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.

Understanding the problem

The first task to address when faced with a problem is to understand that problem. What exactly is being asked? In everyday life, all too often the wrong problem is addressed because people do not take this step (politicians are particularly bad offenders here). But not just in everyday life. All too often pupils will not read a question properly and will rush off and do pages of maths and not produce the answer that was required of them, simply because they have not taken the time to understand the question (with school children this is forgivable, they are learning still, whereas with politicians there can be no such mercy).
Puppet Maths teaches that understanding the problem is the first step in the process of mathematics.

How does one go about understanding the problem? It contains 3 elements: where are we now? where are we trying to get to? and how might we get there? For pupils of maths the second is usually set out pretty explicity in the questions they are set, this leaves just the first and the third to be addressed. Very often the first is straightforward, but not always. Sometimes the scenario set out in the question is complex and requires a diagram to be drawn to make it more comprehensable. Puppet Maths teaches pupils to draw diagrams, after all why hold all that information in your head when you can put it down on paper. But what if the problem isn't easily drawn? Puppet Maths teaches its pupils to think analogously, to use their imagination so that they can represent complex scenarios simply and understanably.
The third question is the one that most find the most difficult. How might we get from the initial problem to the final solution? Notice the wording here... it's "how might" not "how do". Often pupils think that there is one correct route to get a solution, and if they do not know it at the start of the question, then they are lost and there is no opportunity of solving the problem. It is a tragedy that so many pupils think like this, and it is a consequence of the way in which they have been taught maths in school. Maths is about the application of logic and reason, a maths problem is an adventure in which the pupils explore various paths through it in an attempt to come out the other side. Sometimes they will make a wrong turning and have to stop and go back. Sometimes they will go off in completely the wrong direction and have to start again. Sometimes they will recognise a path, something they have seen before, and consequently know the way out of the maze. If more pupils thought of maths in these terms then more pupils would find maths easy, more pupils would find maths fun. At Puppet Maths, we inculcate this view of maths, because we want to make maths easy and make maths fun.

Mathematics should not be divorced from life

Maths exists to solve problems in everyday life. Unfortunately, in schools maths has been taught not as a tool for use outside the classroom, but as a mystery existing in an ivory tower. Is it any wonder that so many pupils give up trying? They cannot see the relevance of the material they're being taught.

When schools were first set up they had a very pragmatic approach to education. Often they were set up by churches or other religious bodies with the purpose of promoting a particular religious philosophy. In the UK, in the 19th century, rival school systems were set up by the Church of England, the Roman Catholic church and the Methodists. To be able to learn and understand the religious teachings, the pupils had to learn to read, so literacy was taught. Since scholars would not be respected and the religion would lose kudos if the scholars could not match the numerical abilities of ordinary people, arithmetic was taught; and since there were no printing machines handwriting was taught. It was all very practical. As the industrial revolution created ever more sophisticated products, ever higher skills were required of the workforce, this brought the state into the education business. But still education was pragmatically focused, it was about allowing people to read work instructions, about enabling them to do the calculations needed by their employers, but then educational theorists got involved. "What is the point of education?" they asked. They came to the conclusion that it was to produce "a well rounded individual". What on Earth does that mean? In the words of my P5 teacher, Miss Naylor, "Don't beg the question, boy!" There is no kudos in doing mundane things, so teachers shyed away from the practical aspects of their subjects, and over time each subject became more and more rarified and less aligned to the needs of the real world. A comparison of my great uncle's French primer and that I suffered under demonstrates the progression. His text book is full of phrases such as "in reply to your communication of the 16th inst." and "delivery of your goods is scheduled for"... that is, solid commercial language that would be of use to an employer and would help the pupil make a living. My french text book actually attempted to teach me the french for the verb "to consecrate", a verb that I have never used in English let alone in French. The same process has happened in Maths.

Whereas, once upon a time, maths was taught with the specific aim of enabling people to perform better at work, it is now taught as an abstract subject, and consequently many pupils are alienated from it. At Puppet Maths we link the maths we teach to the real world, and make it relevant to the lives of our pupils.






