Understanding How Undergraduate Students Formalise Their Written Argumentations in Mathematics

Abstract

All students who take an undergraduate university course in mathematics will confront the task of having to present a written argument to someone else. This could be one or more of their colleagues, or teaching staff who are also likely to be the markers of their course assignments, tutorials, tests and examinations. During lectures and tutorials, the lecturer and tutor may provide justifications for various aspects of mathematics presented – which might include elementary proofs, or give examples of how various theorems or known results contribute to solving a range of problems. Students will very likely be exposed to modes of argumentation that encourage certain habits of mind and speech, and at the same time, foster a disposition suited to engaging successfully with mathematics. These combine to shape the first experiences students have with argumentation in mathematics. Usually, by their second year, many students will have been introduced to formal mathematics through linear algebra, differential and integral calculus, or courses that are specifically designed to teach logic, proof and formal thinking in mathematics. Certainly, by their third year, reading and writing formal arguments of proof is an established part of the student’s life-world. However, literature from mathematics education shows that proof is a difficult mathematical concept for students to master. Underlying this research, is an assertion that argumentation plays an important role in developing the thinking, reasoning and communicative skills required to produce formal mathematical arguments. One way to investigate this is to see how students adapt their written argumentations from ones that give a personal conviction to ones aimed at convincing an other. This research applied Tall’s model of Three Worlds of Mathematics, Toulmin’s layout of an argument, and Habermas’ three dimensions of rationality to create a framework that was used to examine the argumentations of students, investigate how they formalised them, and describe some obstacles that may have prevented them from doing so. Students who were taking a first year mathematics course, and a smaller number taking a second year mathematics course participated in this research by completing tasks that were designed to promote the production of written argumentations. The argumentation of four students were tracked over three years in an attempt to capture, and explain, general changes that occurred over a longer term. These were analysed using the research framework developed in this study. The results of this study suggest that the first year mathematics students had the readiness to engage successfully with writing formal mathematical arguments. When they perceived it necessary to do so, they could select the necessary cognitive and mathematical tools that allowed them to reason algebraically and deductively. By the second year, students’ mathematical reasoning were well established, and they could confidently use methods of argument and logic usually associated with proof. A significant inclusion in second year students’ argumentations was the use of definitions. Over time, the students used increasingly sophisticated symbols, concepts and mathematical theory to produce more complex argumentations. Although the primary form of reasoning in formalising argumentations was deductive, other forms of reasoning that provided elaboration, understanding, and further support for personal conviction often supported it. It is anticipated that this research might provide further support to the assertion that written argumentation has an important role in learning mathematics, particularly in developing an understanding of proof.