Pupils should develop confidence in their own abilities

Perseverance is a quality that is not prized as much as it should be. Today, we live in a society that seeks instant gratification. If something can’t be completed immediately it’s not worth doing, it should be abandoned, in favour of something that can. Another word that isn’t used as commonly as heretofore, is “endeavour”. In our culture of immediacy, why should anyone be expected to struggle on to achieve a long term goal? This is what has bred the cult of television talent and reality TV shows. Why strive when one can gain celebrity and fortune by appearing on the television. Why learn maths, when all you have to do is learn to sing (not necessarily particularly well) or behave outrageously? In life, two of the keys to success are persistence and endeavour, but these are not properly acknowledged either in popular culture or at school. In English schools the government has placed a timetable that says what should be taught by what age, and the schools are running to keep up with the timetable, for should they fail to do so they are adjudged to be failing. This forced pace at which the topics within maths must be covered, removes the time required to allow pupils to practice persistence, for them to learn that they can master a maths topic themselves by repeated application. It also removes the opportunity for pupils to practice endeavour, for their teachers, fearful of criticism themselves and the consequences that such criticism might have on their careers and livelihoods, meet their targets by spoon feeding their pupils rather than by requiring them to think for themselves. At Puppet Maths, we encourage children to think for themselves. We want them to find new ways to approach problems, to learn by trial and error, to succeed by their own efforts so that they learn confidence in their own abilities to succeed at maths.

Don't be put off by the vocabulary

The use of words gets in the way of doing maths. In one study, they found that children could not do calculations based on Pythagoras’s Theorem, but they could do calculations using “three, four, five”. Apparently, children were frightened off by the word “Pythagoras”. For young children, it is a long and a strange word. They try to remember it, and fail because it is such a strange word. They think that because the name of the theorem is hard to pronounce and remember, that what the theorem says must be hard to follow (and what is a theorem anyway, anything that is a “Theorem” must be hard). The phrase “three, four, five” comes from the building industry. Naturally, when bricklayers are building buildings they need to get the corners at 90°. To do this they have a loop of string marked into 12 equal sections. They are taught that they if they peg a length three long to the ground, and then take hold of the string at the point where there are four sections on one side and five on the other, and pull it tight, they will get a right angle where the pegged three sections meet the four sections they are holding. When pupils are taught about “three, four, five”, three words that they recognise and aren’t intimidated by, they find no difficulty with the concept embodied in Pythagoras’s Theorem. Once again we observe how the use of words gets in the way of maths. At Puppet Maths, our puppets introduce our pupils to strange words like “theorem” in a fun and non-threatening way. As a result they come to accept these words as just another word in their vocabulary. They are not put off or intimidated by them anymore and as a result they can focus on the important thing - the underlying maths. This way, we help them succeed rather than be distracted and put off by nomenclature.

Maths should not be torture

What makes anyone succeed at anything? What is it that made David Beckham a superb footballer? What is it that made Bill Gates a great computer programmer? The answer is practice. According to Malcolm Gladwell’s book “Outliers” it is 10000 hours of practice. Would these people have spent so much time practicing their skills in their respective areas if they had not enjoyed doing so? Of course not. Children are capable of speaking English, which has rules that are much more complex than those of mathematics, because they practice it. They enjoy doing so because language is what gets them the things they want. But where is the incentive for undertaking maths? So often, maths is not fun. For many children, maths is incomprehensible and therefore boring. For many children, maths is something that makes them feel powerless and only serves to provide an opportunity for them to earn criticism. I remember myself, aged 7, hiding the fact that I couldn’t do some mathematical calculation or other. During the subsequent lesson, the teacher required us to continue on with the same type of calculations and I had to pretend to be busy and be working diligently even though I was doing nothing. All the time I was under the stress of being found out and getting told off. Worse still, the teacher continued with the exercise during the next lesson - another period of having to hide my weakness! At the end of this third session, the teacher wanted to see my work. Agony. What work? I’d done none! I had to come clean, and admit that I couldn’t do the sums. I expected to be told off. I wasn’t, the teacher didn’t seem to care. I was amazed. As an adult I can see that the teacher was simply happy to know which of her pupils could do the work and which couldn’t, and wasn’t about to punish those who still needed to learn. But the point of view of the young child is so different. I endured the fear of failure at maths for 3 lessons. Other pupils endured it each day and every day. For them maths was nothing less than a daily torture, whose purpose was to expose them as inadequate. At Puppet Maths we want to banish such terrors from the minds of our pupils. We want maths to be unthreatening. We want maths to be fun, so that children will want to practice it, and through practice will become good at it. We want to make maths somewhere where children can feel safe, somewhere where they can feel at home.

