, L., & Stephens, A. C. (2006). Algebraic thinking and geometry. *Mathematics Teaching in the Middle School, 11*(7), 344–349.

ABSTRACT: This article describes how two middle school teachers incorporated algebraic thinking into their textbook-based geometry lessons. One teacher embedded algebraic concepts within an existing textbook lesson while the other teacher elicited algebraic thinking by extending a textbook lesson.

Griffin, L., Evans, A., Timms, T., Trowell, T. (2000). *Arkansas Grade 8 Benchmark Exam: How do Connected Mathematics schools compare to state data?* Little Rock, AR: Arkansas State Department of Education.

Grunow, J. E. (1998). *Using concept maps in a professional development program to assess and enhance teachers’ understanding of rational numbers.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 60(3). (ProQuest ID No. 734420161)

ABSTRACT: This professional-development institute was designed to look at a little researched component, adult learning in a specific content area. Rational-number understanding was the domain addressed. Teachers were selected to participate in a two-week institute and were supported the following year with on-line mentoring.

Assessment of teacher rational-number knowledge was done using concept maps, tools chosen because of their congruence with the domain. Concept maps, as an alternative assessment measure, had potential to satisfy another need, authentic assessment of the professional-development experience.

The study investigated three questions: (1) Will middle-school teachers' understanding of rational number, as reflected on concept maps, be enhanced as a result of participation in a professional-development institute designed specifically to develop understanding of a domain? (2) Will middle-school teachers' understanding of interrelationships among concepts and awareness of contexts that facilitate construction of conceptual knowledge, as assessed through concept maps, be increased as a result of participation designed with a focus on reform curricula, authentic pedagogy, and learner cognitions to facilitate decision-making? (3) Will middle-school teachers communicate knowledge growth through well-elaborated graphic displays using concept maps?

The research design used was both qualitative and quantitative. Participants designed preinstitute concept maps of rational-number understanding. Following instruction, participants designed postinstitute concept maps to reflect their learning. Quantitative analysis of the concept maps was achieved by scoring participant maps against an expert criterion map and a convergence score was used. Qualitative analysis of the maps was done using holistic techniques to determine overall proficiency.

The Wilcoxon Signed-Ranks Test was used to analyze the data obtained from scoring the concept maps. Three areas were examined: (a) knowledge of concepts and terminology; (b) knowledge of conceptual relationships; and (c) ability to communicate through concept maps. Results of scoring in all areas yielded significant gains. Holistic scoring showed all participants attaining proficiency with regard to rational-number understanding.

It was concluded that teacher knowledge of a content domain can be enhanced significantly in a professional-development experience designed to concentrate on such growth, that teachers can become aware of contexts that facilitate development of content knowledge, and that concept maps can be valid, reliable measures.

Gutstein, E. (2006). "The real world as we have seen it": Latino/a parents' voices on teaching mathematics for social justice. *Mathematical Thinking and Learning, 8*(3), 331-358.

ABSTRACT: This article describes the views of Latino/a parents who supported social justice mathematics curriculum for their children in a 7th-grade Chicago public school classroom in which I was the teacher. The parents viewed dealing with and resisting oppression as necessary parts of their lives; they also saw mathematics as integral and important. Because (mathematics) education should prepare one for life -and injustice, resistance, and mathematics were all interconnected parts of life -an education made sense if it prepared children to be aware of and respond to injustices that they faced as members of marginalized communities. Such education may be unusual, but it was congruent with the parents' core values and worth standing up for.

Haile, T. K. (2012). A* study on the use of history in middle school mathematics: The case of Connected Mathematics Curriculum.* (Unpublished doctoral dissertation). The University of Texas at Austin, Austin, TX.

ABSTRACT: This dissertation explores the use of history of mathematics in middle school mathematics. A rationale for the importance of the incorporation of historical dimensions (HD) of mathematics is provided through a review of the literature. The literature covers pedagogical, philosophical, psychological, and social issues and provides arguments for the use of history. The central argument is that history can help reveal significant aspects regarding the origins and evolutions of ideas that provide contexts for understanding the mathematical ideas. History can be used as a means to reflect on significant aspects—errors, contractions, challenges, breakthroughs, and changes—of mathematical developments. Noting recent NCTM (2000) calls for school math to include so-called process standards, I contend that incorporating the history of mathematics can be considered as part of this standard. This study examines how HD is addressed in a contemporary mathematics curriculum. Specifically, the study examines the Connected Mathematics Project (CMP) as a case. This curriculum has some historical references which triggered further exploration on how seriously the historical aspects are incorporated. The analysis and discussion focus on four CMP units and interviews with three curriculum experts, eight teachers, and 11 middle school students. The analysis of textbooks and interviews with the experts explore the nature and purpose of historical references in the curriculum. The interviews with teachers and students focus on their perspectives on the importance of HD in learning mathematics. This study examines specifically historical incorporations of the concepts of fractions, negative numbers, the Pythagorean Theorem, and irrational numbers . The analysis reveals that CMP exhibits some level of historical awareness, but the incorporation of HD was not systematically or seriously considered in the development of the curriculum. The interviews suggest that the teachers did not seriously use the limited historical aspects available in the textbooks. The experts’ and teachers’ interviews suggest skepticism about the relevance of HD for middle school mathematics. The teachers’ accounts indicate that students are most interested in topics that are related to their experience and to future applications. The students’ accounts do not fully support the teachers’ assessment of students’ interest in history. I contend that incorporating HD can complement instruction in ways that relate to students’ experiences and to applications besides adding an inquiry dimension to instruction.

Halat, E. (2006). Sex-related differences in the acquisition of the Van Hiele levels and motivation in learning Geometry.* Asia Pacific Education Review, 7*(2), 173-183.

ABSTRACT: The purpose of this study was to examine the acquisition of the van Hiele levels and motivation of sixth-grade students engaged in instruction using van Hiele theory-based mathematics curricula. There were 150 sixth-grade students, 66 boys and 84 girls, involved in the study. The researcher employed a multiple-choice geometry test to find out students’ reasoning stages and a questionnaire to detect students’ motivation in regards to the instruction. These instruments were administered to the students before and after a five-week period of instruction. The paired-samples t-test, the independent-samples t-test, and ANCOVA with α = .05 were used to analyze the quantitative data. The study demonstrated that there was no statistically significant difference as in motivation between boys and girls, and that no significant difference was detected in the acquisition of the levels between boys and girls. In other words, gender was not a factor in learning geometry.

Halat, E. (2007). Reform-based curriculum & acquisition of the levels. *Eurasia Journal of Mathematics, Science & Technology Education, 3*(1), 41–49.

ABSTRACT: The aim of this study was to compare the acquisition of the van Hiele levels of sixth- grade students engaged in instruction using a reform-based curriculum with sixth-grade students engaged in instruction using a traditional curriculum. There were 273 sixth-grade mathematics students, 123 in the control group and 150 in the treatment group, involved in the study. The researcher administered a multiple-choice geometry test to the students before and after a five-week of instruction. The test was designed to detect students’ reasoning stages in geometry. The independent-samples t-test, the paired- samples t-test and ANCOVA with α = .05 were used to analyze the data. The study demonstrated that although both types of instructions had positive impacts on the students’ progress, there was no statistical significant difference detected in the acquisition of the levels between the groups.

Hallagan, J. E. (2003). *Teachers' models of student responses to middle school algebraic tasks.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 64(2). (ProQuest ID No. 765247341)

ABSTRACT: Often, the difficulties of students to make the cognitive leap from arithmetic to algebra is related to instructional strategies. The way teachers make sense of their practice, in turn, informs pre-service and in-service algebraic instruction. Algebraic instruction is also of current interest due to recent national initiatives calling for all students to learn high school algebra.

The purpose of this study was to describe middle school mathematics teachers' models or interpretations of students' responses to middle school algebraic tasks. The research questions focused on the nature of the teachers' developing ideas and interpretations of student responses from selected algebraic tasks involving the distributive property and equivalent expressions. The core research questions were: (a) What information do middle school mathematics teachers acquire about their students' algebraic thinking? and (b) How do middle school mathematics teachers interpret their students' algebraic thinking? A models and modeling framework guided the study's design. Model-eliciting activities were used to perturb and at the same time reveal their thinking. These activities consisted of asking the teachers to create a "Ways of Thinking" sheet based upon students' responses to the selected algebraic tasks, and to select, analyze and interpret samples of student work. Five teachers participated from two middle schools. Data collection included classroom observation, artifact collection from the model-eliciting activities, semi-structured interviews, and team discussions.

Two sets of findings emerged from this study. First, I concluded that the models and modeling perspective is indeed an effective methodology to elicit teachers' models of their students' algebraic thinking. Second, I found the following five aspects are central to teachers' models of student responses to tasks with equivalent expressions and the distributive property. Teachers recognized that students: (a) tended to conjoin expressions, (b) desired a numerical answer, and (c) had difficulty writing algebraic generalizations. In addition, teachers identified that (d) visual representations were highly useful as instructional tools. And finally, (e) the teachers in this study needed more experience in analyzing and interpreting student work. The findings from this study revealed consistent information across the Ways of Thinking sheets, library of student work, individual and team interviews, and classroom observations.

Hallagan, J. E. (2004). *A teacher’s model of students’ algebraic thinking about equivalent expressions.* Paper presented at the 28th Conference of the International Group of the Psychology of Mathematics Education, Bergen, Norway.

ABSTRACT : This research report describes the findings of a study on teachers' ways of interpreting student responses to tasks involving equivalent expressions. The teachers in this study were engaged in model-eliciting activities designed to promote the development of their knowledge and reveal their models (or interpretations) of their students' algebraic thinking about equivalent expressions by creating a library of their students' work. This report focuses on one teacher's model of his algebraic practice. Results showed that this teacher devoted a significant amount of time to the implementation of the algebraic unit. The teacher employed visual strategies for the first time and began to perceive their usefulness in demonstrating the equivalency of two expressions.

Hansen-Thomas, H. (2009). Reform-oriented mathematics in three 6th Grade classes: How teachers draw in ELLs to academic discourse. *Journal of Language, Identity, and Education, 8*(2&3), 88-106.

ABSTRACT: Traditionally, mathematics has been considered easy for English language learners (ELLs) due to the belief that math is a "universal language." At the same time, reform-oriented mathematics curricula, designed to promote mathematical discourse, are increasingly being adopted by schools serving large numbers of ELLs. CMP, the Connected Math Project, is one such reform-oriented curriculum. Taking a community-of-practice approach, this article compares how three 6th grade mathematics teachers in a Spanish/English community utilized language to draw ELLs into content and classroom participation. Teacher use of standard language fell into 2 categories: (a) modeling and (b) eliciting student practice. In the teacher's class that regularly elicited language, ELLs were successful on academic assessments; whereas students in the other 2 classes were not. Results suggest that CMP facilitates ELLs' learning and that a focus on mathematical language and elicitation benefits the development of mathematical discourse and content knowledge.

Harris, K., Marcus, R., McLaren, K., & Fey, J. (2001). Curriculum materials supporting problem-based teaching. *Journal of School Science and Mathematics, 101*(6), 310-318.

ABSTRACT: The vision for school mathematics described by the National Council of Teachers of Mathematics (NCTM) suggests a need for new approaches to the teaching and learning of mathematics, as well as new curriculum materials to support such change. This article discusses implications of the NCTM standards for mathematics curriculum and instruction and provides three examples of lessons from problem-based curricula for various grade levels. These examples illustrate how the teaching of important mathematics through student exploration of interesting problems might unfold, and they highlight the differences between a problem-based approach and more traditional approaches. Considerations for teaching through a problem-based approach are raised, as well as reflections on the potential impact on student learning.

Hartmann, C. (2004). Using teacher portfolios to enrich the methods course experiences of prospective mathematics teachers. School Science and Mathematics, 104(8), 392-407.

ABSTRACT: This paper illustrates ways to employ teacher portfolios to improve the quality of methods course experiences for prospective mathematics teachers. Based upon research conducted in an undergraduate teacher preparation program, this case study describes how the author used teacher portfolios to mentor prospective teachers in new ways. The case describes the author's experiences through a case study of his assessment of and response to one prospective teacher's portfolio. This portfolio illustrated themes that were present in other teachers' portfolios, but did so in ways that highlighted strategies for change to the methods course. Through the lens of this teacher's portfolio the author identified specific ways that the prospective teacher's beliefs were impacting her teaching practice, a result that enabled him to better help all of the teachers in the methods course reflect on their teaching. By providing a detailed account of the feedback process that led to this result, this paper illustrates how mathematics teacher educators can use prospective teachers' portfolios to enrich the quality of their methods courses.

Hattikudur, S., Prather, R. W., Asquith, P., Alibali, M. W., Knuth, E. J., & Nathan, M. (2012). Constructing graphical representations: Middle schoolers’ intuitions and developing knowledge about slope and y-intercept. *School Science and Mathematics, 112*(4), 230-240.

ABSTRACT: Middle-school students are expected to understand key components of graphs, such as slope and y-intercept. However, constructing graphs is a skill that has received relatively little research attention. This study examined students’ construction of graphs of linear functions, focusing specifically on the relative difficulties of graphing slope and y-intercept. Sixth-graders’ responses prior to formal instruction in graphing reveal their intuitions about slope and y-intercept, and seventh- and eighth-graders’ performance indicates how instruction shapes understanding. Students’ performance in graphing slope and y-intercept from verbally presented linear functions was assessed both for graphs with quantitative features and graphs with qualitative features. Students had more difficulty graphing y-intercept than slope, particularly in graphs with qualitative features. Errors also differed between contexts. The findings suggest that it would be valuable for additional instructional time to be devoted to y-intercept and to qualitative contexts.

Heck, D. J., Banilower, E. R., Weiss, I. R., & Rosenberg, S. L. (2008). Studying the effects of professional development: The case of the NSF's local systemic change through teacher enhancement initiative. *Journal for Research in Mathematics Education, 39*(2), 113-152.

ABSTRACT: Enacting the vision of NCTM's Principles and Standards for School Mathematics depends on effective teacher professional development. This 7-year study of 48 projects in the National Science Foundation's Local Systemic Change Through Teacher Enhancement Initiative investigates the relationship between professional development and teachers' attitudes, preparedness, and classroom practices in mathematics. These programs included many features considered to characterize effective professional development: content focus, extensive and sustained duration, and connection to practice and to influences on teachers' practice. Results provide evidence of positive impact on teacher-reported attitudes toward, preparedness for, and practice of Standards-based teaching, despite the fact that many teachers did not participate in professional development to the extent intended. Teachers' perception of their principals' support for Standards-based mathematics instruction was also positively related to these outcomes.

Herbel-Eisenmann, B. A. (2000). *How discourse structures norms: A tale of two middle school mathematics classrooms.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62 (1). (ProQuest ID No. 727910361)

ABSTRACT: My experiences as a student and a teacher of mathematics have led me to pursue the topic of this dissertation--discourse patterns and norms in two "reform-oriented" mathematics classrooms. The two 8th grade classrooms that form the focus of this dissertation were using the Connected Mathematics Project, an NSF-funded curriculum project. I was intrigued by the teachers and their teaching because I noticed the students seemed to have similar understandings, but each classroom felt different to me as a participant-observer.

These classrooms offered a context that allowed me to study differences in the context of similarity. The teachers had many attributes in common (detailed in Chapter 5): similar academic backgrounds and professional development activities, same certification, same school, same curriculum and similar enthusiasm for it, same heterogeneous group of students, similar student-understandings, etc. However, the teaching in the two classrooms was different. Drawing from the sociolinguistics and mathematics education literatures, I describe the social and sociomathematical norms of the two classrooms in terms of the classroom discourse which they were embedded in and carried by. I also interpret student understandings whenever possible throughout the thesis, taking a social constructivist perspective. In the year prior to commencing my dissertation study (1997-1998), I completed classroom observations and student interviews as part of my practicum work and research assistantship, which were used to form preliminary hypothesis about student understandings and the classroom environment. The data used for this dissertation was collected over the next two years (1998-2000). During the first, I observed and audio-and video-taped students on a weekly basis. In addition, students were interviewed about their algebraic understandings and their classroom experience. The second year, one of the classrooms was observed to trace the formation of the norms in the classroom. The teachers took part in four extensive interviews, in which they were asked about influencing experiences related to their teaching and the norms in their classroom (in terms of the expectations, rights and roles of themselves and their students).

The ideas I investigate in this dissertation include how social and sociomathematical norms are embedded in and carried by the classroom discourse in each classroom (Chapters 6 and 7). I also discuss aspects in the teachers' professional lives that influenced the ways they think about and work to establish and maintain the norms in their classrooms (Chapter 5). In Chapter 8, I look across the two classrooms to offer what I see as being similar and different, which has allowed me to locate differences in: the overall structure of teacher talk, the positioning of the teacher with respect to the locus of authority, the way each teacher draws from potential other knowledge sources in the classroom (i.e. students and the textbook), and the way each teacher draws attention to the common knowledge constructed in the classroom.

Herbel-Eisenmann, B. A. (2002). Using student contributions and multiple representations to develop mathematical language. *Mathematics Teaching in the Middle School, 8*(2), 100-105.

ABSTRACT: Describes a way to introduce and use mathematical language as an alternative to using vocabulary lists to introduce students to mathematical language in mathematics classrooms. Draws on multiple representations and student language.

Herbel-Eisenmann, B. A. (2004). An examination of textbook “voice”: How might discursive choice undermine some of the goals of the reform? In D. McDougall & J. Ross (Eds.),* Proceedings of the 26th Meeting of the North American Group for the Psychology of Mathematics Education* (Vol. 2, pp. 862-870). Toronto, Canada: PME-NA.

Herbel-Eisenmann, B. A. (2007). From intended curriculum to written curriculum: Examining the "voice" of a mathematics textbook. *Journal for Research in Mathematics Education, 38*(4), 344-369.

ABSTRACT: In this article, I used a discourse analytic framework to examine the "voice" of a middle school mathematics unit. I attended to the text's voice, which helped to illuminate the construction of the roles of the authors and readers and the expected relationships between them. The discursive framework I used focused my attention on particular language forms. The aim of the analysis was to see whether the authors of the unit achieved the ideological goal (i.e., the intended curriculum) put forth by the NCTM's Standards (1991) to shift the locus of authority away from the teacher and the textbook and toward student mathematical reasoning and justification. The findings indicate that achieving this goal is more difficult than the authors of the Standards documents may have realized and that there may be a mismatch between this goal and conventional textbook forms.

Herbel-Eisenmann, B. A., & Phillips, E. (2008). Analyzing student work: A context for connecting and extending algebraic knowledge for teachers. In C. E. Greenes & R. Rubenstein (Eds.), *Algebra and algebraic thinking in school mathematics, 70th yearbook* (pp. 295-311). Reston, VA: National Council of Teachers of Mathematics.

Herbel-Eisenmann, B. A., Smith III, J. P., and Star, J. R. (1999). *Middle school students’ algebra learning: Understanding linear relationships in context.* Paper presented at the annual meeting of American Educational Research Association, Montreal, Canada.

Herbel-Eisenmann, B., Wagner, D., & Cortes, V. (2010). Lexical bundle analysis in mathematics classroom discourse: the significance of stance. *Educational Studies in Mathematics*, 75, 23-42.

ABSTRACT: In this article, we introduce the lexical bundle, defined by corpus linguists as a group of three or more words that frequently recur together, in a single group, in a particular register (Biber, Johansson, Leech, Conrad, & Finegan, 2006; Cortes, English for Specific Purposes 23:397–423, 2004). Attention to lexical bundles helps to explore hegemonic practices in mathematics classrooms because lexical bundles play an important role in structuring discourse and are often treated as “common sense” ways of interacting. We narrow our findings and discussion to a particular type of lexical bundle (called a “stance bundle” or bundles that relate to feelings, attitudes, value judgments, or assessments) because it was the most significant type found. Through comparing our corpus from secondary mathematics classrooms with two other corpora (one from university classrooms (not including mathematics classrooms) and one from conversations), we show that most of the stance bundles were particular to secondary mathematics classrooms. The stance bundles are interpreted through the lens of interpersonal positioning, drawing on ideas from systemic functional linguistics. We conclude by suggesting additional research that might be done, discussing limitations of this work, and pointing out that the findings

Herron-Thorpe, F. L., Olson, J. C., & Davis, D. (2010). Shrinking your class. *Mathematics Teaching in the Middle School, 15*(7), 386-391.

Hill, H. C. (2007). Mathematical knowledge of middle school teachers: Implications for the No Child Left Behind policy initiative. *Educational Evaluation and Policy Analysis, 29*(2), 95-114.

ABSTRACT: This article explores middle school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teachers' subject matter preparation, certification type, teaching experience, and their students' poverty status. The author administered multiple-choice measures to a nationally representative sample of teachers and found that those with more mathematical course work, a subject-specific certification, and high school teaching experience tended to possess higher levels of teaching-specific mathematical knowledge. However teachers with strong mathematical knowledge for teaching are, like those with full credentials and preparation, distributed unequally across the population of U.S. students. Specifically, more affluent students are more likely to encounter more knowledgeable teachers. The author discusses the implications of this for current U.S. policies aimed at improving teacher quality.

Hill, H. C., & Charalambos, C. Y. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Lessons learned and open issues. *Journal of Curriculum Studies, 44*(4), 559-576.

ABSTRACT: This paper draws on four case studies to perform a cross-case analysis investigating the unique and joint contribution of mathematical knowledge for teaching (MKT) and curriculum materials to instructional quality. As expected, it was found that both MKT and curriculum materials matter for instruction. The contribution of MKT was more prevalent in the richness of the mathematical language employed during instruction, the explanations offered, the avoidance of errors, and teachers' capacity to highlight key mathematical ideas and use them to weave the lesson activities. By virtue of being ambitious, the curriculum materials set the stage for engaging students in mathematical thinking and reasoning; at the same time, they amplified the demands for enactment, especially for the low-MKT teachers. The analysis also helped develop three tentative hypotheses regarding the joint contribution of MKT and the curriculum materials: when supportive and when followed closely, curriculum materials can lead to high-quality instruction, even for low-MKT teachers; in contrast, when unsupportive, they can lead to problematic instruction, particularly for low-MKT teachers; high-MKT teachers, on the other hand, might be able to compensate for some of the limitations of the curriculum materials and offer high-quality instruction. This paper discusses the policy implications of these findings and points to open issues warranting further investigation.

Hill, H. C., & Charalambos, C. Y. (2012). Teaching (un)Connected Mathematics: Two teachers’ enactment of the Pizza Problem. *Journal of Curriculum Studies, 44(*4), 467-487.

ABSTRACT: This paper documents the ways mathematical knowledge for teaching (MKT) and curriculum materials appear to contribute to the enactment of a 7th grade Connected Mathematics Project lesson on comparing ratios. Two teachers with widely differing MKT scores are compared teaching this lesson. The comparison of the teachers' lesson enactments suggests that MKT appears to contribute to the mathematical richness of the lesson, teacher ability to capitalize on student ideas, and capacity to emphasize and link key mathematical ideas; yet the relationship of MKT to whether and how students participated in mathematical reasoning was more equivocal. Curriculum materials seemed to contribute to instructional quality, in that the novel tasks contained in the curriculum laid the groundwork for in-depth student problem-solving experiences; they also prevented the low-MKT teacher from making a mathematical error. At the same time, these ambitious materials influenced enactment because of the difficulties they caused teachers: the lesson's tasks needed to be ‘repaired' to enable students to engage with the main mathematical ideas, and off-track student responses to these tasks required remediation. Only the higher-MKT teacher was successfully able to meet the challenge, a finding suggestive of the confluence of MKT and the curriculum materials in informing instructional quality.

Hirsch, C. R. & Reys, B. J. (2009). Mathematics curriculum: A Vehicle for school improvement. *International Journal on Mathematics Education, 41*(6), 749-761.

