This study sought to understand how aspects of middle school mathematics teachers' knowledge and conceptions are related to their enactment of cognitively demanding tasks. The author 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.

# Search Results

### Margaret Mohr-Schroeder, Robert N. Ronau, Susan Peters, Carl W. Lee, and William S. Bush

This article describes the development and validation of two forms of the Geometry Assessments for Secondary Teachers (GAST), which were designed to assess teachers' knowledge for teaching geometry. Both forms were developed by teams of mathematicians, mathematics educators, psychometricians, and secondary classroom geometry teachers. Predictive validity for the GAST assessment was explored by observing and testing 157 teachers as well as administering pre– and post–tests to 3,698 students. The reliability coefficient for both GAST assessment forms was acceptable (r = .79). GAST assessment scores explained a statistically significant but small amount of the variance of student scores, demonstrating an effect that was greater than the number of years of teaching experience but smaller than the effect of having an advanced degree.

### Courtney A. Bell, Suzanne Wilson, Traci Higgins, and D. Betsy McCoach

This study examines the impact of a nationally disseminated professional development program, Developing Mathematical Ideas (DMI), on teachers' specialized knowledge for teaching mathematics and illustrates how such research could be conducted. Participants completing 2 DMI modules were compared with similar colleagues who had not taken DMI. Teacher knowledge was measured with multiple-choice items developed by the Learning Mathematics for Teaching project and open-ended items based on problems initially developed by DMI experts. After controlling for pretest scores, a hierarchical linear model identified statistically significant differences: The DMI group outperformed the comparison group on both assessments. Gains in teachers' scores on the more closely aligned measure were related to the degree of facilitator experience with DMI. This study adds to our understanding of the ways in which professional development program features, facilitators, and issues of scale interact in the development of teachers' mathematical knowledge for teaching. Study limitations and challenges are discussed.

### Martin A. Simon

Prospective teachers' knowledge of division was investigated through an open-response written instrument and through individual interviews. Problems were designed to focus on two aspects of understanding division: connectedness within and between procedural and conceptual knowledge and knowledge of units. Results indicated that the prospective teachers' conceptual knowledge was weak in a number of areas including the conceptual underpinnings of familiar algorithms, the relationship between partitive and quotitive division, the relationship between symbolic division and real-world problems, and identification of the units of quantities encountered in division computations. The research also characterized aspects of individual conceptual differences. The research results suggest conceptual areas of emphasis for the mathematical preparation of elementary teachers.

### Theodore A. Eisenberg

This study investigated the relationship between a teacher's knowledge of algebra and student performance. Twenty-eight Algebra I teachers and their 807 students took part in the study. Regression analysis was used to predict for each teacher an “expected” score for teaching algebraic concepts and skills. The difference between the observed score and the expected score was called “teacher effect.” The effect scores (one for algebraic concepts, the other for skills) were correlated with the teacher's knowledge of the real number system and other related algebraic structures, length of service, college mathematics grade point average, and the number of postcalculus courses taken. The only significant correlation obtained was between the teacher's knowledge of algebraic structures and the number of postcalculus courses.

### Yasemin Copur-Gencturk

In this study, I examined the relationship between teachers' mathematical knowledge and instruction. Twenty-one K—8 teachers who were enrolled in a master's program were followed for 3 years to study how their mathematical knowledge and teaching changed over time. The results of multilevel growth models indicated that gains in teachers' mathematical knowledge predicted changes in the quality of their lesson design, their mathematical agenda, and the classroom climate. Analyses of interviews and classroom observation data conducted with a subgroup of teachers revealed that in addition to the gains teachers made in their mathematical knowledge, their exit level of knowledge played a significant role in the quality of the changes in their practices.

### Richard Lehrer and Megan Loef Franke

Personal construct psychology provides a coherent theoretical and methodological framework for the examination of teachers' knowledge. We report case studies of two teachers who varied in their knowledge about fractions and mathematical pedagogy. We used personal construct psychology and the logic of fuzzy sets to elucidate the content and organization of the teachers' knowledge of fractions. The approach proved especially useful for describing conditional relationships among content, general pedagogical, and pedagogical content knowledge frames. We also explored associations between teachers' personal constructions and their classroom teaching. These associations suggested that personal construct psychology shows considerable promise as a way of addressing issues of teacher knowledge in the context of the classroom.

### Beth A. Herbel-Eisenmann and Elizabeth Difanis Phillips

Recent literature has shown that having teachers examine student work can enhance teachers' thinking about what constitutes mathematical understanding (Crespo 2000; Crockett 2001). There is also evidence that teachers need to experience unconventional mathematics problems to see the value of using them in their own classrooms (Ball 1988; Crespo 2003).

### E. Harold Harper

Many articles in professional jounals have pointed up the need for teachers to be well grounded in basic skills and understandings in mathematics to teach properly the arithmetic concepts children need to know. With added emphasis on “modern mathematics” in the elementary school curriculum in the past few year, teachers in service are becoming more and more concerned about their own inadequacies and lack of understanding of basic arithmetic concepts. Many have expressed a desire for in-service programs that would bring them up-to-date. Many communities have indicated a desire to have in-service programs in modern mathematics and methods of teaching a rithmetic. There are not enough people in the schools who have had training in “modern mathematics” to meet the needs for in-service instructors. This, coupled with the fact that many schools are adopting “modern mathematics” textbook series before their teachers have had training in this subject, makes for frustrated and worried elementary school teachers.

### Timothy A. Boerst

In rooms a bit smaller or larger than 24' × 30', millions of schoolchildren across the country learn valuable mathematics from teachers who masterfully integrate knowledge of mathematics, curriculum, and students to enact high-quality instruction. One of the biggest challenges to improving mathematics education may lie not in standard-setting, teacher recruitment, or accountability but in encouraging mathematics teachers to share their knowledge and practices with those outside the 24' × 30' space in which they teach every day. Teachers tend to treat their knowledge of teaching in ways that remove it from benefiting the profession (Shulman 1993). What teachers know and can do certainly impacts the hundreds, and possibly thousands, of students they teach over the course of their careers. Imagine the impact if mathematics teachers routinely shared their complex, refined understandings of teaching in ways that could be built on to benefit other students and teachers. If we hope to ensure NCTM's vision of quality mathematics instruction for all students, we must move beyond the private production and possession of mathematics teaching knowledge.