When does an argument use a generic example?

Yopp, David A.; Ely, Rob

doi:10.1007/s10649-015-9633-z

Cited by 24 publications

(14 citation statements)

References 21 publications

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“…That is, the given diagram is used as a generic example, defined as "an actual example, but one presented in such a way as to bring out its intended role as the carrier of the general" (Mason & Pimm, 1984;p. 287; for a recent review of generic examples, see Yopp & Ely, 2016). However, the proof may suffer from "one-case concreteness" (e.g., Dvora & Dreyfus, 2014;Presmeg, 1986;Yerushalmy & Chazan, 1990) in the sense that the proof based on the single concrete diagram may be applicable only to a subset of all cases considered in the statement.…”

Section: The Meanings Of Proof Validation and Proof Modificationmentioning

confidence: 99%

Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

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Proof validation is important in school mathematics because it can provide a basis upon which to critique mathematical arguments. While there has been some previous research on proof validation, the need for studies with school students is pressing. For this paper, we focus on proof validation and modification during secondary school geometry. For that purpose, we employ Lakatos' notion of local counterexample that rejects a specific step in a proof. By using Toulmin's framework to analyze data from a task-based questionnaire completed by 32 ninth-grade students in a class in Japan, we identify what attempts the students made in producing local counterexamples to their proofs and modifying their proofs to deal with local counterexamples. We found that student difficulties related to producing diagrams that satisfied the condition of the set proof problem and to generating acceptable warrants for claims. The classroom use of tasks that entail student discovery of local counterexamples may help to improve students' learning of proof and proving. first category encompasses studies in which different types of mathematical argument (visual, inductive, generic, or deductive) are presented to students and the students are asked to evaluate whether the arguments are personally convincing and whether the arguments can be considered to constitute proofs (e.g., Healy & Hoyles, 2000; Martin & Harel, 1989; Segal, 1999). A second category relates to students' understanding of given correct proofs. Based on models of proof comprehension (Mejia-Ramos, Fuller, Weber, Rhoads, & Samkoff, 2012; Yang & Lin, 2008), studies in this category seek to identify effective proof comprehension strategies and examine the effectiveness of specific training to improve students' proof comprehension (Hodds, Alcock, & Inglis, 2014; Samkoff & Weber, 2015). A third category involves proof validation; here, researchers show students purported deductive proofs and ask them to determine whether the proofs are valid or invalid (

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Section: The Meanings Of Proof Validation and Proof Modificationmentioning

confidence: 99%

Proof validation and modification in secondary school geometry

Komatsu

Jones

Ikeda³

et al. 2017

The Journal of Mathematical Behavior

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“…I view GEs as less rigorous than a formal proof, but I agree with A. Stylianides (2007), Reid & Vallejo Vargas (2018) and Rowland (2002) that generic examples are valid mathematical arguments (Yopp & Ely, 2016). An important consequence of this view is that mathematics learners can still authentically engage in mathematical proving practices even without facility with formal proof representation.…”

Section: Definition Of Proofmentioning

confidence: 65%

“…Ultimately the audience can conceive of no possible instance in which the analogy could not be achieved. (p. 161, italics original) Similarly, Yopp and Ely (2016) state that "a formal proof often uses general representations, such as quantified variables or symbolic placeholders. But, in a generic example argument, the generality lies not in the representation but in the way the example is appealed to" (p. 41).…”

Section: Definition Of Proofmentioning

confidence: 99%

“…Generic examples have been used primarily to categorize different types of mathematical reasoning and arguments (Balacheff, 1988), and while researchers have recommended generic examples as a powerful tool for supporting proof (e.g., Rowland, 2002;Stylianides, 2007;Yopp & Ely, 2016), there is little agreement on what constitutes a generic example. I follow Yopp & Ely (2016) in their definition of a generic example, which they describe as reasoning around a particular example that nonetheless can be applied to all cases in the example's domain. Balacheff (1988) highlights the importance of the actions performed upon the generic example as determining whether or not the example is generic -for example, an action taken upon an example that relies upon a particular characteristic of the example that is not common to the entire domain identifies an example as non-generic.…”

Section: Definition Of Proofmentioning

confidence: 99%

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Pre-Service Teachers’ Criteria for Evaluating Mathematical Arguments That Include Generic Examples

Doğan

2022

Int. J. of Cont. Edu. Res.

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“…Someone who can write valid proof shows that the person has a thorough understanding of the problem. General representations such as mathematical symbols or quantified variables are often used in formal proof (Yopp & Ely, 2016).…”

Section: Introductionmentioning

confidence: 99%

Mental Structure Constructed by Field Dependent Student on ACE Learning

Wijayanti¹,

Waluya²,

Kartono³

et al. 2020

Proceedings of the International Conference on Science and Education and Technology (ISET 2019)

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This study aimed to describe the mental structure constructed by field-dependent (FD) students in the concept of group theory on ACE learning. This study was a descriptive qualitative. We discuss our findings based on the genetic decomposition of the group. The participants of this study were three (3) undergraduate students. The data was collected by a written test, videotape, and interviews. Instrument validation was obtained through expert judgment for the test of the proof ability and experiment for the genetic decomposition of a group. The credibility of the data was done by using triangulation: test and depth interview. Data analysis used Miles and Huberman's model. The FD-1 participant constructed the mental structure Action for the axioms, Action for the set, and Process for binary operation. The FD-1 participant constructed the mental structure of Action for a group. For the FD-2 participant constructed the mental structure of Process for axioms, Object for a set, and Object for the binary operation. The FD-2 participant constructed the mental structure of Process for a group. The FD-3 participant constructed the mental structure of Schema for axioms, Object for a set, and Object for the binary operation. The FD-3 participant constructed the mental structure of Schema for a group.

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When does an argument use a generic example?

Cited by 24 publications

References 21 publications

Proof validation and modification in secondary school geometry

Proof validation and modification in secondary school geometry

Pre-Service Teachers’ Criteria for Evaluating Mathematical Arguments That Include Generic Examples

Mental Structure Constructed by Field Dependent Student on ACE Learning

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