Use of words

When subtracting using the formal paper and pencil method, when the size of the digit that's being taken away is larger than that of the number its being taken away from, ten more have to be moved across from the next, bigger column. Just moving numbers about is not likely to inspire a child's imagination. How much better it would be to "steal" ten from the next column. Now there's an exciting word, full of conspiracy and danger. How much more fun can that be than just moving a number about? Children love stories, so when teaching maths, give them stories. Give them something that interests them. So much of the language of maths is dry and boring. This is a necessity in the adult world, where precision and certainty is needed. But is a big disadvantage when viewed from a child's perspective, they want romance and adventure. If this is what children want, lets give it to them, and have them learn arithmetic at the same time. This is what Puppet Maths aims to do.

The easy to take away

Once a child has grasped the concept of the difference between two numbers, he or she can understand the easy way of subtracting two numbers. Of course, a child can perform subtraction this way even if they don't understand why it works, but understanding is better.
It is based on the premise that adding is easier than subtracting. So to find the difference between two number all one has to do is start at the smaller number and count up to the bigger number. The quantity that is counted during in this process is the answer to the subtraction.
For example lets take 45 away from 224. If we were to subtract the conventional way, since, in the units column, we are trying to take a 5 from a 4, we couldn't do it without stealing a ten from the tens column and putting it as ten units into the units column, so as to have enough units make the subtraction. Similarly, in the tens column, we would have to steal a hundred from the hundreds column and put it as ten tens in the tens column to have enough tens available to make the subtraction. But the counting up method avoids all this palaver.
Let's do the subtraction using the counting up method. We start at 45, and we want to end up with a 4 in the units column. So we add units onto our original 45 until we get to a number with a 4 in the units column. The first number which gives us a 4 there is 54, at which point we have added 9. This is the value of the unit in our answer so we write it down in the units column of our answer. Now we've got to 54 and we've got to get to 224, so we count up in tens until we get a 2 in the tens columns, so 54 and ten makes 64, 74, 84, 94, 104, 114, 124... right we've got a 2 in the tens column now so how many tens did we have to add? 7, right? (You can count the number of tens on your fingers as go up, that way you don't have to keep two numbers in your head at the same time). This 7 is the number of tens in the answer so we write that down in the tens column of the answer. So we're now at 124 and we want to get to 224, so we count up in hundreds. It takes one hundred to get there. So we write this down in the hundreds column of the answer. So the answer is 179, and the only paper and pencil that was needed was the ones we wrote the answer down with.
Puppet Maths teaches children to do their calculations the easy way, as well as more formal methods, so the child can select whichever method is most appropriate for the question they face.

The difference between two numbers

"The difference between", that's a funny phrase to use with respect to a number. What's the difference between 6 and 9? Actually they're very similar to each other. Well, they're the same shape but one's upside down. What's the difference between 3 and 7, well they're completely different shapes, one's sort of made up of 2 curves, the other is made up of 2 straight lines. But in maths, "the difference between" means "take away". How on Earth did this phrase come to mean "take away"? If you represent the two numbers either by patterns of dice dots or by number lines, and then compare them, then it is quickly apparent where the phrase has come from. With dice dots some part of the dice dot pattern will be common to both numbers, and some part of the pattern will be present in one number but not in the other. It is this difference in pattern which represents what's left over when one number is taken away from the other. The effect is probably seen even more clearly when the number line is used. The number line for both numbers will start at 0 and extend, counting up, until the particular number is reached. This means that the whole of the number line of the smaller number will be common to both numbers. So the difference between the smaller number and the larger number will be that portion of the larger number's numberline that extends beyond the end of the smaller number's numberline. This is the bit that's left over when the smaller number is taken away from the larger number.
To understand the phrase one has to be able to imagine what numbers represent, the things that the numbers stand for (and notice that the number represents the numberline, not the other way round). It is this mental picture that will help a child succeed at maths and avoid the confusion of apparently meaningless abstract concepts. Puppet Maths is dedicated to helping children visualise numbers in a way that will make aritmetic straightforward.

