# 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.

Celedon, S. (1998). *An analysis of a teacher's and students' language use to negotiate meaning in an ESL/mathematics classroom. *(Doctoral dissertation). Retrieved from Dissertation Abstracts International, 69(9). (ProQuest ID No.732855961)

ABSTRACT: The research reviewed indicates a paucity of studies addressing issues regarding language as used by linguistically diverse students and its role in mathematics problem solving, especially at the secondary level. The purpose of this qualitative study was threefold: (1) to describe how English as a second language (ESL) students and their teacher used language (Spanish and English) to negotiate mathematical meaning in an ESL/Mathematics classroom, (2) to explore problem-solving strategies used by ESL students and examine how these connect, or not, to those presented by their teacher, and (3) to generate a theory about the use of language to teach mathematics to ESL students. Research was conducted in a self-contained ESL/Mathematics classroom at the middle school level (6th-8th grade). The study included participant observations, in-depth interviews with a representative sample of nine students and the teacher, and written documents.

Analysis of the data collected throughout a nineteen-week period indicated that Spanish was the language used by most ESL students to express themselves when they needed to elaborate on their responses orally or in written form as they engaged in a curriculum, the Connected Mathematics Project(CMP), that promoted higher order thinking skills. From the teacher-student discourse samples, it was evident that using Spanish created more opportunities for students to participate in discussions where an explanation of their responses was needed. Furthermore, these students felt comfortable expressing themselves in their first language when explaining their problem-solving strategies during think-aloud protocols. Overall, the accuracy of these nine students improved by one or two word problems (out of five)in the Spanish version. These results indicate the importance of making both languages accessible to students during mathematics problem solving. While I am not advocating that Spanish be used as the only language of instruction, I am suggesting that students' sociocultural and linguistic experiences be used to make the mathematical connections between the everyday use of English and the language that is specific to mathematics.

Studying how ESL students used language when engaged in mathematical problem solving provides educators insight as to how they can help students make connections between their existing everyday language and the mathematical language necessary for problem solving. In addition, these findings provide both ESL and mathematics teachers with detailed information regarding the variety of problem-solving strategies used by ESL students.

Choppin, J. (2006). *Design rationale: Role of curricula in providing opportunities for teachers to develop complex practices.* Paper presented at the 28th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, Mèrida, Mèxico.

ABSTRACT: This study analyzes the potential of two similar tasks to generate dialogic classroom interactions. Although both tasks were similar in context and outcome, one affords teachers’ actions to elicit and build from diverse student explanations. This would require greater teacher expertise – both mathematically and pedagogically – and an articulation of conditions when more potentially dialogical tasks should be implemented.

Danielson, C. (2005).* Walking a straight line: Introductory discourse on linearity in classrooms and curriculum.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 67(2). (ProQuest ID No. 1095417771)

ABSTRACT: The current curricular reform in US mathematics education has changed many aspects of classroom teaching. Commonly, discussions about this curricular reform presume an unproblematic relationship between textbooks and classroom instruction. This study contributes to the understanding of the relationship between one published reform curriculum, Connected Mathematics (CMP) (Lappan, Fey, Fitzgerald, Friel & Phillips, 2001) and classroom instruction. The study characterizes teaching and learning in terms of communication patterns---discourse ---and analyzes the discourse of CMP, of a traditional US curriculum, Mathematics, Structure and Method (Dolciam, Sorgenfrey & Graham, 1992), and of two teachers in urban classrooms---focusing on the introductory lessons on linear relationships in each case. Results include full descriptions of the introductory discourse on linearity in the textbooks and changes that the CMP textbook discourse undergoes as the curriculum is implemented in these two classrooms.

De Groot, C. (2000). *Three female voices: The transition to high school mathematics from a reform middle school mathematics program.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 61(4). (ProQuest ID No. 731933601)

ABSTRACT: In this ethnographic study, the transition experiences and coping mechanisms of three female students are reported. These students were members of a cohort in grades 6, 7, and 8 (ages 12-14) that participated in the field testing of the Connected Mathematics Project (1990-1995), a middle school curriculum closely reflecting recommendations of the National Council of Teachers of Mathematics. The participants of the study were in the same mathematics class during their grade 8 experience, but went to different high schools.

