Imagining the puzzle

In maths the ability to imagine the problem is key. A couple of years ago I was studying in the company of a group of postgraduate students the eldest of whom was 14 years my junior. As a result all were too young to remember when the UK currency consisted of pounds, shillings and pence. These were all highly intelligent people. I proposed a pub challenge, in which I would ask the questions and they would compete against the clever clogs [this guy was a Physics PhD, who got 780 (out of 800) on his GMAT exam score] to do sums in pounds shillings and pence, the winner being the one who could furnish the correct answers first. My proposed quiz never took place because the PhD was unwilling to try his brain at calculating in bases 12 and 20 (there being 12 pennies in a shilling and 20 shillings in a pound) in such a public arena, where he might lose face. What he lacked, because he had never been previously required to possess it, was the ability to imagine counting in these bases. If he had tried, he would quickly have been able to imagine piles of copper pennies growing in height until they were 12 high when they turned into a silver shilling, and piles of shillings growing in height until they were 20 coins high, when they turned into a pound. Had he developed this picture in his mind he would have been able to solve any questions that I would ask involving the old UK currency, but until he did, until he could imagine the problem, he was not in a position to attempt the questions I might ask. So often, maths is not related to a real situation. As a result children are unable to imagine the problem. Like the Physics PhD, they just cannot get their heads it. At Puppet Maths we couch the puzzles we give our pupils in terms of familiar situations, so that they feel at home with the puzzle and can imagine what’s being asked of them and apply their minds to finding a solution. We mean to help them develop their ability to visualise in their mind’s eye what needs to be done to arrive at a solution.

Problems and puzzles

What’s the difference between a problem and a puzzle? The answer is… the consequences. If there are no dire consequences to getting the answer wrong, then you’re dealing with a puzzle, and it is an enjoyable mental challenge. If there are unfortunate consequences then you’ve got a problem, which is an unpleasant experience. But what constitutes unfortunate consequences? Well that rather depends on your position in life, however, for children, things that an adult would find trivial can take on great significance. Not being able to do an exercise of sums at school because the child has not understood what is required of them, can be a disaster. They may not want to ask for help, as that would mean admitting their inability to do the work, and they expect to be told off for it. They may fear being teased by other children if word gets out that they can’t do it. So they hide their inability, and what should be a maths puzzle and fun to do becomes a maths problem and a headache. This can turn them off maths for life. At Puppet Maths we don’t give our pupils maths problems, we don’t want them to experience maths adversely, Puppet Maths is the home of maths puzzles, puzzles that will develop our pupils and teach them to enjoy maths.

School as a parallel world

Children used to learn in the real world. My own grandfathers left school at 12 and went to work. One grandfather said that he learnt nothing of importance until he left school. I think that that is not quite true. He would have learnt to read, write and do arithmetic at school, which was all that was expected of schools in those days, but he was taking those skills for granted when he made that comment. However, there is some truth in what he said. As a practical man he valued skills that he could apply to making a living, to bettering his family, and these were not taught in the world of the schoolroom. Indeed today, 100 years later, schools do not teach children how to survive and prosper in the real world. Schools have become a parallel existence and when young people leave education the realities of life in the real world often come as quite a shock. This parallel existence is true in maths education. We ask pupils to calculate 3 + 5. This is an abstract calculation. It has not been put into any context. They are not being asked to calculate the sum of £3 and £5, or 3 boxes and 5 boxes. Such examples belong to the real world. Without context maths loses its meaning. If maths loses its meaning then it becomes simply an exercise that children feel they have to do to please their teacher and stay out of trouble. If we expect children to calculate in an abstract manner, then they will do so, but they will not always appreciate the uses of what they are doing, it will not catch their imagination, and they will not enjoy doing it because they fail to understand the purpose for doing it. This is especially true as children become teenagers. At Puppet Maths we put the maths we teach in the context of solving real world problems. We home in on the relevance of the maths to situations that the pupils can imagine either at home or in an everyday situation. This way they will engage with the subject, and achieve good results.