The number line

The number line is one of the more powerful tools in teaching arithmetic. It conisists of a line with little graduations on it, starting with the smallest number of interest and counting up to the largest number of interest. Until the child has learnt about negative numbers, the line will usually start at 0, but thereafter it will start at whatever negative number is appropriate. The power of the number line is that it gives the child something solid to use when imagining the relationship of numbers one to another. The number line is very useful because it allow the childs to count up and down it, and to compare numbers on one number line with those on another. It provides the framework on which they can hang their entire concept of the number system. The importance of integrating these ideas in children's imaginations is often overlooked in schools where the emphasis is on teaching the mechanics of manipulating numbers in order to get the right answer. At Puppet Maths we are concerned with capturing the child's imagination as well as giving them the skills to get the right answers.

Counting down

Looked at with an adult eye, subtraction is pretty much the same thing as addition, it's just counting backwards. But for the child, subtraction can be very confusing. It starts with the plethora of different names that the operation carries. Subtraction is not only called subtraction, but also: taking away, difference, and minus, which doesn't make life any more straightforward.
Children also find counting backwards difficult. It was not always thus. In the early 1960s many children learnt to count down from 10 before they learnt to count up to 10, such was the effect of the space race on their imaginations. Needless to say, it wasn't in school that they learnt to do this, it was in play. Also back in the 1960s TV and radio used to play the "Hit Parade" which typically counted down either the top 10 or the top 20 all the way to "Number 1", the best selling song of the week. Neither of these are a fixture in today's children's lives, as a consequence they don't learn to count backwards, and that makes subtraction more difficult for them. It's a good idea to play rocket launching with young children so that they learn to count backwards. At Puppet Maths, we involve children in playing rocket launching specifically to teach this important skill.

Adding more numbers

Piles of coins are a good analogy when one is doing addition, and one is moving quanntities from one column of numbers to the next, but it is not the best analogy. The problem with piles of coins is that it is hard to see quickly just how many coins there are in the pile. The spots on a die are much better because the number of spots can be instantly recognised. The challenge is to combine the two to create a system that combines the good qualities of both. This is what we have done at Puppet Maths. Once children recognise the quantity represented by the arrangement of the spots on a die, then they can learn to rearrange those spots in conjunction with other spots that they wish to add the number to, to form a pile, which can then be easily separated into those that are numerous enough to turn into a quantity that will fit in the next column, and those that are left behind in the column they already find themselve in. By using spots from a die, we have developed a mechanical analogy, that children can hold in their imagination, and instead of struggling with manipulating numbers they can just let their imagination tell them what the answer will be. This is a much easier way of getting the right answer, and it is the secret that children who are good at maths already know, which those who struggle with maths do not. We at Puppet Maths want all children to know these secrets, we want them all to get their sums right simply by applying their imagination to their maths problems.

Adding numbers

When I was 7 years old, I was required to learn about money. In those days the United Kingdom (along with most of what had been the British Empire) used pounds, shillings and pence as their currency system. There were 12 pennies in a shilling, and 20 shillings in a pound. Suddenly I had to learn to count up to 12 before my pennies turned into a shilling. I was taught to imagine this as a pile of copper pennies which grew taller as more were added, and when they were 12 high, those 12 were slid through a gate into the next column where they changed into a single silver shilling coin. Any copper coins which were above the lintel at the top of the gate (because there were more than 12 pennies there) didn't get into the shillings column, and fell back down to the bottom of the pennies column ready to have more placed on top of them later.
This mechanical analogy made the abstract concept of number base real, because at that age I was still a concrete thinker. It didn't matter what the number base was, all I had to do was adjust the height of the gate lintel. So for converting shillings into pounds, I just used a similar system, but with a bigger gate.
What is important in today's world is that the technique is just as applicable for adding when using base 10 as it is for any other base. When adding numbers, children can imagine that they have got a big pile of pennies (or perhaps I should say "new pennies" to distinguish them from real pennies, which had Britannia seated majestically on the reverse, were large and were capable of buying a bread roll all on their own) which slide through a gate whose height is 10, and those that get through turn mmagically into a ten penny piece, whereas those that get wiped off by the lintel stay in the pennies column, and fall down to the bottom of that colummn with a mighty crash. How much more fun is that than just adding numbers? If it is more fun it'll keep the child's interest longer, they'll enjoy maths more, they will become better at the subject. This is what we aim to do at Puppet Maths. We aim to make maths fun by appealing to children's imaginations. Once we have their imagination, then we can make them do well at maths.