ABSTRACT: Different forms of curriculum determine what is taught and learned in US classrooms and have been used to stimulate school improvement and to hold school systems accountable for progress. For example, the intended curriculum reflected in standards or learning expectations increasingly influences how instructional time is spent in classrooms. Curriculum materials such as textbooks, instructional units, and computer software constitute the textbook curriculum, which continues to play a dominant role in teachers’ instructional decisions. These decisions influence the actual implemented curriculum in classrooms. Various curriculum policies, including mandated end-of course assessments (the assessed curriculum) and requirements for all students to complete particular courses (e.g., year-long courses in algebra, geometry, and advanced algebra or equivalent integrated mathematics courses) are also being implemented in increasing numbers of states. The wide variation across states in their intended curriculum documents and requirements has led to a historic and precedent-setting effort by the Council of Chief State School Officers and the National Governors Association Council for Best Practices to assist states in the development and adoption of common College and Career Readiness Standards for Mathematics. Also under development by this coalition is a set of common core state mathematics standards for grades K-12. These sets of standards, together with advances in information technologies, may have a significant influence on the textbook curriculum, the implemented curriculum, and the assessed curriculum in US classrooms in the near future.

Hodges, T. & Cady, J. A. (2012). Negotiating contexts to construct an identify as a mathematics teacher.* The Journal of Educational Research, 105*(2), 112-122.

ABSTRACT: The authors focused on 1 middle-grades mathematics teacher's identity and her efforts to implement standards-based instructional practices. As professionals, teachers participate in multiple professional communities and must negotiate and manage conflicting agendas. The authors analyze how the contexts of these communities influence the teacher's identity and thus her teaching of mathematics.

Hoffmann, A. J. (2004). Middle school mathematics students' motivations for participating in whole-class discussions: Their beliefs, goals, and involvement. (Doctoral dissertation). *Retrieved from Dissertation Abstracts International, 65*(9). (ProQuest ID No. 795927881)

ABSTRACT: Whole-class discussions in mathematics classrooms are considered to foster active sense-making and intellectual autonomy among students. Through participating in these discussions, students have the opportunity to develop skills of mathematical communication, reasoning, and justification. However, middle school students may resist participating in whole-class discussions if they perceive social consequences resulting from this activity.

Research on mathematics classroom discourse typically focuses on the role of the teacher in discourse, examining student variables as outcomes to measure the effectiveness of the teachers' strategies. Alternatively, in this study, students' beliefs and goals are examined for how they influence students' participation in classroom discourse rather than as outcomes.

I assessed beliefs and goals of 15 target students from two seventh grade mathematics classrooms through one-on-one interviews and a Likert-scale survey instrument. Students' talk in interviews was analyzed through the use of a framework that included imperative verbs to capture idealized states, repetition to capture emphasis, and connections to affect to capture relative importance to the student. This framework allowed for a more rigorous analysis of students' beliefs in contrast to reporting any and all of their responses to interview questions.

Students' involvement in classroom discourse was described based on an analyses of videotaped classroom discussions about four investigation problems from the Connected Mathematics Project Standards-based mathematics curriculum.

Results from this study indicate that students' involvement in classroom discussions is influenced by their social goals and epistemological beliefs. Students who believed they learned mathematics through a process of negotiation and associated a low level of risk with participating in discussion were more likely to extend their participation during an interaction, critique the thinking of their classmates, and talk about mathematics at a high level of explicit meaning. There were also differences in students' involvement between the target students based on their classrooms.

This study illustrates how adolescence intersects with the mathematics reform movement by taking into account students' perspectives. Future research investigating how beliefs and goals relate to students' involvement in discussions may explain how a classroom of students together supports the development of effective classroom discussions.

Hoover, M. N., J. S. Zawojewski, & J. Ridgeway. (1997). *Effects of the Connected Mathematics Project on student attainment. *Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Hughes, E. K. (2006).* Lesson planning as a vehicle for developing pre-service secondary teachers’ capacity to focus on students’ mathematical thinking.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 68(1). (ProQuest ID No. 1288646801)

ABSTRACT: This study investigated the extent to and ways in which attention to students’ mathematical thinking was evident in the written lesson plans or lesson planning process of ten pre-service secondary mathematics teachers at various points during their teacher education program: prior to and immediately after participation in a course (the Teaching Lab) that emphasized students’ mathematical thinking as a key element of planning, during teachers’ first semester of their field experience as they planned lessons in their actual practice of teaching, and near the end of the first semester of their field experience as they planned lessons on demand and for university assignments.

With respect to learning from the Teaching Lab, the study shows that the teachers demonstrated significant growth on pre to post course measures in their ability to attend to students’ thinking when planning a lesson on demand and for a university assignment. Furthermore, teachers continued to be able to apply these ideas when planning on demand and for university assignments several months later. When investigating whether or not teachers would apply the ideas they had learned when planning in their own practice, the study suggests three findings. First, teachers’ attention to students’ thinking when planning lessons that used tasks with a high level of cognitive demand was not significantly different from their planning for a lesson on demand or the lesson plan they produced for the Teaching Lab assignment. Furthermore, teachers were more likely to attend to students’ thinking when planning a lesson that used a high-level task compared to a lesson that used a low-level task. Second, for some teachers, written lesson plans significantly under-represented their attention to students’ thinking in their planning process. Finally, the study suggests that support from the mentor teacher and/or university supervisor may be an important factor in determining whether or not the teacher applies their knowledge of attention to student’s thinking to their planning in practice.

Hull, L. S. H. (2000). *Teachers' mathematical understanding of proportionality: Links to curriculum, professional development, and support.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62(2). (ProQuest ID No. 727942411)

ABSTRACT: The Proportional Relationship Study was designed to investigate whether using a standards-based middle school mathematics curriculum, together with professional development and followup support, can lead to increased teacher content knowledge and pedagogical content knowledge of proportionality. From the literature, it is clear that what teachers do in the classroom affects what students learn, and that what teachers know affects their actions in the classroom. Teachers need strong personal content knowledge and pedagogical content knowledge in order to teach mathematics well; therefore, the question is an important one.

Seven sites participated in a statewide implementation effort during 1996-1999 that included Connected Mathematics Project (CMP) curriculum professional development experiences for teachers plus additional district and/or campus support. As part of this study, the Proportional Reasoning Exercise (PRE) was given to seventh-grade teachers three times: before CMP professional development, after a year of teaching with CMP materials, and again after a second year of teaching with the materials. Teacher responses were coded for correctness and for problem-solving strategy; group responses were compared for all three PREs. In addition, group and individual interviews were conducted with CMP teachers.

Data from the three PREs anti group and individual interviews of seventh-grade teachers showed growth in performance and understanding of proportional relationships over the two-year period. Analysis of each of the PRE problems revealed an increase in the percent of teachers who correctly answered the problems and a tendency toward using more sophisticated proportional relationship strategies. However, choice of strategy appeared to depend on the context of the problem. Participants also tended over time to record multiple and more diverse strategies, increase the depth and detail of their written explanations, and include units along with numbers.

Interviews after the first year confirmed that experienced teachers placed in a new situation, with new curriculum and expectations of using new instructional approaches, often revert to "novice" status, concerned primarily with survival (Borko & Livingston, 1989). However, individual interviews conducted after the second year showed that teachers were then ready to focus on student understanding of mathematics and were themselves learning new and important mathematics.

Hunter, M. A. (2006). Opportunities for environmental science and engineering outreach through K-12 mathematics programs. *Environmental Engineering Science, 23*(3), 461-471.

ABSTRACT: Programs to improve mathematics education provide an opportunity to educate K-12 students about environmental science and engineering. Many professional organizations as well as the National Science Foundation have developed activities for middle school and high school teachers that can be utilized by higher education faculty when participating in such programs. A hands-on workshop, provided a discussion of environmental and civil engineering as a career for young women whom participated in a girls mathematics day called "Y2M, Yes to Mathematics" hosted at a local community college. Another project involving 10 school districts on Long Island, provided the opportunity to incorporate environmental science and engineering outreach to middle school students. The project goal is to increase the time students spend on mathematics in mathematics, science, and technology classes using suitable pedagogy and curricula. The first year of the 5-year program involved organizing and training of district teams, then developing a district plan for increasing the math content across the curriculum. The second year involved training of additional middle school teachers and piloting exemplary materials. The second year of this program has been completed and progress towards meeting the expected goals and benchmarks such as improved performance on the NY state Mathematics assessment and increased use of mathematics in the science classroom has occurred. Incorporation of mathematics into the science curricula can occur through environmental science or engineering activities. The program should, in turn, significantly improve the students' understanding of mathematics and increase their interest in environmental science and engineering.

Huntley, Mary Ann. (2008). A framework for analyzing differences across mathematics curricula. *National Council of Supervisors of Mathematics Journal, 10*(2), 10-27.

Hwang, S., Cai, J., Shih, J., Moyer, J. C., Wang, N., & Nie, B. (2015). Longitudinally investigating the impact of curricula and classroom emphases of the algebra learning of students of different ethnicities. *In Large-Scale Studies in Mathematics Education*(pp. 45–60). doi:10.1007/978-3-319-07716-1

Institute of Education Sciences (2010). Connected Mathematics Project (CMP). What Works Clearinghouse Intervention Report. What Works Clearinghouse.

ABSTRACT: The "Connected Mathematics Project" ("CMP") is a mathematics curriculum designed for students in grades 6-8. Each grade level of the curriculum is a full-year program and covers numbers, algebra, geometry/measurement, probability, and statistics. The curriculum uses an investigative approach, and students utilize interactive problems and everyday situations to learn math concepts. The What Works Clearinghouse (WWC) reviewed 79 studies of "CMP." No studies of "CMP" meet WWC evidence standards, and one study meets WWC evidence standards with reservations. The one study included more than 12,000 students from grades 6-8 in Texas. Based on this study, the WWC considers the extent of evidence for "CMP" to be small for math achievement. "CMP" was found to have no discernible effects on math achievement. Appended to this report are: (1) Study characteristics: Schneider, 2000 (quasi-experimental design); (2) Outcome measure for the math achievement domain; (3) Summary of study findings included in the rating for the math achievement domain; (4) Summary of cohort findings for the math achievement domain; (5) "CMP" rating for the math achievement domain; and (6) Extent of evidence by domain. (Contains 9 notes.) [The following study is reviewed in this intervention report: Schneider, C. L. (2000). "Connected Mathematics and the Texas Assessment of Academic Skills" (Doctoral dissertation, University of Texas at Austin, 2000). Dissertation Abstracts International, 62(02), 503A. (UMI No. 3004373). For previous WWC intervention reports on the "Connected Mathematics Project," see ED499297 (2007) and ED485389 (2004).

Izsák, A. (2000). Inscribing the winch: Mechanisms by which students develop knowledge structures for representing the physical world with algebra. *Journal of the Learning Sciences, 9*(1), 31-74.

ABSTRACT: I propose and test an account of mechanisms by which students develop knowledge structures for modeling the physical world with algebra. The account begins to bridge the gap between current mathematics curricula, in which modeling activities play an important role, and theoretical accounts of how students learn to model, which lag behind. After describing the larger study, in which I observed 12 pairs of 8th-grade students introduce and refine algebraic representations of a physical device called a winch, I then focus on 1 pair that generated an unconventional yet sound equation. Because the prevailing genetic accounts of knowledge structures in mathematics education, cognitive science, and information-processing psychology do not explain key characteristics of the data, I begin to construct a new developmental account that does. To do so, I use forms, a class of schemata that combine patterns of algebra symbols with patterns of experience in the physical world, and 2 mechanisms, notation variation and mapping variation. I then use forms and the 2 mechanisms to analyze how the selected pair of students introduced and refined initial, faulty algebraic representations of the winch into an unconventional yet sound equation.

Izsák, A. (2003). “We want a statement that is always true”: Criteria for good algebraic representations and the development of modeling knowledge. *Journal for Research in Mathematics Education, 34*(3), 191-227.

ABSTRACT: Presents a case study in which two 8th grade students developed knowledge for modeling a physical device called a winch. Demonstrates that students have and can use criteria for evaluating algebraic representations. Explains how students can develop modeling knowledge by coordinating criteria with knowledge for generating and using algebraic representations.

Izsák, A. (2004). Students' coordination of knowledge when learning to model physical situations. *Cognition and Instruction, 22*(1), 81-128.

ABSTRACT: In this article, I present a study in which 12 pairs of 8th-grade students solved problems about a physical device with algebra. The device, called a winch, instantiates motions that can be modeled by pairs of simultaneous linear functions. The following question motivated the study: How can students generate algebraic models without direct instruction from more experienced others? The first main result of the study is that students have and can use criteria for judging when I algebraic expression is better than another. Thus, students can use criteria to regulate their problem-solving activity. The second main result is that constructing knowledge for modeling with algebra can require students to coordinate criteria for algebraic representations with several other types of knowledge that I also identify in the article. These results contribute to research on students' algebraic modeling, cognitive processes and knowledge structures for using mathematical representations, and the development of mathematical knowledge.

Izsák, A. (2005). "You have to count the squares": Applying knowledge in pieces to learning rectangular area. *Journal of the Learning Sciences, 14*(3), 361-403.

ABSTRACT: This article extends and strengthens the knowledge in pieces perspective (diSessa, 1988, 1993) by applying core components to analyze how 5th-grade students with computational knowledge of whole-number multiplication and connections between multiplication and discrete arrays constructed understandings of area and ways of using representations to solve area problems. The results complement past research by demonstrating that important components of the knowledge in pieces perspective are not tied to physics, more advanced mathematics, or the teaming of older students. Furthermore, the study elaborates the perspective in a particular context by proposing knowledge for selecting attributes, using representations, and evaluating representations as analytic categories useful for highlighting some coordination and refinement processes that can arise when students learn to use external representations to solve problems. The results suggest, among other things, that explicitly identifying similarities and differences between students' past experiences using representations to solve problems and demands of new tasks can be central to successful instructional design.

Izsák, A. (2008). Mathematical knowledge for teaching fraction multiplication. *Cognition and Instruction, 26*(1), 95-143.

ABSTRACT: The present study contrasts mathematical knowledge that two sixth-grade teachers apparently used when teaching fraction multiplication with the Connected Mathematics Project materials. The analysis concentrated on those tasks from the materials that use drawings to represent fractions as length or area quantities. Examining the two teachers' explanations and responses to their students' reasoning over extended sequences of lessons led to a theoretical frame that emphasizes relationships between teachers' unit structures and pedagogical purposes for using drawings. In particular, the present study builds on the distinction made in past research between reasoning with two and with three levels of quantitative units and demonstrates that reasoning with three levels of units is necessary but insufficient if teachers are to use students' reasoning with units as the basis for constructing generalized numeric methods for fraction arithmetic. Teachers need also to assemble three-level unit structures with flexibility supported by drawn versions of the distributive property.

Izsák, A., Tillema, E., & Tunc-Pekkan, Z. (2008). Teaching and learning fraction addition on number lines. *Journal for Research in Mathematics Education, 39*(1), 33–62.

ABSTRACT: We present a case study of teaching and learning fraction addition on number lines in one sixth-grade classroom that used the Connected Mathematics Project Bits and Pieces II materials. Our main research questions were (1) What were the primary cognitive structures through which the teacher and students interpreted the lessons? and (2) Were the teacher's and her students' interpretations similar or different, and why? The data afforded particularly detailed analyses of cognitive structures used by the teacher and one student to interpret fractions and their representation on number lines.

Jackson, K. J., Shahan, E. C., Gibbons, L. K., & Cobb, P. (2012). Launching complex tasks. *Mathematics Teaching in the Middle School, 18*(1), 24-29.

Jackson, K., Cobb, P., Wilson, J., Webster, M., Dunlap, C., & Appelgate, M. (2015). Investigating the development of mathematics leaders’ capacity to support teachers’ learning on a large scale. *ZDM*, 47, 93–104. doi:10.1007/s11858-014-0652-5.

ABSTRACT: A key aspect of supporting teachers’ learning on a large scale concerns mathematics leaders’ practices in designing for and leading high-quality professional development. We report on a retrospective analysis of an initial design experiment aimed at supporting the learning of three math leaders who were charged with supporting the learning of middle-grades mathematics teachers across a large US school district. Initial goals for the math leaders’ learning included: (a) viewing teachers’ improvement of their classroom practices as a progression;(b) designing supports for teachers’ learning that were informed by assessments of teachers’ current practices, were oriented towards long-term goals for teachers’ practices, and would enable teachers to attain short-term goals that constituted reasonable next steps; and (c) facilitating professional development by pressing on teachers’ ideas differentially and building on their contributions. Findings suggest that the math leaders increasingly viewed teachers’ improvement of their classroom practices as a developmental progression and began to design connected sequences of activities. However, they struggled to facilitate the activities in ways that would meet their ambitious goals for teachers’ learning. Based on our findings, we indicate potential improvements to our design for supporting math leaders’ learning. More generally, we provide the field with a set of potentially revisable learning goals for math leaders’ learning, a set of principles to guide the design of supports for their learning, and a provisional design to support the development of their practices

Jackson, K., Garrison, A., Gibbons, L., Shahan, E., Wilson, J. (2013). Exploring relationships between setting up complex tasks and opportunities to learn in concluding whole-class 5 discussions in middle-grades mathematics instruction. *Journal for Research in Mathematics Education, 44*(4), 646-682.

ABSTRACT: This article specifies how the setup, or introduction, of cognitively demanding tasks is a crucial phase of middle-grades mathematics instruction. The authors report on an empirical study of 165 middle-grades mathematics teachers' instruction that focused on how they introduced tasks and the relationship between how they introduced tasks and the nature of students' opportunities to learn mathematics in the concluding whole-class discussion.

Jansen, A. & Herbel-Eisenmann, B. A. (2001). *Moving from a reform junior high to a traditional high school: Affective, academic, and adaptive mathematical transitions.* Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA.

Jansen, A. (2006). Seventh graders’ motivations for participating in two discussion-oriented mathematics classrooms. *Elementary School Journal, 106*(5), 409–428.

ABSTRACT: In this study I examined the self-reported motivational beliefs and goals supporting the participation of 15 seventh graders in whole-class discussions in 2 discussion-oriented Connected Mathematics Project classrooms. Through this qualitative investigation using semistructured interviews, I inductively identified and described the students' motivational beliefs and goals and relations among them. Results demonstrated beliefs that constrained students' participation and ones that supported their participation. Students with constraining beliefs were more likely to participate to meet goals of helping their classmates or behaving appropriately, whereas students with beliefs supporting participation were more likely to participate to demonstrate their competence and complete their work. Results illustrated how the experiences of middle school students in discussion-oriented mathematics classrooms involve navigating social relationships as much as participating in opportunities to learn mathematics.

Jansen, A. (2008). An investigation of relationships between seventh-grade students' beliefs and their participation during mathematics discussions in two classrooms. *Mathematical Thinking and Learning, 10*(1), 68-100.

ABSTRACT: As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.

Johanning, D. I. (2005). *Learning to use fractions after learning about fractions: A study of middle school students developing fraction literacy.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 66(4). (ProQuest ID No. 913515271)

ABSTRACT: There is a large body of literature, both empirical and theoretical, that focuses on what is involved in learning fractions when fractions are the focus or goal of instruction. However, there is very little research that explores how students learn to use what they have learned about fractions outside instruction on fractions. The specific goal of this research was to explore how middle school students learned to use fraction knowledge, the fraction concepts and skills studied in formal curriculum units, in mathematical instructional settings where fractions were not the main focus of study, but rather supported the development of other mathematical content.

This study is sociocultural in nature. It is guided by a practice account of literacy (Scribner and Cole, 1981) and Barton's (1994) ecological approach to literacy. Studying literacy involves studying the practices that people engage in as they use knowledge for specific purposes in specific contexts of use. This research describes the practices that grade six and seven students engaged in when they had to use what they learned about fractions to make sense of mathematical contexts such as area and perimeter, decimal operations, probability, similarity, and ratio. In order to understand how the practices students engaged in when learning to use fractions differed from the practices students engaged in when learning about fractions, data collection and analysis focused on identifying and then comparing these two types of practices.

Data collection for this dissertation spanned approximately one and one-half school years. In the fall of 2002 and winter of 2003 I collected data during the two units where one class of sixth-grade students learned about fractions. In the spring of 2003 I began to collect data during three units where these sixth-grade students were using fractions as part of learning about area and perimeter, decimal operations, and probability. Data collection continued into seventh grade as I followed a subset of these sixth-grade students into their seventh-grade year. Data was collected during two seventh grade units were fractions were used in the context of similarity and ratio. Data collection ended in December of 2003. The data collected included field notes, video recordings of whole class discussions, video-recording the small-group interactions of one group of four focus students, interviews with the four focus students, and copies of their written work.

The study's results revealed that students did not simply take the concepts and skills learned in the fractions units and use them. Understanding how to use fractions was tied to understanding situations in which they can be used. Students had to take into account both mathematical and situational contexts when making choices about how to use fractions. This led students to raise questions regarding what was appropriate when using fractions in these new contexts and how fractions and the new context were related. It was clear that the conversations these students had regarding the use of fractions were not only different from the conversations they had when learning about fractions, but potentially may not have occurred when learning about fractions directly. It is argued that providing students the opportunity to use fraction knowledge is critical to the development of fraction literacy.

Johanning, D. I. (2008). Learning to use fractions: Examining middle school students' emerging fraction literacy. *Journal for Research in Mathematics, 39*(3), 281-310.

ABSTRACT: This article describes 1 prevalent practice that a group of 6th-and 7th-grade students engaged in when they used fractions in the context of area and perimeter, decimal operations, similarity, and ratios and proportions. The study's results revealed that students did not simply take the concepts and skills learned in formal fractions units and use them in these other mathematical content areas. Their understanding of how to use fractions was tied to their understanding of situations in which they could be used.

Johanning, D. I. (2010). Designing curricula to grow and extend mathematical knowledge. In R. Reys & B. Reys (Eds.), *K-12 mathematic curriculum: Issues, trends, and future directions, 72nd yearbook* (pp. 171-180). Reston, VA: National Council of Teachers of Mathematics.

Johanning, D. I., & Keusch, T. (2004). Teaching to develop students as learners. In R. N. Rubenstein (Eds.), *Perspectives on the Teaching of Mathematics, 66th yearbook*. Reston: VA: National Council of Teachers of Mathematics.

Jong, C., Pedulla, J. J., Reagan, E. M., Salomon-Fernandez, Y., & Cochran-Smith, M. (2010). Exploring the link between reformed teaching practices and pupil learning in elementary school mathematics. *School Science and Mathematics, 110*(6), 309–326.

ABSTRACT: This study examined the classroom practices of beginning elementary school teachers’ instruction of mathematics and how it connected to their pupils’ learning. The Reformed Teaching Observation Protocol (RTOP) was used to measure the extent to which beginning teachers used reformed teaching practices. As a measure of pupil learning, we utilized assessment scores specific to the mathematics unit observed and correlated them with teachers’ RTOP scores. We found that beginning teachers who implemented reformed teaching practices tended to have pupils who scored higher on the district mathematics test with a statistically significant correlation of 0.56 (p < .05). Implications of these findings and others are discussed in terms of using the RTOP to improve practice at the elementary school level and for future school-based research.

Kaput, J., & Thompson, P. (1994). Technology in mathematics education research: The first 25 years in the JRME. *Journal for Research in Mathematics, 25*(6), 676-684.

Kar, T., & Isik, C. (2015). Comparison of Turkish and American Seventh Grade Mathematics Textbooks in Terms of Addition and Subtraction Operations with Integers. Egitim ve Bilim, 40(177).

ABSTRACT: This study analyzes how addition and subtraction with integers are presented in Turkish and American mathematics textbooks. Analyses focus on how the concepts are given as well as the nature of the presented mathematical problems. It was found that both the Turkish and the American textbooks emphasized the relationships among different representations in teaching addition and subtraction with integers. It was found that the coordination among visual representation, verbal explanations and mathematical sentences was constructed in a more organized manner in the textbook named Connected Mathematics 2. It was found that operational skill oriented problems were proportionately featured more in the Turkish textbooks whereas the problems requiring high-level cognitive skills such as mathematical reasoning and problem posing were featured more in the American textbooks.

Kasmer, L. (2008). *The role of prediction in the teaching and learning of algebra. *(Doctoral dissertation). Retrieved from ProQuest Dissertations and Theses database. (UMI 3303469)

ABSTRACT: Research has shown that including prediction questions within reading and science instruction has been advantageous for students, yet minimal research existed regarding the use of such questions within mathematics instruction. In order to extend and build on our knowledge about the effects of prediction in mathematics instruction, this study explored the impact of this paradigm in the teaching and learning of algebra. Specifically, this study probed whether utilizing prediction questions provided students opportunities for engaging in mathematical thinking, retrieving prior knowledge, and discussing related mathematical ideas, could increase such students' conceptual understanding and mathematical reasoning in the content area of algebra.

