Gifted ninth graders' notions of proof: Investigating parallels in approaches of mathematically gifted students and professional mathematicians

High school students normally encounter the study and use of formal proof in the context of Euclidean geometry. Professional mathematicians typically use an informal trial-and-error approach to a problem, guided by intuition, to arrive at the truth of an idea. Formal proof is pursued only after mathematicians are intuitively convinced about the truth of an idea. Is the use of intuition to arrive at the plausibility of a mathematical truth unique to the professional mathematician? How do mathematically gifted students form the truth of an idea? In this study, 4 mathematically gifted freshmen with no prior exposure to proof nor high school geometry were given the task of establishing the truth or falsity of a nonroutine geometry problem, sometimes referred to as "circumscribing a triangle" problem. This problem asks whether it is true that for every triangle there is a circle that passes through each of the vertices. This paper describes and interprets the processes used by the mathematically gifted students to establish truth and compares these processes to those used by professional mathematicians. All 4 students were able to think flexibly, as evidenced in their ability to reverse the direction of a mental process and arrive at the correct conclusion. This paper further validates the use of Krutetskiian constructs of flexibility and reversibility of mental processes in gifted education as characteristics of the mathematically gifted student.