# 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.

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

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

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

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

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is there a difference between Standards-based curriculum students and non-Standards-based curriculum students?* (Unpublished master’s thesis).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

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

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.

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

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

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.

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

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

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

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

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

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

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

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

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

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

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