A Handbook for Teaching and Learning in Higher Education Enhancing academic and Practice

254 ❘
Teaching in the disciplines
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:
‘X3 or X4 is a root of the equation X 2 7X120’
‘The roots of the equation X 2 7X120 are X3 and X4’
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:
(X3)(X4)0
is recorded as the intersection of two intervals rather than the union, things have probably
gone beyond redemption (see also Brakes, 2001).
Interrogating practice
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?
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.