43 research outputs found
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
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
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
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
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
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
Watching mathematicians read mathematics
This report contributes to the debate about whether expert mathematicians skim-read
mathematical proofs before engaging in detailed line-by-line reading. It reviews the conflicting introspective and behavioural evidence, then reports a new study of expert mathematicians' eye movements as they read both entire research-level mathematics papers and individual proofs within those papers. Our analysis reveals no evidence of skimming, and we discuss the implications of this for research and pedagogy
Representation systems and undergraduate proof production: a comment on Weber
Weber (2009) suggested that counterexamples can be generated by a syntactic proof production, apparently contradicting our earlier assertion (Alcock & Inglis, 2008). Here we point out that this ostensible difference is the result of Weber working with theoretical definitions that differ slightly from ours. We defend our approach by arguing that Weber’s relies upon an as yet unspecific metric for gauging the amount of work conducted in each representation system, and that it does not recognize an important asymmetry between the status of representation systems in the context of undergraduate mathematics
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