Generating Mathematical Knowledge in the Classroom through Proof, Refutation, and Abductive Reasoning

Autor/inn/en	Komatsu, Kotaro; Jones, Keith
Titel	Generating Mathematical Knowledge in the Classroom through Proof, Refutation, and Abductive Reasoning
Quelle	In: Educational Studies in Mathematics, 109 (2022) 3, S.567-591 (25 Seiten)Infoseite zur Zeitschrift PDF als Volltext Verfügbarkeit
Zusatzinformation	ORCID (Komatsu, Kotaro) ORCID (Jones, Keith)
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0013-1954
DOI	10.1007/s10649-021-10086-5
Schlagwörter	Mathematics Instruction; Validity; Mathematical Logic; Knowledge Level; Teaching Methods; Teacher Role; Secondary School Students; Grade 9; Secondary School Mathematics; Inferences + Suchen Sie Ihr Suchwort? Mathematics lessons; Mathematikunterricht; Gültigkeit; Mathematical logics; Mathematische Logik; Wissensbasis; Teaching method; Lehrmethode; Unterrichtsmethode; Lehrerrolle; Sekundarschüler; School year 09; 9. Schuljahr; Schuljahr 09; Inference; Inferenz
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. (As Provided).
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Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2024/1/01