Two proving strategies of highly successful mathematics majors

• We examined the proof-writing behavior of six highly successful mathematics majors from a problem-solving perspective.
• We observed that successful mathematics majors constructed proofs in two qualitatively different ways.
• The identification of these two strategies adds to what is known about successful proving strategies.

We examined the proof-writing behaviors of six highly successful mathematics majors on novel proving tasks in calculus. We found two approaches that these students used to write proofs, which we termed the targeted strategy and the shotgun strategy. When using a targeted strategy students would develop a strong understanding of the statement they were proving, choose a plan based on this understanding, develop a graphical argument for why the statement is true, and formalize this graphical argument into a proof. When using a shotgun strategy, students would begin trying different proof plans immediately after reading the statement and would abandon a plan at the first sign of difficulty. The identification of these two strategies adds to the literature on proving by informing how elements of existing problem-solving models interrelate.