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On a generality framework for proving tasks
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Örebro Univ, Örebro, Sweden.ORCID iD: 0000-0002-8520-8200
2015 (English)In: PROCEEDINGS OF THE NINTH CONFERENCE OF THE EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION (CERME9) / [ed] Krainer, K Vondrova, N, CHARLES UNIV, FAC EDUC , 2015, p. 86-92Conference paper, Published paper (Refereed)
Abstract [en]

In this paper I present an analytic framework for generality in textbook proving tasks that involve functions. The framework is discussed in relation to results obtained when analysing tasks in integral calculus. The results show that the frameworks' categories are easily distinguishable if the functions are explicitly described. The results are also promising regarding the possibility to clarify differences between textbooks. The analysed sections exemplify that there is not necessarily a correlation between the number of general proving tasks and the opportunities for students to engage in reasoning about arbitrary functions. Limitations and possible refinements of the framework are also discussed.

Place, publisher, year, edition, pages
CHARLES UNIV, FAC EDUC , 2015. p. 86-92
Keywords [en]
Mathematical proof, mathematics textbook, upper secondary school, undergraduate mathematics, integral calculus
National Category
Educational Sciences
Identifiers
URN: urn:nbn:se:mdh:diva-44545ISI: 000466853900007ISBN: 9788072908448 (print)OAI: oai:DiVA.org:mdh-44545DiVA, id: diva2:1330010
Conference
NINTH CONFERENCE OF THE EUROPEAN SOCIETY FOR RESEARCH IN MATHEMATICS EDUCATION (CERME9)
Available from: 2019-06-25 Created: 2019-06-25 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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Bergwall, Andreas

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