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.

Thesis (Ed. D.)--West Virginia University, 2001. / Title from document title page. Document formatted into pages; contains xi, 156 p. Vita. Includes abstract. Includes bibliographical references (p. 125-138).

Students’ struggle with learning mathematics is often cast in a negative light. Mathematics educators and researchers, however, suggest that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding. In order to investigate the possible connection between struggle and learning, this study examined students’ productive struggle as students worked on tasks of higher cognitive demand in middle school mathematics classrooms. Students’ productive struggle refers to students’ “effort to make sense of mathematics, to figure something out that is not immediately apparent” (Hiebert & Grouws, 2007, p. 287) as opposed to students’ effort made in despair or frustration. As an exploratory case study using embedded multiple cases, the study examined 186 episodes of student‐teacher interactions in order to identify the kinds and nature of student struggles that occurred in a naturalistic classroom setting as students engaged in mathematical tasks focused on proportional reasoning. The study identified the kinds of teacher responses used in the interaction with the students and the types of resolutions that occurred. The participants were 327 6th and 7th grade students and their six mathematics teachers from three middle schools located in mid‐size Texas cities. Findings from the study identified four basic types of student struggles: get started, carry out a process, give a mathematical explanation, and express misconception and errors. Four kinds of teacher responses to these struggles were identified as situated along a continuum: telling, directed guidance, probing guidance, and affordance. The outcomes of the student‐teacher interactions that resolved the students’ struggles were categorized as: productive, productive at a lower level, or unproductive. These categories were based on how the interactions maintained the cognitive level of the implemented task, addressed the externalized student struggle, and built on student thinking. Findings provide evidence that there are aspects of student‐teacher interactions that appear to be productive for student learning of mathematics. The struggle‐response framework developed in the study can be used to further examine the phenomenon of student struggle from initiation, interaction, to its resolution, and measure learning outcomes of students who experience struggle to make sense of mathematics. / text

Educators need clear, actionable data to help them understand students' current levels of performance and students' probable trajectory toward college- and career-readiness in math if they are to make informed programmatic decisions to shape that trajectory. This study explored the relation between CBM-math in Grade 7 as a one-point, teacher accessible measure of student math skill and the students' performance on the Grade 8 EXPLORE-math test, a large-scale achievement test linked to one set of college- and career-readiness benchmarks. Results indicated that a moderate positive correlation and predictive relation exist between CBM-math and EXPLORE-math. Information was disaggregated by gender and for subgroups, including students eligible for special education, free or reduced meals, and English language development services. No difference in means for male and female students on either measure was identified, but eligibility for special education or for free or reduced lunch was associated with lower performance on both measures. Insufficient numbers of ELD students hindered detailed analysis, but none of the ELD students included in the study achieved the EXPLORE benchmark or the CBM normalized cut score based on the 40th percentile. An ROC analysis showed that easyCBM consistently predicted students who did not meet the EXPLORE benchmark, although results indicated that a higher cut score on easyCBM may be a more consistent predictor. The study adds to validity research on CBM and may be useful for educators seeking to identify students at risk of missing achievement benchmarks and make programmatic decisions to ensure students are on track to be college- and career-ready in math.

One of the main reasons that students struggle as they transition from arithmetic to algebra in the middle grades is that they fail to develop the appropriate understanding of the equal sign. Previous research has suggested that students need to move past an operational understanding and develop a relational understanding of the equal sign in order to work with algebraic equations successfully. Other research has suggested that the way that we interpret and utilize the equal sign is based on three main factors: multiple meanings of the equal sign, equation types, and structural conventions. This study extends both areas of research by analyzing two middle grade curricula and looking for what meanings, equation types, and structural conventions appear in both teacher and student materials. The study confirms that students are exposed to three main meanings of the equal sign in the middle grades. The study also describes which meanings of the equal sign are associated with particular equation types and the frequency with which these equation types appear throughout the 7th and 8th grade curricula. Study findings can be used to inform instruction, as they delineate the factors that are attended to while making sense of the equal sign in the middle grades.

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An operational definition of Systemic Reform is used to document a case study of mathematics education reforms occurring in the mathematics classrooms of one middle school in one school division in one state. The middle school had two lead teachers who participated in the training component of a National Science Foundation-funded state-wide Systemic Reform Initiative. Systemic Reform conceptualized reform as combining bottom-up reform with top-down support. Therefore, the research methodology confronted the challenges of the breadth and complexity of Systemic Reform through the use of selective data in the form of artifacts, interviews, and observations from four populations: (1) classroom teachers, (2) building administrators, (3) district administrators, and (4) state-level staff. The study was conducted at four levels over a five-year period to provide the focus for longitudinal data collection to document: (1) the status of mathematics education during the 1995-96 primary data collection year, (2) the evolution of mathematics education reform over the course of the five year period, and (3) the manner in which Systemic Reform occurred. All levels of educators involved made an initial five-year commitment as active participants in the State's Systemic Reform Initiative, but only the Lead Teacher actually carried through with this commitment. After the first year division-level administrators shifted the focus of reform efforts to the elementary schools and discontinued support for the middle schools; after the second year both the division and state-level administrators withdrew all support. Although changes were made at the school level which supported reform in mathematics education (i.e., adoption of constructivist-type instructional materials, purchase of classroom sets of manipulatives and calculators, implementation of block scheduling, and the organization of teachers in interdisciplinary teams) the necessary changes in technology, curriculum and assessment were not in place to support the reform efforts. Through the perseverance of the Lead Teacher some changes in mathematics classrooms were documented, but the lack of consistent administrative leadership/support and emphasis on multiple reforms ended in the all to common bandwagon phenomena at the building, division and state levels so characteristic of change efforts in schools. / Ph. D.