Programmed to succeed at maths

Children are programmed to do whatever the adults around them do. When she was small, my eldest daughter was placed in the care of a lady who came to our home and looked after her while we went to work. This lady, when not caring for her and her baby sister, used to clean the house compulsively. That year, when we went to the Ideal Home Exhibition, we placed her in the children play room, which was being sponsored by the local toyshop. The last thing I saw on leaving her was her taking hold of a toy vacuum cleaner and starting to hoover the carpet. On my return some 90 minutes later, she was just arriving at the opposite corner of the carpet, having spent the whole time playing at vacuuming. Shame it didn’t survive throughout her teenage years. Why do they stop doing these useful things? Because adults do not expect them to continue. Children will learn to do the things that they are expected to do. If they live in a society where they are expected to stand up in a boat and spear fish, then they will learn the balance needed to stand in a boat at a young age. If they are expected to count in bases 12 and 20 (as was required for using UK currency before decimalization in 1971 – there being 12 pennies in a shilling and 20 shillings in a pound) then they will learn to count in bases 12 and 20 when still young. It is common for adults to think of maths as being hard. As a result it is common for people to expect children to find maths hard… so children find maths hard. At Puppet Maths we believe that maths is easy. It is not only easy it is also fun. We make it easy, we make it fun. We expect children to do well in maths, and as a consequence we expect that they will indeed do so.

Children are cleverer than adults used to think 2

After writing yesterday’s post on what researchers have observed about babies ability to do maths, I was reminded of the behaviour of my oldest daughter when she was a baby. When she was 5 weeks old, her mother went into hospital to have her tonsils removed, so the poor child was left in my tender care. To make life as easy as possible for myself, I quickly developed a routine. I brought her home from the child minder’s, and fed her her tea. He clothes then went straight into the washing machine, and she went straight to the bath. I washed her while singing the bathing song. This determined what part of her body I washed… something along the lines of “now we go and wash the feet, and now we go and wash the knees” set to the old Irish folk tune “Let no one steal away your thyme”. I remember that it finished with the line “It’s time to ring the baby out” when I would lift her out of the water and swing her from side to side while saying “Ding, dong, ding, dong, ding dong”. But I digress from the purpose of my script today. The point of this story comes with the drying song. I would lay her on the bed and dry her systematically, while singing the drying song. This process required me to lift her right arm and dry her armpit, then to lift her left arm to dry her left armpit. On the third day of doing this no sooner had I dried her right armpit than she raised her left arm herself. At the age of 5 weeks my daughter knew what was happening to her, and able to anticipate what was going to happen next. To what extent the use of her song important in this? Would she have recognised the routine without it? There is work for the child psychologists to do there, but I believe that it played a crucial part. I think that babies and children respond to the use of music and song, that it gives them patterns to recognise, it helps them learn. Surely this is obvious, don’t we all remember poetry that scans better than prose that is just a connection of words? This is why at Puppet Maths we have adopted the use of the Sands Daniels musical times tables. The times tables comprise 80 facts that children have to learn (or 40 facts if you consider that half of them are the same as the other half - 2x6 being the same as 6 x 2). These are boring to learn by rote. But if they’re set to music, if they scan, then they are memorable and children can pick them up without difficulty. At Puppet Maths our aim is to help children use their natural ability to learn easily.

Children are cleverer than adults used to think

Children are more capable than people used to think. Recent research has shown that babies only weeks old can count. Interestingly this experiment involved the use of puppets. Babies, will stare at a situation whenever something unexpected occurs. This enables researchers to know whenever a baby encounters something they find surprising. The experiment involved the researchers showing two puppets to a baby. They then dropped a screen and very obviously removed one of the puppets from behind the screen, so that the babies could see it being taken away. On some occasions when the screen was raised again there was only one puppet there, but on others the researchers surreptitiously introduced another puppet so that when the screen was raised there were two. The baby was found to be expecting to see only one puppet, because on those occasions when there were two, the babies stared at the occurrence for a significantly longer time than they did when only one puppet was there, indicating that they were surprised. This indicates that even very young children can count. So why do so many children find so much difficulty with maths? It must be a function of the way maths is taught. Maths teachers simply are not connecting and communicating with young children well. This is why we have developed Puppet Maths, to communicate the ideas of maths with children and develop their natural ability to succeed in the subject.