To address the research questions, a longitudinal quasi-experimental study was conducted to explore to what extent and in what ways prediction questions could help students develop mathematical reasoning and conceptual understanding. In this research, instruction and learning for two groups of students were examined whereby prediction questions were infused within the treatment class, while the control group received instruction devoid of such prediction questions. Both groups were taught by the same teacher and curriculum, with no initial significant differences between these two groups. During the course of one school year within this treatment group, the teacher employed prediction questions at the launch of each lesson and then revisited the student predictions at the closure of the lesson. A total of 1,178 unit assessment responses and 494 responses to Mathematical Reflections were examined, along with videotaped sessions from both classes to explore out-come based differences between the two groups. In addition, 491 prediction responses from the treatment class were coded for levels of reasoning and characteristics of prediction responses.

The overall results suggest prediction is a relevant and valid construct with respect to enhanced conceptual understanding and mathematical reasoning. The treatment class outperformed the control class on a number of measures. The benefits from a teacher's perspective were also identified. Prediction questions became a catalyst for classroom discussions, increased student engagement, and an informal assessment tool for the teacher. Through this study, benefits for instruction, professional development, and curriculum design in relation to prediction became apparent.

Kasmer, L. A. & Kim, O-K. (2012). The nature of student predictions and learning opportunities in middle school algebra. *Educational Studies in Mathematics, 79*(2), 175-191.

ABSTRACT: In this article, we describe how using prediction during instruction can create learning opportunities to enhance the understanding and doing of mathematics. In doing so, we characterize the nature of the predictions students made and the levels of sophistication in students’ reasoning within a middle school algebra context. In this study, when linear and exponential functions were taught, prediction questions were posed at the launch of the lessons to reflect the mathematical ideas of each lesson. Students responded in writing along with supportive reasoning individually and then discussed their predictions and rationale. A total of 395 prediction responses were coded using a dual system: sophistication of reasoning, and the mechanism students appeared to utilize to formulate their prediction response. The results indicate that using prediction provoked students to connect among mathematical ideas that they learned. It was apparent that students also visualized mathematical ideas in the problem or the possible results of the problem. These results suggest that using prediction in fact provides learning opportunities for students to engage in mathematical sense making and reasoning, which promotes students’ understanding of the mathematics that they learn.

Kasmer, L., & Kim, O. K. (2011a). Using prediction to motivate personal investment in problem solving. In D. Brahier (Ed.), *Motivation and disposition: Pathways to learning mathematics, 73rd yearbook. *Reston, VA: National Council of Teachers of Mathematics.

Kasmer, L., & Kim, O. K. (2011b). Using prediction to promote mathematical reasoning and understanding. *School Science and Mathematics Journal, 111*(1), 20-33.

ABSTRACT: Research has shown that prediction has the potential to promote the teaching and learning of mathematics because it can be used to enhance students' thinking and reasoning at all grade levels in various topics. This article addresses the effectiveness of using prediction on students' understanding and reasoning of mathematical concepts in a middle school algebra context. In the treatment classroom, prediction questions were utilized at the launch of each algebra lesson, and in the control classroom such questions were not used. Both classrooms were taught by the same teacher and used the same curriculum. After completing each of the linear and exponential units, the two classrooms were compared in terms of their mathematical understanding and reasoning through unit assessments. Overall, the treatment classroom outperformed the control classroom on the unit assessments. This result supports that prediction is a valid construct with respect to enhanced conceptual understanding and mathematical reasoning.

Katwibun, D. (2004). *Middle school students' mathematical dispositions in a problem-based classroom. *(Doctoral dissertation). Dissertation Abstracts International, 65(5). (ProQuest ID No. 766026571)

ABSTRACT: The purpose of this study was to describe middle school students' mathematical dispositions in a problem-based learning [PBL] classroom. Eight volunteer students from one 6th grade mathematics classroom participated in this study. The curriculum used was the Connected Mathematics Project [CMP]. The main sources for data collection were classroom observations, the Attitudes and Beliefs questionnaire, teacher interviews, and student interviews. The CMP class routine consisted of four phases: Warm-up, Launch, Explore, and Summarize. The teacher in this study had her students investigate mathematics problems within cooperative small groups and share their ideas in large group discussions. The teacher acted as a facilitator and encouraged her students to try new ideas without fear of making mistakes. The findings revealed that almost all of the students in this study demonstrated positive mathematical dispositions. They volunteered and shared their ideas, both in small cooperative group investigations and in large group discussions. They believed mathematics was about "learning new ideas" and mathematics was "life" because it was everywhere in their lives. They also mentioned the usefulness of numbers, measurement, and geometry in their daily lives. All eight participants liked hands-on activities and working on a mathematics project. Most of them agreed that they liked mathematics because it was fun and interactive. Most also saw themselves as good at mathematics. All of them agreed that mathematics was useful, and that one's mathematics ability could be increased by effort. They also believed that there were no gender differences in mathematics, even though in their class, they realized that boys outperformed girls. Most of the students agreed that they could solve time-consuming mathematics problems and that it was important to understand mathematical concepts. None of them had negative feelings about group work; they learned from each other.

Finally, an analysis of the participants' mathematical dispositions was discussed. Based on the Taxonomy of Educational Objects: Affective Domains by Krathwohl, Bloom, and Masia (1964), the participants were categorized into three disposition levels: Level 1: "receiving;" Level 2: "responding;" and Level 3: "valuing." Half of the participants demonstrated dispositions at the high level (Level 3: "valuing") because of their willingness to pursue and/or seek to do mathematics outside the classroom. Three of them were in mathematics disposition Level 2.3: "satisfaction in response" because they usually participated in the class activities. They were satisfied and enjoyed doing mathematics. One of them demonstrated mathematical disposition Level 1.2: "willingness to receive" because she listened to the whole class and group discussions without sharing any ideas or asking for help when she needed it.

Keiser, J. M. (1997). *The development of students' understanding of angle in a non-directive learning environment.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 58 (8). (ProQuest ID No. 736600251)

ABSTRACT: Curriculum reform in mathematics shows that geometry is becoming an important part of the middle grades curriculum. This dissertation study looks at the geometry learning of sixth-grade students who were using a newly-drafted unit, Shapes and Designs, from a reformed middle grades curriculum, the Connected Mathematics Project (CMP).

The research focuses on students' understandings of angle concepts. The research questions are as follows: What understandings of angle concepts are revealed by sixth-grade students during their geometry investigations? Which concepts are particularly difficult (easy) for students to grasp? What are some of them is conceptions they hold? How well-connected are their ideas and what are the gaps in their thinking concerning the angle concepts that are presented?

A CMP pilot-testing school in Michigan was chosen as the site for in-class observations since the teachers had been teaching with CMP materials for two years. Two mathematics classrooms were observed daily during the duration of the Shapes and Designs instruction which lasted 5 weeks during the winter of1995-96. The researcher observed and audio-taped all classroom discourse and collected samples of students' work. Data were transcribed and analyzed for important themes in the students' understandings. Results revealed that students' understandings of angle concepts are disconnected and fragile. Students tend to focus on one of three aspects--the angle's vertex, its rays, or its interior region. These unbalanced concept images often exclude many angles from being considered as angles, and can also interfere with other understandings such as angle size. However, these understandings are a very natural part of development given three different influences--the mathematical community's construction of the angle concept throughout history, the students' everyday experiences and language, and the instructional approach--all of which were highly influential factors in the students' development of the angle concept.

Keiser, J. M. (1997). *The role of definition in the mathematics classroom. *Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Keiser, J. M. (1997). *The development of students' understanding of angle in a non-directive learning environment.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 58 (8). (ProQuest ID No. 736600251)

ABSTRACT: Curriculum reform in mathematics shows that geometry is becoming an important part of the middle grades curriculum. This dissertation study looks at the geometry learning of sixth-grade students who were using a newly-drafted unit, Shapes and Designs, from a reformed middle grades curriculum, the Connected Mathematics Project (CMP).

The research focuses on students' understandings of angle concepts. The research questions are as follows: What understandings of angle concepts are revealed by sixth-grade students during their geometry investigations? Which concepts are particularly difficult (easy) for students to grasp? What are some of them is conceptions they hold? How well-connected are their ideas and what are the gaps in their thinking concerning the angle concepts that are presented?

A CMP pilot-testing school in Michigan was chosen as the site for in-class observations since the teachers had been teaching with CMP materials for two years. Two mathematics classrooms were observed daily during the duration of the Shapes and Designs instruction which lasted 5 weeks during the winter of1995-96. The researcher observed and audio-taped all classroom discourse and collected samples of students' work. Data were transcribed and analyzed for important themes in the students' understandings. Results revealed that students' understandings of angle concepts are disconnected and fragile. Students tend to focus on one of three aspects--the angle's vertex, its rays, or its interior region. These unbalanced concept images often exclude many angles from being considered as angles, and can also interfere with other understandings such as angle size. However, these understandings are a very natural part of development given three different influences--the mathematical community's construction of the angle concept throughout history, the students' everyday experiences and language, and the instructional approach--all of which were highly influential factors in the students' development of the angle concept.

Keiser, J. M. (2000). The role of definition. *Mathematics Teaching in the Middle School*, *5*(8), 506–511.

ABSTRACT: The writer examines the role that mathematical definitions can play in the middle grades math classroom, focusing on the concept of angle as it was introduced to sixth-grade students.

Keiser, J. M. (2010). Shifting our computational focus. *Mathematics Teaching in the Middle School, 16*(4), 216-223.

ABSTRACT: Through professional development activities involving action research, middle-grades teachers at this author's school learned how to honor students' prior knowledge and experience by finding out about their K-5 computational development. Rather than complaining about what their students did not know, they learned to appreciate results from their K-5 instruction. These results seem to indicate more conceptual understanding, a strong number sense, and increased computational flexibility than they had seen in the past. In this article, the author shares the data and the process that middle-grades teachers undertook to learn about their students. She describes how middle-grades teachers used the Connected Mathematics Project (CMP) for mathematics instruction. Overall, the process of learning about the computational knowledge that students bring to middle school has highlighted the importance of flexibility.

Keiser, J. M. (2012). Students’ strategies can take us off guard. *Mathematics Teaching in the Middle School, 17*(7), 418-425.

Keller, B. A., Martin, W. G., & Hart, E. W. (2001). *Illuminating NCTM’s Principles and Standards for School Mathematics.* School Science and Mathematics, 101(6), 292 304.

ABSTRACT: Describes electronic resources designed to illuminate the National Council of Teachers of Mathematics' (NCTM) "Principles and Standards for School Mathematics". Provides a vehicle for further discussion of the vision put forth in the Standards.

Kendrick, D. G. (2004). *High school algebra teachers' beliefs and attitudes about the mathematics reform movement and high-stakes testing: Implications for staff development.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 65(7). (ProQuest ID No. 775169461)

ABSTRACT: This study attempts to define urban teacher quality, understand teacher learning, and gauge the success of efforts to develop urban teacher competencies in implementing mathematics education reform in high school algebra classrooms. The Professional Standards for Teaching Mathematics (NCTM, 1991) includes two underlying premises of mathematics education reform: that teachers are the primary facilitators of change, and that teachers need adequate support to make changes. The study focuses on: (a) the extent to which reform-oriented practices are being implemented in high school algebra classrooms; (b) the relationship between teacher preparation and reform-oriented practices; and (c) the effects of teacher beliefs, school-level environment, and staff development on the mathematics education reform practices in high school algebra classrooms. The research includes examination of the responses of high school algebra teachers on a self-administered survey as well as the responses of high school algebra teachers and mathematics chairpersons in one-on-one interviews. The subjects were asked about their beliefs, actions, and needs as related to the NCTM Standards.

Kersaint, G. (1998). *Preservice elementary teachers' ability to generalize functional relationships: The impact of two versions of a mathematics content course.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 59(5). (ProQuest ID No. 1251814391)

ABSTRACT: This study investigated the impact of two versions of a mathematics content course designed for preservice elementary teachers on their growth in algebraic understanding. One section of the course was presented a traditional approach using instructor developed or compiled materials. Another section of the course was presented a function-based approach using algebra materials developed for middle school students by the Connected Mathematics Project (CMP). Specifically, this study examined the influence these materials had on preservice teachers' ability to generalize problem situations, to represent them symbolically, and to use their representation to solve related problems. Achievement gains and obstacles experienced by the students were also analyzed.

Data collection for this study included self-reported background data, instructor and student journals, written pre-and post-assessments, interviews, and observations. Qualitative and quantitative data analysis methods were used to analyze the data. Sfard's (1991) model of conceptual development was used as a lens by which to examine, describe, and interpret the students' conceptual understandings.

Achievement gains on the post-assessment were not statistically significant. Students from both classes performed similarly. Responses from the students in both sections of this course were characterized at the interiorization and condensation phases of Sfard's model. In spite of this, results from the study show differences in the kinds of understandings developed by the students. The section using the CMP materials focused on developing students' conceptual understanding of algebra. While the other section of the course focused on developing students' procedural understanding of algebra. In addition to developing conceptual understandings, students using the CMP algebra units reported that they learned an alternate method for introducing and teaching algebra.

Kilman, T. (2015). *The relationship between students’ applied mathematics skills and students’ attitudes towards mathematics.* (Doctoral Dissertation). Retrieved from The Aquila Digital Community, Paper 54. (Proquest ID No. 1664611515).

ABSTRACT: Mathematics is a subject with which many students struggle. It has been noted that students’ attitudes towards mathematics can often affect their performance in related courses. The goal of this research is to explore the relationship between students’ basic applied mathematics skills and students’ attitudes towards mathematics. That is, do students, as they learn how to use mathematics in the real world, tend to develop a more favorable outlook towards mathematics? Or, on the other hand, do the attitudes towards mathematics of students remain unaffected as their ability to use mathematics in the real world increases? The current research seeks to clarify these propositions in an effort to improve mathematics instruction by providing educators with a better understanding of students’ attitudes towards mathematics.

Multiple linear regression analysis found that attitude toward mathematics was indeed significantly related to students’ basic applied mathematics skill. Attitude toward mathematics explained 29.7% of the variance observed in basic applied mathematics skill. Attitudinal subscales were also analyzed. Student self-confidence and motivation were both significant predictors of basic applied mathematics skill. In a separate analysis, attitude toward mathematics was found not to be significantly related to mathematical achievement in the college classroom.

Kim, O. K., & Kasmer, L. (2007). Using "prediction" to promote mathematical reasoning. *Mathematics Teaching in the Middle School, 12*(6), 294-299.

ABSTRACT: This article introduces prediction as a useful tool to promote mathematical reasoning. First, the article addresses prediction expectations in state standards and gives examples. It also provides a classroom example and activities to illustrate what prediction can look like and how it can serve as a building block for the development of students' reasoning abilities. Second, the article suggests some ideas to teachers that promote reasoning when prediction is incorporated into mathematics lessons.

Kim, R. Y. (2012). The quality of non-textual elements in mathematics textbooks: An exploratory comparison between South Korea and the United States. *ZDM Mathematics Education, 44*(2), 175-187.

ABSTRACT: As an exploratory investigation, this study aims to analyze non-textual elements in some Korean and US mathematics textbooks using a conceptual framework whose components include accuracy, connectivity, contextuality, and conciseness. By analyzing three US textbooks and three Korean ones, the study not only shows patterns in the use of non-textual elements in mathematics textbooks in different contexts but also provides insights into how to assess the quality of non-textual elements in mathematics textbooks, which I hope will contribute to the provision of more meaningful and productive learning opportunities to school children. Overall, the results from this study show that there is significant difference across topics and textbooks, which implies different opportunities to learn through non-textual elements. This study makes a unique contribution to the conceptualization of non-textual elements in mathematics education and has implications for textbook analysis and curriculum development.

King, D. A. (2007). *A study to ascertain the effects of the Connected Mathematics Project on student achievement in the Buffalo public schools. *(Unpublished master’s thesis). State University of New York at Buffalo, Buffalo, NY.

King, K. D., Mitchell, M. B., Tybursky, J., Simic, O., Tobias, R., Barriteau Phaire, C., & Torres, M. (2011). Impact of teachers’ use of Standards-based instructional materials on students’ achievement in an urban district: A multilevel analysis. Paper presented at the Annual Meeting of the American Educational Research Association, New Orleans, LA.

ABSTRACT: This effectiveness study explores the relationship between the use and adaptation of the Connected Mathematics Project instructional materials by middle grades teachers in an urban school district and their students’ achievement. All middle grades mathematics teachers in Newark, NJ Public Schools were surveyed using the Surveys of Enacted Curriculum and the CMP Implementation Survey. The 6th, 7th, and 8th grade students in these teachers’ first period classes completed the New Jersey Assessment of Knowledge and Skills for their grade. Using hierarchical linear modeling with two levels, we found that both increased use and adaptation of the instructional materials were related to increased student achievement. Implications for further research on instructional materials implementation and the design and implementation of materials are discussed.

Kladder, R., Peitz, J., & Faulkner, J. (1998). *On the right track. Middle Ground, 1*(4), 32-34.

Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005). Middle school students' understanding of core algebraic concepts: Equivalence & variable. *ZDM, 37*(1), 68-76.

ABSTRACT: Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6–8). This article focuses on middle school students' understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students' understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed.

Knuth, E. J., Choppin, J. M., & Bieda, K. (2009). Middle school students’ production of mathematical justification. In D. Stylianou, M. Blanton, & E. Knuth (Eds.), *Teaching and learning proof across the grades: A K-16 perspective* (pp. 153-170). New York City, NY: Taylor and Francis.

Kramarski, B., & Mevarech, Z. R. (2003). Enhancing mathematical reasoning in the classroom: The effects of cooperative learning and metacognitive training. *American Educational Research Journal, 40*(1), 281-310.

ABSTRACT: The purpose of this study was to investigate the effects of four instructional methods on students' mathematical reasoning and metacognitive knowledge. The participants were 384 eighth-grade students. The instructional methods were cooperative learning combined with metacognitive training (COOP+META), individualized learning combined with metacognitive training (IND+META), cooperative learning without metacognitive training (COOP), and individualized learning without metacognitive training (IND). Results showed that the COOP+META group significantly outperformed the IND+META group, which in turn significantly outperformed the COOP and IND groups on graph interpretation and various aspects of mathematical explanations. Furthermore, the metacognitive groups (COOP+META and IND+META) outperformed their counterparts (COOP and IND) on graph construction (transfer tasks) and metacognitive knowledge. This article presents theoretical and practical implications of the findings.

Krebs, A. S. (1999). *Students' algebraic understanding: A study of middle grades students' ability to symbolically generalize functions. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 60(6). (ProQuest ID No. 733526481)

ABSTRACT: The publication of the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards in 1989 was pivotal in mathematics reform. The National Science Foundation funded several curriculum projects to address the vision described in the Standards. After these materials were developed and implemented in classrooms, questions arose surrounding students' learning and understanding. This study investigates students' learning in a reform curriculum. Specifically, "What do eighth grade students know about writing symbolic generalizations from patterns which can be represented with functions, after three years in the Connected Mathematics Project curriculum?"

The content, the curriculum, the data, and the site chosen define the study. Initially, the study surrounded students' algebraic understanding, but I focused it to investigate students' ability to symbolically generalize functions. Although this selection is a particular slice of algebra it represents a significant piece of the discipline.

I selected the Connected Mathematics Project (CMP) as the curriculum. I supported the authors' philosophy that the teaching and learning of algebra is an ongoing activity woven through the entire curriculum, rather than being parceled into a single grade level.

The data surrounded the solutions of four performance tasks, completed by five pairs of students. These tasks were posed for students to investigate linear, quadratic, and exponential situations. I collected and analyzed students' written responses, video recordings of the pairs' work, and follow-up interviews. The fourth choice determined the site. I invited Heartland Middle School, a pilot site of the CMP to participate in this study. I approached a successful teacher, Evelyn Howard, who allowed her students to participate. Together, we selected ten students who were typical students in her classroom to participate in this study.

In conclusion, I present two major findings of this study surrounding students' understanding of algebra. First, students who had three years in the Connected Mathematics Project curriculum demonstrated deep understanding of a significant piece of algebra. And second, teachers can learn much more about students' understanding in algebra by drawing on multiple sources of evidence, and not relying solely on students' written work.

Krebs, A. S. (2003). Middle grades students’ algebraic understanding in a reform curriculum. *School Science & Mathematics, 103*(5), 233-245.

Kulm, G., Capraro, R. M., & Capraro, M. M. (2007). Teaching and learning middle grades mathematics with understanding. *Middle Grades Research Journal, 2*(1), 23-48.

ABSTRACT: This study addresses the nexus of two critical challenges for today's mathematics teacher. On the one hand, teaching for understanding for all students is the goal of most mathematics teachers. However, many teachers also must acknowledge and address the requirement that students do well on high stakes tests. This study analyzed data on 6th grade students' performance and achievement after a year-long implementation of Connected Mathematics (CMP). Texas Assessment of Academic Skills (TAAS) data were analyzed, comparing students' achievement from 5th to 6th grade. The variables of at-risk, socio-economic status, and ethnicity were analyzed to determine the nature and practical importance of adopting CMP. The results indicated that the overall gain from using CMP materials over the previous year's mathematics achievement was four points (p less than 0.01). The at-risk students demonstrated a mean 10-point gain (p less than 0.01) while the non at-risk students demonstrated a mean 2-point gain.

Kulm, G., Wilson, L. D., Kitchen, R. (2005). Alignment of content and effectiveness of mathematics assessment items. *Educational Assessment, 10*(4), 333-356.

ABSTRACT: Alignment has taken on increased importance given the current high-stakes nature of assessment. To make well-informed decisions about student learning on the basis of test results, assessment items need to be well aligned with standards. Project 2061 of the American Association for the Advancement of Science (AAAS) has developed a procedure for analyzing the content and quality of assessment items. The authors of this study used this alignment procedure to closely examine 2 mathematics assessment items. Student work on these 2 items was analyzed to determine whether the conclusions reached through the use of the alignment procedure could be validated. It was found that the Project 2061 alignment procedure was effective in providing a tool for in-depth analysis of the mathematical content of the item and a set of standards and in identifying 1 particular content standard that was most closely aligned with the standard. Through analyzing student work samples and student interviews, it was also found that students' thinking may not correspond to the standard identified as best aligned with the learning goals of the item. This finding highlights the potential usefulness of analyzing student work to clarify any additional deficiencies of an assessment item not revealed by an alignment procedure.

Lachance, A., & Confrey, J. (2003). Interconnecting content and community: A qualitative study of secondary mathematics teachers. *Journal of Mathematics Teacher Education, 6*(2), 107- 37.

ABSTRACT: The publication of the National Council of Teachers of Mathematics initial Standards(1989) has acted as a catalyst to begin reforming the way mathematics is taught in the USA. However, the literature regarding reform movements suggests that changing our educational systems requires overcoming many barriers and is thus difficult to achieve. Reform in mathematics education, like reform movements in other areas of education, has thus been slow to take hold. One structure that has been shown to support educational reform, particularly instructional reform, has been teacher community. This paper discusses a professional development intervention that attempted to start a professional community among a group of secondary mathematics teachers through in-service work on mathematical problem solving and technology. The results of this study suggest that the use of mathematical content explorations in professional development settings provides a means to help mathematics teachers build professional communities. Together, these two components –mathematical content explorations and teacher community – provided these secondary mathematics teachers with a strong foundation for engaging in the reform of their mathematics classes.

Lambdin, D. V., & Lappan, G. (1997). *Dilemmas and issues in curriculum reform: Reflections from the Connected Mathematics Project.* Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Lambdin, D. V., Lynch, K., & McDaniel, H. (2000). Algebra in the middle grades. *Mathematics Teaching in the Middle School, 6*(3), 195-198.

ABSTRACT: The writers describe a weeklong series of lessons with their sixth graders that used bicycle racing as both a motivator and a context for thinking about rate of change and the shapes of graphs.

Lambdin, D., & Keiser, J. M. (1996). The clock is ticking: Time constraint issues in mathematics teaching reform. *Journal of Educational Research, 90*(4), 23-32.

