Mathematics Education Literature Research Articles

Equity in mathematics education: unions and intersections of feminist and social justice literature

Traditional models of gender equity incorporating deficit frameworks and creating norms based on male experiences have been challenged by models emphasizing the social construction of gender and positing that women may come to know things in different ways from men. This paper draws on the latter form of feminist theory while treating gender equity in mathematics as intimately interconnected with equity issues by social class and ethnicity. I integrate feminist and social justice literature in mathematics education and argue that to secure a transformative, sustainable impact on equity, we must treat mathematics as an integral component of a larger system producing educated citizens. I argue the need for a mathematics education with tri-fold support for mathematical literacy, critical literacy, and community literacy. Respectively, emphases are on mathematics, social critique, and community relations and actions. Currently, the integration of these three literacies is extremely limited in mathematics.

Mathematical Cognition and the Final Report of the National Mathematics Advisory Panel: A Critical, Cultural-historical Activity Theoretic Analysis

In its Final Report, the National Mathematics Advisory Panel has depicted a stark image of mathematical competencies and achievement among U.S. students. The Advisory Panel notes the lack of research with “truly scientific” rigor and, mentioning Vygotsky’s cultural-historical activity theory in passing, suggests its utility to be untested. In reading the report, I noted the limited understanding of the mathematics education literature it articulates and a complete failure to draw on established, “tried and proven,” theory and practice in mathematics education founded upon an encompassing culturalhistorical activity theory. This theory is comprehensive and encompassing, because it retains activity in its entirety as the unit of analysis, which leads to an integrated and integral consideration of those “factors” that are taken to be external to mathematical cognition in the Final Report. In this article, I articulate the current state-of-the-art understanding of cultural-historical activity theory and then use it to provide a critical perspective on the report, its recommendations, and its conclusions as these pertain to the learning of mathematics.

Supporting students’ participation in authentic proof activities in computer supported collaborative learning (CSCL) environments

In this paper, I review both mathematics education and CSCL literature and discuss how we can better take advantage of CSCL tools for developing mathematical proof skills. I introduce a model of proof in school mathematics that incorporates both empirical and deductive ways of knowing. I argue that two major forces have given rise to this conception of proving: a particular learning perspective promoted in reform documents and a genre of computer tools, namely dynamic geometry software, which affords this perspective of learning within the context of mathematical proof. Tracing the move from absolutism to fallibilism in the philosophy of mathematics, I highlight the vital role of community in the production of mathematical knowledge. This leads me to an examination of a certain CSCL tool whose design is guided by knowledge-building pedagogy. I argue that knowledge building is a suitable pedagogical approach for the proof model presented in this paper. Furthermore, I suggest software modifications that will better support learners’ participation in authentic proof tasks.

Learning opportunities from group discussions: warrants become the objects of debate

In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In analyzing a videotaped episode of eight middle school students discussing a statistical problem, we observed that these students frequently challenged the arguments that their colleagues presented. These challenges invited students to be explicit about what mathematical principles, or warrants, they were implicitly using as a basis for their mathematical claims, in some cases recognize the modes of reasoning they were using were invalid and reject these modes of reasoning, and in other cases, attempt to provide deductive support to justify why their modes of reasoning were appropriate. We then describe what social and environmental conditions allowed the discussion analyzed in this paper to occur.

How Does Language Impact the Learning of Mathematics? Let Me Count the Ways

The role that language plays in the teaching and learning of mathematics is at the forefront of current literature in mathematics education. In this paper, I give particular attention to the manner in which teachers and students engage in the exploration of mathematical concepts and procedures with the goal of revealing how language impacts students’ learning. Through a series of examples of language commonly used in the mathematics classroom, I address specific issues pertaining to language used to describe mathematical processes, to read and interpret notation, and to define mathematical terms. Considering that communication is a key factor in the building of understanding, it is hoped that these examples will motivate teachers to examine and to adapt their own practices in order to cultivate productive and meaningful mathematical discourse in their classrooms.

Using Creative Writing and Literature in Mathematics Classes

Using literature in mathematics education is not a novel idea. Many books and articles have been written on how and why to introduce children's literature in elementary mathematics education over the past ten years, but little has been done at the middle school and high school levels.

Making Sense of Number Sense

Drawing on various approaches to the study of mathematics learning, Gersten, Jordan, and Flojo (in this issue) explore the implications of this research for identifying children at risk for developing mathematical disabilities. One of the key topics Gersten et al. consider in their review is that of "number sense." I expand on their preliminary effort by examining in detail the diverse set of components purported to be encompassed by this construct. My analysis reveals some major differences between the ways in which number sense is defined in the mathematical cognition literature and its definition in the literature in mathematics education. I also present recent empirical evidence and theoretical perspectives bearing on the importance of measuring the speed of making magnitude comparisons. Finally, I discuss how differing conceptions of number sense inform the issue of whether and to what extent it may be teachable.

