Toggle Accessibility Tools

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.

Asquith, P., Stephens, A.C., Knuth, E.J., Alibali, M.W. (2005). Middle school mathematics teachers' knowledge of students' understanding of core algebraic concepts: Equal sign and variable. Mathematical Thinking and Learning, 9(3), 249-272.

ABSTRACT: This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.

Read Article

Ball, D. L. (1996). Teacher learning and the mathematics reforms: What we think we know and what we need to learn. Phi Delta Kappan, 77(7), 500-508.

ABSTRACT: In order to improve mathematics education, a close examination of assumptions about teacher learning and the teaching of mathematics must be made. Teachers and others participating in the reform process will have to learn many new ideas and unlearn many previous assumptions.

Read Teacher Learning and the Mathematics Reform

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

Charalambos, C. Y., & Hill, H. C. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Unpacking a complex relationship. Journal of Curriculum Studies, 44(4), 443- 466.

ABSTRACT: The set of papers presented in this issue comprise a multiple-case study which attends to instructional resources—teacher knowledge and curriculum materials—to understand how they individually and jointly contribute to instructional quality. We approach this inquiry by comparing lessons taught by teachers with differing mathematical knowledge for teaching who were using either the same or different editions of a US Standards-based curriculum. This introductory paper situates the work reported in the next four case-study papers by outlining the analytic framework guiding the exploration and detailing the methods for addressing the research questions.

Read Article

Charalambos, C. Y., Hill, H. C., & Mitchell, R. N. (2012). Two negatives don't always make a positive: Exploring how limitations in teacher knowledge and the curriculum contribute to instructional quality. Journal of Curriculum Studies, 44(4), 489-513.

ABSTRACT: This paper examines the contribution of mathematical knowledge for teaching (MKT) and curriculum materials to the implementation of lessons on integer subtraction. In particular, it investigates the instruction of three teachers with differing MKT levels using two editions of the same set of curriculum materials that provided different levels of support. This variation in MKT level and curriculum support facilitated exploring the distinct and joint contribution of MKT and the curriculum materials to instructional quality. The analyses suggest that MKT relates positively to teachers' use of representations, provision of explanations, precision in language and notation, and ability to capitalize on student contributions and move the mathematics along in a goal-directed manner. Curriculum materials set the stage for attending to the meaning of integer subtraction and appeared to support teachers' use of representations, provision of explanations, and precision in language and notation. More critically, the findings suggest that less educative curriculum materials, coupled with low levels of MKT, can lead to problematic instruction. In contrast, educative materials can help low-MKT teachers provide adequate instruction, while higher MKT levels seem to enable teachers to compensate for curriculum limitations.

Read Article

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.

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

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

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

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

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

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

Read Article

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.

Gencturk, Y. C. (2012). Teachers’ mathematical knowledge for teaching, instructional practices, and student outcomes. (Unpublished doctoral dissertation). University of Illinois at Urbana-Champaign, Champaign, IL.

ABSTRACT: This dissertation examines the relationships among teachers’ mathematical knowledge, their teaching practices, and student achievement. Quantitative and qualitative data collection techniques (content knowledge assessments, surveys, interviews, and classroom observations) were used to collect data from 21 teachers and 873 students. Twenty-one in-service teachers who enrolled in a master’s program designed specifically for the needs of a partnership district were followed for 4 years to study how their mathematical knowledge as well as their teaching changed over time. Of the 21 teachers, 8 teachers were chosen for additional classroom observations and interviews. For the quantitative part of the study, two-level linear growth models were used to examine the effects of the mathematical knowledge of K-8 teachers on their instructional practices. After student-level data were added, three-level growth models were used to analyze the effects of teachers’ knowledge and instructional practices on students’ gain scores. Teachers’ beliefs about teaching and learning mathematics were also included in some analyses. The results indicated that, compared with the initial baseline data, teachers’ mathematical knowledge increased dramatically, and the teachers made statistically significant changes in their lesson design, mathematical agenda of the lessons, task choices, and classroom climate. The gains in teachers’ mathematical knowledge predicted changes in the quality of their lesson design, mathematical agenda, and classroom climate. Teachers’ beliefs were related to the quality of their lesson design, mathematical agenda, and the quality of the tasks chosen. However, only student engagement was significantly related to students’ gain scores. Neither teachers’ mathematical knowledge nor other aspects of instruction (inquiry-oriented teaching, the quality of task choices, and the classroom climate) were associated with students’ gain scores. The qualitative analyses revealed particular strands of the complex relationship between teachers’ mathematical knowledge and their instructional practices. Teachers’ beliefs played a mediating role in the relationship between teachers’ mathematical knowledge and instructional practices. Teachers favoring standards-based views of mathematics tended to teach in more inquiry-oriented ways and ask more questions of students; however, among teachers with limited mathematical knowledge, these practices seemed superficial. Additionally, the teachers’ task choices appeared to be confounded by teachers’ current level of mathematical knowledge and their textbook use.

