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I would like to offer some thoughts on mathematical passion - through what brain processes mathematical passion might arise - because mathematical talent and mathematical passion are dialectically intertwined: each is the genesis of the other.
I would like to begin with an anecdote about the origins of my own mathematical passion. I was about 12 years old, having completed my first year of high school. I was languishing in bed recovering from pneumonia when a neurosurgeon friend of my fathers tossed me a copy of Edward Kasner and James Newman's Mathematics and the Imagination. Zeno, Googels ... I was captivated. But when I came to Euler's exposition of De Moivre's Theorem, , I levitated out of bed with excitement. All of these independently developed areas of mathematics were so elegantly inter-related? I was both cured and smitten at the same instant!
It seems therefore fair to ask: What educational research over the past decade or two has informed our understanding of this phenomenon? The disappointing answer, I would suggest, is very little. However, recent research in the fields of neuropsychology and genetics has produced data which does inform our understanding of mathematical talent (and mathematical passion), and challenges how we provide for its development.
This is not to say that there has been a paucity of educational research into mathematical talent, much less mathematics education. Rather, it is that the post-modern agenda which has dominated educational research over the past decades has produced a plethora of subjective accounts whose qualitative analysis has described the complexities of mathematical learning adequately enough, at least in social and personological terms, but has failed to progress our knowledge of the cognitive underpinnings of that mathematical passion.
As a typical recent example of good educational research, Julie Landvogt, Gilah Leder and Helen Forgasz at La Trobe University in Melbourne carried out a series of studies into the beliefs of teachers, parents and students about gender differences in school mathematics. Their qualitative data and analyses are interesting in that while many respondents believed that girls are more intuitive about maths than boys, many others who were interviewed believed that boys are more intuitive about maths than girls. The researchers do note in passing the dichotomous positions of their respondents' attitudes. Perhaps this dichotomy could be resolved by positing that it is only mathematically talented boys who are the more intuitive? Such a hypothesis is not considered by these researchers, but they do compare the beliefs of self-selected mathematically talent students with their peers.
Clearly, most respondents plumbed for neither position. No inferential testing is reported of these quantitative data, but as is obvious from inspection (and a simple Chi-square analysis), none of the differences in beliefs between mathematically talented and normal students were significant. In other words, surveying beliefs does not get us very far in understanding the phenomenon.
Turning to neuropsychology, however, we find that Richard Haier and Camilla Benbow (1995) in Iowa tested the hypothesis that mathematically talented boys do think differently with a PET imaging study of mathematical reasoning, in which mathematically gifted 13 year-olds were compared with ability-matched college students (both scored 1100/1400 on the SAT-M), and with normal 13 year olds (who bottomed at 200/1400 on the SAT-M).
Extremely mathematically gifted 13 year olds had similar PET profiles to the 20 year old math-major college students. Haier and Benbow speculated that
men who have high SAT-M scores have a different way of mathematical reasoning from everyone else, although this is not strongly apparent in these data.
Of course, neurobiological sex differences that affect mathematical abilities might only to be expected given that differences in prenatal testosterone exposure may differentially influence underlying brain organisation.
Such putative differences were explored by Joel Alexander, Michael O'Boyle and Camilla Benbow (1996) in Iowa with an EEG study of alpha wave power in a similar sample of 13 year-old extremely talented males and females compared to SAT-matched college students aged 20 years.
The results show that the gifted adolescents have superior alpha wave power, especially in non-dominant hemisphere and frontal lobes, over their age peers. The researchers note that:
gifted subjects may have an unusually rapid and high level development of inter-hemispheric interactions,
the area where structural and functional development are most closely related are the frontal lobes - gifted adolescents and college students have a similar level of brain maturation in these regions.
