Proving and Doing Proofs in High School Geometry Classes: What Is It That Is Going On for Students?

Herbst, Patricio; Brach, Catherine

doi:10.1207/s1532690xci2401_2

Cited by 92 publications

(68 citation statements)

References 38 publications

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“…In fact, in school geometry, proofs are often presented in a somewhat formalised form of Btwo-column format^. Yet, as Herbst and Brach (2006) show, such an approach does not necessarily support students through the creative reasoning processes that they need if they are to be able to build up reasoned arguments for themselves. On the contrary, teaching might usefully include a focus on the structural characteristics of deductive reasoning because these characteristics become increasingly important as students develop from elementary school mathematics to secondary school mathematics and beyond.…”

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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Section: Introductionmentioning

confidence: 99%

Students’ understanding of the structure of deductive proof

2016

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“…Teachers of geometry create work contexts in which students have the chance to experience, learn, and demonstrate knowledge of 'proof'. The notion of an instructional situation as a 'frame' (a set of norms regulating 270 who does what and when) for the exchange between work done and knowledge transacted was initially exemplified in what Herbst and associates called the 'doing proofs' situation (Herbst & Brach, 2006;Herbst, Chen, Weiss, González et al, 2009). That work of modelling classroom interaction as a system of norms produced the observation that many of the operations in the work of proving (e.g., those listed in the previous paragraph) are not accommodated in classroom work contexts where knowledge of proof is exchanged.…”

Section: The Theory Of Instructional Exchangesmentioning

confidence: 99%

Proof, Proving, and Teacher-Student Interaction: Theories and Contexts

Jones

Herbst

2012

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Abstract. This chapter focuses on the teachers' role in teaching proof and proving in the mathematics classroom. Within an over-arching theme of diversity (of countries, curricula, student age-levels, teachers' knowledge), the chapter presents a review of three carefully-selected theories: the theory of socio-mathematical norms, the theory of teaching with variation, and the theory of instructional exchanges. We argue that each theory starts by abstracting from observations of school mathematics classrooms. Each then uses those observations to probe into the teachers' rationality in order to understand what sustains those classroom contexts and how these might be changed. Here, we relate each theory to relevant research on the role of the teacher in the teaching and learning of proof and proving. Our review offers evidence and support for mathematics educators meeting the challenge of theorising about proof and proving in mathematics classrooms across diverse contexts worldwide.

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“…However, at the same time, this method of formalization divorces the formal act of proving from the construction of knowledge, and as a result, high school students are implicitly asked to focus only on the former at the expense of the latter. Thus, students can come to believe that proving is a purely formal activity, and may only become engaged in proving when explicitly asked to do so, and even then, only when certain prerequisites are given [29]. Students may or may not be concerned with the questions that mathematicians are concerned with, such as whether the proof is valid or convincing [29].…”

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confidence: 99%

“…Thus, students can come to believe that proving is a purely formal activity, and may only become engaged in proving when explicitly asked to do so, and even then, only when certain prerequisites are given [29]. Students may or may not be concerned with the questions that mathematicians are concerned with, such as whether the proof is valid or convincing [29]. While high-school geometry students may be encouraged to reason using diagrams and other less formal ideas by some instructors, other instructors may discourage these ideas, at least in the discussion of mathe-matical proof.…”

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confidence: 99%

What Do We Mean by Mathematical Proof?

CadwalladerOlsker¹

2011

Journal of Humanistic Mathematics

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Proving and Doing Proofs in High School Geometry Classes: What Is It That Is Going On for Students?

Cited by 92 publications

References 38 publications

Students’ understanding of the structure of deductive proof

Students’ understanding of the structure of deductive proof

Proof, Proving, and Teacher-Student Interaction: Theories and Contexts

What Do We Mean by Mathematical Proof?

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