On the teaching and learing of Dienes' principles

Zoltan Dienes' principles of mathematical learning have been an integral part of mathematics education literature and applied both to the teaching and learning of mathematics as well as research on processes such as abstraction and generalization of mathematical structures. Most extant textbooks of cognitive learning theories in mathematics education include a treatment of Dienes' seminal contributions. Yet, there are no available studies at the tertiary level on how students internalise the meaning of Dienes' principles. This paper explores post-graduate mathematics education student's understanding of Dienes' principles and their ability to reflexively apply the principles to their own thinking on structurally similar problems. Some implications are offered for university educators engaged in the training of future researchers in the field.

Reference framework for describing and assessing students’ understanding in first year calculus

This paper presents aspects of a study that investigates the development of an instrument, a reference framework, to analyse students’ written responses to non-routine problems in a first year calculus course in order to describe the complexities of their understanding and to assess their understanding of particular mathematical concepts. Consistent with methodological procedures of grounded theory, the constructed reference framework, which contains six categories, is an amalgam of established instruments founded in the mathematics education literature and based on emergent categories of understanding founded in the empirical data collected for the study. Results of the study, which also implements a fuzzy measure of the level of students’ understanding, are based on the inscriptions in the form of student mathematical texts that participants produced.

Surprises from Mathematics Education Research: Student (Mis)use of Mathematical Definitions

1. INTRODUCTION. The authors of this paper met at a summer institute sponsored by the Oregon Collaborative for Excellence in the Preparation of Teachers (OCEPT). Edwards is a researcher in undergraduate mathematics education. Ward, a pure mathematician teaching at an undergraduate institution, had had little exposure to mathematics education research prior to the OCEPT program. At the institute, Edwards described to Ward the results of her Ph.D. dissertation [5] on student understanding and use of definitions in undergraduate real analysis. In that study, tasks involving the definitions of “limit” and “continuity,” for example, were problematic for some of the students. Ward’s intuitive reaction was that those words were “loaded” with connotations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, “I’ll bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like ‘group’ and ‘coset,’ are not so loaded.” Eventually, with OCEPT support, the authors studied student understanding and use of definitions in an introductory abstract algebra course populated by undergraduate mathematics majors and taught by Ward. The “surprises” in the title are outcomes that surprised Ward, among others. He was surprised to see his algebra students having difficulties very similar to those of Edwards’s analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students’ understanding of the very nature of mathematical definitions, not just from the content of the definitions. Upon hearing of Edwards’s dissertation work, some other mathematicians who teach undergraduates found those difficulties surprising even when restricted to real analysis. Hereafter, we present a simple two-part theoretical framework borrowed from philosophy and from mathematics education literature. Although it is not our intent to give an extensive report of either study, we next indicate the methodology used in Edwards’s dissertation and in our joint abstract algebra study so that the reader may know the context from which our observations are drawn. We then list the “surprising” difficulties of the two groups of students, documenting them with examples from the studies and using the framework to provide a possible explanation for them. We conclude with what we see as the implications for undergraduate teaching, along with some specific classroom activities that the studies and our experience as teachers suggest might be of value. 2. FRAMEWORK. It is commonly noted in mathematics departments that undergraduate mathematics majors often experience difficulties when trying to write mathematical proofs in their introductory abstract algebra, real analysis, or number theory courses. Some researchers have investigated certain aspects of students’ understanding or success in proof-writing [8], [16], [11]. In particular, Moore [11] notes that, while attempting to write formal proofs, students do not necessarily understand the content of relevant definitions or how to use these definitions in proof-writing. Edwards’s study

Quantitative Estimation: One, Two, or Three Abilities?

The mathematics education literature refers to 3 types of quantitative estimation skill: numerosity, measurement, and computational estimation. The psychometric literature includes a confusing array of tests intended to define quantitative estimation. This study examined relations among tests for numerosity, measurement, and computational estimation, and recognized tests for numerical facility and quantitative reasoning using principal components analysis. 2 components were identified. The first component aligned computational estimation with numerical facility and general quantitative reasoning. The second component included the tests of numerosity and measurement estimation. It was suggested that this second component might be related to spatial ability. Implications for mathematics education and assessment are discussed.

Criteria for authentic assessment of mathematics: Understanding success, failure and inequality