ABSTRACT: Time issues raised by sixth-and seventh-grade teachers involved in field testing an NSF-sponsored investigation-centered mathematics curriculum (the Connected Mathematics Project--CMP) for middle-grades students were examined in this study. Questions investigated included the following: How much scheduled time is actually available for mathematics instruction in elementary and middle schools and how is it configured? How do project teachers and students spend their time in class? What factors influence CMP teachers' pacing through this new curriculum? Findings indicate that teaching in the spirit of the current mathematics education reform movement may be highly dependent upon flexibility in class scheduling. Innovations in teaching mathematics (e.g., increased group work, writing, extended projects, and alternative forms of assessment) seem to require additional time, and new ways of thinking about using class time.

Lambdin, D., & Preston, R. (1995). Caricatures in innovation: Teacher adaptation to an investigation-oriented middle school mathematics curriculum. *Journal of Teacher Education, 46*(2), 130-40.

ABSTRACT: The National Council of Teachers of Mathematics has released guidelines on educational reform in the development of mathematical curriculum, teaching methods and assessment. Some teachers were able to adapt to change without much problems, while others exercised a more cautious disposition to change. Reactions to teaching recommendations were conditioned by course content, instruction method, environmental factors and teacher's desire for a problem-free school day.

Lapan, R. T., Reys, B. J., Barnes, D. E., & Reys, R. E. (1998). Standards-based middle grade mathematics curricula: Impact on student achievement. Paper presented at the annual meeting of the American Association of Educational Research, San Diego, CA.

Lappan, G. (1997). The challenge of implementation: Supporting teachers. *American Journal of Education, 106*(1), 207-239.

ABSTRACT: Reform in mathematics education has been stimulated and propelled by the publication of standards documents by the National Council of Teachers of Mathematics. This article examines the vision of teacher decision making that is portrayed in NCTM Professional Teaching Standards: choosing worthwhile mathematical tasks, orchestrating and monitoring classroom discourse, creating an environment for learning, and analyzing one's practice. The philosophical orientation and the set of commitments to teaching and learning on which the standards are based include stances on equity, curriculum, teaching, and learning. These stances are summarized under the following headings: inclusiveness, depth over coverage, teaching for understanding, active engagement of students, curriculum investigations, applications, and connections.

Lappan, G., & Bouck, M. K. (1998). Developing algorithms for adding and subtracting fractions. In L. J. Morrow & M. J. Kenney (Eds.), *The Teaching and Learning of Algorithms in School Mathematics, 60th yearbook *(pp. 183-197). Reston, VA: National Council of Teachers of Mathematics.

Lappan, G., & Ferrini-Mundy, J. (1993). Knowing and doing mathematics: A new vision for middle grades students. *The Elementary School Journal, 93*(5), 625-639.

ABSTRACT: Research provides characteristics of effective programs for schools that want to restructure their programs to better meet the needs of students in the middle grades. Direction in revising both the curriculum and instruction in mathematics classrooms is provided by the National Council of Teachers of Mathematics in its two documents Curriculum and Evaluation Standards for School Mathematics and the Professional Standards for Teaching Mathematics. In this article we discuss mathematical tasks, classroom environments, and means of assessment that might encourage rich mathematical growth for middle grades students. Proposed changes in the mathematics content and processes emphasized in the middle grades are outlined. Shifts in the culture of the mathematics classroom that support students' development of mathematical power are described, and two problems that involve the mathematics content and processes we advocate are provided. Finally, we acknowledge the complexity of implementing such changes in tasks, environment, and assessment and point to the need for transformative research and structural shifts.

Lappan, G., & Phillips, E. (1998). Teaching and learning in the Connected Mathematics Project. In L. Leutzinger (Ed.), *Mathematics in the Middle.* Reston, VA: National Council of Teachers of Mathematics.

Lappan, G., Phillips, E. (2009). A designer speaks: Glenda Lappan and Elizabeth Phillips. *Educational Designer, Journal of the International Society for Design and Development in Education, 1*(3), 1-19. Retrieved from: http://www.educationaldesigner.org/ed/volume1/issue3/article11

ABSTRACT: The need to improve the teaching and learning of mathematics has been a focus of attention in the US over our entire careers. There have been waves of national interest in mathematics education that have attracted mathematicians and mathematics educators to the work of improving K-12 mathematics education. Today we will focus our remarks in two areas, our own curriculum development work including the story of how we came to engage in and accomplish the work and our comments on the challenges we face in future work to improve mathematics teaching and learning. We expect that many of the challenges we see are also challenges for mathematics education worldwide. First we will share relevant aspects of the work in which our research and development group have engaged for over 35 years. Many of these remarks are based on other papers that we have published about our work. But for this special audience we would like to tell you a bit of our personal stories.

Lappan, G., Phillips, E. D., & Fey, J. T. (2007). The case of Connected Mathematics. In C. R. Hirsch (Ed.), *Perspectives on the design and development of school mathematics curricula* (pp. 67-79). Reston, VA: National Council of Teachers of Mathematics.

Legaspi, A. V. C., & Rickard, A. (2011). A case study of multicultural education and problem-centered mathematics curricula. *National Forum of Multicultural Issues Journal, 9*(1), 1–18.

ABSTRACT: Multicultural education is an important issue in K-12 mathematics education. However, efforts to address multicultural education in K-12 mathematics, including through curriculum materials, are generally perceived as unsuccessful or having limited impact. This case study examines how a problem-centered middle school mathematics curriculum addresses multicultural education and then draws on studies that have investigated the effects of the curriculum on the mathematics achievement of diverse groups of students. The results of this study show that the curriculum incorporates three categories of multicultural elements throughout the curriculum to address multicultural education. Moreover, the work of other researchers shows that the effects of the curriculum on the mathematics achievement of all students, especially diverse groups of students, are positive and well documented. This case study motivates future research into whether the positive effects of the curriculum on the mathematics achievement of diverse students is due, in whole or in part, to the problem-centered structure of the curriculum (e.g., accommodates more diverse learning styles), the multicultural elements in the curriculum (e.g., makes mathematics more meaningful to diverse students), or both. Further research should also examine how other problem-centered mathematics curricula address multicultural education, including the effects of such curricula on the mathematics achievement of diverse groups of students, and to what extent such curricula help students develop positive attitudes and understandings about people from different cultural groups.

Lehrer, R., Kobiela, M., & Weinberg, P. J. (2012). Cultivating inquiry about space in a middle school mathematics classroom. *ZDM Mathematics Education, 45*(3), 365-376.

ABSTRACT: During 46 lessons in Euclidean geometry, sixth-grade students (ages 11, 12) were initiated in the mathematical practice of inquiry. Teachers supported inquiry by soliciting student questions and orienting students to related mathematical habits-of-mind such as generalizing, developing relations, and seeking invariants in light of change, to sustain investigations of their questions. When earlier and later phases of instruction were compared, student questions reflected an increasing disposition to seek generalization and to explore mathematical relations, forms of thinking valued by the discipline. Less prevalent were questions directed toward search for invariants in light of change. But when they were posed, questions about change tended to be oriented toward generalizing and establishing relations among mathematical objects and properties. As instruction proceeded, students developed an aesthetic that emphasized the value of questions oriented toward the collective pursuit of knowledge. Post-instructional interviews revealed that students experienced the forms of inquiry and investigation cultivated in the classroom as self-expressive.

Lepak, J. (2014). Enhancing Students' Written Mathematical Arguments. *Mathematics Teaching in the Middle School, 20*(4), 212-219.

Description: The article shares how one teacher used peer-review activities involving rubrics to support students’ arguments and justifications for the Pool problem in Say it With Symbols, among other tasks.

Lewis, J. M. & Blunk, M. L. (2012). Reading between the lines: Teaching linear algebra. *Journal of Curriculum Studies, 44*(4), 515-536.

ABSTRACT: This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were teaching from the same curriculum materials, the teacher with higher MKT had more complete and concise ways to describe key concepts, had multiple ways to represent ideas about linear equations, could move nimbly among different mathematical expressions of linear relationships, and gave students a larger role in articulating the mathematical ideas of the lesson. However, curriculum materials seem to have moderated what would otherwise have been larger disparities in the quality of instruction between the two teachers. The lower-MKT teacher made minor mathematical errors, stayed on topic, and defined concepts in reasonably accurate ways when he followed the curriculum materials closely.

Lewis, R. M. (2002). *Mathematics for all? The cultural relevance of connected mathematics.* (Masters thesis). Retrieved from Masters Abstracts International, 41(2). (ProQuest ID No. 766358581)

ABSTRACT: Studies have shown that White students consistently achieve higher than students of color. Recent calls for mathematics reforms have made many suggestions for narrowing this gap. One local school district adopted a standards-based mathematics program, Connected Mathematics, for the middle school level. Using theory from Freire, Giroux, Dewey, Tate, and Ladson-Billings, a framework for a culturally relevant curriculum was constructed. This inquiry project identifies the characteristics of a culturally relevant curriculum, examines why a culturally relevant curriculum is important in mathematics, and assesses Connected Mathematics for its cultural relevance based on the framework. Connected Mathematics was found to adequately support two of the five components of a culturally relevant curriculum.

Li, Y., & Lappan, G. (2014). *Mathematics curriculum in school education.* Springer.

Lim, K. H., Buendía, G., Kim, O. K., Cordero, F., & Kasmer, L. (2010). The role of prediction in the teaching and learning of mathematics. *International Journal of Mathematical Education in Science and Technology, 41(*5), 595-608.

ABSTRACT: The prevalence of prediction in grade-level expectations in mathematics curriculum standards signifies the importance of the role prediction plays in the teaching and learning of mathematics. In this article, we discuss benefits of using prediction in mathematics classrooms: (1) students' prediction can reveal their conceptions, (2) prediction plays an important role in reasoning and (3) prediction fosters mathematical learning. To support research on prediction in the context of mathematics education, we present three perspectives on prediction: (1) prediction as a mental act highlights the cognitive aspect and the conceptual basis of one's prediction, (2) prediction as a mathematical activity highlights the spectrum of prediction tasks that are common in mathematics curricula and (3) prediction as a socio-epistemological practice highlights the construction of mathematical knowledge in classrooms. Each perspective supports the claim that prediction when used effectively can foster mathematical learning. Considerations for supporting the use of prediction in mathematics classrooms are offered.

Liu, Y. (2014). Teachers’ in-the-moment noticing of students’ mathematical thinking: A case study of two teachers. (Unpublished doctoral dissertation). University of North Carolina at Chapel Hill, Chapel Hill, NC.

ABSTRACT: The purpose of this research is to access teachers’ in-the-moment noticing of students’ mathematical thinking, in the context of teaching a unit from a reform-based mathematics curriculum, i.e., Covering and Surrounding from Connected Mathematics Project. The focus of the study is to investigate the following research questions:

How and to what extent do teachers notice students’ mathematical thinking in the midst of instruction?

How and to what extent does teachers’ in-the-moment noticing of students’ mathematical thinking influence teachers’ instruction?

Conceptualized as a set of interrelated components in this study, the construct of teachers’ in-the-moment noticing of students’ mathematical thinking includes attending to students’ strategies, interpreting students’ understandings, deciding how to respond on the basis of students’ understandings, and responding in certain ways.

A review of literature reveals that much of the research on teacher noticing does not examine teacher noticing as it occurs in the midst of instruction. Rather, it involves asking teachers to analyze and reflect on videos outside the context and pressure of in-the-moment instruction. Thus, in order to access teachers’ in-the-moment noticing in a more explicit and direct way, the researcher in this study applied a new technology to explore teacher noticing, enabling two teacher participants to capture their noticing through their own perspectives while teaching in real time.

Findings indicate that teacher participants noticed for a variety of reasons, including student thinking, instructional adaptations, assessment, content, and student characteristics, focusing primarily on student thinking and instructional adaptations. Furthermore, these participants noticed student thinking in the midst of instruction to different extents, and made adjustments to instruction in different ways.

Examination of the data also suggests that teachers’ noticing of student thinking was shaped by teachers’ beliefs, knowledge, and goals. Therefore, influenced by these constructs, teachers noticed student thinking to different extents, influencing students’ opportunities to think mathematically in different ways. A diagram that illustrates the paths through which teachers traveled in the process of noticing is presented, as one of the findings.

Lloyd, G. M. (2006). Using K-12 mathematics curriculum materials in preservice teacher education: Rationale, strategies, and teachers' experiences. In K. Lynch-Davis, & R. L. Rider (Eds.), *The work of mathematics teacher educators: Continuing the conversation* (pp. 11-27). San Diego, CA: Association of Mathematics Teacher Educators.

Lloyd, G. M. (2008). Curriculum use while learning to teach: One student teacher's appropriation of mathematics curriculum materials. *Journal for Research in Mathematics Education, 39*(1), 63-94.

ABSTRACT: This article describes one student teacher's interactions with mathematics curriculum materials during her internship in a kindergarten classroom. Anne used curriculum materials from two distinct programs and taught lessons multiple times to different groups of children. Although she used each curriculum in distinct ways, her curriculum use was adaptive in both cases. Anne's specific ways of reading, evaluating, and adapting the curriculum materials contrast with previous results about beginning teachers' curriculum use. Several key factors appeared to contribute to Anne's particular ways of using the curriculum materials: features of her student-teaching placement, her personal resources and background, and characteristics of the materials. Directions for future research about student teachers' and other teachers' curriculum use are suggested in accord with these factors.

Lloyd, G. M., & Behm, S. L. (2005). Preservice elementary teachers' analysis of mathematics instructional materials. *Action in Teacher Education, 26*(4), 48-62.

ABSTRACT: This article explores the practice of engaging prospective elementary teachers in the analysis of lessons from different textbooks. A rationale for such engagement, based on particular teacher education goals, is provided. The article focuses on a specific lesson analysis assignment given to prospective elementary teachers in which portions of mathematics textbooks were compared and contrasted. Examination of 23 preservice teachers' analysis of two textbook lessons (one fairly traditional and one more reform oriented) revealed that, with very few exceptions, the preservice teachers searched the textbook lessons for familiar, mainly traditional instructional components. The teachers' preference for traditional lesson components appeared to contribute to a tendency to make considerable misinterpretations of the two textbook lessons. These tendencies, including ways that the teachers attempted to justify differences between the two lessons, offer important insights into prospective elementary teachers' conceptions of the role of textbooks in the teaching and learning process. In addition, these findings suggest the necessity of involving prospective teachers more extensively in the analysis of textbooks, curriculum materials, and other instruction resources so that richer, more useful conceptions may develop.

Lo, J-J., Cox, D., & Mingus, T. (2006). A conceptual-based curricular analysis of the concept of similarity. In Alatorre, S., Cortina, J.L., Sáiz, M., and Méndez, A. (Eds), Proceedings of the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Mérida, México: Universidad Pedagógica Nacional.

ABSTRACT: As they engage with activities in mathematics textbooks, students have a variety of opportunities to make sense of the concept of similarity. The nature and sequence of these activities have an impact on the development of concept images that support students as they make sense of the terms “similar figures” or “scale drawings” and the properties they hold. In this analysis of the treatment of similarity in three middle grade textbook series, the authors share their analysis of the concept definitions and concept images supported by these texts. The term “curriculum” has different meanings in different contexts. According to the Center for the Study of Mathematics Curriculum, the most familiar terms include the ideal curriculum, the intended curriculum, the enacted curriculum, the achieved curriculum and the assessed curriculum. The focus of the present study was on the intended curriculum, which typically includes teacher’s manuals, student books, and additional resources such as technology, assessment, etc.

Lowe, P. (2004). A new approach to math in the middle grades. *Principal, 84*(2), 34-39.

ABSTRACT: Part of a special section on mathematics teaching and learning. Suggestions for implementing reform programs such as Connected Mathematics Project in the middle grades are provided. The advantages and disadvantages of such research-based reform programs are also discussed.

Lubienski, S. T. (1996). *Mathematics for all? Examining issues of class in mathematics teaching and learning.* (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.

Lubienski, S. T. (1997). *Successes and struggles of striving toward “Mathematics for All”: A closer look at socio-economics.* Paper presented at the annual meeting of the American Education Research Association, Chicago, IL.

Lubienski, S. T. (1997). Class matters: A preliminary exploration. In J. Trentacosta, & M. J. Kenney (Eds.), *Multicultural and gender equity in the mathematics classroom, the gift of diversity, 59th Yearbook* (pp. 46-59). Reston, VA: National Council of Teachers of Mathematics.

ABSTRACT: As a researcher-teacher, I examined 7th-graders' experiences with problem-centered curriculum and pedagogy, focusing on SES differences in students' reactions to learning mathematics through problem solving. Although higher SES students tended to display confidence and solve problems with an eye toward the intended mathematical ideas, the lower SES students preferred more external direction and sometimes approached problems in a way that caused them to miss their intended mathematical points. An examination of sociological literature revealed ways in which these patterns in the data could be related to more than individual differences in temperament or achievement among the children. I suggest that class cultural differences could relate to students' approaches to learning mathematics through solving open, contextualized problems.

Lubienski, S. T. (2000a). Problem solving as a means toward mathematics for all: An exploratory look through a class lens. *Journal for Research in Mathematics Education, 31*(4), 454-482.

ABSTRACT: As a researcher-teacher, I examined 7th-graders' experiences with a problem-centered curriculum and pedagogy, focusing on SES differences in students' reactions to learning mathematics through problem solving. Although higher SES students tended to display confidence and solve problems with an eye toward the intended mathematical ideas, the lower SES students preferred more external direction and sometimes approached problems in a way that caused them to miss their intended mathematical points. An examination of sociological literature revealed ways in which these patterns in the data could be related to more than individual differences in temperament or achievement among the children. I suggest that class cultural differences could relate to students' approaches to learning mathematics through solving open, contextualized problems.

Lubienski, S. T. (2000b). A clash of social class cultures? Students’ experiences in a discussionintensive seventh-grade mathematics classroom. *The Elementary School Journal, 100*(4), 377–403.

ABSTRACT: Examined how a curriculum development project, aligned with current mathematics education reforms, affected 18 students in socially diverse mathematics classroom. Found that students of lower socioeconomic status preferred direct teacher intervention as opposed to open discussions among classmates. Higher socioeconomic status students displayed more comfort with abstract mathematical concepts. Findings suggest that discussion-intensive classrooms align more with middle-class cultures.

Mac Iver, M. A., & Mac Iver, D. J. (2009). Urban middle-grade student mathematics achievement growth under comprehensive school reform. *Journal of Educational Research, 102*(3), 223–236.

ABSTRACT: Recognizing the need to implement standards based instructional materials with school wide coherence led some Philadelphia schools to adopt whole-school reform (WSR) models during the late 1990s. The authors report on the relation between mathematics achievement growth for middle-grade students on the Pennsylvania System of School Assessments and the number of years schools implemented either a WSR model with National Science Foundation-supported mathematics curriculum or a WSR model without a mathematics curriculum component, from 1997 to 2000. As the authors hypothesized, mathematics achievement gains (Grades 5–8) were positively related to the number of years those schools were implementing a specific mathematics curricular reform. Additional analyses indicated that the relation held for both computation skills and ability to apply mathematics concepts.

Maccini, P., & Gagnon, J. (2002). Perceptions and application of NCTM standards by special and general education teachers. *Exceptional Children, 68*(3), 325-344.

ABSTRACT: This study determined teachers' perceptions related to application of and barriers to implementation of the National Council of Teachers of Mathematics (NCTM) Standards with students labeled learning disabled (LD) and emotionally disturbed (ED). A stratified random sample of 129 secondary general education math and special education teachers responded to a mail survey. A majority of special education teachers indicated they had not beard of the NCTM Standards. Respondents reported teaching mostly basic skills/general math to secondary students with LD and ED, versus higher-level math, such as algebra and geometry. Teachers identified lack of adequate materials as a considerable barrier to successful implementation of activities based on the Standards. Implications for practice and future research are also provided.

Males, L. (2011). Educative supports for teachers in middle school mathematics curriculum materials: What is offered and how is it expressed? (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.

ABSTRACT: Teaching can have a substantial impact on student learning (Darling-Hammond, 1999). However, teaching excellence depends on many factors, including the need for high quality teachers and their continued education, and high quality materials (Cohen, Raudenbush, & Ball, 2002; Putnam & Borko, 2000). This learning includes learning to plan and enact lessons that are appropriate for all students, which requires learning to interpret and understand student thinking and learning instructional routines and practices that will enable them to use student thinking productively. As we enter into the era of the Common Core State Standards for Mathematics this learning is even more critical, as the standards may require teachers to not only learn to understand and unpack the standards themselves, but may also require teachers to learn new content and learn to teach in different ways (Lappan, McCallum, Kepner, 2010).

Due to the complex nature of teaching and the myriad of demands placed on teachers, mathematics educators need to consider all possible venues for teacher learning. In this paper, I discuss my examination of the opportunities for teacher learning embedded within written curriculum materials. Research indicates that teachers can and do learn from curriculum materials. Curriculum materials, particularly educative ones, emerge as a potential source for opportunities for teacher learning in ways that set them apart from more traditional professional development, which is often criticized for being decontextualized, contrived, short-term, fragmented, discontinuous, and disconnected (Ball & Cohen, 1999; Little, 1994; Lord, 1994; Wilson & Berne, 1999). Educative curriculum materials are materials for Grades K-12 that are ?intended to promote teacher learning in addition to students‘ ? (Davis & Krajcik, 2005, p. 3).

I investigated the opportunities to learn embedded in four middle school curricular series in the areas on introduction to variables and geometric transformations, by examining the content and its expression in the teachers' guides. I developed and used two analytical frameworks; one to code the content support derived from work in science education (Beyer et al., 2009) and a second framework to describe the expression of text developed by Morgan (1996) and augmented by Herbel-Eisenmann (2007).

My results indicated that all four curricular series included opportunities for teacher learning (mostly related to Pedagogical Content Support for Practices and Curricular Knowledge, depending on the curriculum) in both the variable and the transformations units, but these opportunities were quite minimal and focused heavily on particular types of supports. This lack of support was particularly true for Rationale Guidance for teachers. In addition to the content support, my analysis of aspects of voice indicated that although these four series provided opportunities for teacher learning, they also may hinder teachers' learning by speaking "through" teachers rather than "to" teachers (Remillard, 2000), as evidenced by the ways in which personal pronouns were used and the frequencies of imperatives and modal verbs. I discuss implications for curriculum development, teacher education, and research.

Martin, T., Brasiel, S. J., & Turner, H. (2012).* Effects of the Connected Mathematics Project 2 (CMP2) on the Mathematics Achievement of Grade 6 Students in the Mid-Atlantic Region. Final Report.* (NCEE 2012-4017). National Center for Education Evaluation and Regional Assistance. U.S. Department of Education.

ABSTRACT: This study examines the effects of Connected Mathematics Project 2 (CMP2) on grade 6 student mathematics achievement and engagement using a cluster randomized controlled trial (RCT) design. It responds to a need to improve mathematics learning in the Mid-Atlantic Region (Delaware, Maryland, New Jersey, Pennsylvania, and Washington, DC). Findings reveal that the type of instructional activity taking place in intervention schools differed from that in control schools, and the activity observed in intervention schools was the type expected when implementing CMP2. Sixty-four percent of intervention teachers reported implementing the curriculum at a level consistent with the publishers' recommendations on the number of units completed per school year (six), and 68 percent of them reported implementing the curriculum consistent with the recommended amount of class time per week. But CMP2 did not have a statistically significant effect on grade 6 mathematics achievement as measured by the TerraNova, which answered the primary research question.12 Indeed, grade 6 mathematics students in schools using CMP2 performed no better or worse on a standardized mathematics test than did their peers in schools not using it. The results for the secondary research question were similar. There was no statistically significant difference between groups in PTV, and the small effect size is unlikely meaningful. These results were insensitive to alternative model specifications. The lack of statistically significant effects is consistent with prior research on CMP2 rated in the 2010 WWC review as meeting standards "with reservations" (Schneider 2000) and the Eddy et al. (2008) RCT. The intent-to-treat analytical approach used in this study, which analyzes participants based on how they are randomly assigned, yielded unbiased estimates of program effectiveness as implemented. To estimate the effect of CMP2 under typical conditions, teachers were provided all the typical materials and PD that a normal school adopting CMP2 would have. However, while CMP2 use was tracked, the study team did not ensure a particular amount or quality of CMP2 instruction. So, the curriculum impact reflects the effect of a school being assigned to use CMP2 or to continue use of their regular curriculum, not necessarily of actually using CMP2. The results apply to the implementation of the CMP2 curriculum, after typical PD, in schools with grade 6 students. Use of a volunteer sample limits the findings to the schools, teachers, and students that participated in the study in the Mid-Atlantic region. The conclusions drawn in this study about the effects of CMP2 on student math achievement are limited to student math achievement as measured by the TerraNova, and do not generalize to any other standardized test.

