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.

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.

Money for counting

What do young children understand? They understand what money is for. Money can buy them sweets and toys, it’s also used for other things as well, but they are not as important. Most young children can count out money to pay for their purchases. Some, however, do not work out the change they are due, they will pay with a £1 coin, or with a 50p piece, and accept that the change they are given is right. When I was young the shop assistants used to count out the change by counting up. If the purchase cost 33p and you paid with a £1 coin, then the shop assistant would count money into my hand starting at the value of the purchase (33p) and then add pennies “34p, 35p” then add a 5p piece “40p” then a 10p piece “50p” and finally a 50p piece “£1”. This way I knew that the change I was getting was right and incidentally I learnt how to work out the amount of change I was due from observation. But today, the electronic tills tell the shop assistant what change is due, they no longer have to count it up. A learning opportunity is lost, gone forever. At Puppet Maths we want to teach children these old techniques of handling numbers, because they’re useful and proven, tested throughout time. We provide relevance to our maths by associating it with money. Young children are concrete thinkers and they find it easier to imagine coins rather than handle abstract numbers, so we use money as a vehicle to allow children to achieve success in their arithmetic.

Using maths in everyday life

I worked for many years as an engineer. I used maths to solve the problems that I faced in my work. However, when faced with everyday problems I tended to use logic to solve them. Why was this? Why was my mind compartmentalised in this way? Why did I not use the maths skills I had to solve my everyday problems? Many times it would have been easier to have used an algebraic equation to arrive at the answer, than to try and think my way thorough the situation. So why did I not do so? It comes back to the way maths is taught in school. So often it is not related to everyday life. So often it is taught as an academic discipline in an ivory tower of its own. This is how I was taught maths. As a result, when I was doing engineering design work, abstract work, I naturally turned to my maths to solve problems, but when faced with an everyday situation I reverted to my ordinary life mode, and approached my problems using logical reasoning. We here at Puppet Maths relate our maths to everyday life. We show our pupils the use of what they are learning. Thereby we motivate them, because we show the true value of the techniques that we teach.

Fractions

Last night I was visiting a friend. His son is 14 and just starting to study for his GCSE additional maths. He had forgotten to bring his Additional Maths folder home from school and as a result he didn’t have his homework. He spent about 45 minutes trying to get his friend to text him the questions, and eventually, after the intervention of his father, he phoned and spoke to his friend and got him to read the questions out so that he could write them down. Having thus acquired the questions he needed, he set to work to solve them. The problems were solving simultaneous equations in 3 variables. The first question produced results such as x = 2, y=3, z=1; as did the third and forth questions. However, the second problem produced the answers x = -174/39, y = 75/39, z = 17/39. Given the solutions to the other questions were integer numbers it seemed strange that the author of the maths book should include a question with such a bizarre set of numbers for the solution. Having checked the working, I concluded that the boy had written down a wrong digit or sign when copying the question over the phone. But that is not my cause for concern. What concerns me is that this boy, who attends a grammar school and is towards the top of his class in Maths, was unsure when working in fractions. He was put off by having to work in thirty nineths. Why? There is no difference in maths in working in thirds, or working in fifths, or working in thirty nineths or eight sevenths. He was unsure what to do with the fractions he had, or how to manipulate them. He knew that the top was called the “numerator” and that the bottom was called the “denominator”, but he had no idea what they signified.

Fractions were a black art to him. For this I believe the electronic calculator is to blame. Today’s pupils do not work in fractions, they just divide one number by another using the electronic tool and write down the answer without thought as to where it has come from, or what meaning it might carry. A Puppet Maths, we believe that our pupils should have a thorough understanding of fractions, they should be able to do calculations with them without recourse to a calculator; and it should be easy and fun.

Rearranging equations

At the weekend, I was speaking to an ex-student. He is a man of 54 who runs a company selling fresh donuts from vans. However, he wants to study Marine Engineering, but to do so he needs to have an A level in Physics. As a result he enrolled on a Physics course at his local further education college. As a temporary teacher, I only taught him for one term, so on meeting him again, I asked about is progress. He told me that he was OK with the Physics side of things, but that he shed marks in the exams because of his maths. He has a particular blind spot for rearranging equations. He admitted to me that to rearrange an equation he had gone online and asked Google to find a website that would do it for him. This story shows the tragedy of so much of maths education in this country. Here is a man, who is no fool. He is commercially successful, but beyond simple addition, subtraction, multiplication and division maths has remained a mystery to him, and now is providing a block preventing him doing the things he wants to do.
It is not as though rearranging equations is difficult. So often maths teachers make it so, when explaining their way through the process. It is almost as though the teacher has achieved an understanding of the arcane routine, and subconsciously wishes to make it as difficult as they can for others to do so, so that these others may experience a similar sense of achievement when they break through and understand how it works. At Puppet Maths we have no time for such silliness. We are here to make maths easy. The whole point of our using puppets to demonstrate the mechanisms of mathematical computation is to provide images that are easy to interpret and memorable.

Flash powder

When I was a teenager, if I wanted to buy some flash powder all I had to do is walk down to the camera shop and I could buy a tin across the counter. I set out the other day to get some for the Puppet Maths videos that I am shooting. My first problem is that there is no local camera shop anymore. Cameras are sold by big retail multiples, and they only sell the fast moving items where volume and profit margin combine to provide the return they need for their fancy shop fitting and expensive locations. So I turned to the internet to see if I could find any for sale there. There is actually quite a lot about flash powder on the internet, including various recipies for making it in one's own kitchen. There are even some vendors selling the stuff... apparently, but on closer inspection they do not actually have any stock for sale. A phone call to one such vendor supplied me with the answer, no one supplies it because it cannot be transported. There are transport regulations that make it impossible to take delivery of it. Today one can only get it if one orders it in large quantities. I can understand that there should be regulations for the transport of explosives, but surely it is much safer for it to be transported in small quantities rather than large? Clearly the issue is not about safety. The only reason why it only available in large quantities is that the quantities need to be sufficient to pay for the overhead involved in managing the regulation covering the transport. Such is the society that we have engineered for ourselves today.

