It is a truism of the HPM movement that ignorance of the history of ones field can seriously
distort one’s perception of what constitutes a barrier for learners; the story of British algebra is a beautiful example. In this paper I describe some of the twists and turns in the strange saga of the transition to symbolic algebra, which prepared the way for novel (non-arithmetical) algebras, which in turn led to the emergence of abstract algebra and axiomatics. It was (surprisingly) the British, passionately concerned with conceptual clarity, who brought about the great transition, not the more formal, rigour-oriented Continental mathematicians. Yet the British did not and probably could not have gone on to invent abstract algebra. The curiously different way in which the British perceived, developed and agonised over their algebra, has, I believe, much to tell us about our own students’ various predicaments in any classroom and country today. External factors such as culture, world-view, personality, political and moral values, would seem to have a much more profound eff
ect than
is usually admitted, on the making of mathematics, and also on learners’predispositions to embrace and make themselves at home in abstract modern mathematics. It was the peculiarly British concern with the meanings and concrete referents of signs – with exemplifications of the concepts represented by formal symbolic mathematics, that fitted them to be the creators of the new algebras. Augustus De Morgan, in particular, emerges from this story as a paradoxical exemplar and spokesman for the view that the strict logic and axiomatics that mathematics historians tend to associate with his name is actually peripheral to the progress of mathematics. It should not be allowed to displace, in the teaching of mathematics, the gradual, delicate construction of conceptual clarity, which was so important in nerving mathematicians to embark upon the audacious exploration of new mathematical worlds in the nineteenth century.