Maths is a science

Maths is a science. What is a science? How's it differ from an art? The best way to compare the two is to consider cookery. Watching Gordon Ramsay at work in his kitchen, it is obviously a place where artists are at work (and in Mr. Ramsay's case a tempramental artist). The results are variable, and if they are not good enough, they are binned and the creative act is performed again. Each time there is the chance that the product will not meet Mr. Ramsay's high standards. It is a matter of the skill of the individual chef whether or not these standards are met.
A science, on the other hand would be the cooking occuring at a cake factory. The owners have measured everything. They mix the ingredients in known quantities, they stir for a known time, with a known force; a known amount of the mixture is poured into a container, which is placed in an oven at a pre-determined temperature, which is closely monitored, for a particular period of time, and the temperature profile is controlled over that time. The factory owners know that if they repeat everything in exactly the same way, every time, they will arrive at the same results every time. There is no swearing of the shop floor (and very possibly swearing there would be a disciplinary offence), the pressure to perform is off, the system does the work.
Mathematics is a system that is used for problem solving. If one is faced with a problem, one can either follow the Gordon Ramsay model, behave like an artist and rely on one's own skill and flair to work out the logic of the solution; alternatively, one can use the science of mathematics, which has already been used to solve many types of problem successfully. All one has to do is recognise the type of problem one is facing and apply the appropriate proven strategy. The choice is yours.

Different exams different approaches

I have recently been looking at the International "O" level exam for chemistry. Yes, maths is involved in chemistry too. This is the exam that children are expected to take at the age of 16, the same age as children take the GCSE exam in the UK. Of particular interest to me were the calculations regarding the number of moles. In the UK's exam the pupils are given a chemical equation. They are told the weights of reagents (or concentrations and volumes of any solutions used) and asked to find the quantity of product. All very straightforward.
For the International "O" level, which is taken in places like Singapore and Hong Kong, the pupil has to know the chemical formulae for the reagents, be able to put them into a chemical equation and balance that equation, do the moles calculation (which is identical to the GCSE calculation), and then draw a conclusion from the numerical result of that calculation. Basically this is 4 separate GCSE questions all concatenated. These 4 questions are not signposted for the pupil as separate parts of the question, the pupils have to know the route through the problem themselves.
The skills needed to do the "O" level question are all maths skills. Maths reasoning is needed to determine the chemical formulae of compounds. Maths skills are needed to balance the chemical equation, and maths skills are needed to do a moles calculation and draw a quantative conclusion. Are we saying that in the west are children are unable to do the "O" level type question because they lack mathematical ability, and have to be fobbed off with the GCSE question?
I would suggest that the "O" level question more closely represents the skills that an employer would want from a laboratory technician, than does the GCSE, which at best demonstrates a partial ability.
Globalisation means that the western nations are competing head on with the likes of Hong Kong and Singapore. But how will our children be able to compete if they are not equipped at school with the knowledge the need? Are we letting our children down?

Maths using images of real things

Young children do not think in an abstract manner. They like to think of things. This is called concrete thinking. When given a maths problem, for example 2 + 3, they don’t think of an abstract 2 and an abstract 3, they want to think of 2 things and add to them 3 things. This is why they count on their fingers. In primary schools the teachers tell the pupils not to count on their fingers. This is an attempt to force them to abandon concrete thinking and to embrace abstract thought. I believe that this is wrong. Children will start to think in an abstract manner as and when they are ready to do so. If they need their fingers to count on I think that they should use them. Certainly, teachers should hold up abstract thought as a target for the child to aim for, but don’t ban them from using concrete examples. Don’t ban children from counting on their fingers. After all teachers don’t ban children from moving their lips when they read, they let them do so, confident that as the child becomes more proficient at reading, they will cease; similarly, teachers should allow children to count on their fingers until such time that they are ready to count without the bother of using them. At Puppet Maths we encourage children to visualise numbers as sets of dots. They can then count the dots in their mind’s eye, then subsequently, by recognising the patterns formed by the dots, learn to do arithmetic without having to count out each time.

Teaching by stimulating the child's mind

When I was at school, there were still 12 pennies in a shilling (and 20 shillings in a £) so from the age of seven I was expected to count in bases 12 and 20 in addition to base 10. How did I do it? I imagined a big pile of pennies, which, when they reached the height of 12 got turned into a shiny silver shilling (not that I often saw a shilling, they didn’t circulate, they were kept aside for use in electricity meters). As I did my calculations I imagined pennies being put on or taken off the pile.