Mathematics and Science Expert Panel for the U.S. Department of Education. (1999). *Mathematics and science expert panel: Promising and exemplary mathematics programs, evaluation report prepared for the U.S. Department of Education.* Washington, D. C.: U.S. Department of Education.

Mathis, E. (2004). A comparison of two NSF funded middle school mathematics curricula in Delaware's Appoquinimink and Caesar Rodney school districts. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 65(1). (ProQuest ID No. 765270181)

ABSTRACT: This evaluation compares two NSF funded middle school curricula, Math in Context and the Connected Math Project as measured by student achievement on the mathematics portion of the Delaware State Testing Program. The two groups consisted of 7th grade students from the Caesar Rodney and Appoquinimink School Districts who were not classified as receiving special education services nor services for learning English as a second language. The students took the 5th grade math portion of the DSTP in the Spring of 2000 and the 7th grade math portion of the DSTP in the Spring of 2002. The evaluation involved 295 students from the Appoquinimink School District and 291 students from the Caesar Rodney School District.

The findings of the study indicate that the use of different curricula in the Caesar and Appoquinimink School Districts, CMP and MIC, respectively, did not equivocate to a significant difference in math achievement as measured by the math portion of the DSTP. Descriptive data did show that CMP students outperformed MIC students in terms of increasing their scale scores, but again this difference was not significant. It is important to note that the factors of gender and ethnicity did not contribute to any statistically significant differences between the groups.

McNeil, N., Grandau, L., Knuth, E., Alibali, M., Stephens, A., Hattikudur, S., & Krill, D. (2006). Middle-school students' understanding of the equal sign: The books they read can't help. *Cognition and Instruction, 24*(3), 367-385.

ABSTRACT: This study examined how 4 middle school textbook series (2 skills-based, 2 Standards-based) present equal signs. Equal signs were often presented in standard operations equals answer contexts (e.g., 3 + 4 = 7) and were rarely presented in nonstandard operations on both sides contexts (e.g., 3 + 4 = 5 + 2). They were, however, presented in other nonstandard contexts (e.g., 7 = 7). Two follow-up experiments showed that students' interpretations of the equal sign depend on the context. The other nonstandard contexts were better than the operations equals answer context at eliciting a relational understanding of the equal sign, but the operations on both sides context was best. Results suggest that textbooks rarely present equal signs in contexts most likely to elicit a relational interpretation-an interpretation critical to success in algebra.

Meiler, J. (2006). Does a problem-centered curriculum foster positive or negative changes in students' attitude and learning in mathematics? A case study of three sixth grade students. (Masters thesis). Retrieved from Masters Abstracts International, 45(3). (ProQuest ID No. 1251814661)

ABSTRACT: This case study walks you through the educational lives of three students in sixth grade as they journey through learning by "doing" in a newly implemented, problem-centered math curriculum, Connected Math Project. The purpose of this study was to investigate how the learning strategies provided by Connected Math Project impacts students' attitudes and learning in mathematics. The overall confidence in personal mathematical ability, in how good they perceived themselves to be, in math, demonstrated an increase in positive responses over the last year for the case study students. Because of the increase in positive responses over the last year, the achievement level for the students also increased. These gains were impacted by the highly motivating problem-centered curriculum, Connected Math Project.

Mendez, E., Sherin, M., & Louis, D. (2007). Multiple perspectives on the development of an eighth-grade mathematical discourse community. *Elementary School Journal, 108*(1), 41-61.

ABSTRACT: In this article we examine the development, over 1 year, of mathematical discourse communities in 2 eighth-grade mathematics classes in a suburban public middle school. The curriculum topics included probability, functions, graphing, data analysis, and pre-algebra. The 50 students were heterogeneously placed; most were from upper-middle-class families. Data included videotaped classroom observations, field notes, and teacher reflections. We explored both the students' growing competencies with mathematical discourse and changes in how the teacher attended to students' ideas. We present the teacher's impressions of the developing discourse community, and we applied 2 research-based lenses, robust mathematical discussion to assess the strength of student discourse, and professional vision for classroom discourse to analyze the ways in which the teacher paid attention to, and reflected on, ideas students raised during discussion. Applying multiple perspectives highlighted the complex nature of developing a discourse community and the challenges facing the teacher as he worked to orchestrate constructive dialogue for learning mathematics and to become aware of what students were learning in this context. We also provide an analytic tool, the robust mathematical discussion framework, that will be useful for teachers, teacher educators, And researchers to evaluate the evolving nature of classroom discourse.

Meyer, M., Dekker, T., & Querelle, N. (2001). Contexts in mathematics curricula. *Mathematics Teaching in the Middle School, 6*(9), 522 527.

Miller, J. L., & Fey, J. T. (2000). Proportional Reasoning. *Mathematics Teaching in the Middle School, 5*(5), 310-313.

ABSTRACT: Proportional reasoning has long been a problem for students because of the complexity of thinking that it requires. Miller and Fey discuss some new approaches to developing students' proportional reasoning concepts and skills.

Monaghan, S. R. (2013). *Textbooks, teachers, and middle school mathematics student achievement* (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 1469609858)

ABSTRACT: The purpose of this study was to extend the research on textbook effectiveness to a situated investigation of a single large urban school district in which middle schools were given a choice in selecting from three textbooks for mathematics instruction: a reform textbook, a commercially produced textbook developed in response to mathematics standards, and a traditional textbook. Its genesis is rooted in the efforts in the mathematics education community to investigate the interaction of teachers and mathematics curriculum materials, but in light of the shift to an accountability policy climate in public education. In particular, this study sought to determine whether the type of textbook selected by a school, moderated by the human capital of the teachers teaching mathematics, and the interaction of those variables was associated with increased student mathematics achievement on the mathematics portion of the eighth grade statewide standardized test. Hierarchical linear modeling (HLM) was used to investigate models relating to textbook selection, components of teacher human capital, and their interaction. Contrary to the initial hypothesis, the interaction of textbook selection and components of human capital were not found to be significantly associated with student achievement. However, the selection of a reform mathematics textbook (CMP) over other more traditional texts was associated with student achievement, but accounted for very little of the variance in student test scores. To further explicate the interaction of textbook selection with school factors, logistic regression was used to investigate the association between school factors and the selection of a reform textbook. The demographics of the school (i.e. race, SES, ELL) were not associated with the school selecting a reform mathematics textbook. However, one component of teacher human capital, expertise (a component constructed from data about teacher certification, mathematics specialization, and participation in math focused professional development) was associated with the selection of a reform textbook. This study suggests there is a connection between teacher human capital, the use of reform texts and student achievement; however further investigation is needed to understand the mechanisms at work.

Moore, A. J., Gillett, M. R., & Steele, M. D. (2014). Fostering student engagement with the flip. *The Mathematics Teacher, 107*(6), 420–425.

Moschkovich, J. N. (2015). Academic literacy in mathematics for English Learners. The Journal of Mathematical Behavior.

ABSTRACT: This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously.

Mount, R. E., & Schumacker, R. E. (2002). *Improving basic educational programs for minority students.* Paper presented at the annual meeting of the Southwestern Educational Research Association, Austin, TX.

Moyer, J. C., Cai, J., Wang, N., & Nie, B. (2011). Impact of curriculum reform: Evidence of change in classroom practice in the United States. *International Journal of Educational Research, 50*(2), 87–99. doi:10.1016/j.ijer.2011.06.004

ABSTRACT: The purpose of the study reported in this article is to examine the impact of curriculum on instruction. Over a three-year period, we observed 579 algebra-related lessons in grades 6–8. Approximately half the lessons were taught in schools that had adopted a Standards- based mathematics curriculum called the Connected Mathematics Program (CMP), and the remainder of the lessons were taught in schools that used more traditional curricula (non- CMP). We found many significant differences between the CMP and non-CMP lessons. The CMP lessons, emphasized the conceptual aspects of instruction to a greater extent than the non-CMP lessons and the non-CMP lessons emphasized the procedural aspects of instruction to a greater extent than the CMP lessons. About twice as many CMP lessons as non-CMP lessons were structured to use group work as a method of instruction. During lessons, non-CMP students worked individually on homework about three times as often as CMP students. When it came to text usage, CMP teachers were more likely than non- CMP teachers to work problems from the text and to follow lessons as laid out in the text. However, non-CMP students and teachers were more likely than CMP students and teachers to review examples or find formulas in the text. Surprisingly, only small proportions of the CMP lessons utilized calculators (16%) or manipulatives (11%).

Moyer, J. C., Robison, V., & Cai, J. (2018). Attitudes of high-school students taught using traditional and reform mathematics curricula in middle school: a retrospective analysis. *International Studies in Mathematics*, Online First, 1-20.

Abstract: This paper presents findings from a larger research project that provides insight into the attitudes of high-school students who were taught using different types of mathematics curricula when they were in middle school. A total of 44 12th-grade students from 10 high schools in the same urban school district were interviewed. Eighteen (41%) of them had been taught using a reform curriculum in middle school and 26 (59%) had been taught using a more traditional curriculum. Using Di Martino and Zan's three-dimensional model for attitude, we found that the high-school seniors who had been taught using the reform curriculum in middle school harbored attitudes toward mathematics that differed significantly from the attitudes of those who had been taught using a traditional curriculum in middle school. Our analysis of the student interviews culled seven themes that provide fine-grained information about the students' attitudes toward mathematics. Significantly greater percentages of reform students than traditional students had a relational Vision of mathematics as opposed to an instrumental Vision; however, there was no significant difference between the proportions of reform and traditional students who had a positive Emotional Disposition toward mathematics or a positive Perceived Competence in mathematics.

Moyer, J., Cai, J., Laughlin, C., & Wang, N. (2009). The effect of curriculum type on middle grades instruction. In S. L. Swars, D. W. Stinson, & S. Lemons-Smith (Eds.), *Proceedings of the 31st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education* (Vol. 5, pp. 201-209). Atlanta, GA: Georgia State University.

ABSTRACT: In this article, we discuss differences between the mathematics instruction of CMP and non-CMP teachers in the LieCal project. There are three aspects of instruction that 200 6th grade urban classroom observations showed were strongly and differently related to the type of curriculum that teachers were using. These three aspects relate to the teachers' use of (1) group and individual work, (2) written narratives and worked-out examples, and (3) conceptually- and procedurally-focused instruction.

Moyer, J., Robinson, V., & Cai, J. (2014). *Long-term effect of middle school mathematics curricula on students’ attitudes in high school.* Poster presented at the Research Presession of the annual meeting of the National Council of Teacher of Mathematics, New Orleans, LA.

Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., O’Connor, K. M., Chrostowski, S. J., Gregory, K. D, & Smith, T. A. (2001). *Mathematics Benchmarking Report: TIMSS 1999 – Eighth Grade*. Chestnut Hill, MA: International Study Center, Lynch School of Education, Boston College.

Muzheve, M. T. (2008). *Converting among fractions, decimals, and percents: An exploration of representational usage by middle school teachers.* (Unpublished doctoral dissertation). Texas A&M University, College Station, TX.

ABSTRACT: Using both quantitative and qualitative data collection and analyses techniques, this study examined representations used by sixteen (n = 16) teachers while teaching the concepts of converting among fractions, decimals, and percents. The classroom videos used for this study were recorded as part of the Middle School Mathematics Project (MSMP). The study also compared teacher-selected and textbook representations and examined how teachers‘ use of idiosyncratic representations influenced representational choices on the number test by the teachers‘ five hundred eighty-one (N = 581) students.

In addition to using geometric figures and manipulatives, a majority of the teachers used natural language such as the words nanny, north, neighbor, dog, cowboy, and house to characterize fractions and mathematical procedures or algorithms. Coding of teacher-selected representations showed that verbal representations deviated from textbook representations the most. Some teachers used the words or phrases bigger, smaller, doubling, tripling, breaking-down, and building-up in the context of equivalent fractions.

There was widespread use of idiosyncratic representations by teachers, such as equations with missing or double equal signs, numbers and operators written as superscripts, and numbers written above and below the equal sign. Although use of idiosyncratic representations by teachers influenced representational choices by students on the number test, no evidence of a relationship between representational forms and degree of correctness of solutions was found. The study did reveal though that teachers‘ use of idiosyncratic representations can lead to student misconceptions such as thinking that multiplying by a whole number not equal to 1 gives an equivalent fraction.

Statistical tests were done to determine if frequency of representation usage by teachers was related to the textbook, highest degree obtained by teacher, certification, number of years spent teaching mathematics, number of years teaching mathematics at grade level, number of hours completed on professional development related to their textbook, and total number of days spent on the Interagency Education Research Initiative (IERI) professional development. The results showed representation usage was related to all the above variables, except the highest degree obtained and the total number of days spent on the IERI professional development.

Nathan, M. J., & Kim, S. (2007). Pattern generalization with graphs and words: A crosssectional and longitudinal analysis of middle school students' representational fluency. *Mathematical Thinking and Learning, 9*(3), 193-219.

ABSTRACT: Cross-sectional and longitudinal data from students as they advance through the middle school years (grades 6-8) reveal insights into the development of students' pattern generalization abilities. As expected, students show a preference for lower-level tasks such as reading the data, over more distant predictions and generation of abstractions. Performance data also indicate a verbal advantage that shows greater success when working with words than graphs, a replication of earlier findings comparing words to symbolic equations. Surprisingly, students show a marked advantage with patterns presented in a continuous format (line graphs and verbal rules) as compared to those presented as collections of discrete instances (point-wise graphs and lists of exemplars). Student pattern-generalization performance also was higher when words and graphs were combined. Analyses of student performance patterns and strategy use contribute to an emerging developmental model of representational fluency. The model contributes to research on the development of representational fluency and can inform instructional practices and curriculum design in the area of algebraic development. Results also underscore the impact that perceptual aspects of representations have on students' reasoning, as suggested by an Embodied Cognition view.

National Council of Teachers of Mathematics. (1989). *Curriculum and evaluation standards for school mathematics.* Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1991). *Curriculum and evaluation standards for school mathematics.* Professional standards for teaching mathematics. Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (1995). *Assessment standards for school mathematics.* Reston, VA: National Council of Teachers of Mathematics.

National Council of Teachers of Mathematics. (2000). *Principles and standards for school mathematics.* Reston, VA: National Council of Teachers of Mathematics.

Newton, J. A. (2008).* Discourse analysis as a tool to investigate the relationship between the written and enacted curricula: the case of fraction multiplication in a middle school standards-based curriculum.* (Unpublished doctoral dissertation). Michigan State University, East Lansing, MI.

ABSTRACT: In the 1990s, the National Science Foundation (NSF) funded the development of curricula based on the approach to mathematics proposed in Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics, 1989). Controversy over the effectiveness of these curricula and the soundness of the standards on which they were based, often labeled the “math wars,” prompted a plethora of evaluative and comparative curricular studies. Critics of these studies called for mathematics education researchers to document the implementation of these curricula (e.g., National Research Council, 2004; Senk & Thompson, 2003) because “one cannot say that a curriculum is or is not associated with a learning outcome unless one can be reasonably certain that it was implemented as intended by the curriculum developers” (Stein, Remillard, & Smith, 2007, p. 337). Curriculum researchers have used a variety of methods for documenting curricular implementation, including table-of-content implementation records, teacher and student textbook use diaries, teacher and student interviews, and classroom observations. These methods record teacher and student beliefs, extent of content coverage, in-class and out-of-class textbook use, and classroom participation structures, but do little to compare the mathematics presented in the written curriculum (the student and teacher textbooks) and the way in which this mathematics plays out in the enacted curriculum (that which happens in classrooms).

In order to compare the mathematical features in the written and enacted curricula, I utilized Sfard’s Commognition framework (most recently and fully described in Thinking as Communicating: Human Development, the Growth of discourses, and Mathematizing published in 2008). That is, I compared the mathematical words, visual mediators, endorsed narratives, and mathematical routines in the written and enacted curricula. Each of these mathematical features provided a different perspective on the mathematics present in the curricula. The written curriculum in this study was represented by Investigation 3(Multiplying with Fractions) included in Bits and Pieces II: Using Fraction Operations in Connected Mathematics 2 (Lappan, Fey, Fitzgerald, Friel, & Phillips, 2006). Videotapes of this same Investigation recorded in a sixth grade classroom in a small, rural town in the Midwest were used as the enacted curricula for this case.

The study revealed many similarities and differences between the written and enacted curricula; however, most prominent were the findings regarding objectification in the curricula. Sfard defines objectification as “a process in which a noun begins to be used as if it signifies an extradiscursive, self-sustained entity (object), independent of human agency” (Sfard, 2008, p. 412). She proposes that objectifying is an important process for students’ discursive development and that it serves them particularly well in the study of advanced mathematics. Both objectification itself and the opportunities present for objectification were more prevalent in the written curriculum than in the enacted curriculum.

Newton, J. A. (2012). Investigating the mathematical equivalence of written and enacted middle school Standards-based curricula: Focus on rational numbers.* International Journal of Educational Research*, 51-52, 66-85.

ABSTRACT: Although the question of whether written curricula are implemented according to the intentions of curriculum developers has already spurred much research, current methods for documenting curricular implementation seem to be missing a critical piece: the mathematics. To add a mathematical perspective to the discussion of the admittedly controversial and conceptually complex issue of “fidelity of curricular implementation,” this study proposes a method for investigating fidelity that deals with the question of mathematical equivalence of written curricula and their enactments in the classroom. The method rests on the assumption that the curricula, both written and enacted, can be treated as discourses, and that one of the ways to judge their mathematical equivalence is to compare the mathematical objects around which these discourses evolve. As an illustration for how the method works, I analyzed a part of the written Connected Mathematics Project (CMP) curriculum and its enactment in a sixth grade classroom learning about fractions. This analysis showed that the written and enacted versions of the central mathematical objects of the two curricula, rational numbers, differed in almost every aspect: in their ontology, in the relative prominence of their realizations (i.e., symbols, icons and concrete objects) and in the importance attributed to their different properties. These differences may have an impact on the nature of students’ mathematical competence.

Newton, J., Geller, R., Umbeck, L., & Kasmer, L. (2012). Reflections on teaching with a standards-based curriculum: A conversation among mathematics educators. Montana Mathematics Enthusiast, 9(1), 179–192.

Nie, B., Cai, J., & Moyer, J. (2009). How a Standards-based mathematics curriculum differs from a traditional curriculum: with a focus on intended treatments of the ideas of variable. *Zentralblatt fuer Didaktik der Mathematik (International Journal on Mathematics Education), 41*(6), 777-792.

ABSTRACT: Analyzing the important features of different curricula is critical to understand their effects on students’ learning of algebra. Since the concept of variable is fundamental in algebra, this article compares the intended treatments of variable in an NSF-funded standards-based middle school curriculum (CMP) and a more traditionally based curriculum (Glencoe Mathematics). We found that CMP introduces variables as quantities that change or vary, and then it uses them to represent relationships. Glencoe Mathematics, on the other hand, treats variables predominantly as placeholders or unknowns, and then it uses them primarily to represent unknowns in equations. We found strong connections among variables, equation solving, and linear functions in CMP. Glencoe Mathematics, in contrast, emphasizes less on the connections between variables and functions or between algebraic equations and functions, but it does have a strong emphasis on the relation between variables and equation solving.

Nie, B., Freedman, T., Hwang, S., Wang, N., Moyer, J. C., Cai, J. (2013). An investigation of teachers’ intentions and reflections about using Standards-based and traditional textbooks in the classroom. *ZDM*, 45(5), 699-711.

ABSTRACT: This study analyzed teachers’ intentions for and reflections on their use of Standards-based [Connected Mathematics Program (CMP)] textbooks and traditional (non-CMP) mathematics textbooks to guide instruction. In this investigation of the interplay between textbooks and instruction, we focused on learning goals, instructional tasks, teachers’ anticipation of students’ difficulties, and their perceptions of students’ achievement of learning goals. All of these are aspects of teachers’ intentions and reflections that have proved fruitful in comparing the roles of the CMP and non-CMP mathematics textbooks in our Longitudinal Investigation of the Effect of Curriculum on Algebra Learning project. Whereas the cognitive level of the teachers’ intended learning goals appeared generally to reflect the emphases of their respective textbooks, we found that the CMP teachers’ intended learning goals were not as well aligned with the CMP textbooks as the non-CMP teachers’ learning goals were aligned with their non-CMP textbooks. The CMP and non-CMP teachers’ implementations of the lessons seemed to reduce the degree of difference between the cognitive levels of their intended goals. Even so, we found that significantly more CMP lessons than non-CMP lessons were implemented at a high level of cognitive demand. Although the non-CMP teachers’ intended learning goals were better aligned with their textbook’s learning goals, we found that the CMP teachers were more likely than the non-CMP teachers to follow the guidance of their textbooks in designing and selecting instructional tasks for a lesson. Future research should consider other aspects of teachers’ intentions and reflections that may shed a broader light on the role of textbooks and curriculum materials in teachers’ crafting of instructional experiences for their students.

O'Clair, K. K. (2005). Impact on student achievement: Going to scale with a middle school math initiative. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 66(5). (ProQuest Id No. 921030071)

ABSTRACT: To measure the impact of a middle school math initiative on student achievement, a survey research design was used to categorize the levels of implementation by 7th -grade math teachers. The survey targeted the teachers' participation in 4 key components of the middle school math initiative, based on an expanded model of the theory of action of standards-based reform by Elmore & Rothman (1999): district-led professional development, school walkthroughs, site-based team planning, and use of standards-based Connected Mathematics program. In a western urban school district, 18 of the 21 contacted teachers from 2002-2003 completed and returned their selfadministered surveys; 26 of 33 from 2003-2004. The Year 1 teacher sample represented 29% of the total teacher population and their 1,259 students were 24% of the total student population. The Year 2 teacher sample represented 39% of the teachers and their 1,765 students were 33% of the total student population. The scale scores of these students from 18 schools were the dependent variable for analyses of variance. The independent variables were the year and the level of implementation that was determined by weighting the results from the teacher survey against a rubric of implementation created by the researcher.

The major findings showed statistically significant differences by years and by levels of implementation. The 7th -grade student math scale scores of the statewide standards-based assessment positively improved and the strength of the effect was small. Using a 2-way ANOVA to compare the 4 groups of high and low implementation in both years, there was a statistically significant difference between the students' scores who experienced higher versus lower levels of implementation in their 7th grade math classes. The students of the higher implementation group of teachers, who had less teaching experience but attended more professional development and had more team planning, had higher math scale scores.

The research results conclude that there was a statistically significant small improvement, Recommendations for further research suggest investigation of the quality of instructional delivery, not only the quantity of CMP units. More involvement with instructional leaders on-site could support scheduling efforts for grade-level planning and more walkthroughs.

O’Neal, S. W., & Robinson-Singer, C. (1998). *The Arkansas Statewide Systemic Initiative Pilot of the Connected Mathematics Project: An evaluation report.* Report submitted to the National Science Foundation as part of the Connecting Teaching, Learning, and Assessment Project.

Otten, S., & Soria, V. M. (2014). Relationships between students’ learning and their participation during enactment of middle school algebra tasks. *ZDM*, 46(5), 815–827. doi:10.1007/s11858-014-0572-4

ABSTRACT: This study examines a sequence of four middle school algebra tasks through their enactment in three teachers’ classrooms. The analysis centers on the cognitive demand—the kinds of thinking processes entailed in solving the task—and the participatory demand—the kinds of verbal contributions expected of students—of the task as written in the instructional materials, as set up by the three teachers, and as discussed by the teachers and their students. Relationships between the nature of the task enactments and students’ performance on a pre- and post-test are explored. Findings include the fact that the enacted tasks differed from the written tasks with regard to both the cognitive demand and the participatory demand, which related to students’ lack of success on the post-test. Specifically, cognitive demand declined in the enacted curriculum at different points for different classes, and the participatory demand during enactment tended to involve isolated mathematical terms rather than students verbally expressing mathematical relations.