A further series of information processing studies (finger tap - chimeric face recognition - dichotic listening - word pleasantness) of hemispheric preference of gifted vs normal adolescents showed the former to display a non-dominant hemispheric preference for processing information, together with superior frontal activity. Here it was noted that:
precocious individuals were more active in the frontal lobes, suggesting the frontal lobes mediate high-level intelligence.
Importantly, one of the features of human neuro-anatomy is the very strong connections between the frontal lobes and the modules of the limbic system responsible for the generation of emotions. Antonio Damasio has described how the functioning of the frontal regions - our so-called rational thinking - is driven by our feelings and emotions. In other words, performance necessitates emotion. With mathematically talented people, their greater activity in the frontal regions means greater activity in the limbic area. In sum, mathematical talent generates, and is generated by, mathematical passion.
It is comforting that the neuropsychological results are also consistent with those from a wealth of studies into the individual differences in information processing of school children, in Canada, supervised by J. P. Das, University of Alberta, Edmonton, and in Australia, supervised by Don Fitzgerald of the University of New England, Armidale, using psychometric instruments as operationalisations of the neuropsychological model of the eminent Russian Alexander Luria. In these studies, measures on simultaneous synthesis, an operationalisation of non-dominant parietal lobe associative functioning, explains up to 80% of the variance in school mathematics performance, while demonstrably talented children in mathematics and, with my own work, in music, showed significantly superior measures on executive synthesis, an operationalisation of frontal lobe functioning. For example, in one of my studies, superiority in executive synthesis by young 'mozarts' explained nearly 25% of the variance in their perception of algorithmic computer-generated music. Here it is musical passion driving and driven by musical talent.
Just what kind of a differential pedagogy for the mathematically talented and passionate, might be proposed? Doubtless we would all agree with the position that mathematically talented students should engage in problem solving and group work in a mathematically rich environment. I would argue that the best arrangement for the continuing development of mathematical talent and passion is homogeneous grouping of the mathematically talented, on an inclusive, not exclusive basis. Such a position challenges the 1970s orthodoxy that heterogeneous ability groupings are best for all students, including the talented. (The rationale that the gifted can learn through their teaching of the less talented is a leaf straight out of the monitorial schools of early last century - though 200 years ago at least the monitors were taught by their teachers.)
A recent American study lead by Lynn Fuchs of Nashville University compared the mathematical problem solving performances of homogenous and heterogeneous ability-pairs of primary school students. Analysis of videotapes showed that homogeneous pairs interacted more collaboratively and were more productive than the heterogenous parings. Interestingly, both high-ability and normal-ability subjects preferred homogeneous pairings, the high-ability children because they could get on with it, the normal-ability children because they felt less intimidated when working with peers of like mind - they found working with higher-ability children restricted their problem solving performance.
Whether or not to segregate gifted children from the mainstream has been the subject of heated debate in Australia over the last few years. Herbert Marsh, University of Western Sydney, has undertaken a series of large-scale studies which consistently show declines in both total and academic self-esteem of gifted children placed into selective school programs, the Big-Pond-Little-Fish effect. His results have been disputed by Miraca Gross, Director of the Gifted Education Research, Resource and Information Centre at the University of NSW in Sydney, on the grounds that the programs into which these children have been transferred are less than optimal for gifted students as the teachers mostly lack any specialised training in this area.
Gross measured the self-esteem of gifted students in accelerated programs and found that self-esteem declines were less severe than those reported by Marsh. Any slight decline in self-esteem Gross interpreted as an adjustment to reality rather than a negative attribute of streaming. As a case in point, two of the exceptionally gifted children which Gross has been tracking throughout their schooling, and who were radically accelerated, have gone to Cambridge to study mathematics at age 16; and both have fared very successfully.
As Editor of the Australasian Journal of Gifted Education , I have been glad to provide a forum for this debate, although after two years of metaphors about fish, ponds, fish tanks and their optical illusions, fishing for data, fish food and so on, I am beginning to hunger for a change of menu. As a researcher I must confess some misgivings about the exclusive use of ANOVA to test these data. Any sample of gifted children will have a non-normal distribution, and a limited range, and as Keselman et al have taken some pains to point out in a recent issue of Review of Educational Research , ANOVA does not fare well under such conditions.