Imagination and maths

There was an edition of "The News Quiz" on BBC Radio 4 some years ago now which annoyed me. "The News Quiz" is a satirical programme, in which commedians make fun of the events and personalities involved in the week's news stories. However, in the course of this programme there was one story that got to me, such that I remember it to this day. A story was told about someone or other, who had once applied to Oxford University to study maths, but who had been rejected on the grounds that they didn't have the necessary imagination. The audience, led by Barry Took, the compere of the time, found this extremely funny. Now I suppose that I shouldn't have got annoyed that the mickey was taken during a programme of the nature of "The News Quiz", because that is what it does, and I was listening to hear other people and events being sent up, but I was upset to find that so many people failed to realise the importance of imagination in maths.
Many persons find maths boring, and I assume that these are the people who are not using their imaginations when faced with maths. If one doesn't use one's imagination, then naturally whatever it is that one is doing is going to be tedious. Cartoons on the TV stimulate children's imagination, and they are riveted by them, much more so than by live action. Similarly puppets stimulate childrens imaginations. This is why we at Puppet Maths use puppets to teach maths. If we can stimulate children's imaginations when they do maths, then they'll find it much more fun, they'll find it easier, they'll get the right answers more often, and they'll grow in confidence and ability, and they'll enjoy the subject rather than find it a confusing chore to do.

The use of dice

Dice are extremely useful in teaching basic addition. It is a shame that they are 6 sided rather than 9 sided, and perhaps it would be better for the world if it were flooded with 9 sided dice just to help maths education. However, 6 sided dice are what people have played with for thousands of years, and I suppose that we are stuck with them and their limitations. The fact that they are not the perfect tool for teaching maths skills, does not mean that they are not a useful tool.
Dice have many good features. They have dots on them, arranged in specific patterns that allow the child to recognise a number immediately, and this, importantly, gives the child a concrete idea of what the number looks like. The figure "5", for example, is just a shape, but the arrangement of dots on a die is something that is immediately recognisable (after a bit of practice) as being a five with the added bonus that there is something concrete for the child to use to count with. Once the child has mastered the idea of the dots representing numbers, then they can rearrange the pattern of the dots in their mind's eye and combine it with other patterns of dots, and so do their sums mentally.
Puppet Maths encourages the use of dice, the recognition of patterns of dots to represent numbers, and the rearranging of these patterns within a child's immagination to accomplish addition and subtraction.

Developing abstract thinking

Abstract thinking is a sign of maturity. Children do not develop it until they reach a certain age. However, that age is determined by what they are expected to do. Across the board, child development is predicated on what is expected of the child. In western Europe and North America, chidren do not learn the balance required to stand up in a boat and spear a fish until they are about 13 years old, however in the Amazon jungle, where children are encouraged to undertake this activity, it is rare that a child hasn't developed this ability by the age of 7. Similarly, American research found that children do not develop the ability to count in bases other than base 10 until they are 11 years old, and in doing so ignored the millions of people who learnt to count in bases 12 and 20 at the age of 7, simply because there were 12 pennies in a shilling and 20 shillings in a pound. The British monetary system required children to develop this ability earlier, so children developed it earlier.
So if children do activities that require them to undertake abstract thinking, then their ability to think abstractly will develop sooner.
One good way to promote this development is for them to play board games. Board games take real life and make it abstract. Simple games like "Snakes and Ladders" are ideal for teaching a child to count up to 100, to recognise the number of dots on a die, and so count up to 6 in a flash, and how to add numbers up to 6 + 6 = 12. All this should be accomplished in an unthreatening environment. There are no real ladders or real snakes involved, but the concept is there in abstract form, and this analogue develops abstract thinking. As a child gets older, then more complex board games, games like Monopoly and Risk, where tokens represent armies, and a map represents the world, should be played. These help develop abstract thinking. The use of puppets, as analogues for people, as we do in Puppet Maths, encourages abstract thinking by children, as the things that are possible in the immaginary world we create do not happen in the real world, but conceiveably could do.

