Proofs and hypotheses

Abstract

On the basis of an analysis of common features and differences between general statements in every day situations, in physics and in mathematics the paper proposes a didactical approach to proof. It is centred around the idea that inventing hypotheses and testing their consequences is more productive for the understanding of the epistemological nature of proof than forming elaborate chains of deductions. Inventing hypotheses is important within and outside of mathematics. In this approach proving and forming models get in close contact. The idea is exemplified by a teaching unit on the angle sum theorem in Euclidean geometry.

Future teachers’ professional knowledge on argumentation and proof: a case study from universities in three countries

“…Part of this experience should address the plea of Blum and Kirsch (1991), also echoed by Holland (1996), ''for doing mathematics on a pre-formal level'' and hence providing all students with the opportunity to engage deeply with ''pre-formal proofs that are as obvious and natural as possible especially for the mathematically less experienced learner'' (p. 186). However, the findings of the present study in keeping with Jones (2000) and Jahnke (2007) also suggest that even future teachers with strong mathematical backgrounds from tertiary studies are not necessarily experiencing proof in such a manner that they can convey a complete image of proving at the lower secondary level.…”

“…The lure of empirically based validation for secondary students rather than a deductive argument has been described widely (Healy & Hoyles, 2007;Heinze & Kwak, 2002). Jahnke (2007) notes that this preference is found for high performing students and hence he concludes that ''the usual teaching of mathematics is not successful in explaining the epistemological meaning of proof'' (p. 80). Hanna, de Villiers and others (2008) in the ICMI study discussion document ask the question: ''How can we design opportunities for student teachers to acquire the knowledge (skills, understandings and dispositions) necessary to provide effective instruction about proof and proving?''…”

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

“…Another reason for classifying the students' arguments as invalid was that they used numbers to verify the statements instead of proving them. In fact, students at various levels, even undergraduate students, think that numerical values and examples are more convincing than mathematical proofs (Brown, 2014;Jahnke, 2007;Martin & Harel, 1989;Segal, 2000;Stylianides & Stylianides, 2009). However, Weber (2010) stated that most of the mathematics majors who completed a transition to proof course did not accept empirical arguments as proof.…”

Proofs as bearers of mathematical knowledge

“…These areas of emphasis are apparent from the specific issues addressed in much of this recent research. The following is by no means an exhaustive list of issues, but is fairly representative: the epistemological aspects of proof (Balacheff, 2004;Hanna, 1997); the cognitive aspects of proof (Tall, 1998); the role of intuition and schemata in proving (Fischbein, 1982(Fischbein, , 1999; the relationship between proving and reasoning (Yackel & Hanna 2003;Maher & Martino 1996); the usefulness of heuristics for the teaching of proof (Reiss & Renkl, 2002); the emphasis on the logical structures of proofs in teaching at the tertiary level (Selden & Selden, 1995); proof as explanation and justification (Hanna, 1990(Hanna, , 2000Sowder & Harel 2003); proof and hypotheses (Jahnke, 2007); curricular issues (Hoyles, 1997); proof in the context of dynamic software (Jones, Gutiérrez & Mariotti 2000;Moreno & Sriraman 2005); the analysis of mathematical arguments produced by students (Inglis, Mejia-Ramos & Simpson 2007); the relationship between argumentation and proof (Pedemonte, 2007). Understandably, the empirical classroom research on the teaching of proof has focused upon students' difficulties with learning proof and on the design of effective teachers' interventions (see the survey of research in the last 30 years in Mariotti 2006).…”