Any child who didn’t use their imagination in this way that must have been an enormous disadvantage. They must have faced enormous difficulties counting in 3 bases (10, 12 and 20). Later, when I was 8 we were expected to count in bases 14 and 16 (there being 16 ounces in a pound, and 14 pounds in a stone). Then at age 9 we were asked to count in bases 22, and 8, there being 22 yards in a chain and 8 furlongs in a mile (The chain was actually a decimal unit, as there were 100 links in a chain, and 10 chains in a furlong). We were expected to count in all these different bases before we reached the age of ten years! And we did. Why? Because we rose to meet what was expected of us. This is why it is so important not to dumb down maths.
At Puppet Maths we believe in academic excellence. We will not dumb anything down. We believe in making academic ideas and concepts accessible to the young mind, and we have found the use of puppets to be the ideal vehicle to achieve this.
Puppets stimulate children’s imaginations, and once they are imagining then they can see piles of pennies, or lines of numbers, or patterns in their mind’s eye.

Children love animation

Children have vivid imaginations. Maths is not often taught in a manner that allows the child to use their imagination. Maths is considered to be far too serious a subject to let children harness their imagination. What sort of answers would a child come up with if it used its imagination when doing maths?
Q. “What’s 1 + 1?”
A. “Pink bunnies”.
But is this the case? Children want to please the adults in their lives. They know when it is appropriate to be serious and when to be frivolous. They might use frivolity as an excuse to avoid work, but they know exactly what they’re doing, when they do it. The aim of Puppet Maths is to harness children’s imagination to make maths fun. And Maths should be fun, without having to dumb it down.
Children love animation, it simply catches their imagination. The use of puppets immediately propels a child to start using the imaginative part of their mind. When they’re doing this, the child’s mind is open for learning. This is why Puppet Maths is so effective at teaching maths, it engages children’s open minds.

Many ways to solve a maths problem

It is a humbling experience for any maths teacher when a pupil comes up with a better way to solve the problem than that used by the teacher himself. Naturally, the teacher teaches the standard method, and when faced with a maths problem, as a matter of routine, launches into a solution based on applying that standard method. This saves the teacher from having to think about the problem. Then along comes a pupil who has thought about the problem, and using imagination reframes it in terms that are significantly simpler. This allows the pupil to shortcut the route to the answer. This ability to think about a problem and reframe it with simplicity is the mark of a good mathematician. Unfortunately, this is not a skill that is taught in most school maths classes. At Puppet Maths, we think about how maths problems should be approached, and we encourage our pupils to think about how they might be solved. We present our pupils with a number of different ways to view the same data, and a number of different ways to manipulate that data to arrive at an answer. By this means we show our pupils that there is more than one way of considering a maths problem and that there is more than one method reaching a satisfactory answer. We want our pupils to be aware of this diversity, so that they can use their imaginations to arrive at the best path to a solution. We, at Puppet Maths, want to lead children away from the idea that the way that teacher does it is the only good way of doing it, and that it must only be done by that method. Whether at home or at school, we want pupils to develop insight and, yes, cunning into their relationship with numbers.

Don't judge students

An unhappy thing happened today, the connector to my external hard drive got torn from the printed circuit board that it’s mounted on. It needs 9 solder joints to reattach it, but its a surface mount component, so it’ll be hard to get at with a soldering iron. If I had a reflow oven then soldering it would be straightforward, except the disk drive would have to be removed from the circuit board, and no doubt that would require unsoldering. Not so simple. At least maths is easier.
Maths can be solved in your head, no fiddling around with hot tools. The nastiest material you come across with maths is ink (that is unless you think that paper is more offensive). It’s one of the reasons that I gave up semiconductor engineering to teach maths, there’s no hydrofluoric acid or arsene involved in doing sums. When viewed from this perspective, maths really is easy.
If maths is such a good option, why do so many children hate it? My theory is that it’s because, the subject of maths forms a pyramid. Every new thing that a pupil learns depends upon them knowing the preceding stuff. If there is something that they “don’t get”, then from there on, they are at sea. Young children can be very shy of pointing out that they don’t understand something, they’ll pretend that they do, and go to great lengths to hide the fact that they don’t. In this way they act against their own best interests, but, of course, they don’t understand it that way, because they don’t see the bigger picture. This has a lot to do with their status within their peer group, and not being seen as being weaker than the others around them, weaker than those who they are competing with. This is where Puppet Maths helps. Children know that they are superior to puppets, and so are more inclined to admit their inability to do something, because it carries no social stigma. They know they are not going to be picked on by the puppets; they know they are not going to be judged by the puppets.

