Humdrum memorising of procedures does not 'train the mind':we need to instil a love of ideas, invention and problem-solving, says Zoltan Paul Dienes
So much has been written in the past few decades about the teaching of mathematics, that in a way I am not very keen to add to the pile. On the other hand, from attempts so far to remedy the admittedly very poor state of affairs in the average mathematics classroom, it appears that not much more than a bit of jack-knife surgery has been taking place. The whole reason for teaching mathematics at all needs to be re-examined.
Let us look back to the time when compulsory schooling was being introduced in most European countries. That was at the height of the Industrial Revolution when it was suddenly important that the workforce became literate as well as to some extent numerate. The workers had to be able to read signs of warning, instructions, prohibitions and the like. They also needed to be able handle quantities of merchandise, weights and measures and the corresponding money values. And this had to be done for a large number of entrants into industry and it had to be done quickly.
The "educational" result was that people learned to carry out simple calculations, when they were told what was to be calculated. Rote-learned procedures were perfectly satisfactory to meet these needs.
We have inherited some of these ideas. We still teach one digit numbers first, then pass on to two digit numbers, finally graduating to hundreds, while thousands are supposed to be left until much later. It is conveniently forgotten that the "difficulty" does not lie in the number of digits, namely in the number of different powers of ten, to be handled, but in passing from one power to the next power. It is even ignored that the idea of power is involved in the place value notation. My "multibase blocks" have done something to clear up this confusion, although some textbooks still talk about the base ten blocks as "multibase", not realising that they are contradicting themselves. It seems I have not been shouting loud enough that to learn a mathematical concept, it is best to vary all the variables contained in it. The idea of power has the variables base and exponent.
"Understanding" the arithmetical procedures was totally ignored 100 years ago, since understanding was not an economic necessity. Today jokes are cracked about "never mind if you get it right, as long as you understand it", meaning that understanding is just a new fad, which will soon pass and we shall be back to chanting tables. Yet it is more efficient to learn something that you understand than something which you simply learn as a series of mechanical procedures, rather like Charlie Chaplin in Modern Times. Try to learn a paragraph by heart in a language you don't understand, and then a paragraph in your mother tongue. It is obvious which is going to take longer, and which you will remember longer. Mechanical techniques today are best left to machines that are much better at carrying them out.
Let us turn to the differences between the situation in the 1800s and the situation now. Why do we need to know any mathematics today? Is it because it is useful? Perhaps. Maybe 5 to 10 per cent of the population will need to know some mathematics in order to work in fields like physics, chemistry, biology, engineering, and psychology. It often seems that the way mathematics is taught is not even sufficiently well done for such candidates. I knew a physicist who worked in cosmic rays who had taken an Honours degree in mathematics before doing his training in physics, to make sure he knew the tools that he was going to use. He told me that his colleagues often came to him with questions which arose in the course of their physical research, which they could not handle themselves as they had learned their mathematics "by rote" and in a new field they were at a loss to know how to apply it. Psychologists need to know some statistics, but they can make real blunders, because they do not understand the mathematics behind the statistical tests they use. Learning mathematics by rote does not seem to be very useful even to "important" users.
What about the remaining 90 or 95 per cent, the ones who are not going to use any "sophisticated" mathematics, but just need to be able to get the right change, and measure a room to see if the furniture would fit? Will they work out their arithmetic by using their rote-learned procedures when for a small amount of money they can purchase a little calculator that will do the job? Does the store-keeper need any arithmetic when the checkout computer does all the adding, takes off the discount, adds on the tax, and doesn't make mistakes. Do you need to know some to do your accounts? If the problem is not too complex, a hand calculator will do the job; for anything out of the ordinary, you can buy some software and your PC will do the job while you read the paper.
The amount of arithmetic you need to know for actual use is minimal, and it could be learned in a matter of months. You certainly do not need 12 years of arduous learning. When did you last solve a quadratic equation at the superstore? When did you, in the course of your daily life, have to add 313 to 47? Textbooks abound in such exercises, the vast majority of them totally useless for coping with real-life problems. Why do we do this to our children? The answer often is: "It trains their mind." But does it? Does it do anything else than overload the long-term memory, so that at any moment, any additional useless procedures would spill out the entire content of the "knowledge" acquired and reduce it to near zero?
