At the Sixth Form Problem Solving session at Greenwich University on 7 January 2017, my colleague Jo Munday discussed the following problem, as a preliminary to bring various methods to mind before the students tackled STEP questions.
“The line L1 has equation y=mx. The line L2 is perpendicular to the line L1 and goes through the point (0,4). The lines intersect at P. What is the locus of the point P as m varies?”
Two students, Rose and Gunjun, between them, instantly saw that the locus (that means: path) of P is a circle with diameter between the origin and (0,4), because the angle in a semicircle is a right angle.
(Strictly speaking, the locus is the whole circle other than the point (0,0), since that would demand m=∞).
Jo then showed how the same result can be got by algebra, solving the simultaneous equations:
y = mx
and y = (−1/m) (x−4)
Even with Jo’s characteristic very clear exposition, the algebra it was longer and more complicated than Rose’s and Gunjun’s geometric argument.
I asked the students which method they preferred. Some, including Rose, said they preferred the algebra. Why? Because they preferred something written down which makes the result certain.
I thought that the geometric method which enabled Rose and Gunjun to see the answer immediately, without working, was better than the algebraic method in which a neat answer appears as if just by good luck from detailed working.
I draw two conclusions:
One: not only the amount, but also the type, of mathematical ability differs from individual to individual.
Speed and reliability with mental arithmetic helps in mathematics? You’d think so. It does, generally. Great mathematicians such as Leonhard Euler and Carl Friedrich Gauss were also brilliant calculators.
Yet the great 19th century German mathematician Ernst Eduard Kummer, inventor of the theory of ideals in rings, was notoriously inept at mental arithmetic. A widely-told story has him in front of a university lecture hall, needing to know 7×9 for the next step in his working, and forced to appeal to the audience. Is it 61? 67? 69?
What makes that even more odd is that for the first ten years of his mathematical career, while he was already doing ground-breaking research, Kummer had no university post, but was only a high-school teacher. But the story is widely validated.
The leading British mathematicians of a century ago, G H Hardy and J E Littlewood, gave great importance to writing their maths out clearly. If Hardy wanted to understand a proof he was reading, frequently he would do so by writing it out himself. Working on his own research problems, he always made sure that his work was clear and legible. If it became a mess, he would write it out again, from the start, in his beautiful and careful handwriting.
Littlewood also had exceptionally clear handwriting, and his rule to himself was to keep all his working on any one problem on a single sheet of paper.
Yet there have been many mathematicians who have written unclearly. Of one famous mathematician, I forget whom, it is said he once used the symbol “n” with five different meanings in a single line of working.
The eminent probability theorist William Feller was not particularly good at clear writing-out. Of his lectures it is recounted: “Sometimes only one huge formula appeared on the blackboard during the entire period; the rest was hand waving… He took umbrage when someone interrupted his lecturing by pointing out some glaring mistake. He became red in the face and raised his voice, often to full shouting range”.
Terry Tao writes: “A picture is worth a thousand equations”. George Polya, in his well-known book on problem-solving in maths, makes “Draw a picture” one of his first bits of advice.
For thousands of years, right up to and including the time when I was at school, the mathematics of diagrams – Euclidean geometry – was the core of school maths as soon as it went beyond long division and other everyday arithmetic.
Yet the influential Bourbaki group of French mathematicians deliberately included no diagram in their books. In 2B6 I have a copy of Jean Dieudonné’s book “Foundations of Analysis”. Not a single diagram.
Some of the Bourbaki mathematicians thought that diagrams were unrigorous. One of them, Laurent Schwartz, wrote in his autobiography that he had simply never found visual intuition of use in maths. His visual intuition was very poor. He often got lost in the streets. He could never reliably remember even his everyday route from home to university.
The conclusion, I guess, is that wherever possible we should offer a variety of approaches to mathematical ideas – visual, algebraic, arithmetic, and so on.
Two: school mathematics, as it is now, undervalues the visual and geometric, and overvalues algebraic manipulation.
Emmy Noether’s Ph D thesis was a tour de force of “traditional” algebraic manipulation. She soon came to describe it as “crap” and “a mess of formulas” and to develop a whole new approach which won her the description: “the mother of modern algebra”.
In modern algebra – summarised from Noether’s and Artin’s lectures in van der Waerden’s classic book Algebra (there’s a copy of it in 2B6) – the sort of detailed algebraic calculation found in Noether’s thesis is replaced by discussion of concepts.
Skill in algebraic manipulation is useful for mathematicians, as are fluency in mental arithmetic, visual intuition, the ability to write clearly, and, above all, imagination. But, I think, current school mathematics tends hugely to over-value skill in algebraic manipulation.
The switch from the traditional dominance in school mathematics of geometry and the visual has come as part of the same process as the virtual expulsion from school maths of proper mathematical proofs, which I have written about here: https://mathsmartinthomas.wordpress.com/2014/09/05/how-proof-dropped-out-of-school-maths/.
Now there is some geometry in GCSE maths, but pretty much none in AS and A level maths, and not much in many undergraduate courses either. On the other hand, there is more geometry in STEP exam papers, and there is plenty of research in geometry.
In fact, with the publication of Nelsen’s Proofs Without Words, and books such as Tristan Needham’s excellent Visual Complex Analysis (copies of both in 2B6), the visual and the diagram has made a comeback in mathematics outside schools. When Tim Gowers came to speak to us at CoLA, he gave over a large part of his time precisely to “proofs without words”.
Yet students tend to think that getting a result by algebraic manipulation is more solid, respectable, and secure than getting it by geometric reasoning. The opposite is often true. To check through a trail of algebraic manipulation and make sure there is no error in it as it leads to a miraculously simple result is more difficult than to check a (usually simpler) chain of geometrical reasoning.
I remember teaching a trig assignment to do with angles of vision in a cinema at Kelvin Grove State College in Brisbane. The teacher who passed the assignment on to me remarked as a curiosity that the answers were independent of the slope of the cinema floor. This intrigued me, so I found a geometric proof. (While standing at a bus stop in Brisbane city centre, with nothing to write on but an A6 notebook: I couldn’t have done the algebra there and then). The other maths teachers were baffled rather than interested.
Students tend to imagine that more advanced mathematics means mostly more complicated “formulas” and lengthier and more complicated algebraic manipulation. I think they even tend to think that a more complicated solution to a mathematical problem, with more algebraic working, deserves more credit than a quick and neat geometric solution. (They’ve “showed more working”).
Really, to progress in maths students need to develop a drive to find the neatest, clearest way to tackle problems, and an aptitude for understanding new concepts. The approach of dealing with problems by “formulas” and ever-more-complicated algebraic manipulation will break down at some point (depending on the skill of the individual) as they tackle more advanced mathematics. And it will not help students when they come to try to apply mathematics in real-life problems.