Alan H. Schoenfeld

Cognitive science and mathematics education

Contents: A.H. Schoenfeld, Cognitive Science and Mathematics Education: An Overview. E.A. Silver, Foundations of Cognitive Theory and Research for Mathematics Problem-Solving. J.G. Greeno, Instructional Representations Based on Research About Understanding. R.D. Pea, Cognitive Technologies for Mathematics Education. J. Kilpatrick, Problem Formulating: Where Do Good Problems Come From? A. Henderson, From the Teachers Side of the Desk. S.B. Maurer, New Knowledge About Errors and New Views About Learners: What They Mean to Educators and More Educators Would Like to Know. A.H. Schoenfeld, Whats All the Fuss About Metacognition? R.H. Wenger, Cognitive Science and Algebra Learning. H.O. Pollak, Cognitive Science and Mathematics Education: A Mathematicians Perspective. F.J. Crosswhite, Cognitive Science and Mathematics Education: A Mathematics Educators Perspective.

Toward a Theory of Teaching-in-Context.

DOCUMENT RESUME

Improving Educational Research:Toward a More Useful, More Influential, and Better-Funded Enterprise

Educational research is not very influential, useful, or well funded. This article explores why and suggests ways that the situation could be improved. Our focus is on the processes that link the development of good ideas and insights, the development of tools and structures for implementation, and the enabling of robust implementation in realistic practice. We suggest that educational research and development should be restructured so as to be more useful to practitioners and to policymakers, allowing the latter to make better-informed, less-speculative decisions that will improve practice more reliably.

Making Mathematics Work for All Children: Issues of Standards, Testing, and Equity:

“Mathematics Education is a civil rights issue,” says civil rights leader Robert Moses, who argues that children who are not quantitatively literate may be doomed to second-class economic status in our increasingly technological society. The data have been clear for decades: poor children and children of color are consistently shortchanged when it comes to mathematics. More broadly, the type of mathematical sophistication championed in recent reform documents, such as the National Council of Teachers of Mathematics’ (2000) Principles and Standards for School Mathematics, can be seen as a core component of intelligent decision making in everyday life, in the workplace, and in our democratic society. To fail children in mathematics, or to let mathematics fail them, is to close off an important means of access to society’s resources. This article discusses the potential for providing high quality mathematics instruction for all students. It addresses four conditions necessary for achieving this goal: high quality curriculum; a stable, knowledgeable, and professional teaching community; high quality assessment that is aligned with curricular goals; and stability and mechanisms for the evolution of curricula, assessment, and professional development. The goal of this article is to catalyze conversations about how to achieve sustained, beneficial changes.

The Math Wars

During the 1990s, the teaching of mathematics became the subject of heated controversies known as the math wars. The immediate origins of the conflicts can be traced to the “reform” stimulated by the National Council of Teachers of Mathematics’ Curriculum and Evaluation Standards for School Mathematics. Traditionalists fear that reform-oriented, “standards-based” curricula are superficial and undermine classical mathematical values; reformers claim that such curricula reflect a deeper, richer view of mathematics than the traditional curriculum. An historical perspective reveals that the underlying issues being contested—Is mathematics for the elite or for the masses? Are there tensions between “excellence” and “equity”? Should mathematics be seen as a democratizing force or as a vehicle for maintaining the status quo?—are more than a century old. This article describes the context and history, provides details on the current state, and offers suggestions regarding ways to findaproductive middle ground.

How we think : a theory of goal-oriented decision making and its educational applications

Introduction and Acknowledgments 1. The Big Picture 2. Reflections, Caveats, Doubts, and Rationalizations 3. The Structure of the Representations Used in this Book 4. Lesson Analysis I: A beginning teacher carrying out a traditional lesson 5. Lesson Analysis II: An experienced teacher carrying out a non-traditional lesson 6. Lesson Analysis III: Third graders! A non-traditional lesson with an emergent agenda 7. Lesson Analysis IV: The analysis of a doctor-patient Consultation - an act of joint problem solving 8. Next Steps Indices, etc

Mathematical thinking and problem solving.