Use your imagination to be good at maths

Although younger children can confuse the two, older children know the difference between the world of their imagination and reality. At what age does a child start to distinguish between them? At what age does a child compartmentalise the two? This may well be a reason why children, who are to go on and fail at maths, start to do so around the ages of 8 and 9. Maths being a subject studied at school, where you get praised for getting your sums right, and get told off for getting them wrong is clearly part of the real world. So children compartmentalise maths as being in the real world. But this is not helpful. Maths is all about the imagination, though this is a point missed by most people. I well remember an episode of “The News Quiz” on BBC Radio 4, where the chairman Barry Took made a joke about the comment passed by an academic regarding one of his post graduate students, who had subsequently become eminent in another field, “He didn’t have the imagination to be a good mathematician”. This observation was thought by Mr. Took, his scriptwriters and the audience of the show to be an absolute hoot, but I believe that they were all missing a vital point. Mathematics is all about the imagination. That so many people could find the academic’s comment ridiculous demonstrates how sterile and dry the mainstream manner of teaching maths is. We at Puppet Maths are committed to redressing this failing in maths education. Through the use of puppets to teach maths, we put mathematics firmly in the realm of the imaginative realm of a child’s life. When children are working with puppets they know that they have entered an imaginary world. The imaginative areas of their brains become active, and they become susceptible to learning. We tap into children’s imaginations while we teach, because we know that that is how they learn best.

Maths as puzzle solving

Stan Raimes’s third rule for solving differential equations, like his second rule, also had general applicability. Raimes’s third rule was “if you don’t know the answer, find it by hook or by crook”. Here again we get good advice. This rule tells us clearly that we are to think about our maths problems, and we are to use our imaginations to arrive at a solution. It tells us that there is not necessarily a path already plotted, an algorithm that we can simply slot our numbers into, which will yield us the answer we desire. Because at the start of their maths education, when pupils are still learning the equivalent of the maths alphabet, they are taught to learn set methods to arrive at the correct answers, and they pick up the idea that maths is not about inventiveness, or cunning, or lateral thought, but that it is about mindlessly following rules that they have learnt by rote or repetition. This does them a disservice, and although helpful in the short term, it acts as a block to later success in the subject. At Puppet Maths, we recognise this mental block, and we aim to avoid it by teaching our pupils to approach maths from the perspective of puzzle solving rather than maths problem solving. This is also the approach taken by Singapore Maths, so we feel that we are in good company in our approach.

Manipulating maths problems

As an undergraduate at Imperial College, I was one of the many thousands who were taught maths by the great Stan Raimes. As part of his teaching, he taught us “Raimes’s Rules”. These 3 rules were, as it happens, rules for solving differential equations, but apart from rule 1, they had much more general applicability. In this article I wish to consider the second rule. Rule 2 stated “If you already know the answer, you don’t have to work it out”. The point of this rule is that maths is not about just doing a calculation (as so many pupils think it is), it is about manipulating a problem until it is in a soluble form. This is the first bit of solving any maths problem. However, it is the bit that many pupils just don’t get. They look at a maths problem and they can’t see how to solve it immediately and give up. “I can’t do this,” they tell themselves. The part they are missing is an essential part of solving maths problems. It is at this stage that they need to use logic and imagination to convert the problem into an equivalent one that they know how to solve. But, unfortunately, so often, they do not do this. At Puppet Maths we inculcate our pupils with the concept of manipulating the maths problems they get into a form that they can recognise, so that they can solve them. We teach them the process of how to approach maths as well of how to accomplish the mechanics of calculation.

Thinking in pictures

There are persons, autistic persons, known as savants, who can do complex calculations in their heads at a fantastic speed. This shows us just what the human brain is capable of. Most of us cannot compete at mathematical ability with such people. But these people, as autistics, have difficulty communicating. Can it be that the data processing our brains undertake to communicate inhibits our ability to perform maths? Well we have seen that the names we give to numbers affects our ability to do mental arithmetic, why would the way we think about numbers, in verbal, descriptive terms, not also affect our ability? This is why Puppet Maths teaches its pupils to think in pictures when doing maths. A picture says a thousand words. Thinking in pictures helps pupils perform mathematics. Those who stumble upon this technique go on to succeed at the subject, those that do not are handicapped. At Puppet Maths we want introduce all pupils to the techniques that help them succeed.