Sudoku and overcoming the fear of numbers

Numbers are just shapes drawn on paper. What they symbolise is relatively straight forward too. Unless you are 5 or 6 years old that is. Children who are at school but who have not grasped how numbers work, find that they are not getting approval for their work, but they don’t really know why, they don’t know what they’re doing wrong… but there are all these other children around them who seem to know what going on, who seem to be able to get their sums right. This is intimidating. The child is not one of the group, and even worse is not getting approval from the adult in the room. They feel isolated. They hide their inability, because they need to be part of the group, they need approval. The teacher should be identifying these children and helping them to understand, but with a classroom full of children, perhaps the child’s problem get missed for a short while. Unfortunately, that short while is long enough for some children to develop a fear of numbers. Whenever they see numbers they cease to be able to think logically because fear sets in, and their minds close down. These children have learnt that these written shapes can separate them from the social group. This can lead to many years of underachievement. One remedial method to help a child overcome a fear of numbers is to teach the child to play sudoku. No maths is involved in sudoku, the numbers are simply used as different shapes, any other 9 ideograms could be used (take your pick from the Chinese dictionary). However, getting a child to play sudoku teaches them that they can manipulate numbers and get them to do what they want them to do within the game, and so helps them overcome their fear of numbers.

Column discipline

For simple arithmetic column discipline is one of the most important things that a child can learn. If the columns in which they put their numbers are not neat then it is very easy, for example, for the child to add a number which resides in the tens column to a number that is in the units column or is in the hundreds column. If this happens, then no matter how good their skills at adding single digits together are, they will arrive at the wrong answer. This is very dispiriting and can lead a child, who is perfectly capable, to simply give up. In schools pupils are provided with squared paper to encourage pupils to put their numbers in straight columns, but often this is not sufficient, still many pupils allow the figures they write to meander across the page. Pupils should be taught how to lay out columns of figures neatly, so that they can subsequently add them easily. But where does that leave the dyslexic child? Many dyslexic children complain that they cannot tell which column a number is in because “they keep moving about”. The secret to success here lies with the highlighter pen. If all the numbers that should be in the units column are highlighted green, and all the number that should be in the tens column highlighted orange, and all the numbers that should be in the hundreds column are highlighted red (and so on), then all the pupil has to do is add up all the green numbers, then add together all the orange ones, then next the red ones. It doesn’t matter that they move about the page, the background colour moves with the digit and tells the child which numbers should be added to which. This is one of the tricks that we teach at puppet maths to make maths easy.

The meaning of statistical functions.

What is the meaning of a standard deviation? It has a definition which can be taught, but what its meaning is is a different question. This is something that cannot be taught. It is a concept that someone only picks up through usage. A sample’s position relative to the standard deviation and the mean says something about that sample, and where it fits within a distribution… but so what? What are the implications of the sample’s position? That will vary from sample to sample, and be different in the case of a pharmaceutical tablet and in the case of an orange. It is this interpretation of the data, which varies from situation to situation that makes teaching and learning statistics so very challenging. Principally we have a chicken and egg situation. The pupil has to use the concept, to learn it, and only after having used it (for a period of 9 months or so) will they begin to develop the understanding of what it actually means, what its implications are in various scenarios. At Puppet Maths we teach statistics via scenarios which we create with our puppets. This we have found is the optimal way of conveying the meaning of the various concepts that exist in statistics.

Drawing conclusion from statistics

What is the meaning of a mean of 14, a mode of 8 and a median of 7? Children calculate these numbers all the time. They get marks for getting the right numbers, but what do they mean? It’s easy for a child to see that when they add 5 apples to a bowl already containing 3 apples what the purpose of the calculation is. Similarly with calculation of money where they start with a pound and spend 35 pence, they can see the point of knowing how much money they have left. But what conclusion can be drawn from knowing the 3 numbers shown at the start of this blog? Well, sometimes a conclusion can be drawn from these 3 numbers. However, on other occasions these 3 numbers perhaps mean nothing much without further information, such as the number of samples from which these averages were calculated, and the range of these samples. The point is that interpreting statistics is a holistic activity. Numbers have to be looked at in conjunction with each other. Children are not used to that in maths, they are used to a single numerical answer. This discrepancy between the type of thinking needed for the maths that children are used to and that encountered with statistics causes them enormous difficulties. Pupils who cannot easily adapt their thinking are likely to respond by not attempting to understand what the statistics are there to explain, and take refuge in just producing the number required of them, without understanding. They perform the necessary calculations and get right answers but miss the point of the exercise. At Puppet Maths, we teach statistics using life like scenarios. The puppets allow us to create situations where statistics can be put to use and conclusions drawn. We can animate real problems that make statistics relevant and so understandable.

