Transfer, Abstraction & Variation

If I understand it correctly, the question of transfer of knowledge between different domains of application – in particular of mathematical knowledge – has come to the fore in the US mathematical community the last year. I haven’t followed this debate, but just became aware of it during my visit to Beloit College. I will nevertheless offer my view of it based on the little I know.

Let me start with the Is algebra necessary? discussion. In another article What Is Mathematics For?, Underwood Dudley, argues that algebra is never used in the workplace, except in highly specialized jobs (engineers, scientists, … – the usual suspects). A great majority of all the people who study algebra in school never use it after leaving school. I think this is basically correct. One cannot argue for algebra – or mathematics in general – in schools on the grounds that it is explicitly used in many jobs. But then Dudley offers a reason for keeping algebra in the schools: It teaches the human race to reason! It might not always succeed – but it is the best method we have, Dudley argues.

This, however, presupposes that the phenomenon of transfer occur. Namely that knowledge from one domain, or skill, transfers to another domain, or skill. I think it is well known among researchers in pedagogy and learning theory (I have no references at hand – but have heard it pointed out several times) that transfer seldom occurs. It is rare. This was also pointed out in a response Does Algebraic Reasoning Enhance … by Peter Johnson.

Then in another article I got in Beloit: The Advantage of Abstract Examples in Learning Math, by Jennifer A. Kaminisky, Vladimir M. Sloutsky and Andrew F. Heckler argue (based on experiments) that transfer occurs more easily from abstract circumstances to concrete than between different concrete circumstances.

Now, if there is some truth in this, that would be an argument for abstract approaches to mathematics in schools. This also connects with a discussion I had with Dave Ellis at Beloit College who also stressed the importance of abstraction, although it was in a somewhat different context.

Let me throw in one more log in this fire: There is another take on the transfer problem pioneered (among others – I guess) by the Swedish researcher in Learning Theory, Ference Marton. Namely the theory of Variation: Variation Is the Mother of Learning, Marton, F. & Trigwell, K. (2000). Higher Education Research and Development, 19(3), 381-395.

An extremely abbreviated description of their theory is: Schools can only teach what is already known, about the past essentially. At the same time, the purpose of school can be construed as preparing for an unknown future with new problems. The theory claims that that can be done by explicitly varying contexts and circumstances. The more variations, the better. This is not really transfer from one context to another, but rather abstraction/generalization from many examples with subsequent specialization to new contexts.

First many different contexts, then abstraction, then specialization to a new context

I have been very brief here – but perhaps that’s a good thing – it leaves room for leaps of imagination!

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