Mathematical practices, in theory and practice
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ORIGINAL ARTICLE
Mathematical practices, in theory and practice Alan H. Schoenfeld1 Accepted: 2 May 2020 © FIZ Karlsruhe 2020
Abstract Descriptions of mathematical thinking have an extended lineage. Sometimes accurate and sometimes not, sometimes misinterpreted and sometimes not, characterizations of mathematical thought processes have inspired and at times misled people interested in designing or framing mathematics instruction. Challenges the field faces in conceptualizing mathematics instruction include: What can be warranted as legitimate mathematical practices? Which aspects of mathematical practices are relevant and appropriate for K-16 instruction? What kinds of support are necessary? What is viable at scale? This paper provides a description of relevant history from the Western literature, bringing readers up to the present. It then addresses two key issues related to contemporary curricula: the framing of the mathematical enterprise as being fundamentally inquiryoriented and the need for curricula and instruction to reflect such mathematical values; and the characteristics of mathematical classrooms that support students’ development as powerful mathematical thinkers. An emphasis is on problem solving, a major component of “thinking mathematically.” The paper concludes with a description of practices that are currently under-emphasized in instruction and that would profit from greater attention. Keywords Mathematical practices · Goals for instruction · Robust learning environments · Teaching for robust understanding framework
1 Introduction This paper explores historical and contemporary issues related to descriptions of the practices of professional mathematicians and their relevance for mathematics teaching and learning. It begins with a selected review of work by major Western1 mathematicians and philosophers whose tacit or explicit characterizations of mathematical practices have had a significant impact on mathematics education. At the end of each section through Sect. 6, I point to the relevance of each perspective and the issues it raises. A major focus is on problem solving, although the intended scope is broader – “thinking mathematically” includes problem posing, generalizing, and abstracting, for example. Sections 7 and 8 frame an approach aimed at helping students become powerful mathematical thinkers. My point of departure is a conversation with a mathematician who studied under the “Moore Method” (Coppin, Mahavier, May & Parker 2009). In the Moore method’s * Alan H. Schoenfeld [email protected] 1
University of California, Berkeley, Berkeley, CA, USA
purest form, students are presented with mathematical definitions and asked to prove theorems. The students are barred from reading mathematics books or using other resources; the idea is for them to develop the mathematics themselves. Moore’s classes started as follows. 1. At the first meeting of the class Moore would define the basic terms and either challenge the class to discover the relations among them, or, depending on the subjec
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