Foundations for the Future in Mathematics Education

The central question addressed inFoundations for the Future in Mathematics Educationis 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 citizenshipas well as preparing them for career opportunities in learning organizations, in knowledge economies, in an age of increasing globalizationhow 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 recentlyand 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 understandingand 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 Educationis an essential reference for researchers, curriculum developers, assessment experts, and teacher educators across the fields of mathematics and science education.

Richard Lesh, Rudy Distinguished Professor of Learning Sciences at Indiana University, is the Chair of the Department of Learning Sciences in the School of Education and is the associate editor for the International Journal for Mathematical Thinking & Learning. More than 250 journal articles and books reflect his international reputation concerning: learning and problem solving in mathematics education, and research and assessment design in mathematics and science education. He has written books for children through research scientists, and has been a software developer, as well as being the principal scientist for research at the Educational Testing Service. Recent publications include: The Handbook of Design Research in Mathematics & Science Education (Kelly & Lesh, 2006), and Beyond Constructivism: Modeling Perspectives on Mathematics Problem Solving, Learning & Teaching (Lesh & Doerr, 2003). 
Eric Hamilton directs the Center for Research on Learning and Teaching (CRLT) at the US Air Force Academy. CRLT efforts focus on networked pedagogical agents in classrooms, enhancing and assessing complex reasoning skill; and the instructional and cognitive affordances of tablet computing. Dr. Hamilton came to USAFA from NSF, where he was a member of the federal Senior Executive Service as Division Director overseeing NSF's education and learning technology research investments. He was also a Program Director, overseeing comprehensive mathematics and science reform programs in large cities. He came to NSF from Loyola University Chicago, where he was on the Mathematics and Computer Science faculty. While at Loyola, he directed NSF's Urban Systemic Initiative in Chicago and the Access 2000 Chicago Partnership, an award-winning consortium to increase minority participation in science and engineering. He holds US and foreign-patents in pen-based collaboration networks. 
James Kaput was originally trained in mathematics (Category Theory). He became interested in teaching teachers and reforming undergraduate education in 1970, and in the latter 70s, in the representational side of teaching, learning and cognition. Throughout his career he sought ways to bring educational and economic opportunity to those for whom it is too frequently denied. In the 1980s, he became interested in the empowering potential of newly available technologies, especially using new representations, to make subtle or complex ideas newly learnable, including concretely-based learning of ratio, proportion, and elementary functions, and new visual databases. In the late 1980s he began working on the question of how to bring empowering technology into the service of math-ed reform and to understand how the core math curriculum might be fundamentally reorganized to democratize access to powerful systems of notation and thought, such as algebra and calculus. To this end, he led two sets of research projects: (1) on the development of algebraic reasoning in the contexts of elementary mathematics, with special emphasis on building teacher capacity at the district level, and (2) the SimCalc Projects, which involve designing technologies, software, and curricula for the learning of the fundamental ideas underlying calculus beginning at early grade levels, with special focus on contextualizing, organizing, and energizing the learning of traditionally difficult core content, including algebra. Kaput also turned attention to massive implementation of technology-based innovations. To this end, with the University of Massachusetts, he founded SimCalc Technologies, Inc., a new educational technology development company focusing on integration and classroom implementation of widely available hand-held devices and desktop computer technologies.