Showing papers in "Journal for Research in Mathematics Education in 2003"

Teaching and Learning Mathematics for Social Justice in an Urban, Latino School

Abstract: This article reports on a 2-year study about teaching and learning mathematics for social justice in an urban, Latino classroom and about the role of an NCTM Standardsbased curriculum. I was the teacher in the study and moved with the class from seventh to eighth grade. Using qualitative, practitioner-research methodology, I learned that students began to read the world (understand complex issues involving justice and equity) using mathematics, to develop mathematical power, and to change their orientation toward mathematics. A series of real-world projects was fundamental to this change, but the Standards-based curriculum was also important; such curricula can theoretically promote equity, but certain conditions may need to exist. Social justice pedagogy broadens the concept of equity work in mathematics classrooms and may help promote a more just society.

Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem?

Abstract: This article reports on an exploratory study of the way that eight mathematics and secondary education mathematics majors read and reflected on four student-generated arguments purported to be proofs of a single theorem. The results suggest that such undergraduates tend to focus on surface features of arguments and that their ability to determine whether arguments are proofs is very limited-perhaps more so than either they or their instructors recognize. The article begins by discussing arguments (purported proofs) regarded as texts and validations of those arguments, that is, reflections of individuals checking whether such arguments really are proofs of theorems. It relates the mathematics research community's views of proofs and their validations to ideas from reading comprehension and literary theory. Then, a detailed analysis of the four student-generated arguments is given and the eight students' validations of them are analyzed.

Understanding Learning Systems: Mathematics Education and Complexity Science.

Abstract: Complexity science may be described as the science of learning systems, where learning is understood in terms of the adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex learning systems, we explore some of the possible contributions of complexity science to discussions of the teaching of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of mathematical communities and the application of principles of complexity to the teaching of mathematics. Through the course of this writing, we attempt to highlight compatible and complementary discussions that are already represented in the mathematics education literature.

A Meta-Analysis of the Effects of Calculators on Students' Achievement and Attitude Levels in Precollege Mathematics Classes

Abstract: The findings of 54 research studies were integrated through meta-analysis to determine the effects of calculators on student achievement and attitude levels. Effect sizes were generated through Glassian techniques of meta-analysis, and Hedges and Olkin’s (1985) inferential statistical methods were used to test the significance of effect size data. Results revealed that students’ operational skills and problem-solving skills improved when calculators were an integral part of testing and instruction. The results for both skill types were mixed when calculators were not part of assessment, but in all cases, calculator use did not hinder the development of mathematical skills. Students using calculators had better attitudes toward mathematics than their noncalculator counterparts. Further research is needed in the retention of mathematics skills after instruction and transfer of skills to other mathematics-related subjects.

A Modeling Perspective on Students' Mathematical Reasoning about Data.

Abstract: A modeling approach to the teaching and learning of mathematics shifts the focus of the learning activity from finding a solution to a particular problem to creating a system of relationships that is generalizable and reusable. In this article, we discuss the nature of a sequence of tasks that can be used to elicit the development of such systems by middle school students. We report the results of our research with these tasks at two levels. First, we present a detailed analysis of the mathematical reasoning development of one small group of students across the sequence of tasks. Second, we provide a macrolevel analysis of the diversity of thinking patterns identified on two of the problem tasks where we incorporate data from multiple groups of students. Student reasoning about the relationships between and among quantities and their application in related situations is discussed. The results suggest that students were able to create generalizable and reusable systems or models for selecting, ranking, and weighting data. Furthermore, the extent of variations in the approaches that students took suggests that there are multiple paths for the development of ideas about ranking data for decision making.

Assessing the Impact of Standards-Based Middle Grades Mathematics Curriculum Materials on Student Achievement.

Abstract: This study compared the mathematics achievement of eighth graders in the first three school districts in Missouri to adopt NSF-funded Standards-based middle grades mathematics curriculum materials (MATH Thematics or Connected Mathematics Project) with students who had similar prior mathematics achievement and family income levels from other districts. Achievement was measured using the mathematics portion of the Missouri Assessment Program (MAP) administered to all 8th graders in the state annually beginning in the spring of 1997. Significant differences in achievement were identified between students using Standards-based curriculum materials for at least 2 years and students from comparison districts using other curriculum materials. All of the significant differences reflected higher achievement of students using Standards-based materials. Students in each of the three districts using Standards-based materials scored higher in two content areas (data analysis and algebra), and these differences were significant.

