Human ecology
Mathematics education
Psychology of learning mathematics
Project Draft
Draft 18/5/00: Review of literature


The dialogue between biology and mathematics. Top

The exchange of ideas between mathematics and biology was very important during the 20th century. This constructive dialogue has been helpful in passing mathematical ideas to biology and in passing biological ideas to mathematics. On the one hand, the mathematical ideas of S. Wright, R.A. Fisher, and J.B.S. Haldane in the nineteen thirties enabled the synthesis between proponents of Darwin and Mendel. On the other hand, neural networks, genetic algorithms, Lindenmayer systems, and chaotic systems are branches of applied mathematics which have directly emerged from biological examples. Recent work has included approaches that re-apply these branches of mathematics to new areas of biology. Dennis Bray has applied neural nework methods to protein interactions within cells (1995). Denis Noble has used supercomputers to model the functioning of the heart from communication between its individual cells (Noble).

The successes of this dialogue asks us to question the relationship between the two subjects. The successes of the mathematical methods in physics has lead some philosophers to suggest that the universe is fundamentally mathematical. This might lead us to assume that all of biology should be expressed as mathematics. On the other hand, some, such as biologist Ernst Mayr, that there are significant differences between mathematics and biology. Specifically, Mayr suggests that mathematics finds single answers to problems, whereas "it is quite possible that in biology the majority of phenomena and processes must be explained by a plurality of theories" (Mayr, 1997, p.68). Mayr concluded that biology has more in common with history (Note: qualify this caricature). In the middle ground are those that admit both differences and similarities between the subjects. For them, a common discrimination is that mathematics deals with universal answers whereas biology deals with unique examples. (Stewart, Sigmund, etc)

How does this dialogue relate to the mathematics education of biology students? We cannot assume that students who are good at biology will be good at mathematics. If we require biology students to be good mathematicians, we may exclude students with the potential to advance the field of biology. We are looking at two questions:

This essay will concentrate on the second of these to questions. As will be seen in the next section, divergent thinking is one example that distinguishes between the methodologies of mathematics and biology.
Guilford's 'divergent' and 'convergent' operations. Top

Definitions:

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 Guilford 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:

Content

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. Guilford devised tests for these abilities. A criticism of this approach is that these abilities may be dependent on the tests - that the tests only measure an ability to do well at that particular test, rather that any underlying intellectual ability (Gould, 1981: 1996, p.339). Guilford terms this the "operational definition of intelligence" (1967, p.13). However, by finding correlations between many different tests (the approach of construct validity), we may be able to locate underlying abilities (Guilford, 1967, p.13-14).

How do these divergent and convergent abilities affect the relationship between biology and mathematics? Here we can restate Mayr's criticism, that mathematics provides single answers whereas biology needs multiple answers. Based on this criticism we could say that (from Mayr's perspective) mathematics is a subject that requires convergent thinking whereas biology requires divergent thinking. We can shed more light on Mayr's perspective if we look at the work of Liam Hudson.


Liam Hudson: divergence and the arts/science division Top

In the nineteen sixties, Liam Hudson carried out tests which 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.

Hudson uses slightly different definitions to Guilford. On the one hand, Hudson uses convergent and divergent tests to signify I.Q. tests and open-ended tests (this is similar to Guilford's definitions). 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.

“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)

In other words, Hudson uses diverger and converger to express bias towards a particular way of thinking. A pupil may have moderate divergent ability, but as long as this ability is greater than their convergent ability they will be classified as a diverger. (ibid.)

The two open-ended tests Hudson used for his study were the "Use of Objects" test and the "Meanings of words" test. These relate to Guilford's definitions as follows:

For his general observations, Hudson aggregates the results of these two tests to obtain divergent ability score. However, for more detailed work, he deals with the tests separately. (Hudson, 1966, p.55) One criticism of these tests could be that they only test divergent thinking within linguistic (semantic) abilities. There may be significant differences to these results if divergent tests were conducted on mathematical abilities. However, Hudson's earlier work on convergent tests suggested that mathematical or verbal bias was not a significant factor in discriminating between arts and science students.