That not withstanding, the question of whether to stream or not to stream is essentially a re-run of the nature versus nurture debate at a meta-conceptual level and in the context of school timetabling. Here, educational research could profit from the genetics literature where for some years genetic expression and environmental constraint have been regarded as dialectically inter-dependent. Much of this research has focused on very young infants.
Before taking up his present position at the London Institute of Psychiatry, Robert Plomin undertook extensive twin studies at the University of Colorado. The logic of twin studies is to compare the correlations of attributes of monozygotic (MZ) twins (identical twins with 100% shared genes) and dizygotic (DZ) twins (fraternal twins with 50% shared genes). Wherever the correlations for MZ significantly exceed those of DZ a genetic causation can be posited. Stephen Petrill and colleagues studied a group of high cognitive ability twins from 14 to 36 months. Mental development was measured by the Baley Scales of Infant Development to 24 months, with a Stanford-Binet at 36 months. From a comparison of MZ to DZ correlations, Petrill et al found that heritability for high cognitive abilities was zero at 14 months, but rose to 64% at 36 months. That is, the heritable genetic contribution to higher cognitive abilities increases to greater levels of stability with older children. Obviously learning or rehearsal improves skill level, but in doing so it actually increases genetic influence by reducing uncontrolled environmental effects, effects that are generally larger in early childhood. That is, learning enables us to reach our genetically mediated ceilings.
It should be noted that this is counter-intuitive to how educationists usually consider genetic influence, as declining with age due to increasing exposure to environmental influences. But such a linear interpretation ignores the nonlinear and dialectical nature of genetic expression, where genetic influence requires an environment in which it is to be expressed, and environmental influence ultimately effects biological expression mediated through the genes. Moreover, an increasing genetic influence throughout the lifespan is consistent with the neurophysiological evidence for continuing neurogenesis and increasing dendritic stabilisation throughout childhood, and explains the results of adoptive twin studies where adoptees become more like their biological parents than their adopting parents on reaching adolescence.
Such findings beg the question of the human genome project, and have begun to be addressed by Plomin's Quantitative Trait Loci analyses - correlations between known gene locations and behavioural characteristics. Of particular interest, general cognitive ability (the infamous g) - the shared variance of all cognitive tasks - correlates highly with a gene for insulin growth factor on chromosome 6 - a puzzling result until the discovery of an insulin receptor on the hippocampus, the brain module which mediates long term memory! Moreover, whereas there was a higher QTL correlation for higher measures of g, the QTL correlation was lower for specific cognitive abilities including mathematics.
From an evolutionary viewpoint this makes good sense. General intelligence is clearly the most adaptive of human traits, but, equally clearly, humans did not evolve to be good at mathematics. Mathematical talent is a behavioural 'bonus'', the development of which requires a good education, even for, or as I would argue, especially for those with a mathematically disposed brain structure. It is therefore very much to the credit of the NRICH project that it provides such a high level of interactivity, and thus is an exemplar of best educational practice. Bearing in mind a geneticist's conceptualisation of intelligence as a predisposition for learning, gifted children have sometimes been a nuisance for teachers burdened with a large class with their unending and difficult questions, and consequently have often been shunted off to the library for individual work. There is now the danger that demanding gifted students will be shunted off to the computer room to surf the Internet instead. At least with NRICH they can find a quality mentor on-line.
As a closing illustration, a painting by Correggio in the National Gallery, The School of Love , suggests that even the gods - gifted by definition - need a good education.
Here Mercury instructs Cupid. What better deitic patron of the Internet than Mercury, who, to give him a 20th Century spin, is surely the god of Information Communications Technology?
Dr John Geake
Department of Learning and Educational Development
The University of Melbourne
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