Some pedagogical aspects of proof

Hanna, Gila

doi:10.1007/bf01809605

Cited by 223 publications

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“…Ed. (2015) (Dreyfus 1990, p. 126), it also serves as a tool for learning mathematics (Hanna 1990(Hanna , 1995Hersh 1993 MAA 2000), has emphasized this goal and has recommended that proof should be a part of students' mathematical experiences-especially of those students in mathematics-related programs. The report states that Bstudents should understand and appreciate the core of mathematical culture: the value and validity of careful reasoning, precise definition and close argument^(p. 6).…”

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“…3 In line with Knuth (2002a, b), we found that, while these roles were proposed in regard to the role of proof in the discipline of mathematics, they provide a useful framework for thinking about school mathematics in general, and students' conceptions of mathematical proof in particular. Below we explicate three of these roles as a framework for our study.The first role of proof-verification as a means to demonstrate the truth of a mathematical conjecture or an assertion-is the most widely held view of proof (Hanna 1990). Indeed, most people who have some understanding of the nature of mathematics as a science expect that for each correct conjecture there is a Bsequence of logical transformations moving from hypothesis to conclusion^(DeVilliers 1990) whose role is to guarantee the truth of the conjecture in an absolute and definite manner.…”

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confidence: 99%

“…, it also serves as a tool for learning mathematics (Hanna 1990(Hanna , 1995Hersh 1993). As such, mathematicians and mathematics educators have called for the learning of proof to become a central goal of mathematics learning and instruction (NCTM 2000;RAND Mathematics Study Panel 2003).…”

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Undergraduate Students’ Understanding of Proof: Relationships Between Proof Conceptions, Beliefs, and Classroom Experiences with Learning Proof

Stylianou

Blanton

Rotou

2015

Int. J. Res. Undergrad. Math. Ed.

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This article presents the results from a study of 535 early undergraduate students at six universities that was designed to describe their views of the meaning of proof and how these views relate to their attitudes and beliefs towards proof and their classroom experiences with learning proof. Results show that early undergraduate students have difficulty with mathematical proof. In particular, the study showed that students' proof choices were strongly influenced by surface characteristics of the tasks. However, a large number of students appear to appreciate and acknowledge the rigor and central role of deductive proof in mathematics despite the difficulties they may face in producing proofs. Further, the study showed a strong positive relationship between students' beliefs about the role of proof and themselves as learners of proof, but weak relationship between proof ability and self-reported experiences with learning proof. Keywords Mathematical proof . Deduction . AttitudesArguments that the Bessence of mathematics lies in proofs^ (Ross 1998, p. 2) and that Bproof is not a thing separable from mathematics…. [but] is an essential component of doing, communicating, and recording mathematics^ (Schoenfeld 1994, p. 76) reinforce the centrality of proof in mathematical thinking. Moreover, not only does the act of proof Bdistinguish mathematical behavior from scientific behavior in other disciplinesÎ nt. J. Res. Undergrad. Math. Ed. (2015) (Dreyfus 1990, p. 126), it also serves as a tool for learning mathematics (Hanna 1990(Hanna , 1995Hersh 1993 MAA 2000), has emphasized this goal and has recommended that proof should be a part of students' mathematical experiences-especially of those students in mathematics-related programs. The report states that Bstudents should understand and appreciate the core of mathematical culture: the value and validity of careful reasoning, precise definition and close argument^(p. 6). Yet, research in mathematics education indicates that most students, in particular at the secondary school level, face substantial difficulties with proof (see Harel and Sowder 2007;Stylianou et al. 2009 for reviews). There also is evidence that college students face similar difficulties with proof as their high school counterparts. Most notably, Harel and Sowder (1998) showed that college students focus their attention on the format of the proof rather than the content. However, we know little about the conceptions of students as they are beginning to immerse themselves in college mathematics. Perhaps more importantly, we have not yet explored the relationship between students' understanding of proof and their beliefs towards proof. Indeed, we now know that mathematics performance in general, and the reading and writing of proofs in mathematics in particular, is a complex one that depends on a wide expanse of beliefs, knowledge, and cognitive skills and that is uniquely shaped by the realm in which learning occurs (Heinze et al. 2005). It is not at all clear, however, which of these beli...