Two interviews were conducted toward the end of their grade 9 experience and six interviews were conducted during their grade 10 experience. Middle school mathematics teachers and high school mathematics teachers were interviewed as well as one parent. One observation of each of their tenth grade mathematics classes was conducted. The reported characteristics of transition in this study focus mainly on changes or discontinuities in the learning of mathematics. Data were analyzed by coding processes and presented in narratives and Qualitative Schematics of Dimensions of Transition in Learning Mathematics Thematic interpretations are given with respect to coping mechanisms that were revealed.

One of the major findings of this study is that early in grade 9 these three students related their learning of mathematics in high school closer to their (traditional) elementary experience, which was termed as regular mathematics, than to their reform middle school experience, which was more constructivist in design. In grade 10 they seemed to connect more with their middle school experience, for example, while doing proofs and related this to "explaining your thinking." Another major finding was that these three students experienced a gradual individualization during this transition together with increased in-class competition among students, particularly for attention from the teacher. In high school, they appeared to cope with this lack of student-to-student discourse by forming out of-class support networks.

Suggestions for future research are made regarding the transition discontinuity from learning in a reform environment to learning in a traditional environment, as well as the need to investigate how transitional standards-based curricula, steeped in problem solving, supports students' development of mathematical proof.

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.

Ellis, A. (2007a). A taxonomy for categorizing generalizations: Generalizing actions and reflection generalizations. *Journal of the Learning Sciences, 16(*2), 221-262.

ABSTRACT: This article presents a cohesive, empirically grounded categorization system differentiating the types of generalizations students constructed when reasoning mathematically. The generalization taxonomy developed out of an empirical study conducted during a 3-week teaching experiment and a series of individual interviews. Qualitative analysis of data from teaching sessions with 7 seventh-graders and individual interviews with 7 eighth-graders resulted in a taxonomy that distinguishes between students' activity as they generalize, or generalizing actions, and students' final statements of generalization, or reflection generalizations. The three major generalizing action categories that emerged from analysis are (a) relating, in which one forms an association between two or more problems or objects, (b) searching, in which one repeats an action to locate an element of similarity, and (c) extending, in which one expands a pattern or relation into a more general structure. Reflection generalizations took the form of identifications or statements, definitions, and the influence of prior ideas or strategies. By locating generalization within the learner's viewpoint, the taxonomy moves beyond casting it as an activity at which students either fail or Succeed to allow researchers to identify what students see as general, and how they engage in the act of generalizing.

Ellis, A. 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.

Garrison, A. L. (2013). *Understanding teacher and contextual factors that influence the enactment of cognitively demanding mathematics tasks.* (Unpublished doctoral dissertation). Vanderbilt University, Nashville, TN.

ABSTRACT: The level of challenge, or cognitive demand, of the tasks students solve is the foundation for their learning opportunities in mathematics classrooms. Unfortunately, it is difficult for teachers to effectively use cognitively demanding tasks (CDTs). I seek to understand how to support and improve mathematics teachers’ enactment of CDTs at scale. The purpose of this three-paper dissertation is to address some of the key unresolved questions and to set a direction for future research.

In paper 1, based on a comprehensive literature review, I identify 13 potentially relevant factors and elaborate a method for building on results from small-scale studies to better understand the enactment of CDTs across large samples of teachers.

Paper 2 investigates how teachers’ mathematical knowledge for teaching and their beliefs about teaching and learning mathematics are related to their enactment of CDTs. I found that aspects of teachers’ knowledge and beliefs are interconnected and are significantly related to their enactment of CDTs.

Paper 3 investigates changes in teachers’ enactment of CDTs over time and whether their interactions with colleagues (e.g., work with a math coach, advice-seeking interactions) are related to these changes. I found that the mere occurrence of interactions was generally not sufficient to support teachers’ development, and expertise available within interactions did not influence the productivity of those interactions. However, advice-seeking interactions were significantly related to teachers’ development. Further, the lack of expertise within interactions might have contributed to these findings.

These three studies suggest that there is much more to be understood about supporting teachers’ enactment of CDTs. There is, however, evidence that teachers’ mathematical knowledge for teaching and their beliefs about teaching and learning mathematics are integral to their enactment of CDTs, and that they are interrelated. In addition, it is clear that in designing supports for teachers’ enactment of CDTs, schools and districts should go beyond policies that provide only opportunities for interaction, and should specifically plan productive activities and enhance the available expertise within those interactions.