It might "train the mind" if indeed we gave the mind something to work on which went beyond memorising complex procedures. Is it conceivable that we could present mathematics as an interesting set of interrelated ideas and relationships, whose architecture often gives us gasps of joy and of surprise? It would be of no use to learn to recite "To be or not to be", if we did not at the same time put that masterpiece of philosophising into the dramatic context which is the tragedy of Hamlet. There are many ways of writing a play, constructing a structure that moves forward to its climax:substitute Moliere's Les femmes savantes or parts of Dante's Inferno for Hamlet. The purpose would be the same. Why don't we regard quadratic equations or "fractions" as parts of an exciting structure, and get children to work on all the surrounding ideas and relationships? Why don't we encourage them to invent different algebras? Different rules from the usual ones, and get children to see how we "pay" for certain ideas? When a child adds two fractions by adding the numerators and adding the denominators, he does not really make a "mistake", he merely answers a different question from the one we put to him; in this case he regards the ordered pairs as vectors and uses vector addition. Such "playing about with ideas" would fling open the doors to all sorts of interesting thinking, at this time not possible, because we restrict children to what we want them to hear.
Mathematical thinking is somewhat different from ordinary "everyday" thinking. It is more structured, more architectural. There are also some unexpected twists and turns to be enjoyed, much symmetry and anti-symmetry to be discovered and appreciated. It does not really matter what mathematics is taught from the point of view of usefulness, so we might as well concentrate on the cultural aspect and bring in all the possible fascinating games that can be played, using mathematical ideas of various kinds. Some of the games could simply be puzzles, others could be played competitively as "win or lose" games, incorporating mathematical ideas.
Here is a very simple one: take some red, blue and yellow counters. Make a pile of each colour. Two players play. Each player takes a turn in removing some counters by putting some in a box, as many as they like, but only of one colour at a time. The last person to put any counters into the box is the winner. You can play with as many colours as you like. If you have five colours, there will be five piles, always as many piles, to start with, as there are colours. The winner is always the last player to "play".
The basis for winning is to arrange, mentally, all the counters in little piles of 1, of 2, of 4, of 8, of 16 and so on, of course not mixing the colours. It is always possible to remove a certain number of counters of just one colour, so that the opponent player has an even number of each size pile to play against, unless of course this is already the situation at the start. The player continually forced to play against an even number of each size pile is bound to lose.
A passenger on a liner played poker with a millionaire. After he lost several thousand dollars he said: "I have played your game, now will you play mine?" Our friend let the millionaire win sometimes, but when he won his money back and asked the millionaire if he wanted to go on, the latter politely declined. So you see, these games do have some "amusement value".
Here is an example of a puzzle that you might want to try and solve. The diagram below is a garden.The arrows show a way in which somebody wants to walk round the garden, so as to visit all eight sections. What is a good "mathematical rule" which tells the walker where to go from where he or she happens to be? Given the number of red flowers and the number of white flowers where you are, the rule should be able to direct you to a part of the garden shown by the arrow, telling you how many red flowers and how many white flowers you will find there.
The solution is shown in the lower diagram: you "enter" the number of red flowers and the number of white flowers from the left, and the calculations suggested will produce the number of red flowers and white flowers at the next spot of the walk.
There are many questions that will arise: How many different walks can you plan using the above type of diagram? Do any walks take you back to the house? Do they all get you to visit all parts of the garden? If not, how many parts can you visit by planning such walks? Do some rules tell you not to go anywhere? What would happen if you had bigger gardens, for example by allowing more than two kinds of flower or more flowers of the same colour in the same part?
Could you still visit every part by making an appropriate rule? What kind of mathematics are we doing here? If my friend mapped out a walk with a rule that is hidden from me, how could I discover what the rule was?
Such an approach will stimulate a very different kind of thinking from the humdrum memorising of procedures that are the subjects of many lessons. It is also more suitable for group work. Discussion between peers as to the road to take in solving a problem is a much better preparation for solving problems than if the teacher andor the textbook does all the solving.
My suggestions will be unwelcome, particularly in Britain, where a massive amount of work has gone into creating a national curriculum to coerce everyone into learning everything just the same. If curricula are for the birds, this will all have been a huge waste of taxpayers' money.
Eventually, however, reason will prevail. For the vast majority of us it is really quite irrelevant what we put in the mathematics curriculum. What is important is how the subject is presented. If it were presented as a cultural gem easily adapted for the invention of fascinating games, we could hope to raise a generation of participating players in the colossal achievements of mankind in constructing the wonderful architectural structures known as mathematics, instead of making children miserable through trying to get them to learn what to them is unintelligible gibberish.
Professor Zoltan Paul Dienes is honorary research fellow at Exeter University and at Acadia University, Canada. Anyone interested in information, explanation can email him at zdieneslinx.com (or via his university email at zoltan.dienescadiau.ca)