Contents: Preface. J.L. Schwartz, The Role of Research in Reforming Mathematics Education: A Different Approach. M. Linn, R. Pea, A Discussion of Judah Schwartzs Chapter. B. Reznick, Some Thoughts on Writing for the Putnam. L.C. Larson, Comments on Bruce Reznicks Chapter. I. Olkin, A.H. Schoenfeld, A Discussion of Bruce Reznicks Chapter. A.H. Schoenfeld, Reflections on Doing and Teaching Mathematics. L. Henkin, J.L. Schwartz, A Discussion of Alan Schoenfelds Chapter. J.J. Kaput, Democratizing Access to Calculus: New Routes to Old Roots. E. Dubinsky, Comments on James Kaputs Chapter. J. Confrey, E. Smith, Comments on James Kaputs Chapter. M. Cohen, A. Knoebel, D.S. Kurtz, D.J. Pengelley, Making Calculus Students Think with Research Projects. B.Y. White, R.G. Douglas, A Discussion of Cohen, Knoebel, Kurtz, and Pengelleys Chapter. E. Dubinsky, A Theory and Practice of Learning College Mathematics. R.G. Wenger, Comments on Ed Dubinskys Chapter. A.A. diSessa, Comments on Ed Dubinskys Chapter. S.S. Epp, The Role of Proof in Problem Solving. J.G. Greeno, Comments on Susanna Epps Chapter. J. Addison, A Discussion of Susanna Epps Chapter. T.A. Romberg, Classroom Instruction that Fosters Mathemical Thinking and Problem Solving: Connections Between Theory and Practice. G. Leinhardt, Comments of Thomas Rombergs Chapter. R.B. Davis, Comments on Thomas Rombergs Chapter. A.H. Schoenfeld, Epilogue.

What Do We Know about Mathematics Curricula

The subject of this essay is the mathematics curriculum: What should we be teaching in mathematics, and in what ways? This issue has been a focus of my problem solving work for nearly two decades, and I have written about it at length from that perspective. However, I am going to take a different point of view in this essay. Here I shall take a distanced perspective, in order to reflect on some difficult issues. Mathematics education is at a turning point. Some radically new programs are being proposed, and the abolition of some familiar programs is being proposed as well. This is a good time to ask, What do we really know? How much of what we think we know is based on a firm knowledge base, how much on informed guesswork, how much is really just opinion? How much of what we plan to do reflects cultural biases, rather than established fact? These are thorny questions. I shall explore the following four major issues related to curriculum: questions of content, tracking, problem-based curricula, and the role of proof. My goal is to be as honest about what I know, and what I don’t know, as I can be.

What Doesn’t Work: The Challenge and Failure of the What Works Clearinghouse to Conduct Meaningful Reviews of Studies of Mathematics Curricula:

An early version of this article, discussing curricular interventions in mathematics, was written for the What Works Clearinghouse (WWC). The Institute of Education Sciences (IES), which funds WWC, instructed WWC not to publish it. An expanded version, written at WWC’s invitation for a special issue of an independent electronic journal and a book to be published by WWC, argued that methodological problems rendered some WWC mathematics reports potentially misleading and/or uninterpretable. IES instructed WWC staff not to publish their chapters—thus canceling the publication of the special issue and the book. Those actions, chronicled here, raise important issues concerning the role of federal agencies and their contracting organizations in suppressing scientific research that casts doubt on current or intended federal policy.

Purposes and Methods of Research in Mathematics Education

T he first quotation above is humorous; the second serious. Both, however, serve to highlight some of the major differences between mathematics and mathematics education—differences that must be understood if one is to understand the nature of methods and results in mathematics education. The Cohen quotation does point to some serious aspects of mathematics. In describing various geometries, for example, we start with undefined terms. Then, following the rules of logic, we prove that if certain things are true, other results must follow. On the one hand, the terms are undefined; i.e., “we never know what we are talking about.” On the other hand, the results are definitive. As Gertrude Stein might have said, a proof is a proof is a proof. Other disciplines work in other ways. Pollak’s statement was not meant as a dismissal of mathematics education, but as a pointer to the fact that the nature of evidence and argument in mathematics education is quite unlike the nature of evidence and argument in mathematics. Indeed, the kinds of questions one can ask (and expect to be able to answer) in educational research are not the kinds of questions that mathematicians might expect. Beyond that, mathematicians and education researchers tend to have different views of the purposes and goals of research in mathematics education. This article begins with an attempt to lay out some of the relevant perspectives and to provide background regarding the nature of inquiry within mathematics education. Among the questions explored are the following: Just what is the enterprise? That is, what are the purposes of research in mathematics education? What do theories and models look like in education as opposed to those in mathematics and the physical sciences? What kinds of questions can educational research answer? Given such questions, what constitute reasonable answers? What kinds of evidence are appropriate to back up educational claims? What kinds of methods can generate such evidence? What standards might one have for judging claims, models, and theories? As will be seen, there are significant differences between mathematics and education with regard to all of these questions.