Categories, definitions and mathematics : student reasoning about objects in analysis

This thesis has two integrated components, one theoretical and one investigative. The theoretical component considers human reason about categories of objects. First, it proposes that the standards of argumentation in everyday life are variable, with emphasis on direct generalisation, whereas standards in mathematics are more fixed and require abstraction of properties. Second, it accounts for the difficulty of the transition to university mathematics by considering the impact of choosing formal definitions upon the nature of categories and argumentation. Through this it unifies established theories and observations regarding student behaviours at this level. Finally, it addresses the question of why Analysis seems particularly difficult, by considering the relative accessibility of its visual representations and its formal definitions. The investigative component is centred on a qualitative study, the main element of which is a series of interviews with students attending two different first courses in Real Analysis. One of these courses is a standard lecture course, the other involves a classroom-based, problem-solving approach. Grounded theory data analysis methods are used to interpret the data, identifying behaviours exhibited when students reason about specific objects and whole categories. These behaviours are linked to types of understanding as distinguished in the mathematics education literature. The student's visual or nonvisual reasoning style and their sense of authority, whether "internal" or "external" are identified as causal factors in the types of understanding a student develops. The course attended appears as an intervening factor. A substantive theory is developed to explain the contributions of these factors. This leads to improvement of the theory developed in the theoretical component. Finally, the study is reviewed and the implications of its findings for the teaching and learning of mathematics at this level are considered

Tilting the classroom

This article describes and illustrates 12 simple ways to make large mathematics lectures more engaging. These include a variety of short-and-snappy activities, framed by organisational practices that support concentration and maintain a positive atmosphere. These practices can be implemented individually or in combination, with no need for a wholesale classroom restructure

Mathematicians' perspectives on the teaching and learning of proof

This paper reports on an exploratory study of mathematicians' views on the teaching and learning that occurs in a course designed to introduce students to mathematical reasoning and proof. Based on a sequence of interviews with five mathematicians experienced in teaching the course, I identify four modes of thinking that these professors indicate are used by successful provers. I term these instantiation, structural thinking, creative thinking and critical thinking. Through the mathematicians' comments, I explain these modes and highlight ways in which students sometimes fail to use them effectively. I then discuss teaching strategies described by the participants, relating these to the four modes of thinking. I argue that teaching aimed at improving structural thinking tends to dominate, and that courses that introduce proof, regardless of classroom organization, should address all four modes in a balanced and integrated way

Teaching proof to undergraduates: semantic and syntactic approaches

This paper contrast the rationales behind semantic and syntactic approaches to teaching an undergraduate transition-to-proof course, using data from interviews with two mathematicians. It addresses the ICMI theme of teachers’ views and beliefs, with particular focus on (1) instructors’ expectations in proofbased courses and (2) both example-based and logical structure-based skills that we would like students to develop before arriving at university

Referential and syntactic approaches to proving: case studies from a transition-to-proof course

The goal of this paper is to increase our understanding of different approaches to proving in advanced mathematics. We present two case studies from an interview-based investigation in which students were asked to complete proof-related tasks. The first student consistently took what we call a referential approach toward these tasks, examining examples of the objects to which the mathematical statements referred, and using these to guide reasoning. The second consistently took what we call a syntactic approach toward these tasks, working logically with definitions and proof structures without reference to examples. Both students made substantial progress on each of the tasks, but they exhibited different strengths and experienced different difficulties. In this paper we: demonstrate consistency in these students' approaches across a range of tasks, examine the different strengths and difficulties associated with their approaches to proving, and consider the pedagogical issues raised by these apparent student preferences for reasoning in certain ways

Peer assessment without assessment criteria

Peer assessment typically requires students to judge peers' work against assessment criteria. We tested an alternative approach in which students judged pairs of scripts against one another in the absence of assessment criteria. First year mathematics undergraduates (N = 194) sat a written test on conceptual understanding of multivariable calculus, then assessed their peers' responses using pairwise comparative judgement. Inter-rater reliability was investigated by randomly assigning the students to two groups and correlating the two groups' assessments. Validity was investigated by correlating the peers' assessments with (i) expert assessments, (ii) novice assessments, and (iii) marks from other module tests. We found high validity and inter-rater reliability, suggesting that the students performed well as peer assessors. We interpret the results in the light of survey and interview feedback, and discuss directions for further research into the benefits and drawbacks of peer assessment without assessment criteria

Time versus line number fixation plots

Time versus line number fixation plot

Undergraduates’ example use in proof construction: purposes and effectiveness

In this paper, we present data from an exploratory study that aimed to investigate the ways in which, and the extent to which, undergraduates enrolled in a transition-to-proof course considered examples in their attempted proof constructions. We illustrate how some undergraduates can and do use examples for specific purposes while successfully constructing proofs, and that these purposes are consistent with those described by mathematicians. We then examine other cases in which students used examples ineffectively. We note that in these cases, the purposes for which the students attempted to use examples are again appropriate, but the implementation of their strategies is inadequate in one of two specific ways. On this basis we identify points that should be borne in mind by a university teacher who wishes to teach students to use examples effectively in proof-based mathematics courses

Doctoral students’ use of examples in evaluating and proving conjectures

This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and semantic reasoning strategies relative to a representation system of proof. We then use this conceptualisation as a basis for contrasting the behaviour of two successful mathematics research students whilst they evaluated and proved number theory conjectures. We observe that the students exhibited strikingly different degrees of example use, and argue that previously observed individual differences in reasoning strategies may exist at the expert level. We conclude by discussing implications for pedagogy and for future research

Classification and concept consistency

This article investigates the extent to which undergraduates consistently use a single mechanism as a basis for classifying mathematical objects. We argue that the concept image/concept definition distinction focuses on whether students use an accepted definition but does not necessarily capture the more basic notion that there should be a fixed basis for classification. We examine students’ classifications of real sequences before and after exposure to definitions of increasing and decreasing; we develop an abductive plausible explanations method to estimate the consistency within the participants’ responses and suggest that this provides evidence that many students may lack what we call concept consistency