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.

Hansen-Thomas, H. (2009). Reform-oriented mathematics in three 6th Grade classes: How teachers draw in ELLs to academic discourse. *Journal of Language, Identity, and Education, 8*(2&3), 88-106.

ABSTRACT: Traditionally, mathematics has been considered easy for English language learners (ELLs) due to the belief that math is a "universal language." At the same time, reform-oriented mathematics curricula, designed to promote mathematical discourse, are increasingly being adopted by schools serving large numbers of ELLs. CMP, the Connected Math Project, is one such reform-oriented curriculum. Taking a community-of-practice approach, this article compares how three 6th grade mathematics teachers in a Spanish/English community utilized language to draw ELLs into content and classroom participation. Teacher use of standard language fell into 2 categories: (a) modeling and (b) eliciting student practice. In the teacher's class that regularly elicited language, ELLs were successful on academic assessments; whereas students in the other 2 classes were not. Results suggest that CMP facilitates ELLs' learning and that a focus on mathematical language and elicitation benefits the development of mathematical discourse and content knowledge.

Herbel-Eisenmann, B. A. (2000). *How discourse structures norms: A tale of two middle school mathematics classrooms.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62 (1). (ProQuest ID No. 727910361)

ABSTRACT: My experiences as a student and a teacher of mathematics have led me to pursue the topic of this dissertation--discourse patterns and norms in two "reform-oriented" mathematics classrooms. The two 8th grade classrooms that form the focus of this dissertation were using the Connected Mathematics Project, an NSF-funded curriculum project. I was intrigued by the teachers and their teaching because I noticed the students seemed to have similar understandings, but each classroom felt different to me as a participant-observer.

These classrooms offered a context that allowed me to study differences in the context of similarity. The teachers had many attributes in common (detailed in Chapter 5): similar academic backgrounds and professional development activities, same certification, same school, same curriculum and similar enthusiasm for it, same heterogeneous group of students, similar student-understandings, etc. However, the teaching in the two classrooms was different. Drawing from the sociolinguistics and mathematics education literatures, I describe the social and sociomathematical norms of the two classrooms in terms of the classroom discourse which they were embedded in and carried by. I also interpret student understandings whenever possible throughout the thesis, taking a social constructivist perspective. In the year prior to commencing my dissertation study (1997-1998), I completed classroom observations and student interviews as part of my practicum work and research assistantship, which were used to form preliminary hypothesis about student understandings and the classroom environment. The data used for this dissertation was collected over the next two years (1998-2000). During the first, I observed and audio-and video-taped students on a weekly basis. In addition, students were interviewed about their algebraic understandings and their classroom experience. The second year, one of the classrooms was observed to trace the formation of the norms in the classroom. The teachers took part in four extensive interviews, in which they were asked about influencing experiences related to their teaching and the norms in their classroom (in terms of the expectations, rights and roles of themselves and their students).

The ideas I investigate in this dissertation include how social and sociomathematical norms are embedded in and carried by the classroom discourse in each classroom (Chapters 6 and 7). I also discuss aspects in the teachers' professional lives that influenced the ways they think about and work to establish and maintain the norms in their classrooms (Chapter 5). In Chapter 8, I look across the two classrooms to offer what I see as being similar and different, which has allowed me to locate differences in: the overall structure of teacher talk, the positioning of the teacher with respect to the locus of authority, the way each teacher draws from potential other knowledge sources in the classroom (i.e. students and the textbook), and the way each teacher draws attention to the common knowledge constructed in the classroom.

Herbel-Eisenmann, B. A. (2002). Using student contributions and multiple representations to develop mathematical language. *Mathematics Teaching in the Middle School, 8*(2), 100-105.

ABSTRACT: Describes a way to introduce and use mathematical language as an alternative to using vocabulary lists to introduce students to mathematical language in mathematics classrooms. Draws on multiple representations and student language.

Herbel-Eisenmann, B. A. (2007). From intended curriculum to written curriculum: Examining the "voice" of a mathematics textbook. *Journal for Research in Mathematics Education, 38*(4), 344-369.

