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.