Introduction: Many students tend to see mathematics as a disparate collection of rules, procedures, theorems, definitions, formulae or applications. Such students believe that mathematical ideas should be memorised, and then used to solve problems. Clearly, though, such a disjointed approach to learning mathematics becomes increasingly untenable as the level of complexity and abstraction increases. But are these students ever likely to appreciate how mathematics fi ts together as an abstract system of thought if they merely attend lectures and work away on solving problems? It remains difficult to shift ingrained perceptions of our discipline, and it seems unreasonable to expect that it will simply be sufficient to present students with the finished products of mathematical rigour.
Study after study makes it clear [1-3] that we cannot expect an increasingly diverse body of students to make the transition to university-level study without further support. One approach is to help students pick up the strategies that expert mathematicians employ in making sense of a proof or coping with further abstraction (see for instance [4,5, 6]). This can provide a basis for students to approach the varied mathematical tasks they face in a more consistent or coherent fashion. When looking to shift deep-seated attitudes towards mathematics, we need more than an occasional reminder or an introductory course. If one begins to realise ways in which the same logical principles or approaches to exemplifying mathematical concepts apply to a wide range of contexts, then one is a good way to appreciating how mathematics fits together as a discipline. The challenge is to help students engage at a substantive mathematical level.
Introduction
Many students tend to see mathematics as a disparate collection of rules, procedures, theorems, definitions, formulae or applications. Such students believe that mathematical ideas should be memorised, and then used to solve problems. Clearly, though, such a disjointed approach to learning mathematics becomes increasingly untenable as the level of complexity and abstraction increases. But are these students ever likely to appreciate how mathematics fi ts together as an abstract system of thought if they merely attend lectures and work away on solving problems? It remains difficult to shift ingrained perceptions of our discipline, and it seems unreasonable to expect that it will simply be sufficient to present students with the finished products of mathematical rigour.
Study after study makes it clear [1-3] that we cannot expect an increasingly diverse body of students to make the transition to university-level study without further support. One approach is to help students pick up the strategies that expert mathematicians employ in making sense of a proof or coping with further abstraction (see for instance [4,5, 6]). This can provide a basis for students to approach the varied mathematical tasks they face in a more consistent or coherent fashion. When looking to shift deep-seated attitudes towards mathematics, we need more than an occasional reminder or an introductory course. If one begins to realise ways in which the same logical principles or approaches to exemplifying mathematical concepts apply to a wide range of contexts, then one is a good way to appreciating how mathematics fits together as a discipline. The challenge is to help students engage at a substantive mathematical level...