Patel, N., Franco, S., Miura, Y., & Boyd, B. (2012). Including curriculum focus in mathematics professional development for middle-school mathematics teachers. *School Science and Mathematics, 112*(5), 300-309.

ABSTRACT: This paper examines professional development workshops focused on Connected Math, a particular curriculum utilized or being considered by the middle-school mathematics teachers involved in the study. The hope was that as teachers better understood the curriculum used in their classrooms, i.e., Connected Math, they would simultaneously deepen their own understanding of the corresponding mathematics content. By focusing on the curriculum materials and the student thought process, teachers would be better able to recognize and examine common student misunderstandings of mathematical content and develop pedagogically sound practices, thus improving their own pedagogical content knowledge. Pre- and post-mathematics content knowledge assessments indicated that engaging middle-school teachers in the curriculum materials using pedagogy that can be used with their middle-school students not only solidified teachers’ familiarity with such strategies, but also contributed to their understanding of the mathematics content.

Patrick, H., Turner, J., Meyer, D., & Midgley, C. (2003). How teachers establish psychological environments during the first days of school: Associations with avoidance in mathematics. *Teachers College Record, 105*(8), 1521-1558.

ABSTRACT: Observations of the first days of school in eight sixth-grade classrooms identified three different classroom environments. In supportive environments teachers expressed enthusiasm for learning, were respectful, used humor, and voiced expectations that all students would learn. In ambiguous environments teachers were inconsistent in their support and focus on learning and exercised contradictory forms of management. In nonsupportive environments teachers emphasized extrinsic reasons for learning, forewarned that learning would be difficult and that students might cheat or misbehave, and exercised authoritarian control. Teachers' patterns of motivational and organizational discourse during math classes near the end of the year were consistent with the messages they expressed at the beginning of the year. When student reports of avoidance behaviors in math from fall and spring were compared with the qualitative analyses of these environments, students in supportive classrooms reported engaging in significantly less avoidance behavior than students in ambiguous or nonsupportive environments.

Perda, D., Noyce, P. E., & Riordan, J. E. (2003). Algebra in Massachusetts middle schools: Access, achievement, and implications. Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Philips, E. (2019). Promoting Productive Disciplinary Engagement and Learning With the CMP STEM Problem Format and “Just-in-Time” Supports in Middle School Mathematics. Poster Presentation, *International Society of Design and Development Conference*. Pittsburgh, Pennsylvania: University of Pittsburgh.

**Philips, E**. (2019). Promoting Productive Disciplinary Engagement and Learning With the CMP STEM Problem Format and “Just-in-Time” Supports in Middle School Mathematics. Poster Presentation, *International Society of Design and Development Conference*. Pittsburgh, Pennsylvania: University of Pittsburgh.

Phillips, E. & Lappan, G. (1998). Algebra: The first gate. In L. Leutzinger (Ed.), *Mathematics in the Middle *(pp. 10-19). Reston, VA: National Council of Teachers of Mathematics.

Phillips, E. (1995). *A response to “A research base supporting long-term algebra reform?”* Paper presented at the 17th Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Columbus, OH.

ABSTRACT: This paper is a reaction to a plenary address, "A Research Base Supporting Long Term Algebra Reform?" by James Kaput (SE 057 182). The reactions fall into three categories: comments on Kaput's dimensions of algebra reform, a brief discussion of algebra and algebra reform from the viewpoint of a curriculum developer of the Connected Mathematics Project (CMP), and some concerns about Kaput's three stages of reform.

Phillips, E. (1998). Developing a coherent and focused K-12 algebra curriculum. In National Research Council (Ed.), *The nature and role of algebra in the K-14 curriculum: Proceedings of a national symposium*, (pp. 27-29). Washington, D.C.: National Academy Press.

**Phillips, E**. (2019). Mathematical Reasoning and Problem Posing- The Case of Connected Mathematics Project. *Proceedings of International Research Forum on Mathematics Curriculum and Teaching Materials in Secondary School* (p. 21). Beijing, China: People’s Education Press and Beijing Normal University.

Phillips, E. A., Smith III, J. P., Star, J. R., & Herbel-Eisenmann, B. (1998). Algebra in the middle grades. *The New England Mathematics Journal, 30*(2), 48-60.

Polly, D. & Orrill, C. (2012). CCSSM: Examining the critical areas in grades 5 and 6. *Teaching Children Mathematics, 18*(9), 566-573).

Post, R. A. (2004). *Generation of mathematical knowledge through teacher practice: Study of a novice elementary teacher. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 65(12). (ProQuest ID No. 845705381)

ABSTRACT: Research on teachers' knowledge has shown that elementary teachers often lack the deep, flexible, and conceptual mathematical understandings necessary for reform efforts in mathematics education to be realized in classroom practice. In order to meet the complex demands of developing a reform-oriented teacher practice, a considerable amount of teacher learning must take place through participation in the activity of teacher practice.

Using case study methods, this research analyzed the practice of one 1st-year elementary teacher as she implemented a reform-based curriculum program (Connected Mathematic Project) and participated in the school, classroom, and reform (i.e., curriculum materials and professional development) communities of practice. Data were collected from observations of three units of instruction, professional development sessions, concept maps, and interviews with the case study teacher and members of the school community. Analysis revealed the key role curriculum materials played in the generation of mathematical knowledge. The classroom and reform communities acted as catalysts in the teacher's participatory practices, which generated expanded, connected, and unresolved mathematical knowledge.

Post, T. R., Harwell, M. R., Davis, J. D., Maeda, Y., Cutler, A., Andersen, E., Norman, K. W. (2008). Standards-based mathematics curricula and middle-grades students' performance on standardized achievement tests.* Journal for Research in Mathematics Education, 39*(2), 184- 212.

ABSTRACT: Approximately 1400 middle-grades students who had used either the Connected Mathematics Project (CMP) or the MATH Thematics (STEM or MT) program for at least 3 years were assessed on two widely used tests, the Stanford Achievement Test, Ninth Edition (Stanford 9) and the New Standards Reference Exam in Mathematics (NSRE). Hierarchical Linear Modeling (HLM) was used to analyze subtest results following methods described by Raudenbush and Bryk (2002). When Standards-based students' achievement patterns are analyzed, traditional topics were learned. Students' achievement levels on the Open Ended and Problem Solving subtests were greater than those on the Procedures subtest. This finding is consistent with results documented in many of the studies reported in Senk and Thompson (2003), and other sources.

Pradere, S. (2007). *Effective staff development connected to increased student achievement. *(Doctoral dissertation). Dissertation Abstracts International, 68(3). (ProQuest ID No. 1310415901)

ABSTRACT: School districts expend considerable resources to establish effective staff development opportunities that lead to increased student achievement. This research project focused on the development, implementation, and evaluation of a school based staff development project built around four distinct instructional elements: teacher instructional practices, student engagement, stating the objective, and developing a literacy rich environment. Specifically, the study addressed the following research questions: (1) Did teachers' instructional practices change as a result of participating in the professional development program? (2) If teachers adopted the new instructional practices, did the changes have an impact on student performance as calculated on the Measures of Academic Progress (MAP) assessment? The study was established within a mixed methodology design in which changes in teacher performance were measured using qualitative research methods including survey, interview, and classroom observation data sets. To measure changes in student performance, traditional univariate and multivariate statistical techniques were utilized including a t-test, analysis of variance, and K-means cluster analysis. To conduct the study the researcher: (1) facilitated the design of the staff development model; (2) provided or facilitated foundation staff development on four key instructional elements to both teachers and administrators; (3) provided guidance to administrators on gathering key data; (4) provided support to teacher leaders and administrators on methods for coaching teachers adopting new practices; (5) observed teachers utilizing skills in practice; (6) gathered both perception and observation data related to teacher implementation of four key elements; (7) gathered and processed student academic performance data; and (8) studied the results measuring the impact of staff development on both teacher practices and student performance. The results of the study verified: (1) that teachers adopted or maintained teacher skill levels related to key instructional practices; (2) students' actual mean growth rates on MAP assessment exceeded projected mean growth rates in reading and language but not mathematics; and (3) students whose teachers exceeded proficient skill levels in instructional practices and student engagement demonstrated higher performance levels in reading and language on the MAP assessment than those students whose teachers met or approached desired skill levels in those two areas.

Prentice Hall. (2006). *CMP: Research and evaluation summary*. Upper Saddle River, NJ: Prentice Hall.

Preston, R. V., & Lambdin, D. V. (1997). *Teachers changing in changing times: Using stages of concern to understand changes resulting from the use of an innovative mathematics curriculum.* Paper presented at the annual meeting of the American Educational Research Association, Chicago, IL.

Purnamasari, W. (2013). *Application of the model Connected Mathematics Project (CMP) in an effort to improve students’ adaptive reasoning abilities: A quasi-experimental study in grade 45 Public VIII SMP Duo.* (Unpublished doctoral dissertation). Indonesia University of Education, Bandung, Indonesia.

ABSTRACT: This study was motivated by the relatively low student achievement in adaptive reasoning ability. One of ways to help students develop adaptive reasoning ability is applying the model of Connected Mathematics Project (CMP). The aim of this study was to determine the adaptive reasoning ability improvement among students who had learning mathematic with CMP model compared with conventional model, determine the increase of adaptive reasoning ability of students in the high group and low group who get the learning of mathematic by CMP model compared with students in the high group and low group with conventional model and determine students' attitudes to learning mathematic with CMP model. The method used in this study was quasi-experimental. The population in this study is the eighth grade students of SMP Negeri 45 Bandung with two samples of the entire eighth grade class available. The instruments used were the adaptive reasoning ability test instruments, questionnaires, observation sheets and daily journals of students. The results of this study showed that the improvement of adaptive reasoning ability of students with CMP model better than students who had learning mathematic with conventional model. Furthermore, students in the high group and low group who get the learning of mathematic by CMP model have the adaptive reasoning ability better than students in the high group and low group with conventional model. In addition, students responded positively to the learning of mathematic with CMP model. Key Words: Adaptive reasoning ability, Connected Mathematics Project (CMP).

Quigley, D. (2010). *Project-based learning and student achievement* (Doctoral dissertation). Available from ProQuest Dissertations and Theses database. (UMI No. 741546820)

ABSTRACT: More evidence is needed to support the soundness of project-based learning in addressing student achievement as measured by assessments used by one northeastern U.S. state, especially since the state is creating project-focused schools. Project-based learning encompasses various strategies designed to enhance engagement and performance. This quasi-experimental, nonequivalent control-group study investigated the effectiveness of project-based instruction and of mixed strategy instruction related to changes in student achievement. Brain-based research, curriculum integration literature, and motivation theory informed this work. Forty-four 6th grade students comprised the sample. The student group was a stratified, single stage sampled, leveled mix with consideration given to age, gender, and ability level. Half of the participants were in a project-based learning environment and the other half were in an environment with a mix of collaborative and investigative strategies. Students explored concepts in the 6th grade Curriculum Frameworks in mathematics and used the aligned Connected Mathematics program. Hypotheses were tested using independent samples t test. No statistically significant difference in the students’ math performance was found. The potential for positive social change is related to a higher level of awareness of integration strategies, and more informed practice, both of which can lead to more opportunities for raising student achievement.

Raymond, A. (2004). “Doing math” in Austin. *Teaching Pre K-8, 34*(4), 42-45.

ABSTRACT: Since 1990, the January issue of "Teaching Pre K-8" has highlighted a school visit by the president of the National Council of Teachers of Mathematics. This article discusses Cathy Seeley's visit to a 6th grade classroom at the J. E. Pearce Middle School in Austin, Texas, where she participated in a math activity from the Connected Mathematics Project, a complete middle school mathematics curriculum for grades 6, 7, and 8. Funded by the National Science Foundation between 1991 and 1997, the program includes eight units for each grade, "built around mathematical problems that help students develop understanding of important concepts and skills in number, geometry, measurement, algebra, probability and statistics."

Reinhart, S. C. (2000). Never say anything a kid can say! *Mathematics Teaching in the Middle School, 5*(8), 478-483.

ABSTRACT: Reinhart discusses teaching mathematics to middle school students. To help students engage in real learning, Reinhart asks good questions, allows students to struggle, and places the responsibility for learning directly on their shoulders.

Remillard, J. T., Herbel-Eisenmann, B. A., & Lloyd, G. M. (2008). *Perspectives on teachers' use of mathematics curriculum materials*. New York, NY: Routledge.

Reys, R., Reys, B., Lapan, R., Holliday, G., & Wasman, D. (2004). Assessing the impact of Standards-based middle grades mathematics curriculum materials on student achievement: Corrections. *Journal for Research in Mathematics Education, 35*(2), 152.

Reys, R., Reys, B., Tarr, J., & Chavez, O. (2006). *Assessing the impact of Standards-based middle school mathematics curricula on student achievement and the classroom learning environment.* Washington, DC: National Center for Education Research.

Richards, K. T. (2004). *Communications in mathematics.* (Masters thesis). Masters Abstracts International, 43(2). (ProQuest ID No. 813818281)

ABSTRACT: The mathematics classroom is evolving to include more writing and discourse as a means of deepening student understanding of mathematical concepts. Traditionally, math has been taught as a set of procedures that when you plug in the right numbers in to the right equations, you get the right answers. My research in a middle school setting using the Connected Math Project curriculum required students to think more deeply and reflect on their math knowledge to write and discuss mathematical concepts with their classmates. In the process the students were more engaged, took more ownership, and constructed knowledge themselves. Through student observations, samples of writing work, Socratic Seminars, and student surveys and interviews, I discovered most students enjoy and value both writing and discourse. They see themselves as benefiting from both, writing and discourse, gaining better understanding and clarity of thought. Teachers are also able to assess their students understanding more accurately.

Rickard, A. (1993). *Teachers’ use of a problem-solving oriented sixth-grade mathematics unit: Two case studies.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 54 (10), (ProQuest Id No. 745239291)

ABSTRACT: Problem solving is a central issue in current reform initiatives in mathematics education. However, while curriculum developers design problem-solving oriented curricula to help move reforms into K-12 mathematics classrooms, little is known about how teachers actually use problem-solving oriented mathematics curricula to teach.

This study investigates how two sixth-grade mathematics teachers used a problem-solving oriented unit on perimeter and area. A four-dimensional framework is developed and employed to explore how each teacher's knowledge, views, and beliefs shaped her use of the unit. Using data collected through interviews, classroom observations, conversations with teachers and their students, samples of students' work, teachers' lesson plans, and the unit on perimeter and area, two case studies are presented to portray how each teacher used the unit in her classroom.

This study shows that each teacher's use of the unit was consistent with her underlying views and beliefs, and with some aspects of the intentions of the curriculum developers who designed the unit. However, other aspects of the teachers' use of the unit varied from the intentions of the curriculum developers. This study shows further that each teacher's use of the unit was shaped by interplay between her own views, beliefs, and knowledge, and the unit. Therefore, both the perimeter and area unit and the teachers shaped the teaching which occurred in their classrooms.

This study suggests that while problem-solving oriented curriculum can play a role in shaping mathematics teaching, the views, beliefs, and knowledge of teachers should be addressed in curriculum. This study also points to issues for future research that are connected to teachers' use of problem-solving oriented curricula.

Rickard, A. (1995a). Problem solving and computation in school mathematics: Tensions between reforms and practice. *National Forum of Applied Educational Research Journal, 8*(2), 41-51.

Rickard, A. (1995b). Teaching with problem-oriented curricula: A case study of middle school mathematics instruction. *Journal of Experimental Education, 64*(1), 5-26.

Rickard, A. (1996). Connections and confusion: Teaching perimeter and area with a problem solving oriented unit. *Journal of Mathematical Behavior, 15*(3), 303-327.

ABSTRACT: Problem-solving-oriented mathematics curricula are viewed as important vehicles to help achieve K-12 mathematics education reform goals. Although mathematics curriculum projects are currently underway to develop such materials, little is known about how teachers actually use problem-solving oriented curricula in their classrooms. This article profiles a middle-school mathematics teacher and examines her use of two problems from a pilot version of a sixth-grade unit developed by a mathematics curriculum project. The teacher's use of the two problems reveals that although problem-solving-oriented curricula can be used to yield rich opportunities for problem solving and making mathematical connections, such materials can also provide sites for student confusion and uncertainty. Examination of this variance suggests that further attention should be devoted to learning about teachers' use of problem-solving oriented mathematics curricula. Such inquiry could inform the increasing development and use of problem solving-oriented curricula.

Rickard, A. (1998). Conceptual and procedural understanding in middle school mathematics. In L. Leutzinger (Ed.), *Mathematics in the Middle.* Reston, VA: National Council of Teachers of Mathematics.

Ridgeway, J. E., Zawojewski, J. S., Hover, M. N., & Lambdin, D. V. (2003). Student attainment in the *Connected Mathematics* Curriculum. In S. L. Senk & D. R. Thompson (Eds.) *Standards-based school mathematics curricula: What are they? What do students learn?* (pp. 193-224). Hillsdale, NJ: Lawrence Erlbaum Associates.

Ridgeway, J. E., Zawojewski, J., & Hoover, M. (2000). Problematising evidence-based policy and practice. *Evaluation and Research in Education, 14*(3 & 4), 181-192.

ABSTRACT: Evidence-based policy and practice (EBPP) is widely advocated, and for good reason. Here, some challenges to EBPP are identified, illustrated by a large-scale evaluation of a major curriculum development project. Problems include: changes in educational goals, which necessitate the development of new measures of attainment; different time lines over which different patterns of result emerge; the challenge of defining a complex treatment, such as a new curriculum; and the variability of effect size in different classrooms. Several approaches are offered as responses to these challenges. The paper argues that much of the work on EBPP has focused on practice rather than on policy. Evidence-based policy will require detailed work on descriptions of systems and on systems change; more significantly, it will require the development of a new field of endeavor, associated with macro-systemic change, that is to say, the study of systems undergoing radical change.

Riordan, J., & Noyce, P. (2001). The impact of two standards-based mathematics curricula on student achievement in Massachusetts. *Journal for Research in Mathematics, 32*(4), 368-398.

ABSTRACT: Since the passage of the Education Reform Act in 1993, Massachusetts, has developed curriculum frameworks and a new statewide testing system. As school districts align curriculum and teaching practices with the frameworks, standards-based mathematics programs are beginning to replace more traditional curricula. This paper presents a quasi-experimental study using matched comparison groups to investigate the impact of one elementary and one middle school standards-based mathematics program in Massachusetts on student achievement. The study compares statewide standardized test scores of fourth-grade students using Everyday Mathematics and eighth-grade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standards-based programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standards-based program, remained consistent across mathematical strands, question types, and student subpopulations.

Rittle-Johnson, B., & Koedinger, K. (2005). Designing knowledge scaffolds to support mathematical problem solving. *Cognition and Instruction, 23*(3), 313-349.

ABSTRACT: We present a methodology for designing better learning environments. In Phase 1, 6th-grade students' (n = 223) prior knowledge was assessed using a difficulty factors assessment (DFA). The assessment revealed that scaffolds designed to elicit contextual, conceptual, or procedural knowledge each improved students' ability to add and subtract fractions. Analyses of errors and strategies along with cognitive modeling suggested potential mechanisms underlying these effects. In Phase 2, we designed an intervention based on scaffolding this prior knowledge and implemented the computer-based lessons in mathematics classes. In Phase 3, we used the DFA and supporting analyses to assess student learning from the intervention. The posttest results suggest that scaffolding conceptual, contextual, and procedural knowledge are promising tools for improving student learning.

Rohendi, D. & Dulpaja, J. (2013). Connected Mathematics Project (CMP) model based on presentation media to the mathematical connection ability of junior high school student. *Journal of Education and Practice, 4*(4), 17-22.

ABSTRACT: Connection mathematics ability will be greatly needed by students, especially to solve the problems that need the relation between mathematical concepts with other concepts in mathematics and other disciplines or in everyday life. To get that mathematics ability in this research used Connected Mathematics Project (CMP) model based on Presentation Media. CMP model based on presentation media was a student-centered learning model that involved student more; student not only did the problem but also sought the solution actively that enabled student to explore the relation of mathematical concept in real life. This research was a quasi experiment research with the student of 7th grade of Junior High School of Ujungjaya 2 of Sumedang district, Indonesia as the research sample. After the learning in the experiment class was conducted, the data description by using instrument of pre-test and post-test were collected to find out the student’s ability of mathematical connection, as well as observation sheet to find out the activity and condition of student during mathematical learning. The result of the research showed that the student’s mathematical connection ability by using Connected Mathematics Project (CMP) model based on presentation media was better than the conventional one. Besides, student’s activity in the learning process by using Connected Mathematics Project (CMP) based on presentation media was really positive and they became very active.

Rubenstein, R. N., Lappan, G., Phillips, E., & Fitzgerald, W. (1993). Angle sense: A valuable connection. *Arithmetic Teacher, 40*(6), 352-358.

Sahin, A. (2015). The Effects of Quantity and Quality of Teachers’ Probing and Guiding Questions on Student Performance. *Sakarya University Journal of Education, 5*(1), 95-113.

ABSTRACT: This study investigated the types, quantity, and quality of teacher questions and their impact on student understanding. In contrast to previous studies, in order to obtain optimum effects of question types, quantity, and quality, this study controlled for variables such as teachers’ experience, textbooks used, and teachers’ mathematics preparation knowledge, all of which may affect student achievement. The data were collected from 33 7th- and 8th-grade teachers in 2 different states, Texas and Delaware, who participated in a longitudinal project. A total of 103 videotapes were obtained. For the 1st research question, Hierarchical Linear Modeling (HLM) was run with 2 levels; student and teacher. For the 2nd question, inter-correlations were computed between the variables. We found that the quality teachers’ probing questions significantly predicted student performance when other variables were controlled. We also found that the quality and quantity of guiding questions and probing questions significantly correlated.

Schneider, C. (1998). *Connected Mathematics Project: Texas Statewide Systemic Initiative Implementation Pilot.* Report submitted to the National Science Foundation as part of the Connecting Teaching, Learning, and Assessment Project.

Schneider, C. (2000). *Connected Mathematics and the Texas Assessment of Academic Skills.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62(2). (ProQuest Id No. 727941391)

ABSTRACT: This study determined if the use of Connected Mathematics (CMP), a middle school curriculum based on the reform standards called for by the National Council of Teachers of Mathematics in 1989, impacted student performance measured by the state mandated Texas Assessment of Academic Skills (TAAS) test. Did Texas campuses involved in the CMP pilot from 1997 to 1999 have different TAAS results compared to similar Texas campuses that did not use CMP?

In this study campuses were not randomly selected to use the curriculum. CMP and non-CMP campuses were matched using a regression analysis of the significant variables predicting 1996 pre-CMP TAAS rates. Campus level TARS passing rates and student Texas Learning Index (TLI) scores were analyzed using mixed model methodology. There were 48 campuses represented in the campus level analysis and 19,501 students from 32 of these campuses in the student level analysis. Based upon an implementation survey, a high use subset of campuses was identified from teachers' reporting that at least one-third of the total possible curriculum at every grade and year during the pilot was taught. The data were partitioned into cohorts; Cohort 1 represented observations from sixth, seventh, and eighth grades, from 1996-97 to 1998-99. Cohort 2 included data from sixth and seventh grades, 1997-98 to 1998-99. Cohort 3 had data for sixth grade, 1998-99.

For the analyses on TAAS percent passing and student TLI for all campuses and cohorts combined there is no difference between CMP and non-CMP campuses. When disaggregating the analyses by cohort, there is no difference between CMP and non-CMP campuses for either type of data for any individual cohort using all campuses. For the high use subset of campuses with all cohorts combined there is no difference between CMP and non-CMP campuses for either TAAS passing rates or student TLI scores. For the high use subset of campuses and students disaggregated by cohort, differences may be found, but they are not consistent. Research in this study indicates that the use of the CMP curriculum does not make a difference on TAAS passing rates or student level TLI scores.

Schoenfeld, A., Burkhardt, H., Daro, P., Ridgeway, J., Schwartz, J., & Wilcox, S. (1999). *Balanced Assessment: Middle Grades Assessment*. New York, NY: Dale Seymour Publications.

