Understanding and problem solving

In today's world where pupils face a crowded curriculum and paper qualifications are in such demand, where schools are judged by government on the basis of the grades their pupils achieve, schools have become crammers rather than educators. Do it this way, follow the routines... these are the questions you are likely to be asked, these are the ways to solve those questions. You don't have time to think about the questions when you're in an exam and working against the clock, learn the questions, learn the ways to solve the questions, get a good grade.

In the days before electronic calculators were common, calculations had to be done mentally. To do these quickly and effectively it was necessary to think a problem solving manner. The result was that pupils learned the techniques of problem solving. Since the introduction of calculators both teachers and pupils have relied this machine as a crutch and have abandoned practicing problem solving as a routine activity. Instead they just press buttons and an answer magically appears. Maths has stopped being about problem solving, and become about routine following and button pushing.
Mathematics should be about imagination, about tackling problems, being inventive. These are the attributes that Puppet Maths put back into mathematics teaching.

No one way to skin a cat

If mathematics is all about problem solving, then most problems can be approached from a variety of different directions. There is no one single way of solving most problems, usually there are a number of valid paths for arriving at a solution. This is not how maths is taught in school. The teacher, faced with a class of 30 pupils and a time constraint cannot normally afford to let the pupils develop different ways of solving problems, the teacher simply does not have the time to devote to understanding how each pupil approached the problem and to judge if it is a valid one based on sound logic, or if it is flawed, working out in what way it is so. As a result, the teacher enforces a standardised way of tackling problems, thereby crushing the imagination that is the essence of mathematics.
Sir Barnes Wallis, designer of the Wellington bomber and inventor of the bouncing bomb which was used on the Dambusters raid in 1943, used to praise his mathematics education as the foundation of his whole career. This was not just because it gave him the tools to work at the forefront of engineerinng, but because it taught him how to think about problems, how to approach them and how to solve them. He was taught maths by means of a heuristic method. For example, the class he was in was taught how to use a pair of compasses and then armed with that instrument and a ruler instructed to go away and find out everything they could about circles. Two weeks later the pupils came back with a number that was just over 3, which was a constant relating to all circles. Unfortunately, schools do not have the luxury of the time that is needed to undertake heuristic education, today's crowded curriculum does not allow for pupils to learn problem solving through exploration.
Puppet Maths attempts to reverse this trend by teaching maths through logical reasoning and imaginative visualisation. It presents many different ways of approaching problems to let the pupil know that there is no single correct way of tackling a problem, that they are not on a tightrope from which a false step will cause disaster, but that there are many appropriate ways to solve maths problems.

A picture paints a thousand words

Mathematics is about applying logical reasoning to solve problems. But logical reasoning can be hard work. No one likes hard work. That's why tasks that have been thought out already are avoided and the known solutions applied. However, just memorising and applying these known solutions is not interesting or stimulating, nor does it teach the child to tackle new types of problem. Puppet Maths encourages pupils to use logical reasoning when approaching a maths problem. So how do we avoid the hard work? We do it by teaching pupils to visualise the problem. Rearranging counters into patterns and recognising these patterns is so much easier than counting; creating a graphical representation is so much easier than reading a table of figures (which leads to the question of why company accounts are presented as a table of figures rather than in some diagramatic format, are they being made deliberately difficult to interpret?).
When pupils learn to visualise the problems and the routines that they are using to solve their maths problems, they are learning to apply logic. This stands them in good stead for when a problem is presented to them that is in a different format from those which they are used to. It is this ability to apply logic that represents the real skill of mathematics. It is this skill that is so often not taught, or taught badly in schools.

Maths is about logical reasoning

Maths is all about logical reasoning. If we have already solved a problem, then we already know the answer and we don't have to solve it again. If we have already worked out what to do to solve a problem, we don't have to work out what to do a subsequent time, we just do the same things that we did last time. This is the basis of the routines that are taught in schools as mathematics. But they are not mathematics. Learning and using these routines may be variously described as "arithmatic", "algebra", or "geometry", but they are not mathematics, they were once, when they were being thought out and developed, but they ceased to be mathematics once their development was completed and they became established routines. Mathematics should refer to the application of logical reasoning to a problem.
So maths education should be about learning how to interpret the facts presented within a problem, and how to apply sound reasoning to manipulate those facts. Calculation or manipulation of data using established routines is not mathematics. That it has become to be to termed as such is simply terminological inflation (in the same way that over the last 30 years school "pupils" have become school "students").

