The brain

The rapid and dramatic advances in ways that the brain can be studied have changed much of what we thought we knew in the earlier editions of this book. The broad characterisation of behaviours as left brain and right brain may well have to be abandoned. So we will take what we wrote in those editions as an historical introduction.

Some writers have linked cognitive style to hemispheric specialisation. Kane and Kane (1979) suggested the roles played by each hemisphere in a variety of different modes. For thinking, the right brain is described as deductive, divergent, intuitive, holistic, relating to concepts, simultaneity and geometry, whilst the left brain is described as inductive, convergent, segmented, logical and algebraic. Wheatley (1977) and Wheatley et al. (1978) linked problem‐solving styles with left and right brain specialisations. He described the right brain as all‐at‐once and gestalt and the left as one‐at‐a‐time and serial. Wheatley also concluded that a good problem solver achieves a smooth integration of the two modes of thinking.

The interpretation (or speculation) as to what the brain is doing is of interest, in that it gives a neurological perspective on different thinking styles. It seems somewhat dismissive to describe the ‘global’ thinker as intuitive, which seems to infer little conscious thought (though see Kahneman’s System 1), when the intuitive process is probably based on rapid consideration of possibilities, experiences and knowledge rather than being a sort of inspired, unconsidered guess.

Even when our knowledge of the brain was in its infancy Springer and Deutsch (1993) added notes of caution to those who claim that schools fail to educate the right side of the brain: ‘But are these right‐hemisphere functions? We do not think it is as simple as that and there is certainly no conclusive evidence to that effect. Our educational system may miss training or developing half of the brain, but it probably does so by missing out on the talents of both hemispheres.’ Pioneering work by cognitive neuroscientists such as Bugden and Ansari (2015) is now offering the possibilities of detailed and accurate studies of the brain at work and the scope of neuronal plasticity.

Although the location of brain activity is a fascinating area which modern technology is making more accessible to investigation, the way the student presents in class, his observable behaviours, may provide enough information for a teacher when planning appropriate responses and interventions. Cognitive style is often easy to identify if teachers just take time to observe the student as he works and possibly to reinforce those observations with some carefully chosen diagnostic questions and comments.

The teacher’s role

Bath et al. (1986) investigated cognitive style by classifying answers to a series of carefully designed mathematics questions, thus directly taking into account how a child actually does mathematics. One of the main recommendations of our book is the necessity for teacher and child to be flexible in their approach to mathematics – Krutetskii (1976) uses the wonderful word ‘harmonious’ to describe the blending of styles – yet to be aware that learners (and teachers) may not always achieve this goal. Sharma (1989) commented on the need for teachers to be aware of cognitive style:

All of us show and use different and unique mixtures of the two (personalities) but one approach is more dominant than the other in different individuals. And that is what the teacher should be aware of almost constantly. Since you, the teacher, are usually in the controlling role, then the source and sanction for this flexibility must come predominantly from you.

This situation is well expressed by Cobb (1991): ‘We do not mean to imply that the teacher’s beliefs are simply transferred to the student. Rather, the teacher has the authority to legitimise what is acceptable and to sanction what is not acceptable.’

The structure of mathematical abilities

Krutetskii (1976), in presenting a broad outline of the structure of mathematical abilities during school age, specifies a need for flexible thinking and some other skills which dyslexics and dyscalculics may find difficult. He specifies:

• The ability for logical thought in the sphere of quantitative and spatial relationships, number and letter symbols; the ability to think in mathematical symbols.
• The ability for rapid and broad generalisation of mathematical objects, relations and operations.
• Flexibility of mental processes in mathematical activity.
• Striving for clarity, simplicity, economy and rationality of solutions.
• The ability for rapid and free reconstruction of the direction of a mental process, switching from a direct to a reverse train of thought.
• Mathematical memory (generalised memory for mathematical relationships), and for methods of problem solving and principles of approach.
• These components are closely interrelated, influencing one another and forming in their aggregate a single integral syndrome of mathematical giftedness.