In many ways schools have lost t to which pupils are introduced but of whom only a few can become masters of

Logic in puzzle solving

Maths is a method for solving problems and puzzles using logic. When I was at school we were encouraged to reason our way through a maths problem, and taught to write out the problem again and again as we reasoned our way through them. I would like to show an example of a maths problem from Chemistry. It deals with mols. ( The weight in grams of Avogadro's number of atoms of a chemical). This is considered to be so difficult that it has been removed from the double award science specification (syllabus). But approached in the right way it is simplicity itself. All that the student must learn is that a chemical formula is not a shorthand for the name of the chemical but represents a SPECIFIC QUANTITY of that chemical. Then using the chemical equation for a reaction, the pupil can determine what weights of the products are created by what weights of the reagents, a feat that is accomplished simply by looking up the weights of each element involved on the Periodic Table and adding them together. Thereafter it is just verbal reasoning along the lines of:
100g of Calcium Carbonate produces 44g of Carbon dioxide
therefore: 1 g of Calcium carbonate produces 44/100 g of Carbon dioxide
therefore: x g of Calcium carbonate produces 44x/100g of Carbon dioxide.

Once when I was tutoring a pupil in chemistry the pupil claimed inability to do these calculations. But she was not writing any English along with her numbers. This told me that she wasn't reasoning logically, because doing so involves the use of language. Once I prevailed on her to write out the small amount of English that is contained in the sentences above, she immediately found that she could solve this type of problem. At Puppet Maths we teach verbal reasoning as part of maths. We believe that maths is there to solve real life problems not as some mystery that pupils must be subjected to.

How to approach a problem.

Life is full of difficulties. That is its nature, even without governments producing their best efforts! How to overocme these difficulties? To do so we must plot our way through to a solution, which involves the steps of understanding the problem, understanding the solution we wish to attain, and the use of logic and reasoning to get us from the start to the finish. Maths should be teaching us these steps. Maths is not just about performing sterile manipulation of numbers, it is about solving problems. It should teach us a robust methodology for approaching problems. Unfortunately, all to often in schools it fails to do that. Judging by the sales of puzzle books, people like to solve problems, but would the same people who buy these books say that they like maths? In many cases the answer would be a resounding "No". This has much to do with the way maths is taught at school. On the other hand Puppet Maths puts the fun back into maths. It makes maths immaginative so that the exercises become maths puzzles rather than maths problems.

Maths is a problem solving tool

What is maths for? It's for solving problems. This is not always apparent to a school child. They are taught how to perform a particular mathematical routine, and then they are given an exercise to do which practices that routine, but appears to have no relevance to everyday life. For example, what is the point of being able to simplify 3f + 6g + 18h ? Having to do these exercises makes the subject of maths into a mystery to which the pupil just cannot relate. The result of this is that the pupils begin to resent having to study maths, and view it simply as an imposition that they have to tolerate. This alienation causes the pupils to switch off. They'll do what they're asked just to stay out of trouble, but they do not engage with the subject, and consequently, they do not learn it. Myself, as a schoolboy, I would solve practical problems using reasoning and logic rather than convert them into mathematical notation (which would have been a much more efficient way of solving the problem). Why? Because at that time I thought of mathematics in terms of it being an ivory tower subject not for application in the real world. This is a barrier which Puppet Maths addresses. Puppet Maths puts mathematics into a real life context.

Real world problems

Real world problems are not the same as mathematical exercises. There is usually a trick involved. An example is the following puzzle:
"A cat is at the bottom of a well 30 metres deep. Every hour it climbs up 3 metres, but then it slides back down 2 metres. How long will it take for the cat to climb out of the well?"
The person schooled in mathematical routines will notice that for every 3 metres the cat climbs it slides back 2 metres, so it is climbing 1 metre per hour. It has 30 metres to climb so it will get out after 30 hours. Unfortunately, this is not the correct answer. The cat gets out in 28 hours. This is because it does not slide back as it climbs, it only slides back after it has traversed 3 metres. This means that it gets to the top and out of the well after 28 hours, and does not slide back that last time because it is no longer in the well. This seems to be a sneaky trick, but it actually an important learning event. In the real world boundary conditions are important, they have to be considered. In engineering they are often the main focus of attention. Unless children as subjected to this type of problem rather than idealised questions, they will not learn to how to apply mathematics to the real world. At Puppet Maths we engage pupils with these problems which engage imagination with mathematical

Teaching something is the best way to learn

Teaching something is the best way to learn that thing. A person requires to have understanding to be able to tell someone else how to do something. This is especially true when the person learning the new skill might ask questions. One of the learning strategies we encourage at Puppet Maths is to ask our pupils to explain what they have learnt to others. These might be their friends, or their parents. Parents are often indulgent and don 't press their children and test their understanding, this is a mistake. It should not be seen as a weakness or a failure if the child cannot answer such searching questions, but as an opportunity for the child to learn more, to address those points on which they are not confident. By this means, asking pertanent questions stretches the pupils but does not humiliate them. It makes them face those parts of the subject that they don't know as well as they might, and thereby overcome their weaknesses. It also teaches the value of standing back and getting an overview of what you're learing, rather than getting bogged down in the detail. So very often pupils make heavy weather of something that is simple because instead of looking at the principle of what they are doing they are too busy concentrating on the detailed mechanism of doing it. They fail to see the wood because of the trees. Making the pupils teach what they've learnt gets them to analyse the concepts that they're using and fixes those concepts in their minds.