When I sat my A level maths, nearly 40 years ago, I was advised that, were I running out of time, I should do the logical reasoning part of the question, convert the problem into mathematical form, and calculate it until I arrived at a numerical expression, whereupon it would be acceptable to write "hereafter arithmatic", and leave it to go onto the next question. This advise underlined the distinction made then between maths and calculation, a distinction that has since become blurred.

Puppet Maths encourages its pupils to use logic to solve problems. We teach the routines that are the known solutions to problems, and in doing so we show the logical reasoning behind the routines, and encourage the pupils to use that logical reasoning when using the routines, rather than just remember the routines and and regurgitate them (which is what so many traditionally taught pupils do).

Maths is about problem solving

Maths teaching in many schools relies on pupils learning the routines for completing calculations. While there is nothing wrong with this, it represents a narrow approach to learning maths. Maths is about solving problems based on reasoning and logic. The routines are simply known methods for taking particular types of information and producing from them a logical result, that is, they are short cuts to save the mathematician from having to do all the hard work of reasoning. However, for the children who are learning these routines, it hardly seems that they are there to make life easier, from the child's point of view the routines themselves are hard work. Knowinng no diffferently, the child believes that the routines comprise maths. Thus maths becomes divorced from both logical reasoning and the real world, it becomes an abstract discipline that the child is tortured with for no apparent good reason, and they learn to hate the subject. At Puppet Maths we intend to reverse this process, our purpose is to get children reasoning, using logic and using maths to solve problems. Yes, they have to learn the routines as well, but we explain them using reason and logic and methods of visualisation that are possible in our magical puppet world, but which are difficult to explain on a board by a teacher in a classroom, or which would appear abstract if explained without the characterisation provided by the puppets.

Characters in Maths Puppet World - Monsters

The only really horrid character inhabiting the Maths Puppets’ world is the Maths Monster. He is a really scary creature. He appears whenever the pupils have got a problem that they do not know how to tackle. He terrorises them and makes them fret. However, once the pupils find out how to do the questions that conjure him up, he shrinks in size and becomes insignificant. He represents the fear that we all have of not doing well and also the mountain that a subject like maths appears to be when we start out, but turns out to be trivially tiny once we have mastered it all, and we ask ourselves what all the fuss was about. This personification of the fear engendered by maths and the way it is belittled and conquered help instill in children the idea that maths is not insurmountable, that it can be mastered and that the difficulties that they might encounnter along the way do reduce in size and eventually go away.

Characters in Maths Puppet World - Authority figures

The authority figures in the Maths Puppets world are not allowed to make mistakes. After all, we wouldn’t want the children who are watching and learning maths from the puppets to learn something that was wrong. However, sometimes it is useful to show how not to do something. Since neither the Teacher nor the Wizard are able to do this, Maths Puppets has enlisted the capabilitites of Mike, who is an orangutan. He doesn’t do maths properly, he makes mistakes at every turn and acts as a foil for the authority figures, who can then show the right way to do the calculations. Naturally, Mike is never punished or mocked for getting his maths wrong (who could ever be unnpleasant to such an attractive character as Mike?) so children can see that it’s OK to get things wrong... as long as you put them right when you learn how.

Characters in Maths Puppet World - Monsters

The Maths Puppets world is also inhabited by a couple of friendly monsters called Dani and Flic (after two of my colleagues from when I taught at Andover College). These two characters are essentially a comedy duo whose antics are used to illustrate mathematical points and to provide a narrative for the mathematical operations that are being undertaken. They play the part that used to be played by pupils themselves in maths lessons before political correctness made the interactions between teachers and pupils as formal and po faced as they have become today. By being memorable in their behaviour these monsters provide a handle for the pupils to recall the meaning of mathematical routines and the best method for them to select in any particular set of circumstances.