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Undergraduate Students’ Understanding of Proof: Relationships Between Proof Conceptions, Beliefs, and Classroom Experiences with Learning Proof

Stylianou

Blanton

Rotou

2015

Int. J. Res. Undergrad. Math. Ed.

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“…Several roles of proof have been identified and discussed in the literature. Among these are to verify that a statement is true, to explain why a statement is true, to communicate mathematical knowledge, to discover or create new mathematics and to systematize statements into an axiomatic system (deVilliers, 1990;Hanna, 1990). To more closely align with disciplinary practices, proof in school should encompass all these roles.…”

Section: Subject Matter Knowledge For Teaching Proofmentioning

confidence: 99%

“…However, research shows that teachers often have a limited view of proof as merely a tool for verification and reserve proof as a topic of study for a select group of students (Knuth, 2002). To capitalize on proof as a tool for learning, teachers need to better understand how proof can serve as a vehicle for explaining why something is true and providing insight into underlying mathematical concepts (Hanna, 1990).…”

Section: Subject Matter Knowledge For Teaching Proofmentioning

confidence: 99%

Investigating Mathematical Knowledge for Teaching Proof in Professional Development

Lesseig

2016

IJRES

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Research documenting teachers' fragile understanding of proof and how it is advanced suggests that enhancing the role of proof in school mathematics will demand substantial teacher learning. To date, there is little research detailing what mathematical knowledge might support the teaching of proof or how professional development (PD) might afford such learning. This paper advances a framework for Mathematical Knowledge for Teaching Proof (MKT for Proof) that specifies proof knowledge across subject matter and pedagogical domains. To explore the utility of the framework, this paper examines data from four different PD settings in which teachers and leaders worked on the same proof-related task. Analysis of discussions across these four settings revealed similarities and differences in knowledge of proof made available in each setting. These findings provide a means to detail the MKT for Proof framework and demonstrate its usefulness as a tool for designing, analyzing and evaluating proof experiences in PD.

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Positive Behaviors, Problem Behaviors, and Resiliency in Adolescence

Perkins

Borden

2003

Handbook of Psychology

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This chapter examines risk and protective factors within adolescents and their environments and the manifestation of those factors in terms of risk behaviors or resiliency. A detailed exploration of four major risk behaviors in adolescents (i.e., teenage sexual activity, alcohol and/or substance use/abuse, delinquency and antisocial behavior, and school failure) is presented along with the risk factors found to be associated with them. Evidence is presented with regard to the co‐morbidity of those risk behaviors and risk factors. Understanding risk behaviors forms a foundation to understand the influence of risk behaviors on the developmental trajectory of young people. Moreover, specific attention to major risk behaviors also offers a basis for understanding the importance of multiple ecological influences, multiple constitutional influences, and the interactions that occur between them. The chapter then investigates the beginning of resiliency research and follows the field's development through to its current status and theoretical understanding of resiliency. Empirical findings related to protective factors and their interaction with risk factors are also presented. These interactions are critical to the overall development of adolescents, both in positive and negative directions. Understanding why so many are able to overcome adversity provides us with some direction in intervention efforts.

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Some pedagogical aspects of proof

Cited by 223 publications

References 4 publications

Undergraduate Students’ Understanding of Proof: Relationships Between Proof Conceptions, Beliefs, and Classroom Experiences with Learning Proof

Undergraduate Students’ Understanding of Proof: Relationships Between Proof Conceptions, Beliefs, and Classroom Experiences with Learning Proof

Investigating Mathematical Knowledge for Teaching Proof in Professional Development

Positive Behaviors, Problem Behaviors, and Resiliency in Adolescence

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