Concrete Thinking

If abstract thinking is a sign of maturity, then it follows that young children prefer concrete thinking, thinking about actual things rather than ideas. When children start to add and subtract they use their fingers to count on. This is a perfectly legitimate way for them to aid their counting. Unfortunately, it can be slow, and it introduces a limit in their minds that it is easy to add on any number up to 5, but that it's hard to add a bigger one - an impression created solely by the need to put down their pencil to use both hands to counnt on, and having no free hand to point to the fingers while counting.
To encourge children to move on and do their sums without using their fingers, pupils are told not to count on their fingers. Many children, at this point, think that they are expected to add and subtract numbers without using any sort of framework to count up/down the numbers. They can remember the first few like 2+3 which they know add up to 5, but they do not remember bigger ones like 7+8 (which was more than hard enough when they were using their fingers to count on). "How on Earth,"? they ask themselves, "do I add these big nummbers?". What's more, numbers go on up for ever and ever, and the child, quite sensibly, realises that they won't be able to learn all the combinations of all of them. And they give up.

Children need to be given some other way of counting something concrete, which they can do in their mind's eye (or on a scrap of paper) that will enable them to add/subbtract numbers up to 9. Because of the way the number system has been built, addition and subtraction only requires manipulation of numbers up to 19, another fact that isn't immediately apparent to the child. At Puppet Maths, by using puppets, we tap into the children's imagination, we get them to use that part of their brain which is so important when they're doing maths.

Multiplying fractions is easier than adding them

Why can you multiply two fractions together just by multiplying the tops together and by multiplying the bottoms together; but you cannot add fractions by adding the tops together and the bottoms together? The answer goes back to the meaning of multiply and divide. Whereas you cannot add different things together, you can multiply different things. You cannot add one ten and two units together, when you try you end up with 12 – that is one ten (a 1 in the tens column) and two units (a 2 in the units column) – that’s right a ten and a two… right back where you started. But you can multiply a ten by a two. Two times ten is two lots of ten, which we write as a 2 in the tens column. Similarly, when we divide, we can divide one ten by two units. So if we are multiplying a third by a half, we can either understand it as a half split into 3 (which is a sixth) or we can understand it as a third split in half, (which is a sixth). Either way the resulting quantity can be found by multiplying the denominators of the fraction together. This leads to the paradox that multiplying fractions is easier than adding fractions… when most children think that multiplication is hard and adding is easier. But it’s all easy when you learn with Puppet Maths, because Puppet Maths makes arithmetic accessible to children. Puppet Maths explains all the basic ideas which allows children to understand exactly what they’re doing rather than become lost trying to solve problems in a language (the language of maths) they don’t understand.

The development of abstract thinking

Abstraction is difficult for many pupils to comprehend. This is a function of brain development. But children are adaptable, they will develop to do the things that they are expected to do. In societies where children are encouraged to stand up in a boat and spear fish, they will learn the balance necessary to stand up in a boat to spear fish, at an early age. In societies where they are expected to learn to count in bases 12 and 20, they will develop the ability to count in bases 12 and 20 earlier than will children whose only need is to count in base 10. Hence the sooner that children are exposed to abstraction, the sooner they will achieve the ability to deal with it. But one cannot just throw them in at the deep end and expert them to swim, they have to be led to abstract thought. This is what Puppet Maths does. It provides mechanical analogies that will help the child visualise what the maths is attempting to achieve, these help the child succeed in getting the sums right, this gives the child confidence in their ability, and as they practice what they’ve learnt they relinquish the concrete visualisation as they learn to take short cuts – short cuts which rely on their ability to think abstractly.

Representating the denominator as a thing

The number at the bottom of the fraction is the “sort of thing” that the fraction represents. If that number is a 2, then the “thing” is a semicircle; if that number is a 4, then the “thing” is a quadrant; if that number is a 8, then the “thing” is a pizza slice; if the number is a 10, then the “thing” is a slimmer pizza slice.
Teaching children to imagine fraction in terms of these shapes relieves them from the abstract nature of numbers. When adding fractions this approach allows the pupils to see why it is allowable to have some numbers in the denominator of the answer, but not others. Puppet Maths presents children with these pictures so that they can imagine what their calculations are asking of them.

The number system

Most children do not understand the number system we use. It is never explained to them. This is not surprising, because children are expected to start using numbers at a very young age when a description of how the number system works would be incomprehensibly difficult for them. However, the consequences for these pupils of not understanding the system can be quite substantial. They lose confidence in their ability to do maths, they start to think that it’s a hard subject, so they find it hard to do. In extreme cases they give up trying. This is exacerbated by competition in the classroom. If a pupil cannot do something that others around them can, then they lose face from their lack of ability. The result of this is that they switch off from maths, they dismiss it as being of no importance and inconsequential, purely as a mechanism to protect their self esteem.
My eldest daughter could add 3 and 3 to get 6, but couldn’t add 300 and 300 to get 600, simply because she didn’t understand how the number system works. If you don’t understand the number system then you will struggle to manipulate numbers. Puppet Maths explains the number system to children in an accessable manner, allowing them to avoid this pitfall in the study of aritmetic.

