# 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.

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

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

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

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

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

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.

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.”

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.

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.

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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

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

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

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