Generating and using examples in the proving process

Sandefur, James T.; Mason, John; Stylianides, Gabriel J.; Watson, Anne

doi:10.1007/s10649-012-9459-x

Cited by 53 publications

(23 citation statements)

References 12 publications

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“…Since concept formation is not necessarily done using pure definitions, Watson and Mason (2005) emphasized the need to go beyond prototypes and to search for the boundaries of a concept. In this way one becomes aware of the dimensions of possible variation and in each dimension of the range of permissible change (Bills et al, 2006;Sandefur, Mason, Stylianides, & Watson, 2013;Watson & Mason, 2005). The personal example space, the collection of examples and the interconnection between the examples a person has at his/her disposal (the accessible example space), play a major role in how a person makes sense of the tasks he/she is confronted with (Watson & Mason, 2005;Goldenberg & Mason, 2008).…”

Section: Cognitive Units As Building Blocksmentioning

confidence: 99%

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Graphing formulas: Unraveling experts’ recognition processes

Kop

Janssen

Drijvers

et al. 2017

The Journal of Mathematical Behavior

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a b s t r a c tAn instantly graphable formula (IGF) is a formula that a person can instantly visualize using a graph. These IGFs are personal and serve as building blocks for graphing formulas by hand. The questions addressed in this paper are what experts' repertoires of IGFs are and what experts attend to while recognizing these formulas. Three tasks were designed and administered to five experts. The data analysis, which was based on Barsalou and Schwarz and Hershkowitz, showed that experts' repertoires of IGFs could be described using function families that reflect the basic functions in secondary school curricula and revealed that experts' recognition could be described in terms of prototype, attribute, and part-whole reasoning. We give suggestions for teaching graphing formulas to students.

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Section: Cognitive Units As Building Blocksmentioning

confidence: 99%

“…Card-sorting tasks are often used in eliciting structured knowledge (Chi et al, 1981;Jonassen, Beissner, & Yacci, 1993;de Jong & Ferguson-Hessler, 1986;Goldenberg & Mason, 2008;Sandefur et al, 2013).…”

Section: Tasksmentioning

confidence: 99%

Graphing formulas: Unraveling experts’ recognition processes

Kop

Janssen

Drijvers

et al. 2017

The Journal of Mathematical Behavior

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“…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.…”

Section: Background Literature On Examplesmentioning

confidence: 99%

Mathematicians’ Example-Related Activity when Exploring and Proving Conjectures

Lockwood

Ellis

Lynch

2016

Int. J. Res. Undergrad. Math. Ed.

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Examples can play a critical role in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, there is more to learn about the precise ways in which mathematicians leverage examples as they formulate proofs. In this paper, we present results from surveys and interviews with mathematicians that targets the role of examples in mathematicians' proof-related activity. Their responses shed light on specific example-related activity (including strategic example selection and use) and on the overarching ways in which they engage in such activity (including a focus on generalization and metacognition). We share illustrative excerpts from the surveys and interviews and discuss educational implications of the results.

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“…There has, for instance, been some debate on the value of example generation for supporting reasoning (Dahlberg & Housman, 1997;Iannone, Inglis, Mejía-Ramos, Simpson, & Weber, 2011;Watson & Mason, 2005). While spontaneous example generation could well differ in its effects from non-spontaneous example generation (Sandefur, Mason, Stylianides, & Watson, 2013;Yopp, 2014), it would be interesting to compare ways in which defining tasks and example generation tasks influence further reasoning, and to investigate the impact of different defining tasks on example generation. Further studies exploring interactions among factors present in a variety of pedagogical situations could both aid our understanding and help us design tasks and instructional sequences to assist students in making an effective transition to advanced mathematics.…”

Section: Implications For Teaching and Learningmentioning

confidence: 99%

Interactions between defining, explaining and classifying: the case of increasing and decreasing sequences

Alcock

Simpson

2016

Educ Stud Math

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. (2017) 'Interactions between dening, explaining and classifying : the case of increasing and decreasing sequences.', Educational studies in mathematics., 94 (1). pp. 5-19. Further information on publisher's website:https://doi.org/10.1007/s10649-016-9709-4Publisher's copyright statement:The nal publication is available at Springer via http://dx.doi.org/10.1007/s10649-016-9709-4Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract This paper describes a study in which we investigated relationships between defining mathematical concepts -increasing and decreasing infinite sequences -explaining their meanings, and classifying consistently with formal definitions. We explored the effect of defining, explaining, or studying a definition on subsequent classification, and the effect of classifying on subsequent explaining and defining. We report that 1) student-generated definitions and explanations were highly variable in content and quality; 2) explicitly considering the meaning of the concept facilitated subsequent classification, and giving a personal definition or explanation had a greater effect than studying a given definition; 3) classifying before defining or explaining resulted in significantly poorer definitions and explanations. We discuss the implications of these results for the teaching of abstract pure mathematics, relating our discussion to existing work on the concept image/concept definition distinction and on working with examples.

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Generating and using examples in the proving process

Cited by 53 publications

References 12 publications

Graphing formulas: Unraveling experts’ recognition processes

Graphing formulas: Unraveling experts’ recognition processes

Mathematicians’ Example-Related Activity when Exploring and Proving Conjectures

Interactions between defining, explaining and classifying: the case of increasing and decreasing sequences

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