Mathematics appears in the most unlikely places. Part of the teaching challenge is to exploit children's imagination and sense of wonder and make the subject more appealing. Two masters of imagination and wonder were Lewis Carroll (1832-1892) and M C Escher (1898-1972) - both exceptionally individual mathematicians.
Carroll is well known as the creator of Alice in Wonderland and other adventures. Less well known is the fact that Carroll was an Oxford mathematics don and that Alice contains a wealth of mathematical allusions.
Escher has become a cult figure among mathematicians. His "Drawing hands" design provokes subtle questions of start and end, chicken and egg, and distances and infinity. I recently heard a child point out similarities between this and a book which has a photo of itself on the front cover, and with a pattern of 16 triangular polydron tiles which contained its own pattern within itself - what mathematicians call "self-similarity". The main difficulty for this seven-year-old was not the concept, but the vocabulary needed to describe what had been observed. A "self-similar" object is one which contains smaller images of itself, ad infinitum. Thus a straight line is self-similar; so are fractals; and so are many images of Escher as well as ideas in Carroll.
The fantasies of Carroll and Escher are not unlinked. Recent research suggests that Escher was influenced as a child by Carroll's writing. Both have centenaries next year: one died within a few months of the other's birth. Serendipity demands that we explore what happens if, when Alice falls down the rabbit hole, she arrives not in Carroll's Wonderland of Tweedledums, Red Queens and other mock-symmetries, but in an "Escherland" of tessellating mazes, impossible objects and visual paradox. There is marvellous scope here for imaginative cross-curricular groupwork in the classroom. Here are some ideas culled from Carroll and Escher: The number 42
This number was one of Carroll's favourites. There are many obvious references, for instance in The Hunting of the Snark, as well as more hidden examples, as in Alice where the three playing cards at the Queen's croquet match multiply together to make 42. 42 is a good number for exploring how many factors a given number has. (42 has eight factors: 1, 2, 3, 6, 7, 14, 21, 42 because it is a product of three primes, like 30, 66, 105, . . .). No one knows why 42 held such a fascination for Carroll. The most Carrollean explanation I have heard is that his young photographic subjects were invited "for tea two".
Carroll published a book with this title in 1895 - supposedly he devised (and solved) the problems one night when unable to sleep. Here is an example: A bag contains one counter, known to be either black or white. A white counter is put in, the bag shaken, and a counter drawn out, which proves to be white. What is now the chance of drawing a white counter? (Answer 2 in 3 chances.)
The next problem is more complicated, and is adapted here to fit in with classroom work:
Several children are sitting in a circle, each with a certain amount of money. Counting round the circle, the first child has one penny more than the second, who has one penny more than the third, and so on. The first child gives one penny to the second child, who gives two pennies to the third, and so on, each giving one penny more than received, as long as possible. After this there are two neighbours, one of whom has four times as much as the other.
1 How many children are there?
2 How much money do they have in total?
(Answers: 7 children; 35 pence.) Tessellation workshops
The software Tesselmania and Escher Interactive produce tessellating patterns of characters from Alice. Tesselmania, accessible to six-year-olds, introduces a classification notation for tessellating patterns reminiscent of that used by Escher. It is an ideal introduction to ideas of symmetry at a range of different levels. Geometer's Sketchpad can be used in a similar way with older pupils.
Carroll also had ingenious ways of dividing by 9. For example, to divide 2,831 by 9, first find the remainder: this is the digit-sum remainder: 2 + 8 + 3 + 1 = 14 which has remainder 5. His method and words are summarised as follows:
5 45 145 2145 2831 2831 2831 2831
4 14 314 314
"To find the quotient, we write the remainder 5 over the units-digit of 2831, and subtract; 5 minus 1 is 4. The figure 4 is written below the 1, and is the least digit in the answer. The figure 4 is also written above the 3 in the tens-column. Subtracting in the tens-column; 4 minus 3 is 1. The figure 1 is also written above the 8 in the hundreds column. Subtracting in the hundreds-column; 1 minus 8, borrowing 10 from the thousands-column; is 3. The figure 3 is written below the 8 in the hundreds-column, and is the hundreds-digit in the answer. The figure 3 is also written above the 2 in the thousands-column, but since a number has previously been borrowed from this column we change the number to 2. The subtraction is now complete."
Nothing ruins jokes like over-analysis, but if we ask for major conceptual links between Carroll and Escher, the answers much be a) symmetry and b) self-referencing. The latter is central to Escher's designs and Carroll's paradoxes, as examined extensively in Hofstadter's Godel, Escher, Bach. Symmetry, on the other hand (and symmetry-breaking) is central to all forms of art, especially those with strong geometrical structuring like Escher's. It is also central in Carroll's writing - the looking-glass is an obvious source, as are chess-pieces, playing cards and Tweedledum and Tweedledee.
John Bibby is a retired university lecturer who now runs QED Books, specialising in maths and numeracy