Characters in Math's Puppets world - Pupils

There are a number of pupils among the puppets that inhabit the Maths Puppets classroom. They have differing abilities, so that there is always one puppet that a child can identify with as having the same level of understanding of the topic in question as the child has. Such identification between child and puppet is important for learning, because it leads the child to have an open mind. As the puppet learns the techniques necessary to master a topic, so the child learns alongside, letting the puppet make the mistakes which many children find so discouraging. Children can learn from the puppet’s mistakes without having to make them themselves. They also learn that there is no shame in making mistakes, that it is part of the process of learning, and thereby lose the self-conciousness that can deter them from admitting weakness or lack of understanding and thus which holds them back.

Eradication of boredom

Puppetry can make what otherwise would be a boring experience interesting for children. Things that are unmemorable become memorable. Children love stories, they understand the world through stories, yet traditionally maths is not taught using stories. Yet how does one utilise the skills of the storyteller when teaching maths? Is not maths a subject that is so remote from the realm of storytelling that the task is impossible? We, at Puppet Maths, do not think so. We have created a world where we can make maths magical, where numbers and the things that they represent come alive. To this end amongst the characters in our imaginary land, we have created a Mathematical Wizard. This character uses magic to show how the number system works, how calculations can be accomplished through visualising the problem. This wise character is called upon whenever a description is complex and words don’t suffice to explain the concept or the mechanism by which maths works. He causes numbers to move about, to turn into coins, to resemble the dots on a dice - he explains from a number of different perspectives so that children can follow the logic of the subject no matter what their existing background and outlook may be.

Visualising maths problems

Visualising mathematical problems is half the battle in getting a solution. In the days before the invention of the pocket calculator, bank employees in Singapore and Hong Kong used the abacus to perform the calculations they needed to do. After a few months calculating on an abacus, the bank employees did not need to use the apparatus any more. They could solve their problems in thier heads, because they could visualise in their minds’ eye how they would be moving the beads on an abacus were they actually using one. The employees did continue to use their abacuses, because to do so was less effort than doing it all in their heads, but the point is that because they could visualise the problem and the method of its solution, the didn’t need to, and they could, if they chose, solve the problem as fast in their heads. Maths Puppets aims to give children a way to visualise maths problems and the method of their solution, thereby giving them the basis for generating the correct answers, and knowing intuitively when they are getting it right and when something is not quite as it should be and therefore requiring a second look.

Using multiple strategies

Children, like the rest of us, like to put things into categories. They like to think of addition as being addition, subtraction as subtraction and so on. So when they are asked to add 9 to 12, for example, they immediately think of counting up. Counting down does not occur to most children in this circumstance. But as an adult, I know that the easiest way to add nine is to add ten and then take one away. Most children will eventually stumble across this tactic, but before they do, many will give up. They will see that they have been asked to add nine, consider this to be a large number, and think that doing so is hard, and so not try. It is tragic that these children will start the process of falling behind in maths because they are not aware that the addition of nine just involves an addition and a subtraction of one. Puppet Maths aims to show children that maths is EASY, so we teach children the tricks that enable them to succeed, and so build their confidence that they can indeed do maths.

The use of puppets makes maths safe.

Children relate with puppets. They know that, as toys, they are harmless, that they are not going to get angry with them, and are not going to attack them physically or emotionally, or hold a grudge against them, or tease them if they get something wrong. Toys are for fun, and they're not going to do anything nasty to the child. This removes all the danger from trying to do maths. In the world of Puppet Maths, when a puppet gives a wrong answer there are no adverse consequences. So the child learns that in this world, it's all right to fail. If it's alright to fail, then it is worth the risk to try. So many children do poorly at maths simply because they are afraid to try. They start to shy away from maths at an early age, and they never engage with it subsequently. As time goes by, they realise that their ability has fallen behind what might be expected of them at their age, and they become embarrassed by their lack of ability, this puts them off the subject even more. However, they are prepared to admit their lack of ability to puppets, because the puppets do not exist in the real world, so puppets are safe, and once children have entered the imagingary world of Puppet Maths their inhibitions dissolve, their minds open up and they find that can "do" maths after all.