Based on Hudson's tests and definitions, 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) quoted in Orton (1992, p.112)

Furthermore:

"As far as one can tell with the samples available, ... biology, geography, [and] economics ... courses attract convergers and divergers in roughly equal proportions." (Hudson, 1966, p.57)

As was said before, Hudson was looking for a way to predict which pupils would study arts or science. We, on the other hand, are looking to find the distribution of convergers, all-rounders and divergers particular to maths and biology. From the slightly limited data that Hudson gives (1966, p.180) we can construct a table of his results. (Note: as Hudson does not give the data on mathematics, I will use the physical science data in its place.)

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.

Notes: (Statistical tests: Kolomogorov - Smirnov test. Limitation of use of K-S test in this case. Degree of independence of disciplines.)

From the above data it would appear that biology occupies an intermediate position between physical science and history. By inference, we could say that (from Hudson's perspective) mathematics has a bias towards convergent thinking, history has a bias towards divergent thinking, whereas biology shows little bias towards either convergent or divergent thinking. To relate this back to mathematics education, the next section will look at Orton's treatment of Hudson's results.


Orton: divergent thinking and mathematics education. Top

Orton examines divergent thinking as it affects the question "Why do some pupils achieve more than others?" (1992, p.110-113). Although he notes that divergent thinking raises more questions than it answers, I will catagorise his observations into three main topics.

Orton notes that tests in mathematics largely require single answers and therefore convergent thinking. Furthermore, when compared to other disciplines (subjects), 'the opportunity for divergence is not as great' (p.112). Although it is my feeling that better examples of divergence may be found in mathematics, this is outside of the scope of this essay. In passing I will mention that Guilford could not find a numerical example of a divergent production of symbolic classes (DSC) test whereas for divergent production of symbolic relations (DSR), several numerical tests existed but no letter tests (1967, pp.145 & 147). I will return to this question in a future essay.

As there seems litte evidence from divergent thinking in the content of mathematics examinations, Orton looks at the methodological implications of divergent thinking. The main inspirations behind divergent thinking in mathematics seem to be positive moves towards creativity, and negative moves away from absolutist mathematics. Another perspective on this direction of teach seem to be moves towards transferable mathematical skills and away from pure mathematics. This question is less important when dealing with mathematics for biology students (where the interdisciplinary aspects of mathematics are required). On the other hand, it may be noted that blue sky research has a mythos (new word?) of providing new insights and new solutions. (Example: how these ideas have been embodied in the National curriculum)

These first two aspects of Orton's work have greater applicability to 'pure' mathematics. On the whole, mathematics seems to be largely convergent. However, there seems to be at least a limited forms of divergent mathematics and divergent teaching may be useful for interdisciplinary work.

The final aspect of Ortons examination is perhaps the most important to mathematics teaching of biology students. How these distributions of divergent and convergent thinking express / embody themselves in the abilities of individual pupils? Orton takes a slightly different reading of Hudson from the one I have outlined above. The first difference in our readings is that 'only a minority of students coped equally well with convergent and divergent items' (ibid. p.113). The second is that 'mathematics students are predominantly convergent thinkers' (ibid. p.113).

The first difference is the most important. I can find no reference to this in Hudson's work. Hudson was measuring convergent / divergent bias and defined an interval of 'all-rounders' along with his intervals of 'converger' and 'diverger' (see above). Hudson selected the polarised positions of converger and diverger to show the differences in arts / science choices. Much of Hudson's work analysed differences between these polarised positions, and neglects the middle ground of all-rounders. This may have led to Orton's reading that there was only a minority of all-rounders. Elsewhere, Orton suggests that there may be a spectrum of abilities from converger to diverger (ibid, p.110)

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. 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 others.” These two...

[Work in progress]

...work suggests that biology requires both convergent and divergent thinking.


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Created 22/5/00
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