Proofs as Bearers of Mathematical Knowledge

Hanna, Gila; Barbeau, Edward J.

doi:10.1007/978-1-4419-0576-5_7

Cited by 25 publications

(21 citation statements)

References 19 publications

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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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“…Thus, the result of a deduction-a 'making explicit of a special application area' of a more general premise (the rule)-will not be considered as a discovery in this article. Hanna and Barbeau (2008) described how a (chain of) deduction(s) could be the bearer of mathematical methods and strategies. The interpretative process of recognising strategies and methods within these deductions cannot be a deductive process because this would imply that the interpretation of any statement or text would not be ambiguous, but this is of course not possible (cf.…”

Section: Deductionmentioning

confidence: 99%

Abduction—A logical view for investigating and initiating processes of discovering mathematical coherences

Meyer

2010

Educ Stud Math

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According to theoretical concepts like constructivism, each learner has to build up knowledge on his or her own. The learner creates hypotheses in order to explain 'facts'. Hypotheses do not guarantee certainty. They have to be verified. In this article, a theoretical framework will be presented which can help to understand and analyse the processes of creativity and reasoning. Peirce's theory of abduction plays a decisive role in this framework. He elaborated abduction as the third elementary inference, besides deduction and induction. While abduction will be used to describe and analyse the process of discovering an explanatory hypothesis, deduction, induction and their combinations will be used to describe the different processes of verifying knowledge. Theoretical and empirical ways of discovering and verifying mathematical coherences will be presented and illustrated by fictive and empirical examples.

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“…From the perspective of Lave and Wenger (1991) proof can be seen as an artefact with several important functions in the mathematical practice (e.g., Weber, 2002;Hanna & Barbeau, 2008). Lave and Wenger introduce the concept of transparency of the artefacts in connection to technology but in this study it is used for describing proof as a symbolic and intellectual artefact in the teaching and learning of mathematics.…”

Section: Employing the Community Approach Into Researchmentioning

confidence: 99%

Communities in university mathematics

Biza

Jaworski

Hemmi

2014

Research in Mathematics Education

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Communities in University MathematicsThis paper regards communities of learners and teachers that are formed, develop and interact in university mathematics environments through the theoretical lenses of the work of Lave and Wenger (1991) and Wenger (1998) on the Community of Practice. In this perspective learning is drawn on the participation in a community. In addition, when inquiry is considered as a fundamental way of participation, the community becomes a Community of Inquiry. The theoretical underpinnings of the above approaches with examples of their application in research in university mathematics education are discussed in the sections of this paper. The paper concludes with a critical reflection on the theorising of the role of communities at university level teaching and learning as well as ways forward for future research.Keywords: community of practice; community of inquiry; legitimate peripheral participation; identity, critical alignment, university mathematics education IntroductionExperience in university mathematics teaching indicates that there is no clear consensus between university teachers 1 and students on the meaning and the value of mathematics (e.g. Solomon, 2006). This observation attracted the interest of mathematics education researchers to investigate the takes on the meaning and values of mathematics in different communities -such as researcher mathematicians, teachers of mathematics, undergraduate and postgraduate students -that are involved in practices within university especially in relation to the teaching and learning (e.g., Burton, 2004;Herzig, 2002;Solomon, 2007). In this endeavour, research has been drawn on the theoretical 1 We use (university) teacher to describe all those involved in the teaching of mathematics at university level. We describe other identities with specific characterisations, such as researcher mathematician or mathematics educator, when it is necessary. construct of the Community of Practice (henceforth, CoP) based on the work of Lave and Wenger (1991) and Wenger (1998) and the Community of Inquiry (henceforth, CoIParticularly within an developmental environment and based on the work of Jaworski, Goodchild, and many others (e.g., Goodchild, Fuglestad and Jaworski, 2013).Our aim in this paper is to stress and give more insight into the theorisation of the role of these communities in the learning and teaching of mathematics at university level and to take this theorisation forward in future research. Our point is that mathematical practices at university level are distinguished from those at secondary or primary level for reasons related to the mathematical content, the teachers and the students involved. At university level the mathematical theory becomes a language of communication with very specific and rigorous rules and processes (such as theorems, In the following sections we present the main theoretical underpinnings of the CoP and CoI and we exemplify how their theoretical constructs have been used in university mathematics education researc...

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Proofs as Bearers of Mathematical Knowledge

Cited by 25 publications

References 19 publications

Key Mathematical Concepts in the Transition from Secondary School to University

Key Mathematical Concepts in the Transition from Secondary School to University

Abduction—A logical view for investigating and initiating processes of discovering mathematical coherences

Communities in university mathematics

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