For rather more than three years a small group of gifted students have been coming to our home on Saturday mornings
to work at mathematics.
Two principles are involved in choosing topics for such a group. The first, and most important. is that there must
be sufficient intellectual content to excite interest. The second is that the activity should in some way assist
the students in achieving their lifegoals. GH. Hardy expressed it well when he wrote that a good mathematical discovery
is simple, surprising and fruitful.
Of the original group of four students, two are still with me. They are Ursula and David, who were 15 when we started.
James joined us in 1978 at the age of 14½ and Andrew in 1979 at the age of 14. My students are not primarily
mathematicians though they obviously enjoy mathematics. Ursula's interests lie in the direction of literature and
painting. David plans to do veterinary work: he has already spent a couple of weeks at the Pasteur Institute in
Paris and conducted a chemical experiment aimed at reproducing the conditions on earth when the first steps towards
life were taken. James has an expert knowledge of computing. These three came from Chesterton comprehensive school,
where there is a great concern to let the gifted forge ahead. Andrew is from Manor comprehensive school. Already
at the age of eleven he had been recognised as a musical composer of promise. An alert teacher, Miss Sinkinson,
brought him to my attention after he had produced an original mathematical idea.
A question of principle arose at the outset. In the two decades prior to 1977 there had been a great reaction against
manipulation. The effect of this was apparent; in spite of their ability and wide knowledge, the students knew
essentially nothing of
traditional elementary algebra and were painfully slow at any work involving operations on symbols. Some would
say this does not matter, that higher mathematics does not use the older techniques. My own view has always been
that this is totally false: classical and recent mathematics stand in the same relation that fruit and vegetables
do to vitamin extracts: familiarity with the former aids understanding of the latter. And certainly for applications
to science, elementary algebra remains central. I believe that any child, at all bright, should receive a stimulating
introduction to algebra at the age of 9 and should work frequently at interesting applications of the basic techniques
until these become as automatic and effortless as reading and writing. It is rather like imprinting in animals:
I am doubtful if any later treatment will fully compensate for early neglect in this respect.
Needless to say, I did not inflict mind-destroying drill on my little group. I knew there was some interest in
astronomy and raised the question;
- the equation y = x2 gives
the correct shape for a reflecting telescope; if light falls on it from the appropriate direction, where will the
reflected light come to focus? Shortly afterwards David raised the question:
- there is a standard equation for an ellipse; how can we connect this with the
construction that uses a string around two drawing pins? This is not difficult to do but it ca1ls for considerably
more complicated and sustained work than anything in the S.M.P. programme used in Cambridgeshire.
Co-ordinate geometry, with its linking of algebra to diagrams, offers great possibilities for exploration. Later
on, the group seemed to find artistic interest in Newton's Diagram of Squares.
This allows one to sketch without difficulty curves such as-:
x5
+ y5 - 5x2y = 0
and
y2 =
2x3y+x7 = 0
in which loops appear that suggest some strange apparatus for an alchemist.
Some historical studies were used:
- how Bombelli and Poncelet arrived at the 'impossible' ideas of imaginary numbers
and imaginary points:
- Euler's remarkable fusion of trigonometry with expotentials:
- how the work of Wallis and Newton led to the binomial theorem and culminated
in Euler's discovery that you could define the factorial of a fraction.
In 1979 we saw how Fourier series led to Riemann's concept of integration and how
a paradox of Borel paved the way for Lebesgue integration. All of these gave the stimulus of novelty, as did also
Cantor's idea that one infinity could be larger than another.
We had no set system of working. Sometimes I would explain an idea; sometimes they
would read and work examples from a variety of books: at one stage James assumed the role of teacher, and took
them section by section through Piaggio's Differential Equations as far as the solution of partial differential
equations by Fourier series. Sometimes the students themselves raised questions or asked for a particular topic,
and this was always looked into if any of us knew anything about it. One puzzle that they produced was new to me:
- a cylindrical hole is bored centrally through a ball, you are told the length
of the hole: find the volume of the remaining material. The surprising thing is that the data is sufficient - you
do not need to know the radius of the ball.
Another problem came from an original observation: they had noticed that if you put a number into a calculator
and keep pressing the key that gives the cosine of a number of degrees. you always seem to arrive at 0.999 847
742. Why was this? This led to an interesting discussion on solving equations by iteration.
The group also raised a question about secret codes that eventually led us to study finite fields.