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Students’ understanding of the structure of deductive proof

Students’ understanding of the structure of deductive proof
Students’ understanding of the structure of deductive proof
While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.
proof, understanding, structure, universal instantiation, hypothetical syllogism
0013-1954
223-239
Miyazaki, Mikio
812d859a-9bc0-41af-9577-2c8a7fef9e58
Fujita, Taro
8a05b8fc-a1ce-4a7b-9399-3fb00639a3cc
Jones, Keith
ea790452-883e-419b-87c1-cffad17f868f
Miyazaki, Mikio
812d859a-9bc0-41af-9577-2c8a7fef9e58
Fujita, Taro
8a05b8fc-a1ce-4a7b-9399-3fb00639a3cc
Jones, Keith
ea790452-883e-419b-87c1-cffad17f868f

Miyazaki, Mikio, Fujita, Taro and Jones, Keith (2017) Students’ understanding of the structure of deductive proof. Educational Studies in Mathematics, 94 (2), 223-239. (doi:10.1007/s10649-016-9720-9).

Record type: Article

Abstract

While proof is central to mathematics, difficulties in the teaching and learning of proof are well-recognised internationally. Within the research literature, a number of theoretical frameworks relating to the teaching of different aspects of proof and proving are evident. In our work, we are focusing on secondary school students learning the structure of deductive proofs and, in this paper, we propose a theoretical framework based on this aspect of proof education. In our framework, we capture students’ understanding of the structure of deductive proofs in terms of three levels of increasing sophistication: Pre-structural, Partial-structural, and Holistic-structural, with the Partial-structural level further divided into two sub-levels: Elemental and Relational. In this paper, we apply the framework to data from our classroom research in which secondary school students (aged 14) tackled a series of lessons that provided an introduction to proof problems involving congruent triangles. Using data from the transcribed lessons, we focus in particular on students who displayed the tendency to accept a proof that contained logical circularity. From the perspective of our framework, we illustrate what we argue are two independent aspects of Relational understanding of the Partial-structural level, those of universal instantiation and hypothetical syllogism, and contend that accepting logical circularity can be an indicator of lack of understanding of syllogism. These findings can inform how teaching approaches might be improved so that students develop a more secure understanding of deductive proofs and proving in geometry.

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More information

Accepted/In Press date: 11 July 2016
e-pub ahead of print date: 16 September 2016
Published date: February 2017
Keywords: proof, understanding, structure, universal instantiation, hypothetical syllogism
Organisations: Mathematics, Science & Health Education

Identifiers

Local EPrints ID: 400533
URI: http://eprints.soton.ac.uk/id/eprint/400533
ISSN: 0013-1954
PURE UUID: ec4b9195-0776-4e7f-80d8-13d327bff06d
ORCID for Keith Jones: ORCID iD orcid.org/0000-0003-3677-8802

Catalogue record

Date deposited: 19 Sep 2016 10:28
Last modified: 21 Nov 2021 03:35

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Contributors

Author: Mikio Miyazaki
Author: Taro Fujita
Author: Keith Jones ORCID iD

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