How pupils perceive maths

At Puppet Maths we want our pupils to use their imagination when solving maths problems. This means representing the problem in a variety of ways that make it easier to solve. Research has shown that one of the major differences between pupils who achieve highly at maths and those that don't is in their "World View" of the subject. The highly achieving pupils see the subject as open ended, where any experience can be brought to bear to solve any given problem, the pupils who fail to achieve highly are those who see it as a closed set of options, this is the way teacher did it, I must remember that way and use it again whenever a similar problem occurs. Of course with this approach, if a new problem arises, the pupil is ill equipped to tackle it, as teacher hasn't shown them how to do that particular problem yet. Puppet Maths has set itself the task of expanding pupils' perception of maths, to use the understanding that pupils have from every facet of their lives to reason their way through, so that they succeed.

Number talk

A powerful way of getting pupils to think their way through maths is to set them mental arithmatic problems. The pupil then has to talk their way through the problem. Such a problem might be "What is 18 x 5?". Do that in your head! It's not hard and there are many ways to do it. What it does require is the ability to change the problem into a series of easier ones and combining them to arrive at the answer. This concept of changing a problem into a series of easier ones is central to maths, and it is important for pupils to become familiar with it. In this case, perhaps the easiest approach is to divide the 18 by 2 to get 9; and multiply the 5 by 2 to get 10; then multiply 9 and 10 to arrive at the answer of 90. But it would be equally valid, since 18 is 20 minus 2, to multiply 20 by 5 to get 100; then to multiply 2 by 5 to get 10; then to take the 10 from 100 to arrive at 90. Alternatively, one might do the straight forward formal way of multiplying 5 and 8 to get 40, and multiplying 5 and 10 to get 50, and adding these two numbers together to arrive at 90. The point of exercises like these is to get the pupil to think of novel ways of achieving the desired outcome, so that they learn to apply logic and reason to the maths and to become clever and inventive in how they approach maths problems. At Puppet Maths we promote this way of thinking about maths.

Reasoning in maths.

Pupils who think that maths is about remembering set routines, then identifying each problem for the application of a set routine, then applying it are doing a harder sort of maths than the pupil who has learnt to apply reason and logic to solving the problem. This is most clearly shown by the calculation (which is done in Chemistry) of moles. So difficult are these calculations perceived to be that they have been removed from the Science dual award GCSE syllabus, and are only present in the Chemistry single award syllabus. But the calculations are EASY. Why are they perceived as being so difficult? Because they are taught badly. They are taught as a routine that the pupils have to learn, and then apply. If they were taught as logical reasoning then they suddenly become maths that 8 years olds would be familiar with. Maths is about logical reasoning. At Puppet Maths we teach logical reasoning. Why make things harder than they need be? Let's do things the easy way. Let's achieve.

Working together

There are lots of ways of working at maths. One is to do it on your own, in silence; another is to do it as a member of a group, with discussion; one is to do it against the clock; another is to solve open ended problems over an extended period of time; one is to listen to another tell how it should be done; another is to work out for oneself how it might be done. There is a huge variety of different ways in which maths can be learnt, and variety is the spice of life. So often in maths classes in school pupils are required to listen to the teacher and then repeat the procedure he has demonstrated in silence working alone. No wonder pupils become bored and dislike maths. At Puppet Maths, because we are remote from the place where our pupils work, we do not control the environment in which they study, or how they study, but we would ask parents and guardians of our pupils not to leave them to work alone. We would ask them to take the time to work alongside the pupil from time to time, to take interest in the pupil's work and to discuss maths with them. Of particular value would be getting the pupil to teach them how to do it. Nothing develops understanding better than having to teach it. When a pupil is asked to teach what they've just learnt, and fails to explain a point, then it is not a point of failure but an opportunity for a joint investigation to find out what the answer is. Both parent and pupil can look at the explanations given by Puppet Maths, look at school text books, and perhaps most importantly apply logic and reasoning to clarify what is happening. In this way, the pupil can patch up any gap in their understanding but importantly gets social interaction and recognition of their work they're doing as they do so.

