Where will the next generation of UK mathematicians come from? 18–19 March 2005, Manchester www.ma.umist.ac.uk/avb/wherefrom.html The meeting is sponsored by Manchester Institute for Mathematical Sciences and London Mathematical Society What is It That Makes a Mathematician? Alexandre V. Borovik Introduction: what is the purpose of this text? Our meeting Where will the next generation of UK mathematicians come from? will concentrate on the education policy issues arising from our desire to nurture future math- ematical talent. However, a brief look at the programme of the meeting shows that no discussion of what mathematical abilities and talent are is scheduled. I hope that we have a shared understanding sufficient for a meaningful conversation. Nevertheless I believe that some coffee break chats about the nature of mathematical abilities and their early manifes- tations in children might be useful. To facilitate an informal discussion of a highly elusive topic, I have decided to offer my notes on mathematical thinking for the attention of the participants of the meeting. At this point, a disclaimer is necessary. I emphasise that I am not a psychologist nor a specialist in educational theory. My notes are highly personal and very subjective. They do not represent results of any systematic study. The notes are mostly based on my teaching experience in Russia in the 1970s and 1980s, in a social and cultural environment very distant from the modern British landscape. If so, why did I bother to write this text? I teach at a university; I am concerned that our mathematics students frequently lack (and are not being trained in) specific cognitive skills which are crucially important for the profession. But I believe that these very skills (although “traits” is a better word since they might be still undeveloped) can be found in able children at as early stages of education as pre-GCSE. I believe that mathematical cognitive traits should be supported and developed as soon as they first appear in a child. If I formulate my views in a few words, I believe in the unity of mathematics, in- cluding its vertical unity. For me, “recreational”, “elementary”, “undergraduate” and “re- search” mathematics are no more than artificial subdivisions of a single continuous spec- trum. Writing from the position of a university teacher and PhD supervisor, I freely move through the whole range – but here my emphasis is firmly on the early stages of school mathematics. I believe that university mathematicians should be concerned more about mathematics teaching as a system, from primary school to A levels to PhD studies (and do not forget such crucially important branches as teacher training or the teaching of mathematics to engineers). The ultimate aim of this text is to help to persuade my university colleagues that they have to start helping schools in some serious way – for example, by helping to run enrichment and extension activities. c  2005 Alexandre Borovik, borovik@manchester.ac.uk . 1