TY - JOUR

T1 - Is this a coincidence? The role of examples in fostering a need for proof

AU - Buchbinder, Orly

AU - Zaslavsky, Orit

N1 - Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2011/5

Y1 - 2011/5

N2 - It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term "Is this a coincidence?". In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for "proof" was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.

AB - It is widely known that students often treat examples that satisfy a certain universal statement as sufficient for showing that the statement is true without recognizing the conventional need for a general proof. Our study focuses on special cases in which examples satisfy certain universal statements, either true or false in a special type of mathematical task, which we term "Is this a coincidence?". In each task of this type, a geometrical example was chosen carefully in a way that appears to illustrate a more general and potentially surprising phenomenon, which can be seen as a conjecture. In this paper, we articulate some design principles underlying the choice of examples for this type of task, and examine how such tasks may trigger a need for proof. Our findings point to two different kinds of ways of dealing with the task. One is characterized by a doubtful disposition regarding the generality of the observed phenomenon. The other kind of response was overconfidence in the conjecture even when it was false. In both cases, a need for "proof" was evoked; however, this need did not necessarily lead to a valid proof. We used this type of task with two different groups: capable high school students and experienced secondary mathematics teachers. The findings were similar in both groups.

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U2 - 10.1007/s11858-011-0324-7

DO - 10.1007/s11858-011-0324-7

M3 - Article

AN - SCOPUS:84867444498

SN - 1863-9690

VL - 43

SP - 269

EP - 281

JO - ZDM - International Journal on Mathematics Education

JF - ZDM - International Journal on Mathematics Education

IS - 2

ER -