This article explores how issues of power and identity play out in mathematical practices and offers a perspective on how we might better understand the sociopolitical nature of teaching and learning mathematics. We present data from studies of mathematics teaching and learning in out-of-school settings, offering a sociocultural, then a sociopolitical analysis (attending to race, identity, and power), noting the value of the latter. In doing so, we develop a set of theoretical tools that move us from the sociocultural to the sociopolitical in studies of mathematics teaching and learning.
Na'ilah Suad Nasir and Maxine McKinney de Royston
Harold L. Shoen, Kristen J. Cebulla, Kelly F. Finn, and Cos Fi
We report results from a study of instructional practices that relate to student achievement in high school classrooms in which a standards-based curriculum (Core-Plus) was used. We used regression techniques to identify teachers' background characteristics, behaviors, and concerns that are associated with growth in student achievement and further described these associations via graphical representations and logical analysis. The sample consisted of 40 teachers and their 1,466 students in 26 schools. Findings support the importance of professional development specifically aimed at preparing to teach the curriculum. Generally, teaching behaviors that are consistent with the standards' recommendations and that reflect high mathematical expectations were positively related to growth in student achievement.
Daniel F. McGaffrey, Laura S. Hamilton, Brian M. Stecher, Stephen P. Klein, Delia Bugliari, and Abby Robyn
A number of recent efforts to improve mathematics instruction have focused on professional development activities designed to promote instruction that is consistent with professional standards such as those published by the National Council of Teachers of Mathematics. This paper describes the results of a study investigating the degree to which teachers' use of instructional practices aligned with these reforms is related to improved student achievement, after controlling for student background characteristics and prior achievement. In particular we focus on the effects of curriculum on the relationship between instructional practices and student outcomes. We collected data on tenth-grade students during the 1997–98 academic year. Some students were enrolled in integrated math courses designed to be consistent with the reforms, whereas others took the more traditional algebra and geometry sequence. Use of instructional practices was measured through a teacher questionnaire, and student achievement was measured using both the multiple-choice and open-ended components of the Stanford achievement tests. Use of standards-based or reform practices was positively related to achievement on both tests for students in integrated math courses, whereas use of reform practices was unrelated to achievement in the more traditional algebra and geometry courses. These results suggest that changes to instructional practices may need to be coupled with changes in curriculum to realize effects on student achievement.
Indigo Esmonde and Jennifer M. Langer-Osuna
In this article, mathematics classrooms are conceptualized as heterogeneous spaces in which multiple figured worlds come into contact. The study explores how a group of high school students drew upon several figured worlds as they navigated mathematical discussions. Results highlight 3 major points. First, the students drew on 2 primary figured worlds: a mathematics learning figured world and a figured world of friendship and romance. Both of these figured worlds were racialized and gendered, and were actively constructed and contested by the students. Second, these figured worlds offered resources for 1 African American student, Dawn, to position herself powerfully within classroom hierarchies. Third, these acts of positioning allowed Dawn to engage in mathematical practices such as conjecturing, clarifying ideas, and providing evidence.
Thomas W. Judson and Toshiyuki Nishimori
In this study we investigated above-average high school calculus students from Japan and the United States in order to determine any differences in their conceptual understanding of calculus and their ability to use algebra to solve traditional calculus problems. We examined and interviewed 18 Calculus BC students in the United States and 26 Suugaku 3 (calculus) students in Japan. Each student completed two parts of a written examination. The first part (Part I) consisted of problems emphasizing conceptual understanding but requiring little or no algebraic computation. Problems on the second part (Part II) required sound algebraic skills in addition to good conceptual understanding. Following the examination, we interviewed each student in order to assess their mathematical and educational background, their college and career plans, their thinking on the examination problems, their understanding of concepts, and their computational and reasoning skills. We found little difference in the conceptual understanding of calculus between the two groups of students, but the Japanese students demonstrated much stronger algebra skills than their American counterparts.
John P. Smith III and Jon R. Star
Research on the impact of Standards-based, K–12 mathematics programs (i.e., written curricula and associated teaching practices) and of reform calculus programs has focused primarily on student achievement and secondarily, and rather ineffectively, on student attitudes. This research has shown that reform programs have competed well with traditional programs in terms of student achievement. Results for attitude change have been much less conclusive because of conceptual and methodological problems. We critically review this literature to argue for broader conceptions of impact that target new dimensions of program effect and examine interactions between dimensions. We also briefly present the conceptualization, design, and broad results of one study, the Mathematical Transitions Project (MTP), which expanded the range of impact along those lines. The MTP results reveal substantial diversity in students' experience within and between research sites, different patterns of experience between high school and university students, and surprising relationships between achievement and attitude for some students.
Carmen Batanero and Luis Serrano
In the experimental study reported here we intended to examine possible differences in secondary students' conceptions about randomness before and after instruction in probability, which occurs for the Spanish students between the ages of 14 and 17. To achieve this aim, we gave 277 secondary students a written questionnaire with some items taken from Green (1989, 1991). With our results we extend Green's previous research to 17-year-old students and complement his results with the analysis of students' arguments to support randomness in bidimensional distributions. Our results also indicate that students' subjective understanding of randomness is close to some interpretations of randomness throughout history.
Denisse R. Thompson and Sharon L. Senk
We examine the performance of 8 pairs of 2nd-year algebra classes that had been matched on pretest scores. One class in each pair used the UCSMP Advanced Algebra curriculum, and the other used the 2nd-year-algebra text in place at the school. Achievement was measured by a multiple-choice posttest and a free-response posttest. Opportunity-to-learn (OTL) measures were used to ensure that items were fair to both groups of students. UCSMP students generally outperformed comparison students on multistep problems and problems involving applications or graphical representations. Both groups performed comparably on items testing algebraic skills. Hence, concerns that students studying from a Standards-oriented curriculum will achieve less than students studying from a traditional curriculum are not substantiated in this instance.
Gwendolyn M. Lloyd and Melvin Wilson
In this study we investigate the content conceptions of an experienced high school mathematics teacher and link those conceptions to their role in the teacher's first implementation of reform-oriented curricular materials during a 6-week unit on functions. The teacher communicated deep and integrated conceptions of functions, dominated by graphical representations and covariation notions. These themes played crucial roles in the teacher's practice when he emphasized the use of multiple representations to understand dependence patterns in data. The teacher's well-articulated ideas about features of a variety of relationships in different representations supported meaningful discussions with students during the implementation of an unfamiliar classroom approach to functions.
Pessia Tsamir and Dina Tirosh
In this article we demonstrate how research-based knowledge about students' incompatible solutions to various representations of the same problem could be used to raise their awareness of inconsistencies in their reasoning. In the first part of the article we report that students' decisions as to whether 2 given infinite sets have the same number of elements depend on the specific representation of the infinite sets in the problem. We used these findings to construct an infiniteset activity with the aim of encouraging students to reflect on their own thinking about infinity. The findings indicate that taking part in this activity led a number of the participating students to realize that producing 2 contradictory reactions to the same mathematical problem is problematic; yet, few chose to avoid these contradictions by using 1-to-1 correspondence as a criterion for comparing infinite sets.