Maths problems

Why do people do puzzles? They do them for fun. They like the challenge... but only if it is a challenge that they think that they can succeed in. So it is with Maths. Pupils will happily do maths questions if they are intriguing and the pupil can sense that they are soluable. When they succeed in solving such a question they gain a sense of achievement that bolsters their self esteem, their self confidence, and makes them enjoy the subject that has such a positive effect upon them. So many problems in maths are bland calculations that the pupil has to "do". There is a place for these, in so far as they allow the pupil to practice a technique or routine that they've learnt, but this should be the minority of the problems that the pupil undertakes. They should also be given complex many layered problems, that they can approach from different angles, which make use of their learning, their cunning, their insight and their imagination. These are the types of problems that, once solved, give the pupil the greatest satisfaction, these are the types of problems that can make maths fun. At Puppet Maths, we do both types of problems. Some will be to enable the pupil to practice their mathematical techniques, others will be of the more interesting type, but we demand of our pupils that in every case they use their imagination to solve the problems.

Maths is interesting

Many pupils think of Maths as being boring. They are taught routines that they have to remember and then recall, and use to solve abstract problems. If a maths pupil thinks that there is a lot to remember, then they are not learning maths properly. It is of paramount importance that the pupil realises that maths is all about reasoning and applying logic. There are a few rules that they have to remember, but these are indeed few and far between, everything else is reasoning and logic. In my many years of teaching, I have found that the lower achieving pupils have difficulty with applying reasoning and logic to the problems that they're given. They have got through maths by memorising routines and then using them, and as a result they are underpracticed at reasoning and logic. Once they start to approach their maths from this perspective they often blossom as if a revelation has opened up before them, and what was hard has suddenly become much easier.

holding post

holding post

Problems vs. Puzzles: Work vs. Play

So the best way to engage children with maths is to teach them the strategies for problem solving and then present them with interesting problems to solve. Mathematics is about solving problems not about performing routines. Indeed the pupil must use routine to gain the answers, but that is just arithmetic, or algebra, or geometry, mathematics involves solving problems. But at Puppet Maths we don’t like problems. Life is hard enough anyway without problems, we like puzzles. It is amazing how one’s approach changes one’s outlook. Problems are difficult, puzzles are fun. Both require thought and strategy to produce a solution, but whereas the former is work, the latter is play. We puppet mathematicians prefer to play… we’ll get to the same place at the end of our journey, but we’ll have a better time en route.

Reviewing one's work for mistakes

The process of making sure that one is on the right path to solving a problem is called reviewing. It is important to be humble and accept that one can make mistakes and that there is nothing wrong with making mistakes as long as one learns from them. In mathematics one can go back and correct one’s mistakes… it’s not like woodwork where a cut irrevocably changes the shape of the piece of wood and cannot be undone. In maths a simple line through the offending number or expression is sufficient. Of course sometimes, as in woodwork, it is better to start all over again, but it is not forced on the mathematician in the same way, and there isn’t the loss of resources that losing a piece of wood entails. At Puppet Maths we teach that it is OK to make mistakes, providing that you learn from them. We teach that mistakes are a natural part of doing maths, and that the pupils should check their work for mistakes as they go along. We teach that when a mistake is found, that the pupil shouldn’t lose heart and give up, because maths is a puzzle solving activity and very few people navigate their way through puzzles flawlessly every time.

Carrying out the plan of action

Having determined on a plan of action, the mathematician sets about solving the problem. At every step, the mathematician check the work to ensure that the processes that have been chosen are making the problem more tractable to solution. Sometimes the mathematician will have done something unnecessary, sometimes the mathematician will have gone down a blind alley and applied a process that is moving away from a solution, requiring a backtrack. Like a journey through jungle the mathematician has to feel a way through the problem. The willingness to accept that they might be in error and to be flexible is the identifying feature of the good mathematician. Pupils who fail to succeed at maths tend to decide on an approach to a problem and then to apply it ridgidly, and reject any suggestion that they might be barking up the wrong tree. They tend to take criticism of their approach to the problem as a personal insult. The important thing to recognise is that to be successful in maths problem solving needs to be an adventure, maths problems need to be puzzles that are decoded through cunning and guile. Once this is understood, then maths really can become fun. At Puppet Maths we want maths to be fun, so our aim is to inculcate this outlook upon maths within our pupils.