Secondary School Students’ Understanding of Mathematical Induction: Structural Characteristics and the Process of Proof Construction

Palla, Marina; Potari, Despina; Spyrou, Panagiotis

doi:10.1007/s10763-011-9311-2

Cited by 14 publications

(9 citation statements)

References 13 publications

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“…A possible introduction to proof, suggested by Harel (2008) and Palla et al (2012) is proof by mathematical induction. However, they propose that it should be introduced slowly, building on students' own pre-existing epistemological resources (Solomon 2006) valuing both ways of understanding and thinking (Harel 2008), and distinguishing between proof schemes and proofs.…”

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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show abstract

“…In the proof literature, many studies have investigated students' ability in conducting proof; their reasoning skills at different levels of education such as elementary, secondary, and undergraduate (Mejia-Ramos & Inglis, 2009;Palla, Potari & Spyrou, 2011;Stylianides, Stylianides & Philippou, 2004;Weber, 2001); and difficulties that they encounter while proving (Alcock & Weber, 2010;Harel & Sowder, 1998;Ko & Knuth, 2009;Moore, 1994;Weber, 2001). According to results of these studies, the ability in conducting proof depends on issues such as the education level of participants, the structure and content of questions, and the context in which they were taught.…”

Section: Literature Reviewmentioning

confidence: 99%

An Investigation of Pre-service Middle School Mathematics Teachers’ Ability to Conduct Valid Proofs, Methods Used, and Reasons for Invalid Arguments

Demiray

Işıksal-Bostan

2015

Int J of Sci and Math Educ

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The purposes of this study are to investigate Turkish pre-service middle school mathematics teachers' ability in conducting valid proofs for statements regarding numbers and algebra in terms of their year of enrollment in a teacher education program, to determine the proof methods used in their valid proofs, and to examine the reasons for their invalid arguments. A proof questionnaire containing three proof statements was administered to 115 pre-service middle school mathematics teachers in a large state university in Ankara, Turkey. The results showed that more than half of the pre-service teachers were able to conduct valid proofs for the given statements. In terms of year levels, it was seen that the seniors were the least successful group in conducting valid proofs for each statement. When pre-service teachers' valid proofs were analyzed, it was concluded that mathematical induction and direct proof were the mostly used methods for the given statements. When pre-service teachers' invalid arguments were analyzed, it was seen that "inserting numbers to verify the given statement" and "rewriting the givens in the statement" were the common reasons for stating invalid arguments.

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“…Usually, these are students that have taken a program to become school teachers in mathematics, or mathematics majors who have later on taken interest in teaching at high schools. Their capacities relating to proof are, usually, either authoritative, or empirical (Almeida 2001;Basturk 2010;Harel 2002;Muis 2004;Palla et al 2012). In the authoritative case, students are unable to decide for themselves whether or not something is a proof.…”

Section: Ordinary People and Proofmentioning

confidence: 99%

Accessibility of reformulated mathematical content

Buijsman

2016

Synthese

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I challenge a claim that seems to be made when nominalists offer reformulations of the content of mathematical beliefs, namely that these reformulations are accessible to everyone. By doing so, I argue that these theories cannot account for the mathematical knowledge that ordinary people have. In the first part of the paper I look at reformulations that employ the concept of proof, such as those of Mary Leng and Ottavio Bueno. I argue that ordinary people don't have many beliefs about proofs, and that they are not in a position to acquire knowledge about proofs autonomously. The second part of the paper is concerned with other reformulations of content, such as those of Hartry Field and Stephen Yablo. There too, the problem is that people are not able to acquire knowledge of the reformulated propositions autonomously. Ordinary people simply do not have beliefs with the kind of content that the nominalists need, for their theory to account for the mathematical knowledge of ordinary people. All in all then, the conclusion is that a large number of theories that suggest reformulations of mathematical content yield contents that are inaccessible for most people. Thus, these theories are limited, in that they cannot account for the mathematical knowledge of ordinary people.

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Secondary School Students’ Understanding of Mathematical Induction: Structural Characteristics and the Process of Proof Construction

Cited by 14 publications

References 13 publications

Key Mathematical Concepts in the Transition from Secondary School to University

Key Mathematical Concepts in the Transition from Secondary School to University

An Investigation of Pre-service Middle School Mathematics Teachers’ Ability to Conduct Valid Proofs, Methods Used, and Reasons for Invalid Arguments

Accessibility of reformulated mathematical content

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