Although Krutetskii makes these observations concerning giftedness in mathematics, they are equally apposite for competence and for avoiding failure. The reader can see within these observations where dyslexics and dyscalculics may typically be at a disadvantage and where learning difficulties may create problems.

There are other sources of support for different learning styles (e.g. and in more general terms, de Bono, 1999), but the emphasis of the remainder of this chapter is to paint a clearer picture of the background reasons for, and consequences of, different cognitive styles in mathematics.

Thinking styles

To expand and clarify the picture of the two extremes of the cognitive style continuum, consider some mathematics problems and the solutions which an inchworm and a grasshopper might use. There is no implied value judgement on the two (extremes) of style. Indeed Kubrick and Rudnick (1980) suggested that teachers should encourage a wide variety of approaches, ideas and solutions. As has been already quoted, Krutetskii looks for a ‘harmonious’ approach. If for no other reason, the ability to generate more than one way of solving a problem allows the student to be more effective when checking his own work.

Inferences from the examples

These examples are used to show how learners with the two styles approach problems. Our experience of teaching dyslexic and dyscalculic students leads us to some observations:

• There are some learners at the extremes of the continuum.
• An individual learner may (and should) use both styles sometimes for different questions and sometimes for the same question.
• The style an individual learner uses can depend on the type of question or even on the level of difficulty of the same type of question.
• The type of compensatory strategy (e.g. finger counting or inter‐relating facts) that is used relates to cognitive style.
• Prescriptive curricula create more inchworms than grasshoppers.
• Inchworms with a poor memory for basic facts and procedures are at risk in mathematics.
• Inchworms with a poor working memory are at risk in mathematics.
• Insecure learners are more likely to favour the inchworm style.
• Grasshoppers in the school system need to learn how to document their work.
• Inaccurate grasshoppers are at high risk in mathematics.
• Some questions favour grasshoppers, others favour inchworms, particularly multiple choice questions for grasshoppers.
• Having a dominant style does not mean that the learner is successful in using that style.
• Inchworms want to know ‘How?’ Grasshoppers want to know ‘Why?’

Chinn et al. (2001) in a European study of dyslexic pupils found that, overall, there is a greater tendency for dyslexics to use the inchworm strategies, not because they relate to this thinking style better, but for security and for a perceived minimisation of risk in another manifestation of the tendency to try and avoid errors and thus negative feedback. The study also showed that at that time, before the new Primary Curriculum was introduced in Ireland, the prescriptive nature of the maths curriculum gave the Irish sample a heavy inchworm bias. This would suggest that children can be indoctrinated towards an inchworm thinking style in mathematics by a prescriptive, formula‐oriented curriculum.

You, the teacher, have to be aware of these different styles and the fact that the child may not use the style he has been taught (Hart, 1978) or may, as Duffin (1991) observed, use his own method first and then diligently reproduce for the teacher the standard method he has been taught. Some children may have their enthusiasm for maths crushed if an inflexible cognitive style approach is demanded by the teacher or the curriculum. This is yet another characteristic of learners as individuals, which should warn us that there is no such thing as one method for all.

We could speculate which of the sub‐skill deficits listed in Chapter 2 affect the way a child would solve such problems. For example, poor long‐term memory for basic facts could contribute to a grasshopper style, in that a child may have to overview and combine data in order to avoid facts that he cannot recall. What is clear is that the way a child (or adult) solves a question depends on the blend of deficits and strengths he brings to the problem. You can usually go a long way to finding out how a child solves a problem by asking the simple question, ‘How did you do that?’ This question also encourages meta‐cognition. An interest in how the student is working, rather than a judgement, will be a major source of help for many learners, especially when combined with an awareness of what the child brings to the question.

Contrary to their natural inclination, particularly for examinations, grasshoppers need to learn to know and communicate how they achieved an answer, not just what the answer is. Contrary to their natural inclination, inchworms need to learn, where appropriate, the appropriateness of an answer and why a procedure works.