The purpose of Statistics

The purpose of statistics give a “gut feel” for what’s happening in a complex system which is intractable to analytic mathematics. The subject is by its very nature imprecise. Children who have been brought up on arithmetic often don’t have this concept. They cannot interpret meaning from the numbers. For so many of them, numbers don’t have meanings. Numbers are numbers and that’s all they are. They are either right or they are wrong. They are not there to be interpreted. They do not exist to provide information. For these children maths is an ivory tower divorced from real life, something you do at school, because that’s what you do at school. This puts them at a great disadvantage when confronted by statistics. Their previous maths ways of thinking do not help them. Many do not attempt to understand what the statistics are trying to explain, and take refuge in the number crunching that statistics involves. They can understand that, they can do that, they can get the right answers for that. In doing so they miss the point of what statistics are for. The fault for this situation lies with the way that maths is taught and the exercises that are given to pupils to practice mathematical operations. Most often they are simply numerical exercises not related to any particular problem, so pupils lose sight that the maths might be for something. Maths is treated by them as being for its own sake, so the calculation of a mean, a mode or a median is the end in itself and the interpretation of these numbers is left unconsidered. Is it any wonder that children find the whole thing confusing and sterile? At Puppet Maths we put our maths into context. We teach maths in the frame of scenarios where there are problems to be solved. The puppets act out these scenarios and make the maths tangible and real for children.

Statistics

When children first encounter maths, they learn arithmetic. Arithmetic requires them to be precise. There are no rewards for rough answers or approximate answers. Consequently, children learn that maths is a precise science and that they have to think precisely to get it right. Then at the age of 13, after they have been indoctrinated for about 8 years, they are presented with statistics. Statistics is all about approximate answers, rough quantities, ball park figures. The output of statistical analysis is what the likely answer might be. Children find statistics so very difficult because the turn around in mindset needed to understand what it is about is so very massive.
The mean is a real maths concept, because it involves calculation which can be done with precision… but what about the mode? So many who start on statistics consider finding the mode to be not maths. There is no arithmetic involved, all one has to do is arrange the numbers and see how many of them there are. They feel that this is for kindergarten, not for serious mathematicians. It is clearly too simplistic to be of any practical use. What can its purpose possibly be? And the median? How can it be scientific just to pick the middle number irrespective of their spread and distribution?
What all these attitudes display is lack of understanding of the purpose of maths. Maths exists to provide an understanding of things that happen in the real world. Maths concepts are used where they are useful and left alone when they are not. Mean, mode and median all exist to give the mathematician a feeling for what a typical number in a distribution of numbers might be. Depending upon the distribution these values might all be similar, or they might vary dramatically. Whichever case they fall into, their relative sizes tells us something about the nature of the distribution, but it requires experience to understand what that nature.

Relevance of mathematical operations.

The school I attended taught one year of what was in those days called “modern maths”. We pupils were scornful of the subject because it was not in any way intellectually challenging. I remember we were required to do calculations in duodecimal (base 12) using the characters “@” and “*” for the digits 10 and 11. This we found to be gross stupidity, as at the time the UK currency had 12 pennies in a shilling, and we were adept at writing “10” and “11” in the units (pennies) column. We could count in 12s and 20s (there being 20 shillings in a pound) with facility, and being subsequently asked to count in bases 8, 16 and 2 were simply annoying variations on a theme (counting in different bases) we had already mastered. Also I remember a homework in which we had to draw concentric circles around a point and colour them in, the reason for doing so escapes me, I seem to think that is was the various fields of damage from an exploding atom bomb, but there may well have been a less warlike excuse for making the drawing.
What the modern maths project was really about was linking maths to real world situations, and making it relevant to pupils. In this it failed mainly because the real world situations that it utilised were too abstracted from our realities. The counting in octal (base 8) hexadecimal (base 16) and binary (base 2) was designed to prepare pupils for work in digital electronics, which was the developing technology of the day, but this was not explained to us pupils, we had explanations about octal being the base spiders count in! At Puppet Maths we aim to make our maths relevant to the world in which our pupils inhabit. We take everyday activities to explain what the maths are for.

