A DNR Perspective on Mathematics Curriculum and Instruction

Guershon Harel
Wednesday 19th November 2008
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The goal of this talk is to contribute to the debate on a pair of questions that are on the mind of many mathematics educators—teachers, teacher leaders, curriculum developers, and researchers who study the processes of learning and teaching—namely: (1) What is the mathematics that we should teach in school? (2) How should we teach it?

Clearly, one presentation is not sufficient to address these colossal questions, which are inextricably linked to other difficult questions—about student learning, teacher knowledge, school culture, societal need, and educational policy, to mention a few. The goal the talk, thus, is merely to articulate a pedagogical stance on these two questions. The stance is not limited to a particular mathematical area or grade level; rather, it encompasses the learning and teaching of mathematics in general. This stance is oriented within a theoretical framework, called DNR-based instruction in mathematics (DNR, for short). The initials D, N, ard R stand for three foundational instructional principles in the framework: duality, necessity, and repeated reasoning. DNR can be thought of as a system consisting of three categories of constructs: premises—explicit assumptions underlying the DNR concepts and claims; concepts—constructs defined and oriented within these premises; and instructional principles—claims about the potential effect of teaching actions on student learning.

Biographical Sketch:
Dr Guershon Harel is Professor of Mathematics at the University of California, San Diego, where he has taught for the past 8 years. Prior to UCSD, he taught at Purdue University and Northern Illinois University. He is a Principal Investigator for the Rational Number Project, a project that has received funding from the National Science Foundation to create a companion module to the current RNP Fraction Lessons for the Middle Grades. Other research projects include the Algebraic Thinking Institute (ATI) at UCSD, Proof Understanding, Production, and Appreciation (PUPA), and Development of Mathematics Teachers' Knowledge Base Through DNR-Based Instruction: Focus on Proofs in Algebra. He holds a BS, MS and PhD in Mathematics from Ben-Gurion University of the Negev. His areas of interest include: cognition and epsitemology of mathematics and their implications to mathematics curricula and teacher education; advanced mathematical thinking, particularly the concept of proof, the learning and teaching of linear algebra, and the development of the multiplicative conceptual field.