Writing and maths

The maze walker does two things that the routine matching pupil does not do. The maze walker is prepared to write out the expression she is working on over and over again with minor changes in each iteration, and also the maze walker is prepared to write out English explanations of what is going on as they make a logical progression from one point in their work to the next. So often I find that pupils do not want to explain the work they've done, and their calculation is just a jumble of numbers whose meaning is far from transparent as far as I, an experienced teacher is concerned. When I see such a jumble of numbers I know that the pupil has not grasped a basic tenet of mathematics, that is, that mathematical working should show a logical argument of how to get from one position to another. So many pupils think that maths is all about manipulating numbers. It is true, number manipulation is involved in maths, but that is not what it is about. Maths is about thinking logically, about reasoning and drawing conclusions, and it is about explaining how conclusions were arrived at. It is not just about getting the right answers - it is the process of arriving at those answers that matters.

Next in a sequence

Here is a maths puzzle. What is the next in the sequence?
3,5,7...
well it could be 9 (series of odd numbers), or it could be 11 (series of prime numbers)... or it could be anything. The series might be 3,5,7,301,303,305,307,601. You cannot tell the next number in a series from the preceeding numbers. If it were possible, and I knew how it were done, then I'd have made my fortune on the stock market already and retired. But spotting patterns and rules is an important skill, for many processes in life follow relatively simple rules. So we ask pupils to predict the next number in the series. Therein lies the mistake. We should be asking the pupils to determine the next possible numbers in the series, and to explain the rules that would cause those numbers to appear in the sequence. That transforms the process from a right/wrong answer to one of exploration and inventiveness. Therein lies one of the great problems with maths as it is taught in our schools. It is taught as a subject of boundaries and rules which are there to constrain the pupil rather than an opportunity for exploration and for the pupil to show how clever they are. Boys are particularly disposed to getting one over on those around them. They want to show how clever they are by tricking or outsmarting others, but they are often the most alienated from maths. At Puppet Maths we give our pupils the opportunity to show off how clever they are, we provide them with open ended questioning that allows them to think beyond the parameters of a narrow maths question and get one over on the teacher.

Equals and therefore

As a supply teacher, I always realised when pupils had not been taught maths well when they asked me "what do those 3 dots mean". When demonstrating to them how to do some routine or other I would invariably start subsequent lines of my maths with 3 dots arranged as an equilateral triangle, the symbol for "therefore". The fact that the pupils did not recognise this symbol showed me that they did not use it. If they did not use it, then they were not reasoning their way through the calculation.
This is not to say that they were not using mathematical routines. They might well have been following the rules of mathematics and rearranging their expressions in a manner that would eventually lead them to the desired solution, but they were not reasoning their way through. Maths should be about reasoning. At Puppet Maths we stress the use of logical reason within maths problems.
Instead of using "therefore" to start each line of their calculations pupils would often use the equals sign at the start of the line. They would do this even when the new line of working was not the same as the preceeding one. They might divide all the expressions on the previous line by 2, for example, and write it out again starting with the "=" sign. This is not true, and shows a lack of proper understanding of what "=" means. When challenged the pupils would say "you know what I mean", indeed I did (they meant "therefore", but that is no compensation for the lack of precision in the pupils' work. This lack of precision is indicative of pupils regurgitation maths routines rather than engaging in logical thinking on their own part. They do this... then do this... then do that, because that's what they've been taught to do, and they are not reasoning their way through the problem. At Puppet Maths we want our pupils to reason their way through their problems, as this will make maths easier for them, it will turn problems into puzzles, and it will make maths fun.

The maze

Solving a maths problem is like working one's way through maze. It is exploratory. In maths one has to rearrange the expressions constantly, simplifying them, trying to spot something one has seen before that might give a clue to the route thereafter. Many pupils of mathematics do not share this conceptual model of a maths problem. To them maths consists of the statement of a problem to which they have to match up the routine which will solve that problem. This is a huge burden for the pupil. They have to remember so many routines, and they have to be able to remember the occasions on which they are to be used. As they progress through their education the mathematical problems they encounter become more and more complex and there is no single routine that can be applied to provide a solution. At this point the pupil finds that they cannot do maths anymore. This method of doing maths probles is a much more difficult way than that encountered by the pupil who imagines the maths problems from the point of view of the maze walker. All this latter pupil has to do is simply rearrange whatever expression he has in front of him over and over again, simplifying it and trying to shape it into something that is able to produce a solution. On occasion the maze walking pupil will change the expression in a way that is unhelpful and have to stop and go back to a point where he is confident that he is at a reasonably sound position and then rearrange the mathematical expression differently to take another path. The problems he has to solve are the smaller immediate problems of what rearrangement to do next. He can ignore the bigger picture because he knows that as long as he obeys the rules of maths whatever expression he comes up with will be valid (whether or not it is helpful in producing a solution).
This trial and error approach makes maths easier, but it requires sufficient confidence on the part of the pupil to make mistakes, and then go on and correct them. At Puppet Maths we teach pupils to treat maths problems as a maze or a jungle through which they have to find their path, by skill, by guile or even through inspiration.