Schrauth, M. A. (2014). *Fostering Mathematical Creativity in the Middle Grades: Pedagogical and Mathematical Practices *(Doctoral dissertation, Texas State University). Retrieved from ProQuest Dissertations & Theses Global. (Proquest ID No. 1634513917).

ABSTRACT: Increased automation and outsourcing have increased the need for creativity in many domestic jobs, so the purpose of this study is to explore middle school students’ opportunity to be mathematically creative. The process standards of the Texas Essential Knowledge and Skills (TEKS) and National Council of Teachers of Mathematics (NCTM) and the Standards for Mathematical Practice of the Common Core State Standards (CCSS) can be inferred to indicate that mathematics content should be taught in a way that develops mathematical creativity. A qualitative case study was done to describe ways that three teachers fostered mathematical creativity in the middle grades. Classroom observations were triangulated with teacher and student interviews, researcher’s log, and documents. Transcripts for whole class discussions of 40 hours of observation and transcripts for teacher and student interviews were open coded initially before categories were standardized and themes emerged. The three themes that emerged were that the teachers helped the students make mathematics personally meaningful, the teachers helped create an environment where students were comfortable expressing their personally meaningful understanding of mathematics and making mistakes, and they maintained expectations of mathematics practices. The teachers helped students make mathematics personally meaningful by allowing students to make some choices in how they do mathematics (use alternative methods, use alternative answer forms, solve problems with multiple correct answers, and flexibility with creating graphs and tables), to use their own words to describe mathematical concepts rather than emphasizing memorization from a textbook, and to make connections (students’ interests and experiences, school experiences and other content areas, and other real world experiences through the eyes of the teacher). A safe environment was created by allowing students adequate thinking time, making it clear that the students’ voices were important (ask questions, share ideas and experiences, differentiate between off-task conversations and enthusiasm, insist students respect each other, and ensure all students participated in whole class discussion), promoting the idea that mistakes are okay (okay for students and teacher, provide a learning experience, and point out silver lining in incorrect or incomplete solutions), encouraging the use of resources, and emphasizing effort over perfection. Finally, they maintained mathematics practices such as explaining reasoning, using appropriate terminology and notation, and using estimation to determine reasonableness of answers.

Seifer, M. D. (2005). *Collaborating with colleagues to improve student learning using the Connected Mathematics Project.* (Unpublished master’s thesis). Bank Street College of Education, New York, NY.

Sherin, B. (2001). How students understand physics equations. *Cognition and Instruction, 19*(4), 479-541.

ABSTRACT: What does it mean to understand a physics equation? The use of formal expressions in physics is not just a matter of the rigorous and routinized application of principles, followed by the formal manipulation of expressions to obtain an answer. Rather, successful students learn to understand what equations say in a fundamental sense; they have a feel for expressions, and this guides their work. More specifically, students learn to understand physics equations in terms of a vocabulary of elements that I call symbolic forms. Each symbolic form associates a simple conceptual schema with a pattern of symbols in an equation. This hypothesis has implications for how we should understand what must be taught and learned in physics classrooms. From the point of view of improving instruction, it is absolutely critical to acknowledge that physics expertise involves this more flexible and generative understanding of equations, and Our instruction should be geared toward helping students to acquire this understanding, The work described here is based on an analysis of a corpus of videotapes in which university students solve physics problems.

Sherin, M. G. (2002). A balancing act: Developing a discourse community in a mathematical classroom. *Journal of Mathematics Teacher Education*, 5, 205-233.

ABSTRACT: This article examines the pedagogical tensions involved in trying to use students' ideas as the basis for class discussion while also ensuring that discussion is productive mathematically. The data for this study of the teaching of one middle-school teacher come from observations and videotapes of instruction across a school year as well as interviews with the participating teacher. Specifically, the article describes the teacher's attempts to support a student-centered process of mathematical discourse and, at the same time, facilitate discussions of significant mathematical content. This tension in teaching was not easily resolved; throughout the school year the teacher shifted his emphasis between maintaining the process and the content of the classroom discourse. Nevertheless, at times, the teacher balanced these competing goals by using a ``filtering approach'' to classroom discourse. First multiple ideas are solicited from students to facilitate the process of student-centered mathematical discourse. Students are encouraged to elaborate their thinking, and to compare and evaluate their ideas with those that have already been suggested. Then, to bring the content to the fore, the teacher filters the ideas, focusing students' attention on a subset of the mathematical ideas that have been raised. Finally, the teacher encourages student-centered discourse about these ideas, thus maintaining a balance between process and content.

Sherman, M. (2014). The Role of Technology in Supporting Students’ Mathematical Thinking: Extending the Metaphors of Amplifier and Reorganizer. *Contemporary Issues in Technology and Teacher Education, 14*(3), 220-246.

ABSTRACT: The use of instructional technology in secondary mathematics education has proliferated in the last decade, and students’ mathematical thinking and reasoning has received more attention during this time as well. However, few studies have investigated the role of instructional technology in supporting students’ mathematical thinking. In this study, the implementation of 63 mathematical tasks was documented in three secondary and one middle school mathematics classroom, and the Mathematical Tasks Framework (Stein & Smith, 1998) was used to correlate the cognitive demand of mathematical tasks with the use of technology as an amplifier or reorganizer of students’ mental activity (Pea, 1985, 1987). Results indicate that the use of technology generally aligned with teachers’ current practice in terms of the distribution of low- and high-level tasks enacted in their classrooms. However, the use of technology as a reorganizer of students’ thinking was strongly correlated with these teachers’ attempts to engage their students with high-level tasks. The distinction between using technology as an amplifier or a reorganizer is refined and extended through its application at the grain size of mathematical tasks, and implications for mathematics teacher education are discussed.

Shute, V. J., Graf, E. A., & Hansen, E. (2005). Designing adaptive, diagnostic math assessments for individuals with and without visual disabilities. In L. PytlikZillig, M. Bodvarsson, & R. Bruning (Eds.), *Technology-based education: Bringing researchers and practitioners together* (pp. 169–202). Greenwich, CT: Information Age Publishing.

Sinclair, N., & Armstrong, A. (2011). Tell a piecewise story. *Mathematics Teaching in the Middle School, 16*(6), 346–353

Sjoberg, C. A., Slavit, D., Coon, T., & Bay-Williams, J. (2004). Improving writing prompts to improve student reflection. *Mathematics Teaching in the Middle School, 9*(9), 490-495.

ABSTRACT: The teaching of mathematics continues to move away from a sole focus on correctness and a finished product to include a focus on process, context, and understanding. Writing tasks can be ideal tools for supporting student expression of ideas as a learning activity.

Slavin, R., Lake, C., & Groff, C. (2007). Effective programs in middle and high school mathematics: A best-evidence synthesis. *Review of Educational Research, 79*(2), 839-911.

ABSTRACT: This article reviews research on the achievement outcomes of mathematics programs for middle and high schools. Study inclusion requirements include use of a randomized or matched control group, a study duration of at least 12 weeks, and equality at pretest. There were 100 qualifying studies, 26 of which used random assignment to treatments. Effect sizes were very small for mathematics curricula and for computer-assisted instruction. Positive effects were found for two cooperative learning programs. Outcomes were similar for disadvantaged and nondisadvantaged students and for students of different ethnicities. Consistent with an earlier review of elementary programs, this article concludes that programs that affect daily teaching practices and student interactions have more promise than those emphasizing textbooks or technology alone.

Sleep, L., & Eskelson, S. L. (2012). MKT and curriculum materials are only part of the story: Insights from a lesson on fractions. *Journal of Curriculum Studies, 44*(4), 537-558.

ABSTRACT: This paper investigates the contribution of mathematical knowledge for teaching (MKT) and curriculum materials to the mathematical quality of instruction by comparing the enactment of a fractions problem taught by two teachers with differing MKT. It was found that MKT seem to support teachers’ precise use of mathematical language and to prevent errors; the curriculum materials provided a rich representational context for mathematical work. However, teachers’ orientations toward mathematics and mathematics teaching and their goals for student learning also seemed to contribute to their use of curriculum materials to engage students with rich mathematics and to support students’ participation in the development of the mathematics. Although orientations and goals made it more likely for a teacher to use multiple representations and elicit multiple solution methods, MKT was needed to productively use these elements in instruction. Based on this analysis, it is argued that there are aspects of developing orientations and goals that are related to MKT.

Smith III, J. P., & Berk, D. (2001). The “Navigating Mathematical Transitions Project”: Background, conceptual frame, and methodology. Paper presented at the annual meeting of the American Educational Research Association, Seattle, WA.

Smith III, J. P., & Phillips, E. A. (2000). Listening to middle school students’ algebraic thinking. *Mathematics Teaching in the Middle School, 6*(3), 156-161.

Smith III, J. P., & Star, J. R. (2007). Expanding the notion of impact of K-12 Standards-based mathematics and reform calculus programs. *Journal for Research in Mathematics Education, 38*(1), 3-34

ABSTRACT: Research on the impact of Standards-based, K-12 mathematics programs (i.e., written curricula and associated teaching practices) and of reform calculus programs has focused primarily on student achievement and secondarily, and rather ineffectively, on student attitudes. This research has shown that reform programs have competed well with traditional programs in terms of student achievement. Results for attitude change have been much less conclusive because of conceptual and methodological problems. We critically review this literature to argue for broader conceptions of impact that target new dimensions of program effect and examine interactions between dimensions. We also briefly present the conceptualization, design, and broad results of one study, the Mathematical Transitions Project (MTP), which expanded the range of impact along those lines. The MTP results reveal substantial diversity in students' experience within and between research sites, different patterns of experience between high school and university students, and surprising relationships between achievement and attitude for some students.

Smith III, J. P., Herbel-Eisenmann, B., Star, J. R., & Jansen, A. (2000). Quantitative pathways to understanding and using algebra: Possibilities, transitions, and disconnects. Paper presented at the Research Presession of the annual meeting of the National Council of Teachers of Mathematics, Chicago, IL.

Smith III, J. P., Phillips, E. A., & Herbel-Eisenmann, B. (1998). *Middle school students’ algebraic reasoning: New skills and understandings from a reform curriculum.* Paper presented at the 20th Annual meeting of the PME, North American Chapter, Raleigh, NC.

Smith III, J. P., Star, J. R., & Herbel-Eisenmann, B. (2000). Studying mathematical transitions: How do students navigate fundamental changes in curriculum and pedagogy? Paper presented at the annual meeting of the American Educational Research Association, New Orleans, LA.

Spielman, L. J., & Lloyd, G. M. (2004). The impact of enacted mathematics curriculum models on prospective elementary teachers’ course perceptions and beliefs. *School Science and Mathematics, 104*(1), 32-44.

ABSTRACT: This paper communicates the impact of prospective teachers' learning of mathematics using novel curriculum materials in an innovative classroom setting. Two sections of a mathematics content course for prospective elementary teachers used different text materials and instructional approaches. The primary mathematical authorities were the instructor and text in the textbook section and the prospective teachers in the curriculum materials section. After one semester, teachers in the curriculum materials section (n= 34) placed significantly more importance on classroom group work and discussions, less on instructor lecture and explanation, and less on textbooks having practice problems, examples, and explanations. They valued student exploration over practice. In the textbook section (n= 19), there was little change in the teachers' beliefs, in which practice was valued over exploration. These results highlight the positive impact of experiences with innovative curriculum materials on prospective elementary teachers' beliefs about mathematics instruction.

Stancavage, F., Shepard, L., McLaughlin, D., Holtzman, D., Blankenship, C., & Zhang, Y. (2009). *Sensitivity of NAEP to the effects of reform-based teaching and learning in middle school mathematics.* Washington, D. C.: American Institutes for Research.

ABSTRACT: This study is a validity study of the National Assessment of Educational Progress (NAEP), intended to test the adequacy of NAEP for detecting and monitoring the effects of mathematics education reform. The current study design was intended to support a comparison of the relative effectiveness of three different types of large-scale assessments--"Balanced Assessment in Mathematics" (BAM), NAEP, and state assessments--for measuring the learning gains of students participating in a well-implemented reform mathematics curriculum. To provide a context for assessing student learning where the authors could be reasonably certain of observing substantial learning gains in mathematics over the course of a school year, they selected National Science Foundation's (NSF's) Connected Mathematics Project (CMP). Although the authors had initially hypothesized that BAM, being more closely aligned with the reform curriculum, would reveal larger gains than NAEP, they found that both assessments were equally sensitive to the gains of their sample of students in CMP classrooms, and NAEP appeared better able to detect gains in the algebra classrooms. This was true even though the BAM test required twice as much time to administer as the NAEP test. Three appendices are included: (1) Sample NAEP Items; (2) Sample BAM Task; and (3) Analyses Using Booklet Percent Correct Metric.

Stancavage, V. B., Shepard, L., McLaughlin, D., Holtzman, D., Blankenship, C., & Zhang, Y . (2009). *Sensitivity of NAEP to the effects of reform-based teaching and learning in middle school mathematics*. A publication of the NAEP Validity Studies Panel. Palo Alto, CA: American Institutes for Research.

Star, J. R. (2001). *Re-conceptualizing procedural knowledge: Innovation and flexibility in equation solving.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62(10). (ProQuest ID No. 726024271)

ABSTRACT: The studies described in this thesis explore the development of students' knowledge of mathematical procedures. Students' tendency to develop rote knowledge of procedures has been widely commented on and is generally attributed to a lack of connection to principled knowledge. I postulate an alternative endpoint for the development of procedural knowledge, one that Ryle (1949) called an “intelligent performance" and Skemp (1976) described as "relational." Students demonstrate this capacity when they are able to flexibly use mathematical procedures, especially when they choose to deviate from established solving patterns on particular problems for greater efficiency. The purposes of these studies were (a) to demonstrate that students could develop the ability to execute mathematical procedures "intelligently," and (b) to explore the instructional conditions that facilitate the emergence of this outcome. In three studies, students with no prior knowledge of formal linear equation solving techniques were taught the basic transformations of this domain. After instruction, students engaged in problem-solving sessions in two conditions. In the treatment group, students completed "alternative ordering tasks," where they were asked to re-solve previously completed problems but using a different ordering of steps. When the treatment group engaged in such tasks, the control group instead solved structurally isomorphic equations. In Study 1, 10 students worked individually with the experimenter for 4, 30-minute problem-solving sessions. Eight of the 10 students became very successful solvers of linear equations, discovering sub-goal knowledge and also developing an efficient and consistently used solving heuristic. In Study 2, 36 students engaged in 4 one-hour individual problem-solving sessions. The treatment group students became more innovative and more flexible solvers. Study 3 replicated Study 2 using a classroom rather than an individualized learning environment; similar results were obtained.

These results suggest that alternative ordering tasks may help to support the development of flexible knowledge of procedures. Flexibility is an advantage for acquiring more advanced knowledge and also for solving specific problems most efficiently. With training on considering alternative solutions, students can be assisted in avoiding rote learning of procedures and in developing a rich understanding of why procedures work.

Star, J. R., & Hoffmann, A. J. (2002). Assessing students' conceptions of reform mathematics. In D. Mewborn, P. Sztajn, D. White, H. Wiegel, R. Bryant, & K. Nooney (Eds.), *Proceedings of the twenty-fourth annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education* (pp. 1729-1732). Columbus, OH: ERIC Clearinghouse for Science, Mathematics, and Environmental Education.

ABSTRACT: As the use of NSF-sponsored, reform-oriented mathematics curricula has become more prevalent across the US, an increasing number of researchers are attempting to study the "impact" of reform. In particular, mathematics educators are interested in determining whether reforms are having the desired effects on students, particularly with respect to the learning of mathematical content and the improvement of attitudes about mathematics. In this effort, researchers have used a variety of methods, and have looked at a variety of variables, in order to assess the impact of reform. In many cases, such research assesses reform by looking closely at students' scores on tests or their strategies for solving certain kinds of problems. For example, Riordan & Noyce (2001) assessed reform's impact by comparing students' scores on standardized achievement tests. Other researchers have used structured interviews, classroom observations, and more interpretive or ethnographic methods to assess the impact of reform (e.g., Boaler,1997). Both of these methodologies are useful in assessing the impact that reform mathematics curricula are having on students. An alternative evaluation of the impact of reform that has not been as widely used is through the use of survey instruments. Surveys have been widely and reliably used to assess students' motivation (Pintrich, Smith, Garcia, & McKeachie, 1993), beliefs and attitudes (Kenney & Silver, 1997), and interest (Köller, Baumert, & Schnabel, 2001). We propose to add to this literature by using a survey to study the impact of reform on students' conceptions of mathematics.

Star, J. R., & Hoffmann, A. J. (2005). Assessing the impact of Standards-based curricula: Investigating students’ epistemological conceptions of mathematics. *The Mathematics Educator, 15*(2), 25-34.

ABSTRACT: Since the advent of the NCTM Standards (1989), mathematics educators have been faced with the challenge of assessing the impact of Standards-based (or “reform”) curricula. Research on the impact of Standards-based curricula has predominantly focused on student achievement; here we consider an alternative: Students’ epistemological conceptions of mathematics. 297 participants were administered a Likert-scale survey instrument, the Conceptions of Mathematics Inventory. Of these, 163 had not experienced Standards-based curricula, while the rest had used a Standards-based curriculum for over three years. Our results indicate that students at the Standards-based site expressed more sophisticated epistemological conceptions of mathematics than those of the students from the non-Standards-based site. We interpret this result to suggest that implementation of Standards-based curricula may be having an effect on students’ epistemological conceptions of mathematics.

Star, J. R., Herbel-Eisenmann, B. A., & Smith III, J. P. (2000) Algebraic concepts: What's really new in new curricula?. *Mathematics Teaching in the Middle School, 5*(7). 446-451.

ABSTRACT: Examines 8th grade units from the Connected Mathematics Project (CMP). Identifies differences in older and newer conceptions, fundamental objects of study, typical problems, and typical solution methods in algebra. Also discusses where the issue of what is new in algebra is relevant to many other innovative middle school curricula.

Star, J. R., Smith III, J. P., & Hoffmann, A. J. (2004). *Students’ perceptions of difference between traditional and Standards-based mathematics curricula.* Paper presented at the Research Presession of the annual meeting of the National Council of Teacher of Mathematics, Philadelphia, PA.

Star, J. R., Smith III, J. P., & Jansen, A. J. (2008). What students notice as different between reform and traditional mathematics programs. *Journal for Research in Mathematics Education, 39*(1), 9-32.

ABSTRACT: Research on the impact of Standards-based mathematics and reform calculus curricula has largely focused on changes in achievement and attitudes, generally ignoring how students experience these new programs. This study was designed to address that deficit. As part of a larger effort to characterize students' transitions into and out of reform programs, we analyzed how 93 high school and college students perceived Standards-based and reform calculus programs as different from traditional ones. Results show considerable diversity across and even within sites. Nearly all students reported differences, but high-impact differences, like Content, were not always related to curriculum type (reform or traditional). Students' perceptions aligned moderately well with those of reform curriculum authors, e.g., concerning Typical Problems. These results show that students' responses to reform programs can be quite diverse and only partially aligned with adults' views.

Stauffer, T. C. (2011). More of sixth graders flip for breakfast. *Teaching Children Mathematics, 18*(5), 328-330.

ABSTRACT: A coin-flipping activity is meant to show students that a small number of trials may produce a wide variation in results.

Stevens, B. B. A. (2005). *The development of pedagogical content knowledge of a mathematics teaching intern: The role of collaboration, curriculum, and classroom context.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 67(9). (ProQuest ID No. 1212777591)

ABSTRACT: In this study I examined the role of collaboration, curriculum, and the classroom context in the development of pedagogical content knowledge of a mathematics teaching intern. Additionally, I investigated the nature of the collaborative process between the teaching intern and his mentor teacher as they collaborated on action (during structured planning time) and in action (while students were present). The teaching internship resided in a seventh-grade mathematics classroom during the teaching of a probability unit from a standards-based curriculum, Connected Mathematics Project.

Using existing research, a conceptual framework was developed and multiple data sources (audio taped collaborations, observations of the intern's teaching practices, semi structured interviews, and a mathematics pedagogy assessment) were analyzed in order to understand the teaching intern's development of knowledge of instructional strategies, knowledge of student understandings, curricular knowledge, and conceptions of purpose for teaching probability.

Results identified numerous dilemmas related to planning and implementing instruction. Although the teaching intern developed pedagogical content knowledge, he often experienced difficulty accessing it while teaching. Through collaboration, curriculum, and the classroom context, the teaching intern learned to incorporate his pedagogical content knowledge in instruction. Analysis revealed that as he gained new knowledge he was able to shift his focus from content to the use of instructional strategies for teaching and learning. The curriculum was the primary focus of collaboration and initiated the intern's examination of the learning-to-teach process.

Collaboration on action and collaboration in action proved to be essential elements in the development of pedagogical content knowledge.

Stylianides, G. J. (2007). Investigating the guidance offered to teachers in curriculum materials: The case of proof in mathematics. *International Journal of Science and Mathematics Education, 6*(1), 191 -215.

ABSTRACT: Despite widespread agreement that proof should be central to all students’ mathematical experiences, many students demonstrate poor ability with it. The curriculum can play an important role in enhancing students’ proof capabilities: teachers’ decisions about what to implement in their classrooms, and how to implement it, are mediated through the curriculum materials they use. Yet, little research has focused on how proof is promoted in mathematics curriculum materials and, more specifically, on the guidance that curriculum materials offer to teachers to enact the proof opportunities designed in the curriculum. This paper presents an analytic approach that can be used in the examination of the guidance curriculum materials offer to teachers to implement in their classrooms the proof opportunities designed in the curriculum. Also, it presents findings obtained from application of this approach to an analysis of a popular US reform-based mathematics curriculum. Implications for curriculum design and research are discussed.

Stylianides, G. J. (2009). Reasoning-and-proving in school mathematics textbooks. *Mathematical Thinking and Learning, 11*, 258–288.

ABSTRACT: Despite widespread agreement that the activity of reasoning-and-proving should be central to all students' mathematical experiences, many students face serious difficulties with this activity. Mathematics textbooks can play an important role in students' opportunities to engage in reasoning-and-proving: research suggests that many decisions that teachers make about what tasks to implement in their classrooms and when and how to implement them are mediated by the textbooks they use. Yet, little is known about how reasoning-and-proving is promoted in school mathematics textbooks. In this article, I present an analytic/methodological approach for the examination of the opportunities designed in mathematics textbooks for students to engage in reasoning-and-proving. In addition, I exemplify the utility of the approach in an examination of a strategically selected American mathematics textbook series. I use the findings from this examination as a context to discuss issues of textbook design in the domain of reasoning-and-proving that pertain to any textbook series.

Tarr, J. E., Reys, R. E., Reys, B. J., Chavez, O., Shih, J., & Osterlind, S. J. (2008). The impact of middle grades mathematics curricula on student achievement and the classroom learning environment. *Journal for Research in Mathematics Education, 39*(3), 247-280.

ABSTRACT: We examine student achievement of 2533 students in 10 middle schools in relation to the implementation of textbooks developed with funding from the National Science Foundation (NSF) or publisher developed textbooks. Using hierarchical linear modeling (HLM), curriculum type was not a significant predictor of student achievement on the Balanced Assessment in Mathematics (BAM) or TerraNova Survey (TNS) after controlling for student-level variables. However, the Standards-Based Learning Environment (SBLE) moderated the effect of curriculum type. Students were positively impacted on the BAM by NSF-funded curricula when coupled with either Moderate or High levels of SBLE. There was no statistically significant impact of NSF- funded curricula on students in classrooms with a Low level of SBLE, and the relationship between publisher-developed textbooks and SBLE was not statistically significant. Moreover, there was no significant impact of either curriculum type when coupled with varying levels of SBLE on the TNS as the dependent measure.

Tarr, J., Chavez, O., Appova, A., & Regis, T. (2005). *Discordant implementation of mathematics curricula by middle school mathematics teachers.* In G. M. Lloyd, M. Wilson, J. L. M Wilkins, & S. L. Behm (Eds.), *Proceedings of the Twenty-Seventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education.* ERIC Clearinghouse for Science, Mathematics, and Environmental Education: Roanoke, Virginia.

