By Richard A. Lesh, Helen M. Doerr

This publication has fundamental ambitions. at the point of idea improvement, the booklet clarifies the character of an rising "models and modeling point of view" approximately instructing, studying, and challenge fixing in arithmetic and technological know-how schooling. at the point of emphasizing sensible difficulties, it clarifies the character of a few of crucial elementary-but-powerful mathematical or medical understandings and skills that americans are inclined to desire as foundations for fulfillment within the current and destiny technology-based details age. past Constructivism: types and Modeling views on arithmetic challenge fixing, studying, and educating positive aspects an cutting edge website housing on-line appendices for every bankruptcy, designed to complement the print chapters with electronic assets that come with instance difficulties, appropriate learn instruments and videos, in addition to transcripts and different samples of scholars' paintings: modeling/ this can be a vital quantity for graduate-level classes in arithmetic and technological know-how schooling, cognition and studying, and significant and inventive pondering, in addition to a invaluable source for researchers and practitioners in those components.

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Additional info for Beyond Constructivism: Models and Modeling Perspectives on Mathematics Problem Solving, Learning, and Teaching

Sample text

MODEL DEVELOPMENT SEQUENCES 39 dynamic systems of operations, relations, and transformations, and (d) go beyond thinking with a given model to also think about it for a variety of problem-solving functions. For example, as Fig. 2 suggests, if Dienes' principles are used with elementary school children to focus on constructs that underlie our base-ten numeration system, a given child might investigate structural similarities among activities involving three different sets of concrete materials—such as arithmetic blocks, bundling sticks (popsicle sticks), and a counting frame abacus—and the relevant conceptual system is the system of operations and relations that is common to all three embodiments.

18 LESHANDDOEKR. FIG. 1. 9. Modeling may involve sequences of iterative cycles. Unfortunately, when problem solvers work to develop productive interpretations of complex problem solving situations, a single modeling cycle often is not enough. As Fig. 9 suggests, if the mathematical interpretation of a problem is not trivial, solutions typically involve several "modeling cycles" in which descriptions, explanations, and predictions are gradually refined, revised, or rejected—based on feedback from trial testing.

For example, traditionally, problem solving has been defined as getting from givens to goals when the path is not immediately obvious (or it is blocked); and heuristics have been conceived to be answers to the question: What can you do when you are stuck? But in model-eliciting activities, different modeling cycles are characterized by different ways of thinking about givens, goals, and possible solution steps, and, what is problematic is for students to develop more productive ways of thinking.

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Beyond Constructivism: Models and Modeling Perspectives on by Richard A. Lesh, Helen M. Doerr
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