Proof and Problem Solving at University Level

Abstract

Abstract: This paper will be concerned with undergraduate and graduate students' problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld's (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task.Keywords: Hungary, mathematics education, mathematics competition, Olympiads, international comparative mathematics education, problem solving, creativity, mathematically talented students.A Continuum of Tasks from Very Routine to Very Non-routineWhile what is a problem depends on what a solver knows, it is possible for most mathematics teachers to judge what is difficult for most students in a given class. Thus, we see mathematical tasks for a given class, such as a calculus class, on a continuum from those that are very routine to those that are genuinely difficult problems (Seiden, Seiden, Hauk, & Mason, 2000). At one end, there are very routine problems which mimic sample worked problems found in textbooks or lectures, except for minor changes in wording, notation, coefficients, constants, or functions that are incidental to the way the problems are solved. Such problems are often referred to as exercises (and might not be considered to be problems at all in the problem-solving literature).The vast majority of exercises in calculus textbooks are of this nature. Lithner (2004) distinguished three possible solution strategies for typical calculus textbook exercises: identification of similarities (IS), local plausible reasoning (LPR), and global plausible reasoning (GPR). In IS, one identifies surface features of the exercise and looks for a similar textbook situation - an example, a rule, a definition, a theorem. Without consideration of intrinsic mathematical properties, one simply copies the procedure of that situation. In LPR, one identifies a slightly similar textbook situation, but one in which a few local parts may differ. The solution strategy is to copy as much as possible from that similar situation, modifying local steps as needed. In GPR, the strategy is mainly based on analyzing and considering intrinsic mathematical properties of the exercise, and using these, a solution is constructed and supported by plausible reasoning. Lithner selected a textbook used in Sweden [Adams' Calculus: A Complete Course (5th ed.), Addison-Wesley], and worked through and classified solution strategies for 598 single-variable calculus exercises. He found 85% IS, 8% LPR, and 7% GPR. Furthermore, he concluded that "it is possible in about 70% of the exercises to base the solution not only on searching for similar situations, but on searching only the solved examples."Moving toward the middle of the continuum, there are moderately routine problems which, although not exactly like sample worked problems, can be solved by well-practiced methods, for example, ordinary related rates or change of variable integration problems in a calculus course.1 Moving further along the continuum, there are moderately non-routine problems, which are not very similar to problems that students have seen before and require known facts or skills to be combined in a slightly novel way, but are "straightforward" in not requiring, for example, the consideration of multiple sub-problems or novel insights. This is the type of problem we used on the non-routine test in our three studies of undergraduate students' calculus problem solving. One of those problems was: Find values of a and b so that the line 2x+3y=a is tangent to the graph of f (x) = bx2 at the point where x=3. …