Play can be work too

In the UK there is a bottled water company called who are a non-profit making organization. The surplus generated by their sales of bottled water go to sink wells in rural sub-Saharan Africa. But sinking a well to access clean water is only half of the problem. The inhabitants of rural Africa are poor and they cannot afford to buy the fuel to power the pumps that are needed to raise the water from the well. How is the water to be pumped out of the well. This inventive company has solved this problem in a novel way. They have given each community where they have sunk a well a children’s playground roundabout. The children sit on the roundabout and push it into motion with their feet, as children do throughout the world wherever one of these things are found. But in this case the roundabout is not just a roundabout, it is also a water pump. As the children play, swinging round on the play apparatus, they also raise water from the well. It’s what can truly be called a win-win situation. We at Puppet Maths are inspired by this example of lateral thinking. We want children to succeed at maths. How better to get children to practice their maths than to make it into a game that they are eager to play? This is our purpose, to make maths fun, to make children want to do maths, and so practice their maths, and so become good at it.

Summer holidays affect children's ability to do maths

Another surprising finding reported by Malcolm Gladwell in his book “Outliers” is that in the USA, the least well performing pupils make better progress during school year than the best performing pupils do, but their performance falls back over the long summer holiday; whereas the best performing pupils’ performance goes up over the summer, when the children are away from school. The conclusion drawn from these findings is that during the summer the activities that the best performing children undertake enhances their mathematical ability, whereas those undertaken by the least well performing pupils allows their mathematical ability to atrophy. This is linked in the book to the social class and the income of the children’s families. It is proposed that the high earning parents steer their offspring in directions that enhance their maths during the holidays. There are a number of interpretations that can be placed on this deduction. One is that the parents of the high achieving pupils are in some manner hothousing them during the summer, but alternatively, it could just be that they are interacting with their children in a manner that stimulates their thinking and keeps them sharp. But rather than look at what the wealthy parents are doing, let’s think about what the parents of the low achievers are doing. The answer is very possibly that they are simply leaving their children to their own devices. At Puppet Maths, it has occurred to us that if children enjoy maths, then they would choose to practice it themselves without needing prompting from adults. That is the aim at the centre of our vision. We want all children to find maths interesting and find maths fun. We want them to choose to do maths because they enjoy it, and not only because an adult is directing them. We want to make maths interesting, so that children choose to practice maths themselves.

Short names for numbers are better

Apparently, the human animal can hold about 2 seconds worth of number data in our heads. So if it takes us one third of a second to speak a number then we can hold 6 numbers in our heads, if we were to be able to say the name of a number in as quarter of a second, then we’d be able to hold 8 numbers in our heads. Numbers have names in Chinese which are remarkably short. Compare the Chinese word with the English word, “qi” versus “seven”, the Chinese is shorter, or even when the number of syllables are the same, “si” versus “four”, again the Chinese word is shorter. According to the work of Sstanislas Dehaene, reported in Malcolm Gladwell’s book “Outliers” the memory gap between English speaking and Chinese speaking people when it comes to remembering numbers is “entirely due to this difference in length.” The Cantonese dialect, spoken in Hong Kong, names numbers with such brevity that residents of Hong Kong have a memory span of 10 digits. In the light of the difficulty that I wrote about a couple of days ago, of pupils being unable to do mental maths because they forget the numbers that they’re supposed to be working with, the increased number retention that the Cantonese dialect gives Hong Kong residents a distinct advantage when it comes to performing mental maths. One way of overcoming the limitation of memorising numbers is to visualise them. A picture says a thousand words and instead of the information being stored in series as they are when you try and memorise some numbers, in a picture they are stored in parallel. Back in August, I wrote of bank tellers from Singapore and Hong Kong who, in the days before electronic calculators, used the abacus to perform their calculations, and how, after a while they could do the calculations without the abacus, because they simply imagined the movement of the beads on the apparatus. At Puppet Maths we encourage children to visualise the maths puzzles that they are asked to do. This is a technique that any child can master to make maths easy. And that is our aim, to make maths easy, to make maths fun, to make children enjoy maths.