What the = in an equation means

An equation is any mathematical expression which contains an equals sign. The equals sign simply means that the expression to the left of the sign equals, that is the same as, the expression on the right hand side of the sign. Equations can be imagined as a balance. The value of the expression in the pan on one side of the balance is the same as the value of the expression in the pan on the other side of the balance. This has an important implication, to wit if a change is made to the value of the expression in the left hand pan of the balance, and if a change of similar value is made to the expression in the right hand pan, then the two pans will continue to balance each other. This way of thinking about the nature of an equation was common 35 years ago, but is surprisingly often absent in the minds of children today. Today children tend to think of the equals as a magical operator that provides an answer to a calculation, which is accomplished by pressing the “=” button on their calculator. This way of thinking robs them of an important insight into the nature of the calculation they are performing. Puppet maths presents the balance model of the equation in an entertaining way, so that pupils have this tool to use in their maths.

Arithmetic incompetence is widespread

Innumeracy is prevalent in our society. A particularly disturbing episode was earlier this year when Ed Balls, at the time Secretary of State for Children Schools and Families, taunted his opposite number Andrew Gove across the floor of the House of Commons by holding up the following calculation as being a hard sum: “What is two and three quarters minus one and two fifths?”. It’s soluble in one’s head in less than a second if one has been taught how to handle numbers. The secret of maths is to manipulate the numbers you have to put them into a form which makes the problem you have easily soluble. Two and three quarters is 2.75, one and two fifths is 1.4; 2.75 minus 1.4 is 1.35 QED. The scandal was compounded by the fact that Andrew Gove did not shout out the answer “1.35” before Mr. Balls smugly instructed him that the answer was “one and seven twentieths”, thereby demonstrating that neither side of the chamber of parliament is accomplished in arithmetic. What was even more disturbing was that before becoming responsible for the nation’s schooling Ed Balls was Chief Secretary to the Treasury, a job for which one would have thought that numeracy was a precondition. Perhaps the subsequent trajectory of the country’s economy can be explained by this event. What a shame that Puppet Maths wasn't available to enable these members of Parliament, at an early age, to become competent in arithmetic.

Division presents a problem because it is viewed simply as a mathematical operation that is carried out by pressing buttons on a calculator, what it means in the real world is not being addressed by the pupil.
When given a question what is 60 divided by 0.05 one pupil of mine could not understand why the answer the got on his calculator (1200) was so much larger than the original number “60”.

He thought of a fraction as something that was smaller than 1. This pupil did not realise that the answer to a division is "the number of times that the bottom number will fit into the top number". Thus it was incomprehensible to him that if the bottom number is less than one, then it will fit into the top number more than once. His difficulty with maths was not related to manipulating numbers, it was rooted in his concept of what a fraction is, what the fraction means.

The top of a fraction means “how many”, the bottom of the fraction means “what sort of thing”. You cannot add two different things together. For example, what have you got if you add two oranges to three oranges? The answer is 5 oranges. You can add the numbers together because you’re adding the same sort of things. However, what do you have if you add two apples to three oranges? You have two apples and three oranges. You cannot add them together, because they’re different things.
Once pupils discover that the number at the bottom of a fraction represents what sort of thing the fraction is then they can understand why they cannot add a half to one fifth easily, why they have to convert both into something other, where the bottom of the fractions has the same number.

Even the very best pupils can be handicapped by calculators

Imaginative thinking is exactly what is tested in Oxford University’s maths entrance examinations. Many sixth form pupils cannot perform well in these examinations because they have not developed their mathematical intelligence. The reason for this is simply because they have relied on the use of their calculator throughout their secondary maths education. Therefore, they have not developed strategies for manipulating numbers, for rearranging them to make the calculation tractable, instead hey have relied on their calculating machine to crunch the numbers for them. Today these exams are testing for something that was unremarkable 40 years ago, before the ubiquity of the handheld calculator. University mathematics is about imagination, and imagination is what puppet maths helps develop. Rather than pupils simply learning routines for manipulating numbers by rote, puppet maths puts these algorithms in an imaginative context.
But this is not just for the pupil who wishes to go onto advanced study, every pupil can benefit from the use of their imagination when doing sums. It’s what makes an otherwise dry subject interesting, indeed fun.

