What, then, are the special and particular problems that lie in the way of effective teaching and learning in pure mathematics? There are, of course, the issues of transition and mathematical preparedness touched upon above. Lack of technical fluency will be a barrier to further work in pure mathematics. However, there are deeper, more 252 ❘ Teaching in the disciplines
Interrogating practice
How does your department address the changing needs of students entering higher education? What else could you do to address ‘gaps’ in knowledge, skills and understanding?
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fundamental issues concerning reasoning skills and students’ attitudes to proof. Certainly, within the context of higher education in the UK, there are a number of signals and signs that can be read. Among these are current issues in mathematical education, reported problems in contemporary literature (a good recent example is Brakes (2001)), and – to a lesser extent – the results of the QAA Subject Review reports.
So what can be done? There is little evidence that a dry course in logic and reasoning itself will solve the problem. Some success may be possible if the skills of correctly reading and writing mathematics together with the tools of correct reasoning can be encouraged through the study of an appropriate ancillary vehicle. The first (and some would still claim, the foremost) area was geometry. The parade of the standard Euclidean theorems was for many the raison d’être of logic and reasoning. However, this type of geometry is essentially absent from the school curriculum, geometry is generally in some state of crisis, and there is little to be gained from an attempt to turn the clock back. It is worth noting, however, that students’ misuse of the <=> symbol and its relatives were less likely through exposure to the traditional proof in geometry – most commonly, lines were linked with ‘therefore’ or ‘because’ (with a resulting improvement in the underlying ‘grammar’
of the proof as well). Mathematicians may not want to bring back classical geometry, but should all regret the near passing of the proper use of ‘therefore’ and ‘because’ which were the standard features of geometrical proofs.
At one time, introductory mathematical analysis was thought to be the ideal vehicle for exposing students to careful and correct reasoning. Indeed, for many the only argument for the inclusion of rigorous analysis early in the curriculum was to provide a good grounding in proper reasoning. Few advance this case now. Indeed, some researchers in mathematics education have cast doubt on the need for proof itself in such introductions to mathematical analysis (see e.g. Tall, 1999). (We see here a case where research in mathematics education and actual practice are in step with each other.
Concerns, though, have been expressed in recent years as to whether the gap between research and practice in mathematics education is widening. Perhaps if we all do our bit to ‘mind the gap’, we can improve the lot of those at the heart of our endeavours – our students.) Recently the focus has passed to algebra. Axiomatic group theory (and related algebraic topics) is thought by some to be less technical and more accessible for the modern student. Unfortunately, there is not the same scope for repeated use over large sets of the logical quantifiers (‘for all’ and ‘there exists’) and little need for contra-positive arguments. Number theory has also been tried with perhaps more success than some other topics.
Working within the comfort zone
However, most success seems to come when the mathematics under discussion is well inside a certain ‘comfort zone’ so that technical failings in newly presented mathematics do not become an obstacle to engagement with the debate on reasoning and proof.
Examples might include simple problems involving whole numbers (as opposed to
formal number theory), quadratic equations and inequalities and trigonometry. (At a simple level we can explore how we record the solutions of a straightforward quadratic equation.) We may see the two statements:
‘X53 or X54 is a root of the equation X227X11250’
‘The roots of the equation X227X11250 are X53 and X54’
as two correct uses of ‘and’ and ‘or’ in describing the same mathematical situation. But do the students see this with us? Is it pedantic to make the difference, or is there a danger of confusing the distinction between ‘or ’ and ‘and’? By the time the solution to the inequality:
(X23)(X24).0
is recorded as the intersection of two intervals rather than the union, things have probably gone beyond redemption (see also Brakes, 2001).
Workshop-style approaches
One possible way forward is to use a workshop-style approach at least in early sessions when exploring these issues. Good evidence exists now for the usefulness of such active approaches, as Prince (2004) argues. But a real danger for such workshops at the start of university life lies in choosing examples or counterexamples that are too elaborate or precious. Equally, it is very easy to puncture student confidence if some early progress is not made. Getting students to debate and justify proofs within a peer group can help here. One way of stimulating this is outlined in the workshop plan given in Kyle and Sangwin (2002).
Students can also be engaged by discussing and interpreting the phraseology of the world of the legal profession. Much legislation, especially in the realm of finance, goes to some lengths to express simple quantitative situations purely, if not simply, in words.
Untangling into symbolic mathematics is a good lesson in structure and connection.
Further, one can always stimulate an interesting debate by comparing and contrasting proof in mathematics with proof as it is understood in a court of law.
254 ❘ Teaching in the disciplines
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What do you do to assist students to move beyond their ‘comfort zone’? What do you do when it becomes apparent that students are floundering with newly presented mathematics? Are there any specific approaches you feel you would like to improve?
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Teaching from the microcosm
If students are to see mathematics as a creative activity it will help to focus on the different strategies they themselves can employ to help make sense of mathematical ideas or problems. It may also help to focus on the process of learning mathematics more broadly.
Specific teaching approaches, though, are required to realise this. Palmer (1998: 120), for instance, argues that every problem or issue can become an opportunity to illustrate the internal logic of a discipline. We can take a straightforward example of this. If you do not understand an initial concept, it will be virtually impossible to understand a more advanced concept that builds on the initial concept: to understand the formal definition of a group, for instance, you first need to understand the concept of a variable (as well as other concepts). But students cannot rely on the tutor always identifying these prior concepts for them. They themselves need to be able to take a look at an advanced concept and identify contributory concepts, so that they can then make sure they understand them. The same applies to other strategies, as Kahn (2001) further explores, whether generating one’s own examples, visualising, connecting ideas or unpacking symbols. In terms of concrete teaching strategies, the tutor can model these strategies alongside a systematic presentation of some mathematics or require students to engage in such strategies as a part of the assessment process. We thus move away from an exclusive focus on the content, to more direct consideration of the process by which we might come to understand that content.