Key ideas in a proof: The case of the irrationality of √2

Yan, Xiaoheng

doi:10.1016/j.jmathb.2019.04.001

Cited by 3 publications

(6 citation statements)

References 14 publications

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“…Several researchers have argued that focusing students' attention on identifying and articulating the key idea of a proof is a productive pedagogical activity (Hanna & Mason, 2014;Robinson, 2000;Yan, 2019). And hence, given the connection between key ideas and proof summaries, we suggest that engaging students in summarizing proofs is likely to be similarly productive.…”

Section: On Proof Key Ideas and The Activity Of Summarizingmentioning

confidence: 79%

“…The educational literature on key ideas refers to, and relies heavily upon, the notion of summarizing. Yan (2019) described this literature as constituting two camps of research: either as "the most important mathematical ideas, methods or strategies used in a proof" or "as an outline, overview, or architecture of a proof" (p. 2). The former implicitly characterizes proofs as active entities constructed by individuals who must implement, or use, an idea.…”

Section: On Proof Key Ideas and The Activity Of Summarizingmentioning

confidence: 99%

“…Before proceeding, we must define what is meant by a 'proof summary'. To this end, we rely on three influential works reviewed above, each contributing their own properties to our definition; Raman (2003), Yan (2019), and Mejia-Ramos and Inglis (2009) whose work we discussed in the previous section.…”

Section: Defining Proof Summariesmentioning

confidence: 99%

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Assessing Proof Reading Comprehension Using Summaries

Davies

Jones

2021

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

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In this paper, we explore the role of mathematical proof summaries as a tool for capturing students’ reading comprehension of a given proof. We present an interview study based on mathematicians’ pairwise evaluations of student-produced summaries of a proof demonstrating the uncountability of the open unit interval. We present a thematic analysis, exploring features of mathematicians’ pairwise decision-making and their priorities in evaluating summaries. We argue that the students’ proof summaries shared several properties with traditional modes of proof-writing and were frequently evaluated against similar conventions. We consider the consequences for research and practice with proof comprehension and conclude that proof summaries have the potential to form the basis of a new approach to assessment in this area.

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Section: On Proof Key Ideas and The Activity Of Summarizingmentioning

confidence: 79%

Section: On Proof Key Ideas and The Activity Of Summarizingmentioning

confidence: 99%

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Assessing Proof Reading Comprehension Using Summaries

Davies

Jones

2021

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

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“…In understanding a proof, it is very important to be able to capture the key idea in proof. The key idea of proof includes the structure, guidelines and direction, as well as the steps and procedures contained in the proof (Yan, 2019). Yan also underlined the 5 focuses of students in capturing key ideas of the proof.…”

Section: Proofmentioning

confidence: 99%

“…This technique is very helpful for researchers, writers and lecturers of algebraic structure in writing formal proof (Hungerford, 2014;Gallian, 2017;Misri, 2017). The more diverse techniques used in each proposition that will be proven, of course, really helps undergraduate students skilled in proofwriting and applying it in algebraic structure courses (Lee, 2016;Bleiler-Baxter & Pair, 2017;Brown, 2017;Yan, 2019), likewise with lecturers/ mathematicians in capturing the contents of undergraduate students' proofs (Miller, Infante, & Weber, 2017). This paper aims to investigate the diversity of proofing techniques and their effects on students' skills in applying them.…”

Section: Introductionmentioning

confidence: 99%

Propositional Proofing Techniques application in Algebraic Structure Research

Misri¹

2020

Eduma : Mathematics Education Learning and Teaching

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This paper investigates proofing techniques which are used in algebraic structure research and how to apply those techniques. This paper also investigates the reasoning behind the application of the techniques and their effects. The data was collected by reviewing some algebraic structure textbooks used recently. Books reviewed were domestic and foreign books. The results of the review process were noted and documented, and then finally transcribed and coded. The review process of each book was carried out twice. The data obtained were then triangulated, to get the valid data, and eventually used for further analytical purposes. The results obtained show that not all the proofing techniques were used in the proofing process in the algebraic structure research. They tended to use the same technique in every proofing activity. This might be due to mastery of proofing techniques, preferences for certain proofing techniques, lack of information about the development of various proofing techniques. The resulting impact can be identified from the steps of proofing, meanwhile, if it is conducted using another technique, it can reduce the steps, is more acceptable also comprehensible.

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Investigation of Prospective Mathematics Teachers’ Proof Completion Processes Supported By Key Ideas

Cetin

Dikici

2023

ENAD

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The aim of this research was to examine prospective mathematics teachers’ completion processes of mathematical proofs supported by key ideas in the field of abstract algebra. Prospective teachers were asked to complete the parts of a proof left incomplete using those key ideas. It was decided together with three academics who are experts in the field of algebra which theorems to use and which proof sections to leave missing in the proof completion form. The proof completion form consisted of 5 proofs and was applied to 5 participants. The participants' processes of completing the proofs were evaluated in semi-structured interviews supported by the think-aloud method. Interviews were recorded by video and transcribed, and descriptive analysis was performed. The findings obtained were analyzed with think-aloud protocols. According to the results, key ideas can be said to play an active role in the proof process and the teaching of proof. In addition, developing activities consisting of proofs supported by key ideas can contribute to prospective teachers’ abilities to prove without memorization and internalize proof processes.

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Key ideas in a proof: The case of the irrationality of √2

Cited by 3 publications

References 14 publications

Assessing Proof Reading Comprehension Using Summaries

Assessing Proof Reading Comprehension Using Summaries

Propositional Proofing Techniques application in Algebraic Structure Research

Investigation of Prospective Mathematics Teachers’ Proof Completion Processes Supported By Key Ideas

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