Examining the role of examples in proving processes through a cognitive lens: the case of triangular numbers

Abstract

In this paper, we analyze the role of examples in the proving process. The context chosen for this study was finding a general rule for triangular numbers. The aim of this paper is to show that examples are effective for the construction of a proof when they allow cognitive unity and structural continuity between argumentation and proof. Continuity in the structure is possible if the inductive argumentation is based on process pattern generalization (PPG), but this is not the case if a generalization is made on the results. Moreover, the PPG favors the development of generic examples that support cognitive unity and structural continuity between the argumentation and proof. The cognitive analysis presented in this paper is performed through Toulmin's model.

Mathematicians’ Example-Related Activity when Exploring and Proving Conjectures

“…Researchers have identified and categorized types of examples (Buchbinder and Zaslavsky 2009;Mason and Watson 2008;Michener 1978;Tsamir et al 2008;Zazkis and Leikin 2008), investigated people's thinking with and use of examples (Antonini 2006;Balacheff 1987;Bills and Rowland 1999;Buchbinder and Zaslavsky 2011;Goldenberg and Mason 2008;Watson and Mason 2005;Watson and Shipman 2008;Zodik and Zaslavsky 2008), and studied the relationships between example use and proof (Antonini 2003;Alcock and Inglis 2008;Chazan 1993;Iannone et al 2011;Pedemonte and Buchbinder 2012;Rowland 2001;Sandefur et al 2013). This body of research as a whole reveals the richness and variety of the types of examples learners are able to produce, examine, and leverage as they make sense of new ideas, consider and formulate conjectures, and develop deductive arguments.…”

“…They found that the student who was able to produce robust and logically consistent justifications also made use of a variety of examples for multiple purposes, including developing understanding, inductive testing of conjectures, checking arguments, and generating counterexamples. Other researchers have similarly identified ways in which example use can be an effective tool to support proof construction (e.g., Antonini 2003;Buchbinder and Zaslavsky 2011;Pedemonte and Buchbinder 2012;Rowland 2001) and to determine an argument's validity (Weber 2008). For instance, Antonini (2003) found that the generation of non-examples can lead to indirect argumentation and proof by contraction, and Pedemonte and Buchbinder (2012) found that examples appear to be particularly helpful for proof development when they allow cognitive unity and structural continuity between argumentation and proof.…”

“…In these instances, the mathematicians grounded their thinking in a particular example, and by manipulating that example they developed an idea for how a more general proof might develop. A number of researchers have suggested that proofs can emerge from work with examples (e.g., Sandefur et al 2013;Weber 2008), and more specifically in terms of a generic example (e.g., Pedemonte and Buchbinder 2012). One survey response demonstrates that this mathematician can move toward generalization based on observations from basic examples: BI might also look for simpler examples that contain enough structure to be generalized.…”

Investigating Mathematical Knowledge for Teaching Proof in Professional Development

“…Given classroom research indicating how generic examples support students' progression from inductive to deductive reasoning (Lannin, Barker & Townsend, 2006;Pedemonte & Buchbinder, 2011), opportunities for teachers to develop this component of SCK seem particularly salient. As demonstrated in the second small group seminar discussion, leaders critically evaluated the array and area models presented in the Halving and Doubling videocase.…”

How do the Undergraduate Students Use an Example in Mathematical Proof Construction: A Study based on Argumentation and Proving Activity

“…Although proving requires a special form of reasoning that is deductive reasoning, other reasoning such as inductive reasoning is also needed. Examples of proving are used to; (1) understand mathematical statements or understand the proof process being read (Ellis, Lockwood, Williams, Dogan, & Knuth, 2012); (2) exploring problems for constructing conjectures or hypotheses (Buchbinder & Pedemonte, 2011;Conner et al, 2014); (3) illustrating new techniques and ideas; (4) identifying general patterns for the generalization process (Watson, & Mason, 2005) and (5) verifying the results obtained (Balacheff, 1988).…”