Maths is to be worked out not learnt by rote

When I was six years old, Miss Creamer, my teacher, gave me a test. She quickly wrote “3 x 3 = ” on a piece of paper and gave it to me to solve. I immediately noticed that the symbol in the middle was a “times” rather than an “add”, but back then I didn’t know my times tables, so I didn’t know the answer. The cross on the paper wasn’t perfectly aligned as a multiply and I decided that the best strategy would be to assume that the “x” was actually a “+” miswritten. So I added the two 3s together and went back to Miss Creamer. To my dismay she was unimpressed with what I’d done and marked it wrong, and in doing so pointed out that the symbol in the middle was a multiply not an add, which rather quashed my ploy of misunderstanding the question. I went back to my desk and spent some more time thinking about it. To me, the shapes “3” and “3” had always combined to produce the shape “6”, what else could the answer be? So I again wrote “6” and took it back to Miss Creamer. Again she marked it wrong. Now I was stuck, so I asked her to give me help. She declined to do so, but this time as she sent me back to attempt the calculation again she told me to “Think”. Here was a revelation. Suddenly I realised that this was not something that I was expected to know, this was something that I had to work out. I sat down again and this time I tried to reason what the expression meant. I knew that it didn’t mean 3 and 3, so what else could it mean? After a while I thought, could it mean three lots of three? Well it was worth a try. I calculated three lots of three to be 9, and wrote that down on the piece of paper, and took it back to Miss Creamer. This time she marked it correct, and praised me for doing well. I went back to my seat with a feeling of great pride and achievement. I had solved a difficult problem.
This story illustrates a number of elements of teaching and learning maths. It shows the reluctance that a child has to go beyond it boundaries in which it feels safe. I was comfortable with 3 + 3, I didn’t want to face the hostile idea that it might be “x” between those two numbers. Also it shows that after 2 failed attempts at answering the question I had given up and I asked for help, expecting to be bailed out. It shows the enormous satisfaction that a young child can get when they don’t give up (in my case because I was compelled not to) and solve a puzzle on their own. It is this latter point that I remembered this incident for, for many years.
But since I started teaching I’ve realised that there is a further aspect to this story. It was at this point in my life that I learnt a much more important lesson. The lesson was that maths is not a set of facts that have to be learnt, that maths can be worked out. But more than that, I solved that problem using logical reasoning. It was on that day that I realised that maths wasn’t just about numbers and the rules that govern them, but that maths was about the use of logic. This is a message that is central to the philosophy by which Puppet Maths teaches maths. Maths is a branch of logical reasoning. Maths is the expression of logic. From our experience teaching maths in schools we know that once children realise that all they have to do is reason their way to an answer, fear of maths falls away. At Puppet Maths we want to banish fear from maths, we want children to enjoy maths.

Use imagination to make sense of maths

In “Outliers” Mr. Gladwell observes “The much storied disenchantment with mathematics among Western children starts in the third and fourth grades, and Fuson [Karen Fuson, a psychologist at Northwestern University] argues that perhaps a part of that disenchantment is due to the fact that math doesn’t make sense; its linguistic structure is clumsy; its basic rules seem arbitrary and complicated.” Third and fourth grades are children aged eight and nine. There is a clear developmental stage that occurs when children turn eight when they become less obedient and more independent. They suddenly become less likely to do something just to please the teacher or their parents. They want to start doing things for themselves. At this age, they need to understand what’s in it for them. Hence it is even more important that children at this age find maths to be something that they can achieve at, and something that is fun. At Puppet Maths we make maths fun, children love the school subjects that they can “do”. How often have we heard a child say “I like this, this is easy?”. So we at Puppet Maths make maths easy, by capturing the child’s imagination. Children like challenges and puzzles, until they get stuck, so our aim is to challenge children and stretch them, but at the same time to give them the tools and support they need so that they don’t get stuck, instead they succeed at maths.