Confidence in the subject

Lack of confidence can undermine a pupils performance. My third daughter felt that she was poor at maths even though her attainment was perfectly respectable. Because of this she did not try as hard as she might have, she tended to give up rather than persevere through to success. We boosted her confidence by using the Sands-Daniels musical multiplication tapes. We played these to her in the car (where she had nothing else to do to distract her) whenever we were driving her around. I required her to sing along with them, and rewarded her at the end with a sweet if she did so. This taught her her times tables and put her ahead of the rest of the class in maths. This had a dramatic effect on her performance even beyond the narrow confines of multiplication calculations. She felt that she was a leader, that this was a subject she could do well at, and that it was worth her while to work at maths. Her confidence soared and it became one of her best subjects. At school learning is a social activity, but unfortunately the society in which schoolchildren exist can be savage. The social pressure experienced by children can undermine their ability to learn. This means that small difficulties can completely undermine a child’s ability to succeed. We at Puppet Maths wish to address those small difficulties and help children overcome them. We teach them maths in a child friendly way, helping them to understand the concepts and enabling them to perform calculations successfully.

Fear of peers

Competition between pupils can be a good thing if it causes them to stretch themselves in order to do better than their peers. But this presupposes that the pupils involved in the competition are competent. If they are not, rather than compete the child gives up. Why compete in a contest which you can’t win? For many children at school, the point of much of the work they undertake is to do better than a rival. This gives them a social status. If they cannot do better than those about them, they choose not to compete rather than lose face, they turn off and just don’t try. For some, when they do this, they become disruptive, which eliminates the boredom of being in a class where they are not working, but also clearly demonstrates to the rest of the class that they are not trying, which also gives them an excuse for not achieving.
Children are notoriously cruel. If one is weaker, they will pick on the weaker one. This causes children to hide their weaknesses. The upshot of this is that children who have difficulty in maths, hide the fact and stop trying. This is a tragedy, because unlike many other subjects, maths is constructed as a pyramid. If you don’t learn the basics, then you cannot learn the subsequent work as it uses prior knowledge. Pupils who lose out at the start cannot make it up later. A sound grounding is essential. Puppet Maths aims to provide a sound grounding in maths, in a non-threatening environment, where a child can learn without fear of mockery from other children, and get to understand how maths works.

Skimming and skipping

In maths class, while I was spinning my spinner and adding the numbers that it fell on together, and while I was spinning out this activity for as long as possible so that I wouldn’t have to go onto doing something harder that would require me to think more, other children in the class were in competition with each other. These children all knew one another. They all lived on the council estate next door to the school. They had developed a pecking order based on who was furthest forward in the maths text book. I regularly heard one or another of the pupils declaring that they had finished exercise number such and such, followed by scathing comments directed at pupils who had not progressed so far. I kept quiet. I was far behind these high fliers, I didn’t want to attract their attention and gain their distain. Also, in a sense, I gave up. I realised that even if I were to finish a whole exercise a day, it’d be weeks before I got to where they were, and by which time they’d have moved on. So I quietly worked away at my own thing. But these children who were so far ahead of me did not better me at maths. By the end of the following year I was ahead of them. What I had learned I had learned thoroughly, whereas they had skimmed through the work with the objective of finishing the exercises rather than inwardly digesting the lessons they were devised to teach. At Puppet Maths we believe that thoroughness is important in learning maths, and whereas we wouldn’t want a pupil to get stuck at any particular point in the course, we wouldn’t want to hurry the pupil on at a faster rate than they can cope with.

Opportunities for learning

Children like doing the things that they can do well. They appreciate praise and when they do something well they get praise. The upshot is that children like the school subjects that they can “do”. On the other hand, children do not like criticism. Criticism is psychologically linked to rejection, and children fear rejection by the family group, upon which they rely for their very survival. Because of their fear of criticism some children are afraid of doing new things. They want to play safe, and stick to the things that they can do. This isn’t necessarily a bad thing. I, myself, when aged 6, was given a spinner, which I was required to spin twice and then add the two numbers that it fell on together. I found that I could do this and get right answers. There was no end to this exercise, I could keep on spinning the spinners and adding the numbers as long as I liked. So I continued to do so. I remember being afraid that the teacher would notice that I was not moving on and criticise me, but the attraction of being able to do the work was greater than my fear of criticism. At the time I thought that the teacher had not noticed the inordinate amount of time I’d spent on that one activity, but now I realise otherwise. She saw that I was gaining confidence in handling numbers, that I was building a sound base for the future. She let me spend my time practicing what I could already do because she realised the value of my gaining mastery of adding small numbers, and she knew that by playing with these numbers I was gaining a feeling for their relative values. I recall that I managed to spin out that activity to the end of term, and it was only after the holidays that I found myself too embarrassed to return to it, and move onto the next exercise in the book. At Puppet Maths we understand that children may wish to spend longer practicing a topic, that they may wish to consolidate their learning before moving on, so we do not march our pupils through the syllabus, instead we offer them opportunities for learning.

