Conceptual blending: Student reasoning when proving “conditional implies conditional” statements

•We use conceptual blending as a tool to illustrate the evolution of student ideas.•Blending provides a coherent framework to address several aspects of proving.•Students create the key idea for the proof by running a blend of two diagrams.•Students struggle with two proving frames, SPF and CICF, for completing the blend.•Students’ proving processes are hindered by blending the premise and conclusion.

Conceptual blending describes how humans condense information, combining it in novel ways. The blending process may create global insight or new detailed connections, but it may also result in a loss of information, causing confusion. In this paper, we describe the proof writing process of a group of four students in a university geometry course proving a statement of the form conditional implies conditional, i.e., (p → q) ⇒ (r → s). We use blending theory to provide insight into three diverse questions relevant for proof writing: (1) Where do key ideas for proofs come from?, (2) How do students structure their proofs and combine those structures with their more intuitive ideas?, and (3) How are students reasoning when they fail to keep track of the implication structure of the statements that they are using? We also use blending theory to describe the evolution of the students’ proof writing process through four episodes each described by a primary blend.