English is not the best language for numbers

In German apart from the irregular numbers “elf” (11) and “zwoelf” (12) the two digit numbers (those between 10 and 100) have names that start with the quantity of units followed by the quantity of tens. Hence 21 is “ein und zwanzig” or one and twenty and 56 is “sechs und fuenfzig” or six and fifty. This makes it easier for the German school child to do mathematics in their head than it is for an English speaking pupil. When performing addition we are taught to add the units first, then carry any tens created across before adding up the tens. Because of the way that German orders the digits with the units coming first, it is easy for the German child to add the units first before moving onto adding the tens. In doing mental maths, the English speaking child faces a different type of problem. In order to focus on the quantity of units, the English speaker has first to ignore the number of tens, and the problem they face is not with the addition, but with remembering what the original numbers were. In the example above, adding twenty one and fifty six, the child (or adult for that matter) has to ignore the “twenty” and focus on the “one”, and then ignore the “fifty” and focus on the “six”. Now these two numbers can be added, six plus one gives us seven. Now to focus on the tens. But what were they? Having discarded the first half of the two digit numbers from memory in order to focus on the units, the difficulty the pupil has is in recalling what the original quantities of tens were. So whereas German pupils are buoyed up by their ability to do two figure sums in their heads, English speaking pupils are discouraged by their failure to do so. But this has nothing to do with their mathematical ability, it is a result of the language that they use. At Puppet Maths we teach mental arithmetic differently. We teach children to visualise numbers as quantities of physical articles, which allows them to add them in any order they like. We encourage them to count up. So in the example above we would encourage the pupils to imagine 21 pence and 56 pence as coins. Then they could add them up adding the fifty and the twenty to get seventy pence, and then the one and the six to get seven pence. This visualisation, and the performance of the addition in the order in which the digits are used in the name of the numbers overcome the memory problem, and allows children to succeed in this aspect of mental arithmetic.

Maths is not for compartmentalising

In Malcolm Gladwell’s book “Outliers”, in exploring the comparison between the Asian manner of naming numbers and that used in English, Mr. Gladwell quotes Karen Fuson, “a Northwestern University psychologist who has closely studied [sic] Asian-Western differences. ‘I think that it makes the whole attitude toward math different. Instead of being a rote learning thing, there’s a pattern I can figure out. There is an expectation that I can do this. There is an expectation that it is sensible.”
The disturbing thing in this comment by Ms. Fuson is the unsaid presumption that maths is viewed as being something that a child cannot do; something that isn’t sensible but arbitrary, and so must be learnt by rote rather than calculated by reason.
But English speaking children have by definition learnt to speak English. There can be no more irregular and arbitrary language than English (even when the spelling has been simplified as in its American version). So if children are capable of mastering this monster, why do they have such difficulty in mastering the regular language of maths? One reason is that maths is taught as a set of rules. These rules apply to the subject of maths, which is taught separately. So children, rather than integrate maths into their everyday experiences compartmentalise maths. The upshot of this is that they don’t learn to translate between numbers and the English language, and therefore they don’t see the connection between the rules of maths and the logic they are derived from. So often in real life, maths is used simply to save having to wade through much logical reasoning, but that is not apparent to a child. Having compartmentalised maths in their minds, children shut off the possibility of linking it to the rest of their lives. At Puppet Maths we use our puppets to create situations where puzzles are solved both using maths and logic. In this way the equivalence of the two approaches is demonstrated inherently in our teaching.

The names of the numbers makes maths hard

I reported the other day that I have been reading Malcolm Gladwell’s book “Outliers”. Consistent with Mr. Gladwell’s other books, this is an excellent treatise on the hidden factors which underlie the various effects that we observe in everyday life. In this book he reflects on some of the hidden factors that lead to success in maths. He describes a feature that I have touched upon previously in this blog, but he has expanded it. Previously, I mentioned how, in German, the manner in which numbers are named separates the tens from the units and explicitly declares that tens are different things from units, an advantage which the English language does not afford to its speakers. “Outliers” makes further related points.
Mr. Gladwell writes that:
“In English, we say fourteen, sixteen, seventeen, eighteen, and nnineteen, son one might expect hat we would also say oneteen, twoteen, threeteen and fiveteen. But we don’t. We use a different form: eleven, twelve, thirteen and fifteen. Similarly we have forty and sixty, which sound like the words they are related to (four and six), but we also say fifty thirty and twenty which sort of sound like five, three and two, but not really.” He then goes on to compare this nomenclature with that in the Chinese, Japanese and Korean languages. “They have a logical counting system. Eleven is ten one. Twelve is ten two. Twenty four is two tens four and so on.
“That difference means that Asian children learn to count much faster than American children. Four-year-old Chinese children can count, on average, to forty. American children at that age can count only to fifteen.”
He further observes that “The regularity of their number system also means that Asian children can perform basic functions such as addition much more easily. Ask an English speaking seven-year-old to add thirty-seven plus twenty-two in her head and she has to convert the words to numbers (37 + 22). Only then can she do the math… Ask an Asian child to add three-tens-seven and two-tens-two and the necessary equation is there embedded in the sentence.”
Here we see that one of the big problems with learning maths is the interplay between maths and language. Those children who master the boundary between English and numbers, get to understand how the number system works, and depending on the language they speak are facilitated in undertaking maths problems. This puts them at a great advantage over those who do not do so. At Puppet Maths we take great care to teach our pupils how to translate from one language to the other, from the language of the spoken and written word, to the language of mathematics. Once children are able to relate the numbers that they are expected to manipulate at home or at school with ideas which are described in spoken language, then the task of studying maths becomes a very much easier proposition. At Puppet Maths we facilitate this.