The Relevance of Maths

Whenever I have had to add fractions I have had to find a common denominator. At school finding the smallest one is considered to be a big deal. Personally, I haven’t cared about finding the lowest common multiple since in real life any common multiple will do. Therefore, this is a topic in mathematics that I have been unable to come to terms with. As a maths teacher, I know how to find them, I have remembered the rules for getting them and I can follow them, but I haven’t developed an intuitive feel for doing so. I have never been motivated to do so.
For some pupils, whose main motivation is to please, simply the fact that teacher is asking for these things to be calculated is sufficient justification for them to learn them; but other pupils require to understand the practical uses that a mathematical technique can be put to before they gain the motivation to apply themselves to learning. The relevance of a mathematical techniques is of prime importance.
I grew up when the “space race” was on, with the Americans and the Russians competing to be the first to put a man on the moon. The television stations covered the launches of each successive rocket and we watched the countdown to lift off. This made counting backwards to zero not only relevant but exciting. The upshot was that children learnt to count down from ten before they learn to count up to ten! They did it without having to be taught… they picked it up organically from their environment. At Puppet Maths we believe that if a teacher can capture the child’s imagination then the child will learn organically, without effort. This is our aim at Puppet Maths.

Maths and large numbers

“What is Mathematics”. For many school pupils it is simply something you do at school. It is apparent to many that addition is useful… when applied to the small numbers that they encounter every day. The consequence of this is that pupils generally understand the addition of small numbers and can “do” it. However, when big numbers are used, very often the context of the numbers isn’t apparent to the child. This is when they find performing the calculations difficult. For many children a thousand is such a huge number that it might just as well be termed “many”, and if this is the case, then ten thousand is also unimaginably big and too might be termed “many”. Hence there is, in the child’s mind, no difference between these two numbers. To get these children to understand big numbers we need contexts in which big numbers are used. The cynic will suggest currency, on the basis that in due course our governments will inflate away our savings and even ice creams will cost millions, but until this happens children, who have an appreciation of the quantity of money required for everyday living, will not respond. At Puppet Maths we have adopted space travel as our vehicle for explaining huge numbers. This is a fertile field for drama in which the child can get involved, and it is a place where their imagination can be given free rein, while they are being introduced to the concept of large numbers. We believe that if we can make the use of numbers fun, and children will pick up the concepts without effort.

Maths and the Law

When I was at school the one thing that I really hated having to do were mathematical proofs. At that time I could not see the purpose of them of having to prove the rule, I could not see why I could not assume the rule was true and then show that it was self consistent. Happily since then I have learnt the reason for the need for the rigour of a formal proof. Formal proofs are needed because, unlike disciplines such as Physics and Chemistry whose rules are determined by Mother Nature, mathematics has been devised by man. When a new mathematical rule is developed it has to be shown to be consistent with all the other mathematical rules already in existence. This is the reason why maths involves proofs. A mathematical proof demonstrates that in all cases the new rule will conform with the rest of mathematics. This is what differentiates maths from that other man made construct – the law. Maths is logical, self consistent and produces comparable results, whereas the law is arbitrary, inconsistent and produces many paradoxes which have to adjudicated by human intervention in the person of a judge. This is why Maths is a science, whereas practicing the Law is an art. At Puppet Maths we don’t just teach the rules of maths, we explain the logic of why the rules have been formulated in the way they have. This way pupils get to learn not just what the constituent parts of the science of maths are, but to understand the reasons why the rules of maths have developed in the way they have.

The purpose of maths.