Multiplication

Multiplication is just accelerated addition. 3 x 5 means "three lots of five". The number contained in three lots of five can be obtained by counting up all the items present in those 3 lots. Whereas that may be an option for small numbers like those in the example, it ceases to be a viable approach when one is trying to identify the number contained in 257 lots of 983. A shortcut is needed. To achieve this shortcut there is one absolutely essential element. The pupil has to learn their times tables. I once taught maths in a technical college, and one of the young adults was a South African man, whose maths education was practically non-existant. When asked to do the relatively straightforward multiplication 254 x 26 he could lay out the numbers in the proper order to do the calculation, and he would know that he now had to multiply the "4" and the "6". But when asked what 4 X 6 was, he would stop and say "Ah". He was stymied simply because he didn't know the fact that 4 x 6 is 24. Learning these facts are the key to doing mathematics, but having to do so puts children off mathematics. They think that there are too many of them to learn. It's boring having to chant them in class, and all the time they do so, they're concerned (of not in terror) of making a mistake; and although the teacher is unlikely to hear it, the pupil standing next to them will and may well use this knowledge in a disparaging way. I gave my adult student the Sands-Daniels Musical Times Tables to play on his stereo, with the instruction to sing along. He learnt the 45 facts contained in the times tables quickly and painlessly. This 20 year old man had put off learning these 45 facts because he retained the childish view that there were so many of them that doing so was an impossibility. Once he actually attempted to learn these facts, and did so in an environment that made doing so less boring, and which helped him get the answers right by providing him with musical prompts, he found that learning the times tables was easily accomplished. Puppet Maths uses the Sands-Daniels Musical Times Tables to teach these vital facts to our pupils. They can learn them without boredom and often seemingly without effort.

Adding long columns of numbers

Children are generally asked to add two numbers together. They come to view this as being the "usual" form of addition. However, in real life, it is sometimes more common to be required to add long columns of numbers. Many children see this as being daunting or as an imposition. They think that adding two numbers together is sufficiently challenging, and that adding a column of five or six numbers is an impossible mountain for them to scale. There is only one way for them to discover that they can do such a calculation, and that is for them to try. If children can be introduced to doing these without being given time to think about it, they will take it for granted; but given time to think about how hard it will be, they will "psyche" themselves out of being able to complete this type of sum. It is not the children's ability that is the limiting factor in so many cases, but the children's belief in their own ability. Puppet maths attempts to bolster children's belief in their own ability by making maths fun, helping them to get their answers right and developing a virtuous circle in which the child does well at maths leading to better self esteem, leading to better results.

The algorithm

Al-Khwārizmī is the name of a Persian mathematician who lived around the start of the 9th century, and who wrote a book on Mathematics in Arabic. Some 300 years later, (such was the speed at which information was disseminated before the advent of the internet) it was translated in to Latin. The translator, in rendering the author's name from Arabic script into Latin characters corrupted it to "Algoritmi". Because this name was foreign to the readers of the translated book, they did not realise that it was a name, and took it to mean "methods of calculation". Thus the world obtained a new word, "algorithm".
The whole of maths is riddled with algorithms. Indeed, that's how we do maths, we learn the method, we practice the method, and we implement it when we need it. We become part of a machine just performing a routine to solve our maths problem. Just performing a routine can be boring, but it is less boring if we realise that we are part of a machine, and we can imagine all the cogs, wheels and belts, or reels of computer tape and flashing lights that are with us and all around us as we do our bit as part of the machine. Get a lift, imagine yourself as the Central Processing Unit of a vast computer while you do the sums. Suddenly, you're important, rather than a kid who is struggling all alone to process some pointless numbers, who might get snapped at if they make a mistake.
Let's make our children heroic and important when they solve maths problems, let's build their confidence and self-esteem, and make them proud of the fact that they can do maths. This is the aim of Puppet Maths.

Adding numbers in columns.

Young children like to do the things that they're expected to do. So when they're expected to do calculations involving the addition of the single digits, such as 5 + 4 = that is what they do. However, they get imprinted with the idea that sums have this form. It is a stress for them when they are asked to add with the numbers presented to them vertically. As this is the way in which numbers are added in the real world, it would make sense to present them to children in this form from the start. Since our number system operates on position values based on the column in which the digit is placed why would we present children with numbers arranged horizontally. Nevertheless, we do, and children will come across additions arranged in this manner, so it is important that they be introduced to this arrangement of numeerals, but it really should be a cursory introduction, and children moved onto numbers arranged in columns as quickly as possible. Puppet Maths will introduce your child to adding numbers presented horizontally, but will quickly move them on to adding in columns before they become imprinted with numbers presented horizontally as being the "right" way for an addition.

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.