David Fowler’s remarkable approach to teaching

David’s history of mathematics course was universally highly regarded, both by students and in all kinds of teaching assessments – for example, the course was singled out for praise by external examiners on numerous occasions. From around 1990 I became involved in the course, first as joint examiner, as an occasional participant in the seminars, and in one year as the course teacher while David was away. David’s approach was completely different to anything in a normal math course. He would set thirty or forty students primary sources to read, and allowed them to make what they could of it, with little intervention. The students then took part in a seminar that consisted more of debate and learning from each other and from their own mistakes than any formal input on his part. When it came to their essay for credit, the student could write on more or less any topic – I double-marked dozens of students’ essays on anything from the theory of music going back to Pythagoras, Greek studies of siege engines, through renaissance accountancy, Galileo, Descartes, the development of rigorous calculus in the 19th century, and much more. In many cases the students took on board the idea of historical rigour (based on primary material) at the same time as they rethought mathematical ideas.

A constant feature of the history course was David’s probing of the students’ understanding of math – it was clear on many occasions that third year students doing history of math were really understanding for the first time ideas such as continuity or basic methods of calculation. The 1998 e-mail from David quoted below (something of a rarity) gives an idea of how he could put conclusions into the students’ minds. David’s sessions with tutorial groups, and to some extent also his lecture courses were coloured by a similar approach, based on respect for the students’ opinions, and the interest and infinite patience required to allow the student to think for himself.

Miles Reid

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Date: Tue, 27 Jan 1998 15:28:59 +0000
To: mastaff@warwick.ac.uk, mastudents@warwick.ac.uk
From: david.fowler@warwick.ac.uk (David Fowler)
Subject: Newton’s solution of a differential equation

In the history of maths class we were working through a nifty procedure that Newton has for – in our terms – solving a range of examples of differential equations dy/dx = f(x,y), where f is a polynomial in x and y, then for more general f. His procedure gives the solution as an infinite series. What, I asked them, in terms of the kind of thing we do today, is Newton doing here? No answer. How today would you go about giving some kind of solution of this equation? No answer, but, I think, genuine puzzlement. Newton, I said, is here extending his repertoire of things you can do with series; does that ring a bell?
Still no comment.

Defeated, I have to say that he’s looking for a solution of the equation as a power series; surely, I ask, you’ve done that. Blank faces. Then, at last, some reaction, as two people piped up: O yes, said one, we did that in a poxy physics course. And, said another, I did it at A-level.
Nobody else would admit to having done or seen it before.

Is this true? Must my history class learn it from Newton? If so, what about the rest?

David Fowler