Zarzycki (2001) explains that almost all school students in Poland are taught only one written method for dealing with the addition, subtraction, multiplication and division of whole numbers. His research supports the claim that there can be detrimental side effects stemming from this restriction in instruction and learning. As mentioned above, our own research with regard to a similar situation in Ireland pre‐1999 also supports this. There is, however, a benefit in having consistency over the years in that parents are not alienated from the way their children are taught mathematics.

Cognitive style and problems

Cognitive style can be used to compensate for some of the difficulties mentioned in Chapter 2. It also can be used to reinforce the connections and links between number facts and between operations and help students interpret word problems. For example, in the question below:

‘Sam wins £96 at bingo and decides to share it equally between his six grandchildren. How much does each child receive?’

At the least sophisticated level of analysis, the word ‘share’ implies that the operation to use is divide. Most children will assume from previous experience that divide will infer a big number divided by a small number, so they will arrive a 96 ÷ 6.

This has now changed a (contrived) real‐life problem into an abstract number statement. An inchworm will try to divide the numbers as they are. It is unlikely to occur to him that both numbers can be divided by 2, making the problem 48 ÷ 3, and, indeed, this may not have made the problem any easier for him.

It is likely that the numbers will be re‐written in the bus stop form 6)96 and a short division attempted (or not attempted, if anxiety takes over). However, if the student has all these skills and facts at his disposal, then the method is effective.

A grasshopper may picture the problem, the £96 being shared out between the six children, visualising the notes and coins being given out in six piles. So, if six £10 notes are given out, then 6 × £10 has been distributed, so £96 − £60 leaves £36 to distribute. Now six £5 notes can be given out, so 6 × £5 has gone, with £36 − £30 leaving £6 to share. So each grandchild has received £10 and £5 and £1, a total of £16. This may be done by variations on this theme, such as sharing £6 as £1 each, which leaves £90 which shares as £15.

The ‘division’ question has been changed into a question about the subtraction of chunks or partial products. It has been re‐interpreted linguistically and mathematically. The student has been able to use facts within his memory bank. An inchworm is highly unlikely to tackle a division problem by involving multiplication.

Teachers and cognitive style

A learner may not have the same cognitive style as his teacher. If neither party is aware of this there are likely to be consequences, not least in communication and in judgement of work. For example, the marks awarded for a method could be affected. Chinn (1995) has carried out a survey of teachers’ marks for three very different methods, which all resulted in a correct answer, used to solve a word problem. The range of marks for each answer varied from 0 to 10 (out of 10), so a child could be with one teacher and receive an encouraging 10 or be with another teacher and score 0 out of 10. The awarding of a particular mark can usually be rationalised by the teachers concerned. This could be a very good reason for a mathematics department to meet and discuss marking policies. Mackay (1994) noted the restricted range of methods accepted for mathematics tests at Key Stage 2 (age 7–10 years) of the English National Curriculum at that time.

Teachers should be aware of their own cognitive style. If they are towards the extreme ends of the continuum, then they will need to make a conscious effort to ensure they are communicating with learners from the other end of the spectrum. This mismatch of cognitive style between learner and teacher could be one of the reasons why students say, ‘I could understand algebra the way Mrs Jones explained it last year, but I can’t understand the way Mrs Smith explains it this year.’

Sometimes when we lecture for teachers we may ask them to try some questions which can identify their own cognitive style. It is interesting to note in that almost every group that we have done this with, internationally, there is no dominant style within the group, rather a close to even split between the two styles.

Sharma (1989) and Marolda and Davidson (2000) have noted that quantitative and qualitative personalities prefer different learning materials, distinguishing between discrete or quantitative materials (e.g. number lines, bead strings) and continuous or qualitative materials (e.g. Cuisenaire rods, empty number lines). This is just one of the factors teachers have to take into account when choosing concrete materials to illustrate a mathematics concept. Choosing appropriate materials is another teaching skill with, once again, a range of responses from the learners.