Robin Hankin (AUT, New Zealand)

Copyright: © 2016
|Pages: 7

DOI: 10.4018/978-1-4666-9924-3.ch014

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TopThe educational value of a traditional lecture has been questioned many times in the literature on various grounds by Biggs and Tang (2011), Bergsten (2007), and others. Criticisms of the lecture format include: the tendency to turn the audience into passive listeners rather than active participants; the dishonest presentation of mathematics as a linear predetermined progression rather than a “social activity coloured by creativity and struggle”; and poor comprehension of the material by the audience.

Mathematics, however, appears to be qualitatively different from other subjects in the sense that the essence of a mathematics lecture is a mathematician *doing*, rather than *talking about,* mathematics. Consider, for example, a lecture on survey engineering. This will comprise the teacher discussing, explaining, and perhaps illustrating various aspects of surveying. At no point does a lecturer actually perform any action that might be described as surveying: he does not even go outdoors.

It is also interesting to observe that, when a mathematics lecturer does talk *about* mathematics (for example, mentioning the history of the subject), such discussion is invariably short, and presented “for interest”; it stands out like a sore thumb; the students stop writing. They know that it’s not real mathematics, it’s just the lecturer making light conversation by way of a break. Hartley and Hawkes (1983) – a standard mathematics textbook – illustrates this perfectly. Chapter Five opens with a gentle and chatty introduction: “a *module* . . . turns up in many seemingly unlikely guises . . . such an apparently all-embracing object will suffer from some of the defects of great generality . . . the reader's progression will be from the specific to the general and back to the specific again”. The chapter introduction culminates in a sharp “Now down to work!” and the style immediately reverts to the default: formal, axiomatic mathematics.

Thus, during a mathematics lecture, the mathematician actually *performs* genuine mathematics. The teacher will actually prove mathematical statements and explicitly creates (or at least verifies) knowledge in front of the students as part of a live performance. It is worth noting that the process of mathematical proof used in a lecture is *identical* to that used by a professional mathematician. It is also worth noting that, when performed correctly, the audience members will perform genuine mathematics along with the lecturer in the sense that they actually prove mathematical statements.

One might characterise lecture-style proof as being more familiar to the lecturer than the proofs used in research, but the idea is the same. The criteria for acceptance are identical. It is here that Bergsten’s (2007) criticism of lectures as pre-formed linear sequences becomes evident; genuine mathematics research as a process is generally characterized as being frustratingly iterative and bedevilled by confusion and other cognitive impairments.

It is a common philosophy of teaching (Shulman 2005) to model the behaviour of a mathematician; this is made easier by the fact that mathematics teaching is, at least in theory, perfectly aligned in the sense of Biggs and Tang (2011). Consider Cauchy’s theorem, a crucial requirement for many branches of modern mathematics; its proof is regarded as the highlight of many undergraduate courses in complex analysis. The Cauchy’s theorem component of a course will have the following features:

*•*Learning objective: prove Cauchy’s theorem.

*•*Teaching activity: prove Cauchy’s theorem.

*•*Assessment task: prove Cauchy’s theorem.

Specimen Answer: Ideal responses to examination questions generated by the tutor, which would receive 100% marks.

Signature Pedagogy: Forms of instruction associated with preparation of members of particular professions.

Modelling: A process whereby a student changes in response to observing an instructor; the emphasis is on the instructor demonstrating appropriate professional behaviour.

Screencast: A video recording of a computer monitor display, together with voice-over.

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