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.

Fear of failure

Maths is part of the real world. It is taught in schools. Children doing well get praised for it. Children doing badly get to feel that they aren’t good enough. This latter is true however supportive the teacher. The child who has got a sum wrong feels diminished in spite of the teacher saying, “Never mind, this is how you do it, try again”. Why? Because, until adolescence, children are programmed to try and please adults. Also they are very self centred. So instead of taking notice of what adults expect of them, they decide for themselves what would please an adult. These expectations generally start off high, children imagine themselves doing the most ambitious things, and very often the most unrealistically ambitious things. So when they fail to achieve at something that falls well short of their ambitions, when they if they fail to meet something much less than their own expectations (which are way above anything any adult expects of them) they feel failure. Puppet Maths aims to avoid these feelings by taking adults out of the experience of maths. Children know that puppets are an inferior species to themselves. Children have no illusions about trying to impress their toys with their brilliance, they know that their toys are already in awe of them. In this world they are persons of importance, and they know that the puppets know it, so in this world, children are not inhibited by the need to put up a front, to appear confident and capable even when they’re not. They can admit their weaknesses and failings and learn how to address them without losing face. Because puppets are not authority figures, but things which are unthreatening, they make ideal companions for children to learn from. At Puppet Maths we harness this effect to enable children to achieve at maths, and get them home and dry as far as the subject of maths is concerned.

Why it's so hard to learn statistics

The purpose of maths is to help us understand what is happening in the world around us. It starts with the pupil learning arithmetic. This can easily be related to everyday life. The concept of what constitutes a “three” or a “five” is relatively easy to convey. Arithmetic is a precise discipline, and naturally emphasis that is put on accuracy when children learn it, however this is unhelpful when children come to learn statistics. Statistics have no precise “right” answers. The subject is all about getting an idea about what is going on. The metrics that are used in statistics only mean something once you’ve used them and “got a feel for them”. Before one has grasped the concepts one just doesn’t know what they mean. Learning statistics involves the pupil “getting their feet wet”, wading in and engaging with the subject before the pupil knows what it all means. This is hard for them to do when statistics is taught by a maths teacher in a maths lesson, because they’ve already been conditioned to think of maths as being all about precision, and nothing to do with gaining a gut feeling of what the data is “sort of” telling us. At Puppet Maths, because the maths is taught by puppets, who are not strict disciplinarians, pupils find it easier to accept the transition from the precision of arithmetic to the overall appreciation that is embodied in the study of statistics.

Story telling

I haven’t posted on this blog for a couple of months now, so it’s time I started again. The reason for this gap is that I spent Christmas helping one of my daughters, Julia, move home. She was in a flat with other people who just wanted to party, but she had to get up early in the mornings to go to work in the hospital and she didn’t appreciated being kept awake at 3 am.
Then in the middle of January I moved to New Zealand. It’s a lovely country and I’m looking forward to the start of the school year, and getting back into the classroom to teach maths.
As a result of these diversions, I’m afraid that the blog went by the board.

Nevertheless back to business.

Storytelling is integral to human beings. We have evolved to learn from stories. Once this was a matter of survival, the way that the human animal discovered that things were good or dangerous without having to experiment for themselves.

In recent years, since the widespread availability of recorded video, pupil studying English literature have spend ever more time watching the plays they study rather than reading the script. Why, because its simply easier to remember the story and the dialogue from experiencing the story than from reading the script. I myself was at one time an expert in Shakespeare’s King Henry IV (part 2) because when school enacted it, I was on stage for just over half of the production... I played a dead body, but that was good as it allowed me to relax and be a spectator of the play. Without effort I learnt everybody’s part. When my children were young, they loved nothing more than watching Disney videos over and over again. They knew them word perfect as well. They learnt it all without effort. That is why, at Puppet Maths, we dramatise maths. We put maths in the context of a story, and enact that story for children to enjoy. As a by product children learn to do maths without fear and without effort.