Using Conceptual Blending to Analyze Student Proving.

We explore ways that university students handle proving statements that have the overall structure of a conditional implies a conditional, i.e., (p # q) ! (r # s). We structure our analysis using the theory of conceptual blending. We find conceptual blending useful for describing the creation of powerful new ideas necessary for proof construction as well as for describing the creation of blends that slow or hinder student efforts at proof construction.

[1]  S. Coulson,et al.  Blending basics , 2000 .

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

[3]  Michelle Zandieh,et al.  A Case of Distributed Cognition (or, Many Heads Make Light Work...) , 2005 .

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

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

[6]  Carolyn A. Maher,et al.  THE DEVELOPMENT OF THE IDEA OF MATHEMATICAL PROOF: A 5-YEAR CASE STUDY , 1996 .

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

[8]  Paul Sambre,et al.  Gilles Fauconnier & Mark Turner, " The way we think: conceptual blending and the mind's hidden complexities" , 2002 .

[9]  V. Durand-Guerrier,et al.  Which notion of implication is the right one? From logical considerations to a didactic perspective , 2003 .

[10]  R. Núñez.  Creating mathematical infinities: Metaphor, blending, and the beauty of transfinite cardinals , 2005 .

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

[12]  Daina Taimina,et al.  Experiencing Geometry: In Euclidean, Spherical and Hyperbolic Spaces , 2000 .

[13]  T. Dreyfus.  Why Johnny Can't Prove , 1999 .

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

[15]  Paul Cobb,et al.  Conducting Teaching Experiments in Collaboration With Teachers , 2000 .

[16]  Guershon Harel,et al.  The Development of Mathematical Induction as a Proof Scheme: A Model for DNR-Based Instruction , 2001 .