ICMI Study 19: Proof and proving in mathematics education

Hanna, Gila; Villiers, Michael De

doi:10.1007/s11858-008-0073-4

Cited by 68 publications

(50 citation statements)

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“…One recommendation is the need for more explicit teaching of proof, both in school and university (Balacheff 2008;Hanna and de Villiers 2008;Hemmi 2008), with some (e.g., Stylianides and Stylianides 2007;Hanna and Barbeau 2008) arguing for it to be made a central topic in both institutions. A possible introduction to proof, suggested by Harel (2008) and Palla et al (2012) is proof by mathematical induction.…”

Section: Proof and Provingmentioning

confidence: 99%

Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

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This report1 from the ICME12 Survey Team 4 examines issues in the transition from secondary school to university mathematics with a particular focus on mathematical concepts and aspects of mathematical thinking. It comprises a survey of the recent research related to: calculus and analysis; the algebra of generalised arithmetic and abstract algebra; linear algebra; reasoning, argumentation and proof; and modelling, applications and applied mathematics. This revealed a multi-faceted web of cognitive, curricular and pedagogical issues both within and across the mathematical topics above. In addition we conducted an international survey of those engaged in teaching in university mathematics departments. Specifically, we aimed to elicit perspectives on: what topics are taught, and how, in the early parts of university-level mathematical studies; whether the transition should be smooth; student preparedness for university mathematics studies; and, what university departments do to assist those with limited preparedness. We present a summary of the survey results from 79 respondents from 21 countries.

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Section: Proof and Provingmentioning

confidence: 99%

Key Mathematical Concepts in the Transition from Secondary School to University

Thomas

Druck

Huillet

et al. 2015

The Proceedings of the 12th International Congress on Mathematical Education

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“…This conceptual distinction is important in order to discuss possible implications for the teaching and learning of mathematics. For example, Hanna and de Villiers (2008) define argumentation as ''a reasoned discourse that is not necessarily deductive but uses arguments of plausibility''. They discuss two different viewpoints in the mathematics education community, one group regarding argumentation and proof as a dichotomy and another group seeing both as two poles of a continuum.…”

Section: Proof Competencies In the Mathematics Classroommentioning

confidence: 99%

Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples

Reiss

Heinze

Renkl

et al. 2008

ZDM Mathematics Education

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We analyze heuristic worked-out examples as a tool for learning argumentation and proof. Their use in the mathematics classroom was motivated by findings on traditional worked-out examples, which turned out to be efficient for learning algorithmic problem solving. The basic idea of heuristic worked-out examples is that they encourage explorative processes and thus reflect explicitly different phases while performing a proof. We tested the hypotheses that teaching with heuristic examples is more effective than usual classroom instruction in an experimental classroom study with 243 grade 8 students. The results suggest that heuristic worked-out examples were more effective than the usual mathematics instruction. In particular, students with an insufficient understanding of proof were able to benefit from this learning environment.

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“…The teaching and learning of proof is recognised internationally as a key component of mathematics education and thus of mathematics curricula (Hanna & de Villiers, 2008). Yet it remains the case that students at the secondary school level (and beyond) experience difficulties in understanding proof in mathematics in general, and in geometry in particular.…”

Section: Introductionmentioning

confidence: 99%

Students’ understanding of the structure of deductive proof

2016

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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: Prestructural, 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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ICMI Study 19: Proof and proving in mathematics education

Cited by 68 publications

References 0 publications

Key Mathematical Concepts in the Transition from Secondary School to University

Key Mathematical Concepts in the Transition from Secondary School to University

Reasoning and proof in geometry: effects of a learning environment based on heuristic worked-out examples

Students’ understanding of the structure of deductive proof

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