The role of examples in forming and refuting generalizations

Abstract

Acknowledging students' difficulty in generalizing in general and expressing generality in particular, we assert that the choice of examples that learners are exposed to plays a crucial role in developing their ability to generalize. We share with the readers experiences in which examples supported generalization, and elucidate the strategies that worked for us in these circumstances, presuming that similar strategies could be helpful with other students in other settings. We further share several pitfalls and call for caution in avoiding them.

Classification and Concept Consistency

“…Van Dormolen & Zaslavsky, 2003), students' beliefs about them (Zaslavsky & Shir, 2005), and students' guided collaborative development of definitions in contexts new to them (Larsen & Zandieh, 2008;Zandieh & Rasmussen, 2010). Studies have suggested that students who spontaneously generate examples in response to new definitions later perform better on tasks involving the defined concepts (Dahlberg & Housman, 1997), and there has been much recent discussion of the way in which example generation tasks might help students to modify their example spaces so that these more closely coincide with the extensions of the defined concepts (Mason, 2002;Watson & Mason, 2005;Zazkis, Liljedahl & Chernoff, 2008;Zazkis & Leikin, 2008). Definitions are also discussed in the extensive literature on proof, which focuses on the role of definitions as a deductive base in arguments about whole classes of objects.…”

Using patterns generically to see structure

“…Another of the ZDM papers that does not focus on sequential examples is by Zazkis, Liljedahl and Chernoff (2008). Their interest is in the role of examples, and they argue that 'the choice of examples that learners are exposed to plays a crucial role in developing their ability to generalize' (p. 131).…”

Investigating the development of prospective mathematics teachers’ pedagogical content knowledge of generalising number patterns through school practicum

“…Students are able to see the structure of the pattern based on consecutive terms and found it easier to state the relationship verbally than algebraically (e.g. Zazkis et al 2008). Lastly, students' uses of representations to make generalisations also might cause some difficulties.…”