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Conceptualizing reasoning-and-proving opportunities in textbook expositions: Cases from secondary calculus
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Örebro universitet, Institutionen för naturvetenskap och teknik, Sweden.ORCID iD: 0000-0002-8520-8200
2017 (English)In: Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10, February 1-5, 2017) / [ed] Dooley, T., & Gueudet, G., Dublin, Ireland: European Society for Research in Mathematics Education , 2017, p. 91-98Conference paper, Published paper (Refereed)
Abstract [en]

Several recent textbook studies focus on opportunities to learn reasoning-and-proving. They typically investigate the extent to which justifications are general proofs and what opportunities exist for learning important elements of mathematical reasoning. In this paper, I discuss how a particular analytical framework for this might be refined. Based on an in-depth analysis of certain textbook passages in upper secondary calculus textbooks, I make an account for analytical issues encountered during this process and identify aspects of reasoning-and-proving in textbooks that might be missed unless the framework is refined. Among them there are characterizations of generality, use of different representations, logical and mathematical structure, and ordering of material and student activities. Finally, implications beyond textbook research are discussed.

Place, publisher, year, edition, pages
Dublin, Ireland: European Society for Research in Mathematics Education , 2017. p. 91-98
Keywords [en]
Reasoning-and-proving, mathematics textbook, upper secondary calculus
National Category
Other Mathematics Educational Sciences
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:mdh:diva-55924ISBN: 978-1-873769-73-7 (print)OAI: oai:DiVA.org:mdh-55924DiVA, id: diva2:1595055
Conference
Tenth Congress of the European Society for Research in Mathematics Education (CERME10), Dublin, Ireland, February 1-5, 2017
Available from: 2021-09-17 Created: 2021-09-17 Last updated: 2021-09-20Bibliographically approved
In thesis
1. Proof-related reasoning in upper secondary mathematics textbooks: Characteristics, comparisons, and conceptualizations
Open this publication in new window or tab >>Proof-related reasoning in upper secondary mathematics textbooks: Characteristics, comparisons, and conceptualizations
2021 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Proofs and proving are difficult to learn and difficult to teach. A common problem is that many students use specific examples as evidence for general statements. Difficulties with proofs are also part of the transition problems that exist between secondary and tertiary schooling in mathematics. As mathematics teaching often follows a textbook, the design of textbooks has been pointed out as one possible cause of the problems, and international textbook research suggests that proofs often have only a marginal place in textbooks.

This thesis focuses on proofs and proving in upper secondary mathematics textbooks. It also addresses theoretical and methodological questions about what marks an opportunity to develop proving competence, and which properties of such opportunities are relevant to investigate and characterize. The thesis is based on data from four Swedish and Finnish textbook series for upper secondary school, and focuses on sections on logarithms, primitive functions, definite integrals, and combinatorics. It examines how addressed mathematical principles are justified, and whether the textbooks’ exercises offer opportunities to develop proof-related skills such as formulating and investigating hypotheses, developing and evaluating arguments, identifying and correcting errors, and finding counterexamples.

The results show that just over half of the mathematical principles addressed in the analyzed textbook material are justified, and that only half of the justifications are general proofs. Few exercises are proof-related (10%), and those that include reasoning about general cases even fewer. General proofs are more common in the Finnish books, but proof-related tasks are more common and of a more varied nature in the Swedish ones. The most common form of proofs are direct derivations of calculation formulas, while reasoning about existence and uniqueness is unusual, as are contrapositive proofs and proofs by contradiction.

Based on the results, explicit suggestions are offered as to what teaching can pay more attention to. For the analysis and design of proof-related activities, a framework consisting of four main categories is proposed: develop a statement, investigate a statement, develop an argument, and investigate an argument. Several properties that such activities may have, regardless of which category they belong to, are discussed. Finally, three areas for future research are suggested: how worked examples can support students’ learning of proof, how textbooks can be designed to stimulate formulation as well as the formal proving of hypotheses, and mapping of differences regarding proof between upper secondary and university textbooks.

Place, publisher, year, edition, pages
Västerås: Mälardalen University, 2021
Series
Mälardalen University Press Dissertations, ISSN 1651-4238 ; 346
Keywords
mathematics textbook, upper secondary school, proof-related reasoning
National Category
Didactics
Research subject
Mathematics/Applied Mathematics
Identifiers
urn:nbn:se:mdh:diva-55922 (URN)978-91-7485-524-1 (ISBN)
Public defence
2021-11-12, Delta samt digitalt via ZOOM, Mälardalens högskola, Västerås, 09:15 (English)
Opponent
Supervisors
Available from: 2021-09-20 Created: 2021-09-20 Last updated: 2021-10-22Bibliographically approved

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Citation style
  • apa
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