The notion of the desirability of authentic assessment has become widespread across many areas of the curriculum and at various levels of education, including higher education (Torrance, 1995). Within mathematics education, there has been considerable international interest in discussing the characteristics of more authentic means of assessment and attempting to design and implement them (see, for example, Niss, 1993a, 1993b; Romberg, 1995). The term authentic, when used in the context of curriculum and assessment, has rich and multiple connotations that are not always distinguished from one another or clearly analysed. Cumming & Maxwell (1999) provide a useful categorisation, identifying and distinguishing four major interpretations of authenticity, each based on different underlying theories of learning: performance and performance assessment; situated learning and situated assessment; complexity of expertise and problem-based assessment; competence and competence-based assessment. Within the mathematics education literature, however, the underlying theory of learning is usually left implicit and the term authentic is used as if it were unambiguous. In this paper I intend to explore the meaning of authenticity in relation to recent changes in mathematics curriculum and assessment. In particular, I shall consider the forms that assessment criteria may take and the implications for both individuals and groups of students. The examples I employ relate specifically to the use of ‘open-ended tasks’, introduced in many curriculum reform movements internationally (Pehkonen, 1997). The principles for interrogating the nature of assessment criteria and their implications are, I believe, more generally applicable. At the level of the assessment task itself, authenticity tends to refer to the nature of the mathematical activity involved — but this may be judged according to some idealised version of the nature of mathematics (for example, ‘Mathematics is a creative activity, so authentic tasks should allow space for students to be creative rather than to repeat memorised procedures’), relative to a view of what students will find meaningful, often involving some ‘realistic’ aspect or, according to Cumming & Maxwell (1999), reflecting adult activity. Authenticity is also often considered at the level of the entire assessment system. In this case, it is usually taken to mean that assessment procedures match the aims, content and breadth of the curriculum; the assessment system is thus an authentic representation of the curriculum it is assessing. This aspect of authenticity is often asso

Understanding Learning Systems: Mathematics Education and Complexity Science

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.

The mathematics of medical imaging in the classroom

The article presents an integrated exposition of aspects of secondary school mathematics and a medical science specialty together with related classroom activities. Clinical medical practice and theoretical and empirical literature in mathematics education and radiology were reviewed to develop and pilot model integrative classroom topics and activities. The techniques of computer-axial tomography (CT) and positron emission tomography (PET) are discussed, followed by a presentation of accessible mathematical applications in numeration and linear algebra for use in a high school classroom. This discussion of the mathematics of a medical speciality, and the related activities, might not only offer teachers and students specific examples of the connections between their everyday study and a professional discipline, but also might foster further investigation into the importance and relevance of mathematics in other technology-based careers.

A whole language approach in mathematics classrooms

This article discusses the possibility of using a ‘whole language’ approach to the teaching of mathematics. As a foundation for the discussion, an epistemological comparison between language and mathematics is made. Considering both similarities and differences between mathematics and language, an analogy is made transferring the main characteristics of ‘whole language’ to what would be a ‘whole mathematics’ pedagogy. Finally, examples of practical experiences reported in the mathematics education literature are presented to aid in the consideration of the possibility and desirability of such a ‘whole maths’ approach to mathematics teaching.

Multiplication of Fractions Through Paper Folding

“One of the most well established facts in all mathematics education literature is that performance on fraction is undesirably low” (Hope and Owens 1987, 37). One example of that performance is that less than half of the seventh graders tested in the 1986 National Assessment of Educational Progress recognized that 5 1/4 equals 5 + 1/4 (Kouba et al. 1988)! We keep trying to comprehend why so many students and intelligent adults don't understand fractions.

Language Activities That Promote Awareness of Mathematics

Developing students' abilities to rcason with mathematics and apply mathematics to the solution of problems occurring in the real world hould be a primary focus of school mathematics (National Council of Teachers of Mathematics 1980). However, most mathemati cal curricula seem to place more emphasis on memorization of fact and algorithm than on reasoning and problem solving (Romberg and Carpenter 1986). The mathematics education literature abound with ideas for reversing the emphasis on memorization and for guiding the teaching of mathematics so that it has real-life meaning for children. Included among the idea are the following:

Hierarchies in mathematics education

In mathematics education literature the term ‘hierarchy’ is used in a number of ways. It is important that the mathematics educator consider the usefulness of the hierarchies presented by various researchers and theorists, in the light of their application to teaching. Current works on mathematical learning hierarchies are illustrated and in particular the work of the mathematics team of the research project ‘Concepts in Secondary Mathematics and Science’ is examined.

Manipulatives in the classroom

Statements concerning the efficacy of manipulative materials in facilitating the learning of mathematical ideas abound in current mathematics education literature. Partly in response to such statements, mathematics laboratories, or activity-based curricula that include a heavy reliance on the use of such materials, are becoming more and more prevalent. This belief that manipulative materials do indeed enhance the learning of mathematics has gained much validity from learning theories such as those suggested by Bruner, Dienes, and Piaget. These theories strongly support the idea that children need physical involvement, such as might be provided by hands-on experiences with manipulative materials, in order to add new ideas to their cognitive structure. However, attempts to translate such theories into classroom practice and to empirically measure the results have not provided evidence that the use of such materials by teachers does indeed result in better learning than the use of symbols alone (Suydam and Weaver 1970; Kieren 1971; Fennema 1972). This difference between theory and research findings indicates that manipulative materials are no panacea; the use of materials does not automatically ensure that mathematics learning will follow. However, even without strong empirical support, a strong case can be built for the inclusion of manipulative materials in an elementary school mathematics program.

What Every Mathematics Teacher Ought to Read (Seventeen Opinions)

THE current wealth of good literature in mathematics education is a great blessing for mathematics teachers at all levels —high school, elementary school, and college. For many teachers, especially the beginners, this horn of plenty presents a perplexing problem; with limited time or financial resources, the mathematics teacher needs guidance as to what literature would be most beneficial for him to read and, it is, hoped to own.