Ability is usually described as a relative concept; we talk about the most able, least able, exceptionally able, and so on. If mathematical ability is similar to other physical differences between individuals then we might expect it to approximate to a normal distribution, with few individuals being at the extreme ends of the spectrum.
The children/students who are the focus of this article are those few at the top end of the ability spectrum. They may not necessarily be the high achievers, but we'll come back to that issue later. For now let's look at what various writers and researchers have to say about the subject.
Nature or nurture?
Simon Baron-Cohen postulates that able mathematicians are systemisers - highly systematic in their thinking - and this is more predominately a characteristic of the male brain. At its extreme he also suggests it is characteristic of autism, and he is undertaking research to see if there is a
Krutetskii has explored mathematical ability in detail and suggest that it can only be identified through offering suitable opportunities to display it. He says '... abilities are always the result of development. They are formed and developed in life, during activity, instruction, and training. Abilities are always abilities for a definite kind of activity, they exist only in a person's specific activity ... Accordingly, mathematical ability exists only in mathematical activity and should be manifested in it.' (Krutetskii in Mason et al 1986 p119.)
The truth is possibly a mixture of the two - mathematical ability does seem to run in some families, but we also need to offer suitable mathematical activity in order to develop and nurture it.
Abilities change over time
Bloom (1985) identified three developmental phases; the playful phase in which there is playful immersion in an interesting topic or field; the precision stage in which the child seeks to gain mastery of technical skills or procedures, and the final creative or personal phase in which the child makes something new or different. Identifying a highly able pupil at 5 will be different from doing it at 11, or 14, partly because they have fewer skills to exhibit and partly because their abilities may change, but we can often see young children who are fascinated by playing around with number or shape and seek to become 'expert' at it. Such children often:
Krutetskii (1976) would have called this having a 'mathematical turn of mind'. He worked with older students to devise a model of mathematical ability based on his observations of problem solving. The characteristics he noted were:
The message here then is that in order to discover or confirm that a student is highly able, we need to offer opportunities for that student to grasp the structure of a problem, generalise, develop chains of reasoning ... and so on.
High performance and high ability
Trafton (1981) suggests a continuum of ability from
Students who do well on statutory assesments may be represented by any of those three statements because, unless an assessment is designed to promote the characteristics Krutetskii and Straker describe above, it sets a ceiling on what students can do. A hard-working student prepared well for an assessment can succeed without being highly able.
Conversely not all highly able mathematicians show their abilities in class, or do well in statutory assessments. This may be because they are bored, unwilling to stand out as being different, or perhaps have a specific learning disability, such as dyslexia, which prevents them from accessing the whole curriculum.
Bloom B. (1956) A Taxonomy of Educational Objectives: Cognitive Domain New York McKay
Krutetskii V.A. (1976) The Psychology of Mathematical Abilities in School Children Chicago
Mason J., Burton L., Stacey K. (1986) Thinking Mathematically Harlow Pearson
Straker A. (1983) Mathematics for Gifted Pupils London Longmans
Trafton P. (1981) Overview of providing for mathematically able students The Arithmetic Teacher 28(6) Cambridge CUP
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