In the US they call it “Math” in England it’s “Maths”… the Americans only have one, the Brits use the word in the plural. In German it is “Mathe” again the plural, so is this a case of the Europeans recognising the plurality of the subject, the fact that it contains many disciplines or is it simply that the Americans are better at abbreviating words? Nevertheless, it raises the question “What is Mathematics”? What is it for? When I was at school, I believed that it was about numerical manipulation. As a result I thought of geometry, for example, as a peripheral activity rather than a central plank of the subject, more a historical overhang that was being studied for completeness rather than for active use. But I was missing the point. What maths is really about is solving problems. Every day people are faced with problems, and the framework provided by maths can help them solve these problems. Mathematics provides proven ways to think about problems, and informs people of the approaches to take to arrive at a solution, approaches that have been shown to be robust, that work each time. Often there are a number of ways to approach the problem, one way might be numerical, another might be graphical, alternatively the solution might be arrived at via an abstract diagram. All these are component parts of the subject that we call mathematics. The more branches of mathematics that a person has knowledge of, the more options that person has to apply to find a solution to their problems. At Puppet Maths we teach the context in which mathematical techniques can be used, to demonstrate to the pupils the real world use that the maths they are learning can be put to and examples of the problems that it can help solve. This shows the pupils that they are studying maths for their own best interests, and not simply because adults like to make their lives difficult.

Visualising maths

Solving maths problems starts with understanding what the problem is. To do this one of the most powerful techniques is to draw the problem. A drawing enables you to visualise the maths problem. In drawing, one is forced to make the abstract maths concepts and numbers concrete... it is impossible to draw an abstract concept, the very act of drawing causes it to take on form (of one sort or another). Puppet Maths is a visual presentation of maths, consequently, by the very nature of the medium we work in, we have to give maths form. The form we give it is that which I was taught as a child (there were always good maths teachers out there). We use dice to get children thinking of numbers as a series of dots. We organise these dots into piles, which, when they reach a height of 10, magically change into a ten. What does that mean? Well to explain we shift our description to one involving coins... each dot becomes a penny piece and ten of these in a pile is equivalent to a 10 penny coin... and in turn ten of these 10p pieces (florins) can turn into a £1 coin. The purpose is to get the pupil thinking about real objects rather than struggle with strange shapes like "3" and "5". [Incidently these two shapes look alike to younger pupils, an appreciation of left and right doesn't necessarily develop until a child is 8 or 9 years old - my own daughter, when she started at school, would write a line of text from left to right, and then write the next line from right to left in mirror lettering - and she would see nothing unusual in it]. When we teach fractions we get the pupils to imagine them as slices of a pizza, or as a position along a line, so that they have a means for visualising what the numbers represent. The Singapore maths course takes a similar approach, making maths visual, so that the pupil can understand the problems they are given. We at Puppet Maths are proud to be working under the same principles as the Singapore maths course.

There are many ways to solve a maths problem. As long as the person solving the problem sticks to the rules of mathematics, then the problem can usually be solved in a variety of ways. There is no single path to the correct solution. This is one of the attributes of maths that makes it EASY. This is also central to the philosophy of Singapore Maths.
By way of example, if we take the multiplication 7 x 9, one way of arriving at the correct answer is to learn the times tables. Learning these is boring, that is why at Puppet Maths we use the Sands-Daniels Musical Times Tables. These uniquely, in my experience, set the times tables to well known tunes that you can hum, and they are without the vast quantities of extraneous verbiage that have nothing to do with multiplying numbers that so many musical times tables are afflicted with. The music triggers the memory of the lyrics and so removes the fear of getting the words wrong thereby making the times tables fun to learn, and fun to recite. We advise that having learnt the songs that the pupil recite them in their head when they require to recall the product of two digits.
However, if one hasn’t been lucky enough to have learnt your times tables using the Sands-Daniels songs, one can still arrive at the correct answer to the calculation by other means. Another way would be to put 7 dots on the page nine times over and then count them up. Alternatively one might write down 7, add 7 to it to get two lots of 7, then add 7 again, and again, and again, until one had added 7 nine times. Both these method would produce the answer.
Another method would be to notice that 9 is almost 10. One could multiply 7 by ten to get 70, and then reason that since we did not want ten lots of 7 only nine lots, so we could then subtract one seven to get our answer.
But there are more approaches… one could look for a pattern in the 9 times table. Whenever 9 is the multiplicand, the product of 9 and some multiplier is such that its 10s digit is one less than the multiplier, and the tens digit and the units digit of the product add up to nine… so in the case of 7 x 9, seven is the multiplier, so the tens digit of the product will be one less, i.e. 6, and the units digit of the product will be whatever added to 6 makes nine, i.e. 3. The product is 63.
Alternatively, one might look for a pattern in the 7 times table. The seven times table follows the pattern on a mobile phone number pad. For this you ignore the 0 button, and use just the other nine buttons. Starting with 7 at the bottom left hand corner, that is one seven. To find two sevens move up the key board. The rule is every time you move up the keypad then you add 10 and the units is given by the number on the key pad. So for two sevens you move one up the keypad, which gives you a ten for moving up the keypad and a four for the units, as that is the number on the keypad button – result 14. For three 7s you move up the keypad again, that adds a ten for moving up, we now have two tens, and the units are given by the number on the button which is now 1 – three sevens are 21. For four sevens we go back down to the bottom key of the middle row of the keypad (that is the button marked 8). Because we have not gone upwards with this move we don’t add another 10, so our tens digit is still 2, but now our units digit, given by the button, is an 8, - four sevens are 28! Five sevens… we move up the keypad so we add a ten giving us 3 tens now and the number on the keypad button is a 5, five sevens are 35, and so on. Doing this we discover that nine sevens are 63.
At Puppet Maths we teach that as long as the pupil sticks to the small number of rules of maths (there aren’t many… and unlike language where there are irregular verbs, there are no irregularities in maths, it always obeys the rules) then they should find the answer by hook or by crook.