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.

Hallagan, J. E. (2003). Teachers' models of student responses to middle school algebraic tasks. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 64(2). (ProQuest ID No. 765247341)

ABSTRACT: Often, the difficulties of students to make the cognitive leap from arithmetic to algebra is related to instructional strategies. The way teachers make sense of their practice, in turn, informs pre-service and in-service algebraic instruction. Algebraic instruction is also of current interest due to recent national initiatives calling for all students to learn high school algebra.

The purpose of this study was to describe middle school mathematics teachers' models or interpretations of students' responses to middle school algebraic tasks. The research questions focused on the nature of the teachers' developing ideas and interpretations of student responses from selected algebraic tasks involving the distributive property and equivalent expressions. The core research questions were: (a) What information do middle school mathematics teachers acquire about their students' algebraic thinking? and (b) How do middle school mathematics teachers interpret their students' algebraic thinking? A models and modeling framework guided the study's design. Model-eliciting activities were used to perturb and at the same time reveal their thinking. These activities consisted of asking the teachers to create a "Ways of Thinking" sheet based upon students' responses to the selected algebraic tasks, and to select, analyze and interpret samples of student work. Five teachers participated from two middle schools. Data collection included classroom observation, artifact collection from the model-eliciting activities, semi-structured interviews, and team discussions.

Two sets of findings emerged from this study. First, I concluded that the models and modeling perspective is indeed an effective methodology to elicit teachers' models of their students' algebraic thinking. Second, I found the following five aspects are central to teachers' models of student responses to tasks with equivalent expressions and the distributive property. Teachers recognized that students: (a) tended to conjoin expressions, (b) desired a numerical answer, and (c) had difficulty writing algebraic generalizations. In addition, teachers identified that (d) visual representations were highly useful as instructional tools. And finally, (e) the teachers in this study needed more experience in analyzing and interpreting student work. The findings from this study revealed consistent information across the Ways of Thinking sheets, library of student work, individual and team interviews, and classroom observations.

Read Dissertation

Hill, H. C. (2007). Mathematical knowledge of middle school teachers: Implications for the No Child Left Behind policy initiative. Educational Evaluation and Policy Analysis, 29(2), 95-114.

ABSTRACT: This article explores middle school teachers' mathematical knowledge for teaching and the relationship between such knowledge and teachers' subject matter preparation, certification type, teaching experience, and their students' poverty status. The author administered multiple-choice measures to a nationally representative sample of teachers and found that those with more mathematical course work, a subject-specific certification, and high school teaching experience tended to possess higher levels of teaching-specific mathematical knowledge. However teachers with strong mathematical knowledge for teaching are, like those with full credentials and preparation, distributed unequally across the population of U.S. students. Specifically, more affluent students are more likely to encounter more knowledgeable teachers. The author discusses the implications of this for current U.S. policies aimed at improving teacher quality.

Hill, H. C., & Charalambos, C. Y. (2012). Teacher knowledge, curriculum materials, and quality of instruction: Lessons learned and open issues. Journal of Curriculum Studies, 44(4), 559-576.

ABSTRACT: This paper draws on four case studies to perform a cross-case analysis investigating the unique and joint contribution of mathematical knowledge for teaching (MKT) and curriculum materials to instructional quality. As expected, it was found that both MKT and curriculum materials matter for instruction. The contribution of MKT was more prevalent in the richness of the mathematical language employed during instruction, the explanations offered, the avoidance of errors, and teachers' capacity to highlight key mathematical ideas and use them to weave the lesson activities. By virtue of being ambitious, the curriculum materials set the stage for engaging students in mathematical thinking and reasoning; at the same time, they amplified the demands for enactment, especially for the low-MKT teachers. The analysis also helped develop three tentative hypotheses regarding the joint contribution of MKT and the curriculum materials: when supportive and when followed closely, curriculum materials can lead to high-quality instruction, even for low-MKT teachers; in contrast, when unsupportive, they can lead to problematic instruction, particularly for low-MKT teachers; high-MKT teachers, on the other hand, might be able to compensate for some of the limitations of the curriculum materials and offer high-quality instruction. This paper discusses the policy implications of these findings and points to open issues warranting further investigation.

Hill, H. C., & Charalambos, C. Y. (2012). Teaching (un)Connected Mathematics: Two teachers’ enactment of the Pizza Problem. Journal of Curriculum Studies, 44(4), 467-487.