Avoiding stress when teaching maths

The brain is a curious thing. It is made up of two parts, the primitive brain and the cerebral cortex. The way our minds behave are a consequence of the structure of our brains. The primitive brain is concerned with our survival, and the cerebral cortex handles the other functions. When we are under stress, the primitive brain takes over control of our mind and shuts down unnecessary brain functions the better to allow us to focus on the problem of survival. This is why it is so much harder for contestants on TV quiz shows to recall the answers to general knowledge questions than it is for the viewer sitting at home. Their primitive brains have shut down the memory function that would allow them to recall obscure facts while it focuses on fight or flight.
A similar effect occurs in children when faced with maths. They know that they are going to need to use the information that they are being given. This puts them under stress. This causes their primitive brains to shut down the very imaginative and memory functions that they need to comprehend what they are being taught. If pupils fear criticism or punishment if they cannot do the work they’re set, their primitive brains will switch off engagement with the subject completely, and the children will not listen to what’s being taught, but instead fret about the dire consequences that area bout to fall upon them. Instead of asking for help that will address their lack, they will try and hide their inability, because this is what their primitive brain is telling them to do. At Puppet Maths we wish to avoid giving pupils stress. Our puppets engage children’s attention allowing us to explain how mathematics works without them realising that they’re actually working. By this means we address their minds without having to overcome the barrier that the mind erects whenever it knows that it is encountering something for which it will be held accountable later. So we can capture their imaginations and their memory. We believe that this is the best way to teach maths whether the child is learning at home or at school.

Perseverence is the secret of success

As Malcolm Gladwell observes, attainment in any field has as much to do with the experience one gains in the field as it has to do with natural ability and flair. This means that children who are having difficulties with maths should spend more time working at the subject and get more attention and teaching in the subject, because it is not a failing in themselves that is holding them back, but simply a failure to be helped around the misunderstandings that block their path, and a subsequent lack of the practice that would give them facility in the subject. This philosophy underlies Puppet Maths. By encouraging children to view maths as being easy and being fun, they can be lured into spending more time practicing the subject and thereby honing their ability in the subject. Children need encouragement, and help to overcome the stumbling blocks that lie in their path as they try to understand maths, and Puppet Maths aims to provide the support they need. Edison, the American inventor said that success is “10% inspiration and 90% perspiration”, and this is true of maths too. Children who get the attention they need to overcome the difficulties they encounter, and who are given the opportunity they need to practice the subject become good at it. If they are not gaining these opportunities at school then they can still achieve if they can access them at home. At Puppet Maths we aim to describe the workings of subject, in a easily understandable way. We are here to help children through the difficulties they encounter as they learn maths, and we help them practice the subject so that they gain the experience they need to succeed.

Peresistance when learning maths

It is never too late to start studying. By taking the time and opportunity to study most people can achieve well in any particular field. The problem that most people face when they wish to study a subject is accessible teaching. When I worked as a research engineer, I was by definition working at the edge of the knowledge base. There were no books or texts that explained how things worked. Indeed there were no books or texts that explained the background to the current state of technology, the only published information was about the underlying fundamentals. So anyone who was working outside of the field, wishing to enter any particular field, was faced with an almost insuperable problem of discovering the nature of the current state of play. But this problem is more general. It occurs at every level of attainment, even the fundamental level, but here the problem is slightly different. The problem is not gaining access to the information, this is freely available, but lies in gaining access to accessible information. Children who would benefit from extra maths tuition have no difficulty finding maths text books at whatever level they require to be taught, but these need to be interpreted to them. Reading a maths text can be one of the most dispiriting and boring activities on the planet. Children aren’t going to be inspired to learn maths from being presented with a maths book. Puppet Maths is designed to address this problem. Puppet Maths is designed not just to explain how maths works, but also to engage and entertain the child, so that maths becomes fun, and maths becomes easy.

