Papers from the Colloquium

Theme 3: Gaps and transitions: thinking in Mathematics
Dr Elena Nardi, University of East Anglia

This response to Margaret Brown’s contribution aims at introducing the views of a group of mathematicians as expressed in a study on the teaching and learning of mathematics at university level, currently in progress at the University of East Anglia and funded by the Learning and Teaching Support Network. The findings reported here can be seen in greater detail in Nardi, E., Iannone, P. & Cooker, M.J. (2003, in press) ‘Pre-eighteen students have lost something major’: Mathematicians on the impact of school mathematics on students’ skills, perceptions and attitudes, Proceedings of the Day Conference of the British Society for Research Into the Learning of Mathematics, Birmingham University, 13 November 2003.

For the aims, methodology and other publications from the study, please contact Elena Nardi or browse at http://www.uea.ac.uk/~m011

According to the participants in this study, students arrive at university with a perception of mathematical thinking as primarily involving calculation, not reasoning. Their school mathematical experiences are seen as responsible for this image of the field: a brief scrutiny of school textbooks shows up an instrumental approach.
Students may complete schooling without having seen a proof, even of Pythagoras’ Theorem, a mathematical fact for which there exist almost language and logic free proofs and which, in the vernacular, is mostly associated with the word ‘theorem’ – perhaps, recently, along with Fermat’s Last Theorem. But, even when exposed to a proof of Pythagoras’ Theorem or of the irrationality of Ö2 – the latter is often the only proof seen at school – students, dwelling on mathematical thinking as primarily involving calculation, often perceive this exposure as a demonstration of a rule, rather than a proof. To them ‘proof’ is still a rather vague term. Clarifying this vague image is not helped by the fact that the students can be exposed to proof on rare occasions, even just once.

The demise of Geometry in school mathematics is another factor that has hindered the presence of mathematical reasoning and proof as, when Geometry is removed, the number of accessible, doable proofs that remains is severely cut. With some Co-ordinate Geometry and plenty of Trigonometry still in the syllabus, only an implicit experience of proving is still available. In fact this experience is mainly associated with algebraic manipulation and therefore it potentially reinforces the above image of mathematics as mostly involving calculation. The reasoning, as well as the linguistic structures underlying mathematical reasoning - and who would doubt the benefit embedded in exchanging mathematical ideas in grammatically sound, full sentences for a student’s overall enhanced power of persuasion in speaking and writing? - are too implicit in the algebraic expressions used in this type of manipulative proof (for example, starting from one side of an equality and ending up on the other).
A simple proving technique, such as Proof by Mathematical Induction, can be used as a demonstration of how fascinating mathematical thinking can be. It is a great loss to students – and with repercussions for the level of mathematics taught in universities and in teacher education – that, along with other forms of reasoning, consistent student failure at this type of exam question has led to its gradual removal from school mathematics (with further selective omissions from an already depleted syllabus when aiming at groups of students with varying perceived abilities, e.g. C/D or B/C at GCSE level).

A utilitarian approach to school mathematics, namely one that has undermined an image of mathematics as an intellectually attractive activity and one that has ignored the views of mathematical scholars, has produced a dry, unappealing school subject. Convincing young people about the attractiveness of an activity that has sometimes a slower turnover in terms of intellectual and emotional gratification is nowadays, in a culture that favours instant gratification, a difficult task. The surrender to a utilitarian image of mathematics has been a destructive move as, by promoting an image of university studies as a mere opportunity for a better salary, it has led to perilously small numbers of students taking up studies in mathematics, engineering, physics and chemistry. Improved social recognition, as well as better earnings, are vital if we are to address the now urgent need to attract better mathematics graduates to the profession.