Eric Hamilton

Richard A. Lesh

James J. Kaput

The central question addressed in Foundations for the Future in Mathematics Education is this: What kind of understandings and abilities should be emphasized to decrease mismatches between the narrow band of mathematical understandings and abilities that are emphasized in mathematics classrooms and tests, and those that are needed for success beyond school in the 21st century? This is an urgent question. In fields ranging from aeronautical engineering to agriculture, and from biotechnologies to business administration, outside advisors to future-oriented university programs increasingly emphasize the fact that, beyond school, the nature of problem-solving activities has changed dramatically during the past twenty years, as powerful tools for computation, conceptualization, and communication have led to fundamental changes in the levels and types of mathematical understandings and abilities that are needed for success in such fields. For K-12 students and teachers, questions about the changing nature of mathematics (and mathematical thinking beyond school) might be rephrased to ask: If the goal is to create a mathematics curriculum that will be adequate to prepare students for informed citizenship—as well as preparing them for career opportunities in learning organizations, in knowledge economies, in an age of increasing globalization—how should traditional conceptions of the 3Rs be extended or reconceived? Overall, this book suggests that it is not enough to simply make incremental changes in the existing curriculum whose traditions developed out of the needs of industrial societies. The authors, beyond simply stating conclusions from their research, use results from it to describe promising directions for a research agenda related to this question. The volume is organized in three sections: *Part I focuses on naturalistic observations aimed at clarifying what kind of “mathematical thinking” people really do when they are engaged in “real life” problem solving or decision making situations beyond school. *Part II shifts attention toward changes that have occurred in kinds of elementary-but-powerful mathematical concepts, topics, and tools that have evolved recently—and that could replace past notions of “basics” by providing new foundations for the future. This section also initiates discussions about what it means to “understand” the preceding ideas and abilities. *Part III extends these discussions about meaning and understanding—and emphasizes teaching experiments aimed at investigating how instructional activities can be designed to facilitate the development of the preceding ideas and abilities. Foundations for the Future in Mathematics Education is an essential reference for researchers, curriculum developers, assessment experts, and teacher educators across the fields of mathematics and science education.

Foundations for the Future in Mathematics Education

Alan H. Schoenfeld

Ed Dubinsky

Volume III of Research in Collegiate Mathematics Education (RCME) presents state-of-the-art research on understanding, teaching, and learning mathematics at the post-secondary level. This volume contains information on methodology and research concentrating on these areas of student learning: Problem solving - included here are three different articles analyzing aspects of Schoenfeld's undergraduate problem-solving instruction. The articles provide new detail and insight on a well-known and widely discussed course taught by Schoenfeld for many years. Understanding concepts - these articles feature a variety of methods used to examine students' understanding of the concept of a function and selected concepts from calculus. The conclusions presented offer unique and interesting perspectives on how students learn concepts.Understanding proofs - this section provides insight from a distinctly psychological framework. Researchers examine how existing practices can foster certain weaknesses. They offer ways to recognize and interpret students' proof behaviors and suggest alternative practices and curricula to build more powerful schemes. The section concludes with a focused look at using diagrams in the course of proving a statement.

Research in Collegiate Mathematics Education III

David William Carraher

Maria L. Blanton

This book-the first to offer a comprehensive, research-based, multi-faceted look at issues in early algebra-provides a rationale for a stronger and more sustained approach to algebra in school, as well as concrete examples of how algebraic re

James J. Kaput

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