Home education

Many years ago, in 1986, I started to learn Mandarin Chinese. I found the language quite hard to learn until I realised that you sing it rather than speak it, at which point it became much easier. At the end of the academic year, I felt that having been introduced to the language I was now ready to start studying it. I decided that I should repeat the year. So I went to enrol for the first year course the following September. Unfortunately, the Chinese language courses were on a two year cycle. One year the college would teach Beginner’s Mandarin and Advanced Cantonese, and the next Beginner’s Cantonese and Advanced Mandarin. So the first year course was not available to me. At this point I gave up, and went off to learn German instead. A couple of years ago, I met a West Indian gentleman who was an interpreter working for the National Health Service. He translated between English and Mandarin, and taught colloquial Mandarin. He explained to me that I should not have given up on my studies, I should have persevered. Had I done so, I would have spent the last 18 years working on the language, and I would by now be fluent. The drip feed over time would have made me competent. The same is true of maths. So many pupils struggle at maths, and then give up. If they were to keep plodding on, over time they would develop. But they don’t get the opportunity. The curriculum at school moves on year by year, and pupils who have not grasped some basic concept are not given the opportunity to practice that basic topic and achieve at that level, they are required to do new work. When that work relies on the foundation of the previous concepts, pupils who lack the basic understanding of the previous concept cannot succeed at the higher level. However, studying is not confined to school. If the school is not providing a child with support at the level the child needs then the child can make amends by studying at home. Unfortunately so often, parents are not well equipped to help a child study, and children are reluctant to do schoolwork that they don’t enjoy in what they consider their free time. Puppet Maths was created to address this problem. At Puppet Maths we provide a resource that supports parents who wish to teach their children at home and we engage pupils in a way that gives them the opportunity to learn in a fun manner, and to practice what they learn.

Maturity and learning maths

The concepts that underlie maths are not complex or difficult. However, those of us who use maths everyday take them for granted. Young children do not necessarily naturally pick up these concepts. They have to be explained to them. Unfortunately, so often they are not explained well, because when someone takes something for granted they lack the ability to explain what it means. [As a test of this, try explaining the meaning of the word “as”]. Children are, by definition, immature. While they are programmed to learn, they are not necessarily mentally equipped to learn abstract concepts. Ability to do this is a sign of maturity. Does this doom the child who is slower to mature to an inability to ever succeed at maths? Unfortunately, the answer to this is “very possibly”. Of course the degree of disadvantage that a child will suffer will depend upon the extent to which their development lags that of the average of their age group, or perhaps even lags that of the most advanced in their class at school. The principal question that arises from this observation is “how can we help a child who is slower to develop to achieve at maths?”. At Puppet Maths we have created an imaginary world in which our puppets experience everyday problems that illustrate in a concrete fashion the abstract ideas that abound in Maths. This was developed from my work in schools with 11-13 year olds who were failing at maths because they had never grasped the basics of the subject. All of these children were late developers, and still had relatively juvenile view of the world, but were old enough to have become embarrassed at their lack of ability at maths. Our aim at Puppet Maths is to engage young minds, at home or at school, and illustrate the principles of maths so that children can use their imagination to grasp the subject.

The Matthew principle

I have been reading Malcolm Gladwell’s book “Outliers”. Malcolm Gladwell is a journalist and economist who writes popular texts on the way the world really works, as opposed to the way we suppose it works. In this book he observes that Canadian ice hockey stars have a strong tendency to be born in January, February or March. This effect is known as the “Matthew Effect” from the Bible verse Matthew 25:29 “For unto everyone that hath shall be given, and he shall have abundance, but from him that hath not shall be taken even that which he hath.” He argues that because of the age group cut off date of January 1st children who are born in the first three month of the year tend to be more physically mature and therefore able than the rest of the children with whom they play hockey. Therefore, they are the best of their year, so they get the encouragement, they get the extra training opportunities, and more practice. By the time they reach puberty, they are better than those children who were born later in the year simply because they have had access to encouragement, better training, and more practice.
The effect of maturity is evident in Maths also. Those pupils who are a little more mentally mature and whose minds capture the essence of how maths works early on, become good at maths. Those that don’t don’t. Maths is a pyramid, so much of the later work is predicated on a knowledge of the early stuff. If the basics are not firmly in place, then not only does the child not receive encouragement, more attention and better training, but also is handicapped by an inability to access the subsequent parts of the maths curriculum.
Puppet Maths is designed to make maths accessible to young children, so that even relatively immature children can relate to the concepts of the subject. We make maths fun.

