*Disclaimer: as I am only starting to read up on these things, there may be some errors below – everything is true to the best of my knowledge, but my knowledge is as yet shallow.*

For the last few weeks, I have been trying to read about mathematics education at the University level. (This is in preparation for my Essay 2, which will be about comparing the first year of University in the US/England, and more importantly my Thesis, due *next* August, plans for which currently revolve around the question “Your first term as a Cambridge mathmo – how does it change your view of mathematics and your view of yourself?”)

One of the things I have come across is Tall’s Three Worlds of Mathematics (link takes you to a page on his website which has lots of further reading). David Tall is a researcher based at Warwick University, which gives him credibility in my eyes … plus he’s coauthored a couple of maths textbooks with Ian Stewart, which shows that he can write for mathmos (as opposed to writing for educationalists).

The idea of the Three Worlds – which all overlap, by the way – is that it’s useful to think about three different sorts of mathematics/contexts for mathematics: Embodied, Symbolic, and Formal. People learn each of these and develop intuition in them in different ways. Here is a quotation from this paper which explains:

In Gray and Tall (2002), we presented the idea that there were three … fundamentally different types of object, those that arise through empirical abstraction (in the sense of Piaget) by which is meant the study of

to discover their properties, those that arise from what Piaget termed pseudo-empirical abstraction from focusing on actions (such as counting) that are symbolised and mentally compressed asobjectsconcepts(such as number), and those that arise from the study ofand the logical deductions that follow from these found in the modern formalist approach to mathematics.properties

Or in slightly more concrete terms: for example with embodied maths, you perceive things and can “see” why they’re true, such as finding properties of triangles. You need not be able to manipulate symbols that correspond to them (symbolic), or to formally prove the properties from some set of axioms (formal), but you can see them.

Or similarly you may be able to do things with symbols, without necessarily being able to “see” any essential meaning behind them (embodied) or do things from axioms (formal) – an example is the early use of complex numbers, when people didn’t know what these numbers *were* but they seemed to help you solve cubics.

I’ll admit I can’t think of a way to have a specific *thing* that’s formal but neither embodied nor symbolic – mostly because to have such a thing you’d have to write it down, which involves symbols … nonetheless it is a different world because instead of starting with things you have and looking for properties and ways to manipulate them, you sort of start with the properties / rules for manipulation you want, and try to create axioms that capture/preserve them. And then move from there.

I like the way that he presents his theory, not claiming that this is the only way to look at things but that it is a *useful* way to look at things, for example (from here)

I began to realise that the notion of three different worlds of mathematics:

- (conceptual) embodied
- (proceptual) symbolic
- (axiomatic) formal
offered a useful categorisation for different kinds of mathematical context. Each has its own individual style of cognitive growth, each has a different way of using language and together they cover a wide range of mathematical activity.

In terms of my own mathematical experience and understanding of maths, this seems to ring true.

*What do you think – does it sound plausible to you? I’d be really interested to hear other people’s thoughts.*

Two vague comments:

1) It seems to me that emipirical abstraction leads to pseudo empirical abstraction which leads to formalism. We are in a time of mathematics when the resulting formalism of many of the most obvious mathematical concepts has been developed (which took hundreds of years). What is nice is that the formalism can, if taken far enough, point to the existence of theoretical objects which, when discovered, start off the whole process again. However, this is well beyond degree level maths and thus not a facet of mathematics encountered any more by most people.

2) Some things in maths are known to exist but it is not known how they exist. A good example is the ordering of the real numbers such that every subset has a least element (see http://en.wikipedia.org/wiki/Well-order if you don’t know what this means). I’d suggest that this ordering is entirely formal. However, if it is ever defined than playing around with it (empirical abstraction) will doubtless lead to many new mathematical discoveries

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