Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning
Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning
Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers’ role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students’ work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice.
Abduction, Counterexample, Mathematical knowledge, Proof, Task design, Teacher role
Komatsu, Kotaro
22446313-5c59-4d33-a7c2-94684615e98f
Jones, Keith
ea790452-883e-419b-87c1-cffad17f868f
Komatsu, Kotaro
22446313-5c59-4d33-a7c2-94684615e98f
Jones, Keith
ea790452-883e-419b-87c1-cffad17f868f
Komatsu, Kotaro and Jones, Keith
(2021)
Generating mathematical knowledge in the classroom through proof, refutation, and abductive reasoning.
Educational Studies in Mathematics.
(doi:10.1007/s10649-021-10086-5).
Abstract
Proving and refuting are fundamental aspects of mathematical practice that are intertwined in mathematical activity in which conjectures and proofs are often produced and improved through the back-and-forth transition between attempts to prove and disprove. One aspect underexplored in the education literature is the connection between this activity and the construction by students of knowledge, such as mathematical concepts and theorems, that is new to them. This issue is significant to seeking a better integration of mathematical practice and content, emphasised in curricula in several countries. In this paper, we address this issue by exploring how students generate mathematical knowledge through discovering and handling refutations. We first explicate a model depicting the generation of mathematical knowledge through heuristic refutation (revising conjectures/proofs through discovering and addressing counterexamples) and draw on a model representing different types of abductive reasoning. We employed both models, together with the literature on the teachers’ role in orchestrating whole-class discussion, to analyse a series of classroom lessons involving secondary school students (aged 14–15 years, Grade 9). Our analysis uncovers the process by which the students discovered a counterexample invalidating their proof and then worked via creative abduction where a certain theorem was produced to cope with the counterexample. The paper highlights the roles played by the teacher in supporting the students’ work and the importance of careful task design. One implication is better insight into the form of activity in which students learn mathematical content while engaging in mathematical practice.
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Komatsu-Jones2021_Article_GeneratingMathematicalKnowledg
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Accepted/In Press date: 1 July 2021
e-pub ahead of print date: 24 August 2021
Additional Information:
Funding Information:
This study is supported by the Japan Society for the Promotion of Science (Nos 19H01668 and 18K18636).
Publisher Copyright:
© 2021, The Author(s).
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.
Keywords:
Abduction, Counterexample, Mathematical knowledge, Proof, Task design, Teacher role
Identifiers
Local EPrints ID: 453789
URI: http://eprints.soton.ac.uk/id/eprint/453789
ISSN: 0013-1954
PURE UUID: 971606fd-d168-42c9-8678-8c6abbad0f4c
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Date deposited: 24 Jan 2022 17:49
Last modified: 17 Mar 2024 12:51
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Author:
Kotaro Komatsu
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