Learning about Triangles

A couple of days ago I told a joke about mathematics. The point of this joke was that mathematicians do not try and solve a problem to arrive at an answer, but to manipulate the problem into resembling one that has been solved before, so that it can be solved routinely without the need for inventive thought. This is particularly clearly seen with geometry. Geometry is necessary for civil engineering and mechanical engineering. The theorems of geometry allow the designer to calculate on paper what needs to be done, rather than have to build whatever it is they are designing and then modify it. Working out the height of a particular structure involves geometry, as does working out the area and volume it occupies. Deciding if a structure is stable, or if it will fall down requires geometry. However, there is no reason for the designer to reinvent the wheel, mathematicians have already worked out just about every attribute of the triangle, and hence every attribute of any shape that can be made up from triangles. This is why, in maths, so much time and energy is expended on learning the properties of triangles. But for so many pupils in school learning about triangles is an irrelevancy. They cannot see the use of this learning and consequently are unmotivated. At Puppet Maths we use our puppets to create situations that demonstrate the relevancy of the maths we teach, so that our pupils are engaged. Once a young mind is engaged with the subject then maths becomes fun, and once maths is fun, then maths becomes easy

Rote learning versus thinking.

Should children learn facts by rote, or should they they be taught to think? In maths which should be teach? The answer is both. I spent a significant period of my life as a quality manager in the medical devices industry. My job was to stop people thinking. When you do something without thinking you don’t make mistakes (and in the medical devices industry mistakes aren’t acceptable). It is only when you think about an activity that you get things wrong. As an example consider when drivers change gear. On a manual shift car, drivers change gear without thinking routinely, and they do so without encountering a problem. However, when they think about changing gear, they get it wrong and grind the gearbox cogs together. Had they done it automatically they would probably have got it right. So lesson for maths is that for basic routines, which people will use often and repeatedly, the best strategy is to train people to do them automatically, to do them without thinking, to know the method and just apply it. This is where rote learning, and repeated practice is important.
However, before we can solve a maths puzzle using standard routines, we have to convert the problem to a standard form which is tractable to a solution using them. This is where thinking is required. If all one can do is manipulate the standard routines, then one is destined to be nothing more than a calculator. So it is important to teach children how to interpret problems and using logic rearrange them into a form which can then be solved using the standard routines. This is where pupils need imagination and insight. This is the fun part of maths. So many children are put off maths because all they are ever expected to do is practice the standard routines, and then as a consequence they find the subject boring. At Puppet Maths we teach pupils to think and use their imagination when solving maths puzzles, but we also, by using puppets introduce an imaginative element to the activity of practicing the standard routines and alleviate the boredom that so many pupils experience when learning them.

The nature of mathematics.

There is a joke about mathematicians. It involves boiling a beaker of water. A physicist and mathematician are applying for a job and the interviewer has prepared a test to see how good they are. In the first test they are to be presented with a Bunsen burner underneath a tripod, to the left of the Bunsen burner is a box of matches and to the right of the Bunsen is a beaker of water. The physicist goes first. The interviewer asks the physicist to boil the water. He thinks for 5 minutes and then picks up the beaker and stands it on top of the tripod, picks up the matches and lights the Bunsen. The interviewer thanks him and he leaves the room. Then the apparatus is replaced in its original positions and the mathematician is called in. He also thinks for 5 minutes and then picks up the beaker, places it on top of the tripod, picks up the matches and lights the Bunsen. The interviewer thanks him and he leaves the room. The second test is then set up. It uses the same equipment as before but this time the beaker is placed to the left of the Bunsen, and the matches to the right. Again the Physicist goes first. Again he is asked to boil the water. Again he thinks for a full 5 minutes then he lifts the beaker and places it on top of the tripod, picks up the matches and lights the Bunsen. The interviewer thanks him and he leaves the room. The apparatus is returned to its original positions and the mathematician is then invited into the test room. He is also asked to boil the water. Once more he looks at the apparatus for a few minutes. Then he lifts the beaker of water and moves it to the right of the Bunsen, and lifts the box of matches and moves them to the left of the Bunsen. Then he turns to the interviewer and says “hence a problem that’s already been solved”.
And here is a central principal of mathematics. Mathematicians strive to reduce a problem to one which has already been solved. So given a maths problem, the mathematician will use logic to modify it and home in on one that has been solved before. This why so much time is spent learning about geometry and trigonometry, because anything mechanical can often be resolved by applying the rules of geometry and trigonometry. It’s why we learn algebra, because if a problem can be resolved into a formula, then we can stop doing all the hard work of thinking logically and just apply the rules of algebra (which have already been worked out) to arrive at a solution. Many pupils do not understand why they are expected to learn obscure mathematical routines because they see no practical use for them. This makes maths boring. At Puppet Maths we put mathematics into context, so that children understand what it can be used for. Our puppets recreate the circumstances where the maths is needed, so that our pupils can understand why they are learning the mathematical routines that we teach them.