Teaching one thing the better to teach another

Significant figures is a confusing concept for most pupils. This is simply because they have been taught to ignore all the leading and trailing zeros in the numbers they write. Because these are not even seen, pupils are not aware of their presence, and ironically, when they are asked to ignore them, they become confused. The solution is to talk about the leading and trailing zeros early on. If one refers the number 100 as "0000100… but we ignore the leading zeros" and talk that talk, then quite quickly pupils will become fed up with you wasting their time by always talking about the leading zeros which we are destined to forget about. Similarly if one refers to 0.21 as "0.210000… but we are going to ignore all the trailing zeros", again pupils will quickly become familiar with the concept of trailing zeros which we ignore. Then, when we start speaking of significant figures, the pupils will be familiar with the concept of insignificant figures (they’ll have had them up to their eyebrows) and they won’t struggle.

Again I am writing about the abstract nature of mathematics. Why would anyone think in the abstract when they can think in terms of things. It's why we draw diagrammes before we set about solving a problem. It's why scientists create analogies when dealing with conceptual ideas... what is an electron? No one has ever seen one, that's for sure. Sometimes it behaves like a little ball, on other occasions it behaves like a wave. Depending on which is the most appropriate way of thinking for the problem in question, the physicist chooses either one description or the other. In maths we should adopt the same strategy. Think of a mechanical analogy. If we're adding up numbers: think of piles of coins, if we're solving equations: think of a balance, if we're performing algebra: think of boxes containing unknown numbers which we move about. Maths is hard when we think of it in an abstract way. So don't think of it that way. At Puppet Maths we teach our pupils to imagine the situations where the maths might be applied. This not only gives relevance to the mathematical routines, but actually makes the maths easier. At Puppet Maths we are dedicated to making maths both easy and fun.

Mechanical analogies allow pupils to imagine whats going on.

The sine function

A pupil was having difficulty with the concept of a sine. What is a sine? Why should the ratio of the length of the line in a triangle opposite an angle, to the length of the line adjacent to the angle matter? Why would anyone bother with it? What was the point? Because of the use of the electronic calculator, to this pupil a sine was just a magical number that appeared when he pressed a button… he was perplexed, where did the calculator get it from? When I learnt about the sine function at school, I was given a book of mathematical tables. When I looked for the sine of an angle I could see what the numbers for other angles were. I got a feeling for the relative values of the function for various sizes of angle. I learnt by observation that the sine function varied from –1 to 1. I realised that calculating the sine function itself was difficult to achieve… that’s why I was using a table containing pre-calculated values to look up the value I needed. But none of these observations are available to the modern pupil, just the magic calculator button.
I explained the sine function as the distance that a shaft connected to rotating wheel moves vertically and that the cosine is the distance that the shaft moves horizontally. As soon as I’d done that, the pupil saw the reason for the function, that it was about knowing where the piston on an engine was relative to the position of the crankshaft, or how the various levers on a loom move as the driving wheel move around. At Puppet Maths we relate maths functions to the real world, so that they become relevant to the pupils.