Academic journal article New Waves

Prospective Teachers' Learning in Geometry: Changes in Discourse and Thinking

Academic journal article New Waves

Prospective Teachers' Learning in Geometry: Changes in Discourse and Thinking

Article excerpt

Introduction

In the mathematics education research community, investigations of how students learn mathematics have defined mathematical learning as actively building new knowledge from experience and prior knowledge, moving to a higher level of thinking, or as changes in discourse. Other researchers have developed methods to measure learning quantitatively. The question that served as the impetus for this study was: "What do prospective teachers learn in geometry from their preparation for the work of teaching geometry?" It can be argued that this study does little to answer the question because of the complexity of participants' learning, and of the context in which these students were observed. However, my effort is to conceptualize these participants' mathematical thinking through their mathematical discourses as evidence of their learning, thereby adding some information to the two perspectives of learning as moving to a higher level of geometric thinking and as changes in discourse.

The term "level of geometric thinking" came from the van Hieles (1959/1985), and they used the term to describe a process of learning a new language because "each level has its own linguistic symbols" (p.4). The van Hiele levels of thinking reveal the importance of language use, and emphasize that language is a critical factor in movement through the levels; however, the word "language" is not clearly defined. Moschkovich (2010) argued that the language of mathematics does not mean a list of vocabulary words or grammar rules, but rather the communicative competence necessary and sufficient for competent participation in mathematical discourse. Sfard (2008) used a discursive approach inspired by Vygotsky to make a distinction between language and discourse - language is a tool, whereas discourse is an activity in which the tool is used or mediates. This perspective provides an understanding of mathematics as a social and discursive accomplishment in which talk, diagrams, representations, and mathematical objects play an important role. Consequently, mathematics learning requires several modes of communication (Sfard, 2002).

Many researchers have attempted to develop frameworks to examine discourse in learning mathematics. As an example, Sfard's (2008) communicational approach to mathematical learning provides a notion of mathematical discourse that distinguishes her framework from others in several ways. In particular, Sfard (2002) argues that the knowing of mathematics is synonymous with the ability to participate in mathematics discourse. From this perspective, conceptualizing mathematical learning as the development of a discourse and investigating learning means getting to know the ways in which children modify their discursive actions in these three respects: "its vocabulary, the visual means with which the communication is mediated, and the meta discursive rules that navigate the flow of communication and tacitly tell the participants what kind of discursive moves would count as suitable for this particular discourse, and which would be deemed inappropriate." (p.4) Therefore, Sfard's discursive framework is grounded in the assumption that thinking is a form of communication and that learning mathematics is learning to modify and extend one's discourse.

Theoretical Framework

In Sfard's (2008) Thinking as Communicating: Human Development, the Growth of Discourses, and Mathematizing, she introduces her discursive framework, a systematic approach to analyzing the discursive features of mathematical thinking. To examine the development of geometric discourse, this study connects Sfard's discursive framework to the van Hiele theory (see van Hiele, 1959/1985; 1986). The van Hiele theory describes the development of students' levels of thinking in geometry. The levels, numbered 1 to 5, are described as visual, descriptive, theoretical, formal logic and rigor. Connecting Sfard's work with the van Hiele theory provides a new perspective with which to revisit the van Hiele levels as the discursive development of geometric discourses. …

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