Richard Lesh

Problem Solving, Modeling, and Local Conceptual Development

The research reported here describes similarities and differences between (a) modeling cycles that students typically go through during 60–90 min solutions to a class of problems thast we refer to as model-eliciting activities, and (b) stages of development that students typically go through during the “natural” development of constructs (conceptual systems, cognitive structures) that cognitive psychologists consider to be relevant to these specific problems. Examples of relevant constructs include those that underlie children’s developing ways of thinking about fractions, ratios, rates, proportions, or other elementary, but deep mathematical ideas. Results show that, when problem solvers go through an iterative sequence of testing and revising cycles to develop productive models (or ways of thinking) about a given problem solving situation, and when the conceptual systems that are needed are similar to those that underlie important constructs in the school mathematics curriculum, then these modeling cycles often appear to be local or situated versions of the general stages of development that developmental psychologists and mathematics educators have observed over time periods of several years for the relevant mathematics constructs. Furthermore, the processes that contribute to local conceptual development in model-eliciting activities are similar in many respects to the processes that contribute to general cognitive development. Applying principles from developmental psychology to problem solving—and vice versa—is a relatively new phenomenon in mathematics education (Lesh & MATHEMATICAL THINKING AND LEARNING, 5(2&3), 157–189 Copyright

Applied Mathematical Problem Solving.

A case is presented for the importance of focusing on (1) average ability students, (2) substantive mathematical content, (3) real problems, and (4) realistic settings and solution procedures for research in problem solving. It is suggested that effective instructional techniques for teaching applied mathematical problem solving resembles “mathematical laboratory” activities, done in small group problem solving settings.The best of these laboratory activities make it possible to concretize and externalize the processes that are linked to important conceptual models, by promoting interaction with concrete materials (or lower-order ideas) and interaction with other people.Suggestions are given about ways to modify existing applied problem solving materials so they will better suit the needs of researchers and teachers.

Models and Modeling Perspectives on the Development of Students and Teachers

This special issue of Mathematical Thinking and Learning describes models and modeling perspectives toward mathematics problem solving, learning, and teaching (Lesh & Doerr, 2003). The term “models” here refers to purposeful mathematical descriptions of situations, embedded within particular systems of practice that feature an epistemology of model fit and revision. That is, “modeling” is a process of developing representational descriptions for specific purposes in specific situations. It usually involves a series iterative testing and revision cycles in which competing interpretations are gradually sorted out or integrated or both—and in which promising trial descriptions and explanations are gradually revised, refined, or rejected. The latter emphasis on the “fitness” of models is critical because it suggests that models are inherently provisional, and it emphasizes that they are developed for specific purposes in specific situations—even though they may endure for longer periods of time, and even though they generally are intended to be sharable and reuseable in a variety of structurally similar situations. The distinction between model and world is not merely a matter of identifying the right symbol-referent matches; rather, it depends intimately on the accumulation of experience and its symbolic representations over time. Models bootstrap MATHEMATICAL THINKING AND LEARNING, 5(2&3), 109–129 Copyright

Modeling Students' Mathematical Modeling Competencies: ICTMA 13

Modeling Students Mathematical Modeling Competencies offers welcome clarity and focus to the international research and professional community in mathematics, science, and engineering education, as well as those involved in the sciences of teaching and learning these subjects.

Trends in the Evolution of Models & Modeling Perspectives on Mathematical Learning and Problem Solving

In this paper we briefly outline the models and modelling (MM and, rather than being preoccupied with the kind of word problems emphasized in textbooks and standardized tests, we focus on (simulations of) problem solving “in the wild.” Also, we give special attention to the fact that, in a technology-basedage of information, significant changes are occurring in the kinds of “mathematical thinking” that is coming to be needed in the everyday lives of ordinary people in the 21st century—as well as in the lives of productive people in future-oriented fields that are heavy users of mathematics, science, and technology.

Modeling conceptions revisited

The previous issue of ZDM raised several fundamental issues on the role of modeling in the school curricula at micro and macro levels. In this paper we complement the approaches described there by discussing some of the issues and the barriers to the implementation of mathematical modeling in school curricula raised there from the perspective of the on going work of the models and modeling research group. In doing so we stress the need for critical literacy as well as the need to initiate a new research agenda based on the fact that we are now living in a fundamentally different world in which reality is characterized by complex systems. This may very well require us to go beyond conventional notions of modeling.

Introduction to the Special Issue: Modeling as Application versus Modeling as a Way to Create Mathematics

This special issue of IJCML is dedicated to our good friend and colleague, Jim Kaput. In many ways, each article was inspired by collaborations that involved Jim; and, they also are examples of some of the most important directions that Kaput’s research was moving at the time of his untimely death. For example, the article by Lehrer, Kim, and Schauble emphasizes ways that expressive media (which may vary from technical languages, to paper-and-pencil notations and sketches, to computer-based tools for communication, collaboration, and conceptualization) shape and empower the ways that people think. The article by Konold, Harradine, and Kazak emphasizes Kaput’s longstanding mission to use technology to provide all children with democratic access to powerful ideas; and, the article by Lesh, Caylor, and Gupta emphasizes the infrastructural nature of the preceding kinds of conceptual tools. That is, the conceptual tools that humans develop to make sense of their experiences typically have both expanding and constraining influences on thinking; and, the same conceptual tools that have been developed to describe and explain existing situations also tend to be used to create new systems, artifacts, and tools. So, as soon as people develop better ways of thinking about things that currently exist, they tend to change these things in ways that make the development of conceptual systems a neverending enterprise. Furthermore, this is especially true in a technology-based age of information. It is well known that Kaput was a futurist who had a deep interest in history. Consequently, when we, like he, investigate how concepts develop in the thinking of students, our research is informed not only by information about how relevant concepts develop logically, psychologically, and pedagogically—but also historically. Yet, a fact that is not as well known about Kaput is that he was once a literature major who especially loved Don Quixote. So, it is not surprising to hear that many people considered him to be a ‘‘knight for the right.’’ For example, his mission was to use the best available technologies to go

Re-conceptualizing Mathematics Education as a Design Science

In this chapter we propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M&M research are drawn to illustrate our notion of mathematics education research as a design science. Our ideas are intended as a framework and do not constitute a “grand” theory. That is, we provide a framework (a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).

Introduction to Part I Modeling: What Is It? Why Do It?

At ICTMA-13, where the chapters in this book were first presented, a variety of views were expressed about an appropriate definition of the term model – and about appropriate ways to think about the nature of modeling activities. So, it is not surprising that some participants would consider this lack of consensus to be a priority problem that should be solved by a research community that claims to be investigating models and modeling

Beyond constructivism: Identifying mathematical abilities that are most needed for success beyond school in an age of information

Purdue University’s Center for Twenty-first Century Conceptual Tools (TCCT) was set up to investigate the types of problems students need to be able to solve in order to succeed in a technology-based age of information and the abilities and understandings that success requires. This paper describes the rationale behind the foundation of the TCCT, its mode of operation, and some preliminary findings.