This study retrofitted a Diagnostic Classification Model (DCM) known as the Fusion model onto non-diagnostic test data from of the University of Chicago School Mathematics Project (UCSMP) Algebra and Geometry Readiness test post-test used with Transition Mathematics (Third Edition, Field-Trial Version). The test contained 24 multiple-choice middle school math items, and was originally given to 95 advanced 6th grade and 293 7th grade students. The use of these test answers for this study was an attempt to show that by using cognitive diagnostic analysis techniques on test items not constructed for that purpose, highly predictable multidimensional cognitive attribute profiles for each test taker could be obtained. These profiles delineated whether a given test taker was a master or non-master for each attribute measured by the test, thus allowing detailed diagnostic feedback to be disseminated to both the test takers and their teachers. The full version of the non-compensatory Fusion model, specifically, along with the Arpeggio software package, was used to estimate test taker profiles on each of the four cognitive attributes found to be intrinsic to the items on this test, because it handled both slips and guesses by test takers and accounted for residual skills not defined by the four attributes and twenty-four items in the Q-matrix. The attributes, one or more of which was needed to correctly answer an item, were defined as: Skills— those procedures that students should master with fluency; e.g., multiplying positive and negative numbers; Properties—which deal with the principles underlying the mathematics concepts being studied, such as being able to recognize and use the Repeated-Addition Property of Multiplication; Uses—which deal with applications of mathematics in real situations ranging from routine "word problems" to the development and use of mathematical models, like finding unknowns in real situations involving multiplication; and, Representations—which deal with pictures, graphs, or objects that illustrate concepts. Ultimately, a Q-matrix was developed from the rating of four content experts, with the attributes needed to answer each item clearly delineated. A validation of this Q-matrix was obtained from the Fusion model Arpeggio application to the data as test taker profiles showed which attributes were mastered by each test taker and which weren’t. Masters of the attributes needed to be acquired to successfully answer a test item had a proportion-correct difference from non-masters of .44, on average. Regression analysis produced an R-squared of .89 for the prediction of total scores on the test items by the attribute mastery probabilities obtained from the Fusion model with the final Q-matrix. Limitations of the study are discussed, along with reasons for the significance of the study.

The Supreme Court of the United States has recently determined that assigning students to schools and classrooms based on racial identity is unconstitutional. However, it also left the door open for further and different rulings. If researchers are able to show that lack of consideration of race has deleterious effects on federally mandated programs and initiatives, the ruling may be modified or opened up to specific circumstances. Among its many consequences, this ruling brings a focus onto the question of student-teacher matching in classrooms. Over the years, there has been a great deal of discussion in the literature about matching teacher and student by race, ethnicity, gender, and language. Some people claim that matching is crucial for student success while others dispute this claim. The current study examines the question of racial and ethnic matching empirically in the context of a large-scale randomized controlled study of an innovation for middle school mathematics learners. It extends the literature by (1) focusing on the relationship between student-teacher match and a specific, heavily documented situation with targeted learning goals, (2) adding information about Hispanic students to the discussion, and (3) helping evaluate factors that may be important in determining the validity of large-scale experiments. Alone and in conjunction with other similar empirical evidence, it will also have a significant effect on federal and state educational policy. The sample consists of the 92 teachers and 1576 7th grade students on 76 school campuses throughout 8 Texas regions who participated in the Scaling-Up SimCalc project. Teachers and students either used SimCalc Mathworlds™ curriculum and technology or a control for a two-week replacement unit. The crux of the current analysis was a match between aggregated and individual teacher and student characteristics and an inquiry into how these matches influence student math performance in the classroom within and between our experimental and control group. Hierarchical Linear Modeling (HLM) analysis was used to investigate the differences in student mathematics performance, modeled as students nested in classrooms nested in schools. / Ph. D.

The purpose of this study was to reveal and define the problems middle school mathematics teachers experienced in applying transformational geometry and the solutions they proposed to overcome these problems. A total of six elementary mathematics teachers (grades 5-8) in Ankara participated in the study. The data were collected by means of one-to-one interviews with the participants. The findings indicated that the participants&rsquo / problems divided into three parts. These problems were problems arising from teachers, problems arising from students and problems arising from resources. The participants expressed challenges in teaching due to lack of materials, textbooks, and visualization ability of teachers, classroom size, and time. According to the findings, rotation was the most problematic issue. The participants claimed insufficient technological materials were the reason of this problem. Participants did not feel confidence enough to implement transformational geometry especially in rotation since they lacked adequate training and support. The participants claimed that the Ministry&rsquo / s support should be increased, concrete and technological materials should be sufficient in number, and the duration of transformational geometry lesson should be increased.