Referential and syntactic approaches to proving: case studies from a transition-to-proof course

The goal of this paper is to increase our understanding of different approaches to proving in advanced mathematics. We present two case studies from an interview-based investigation in which students were asked to complete proof-related tasks. The first student consistently took what we call a referential approach toward these tasks, examining examples of the objects to which the mathematical statements referred, and using these to guide reasoning. The second consistently took what we call a syntactic approach toward these tasks, working logically with definitions and proof structures without reference to examples. Both students made substantial progress on each of the tasks, but they exhibited different strengths and experienced different difficulties. In this paper we: demonstrate consistency in these students' approaches across a range of tasks, examine the different strengths and difficulties associated with their approaches to proving, and consider the pedagogical issues raised by these apparent student preferences for reasoning in certain ways.

[1]  EFRAIM FISCHBEIN.  Intuition and Proof * , .

[2]  Annie Selden,et al.  Validations of proofs considered as texts: Can undergraduates tell whether an argument proves a theorem? , 2003 .

[3]  Keith Weber,et al.  Student difficulty in constructing proofs: The need for strategic knowledge , 2001 .

[4]  Shlomo Vinner,et al.  The Role of Definitions in the Teaching and Learning of Mathematics , 2002 .

[5]  Phillip A. Griffiths.  Mathematics at the Turn of the Millennium , 2000, Am. Math. Mon..

[6]  Alan H. Schoenfeld,et al.  Mathematical Problem Solving , 1985 .

[7]  G. Lakoff,et al.  Where Mathematics Comes From , 2000 .

[8]  Uses of Example Objects in Proving. , 2004 .

[9]  L. Alcock,et al.  Semantic and Syntactic Proof Productions , 2004 .

[10]  Angel M. Recio,et al.  Institutional and personal meanings of mathematical proof , 2001 .

[11]  A. Sfard.  On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin , 1991 .

[12]  J. Baker.  Students' Difficulties with Proof by Mathematical Induction. , 1996 .

[13]  David Tall.  Metaphorical Objects in Advanced Mathematical Thinking , 1997, Int. J. Comput. Math. Learn..

[14]  Robert C. Moore.  Making the transition to formal proof , 1994 .

[15]  Juan Pablo Mejía-Ramos,et al.  Modelling mathematical argumentation: the importance of qualification , 2007 .

[16]  L. Alcock,et al.  Convergence of sequences and series: Interactions between visual reasoning and the learner's beliefs about their own role , 2004 .

[17]  Lara Alcock,et al.  Convergence of Sequences and Series 2: Interactions Between Nonvisual Reasoning and the Learner’s Beliefs about their own Role , 2004 .

[18]  D. Tall,et al.  Student constructions of formal theory: giving and extracting meaning , 1999 .

[19]  G A Miller,et al.  A MATHEMATICAL PROOF. , 1931, Science.

[20]  D. Tall,et al.  Building Formal Mathematics on Visual Imagery: A Case Study and a Theory. , 2002 .

[21]  M. Raman.  Key Ideas: What are they and how can they help us understand how people view proof? , 2003 .

[22]  Annie Selden,et al.  Unpacking the logic of mathematical statements , 1995 .

[23]  Lara Alcock,et al.  Definitions : dealing with categories mathematically. , 2002 .

[24]  Lara Alcock.  Mathematicians' perspectives on the teaching and learning of proof , 2010 .

[25]  Annie Selden,et al.  Making the Connection: Overcoming Students' Difficulties in Learning to Understand and Construct Proofs , 2007 .

[26]  W. Thurston.  On Proof and Progress in Mathematics , 1994, math/9404236.