Mathematics Problems and Thinking Mathematically in Undergraduate Mathematics

Abstract

Mathematics is much more than a formal system of procedures and formulae; it is also a way of thinking built on creativity, precision, reasoning, and representation. I present a model for framing the process of doing mathematics within a constructivist ideology, and I discuss two fundamental parts to this process: mathematical thinking and the design of undergraduate mathematics problems. I highlight the mathematical content and the structuredness of the problem statement and I explain why the initial work of re-formulating an ill-structured problem is especially important in learning mathematics as a mental activity. Furthermore, I propose three fundamental processes of mathematical thinking: Discovery (acts of creation), Structuring (acts of arranging), and Justification (acts of reflection). In the empirical portion of the study, pairs of university students, initially characterized by certain affective variables, were observed working on carefully constructed problems. Their physical and verbal actions, considered as proxies of their mental processes, were recorded and analyzed using a combination of qualitative and quantitative measurement. The results of this research indicate that ill-structured problems provide opportunities for a concentration of Discovery and Structuring. Though all of the identified processes of mathematical thinking were observed, students who are highly metacognitive appear to engage in more frequent and advanced mathematical thinking than their less metacognitive peers. This study highlights pedagogical opportunities, for both highly metacognitive students as well as for those who demonstrate fewer metacognitive actions, arising from the activity of doing ill-structured problems. The implications of this work are both theoretical, providing insight into the relationship between metacognition and student “performance,” and practical, by providing a simple tool for identifying processes of mathematical thinking.