Frank K. Lester

Musings about Mathematical Problem-Solving Research: 1970-1994.

As this quote from a recently published report of the National Assessment of Educational Progress indicates, the situation in American schools with respect to student performance in mathematical problem solving is desperate! Although conference reports, curriculum guides, and textbooks insist that problem solving has become central to instruction at every level, the evidence suggests otherwise. We may have learned quite a lot over the past 25 years or so about how students learn to solve problems and how problem solving can be taught, but we have not learned enough. And yet there are signs that problem solving has begun to receive less attention from researchers. In this article I provide a brief overview of past research in mathematical problem solving, discuss the apparent recent decline in research in this area, and suggest some issues and questions that should be the focus of research.

Self-Confidence, Interest, Beliefs, and Metacognition: Key Influences on Problem-Solving Behavior

Any good mathematics teacher would be quick to point out that students’ success or failure in solving a problem often is as much a matter of self-confidence, motivation, perseverance, and many other noncognitive traits, as the mathematical knowledge they possess. Nevertheless, it is safe to say that the overwhelming majority of problem-solving researchers have been content to restrict their investigations to cognitive aspects of performance. Such a restricted posture may be natural for psychologists and artificial intelligence scientists who are concerned primarily with expert systems or machine intelligence, but it simply will not suffice for the study of problem solving in school contexts.

On the theoretical, conceptual, and philosophical foundations for research in mathematics education

The current infatuation in the U.S. with “what works” studies seems to leave education researchers with less latitude to conduct studies to advance theoretical and model-building goals and they are expected to adopt philosophical perspectives that often run counter to their own. Three basic questions are addressed in this article:What is the role of theory in education research? How does ones philosophical stance influence the sort of research one does? And,What should be the goals of mathematics education research? Special attention is paid to the importance of having a conceptual framework to guide ones research and to the value of acknowledging ones philosophical stance in considering what counts as evidence.

An Evaluation of a Process-Oriented Instructional Program in Mathematical Problem Solving in Grades 5 and 7.

This paper provides an overview of a process-oriented instructional program and reports the results of an evaluation of that program. Twelve fifth-grade and 10 seventh-grade teachers implemented the Mathematical Problem Solving program for 23 weeks. Eleven fifth-grade and 13 seventh-grade teachers taught control classes. The experimental classes scored significantly higher than the control classes on measures of ability to understand problems, plan solution strategies, and get correct results. Trend analyses showed different student growth patterns for the three measures of problem-solving performance. Data from interviews with teachers supported the results of the quantitative analysis and suggested that both students and teachers had changed positively with respect to attitudes toward problem solving. In addition, teachers gained confidence in their ability to teach problem solving.

Implications of Research on Students’ Beliefs for Classroom Practice

This chapter offers a critical analysis of each of the five reports related to students’ beliefs about mathematics, highlighting the most salient aspects of each report. In addition to assessing each report, an argument is presented for the importance of studying students’ beliefs about the nature of mathematics, how it is learned, and how the subject is taught.

A New Vision of the Nature and Purposes of Assessment in the Mathematics Classroom

Publisher Summary This chapter is dedicated to the discussion of the role of assessment in mathematics classroom. As the view of the nature and purpose of school mathematics has changed, so too has the view of the nature and purpose of assessment. The chapter presents certain “alternative” assessment tasks, which have appeared in recent mathematics assessment documents. These tasks illustrate just how radically different contemporary mathematics assessments have become. The chapter focuses on the fact that school mathematics assessment has changed to ensure consistency with the goals of school mathematics curricula and instruction. The chapter is intended as an introduction to tile nature and extent of the changes that have taken place over the past 10 years. It is organized around six themes. It explores the changing nature of school mathematics and mathematics assessment. The forces driving the changes in assessment are described. Various promising classroom assessment techniques are outlined. The chapter also discusses building a theory of mathematics assessment, research findings, and ideas for debate. Finally, the chapter concludes that as we approach the new millennium, the mathematics education community is formulating a new vision of mathematics education. Changes in assessment are an integral part of this new vision.

CONTRIBUTIONS FROM CROSS-NATIONAL COMPARATIVE STUDIES TO THE INTERNATIONALIZATION OF MATHEMATICS EDUCATION: STUDIES OF CHINESE AND U.S. CLASSROOMS

Cross-national studies offer a unique contribution to the internationalization of mathematics education. In particular, they provide mathematics educators with opportunities to situate the teaching and learning mathematics in a wider cultural context and to reflect on generalization of theories and practices of teaching and learning mathematics that have been developed in particular countries. In this chapter, we discuss a series of cross-national studies involving Chinese and U.S. students that illustrate to how cultural differences in Chinese and U.S. teachers’ teaching practices and beliefs affect the nature of their students’ mathematical performance. We do this by showing that the types of mathematical representations teachers present to students strongly influence the choice of representations students use to solve problems. Specifically, the Chinese teachers overwhelmingly used symbolic representations of instructional tasks, whereas the U.S. teachers relied almost exclusively on verbal explanations and pictorial representations, illustrating that mathematics teaching is local practice which takes place in settings that are both socially and culturally constrained. These results demonstrate the social and cultural nature of teachers’ pedagogical practice

Can Mathematical Problem Solving Be Taught? Preliminary Answers from 30 Years of Research

In this chapter, the authors note that during the past 30 years there have been significant advances in our understanding of the affective, cognitive, and metacognitive aspects of problem solving in mathematics and there also has been considerable research on teaching mathematical problem solving in classrooms. However, the authors point out that there remain far more questions than answers about this complex form of activity. The chapter is organized around six questions: (1) Should problem solving be taught as a separate topic in the mathematics curriculum or should it be integrated throughout the curriculum? (2) Doesn’t teaching mathematics through problem require more time than more traditional approaches? (3) What kinds of instructional activities should be used in teaching through problems? (4) How can teachers orchestrate pedagogically sound, problem solving in the classroom? (5) How can productive beliefs toward mathematical problem solving be nurtured? (6) Will students sacrifice basic skills if they are taught mathematics through problem solving?

Negative instances and the acquisition of the mathematical concepts of commutativity and associativity

The effects of negative instances in the acquisition of the mathematical concepts of commutativity and associativity were examined. Two treatment levels for commutativity (positive instances or positive and negative instances) and the same treatment levels for associativity were crossed to form a 2 × 2 factorial design with 21 subjects per cell. Subjects were undergraduate elementary education majors. Criterion variables were number of correct responses, stimulus intervals, and postfeedback intervals. Results supported the contention that negative instances enhance concept acquisition but also appear to require more time during treatments. No evidence for a transfer effect for negative instances from one concept to another was found.

A Framework for Research on Problem-Solving Instruction

Research on mathematical problem solving has provided little specific information about problem-solving instruction. There appear to be four reasons for this unfortunate state of affairs: (1) relatively little attention has been given to the role of the teacher in instruction; (2) there has been little concern for what happens in real classrooms; (3) there has been a focus on individuals rather than small groups or whole classes; and (4) much of the research has been largely atheoretical in nature. This paper discusses each of these reasons and presents a framework for designing research on mathematics problem-solving instruction. Four major components of the framework are discussed: extra-instruction considerations, teacher planning, classroom processes, and instructional outcomes. Special attention is given to factors that may be particularly fruitful as the focal points of future research.