Showing papers by "Orit Zaslavsky published in 2006"
01 Jan 2006
TL;DR: It is argued that paying attention to examples offers both a practically useful and an important theoretical perspective on the design of teaching activities, on the appreciation of learners’ experiences and on the professional development of mathematics teachers.
Abstract: There is evidence from earliest historical records that examples play a central role in both the development of mathematics as a discipline and in the teaching of mathematics. It is not surprising therefore that examples have found a place in many theories of learning mathematics. Many would argue that the use of examples is an integral part of the discipline of mathematics and not just an aid for teaching and learning. The forum takes as its background both the variety of ways in which examples are construed within different theories of learning and the contribution that attention to examples can make to the learning and teaching processes. Consequently the forum can be seen as addressing issues at the very heart of mathematics education, both drawing upon and informing many other research topics. We argue that paying attention to examples offers both a practically useful and an important theoretical perspective on the design of teaching activities, on the appreciation of learners’ experiences and on the professional development of mathematics teachers. The importance of these ideas does not actually depend on the framework used for analysing teachers’ intentions, nor on any terms used to describe forms of teaching, such as: ‘analytic-inductive’ or ‘synthetic-deductive’, ‘traditional’ or ‘reform’, ‘rotelearning’ or ‘teaching for understanding’, ‘authentic’ or ‘investigative’. Issues in exemplification are relevant to all kinds of engagement with mathematics. This paper positions exemplification on the research agenda for the community by giving a historical overview of the way examples have been seen in mathematics education; an account of associated literature; an exploration of how exemplification ‘fits’ with various perspectives on learning mathematics; accounts of issues relating to teachers’ and learners’ use of examples; and directions for future research.
115 citations
01 Jan 2006
TL;DR: The complexity of treatment of examples in an 8 grade pre-algebra course is illustrated and Zaslavsky & Lavie (submitted) point to the possible complex web of considerations underlying teachers’ choice of examples.
Abstract: Examples are an integral part of mathematics and an important component of expert knowledge (Michener, 1978). Examples play a critical role in learning, and in particular form the basis for generalization, abstraction and analogical reasoning. Studies on how people learn from worked out examples suggest that effective instruction should include multiple examples, with varying formats, that support the appreciation of deep structures rather than excessive attention to surface features (Atkinson et al, 2000). Studies related to concept learning suggest that examples and non-examples be introduced in a carefully thought way, to support the distinction between critical and non-critical features and the construction of a rich and appropriate concept image and example spaces (e.g., Vinner, 1983; Watson & Mason, in press; Zaslavsky & Peled, 1996). A number of studies deal with the contribution of carefully sequenced sets of examples on learning (e.g., Petty & Jansson, 1987). In spite of the critical roles examples play in learning and teaching mathematics, studies focusing on teachers’ choice and treatment of examples are scarce. Rowland et al (2003) identify three types of elementary novice teachers’ poor choices of examples, which concur with the concerns raised by Ball et al (2005) regarding the knowledge base teachers need in order to carefully select appropriate examples that are “useful for highlighting salient mathematical issues” (ibid). Clearly, the choice of examples in secondary mathematics is far more complex. Zaslavsky & Lavie (submitted) point to the possible complex web of considerations underlying teachers’ choice of examples. In our study we illustrate the complexity of treatment of examples in an 8 grade pre-algebra course.
16 citations
TL;DR: In this paper, a 12th-grade student's conceptions of a mathematical definition were revealed through individual and group activities in which they were asked to consider a number of possible definitions of four mathematical concepts: two geometric and two analytic.
Abstract: This article deals with 12th-grade students' conceptions of a mathematical definition. Their conceptions of a definition were revealed through individual and group activities in which they were asked to consider a number of possible definitions of four mathematical concepts: two geometric and two analytic. Data consisted of written responses to questionnaires and transcriptions of videotaped group discussions. The findings point to three types of students' arguments: mathematical, communicative, and figurative. In addition, two types of reasoning were identified surrounding the contemplation of alternative definitions: for the geometric concepts, the dominant type of reasoning was a definition-based reasoning; for the analytic concepts, the dominant type was an example-based reasoning. Students' conceptions of a definition are described in terms of the features and roles they attribute to a mathematical definition.
12 citations