Toggle Accessibility Tools

All Published Research and Evaluation on CMP

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.


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.

Read It's a Fird!

Alibali, M. W., Stephens, A. C., Brown, A. N., Yvonne, S., & Nathan, M. J. (2014). Middle 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.

Read Evaluation Report

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.

Read Article

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

Read Three-Year Study

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.

Read Article

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.

Read The Top 10 Elements

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

Read Article

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.

Read Article

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.

Read Article

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.

Read Reasoning about Data Analysis

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

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.

Read Article

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.)

Read van Hiele Revisted

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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. 

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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).

Read Article 

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.

Read Article

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.

Read Article

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.

Read Article

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.

Read Article

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.

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.”

Read Article

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.

Read Study

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.

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.

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.

Read Article

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.

Read A Taxonomy for Categorizing Generalizations

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. 

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.

Read Paper

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.

Read Connections Between Generalizing and Justifying

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) 

Read Quantifying Exponential Growth

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.

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.

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.

Read Dissertation

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.

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.

Read Article

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.

Read Dissertation

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.

Read Study

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.

Read Paper

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. (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.

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.

Read Paper

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.

Read Study

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.

Read Article

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.

Read Study

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.

Read Article

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.

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.

Read Article

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.

Read Study

Keiser, J. M. (2000). The role of definition. Mathematics Teaching in the Middle School5(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.

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. 

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.

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.

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.

Read Study

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.

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.

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.

Read Article

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.

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. 

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.

Read Article

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.

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.

Read Article

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. (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.

Read Article

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.

Read Article

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.

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.

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., 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.

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.

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.

Read Article

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.

Read Article

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.

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.

Read Article

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.

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.

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.

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.

Read Article

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.

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.

Read Article

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.

Read Article

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.

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

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.

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.

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.