Problem Solving and its elements in forming proof

Abstract: The character of the mathematics education traditions on problem solving and proof are compared, and aspects of problem solving that occur in the processes of forming a proof, which are not well represented in the literature, are portrayed.Keywords: heuristics; problem solving; proof(ProQuest: . denotes formulae omitted.)IntroductionMathematics educators tend to compartmentalize the domains 'problem solving' and 'proof and proving'. This detachment seems somehow artificial as both deal with aspects of producing mathematical argumentation. However, problem solving tends to emphasize the thought processes in furthering on-going work; in contrast the proof tradition is concerned more in evaluating the soundness of the complete output. In this paper, we shall respect the distinction made between problem solving and proof, but at the same time we shall discuss issues that are common.We use the words 'culture', 'tradition' and 'agenda' synonymously for general views broadly adopted by the research community on any given educational perspective. Both the problem-solving tradition and the proof tradition are diverse, so we restrict ourselves to particular stances, mostly attending to the upper school and university level. For problem solving, the subject is taken for it's own sake; hence the full weight of selfconscious decision-making becomes the scope of investigation. For proof, we distinguish the case where the practitioner possesses and implements the requisite mathematical tools to fully articulate the proof from the case where he/she does not. The various types of tools needed will be discussed, especially when the context lies in a mathematical theory currently been taught: then tools are adapted and appropriated from techniques that the theory avails. However, such tools have to be designed and then coordinated in the mind, so within the processes in obtaining a proof it is evident that substantial elements of problem solving must occur. The main focus of the paper is to give a preliminary account of these elements.In the next section, we shall present a short, rather personal, description of problem solving. Largely supposing that the reader is familiar with the core principles laid out by Polya, it discusses more practical issues like the role of the teacher, implementation and assessment. The section that follows deals with the problem-solving component in proof making. Here no attempt has been made to give a coherent picture of the proof tradition; one reason is that proof and proving are, as an educational domain, particularly prone to contrasting standpoints. Rather we limit our attention to those facets of proof that differ from the problem-solving tradition but at the same time retain some problem-solving elements. The choice of papers referred to is made with this in mind. The discourse will not be unidirectional; some points made could be read as if the culture of proof is supporting problem-solving activity. The extended example given in the penultimate section illustrates this, as well as other matters. The epilogue, in part, raises the question how well the problem-solving tradition (as it stands presently) is equipped to cover the problem-solving elements in formulating proof.On the problem-solving tradition and allied practical issuesThe perspective of problem solving has a relatively compact core of ideas, mainly centered on heuristics, meta-cognition including executive control, accessing and applying the knowledge base, and identifying patterns of modes of thinking as students' work progresses, following the pioneering work of Polya (e.g., Polya, 1973) and later by Schoenfeld (e.g., Schoenfeld, 1985). However, problem solving, as a domain of mathematical activity is very general; it concerns the student's engagement on any mathematical task that is not judged procedural or the student does not have an initial overall idea how to proceed in solving the task. …