ABSTRACT: In this article, I used a discourse analytic framework to examine the "voice" of a middle school mathematics unit. I attended to the text's voice, which helped to illuminate the construction of the roles of the authors and readers and the expected relationships between them. The discursive framework I used focused my attention on particular language forms. The aim of the analysis was to see whether the authors of the unit achieved the ideological goal (i.e., the intended curriculum) put forth by the NCTM's Standards (1991) to shift the locus of authority away from the teacher and the textbook and toward student mathematical reasoning and justification. The findings indicate that achieving this goal is more difficult than the authors of the Standards documents may have realized and that there may be a mismatch between this goal and conventional textbook forms.

Herbel-Eisenmann, B., Wagner, D., & Cortes, V. (2010). Lexical bundle analysis in mathematics classroom discourse: the significance of stance. *Educational Studies in Mathematics*, 75, 23-42.

ABSTRACT: In this article, we introduce the lexical bundle, defined by corpus linguists as a group of three or more words that frequently recur together, in a single group, in a particular register (Biber, Johansson, Leech, Conrad, & Finegan, 2006; Cortes, English for Specific Purposes 23:397–423, 2004). Attention to lexical bundles helps to explore hegemonic practices in mathematics classrooms because lexical bundles play an important role in structuring discourse and are often treated as “common sense” ways of interacting. We narrow our findings and discussion to a particular type of lexical bundle (called a “stance bundle” or bundles that relate to feelings, attitudes, value judgments, or assessments) because it was the most significant type found. Through comparing our corpus from secondary mathematics classrooms with two other corpora (one from university classrooms (not including mathematics classrooms) and one from conversations), we show that most of the stance bundles were particular to secondary mathematics classrooms. The stance bundles are interpreted through the lens of interpersonal positioning, drawing on ideas from systemic functional linguistics. We conclude by suggesting additional research that might be done, discussing limitations of this work, and pointing out that the findings

Hoffmann, A. J. (2004). Middle school mathematics students' motivations for participating in whole-class discussions: Their beliefs, goals, and involvement. (Doctoral dissertation). *Retrieved from Dissertation Abstracts International, 65*(9). (ProQuest ID No. 795927881)

ABSTRACT: Whole-class discussions in mathematics classrooms are considered to foster active sense-making and intellectual autonomy among students. Through participating in these discussions, students have the opportunity to develop skills of mathematical communication, reasoning, and justification. However, middle school students may resist participating in whole-class discussions if they perceive social consequences resulting from this activity.

Research on mathematics classroom discourse typically focuses on the role of the teacher in discourse, examining student variables as outcomes to measure the effectiveness of the teachers' strategies. Alternatively, in this study, students' beliefs and goals are examined for how they influence students' participation in classroom discourse rather than as outcomes.

I assessed beliefs and goals of 15 target students from two seventh grade mathematics classrooms through one-on-one interviews and a Likert-scale survey instrument. Students' talk in interviews was analyzed through the use of a framework that included imperative verbs to capture idealized states, repetition to capture emphasis, and connections to affect to capture relative importance to the student. This framework allowed for a more rigorous analysis of students' beliefs in contrast to reporting any and all of their responses to interview questions.

Students' involvement in classroom discourse was described based on an analyses of videotaped classroom discussions about four investigation problems from the Connected Mathematics Project Standards-based mathematics curriculum.

Results from this study indicate that students' involvement in classroom discussions is influenced by their social goals and epistemological beliefs. Students who believed they learned mathematics through a process of negotiation and associated a low level of risk with participating in discussion were more likely to extend their participation during an interaction, critique the thinking of their classmates, and talk about mathematics at a high level of explicit meaning. There were also differences in students' involvement between the target students based on their classrooms.

This study illustrates how adolescence intersects with the mathematics reform movement by taking into account students' perspectives. Future research investigating how beliefs and goals relate to students' involvement in discussions may explain how a classroom of students together supports the development of effective classroom discussions.

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.

Mendez, E., Sherin, M., & Louis, D. (2007). Multiple perspectives on the development of an eighth-grade mathematical discourse community. *Elementary School Journal, 108*(1), 41-61.

