Mathematical Abilities and Mathematical Skills

1 World Federation of NationalMathematics CompetitionsConference 2006 Cambridge, England22 28 July 2006 Abilities andMathematical SkillsAlexandre V. Borovik and Tony Gardiner1. Why do we need this discussion?The concept of Mathematical Abilities is not something that is frequently discussed. Atthe personal level, however, almost every mathematician and Mathematical educator hasrelatively firm views on the subject. In this document, we summarise less disputed aspectsof the highly complex phenomenon and some of its immediate implications for participants of this Conference run mathematics outreach programmes and math-ematics competitions in countries with very diverse educational and cultural traditions.

2 Itwould be futile to try to seek a single universal solution to educational and methodologicalproblems that we encounter in our work. However, we have one thing in common: we workwith mathematically able children. It is likely that some of us feel that their work goesagainst the grain of prevailing trends in educational policy and practice. It is likely thatmany of us are familiar with such phenomena as the dumbing down of the curriculum and teaching to test . The task of explaining the importance of proper Mathematical educationto policymakers and to the wider public becomes ever more challenging.

3 It is even morechallenging to explain, to the lay person, the nature of Mathematical Abilities in and developing our work, we have to reach a wider circle of teachers andparents. Given a widespread prejudice against mathematics, what can we say to them aboutmathematics and Mathematical Abilities ?c 2006 Alexandre Borovik, document may be reproduced in whole or in part for non-commercial educational pur-poses, provided that the information quoted is reproduced without adaptation and the sourceand date of publication are document will be placed on the Internet avb/ V.

4 BOROVIK AND A. D. GARDINER2. What can we say to a non-mathematician?When talking to a non-mathematician, Mathematical Abilities can be usefully comparedto musical Abilities : in their developed form they appear highly specific, but are in factquintessentially human, and so are widely spread in the population at large in all socialand ethnic groups. Like music, mathematics is a personality-building activity, it shapes theway the learner thinks and sees the world. As with music, mathematics has a profoundeducational impact even where someone no longer uses their Mathematical training in laterlife.

5 Like music, success in mathematics depends on systematic, cumulative learning, andeach new skill needs to be built on a solid foundation laid at earlier mathematics is often thought to be a cold subject, this is a profound misun-derstanding. As with music, mathematics requires a high level of motivation and emotionalinvolvement on the part of the learner. Understanding is of course vital; but it is also essen-tial for the learner to experience real difficulties: boredom and lack of challenge present fargreater dangers when seeking to nurture Mathematical talent.

6 A degree ofchallenge andfrustrationare has the ability to learn mathematics, although some children learn and makeconnections more quickly than others. Everyone has some Mathematical ability, but somechildren have potential far beyond what most people are prepared to believe. Mathematicalabilities in a child are often dormant and remain unnoticed both by the child and his orher teachers. This potential can be lost forever if it is not discovered and supported atthe appropriate time. It may even be undermined by inappropriate experience: again, acomparison can be made with music, where a dissonant musical toy can seriously damage achild s perception of Mathematical traits appear at different ages.

7 To develop a pupil s mathematicalabilities effectively, one must tap into these Abilities at appropriate times. A comparison canbe made with language learning: almost every 7 year old child can master a foreign languagewith ease, though for an adult it could be an almost impossible task. Similarly, there areperiods in a child s development when he or she is may respond to formal procedures andalgorithms, and there times when they can be excited by the discovery of a new mathemat-ical activity: generalisation. Matching a talented child s Mathematical experience to theircognitive development is a challenge for every teacher since every child will be different!

8 3. StratificationAbilities form a continuous spectrum; for the purpose of this discussion, we identify twogroups of schoolchildren the top 20% and top 1% in the school. The percentiles of 20%and 1% are chosen not for any intrinsic reasons but for purely practical purposes. Thecohort of 20% is likely to include most of those who will end up working, in their adult life,in professions that require some Mathematical background ( beyond mere numeracy );these professions include engineering, information technology, the financial sector, etc.

9 Thegroup of 20% is sufficiently large to warrant the allocation of significant effort and resources toensure that such a group can be realistically supported and nurtured within every school, viaMATHEMATICAL Abilities AND Mathematical SKILLS3a sufficiently enriched curriculum, exciting problems and, of course, teachers (and appropriateprofessional development). Most importantly, we need some of the top 20% (and the best ofthem) to return to school as teachers of top 1% group (in effect, 2 3 pupils per year in an average British school) shouldperhaps be more of concern to the professional academic community than to their in this group have the potential to join the next generation of mathematicians andcomputer scientists; they could become confident users of advanced hardcore mathematicsin science, engineering, biotechnology and the financial sector.

10 Everyone who works withgifted children in Mathematical competitions and other outreach activities knows, that, assoon as such a child is encouraged in his or her interest in mathematics and is given somesupport and sufficiently challenging problems to solve, he or she rapidly outgrows the levelof his or her school. We could not realistically expect that the school can provide sufficientintellectual nourishment for this group of children. But the health of the professional math-ematical community and the needs of society demand that the top 1% group is not lost tomathematics and is nurtured via a network of outreach and enrichment activities run byuniversities and other professional in many countries shows that much of what we say concerning the select 1%group applies to the wider group of 20%.