Maths as a game

A game is fun. It ceases to be fun when it is taken too seriously. One problem with maths education is that it is taken too seriously. Certainly it is a serious subject. Certainly, it is important that children learn to do it right. But if we are to encourage children to like the subject, and therefore practice it and become good at it, then we should make it fun. If we are to make it fun, then we should not be serious about it. There is obviously a delicate balance to be struck here. My second daughter had difficulty learning to read. Her mother used to lose patience with her, and snap at her when she had difficulty in reading words in her reading homework. This put her off reading completely. For her, reading was simply an activity that led to unpleasantness and upset. I remain indebted to J K Rowling, without whose books she might never have started to read, even so, today as an adult, she still reads slowly. At Puppet Maths we want to make maths fun. We aim to make math a game that children will enjoy, and so practice both at school and at home, and consequently become good at it. Children enjoy doing the things that they are good at, that they can show off doing. Unfortunately most often maths is not the activity they choose for this. Our dream is to change this, so that children choose maths as the vehicle for showing off how good they are.

Doing it the easy way

If a pupil is allowed to imagine the problem/puzzle, then they can rearrange the facts to make the puzzle easier. An simple example of this is when a pupil is asked to calculate 4 + 7 + 6. It is an easier sum to add 4 + 6 + 7, because 4 + 6 canbe recognised as being equal to 10, and 10 + 7 is then a trivial sum. But the principle applies to more complex situations and calculations. Another advantage of imagining the problem/puzzle, is that one can use analogous situations to make the calculation easier. When I was at primary school the class was given a nasty question which involved distances in many different units of measurement. As I struggled with it, the boy I sat next to told me it was easy and that I could use my ruler to find the answer. I misunderstood him and looked for the answer in the writing that was stamped on the rulers [it comprised facts like “10 chains = 1 furlong” and “8 furlongs = 1 mile”]. I quickly determined that there was nothing there which I didn’t already know and that he was either mad, or he had a different type of ruler from the ones in my possession, but neither was true. What he had done was use the scale on the edge of the ruler as a number line, which had aided him in working out the answer. So while I was trying to understand the problem and trying to sort out a solution, he just counted up and down his ruler and arrived at an answer in a fraction of the time it took me. (If I remember rightly, I ran out of time and never did solve that particular question). This ability to find and use an analogue for the content of a maths problem is an important skill which aids children’s maths ability enormously. At Puppet Maths, we teach children to use their imaginations, and to use analogies, the better for arriving at the correct solution with facility. We like to think that we are the home of imaginative thinking in maths.

Let your mind roam when solving maths puzzles

My personal approach to solving maths problems/puzzles is to use logical reasoning. By habit I use this in preference to mathematical routines. Only once I have reduced the situation to one which I recognise as having solved previously do I resort to using a tried and tested maths routine (algorithm). This suits me, as I can imagine situations and apply logic to them. People vary in their ability to do this. When I taught at a top Public School [for non English readers who use different terminology, that’s one that charges the parents a small fortune each year for the privilege of giving their offspring an education] I asked my class to design a three dimensional chess board. They had difficulty in doing so. When subsequently teaching at a state school [that is one that is funded from taxation] I thought that this would make a challenging task for my pupils. However, they produced a wide variety of suitable solutions with facility, almost with contempt. They found the problem trivial. The difference was in the imaginative ability of the pupils. The state school pupils could imagine the possibilities, whereas those destined to be the leaders of English society lacked the capability to do so. (What does this tell us about the way the country is run, and how it might be run in years to come?). Only if you can imagine the problem/puzzle in your mind’s eye can you apply logic to it to arrive at a solution. This is why we at Puppet Maths encourage the imagination of our pupils. We want them to allow their minds to roam widely so that they can visualise the problem in their own terms before using logic to home in on a solution.