ABSTRACT: In this article we examine the development, over 1 year, of mathematical discourse communities in 2 eighth-grade mathematics classes in a suburban public middle school. The curriculum topics included probability, functions, graphing, data analysis, and pre-algebra. The 50 students were heterogeneously placed; most were from upper-middle-class families. Data included videotaped classroom observations, field notes, and teacher reflections. We explored both the students' growing competencies with mathematical discourse and changes in how the teacher attended to students' ideas. We present the teacher's impressions of the developing discourse community, and we applied 2 research-based lenses, robust mathematical discussion to assess the strength of student discourse, and professional vision for classroom discourse to analyze the ways in which the teacher paid attention to, and reflected on, ideas students raised during discussion. Applying multiple perspectives highlighted the complex nature of developing a discourse community and the challenges facing the teacher as he worked to orchestrate constructive dialogue for learning mathematics and to become aware of what students were learning in this context. We also provide an analytic tool, the robust mathematical discussion framework, that will be useful for teachers, teacher educators, And researchers to evaluate the evolving nature of classroom discourse.

Moschkovich, J. N. (2015). Academic literacy in mathematics for English Learners. The Journal of Mathematical Behavior.

ABSTRACT: This paper uses a sociocultural conceptual framework to provide an integrated view of academic literacy in mathematics for English Learners. The proposed definition of academic literacy in mathematics includes three integrated components: mathematical proficiency, mathematical practices, and mathematical discourse. The paper uses an analysis of a classroom discussion to illustrate how the three components of academic literacy in mathematics are intertwined, how academic literacy in mathematics is situated, and how participants engaged in academic literacy in mathematics use hybrid resources. The paper closes by describing the implications of this integrated view of academic literacy in mathematics for mathematics instruction for English Learners, arguing that it is important that the three components not be separated when designing instruction in general, and it is essential that mathematics instruction for English Learners address these three components simultaneously.

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.

Otten, S., & Soria, V. M. (2014). Relationships between students’ learning and their participation during enactment of middle school algebra tasks. *ZDM*, 46(5), 815–827. doi:10.1007/s11858-014-0572-4

ABSTRACT: This study examines a sequence of four middle school algebra tasks through their enactment in three teachers’ classrooms. The analysis centers on the cognitive demand—the kinds of thinking processes entailed in solving the task—and the participatory demand—the kinds of verbal contributions expected of students—of the task as written in the instructional materials, as set up by the three teachers, and as discussed by the teachers and their students. Relationships between the nature of the task enactments and students’ performance on a pre- and post-test are explored. Findings include the fact that the enacted tasks differed from the written tasks with regard to both the cognitive demand and the participatory demand, which related to students’ lack of success on the post-test. Specifically, cognitive demand declined in the enacted curriculum at different points for different classes, and the participatory demand during enactment tended to involve isolated mathematical terms rather than students verbally expressing mathematical relations.

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.

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.

Stevens, B. B. A. (2005). *The development of pedagogical content knowledge of a mathematics teaching intern: The role of collaboration, curriculum, and classroom context.* (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 67(9). (ProQuest ID No. 1212777591)

ABSTRACT: In this study I examined the role of collaboration, curriculum, and the classroom context in the development of pedagogical content knowledge of a mathematics teaching intern. Additionally, I investigated the nature of the collaborative process between the teaching intern and his mentor teacher as they collaborated on action (during structured planning time) and in action (while students were present). The teaching internship resided in a seventh-grade mathematics classroom during the teaching of a probability unit from a standards-based curriculum, Connected Mathematics Project.

Using existing research, a conceptual framework was developed and multiple data sources (audio taped collaborations, observations of the intern's teaching practices, semi structured interviews, and a mathematics pedagogy assessment) were analyzed in order to understand the teaching intern's development of knowledge of instructional strategies, knowledge of student understandings, curricular knowledge, and conceptions of purpose for teaching probability.

Results identified numerous dilemmas related to planning and implementing instruction. Although the teaching intern developed pedagogical content knowledge, he often experienced difficulty accessing it while teaching. Through collaboration, curriculum, and the classroom context, the teaching intern learned to incorporate his pedagogical content knowledge in instruction. Analysis revealed that as he gained new knowledge he was able to shift his focus from content to the use of instructional strategies for teaching and learning. The curriculum was the primary focus of collaboration and initiated the intern's examination of the learning-to-teach process.

Collaboration on action and collaboration in action proved to be essential elements in the development of pedagogical content knowledge.

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.