# 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.

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

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

Read Middle School Students Conceptual Understanding of Equations

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.

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

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.

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.

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

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

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

Cai, J., 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.

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

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

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

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

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

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

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

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

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

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.

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.

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.

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

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

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

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

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

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

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.

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

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

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

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

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.

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%).

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.