I posted recently about the macho culture of pure mathematics, and the undue focus that school mathematics education has on problem-solving and competitive games.

I have just encountered an undated essay, *“The Two Cultures of Mathematics”, *by Fields Medallist Timothy Gowers, currently Rouse Ball Professor of Mathematics at Cambridge. Gowers identifies two broad types of research pure mathematicians: *problem-solvers* and *theory-builders*. He cites Paul Erdos as an example of the former (as I did in my earlier post), and Michael Atiyah as an example of the latter. What I find interesting is that Gowers believes the the profession as a whole currently favours theory-builders over problem-solvers. And domains of mathematics where theory-building is currently more important (such as Geometry and Algebraic Topology) are favoured over domains of mathematics where problem-solving is currently more important (such as Combinatorics and Graph Theory).

I agree with Gowers here, and wonder, then, why the teaching of mathematics at school still predominantly favours problem-solving over theory-building activities, despite a century of Hilbertian and Bourbakian axiomatics. Is it because problem-solving was the predominant mode of British mathematics in the 19th century (under the pernicious influence of the Cambridge Mathematics Tripos, which retarded pure mathematics in the Anglophone world for a century) and school educators are slow to catch-on with later trends? Or, is it because the people designing and implementing school mathematics curricula are people out of sympathy with, and/or not competent at, theory-building? Certainly, if your over-riding mantra for school education is *instrumental relevance* than the teaching of abstract mathematical theories may be hard to justify (as indeed is the teaching of music or art or ancient Greek). This perhaps explains how I could learn lots of tricks for elementary arithmetic in day-time classes at primary school, but only discover the rigorous beauty of Euclid’s geometry in special after-school lessons from a sympathetic fifth-grade teacher (Frank Torpie).