ABSTRACT: This paper documents the ways mathematical knowledge for teaching (MKT) and curriculum materials appear to contribute to the enactment of a 7th grade Connected Mathematics Project lesson on comparing ratios. Two teachers with widely differing MKT scores are compared teaching this lesson. The comparison of the teachers' lesson enactments suggests that MKT appears to contribute to the mathematical richness of the lesson, teacher ability to capitalize on student ideas, and capacity to emphasize and link key mathematical ideas; yet the relationship of MKT to whether and how students participated in mathematical reasoning was more equivocal. Curriculum materials seemed to contribute to instructional quality, in that the novel tasks contained in the curriculum laid the groundwork for in-depth student problem-solving experiences; they also prevented the low-MKT teacher from making a mathematical error. At the same time, these ambitious materials influenced enactment because of the difficulties they caused teachers: the lesson's tasks needed to be ‘repaired' to enable students to engage with the main mathematical ideas, and off-track student responses to these tasks required remediation. Only the higher-MKT teacher was successfully able to meet the challenge, a finding suggestive of the confluence of MKT and the curriculum materials in informing instructional quality.

Read Paper

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.

Read Study

Lewis, J. M. & Blunk, M. L. (2012). Reading between the lines: Teaching linear algebra. Journal of Curriculum Studies, 44(4), 515-536.

ABSTRACT: This paper compares lessons on linear equations from the same curriculum materials taught by two teachers of different levels of mathematical knowledge for teaching (MKT). The analysis indicates that the mathematical quality of instruction in these two classrooms appears to be a function of differences in MKT. Although the two teachers were teaching from the same curriculum materials, the teacher with higher MKT had more complete and concise ways to describe key concepts, had multiple ways to represent ideas about linear equations, could move nimbly among different mathematical expressions of linear relationships, and gave students a larger role in articulating the mathematical ideas of the lesson. However, curriculum materials seem to have moderated what would otherwise have been larger disparities in the quality of instruction between the two teachers. The lower-MKT teacher made minor mathematical errors, stayed on topic, and defined concepts in reasonably accurate ways when he followed the curriculum materials closely.

Sleep, L., & Eskelson, S. L. (2012). MKT and curriculum materials are only part of the story: Insights from a lesson on fractions. Journal of Curriculum Studies, 44(4), 537-558.

ABSTRACT: This paper investigates the contribution of mathematical knowledge for teaching (MKT) and curriculum materials to the mathematical quality of instruction by comparing the enactment of a fractions problem taught by two teachers with differing MKT. It was found that MKT seem to support teachers’ precise use of mathematical language and to prevent errors; the curriculum materials provided a rich representational context for mathematical work. However, teachers’ orientations toward mathematics and mathematics teaching and their goals for student learning also seemed to contribute to their use of curriculum materials to engage students with rich mathematics and to support students’ participation in the development of the mathematics. Although orientations and goals made it more likely for a teacher to use multiple representations and elicit multiple solution methods, MKT was needed to productively use these elements in instruction. Based on this analysis, it is argued that there are aspects of developing orientations and goals that are related to MKT.

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.

Wanko, J. J. (2000). Going public: The development of a teacher educator's pedagogical content knowledge. (Doctoral dissertation). Retrieved from Dissertation Abstracts International, 62(1). (ProQuest ID No. 727910571)

ABSTRACT: When Lee Shulman and his colleagues introduced pedagogical content knowledge (PCK) to the education lexicon in the 1980s, they gave teachers and teacher educators some technical language that could be used for talking about the knowledge needed for work that they do in classrooms, thereby helping to establish teaching as a profession. Since that time, the PCK of classroom teachers has been studied and documented across various content areas. But the PCK of teacher educators has remained a largely unexamined area of research, especially in the providing experiences in helping preservice teachers develop their own PCK. This study examines this issue more fully. Specifically, "Can pedagogical content knowledge be a useful framework for a teacher educator in designing and teaching a mathematics content course for preservice teachers and if so in what ways?"

In this study, I use my own teaching and classroom of prospective elementary teachers as the site for investigation. I examine the ways in which my own PCK as a teacher educator influenced and was influenced by my work with students. Data for the study are provided by my teaching journal, lesson and units plans, student work, and audiotapes of class proceedings.

In conclusion, I present three major findings of this study. First, this study highlights and problematizes Shulman's notion of representation that is used in defining pedagogical content knowledge. In mathematics there are mathematical and empirical representations--classifications which do not map easily onto Shulman's use of representation. This study exposes some of those inherent distinctions and seeks to make Shulman's work more applicable to the field of mathematics. Second, this study describes the importance of task design--a process that is particularly essential in teaching mathematics--and finds that Shulman's notions of PCK and the pedagogical reasoning and action cycle miss or obscure its significance. And third, this study introduces the notion of shared reflection to Shulman's model for pedagogical reasoning and action when it is applied to teacher education. It also finds that the act of going public with one's ideas through shared reflection can be a useful tool for teacher educators in the development of their pedagogical content knowledge.

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.