A Survey Measuring Elementary Teachers' Implementation of Standards-Based Mathematics Teaching

Abstract: Intensive case study is an expensive tool for measuring teachers' instructional practice. Previous research suggests that teacher self-report surveys provide a low-cost and relatively accurate picture of classroom practice. To examine the extent to which teachers implement mathematics education reform, we developed a 20-item survey based on nine dimensions of standards-based teaching. In this article, we provide evidence of the reliability (i.e., internal consistency) and validity of the instrument. The evidence consists of correlations of survey scores with a mandated performance assessment in Grade 6 mathematics, congruence with classroom observations of a small sample of teachers, and demonstrations that teachers who are similar in their claims about using a standards-based text series differ in how they use the text in ways predicted by the survey.

Teacher Variables That Relate to Student Achievement When Using a Standards-Based Curriculum.

Abstract: 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.

Low-Performing Students and Teaching Fractions for Understanding: An Interactional Analysis

Abstract: This article presents an analysis of two low-performing students’ experiences in a firstgrade classroom oriented toward teaching mathematics for understanding. Combining constructs from interactional sociolinguistics and developmental task analysis, I investigate the nature of these students’ participation in classroom discourse about fractions. Pre- and post-instruction interviews documenting learning and analysis of classroom interactions suggest mechanisms of that learning. I propose that three main factors account for these two students’ success: use of tasks that elicited the students’ prior understanding, creation of a variety of participant frameworks (Goffman, 1981) in which the students were treated as mathematically competent, and frequency of opportunities for identity-enhancing interactions.

Fairness of Dice: A Longitudinal Study of Students' Beliefs and Strategies for Making Judgments

Abstract: One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was "loaded") and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.

The Use of Symbols, Words, and Diagrams as Indicators of Mathematical Cognition: A Causal Model

Abstract: This article reports on the results of a study that investigated the strategic representation skills of eighth-grade students while they were engaged in a set of tasks that involved applying geometric knowledge and using algebraic equations. The strategies studied were derived from Dual Coding Theory (DCT) (Paivio, 1971, 1990), and they were elicited with task-specific prompts embedded in an assessment developed for the study. The purpose of the study was to test a model that highlights strategic representation as a mediator of the effects of reading ability, spatial ability, and task presentation on problem solving. The proposed model was tested using the linear structural equations modeling approach to causal analysis and the data did not reject the model. The results showed that students' use of symbols, words, and diagrams to communicate about their ideas each contribute in different ways to solving tasks and reflect different kinds of cognitive processes invested in problem solving.

State Testing and Mathematics Teaching in New Jersey: The Effects of a Test Without Other Supports

Abstract: Conflicting findings about the effects of state testing on mathematics teaching have a number of roots, including the strong ideological positions of advocates and opponents of state tests and the fact that state policies vary such that one is likely to find different results in different states. The pressure that students, teachers, and administrators may feel toward high test scores and the opportunities that teachers and administrators may have regarding related professional development can also confound findings on the effects of tests on actual classroom teaching. This article describes the teaching practices of fourth-grade teachers in New Jersey, a state with a fourth-grade mathematics test designed to be aligned with state and national standards. The intent of this test is to challenge conventional practice. However, there is a lack of strong pressure to produce high test scores or effective guidance on the kinds of learning opportunities that must complement those tests in order to lead to fundamental change in teaching. Through interviews and observations of 63 teachers, we found that the teachers reported that they changed their practices in ways compatible with state and national standards and the test. For example, they reported asking their students to solve more open-ended problems and to explain their thinking. However, direct observations suggested that teachers have adopted specific strategies without changing their basic instructional approach. The results from our investigation suggest that in the absence of effective professional development, testing leads to minimal changes in teaching practice.