Calculators for evaluating fractions

The use of the calculator stands in the way of understanding in maths. In the days before calculators became common the divide symbol was practically unused, everyone wrote a division as a fraction. This is vital if pupils are to understand what the calculation is doing. Use of the calculator hides the method and process of division from the pupil, denying them understanding of what they’re doing. In my teaching I had a number of able pupils give me an answer of “4.99” to a particular question I had set. The right answer was “5”. This led to much complaint when I marked them wrong. Had the pupils written the problem as a fraction, they would have been able to cancel out numbers top and bottom. This would have left them with the correct answer without having to do any calculation at all. They could have solved the problem with a few strokes of their pen, in less time than it took them to type the numbers into their calculator, and without rounding errors.

Fractions continued

Understanding of fractions depends upon the way the pupil thinks about them. If children write their fractions on two lines, then that is how they think of a fraction, as two numbers, and they begin to appreciate a relationship between them. If they are using a calculator to divide on fraction by another, then they do not think of fractions as being a complete representation of a quantity, a number comprising of a top part (numerator) and a bottom part (denominator), they think of a fraction as being an uncompleted operation which can be accomplished on a calculator, whose answer is a single number. This diminishes their appreciation of mathematics and their ability to use mathematics.

More on fractions

The next problem with fractions is that pupils are taught the words “numerator” and “denominator” for the top and the bottom of the fraction, which they learn dutifully, but which mean nothing to them. They do not realise what the numbers signify. They do not realise that the word “numerator” means “how many”; they don’t realise that “denominator” means “what sort of thing” (as in denomination relating to currency or religion). This lack of understanding hampers the pupil as the rules they are expected to learn for manipulating the fraction are both opaque and arbitrary. Once they understand the meaning of these two numbers then the reason why they have to jump through the hoops they do to manipulate the fraction becomes apparent, and much more understandable.

Writing fractions

The line in a fraction should not be sloped.

Often when learning maths pupils have difficulty with fractions. This starts with the way they write the fraction on the page. Copying their parents or other adults, they will draw the line between the numerator and the denominator as a slanting slash

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rather than as a flat, horizontal line and when asked to write it with the horizontal line, they are reluctant to do so. The slash looks so much more stylish, suave and sophisticated. They cannot immediately see why they should use the horizontal version.

But it is very important. It is absolutely vital to be able to see at a glance which number is at the top and which is at the bottom.

Multiplication Songs

Have you noticed how, after watching a Disney film, young children are able to pick up and remember the words to the songs that are in those films? They appear to do so effortlessly. Now contrast that with learning the times tables. How boring was that, coupled with the fear of getting a fact wrong. The solution would seem to be to put the times tables to music.
Many attempts have been made at this. Most of the results have been abysmally bad. Yes the times tables are set to music, but the music doesn't pass the "Old Grey Whistle Test". (Composers in New York in the 1920s, where it was hot in the summer and who worked with the windows open, could tell if they had a hit, if the tramps on the street who overheard them composing whistled their tunes). If the music is tuneless and unmemorable, then it adds nothing to the memorability of the words, the times tables.
Putting the times tables to music is a challenging task, because the lyricist is constrained in the words he can use, seven sevens is forty nine whether or not the music has room for 8 syllables or not. The lyricist can change "Paris in May" for "Paris in April" because the tune requires the extra syllable, but such a change with the times tables would make a nonsense of the entire effort.
We have put the times tables to music, music that is memorable and tuneful, and we haven't found it done better elsewhere. We'd love our arrangement to become as widely used as the tune that is used to teach infants the alphabet, or indeed "Happy Birthday". Your children could be among the first to learn their times tables this way if you study maths with our puppets at "Maths Puppets".

Algorithms

When learning maths, children have to remember the routines (mathematical algorithms) that they are taught. This is a feat of memory for many. It is much easier for them if they have stories to prompt them. Instead of remembering an abstract set of rules, they can remember a story played out by puppets, a much more attractive option. This demonstrates the added value achieved by teaching maths using puppets.