# 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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

NOTE: This study was done in CMP classrooms.