The El Barrio-Hunter College PDS Partnership Writing Collective. (2009). On the unique relationship between teacher research and commercial mathematics curriculum development. In J. T. Remillard, B. A. Herbel-Eisenmann, & G. M. Lloyd (Eds.), *Mathematics teachers at work: Connecting curriculum materials and classroom instruction.* London: Routledge.

Theule-Lubienski, S. A. (1996). *Mathematics for all?: Examining issues of class in mathematics teaching and learning. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 58(1). (ProQuest ID No. 739654911)

ABSTRACT: Diversity and equity are popular topics in the mathematics education community today, particularly amidst current reforms intended to "empower all students." Still, little attention is given to socio-economic diversity in relation to mathematics teaching and learning.

In this study, a researcher-teacher explores the ways in which a curriculum and pedagogy aligned with current, mathematics education reforms played out with a socio-economically diverse group of seventh-grade students. Interviews, surveys, teaching journal entries, and daily audio recordings were used to document students' experiences across the 1993-94 school year. Qualitative analyses compared the lower-and higher-SES students' experiences with the whole-class discussions and contextualized, open-ended mathematics problems. The analyses revealed that while the higher-SES students tended to have confidence in their abilities to make sense of the mathematical discussions and problems, the lower-SES students often said they were "confused" by conflicting ideas in the discussions and the open nature of the problems--they desired more specific direction from the teacher and texts. Additionally, while the higher-SES students seemed to approach the problems and discussions with an eye toward the larger, abstract, mathematical ideas, the lower-SES students more often became "stuck" in the contexts of the problems.

ES students more often became "stuck" in the contexts of the problems. The study examines critical links between the current mathematics reforms and literatures on social class, which suggest there might be a mismatch between the culture of lower-SES students and the culture of the mathematics classroom advocated by current reformers. "Cultural confusion" is proposed as an explanation for the struggles the lower-and working-class students faced in the reformed mathematics classroom. The study suggests that a classroom in which taking initiative in solving problems, analyzing and discussing ideas, and abstracting mathematical ideas from contextualized problems, might be more aligned with middle-class students' preferred ways of communicating, thinking and learning.

Dilemmas involved in educating lower-and working-class students are discussed. This study contributes to our understanding of both possibilities and hazards inherent in constructivist-inspired pedagogies and curricula intended to "empower all students," in both mathematics and other fields.

Triantos, L. M. (2005). *The aftermath of implementing a Standards based curriculum in a K–8 district: Is there a correlation between hands-on instruction and math scores?* (Unpublished master’s thesis). Rowan University, Glassboro, NJ.

Turner, J., & Meyer, D. (2004). A classroom perspective on the principle of moderate challenge in mathematics. *Journal of Educational Research, 97*(6), 311-318.

ABSTRACT: The authors reviewed the research on challenge as a motivator, with a view toward application in mathematics classrooms. The authors conclude that traditional motivational research, with its focus on individual differences and decontextualized tasks, is not readily applicable to classrooms. They argue that a combination of challenging instruction and positive affective support is necessary for promoting motivation in mathematics classrooms. The authors describe the kinds of classroom contexts that are likely to support challenge seeking and learning in mathematics and illustrate an example of a teacher who used challenge effectively in her 7th-grade mathematics classes. Finally, the authors suggest that a focus on creating contexts that support challenge seeking offers a powerful application of this motivational tool for all learners.

Turner, J., & Patrick, H. (2004). Motivational influences on student participation in classroom learning activities. *Teachers College Record, 106*(9), 1759-1785.

ABSTRACT: This study examined how one type of student work habit-classroom participation-is related to a combination of both student factors (math achievement, personal achievement goals, perceptions of classroom goal structures, and teacher support) and features of the classroom context (teachers' instructional practices, average perceptions of classroom goal structures). We focused on the participation of two students in mathematics class during both sixth and seventh grades. Differential teacher expectations, calling patterns, and instructional and motivational support and nonsupport interacted with beliefs and behaviors of both students, and those interactions were associated with different patterns of participation each year. Results suggest that student participation is malleable rather than stable and emphasize the potential of teacher practices to both support and undermine the development of student work habits.

Turner, J., Midgley, C., Meyer, D., Gheen, M., Anderman, E., Kang, Y., & Patrick, H. (2002). The classroom environment and students' reports of avoidance strategies in mathematics: A multimethod study. *Journal of Educational Psychology, 94*(1), 88-106.

ABSTRACT: The relation between the learning environment (e.g., students' perceptions of the classroom goal structure and teachers' instructional discourse) and students' reported use of avoidance strategies (vselfhandicapping, avoidance of help seeking) and preference to avoid novelty in mathematics was examined. Quantitative analyses indicated that students' reports of avoidance behaviors varied significantly among classrooms. A perceived emphasis on mastery goals in the classroom was positively related to lower reports of avoidance. Qualitative analyses revealed that teachers in high-mastery/low-avoidance and low mastery/high-avoidance classrooms used distinctively different patterns of instructional and motivational discourse. High incidence of motivational support was uniquely characteristic of high-mastery/low avoidance classrooms, suggesting that mastery goals may include an affective component. Implications of the results for both theory and practice are discussed.

Umbeck, L. M. (2011). Navigating classroom change: A renegotiated classroom culture results in students learning to participate in new ways. *Mathematics Teaching in the Middle School, 17*(2), 88-95.

Van Dyke, C. L. (2001). *The shape of things to come: Mathematics reform in the middle school.* (Masters thesis). Retrieved from Masters Abstracts International, 40(2). (ProQuest ID No. 727357331)

ABSTRACT: In this thesis I investigate the implementation of the Connected Mathematics Project (CMP) at Gallup Middle School in the Holbrook School District. I analyze my experiences and observations at Gallup Middle School during the 2000-2001 school year in the broad context of mathematics education reform. My observations reveal difficulties with implementing CMP. I describe several factors contributing to these problems. It is my goal to strengthen investigation-oriented mathematics by illuminating its weaknesses. I believe CMP fosters a greater understanding of mathematics among students. This understanding creates the foundation for a mathematical perspective on the world. The development of a mathematical perspective is crucial to the economic well-being of our students and, in turn, our country.v

Waite, R.D. (2000).* A study of the effects of Everyday Mathematics on student achievement of third-,fourth-, and fifth-grade students in a large north Texas urban school district.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 61(10). (ProQuest ID No. 1251814391)

ABSTRACT: Data were examined in this study from student records in a large North Texas urban school district who were taught with two different mathematics curricula to determine whether or not they had different effects on student achievement. One of the mathematics curricula, Everyday Mathematics, was developed upon national mathematic standards, written by the National Council of Teachers of Mathematics. The other mathematics curriculum was district-approved, using a textbook from a large publisher, with a more traditional approach.

The students selected for the experimental group came from six schools that had implemented the Everyday Mathematics curriculum for the 1998-99 school year. An experimental group was formed from these students. Twelve schools with similar socioeconomic ratios, ethnic makeup and 1998 Iowa Test of Basic Skills mathematic score profiles were selected. A control group was formed from this population of students that was similar to the experimental group with the exception of having been taught using the district-approved mathematics curriculum.

These two groups were very similar in socioeconomic, ethnic, gender, and grade level makeup. Most importantly, the experimental group and control group were almost identical (there was no statistically significant difference) in their 1998 Iowa Test of Basic Skills mathematics scores, a gauge used to demonstrate that prior mathematics ability was equal going into the 1998-99 school year.

In the statistical analysis, almost all comparisons showed that the experimental group taught with the Everyday Mathematics curriculum had higher scores on the 1999 Texas Assessment of Academic Skills mathematics test. When compared to children with similar mathematics ability at the beginning of the 1998-99 school year, the students in this study who were taught using Everyday Mathematics showed greater achievement gains than students in classes that used the district-approved curriculum.

Wanko, J. J. (2000). *Going public: The development of a teacher educator's pedagogical content knowledge.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62(1). (ProQuest ID No. 727910571)

ABSTRACT: When Lee Shulman and his colleagues introduced pedagogical content knowledge (PCK) to the education lexicon in the 1980s, they gave teachers and teacher educators some technical language that could be used for talking about the knowledge needed for work that they do in classrooms, thereby helping to establish teaching as a profession. Since that time, the PCK of classroom teachers has been studied and documented across various content areas. But the PCK of teacher educators has remained a largely unexamined area of research, especially in the providing experiences in helping preservice teachers develop their own PCK. This study examines this issue more fully. Specifically, "Can pedagogical content knowledge be a useful framework for a teacher educator in designing and teaching a mathematics content course for preservice teachers and if so in what ways?"

In this study, I use my own teaching and classroom of prospective elementary teachers as the site for investigation. I examine the ways in which my own PCK as a teacher educator influenced and was influenced by my work with students. Data for the study are provided by my teaching journal, lesson and units plans, student work, and audiotapes of class proceedings.

In conclusion, I present three major findings of this study. First, this study highlights and problematizes Shulman's notion of representation that is used in defining pedagogical content knowledge. In mathematics there are mathematical and empirical representations--classifications which do not map easily onto Shulman's use of representation. This study exposes some of those inherent distinctions and seeks to make Shulman's work more applicable to the field of mathematics. Second, this study describes the importance of task design--a process that is particularly essential in teaching mathematics--and finds that Shulman's notions of PCK and the pedagogical reasoning and action cycle miss or obscure its significance. And third, this study introduces the notion of shared reflection to Shulman's model for pedagogical reasoning and action when it is applied to teacher education. It also finds that the act of going public with one's ideas through shared reflection can be a useful tool for teacher educators in the development of their pedagogical content knowledge.

Wasman, D. G. (2000). *An investigation of algebraic reasoning of seventh-and eighth-grade students who have studied from the Connected Mathematics Project curriculum.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 61(9). (ProQuest ID No. 727777811)

ABSTRACT: This study investigated algebraic reasoning of seventh and eighth graders' who have studied from the Connected Mathematics Project (CMP) materials. Algebraic reasoning was defined as the process of thinking logically about and applying algebraic concepts as described by NCTM's expectations for grades six through eight students described in the Patterns, Functions, and Algebra Standard outlined in the Principles and Standards for School Mathematics. The seventh and eighth graders represented 75% of the students at their grade level because the other 25% were enrolled in accelerated courses that did not use CMP. In order to document the extent and nature of the use of CMP, all sixth, seventh and eighth grade teachers completed a survey followed by researcher-conducted classroom observations. The Iowa Algebra Aptitude Test (IAAT) was administered to 100-seventh graders and 73-eighth graders. Five-seventh graders and six-eighth graders were randomly selected for individual interviews consisting of a series of twelve algebra tasks.

Students' performance on the IAAT and interview tasks demonstrated the well-developed nature of their understanding and use of algebraic ideas and strategies. Students demonstrated flexibility in their thinking and ability to describe linear relationships in a variety of representations. Students described rate of change arithmetically, algebraically, and/or geometrically in different situations. Students approached problems in a sense-making way, choosing a variety of different strategies (informal and formal) all of which led to correct solutions and reflected strong conceptual understanding of algebraic ideas. Eighth graders were more likely to use symbolic algebra methods to solve problems than the seventh graders, reflecting a natural development of more symbolic strategies. Context played an important role with regard to students' ability to interpret and symbolize mathematical ideas. Students were more likely to represent situations symbolically when they were embedded in a context-rich setting. Some students had difficulty translating from a recursive pattern to an explicit formula and interpreting a graph as a relationship between independent and dependent variables. These same weaknesses have been noted in other research studies indicating that these ideas may require more time or maturity to develop, regardless of the particular curriculum used.

Wasserman, L. (2008). A Marriage Made in Math Class. Teacher Magazine, 2(1).

Wernet, J. L. W. (2015). *What's the story with story problems? Exploring the relationship between contextual mathematics tasks, student engagement, and motivation to learn mathematics in middle school* (Order No. 3689098). Available from Dissertations & Theses @ CIC Institutions; ProQuest Dissertations & Theses A&I; ProQuest Dissertations & Theses Global. (1678945896).

ABSTRACT: Contextual tasks, or tasks that include scenarios described at least in part with nonmathematical language or pictures, are a long-standing part of mathematics education in the United States. These tasks may have potential to promote student engagement and motivation to learn mathematics by highlighting applications of mathematics to everyday matters and generating interest in the content (e.g., van den Heuvel-Panhuizen, 2005). Yet, several scholars have challenged the belief that contextual tasks can serve to motivate students and problematized their role in mathematics curricula (e.g., Chazan, 2000; Gerofsky, 2004). Some theoretical and empirical evidence exists to support both claims.

This study addresses a call for more research on how student motivation and engagement in mathematics are influenced in specific learning situations, namely, working on contextual tasks. Motivation describes a person's choice, persistence, and performance when engaging in an activity (Brophy, 2004), whereas engagement is active involvement in a learning activity (Helme & Clarke, 2001) and the observable manifestation of motivation (Skinner, Kindermann, & Furrer, 2008). The purpose of this multiple-case study was to consider the general questions, Do contextual tasks have potential to engage students, and if so, under what circumstances?, and How do students experience these tasks relative to their motivation to learn mathematics? In particular, I considered enactment of tasks across lessons in two 7th -grade mathematics classrooms. Through analyzing data from observations, lesson-specific teacher and student surveys, and focus group interviews, I identified the most and least engaging lessons for students, then characterized the tasks in these lessons as written and enacted.

I found that students were more likely to show high levels of engagement in contextual tasks than noncontextual tasks. Their engagement in contextual tasks was related, however, to the learning goals of the task, its placement in a unit, and the function of the context in problem solving. In high-engagement lessons, the tasks tended toward open-ended tasks with contexts central in solving the problem. I also found differences in the way students and teachers attended to contextual features of tasks between the high- and low-engagement lessons. Students drew on the context more in the high engagement lessons, and were more likely to connect the context to the main mathematical ideas in the lesson. Teachers also paid more attention to contexts and in more diverse ways across the high-engagement lessons.

I also drew on the data sources using expectancy-value theory to explore in depth how students responded to individual tasks relative to their motivation to learn mathematics. Aspects of tasks students attended to (including contexts) when reflecting on the value of mathematical content and their experiences in lessons was related to their underlying motivation to learn. Trends across groups of students, however, indicate that task contexts play little role in promoting students' valuing of mathematics or beliefs that they can be successful on a task.

Based on these findings, I argue that some contextual tasks engage students by eliciting genuine interest in the context itself, providing entry into and support in solving the problem, and anchoring the instruction to provide students a shared experience on which to develop their understanding of the mathematical concepts. Yet, contextual tasks do not necessarily have the same potential to motivate students to learn. I discuss implications for teachers, curriculum design, and future research regarding the purpose and function of contextual tasks.

Wilhelm, A. G. (2015). Mathematics teachers’ enactment of cognitively demanding tasks: Investigating links to teachers’ knowledge and conceptions. *Journal for Research in Mathematics Education, 45*(5), 636–674.

ABSTRACT: This study sought to understand how aspects of middle school mathematics teachers’ knowledge and conceptions are related to their enactment of cognitively demanding tasks. I defined the enactment of cognitively demanding tasks to involve task selection and maintenance of the cognitive demand of high-level tasks and examined those two dimensions of enactment separately. I used multilevel logistic regression models to investigate how mathematical knowledge for teaching and conceptions of teaching and learning mathematics for 213 middle school mathematics teachers were related to their enactment of cognitively demanding tasks. I found that teachers’ mathematical knowledge for teaching and conceptions of teaching and learning mathematics were contingent on one another and significantly related to teachers’ enactment of cognitively demanding tasks.

Wilson, Nazemi, Jackson, Wilhelm (2019). Investigating Teaching in Conceptually Oriented Mathematics Classrooms Characterized by African American Student Success. *Journal for Research in Mathematics Education. *Vol. 50, No. 4, 362-400

ABSTRACT: This article outlines several forms of instructional practice that distinguished middle-grades mathematics classrooms that were organized around conceptually oriented activity and marked by African American students’ success on state assessments. We identified these forms of practice based on a comparative analysis of teaching in (a) classrooms in which there was evidence of conceptually oriented instruction and in which African American students performed better than predicted by their previous state assessment scores and (b) classrooms in which there was evidence of conceptually oriented instruction but in which African American students did not perform better than predicted on previous state assessment scores. The resulting forms of practice can inform professional learning for preservice and in-service teachers.

NOTE: This study was done in CMP classrooms.

Wilt, B. J. (2007) *Preservice teachers to inservice teachers: Teaching for social justice. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 68(10). (ProQuest ID No. 1251814391)

ABSTRACT: The purpose of this research project was to explore how preservice teachers, who are now currently inservice teachers, who took an undergraduate secondary education course with a focus on teaching for social justice, currently, make sense of what it means to teach for social justice. The participants in this study took the same secondary education course for preservice teachers which focused on critical consciousness raising experiences in order to promote teaching for social justice in classrooms. My participants took this course when they were working on their undergraduate degrees in education. This course was the one course in the education program that brought together students in each of the content specific discipline areas of the program--mathematics, science, social studies, language and literacy, and world languages (Bullock, 2004). The course was taken the semester prior to student teaching and occurred during the first 10 weeks of the semester followed by a five-week practicum (Bullock, 2004). In order to conduct my research project I solicited sixteen secondary education teachers, who were previously enrolled in the same undergraduate teacher education program (mentioned above) at a major university in the Mid-Atlantic region of the United States to volunteer to participate in this study. The inservice teachers all took the same secondary education course as preservice teachers which focused on teaching for social justice.

There were many factors that influenced the participant's perspectives about teaching for social justice as well as the degree to which they taught for social justice. All these factors--the undergraduate course rooted in critical consciousness raising experiences, sociocultural structures, political structures, contextual influences, hidden curriculum, teaching stance, teaching praxis--connected to power, privilege, and oppression through issues such as race, ethnicity, class, culture, sexual preference, language, ability, etc.

Both participants self-admittedly teach for social justice, however, the degree to which this takes place varies depending on their respective perspectives on what it means to teach for social justice. Jim does not teach for social justice and is less inclined to trouble and challenge dominant perspectives because he is uncomfortable with difference. He also does not understand the sociocultural mechanisms that reproduce hegemony and is part of the complicit process rather than part of the solution. Neil teaches for social justice to a certain degree and is more inclined to trouble and challenge the status quo and hegemonic mechanisms. However, he does this by teaching for the other (Kumashiro, 2002) and teaching about the other (Kumashiro, 2002) but does not teach in a manner that is critical of privileging and Othering (Kumashiro, 2002).

This study suggests that more research is needed in order to explore and understand how teachers who have an awareness of teaching for social justice actually teach for social justice. This exploration and understanding needs to look at the broad scope of the influences on teachers and how these influences impact teaching for social justice. In addition, teacher education programs must emotionally and structurally embrace curricula rooted in social justice in order to promote teaching for social justice in a way that preservice teachers can also embrace and incorporate in to their teaching praxis. If preservice teachers are to do this they need to understand approaches like education for the Other, education about the Other, education that is critical of privileging and Othering, and education that changes students and society (Kumashiro, 2002) in order to teach for social justice.

Winking, D. (1998). The Minneapolis Connected Mathematics Project: Year two evaluation. Retrieved from: http://tis.mpls.k12.mn.us/sites/5df1b159-7ce3-4aa3-8e71- 8e60a7b98e6c/uploads/connected_mathematics_2.pdf. Minneapolis, MN: Minneapolis Public Schools.

Winking, D. (2000a). *Minneapolis data: Excerpts from the year two evaluation report. *Connected Mathematics Project, East Lansing, MI.

Winking, D. (2000b). *Minneapolis data: Excerpts from the year one evaluation report.* Connected Mathematics Project, East Lansing, MI.

Winking, D., Bartel, A., & Ford, B. (1998). *The Connected Mathematics Project: Helping Minneapolis middle school students ‘beat the odds’: Year one evaluation report.* Report submitted to the National Science Foundation as part of the Connecting Teaching, Learning, and Assessment Project.

Woodward, J., & Brown, C. (2006). Meeting the curricular needs of academically low-achieving students in middle grade mathematics. *The Journal of Special Education, 40*(3), 151.

ABSTRACT: An important component of the National Council of Teachers of Mathematics Standards is the equity principle: All students should have access to a coherent, challenging mathematics curriculum. Many in the mathematics reform community have maintained that this principle can be achieved through one well-designed curriculum. However, the extant research on equity—which focuses on either ethnic diversity or academic achievement—suggests that this principle is illusive. The current study compares the effectiveness of two curricula in teaching a range of math concepts to 53 (28 male; 25 female) middle school students at risk for special education services in math. The yearlong, quasi-experimental study involved achievement and attitudinal measures. Results indicated that students in the intervention group who used materials designed according to instructional principles described in the special education literature achieved higher academic outcomes (p < .05, p < .001) and had more positive attitudes toward math (p < .001) than did students in the comparison group.

Wu, Z. (2004). *The study of middle school teacher’s understanding and use of mathematical representation in relation to teachers’ zone of proximal development in teaching fractions and algebraic functions.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 65(7). (ProQuest ID No. 775173261)

ABSTRACT: This study examined teachers' learning and understanding of mathematical representation through the Middle School Mathematics Project (MSMP) professional development, investigated teachers' use of mathematics representations in teaching fractions and algebraic functions, and addressed patterns of teachers' changes in learning and using representation corresponding to Teachers' Zone of Proximal Development (TZPD). Using a qualitative research design, data were collected over a 2-year period, from eleven participating 6th and 7th grade mathematics teachers from four school districts in Texas in a research-designed professional development workshop that focused on helping teachers understand and use of mathematical representations. Teachers were given two questionnaires and had lessons videotaped before and after the workshop, a survey before the workshop, and learning and discussion videotapes during the workshop. In addition, ten teachers were interviewed to find out the patterns of their changes in learning and using mathematics representations. The results show that all teachers have levels of TZPD which can move to a higher level with the help of capable others. Teachers' knowledge growth is measurable and follows a sequential order of TZPD. Teachers will make transitions once they grasp the specific content and strategies in mathematics representation. The patterns of teacher change depend on their learning and use of mathematics representations and their beliefs about them. This study advocates teachers using mathematics representations as a tool in making connections between concrete and abstract understanding. Teachers should understand and be able to develop multiple representations to facilitate students' conceptual understanding without relying on any one particular representation. They must focus on the conceptual developmental transformation from one representation to another. They should also understand their students' appropriate development levels in mathematical representations. The findings suggest that TZPD can be used as an approach in professional development to design programs for effecting teacher changes. Professional developers should provide teachers with opportunities to interact with peers and reflect on their teaching. More importantly, teachers' differences in beliefs and backgrounds must be considered when designing professional development. In addition, professional development should focus on roles and strategies of representations, with ongoing and sustained support for teachers as they integrate representation strategies into their daily teaching.

Zawojewski, J. S., Ridgway, J., Hoover, M. N., & Lambdin, D. V. (2002). The Connected Mathematics curriculum: Intentions, experiences and performance. In S. L. Senk & Denisse R. Thompson (Eds.), *Standards oriented school mathematics curricula: What does the research say about student outcomes?* Mahwah, NJ: Lawrence Erlbaum.

Zawojewski, J. S., Robinson, M., & Hoover, M. V. (1999). Reflections on mathematics and the Connected Mathematics Project. *Mathematics Teaching in the Middle School, 4*(5), 324-30.

Zvoch, K., & Stevens, J. (2006). Longitudinal effects of school context and practice on middle school mathematics achievement. *The Journal of Educational Research, 99*(6), 347– 357.

ABSTRACT: The authors analyzed mathematics achievement data from a longitudinally matched student cohort from a large southwestern U.S. school district to investigate school context and practice effects on the academic performance and growth of middle school students. Investigation of the degree to which aspects of the school environment related to mathematics achievement outcomes revealed 2 distinct patterns. School context, as measured by student and school demographic characteristics, related closely to mathematics performance levels but had little relationship with mathematics growth rates. The opposite was true for aspects of school practice. Teacher educational attainment and the mathematics curricula delivered to students were not related to student performance levels but were moderately associated with mathematics growth rates. These results suggest that the effect of some policy-relevant school variables may be difficult to identify when student achievement is studied at a single point in time. However, investigation of school impacts on student achievement may be facilitated when an analytic strategy that takes into account the time-dependent and cumulative nature of schooling is adopted.