Human ecology
Mathematics education
Psychology of learning mathematics
Project Draft
Project Draft - 30/3/00

Divergent thinking and the dialogue between biology and mathematics

Review of literature

The dialogue between biology and mathematics,

Divergent and convergent thinking


These definitions where introduced by Guilford in the nineteen fifties as part of his structure of intellect model. Guilford was unhappy with the one or two factor models of intelligence that where advocated by Pierson, Spearman, Birt, etc. There had been attempts to contrast intelligence (as measured by I.Q. tests) with "creativity", however Guildford found that these terms were unsatifactory. Guilford constructed a multi-factor model of intellect from factors that emerged from intelligence tests and theoretical discussions of intelligence. This model replaced the terms "IQ intelligence" and "Creativity" with the terms "convergent production" and "divergent production", respectively. Guilford factors are as follows:


Operations Products Table XX. Guilford’s "Structure of intellect" model. (Guilford,

These factors are combined to identify one hundred and twenty different abilities. Each ability has a specific content, operation, and product. Guilfords...

How do these abilities affect the relationship between biology and mathematics. In the sixties, Liam Hudson carried out tests the related the convergent and divergent abilities of sixth-form boys against the subjects that they choose to study (Hudson 1966). Hudson was looking for a way to predict which pupils would study arts or science. His main result was that:

"Between three and four divergers go into arts subjects like history, English literature and modern languages for every one that goes into physical science. And vice-versa, between three and four convergers do mathematics, physics and chemistry for every one that goes into the arts." Hudson (1966, pp.56-57)

"The converger is the boys who is substantially better at the intelligence test than he is at the open-ended tests; the diverger is the reverse. In addition are the all-rounders , the boys who are more or less equally good (or bad) on both types of test." (Hudson, 1966, p.55)

Hudson’s data is better represented by a table:
Null hypothesis Physical science Biology History
Extreme divergers 10% 3% 4% 16%
Mild divergers 20% 12% 15% 34%
All-rounders 40% 32% 54% 39%
Mild convergers 20% 36% 15% 7%
Extreme convergers 10% 18% 12% 5%
Number of students 104 26 44
Significance 0.1% >20% 5%
Table 1.

Orton quotes these results in his exploration of theories of mathematical learning:

Orton also says that "only a minority of students coped equally well with convergent and divergent items" (ibid. p.113). However, the terms used by Hudson may be misleading. On the one hand, Hudson uses convergent and divergent tests to signify I.Q. tests and open-ended tests. On the other hand he uses converger and diverger to signify those boys who do better in one style of test that they do in the other style.

We are looking here at two different ways of talking about the data. Firstly there is the method that samples convergers and finds that more convergers study science than arts. Secondly there is the method that looks at a subject and finds the distribution of convergers, all-rounders and divergers particular to that subject. As said before, Hudson was looking for a way to predict which pupils would study arts or science. Orton is looking for explanations of why "some pupils achieve more than other."

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Created 5/4/00
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