Research in Teaching and Learning Mathematics at an Advanced Level

Conclusion

How can we bring together the discussion of what features, if any, are specific to the learning of advanced mathematics in the first part of this chapter, with the examples of research on the learning of advanced mathematics in the second part?

• • One still finds over and over again a certain number of difficulties, mentioned in the first part of this chapter, that are related to the complexity of the contents of advanced mathematics: abstraction and formalization being particular stumbling blocks. All research in mathematical education shows that there seems to be no easy way of avoiding the difficulty of abstraction discussed in earlier chapters by Tall, Dreyfus and Dubinsky. On the contrary it seems essential to develop new ways of approaching it.

• • Putting aside the research directed at particular complexity of mathematical content, examples of which are described in subsequent chapters, the remaining research (of the second and third broad types described in the second part of this chapter) does have a common feature: the attempt to change the scientific environment of the students to give them a new and more authentic relationship to knowledge that is more akin to that of experts (i.e. researchers and practitioners) than to that of school pupils. It is here perhaps that we might find a genuine application of advanced mathematical thinking: having available in full the resources of the scientific spirit to control, create, and systematically introduce methods of learning and even, perhaps, of effective research.

Thanks are extended to Ed Dubinsky for his initial translation of the draft of this chapter into English.