Plan of action

Once a mathematician has understood the problem, what's the next thing he has to do? The answer is develop a plan of action. This can overlap with the task of understanding the problem. It involves drawing, charting, graphing as the mathematician works out what processes need to be implemented to arrive at a solution. The mathematician may not be able to see the way through the problem at this point, and may decide to undertake the processes that have been identified so far, apply them to the problem and see what it looks like then in it's half solved form. Only then might he be able to map the way forward to a complete solution. One of the most powerful methods of understanding a problem and working out a plan for its solution is to attempt to solve a similar, but simpler problem. "How would I solve an easy problem of this type?" is the question that the mathematician asks. Having discerned a method, then the follow up will be "How can I apply this method to the real problem at hand?"

George Polya

In 1957 a Hungarian mathematician called George Polya wrote a book called "How to Solve it". In this book he delineated the strategies that mathematicians use. They don't just wade into the problem, they first spend time trying to understand what the problem is. What is involved in this problem? What are the questions to be answered? In trying to understand the problem they may well sketch it, draw a diagram, make a chart or draw a graph to help them, all the time considering what is a likely path to a solution. This is something that less proficient persons tend not to do. Such people, generally, take a quick look at the problem and classify it according to which routine they think will provide them with a solution, then they apply that mathematical method to the problem. Whereas this approach may work with simple problems given in a school classroom to illustrate a mathematical routine, it is not appropriate to solving more difficult problems with more than one layer of reasoning for which different techniques might be required.

Oxford scholars

How do you get into Oxford University? One way is to have very good grades, a better way is to be enthusiastic about your subject, so enthusiastic that you are teaching yourself it. A school friend of mine, Peter Kirby, was just such an enthusiast. In his case his passion was entomology. He had a comprehensive collection of moths (they all looked alike to me), and had made himself knowledgeable in the field. When he went for interview at Oxford, it was immediately obvious to the admissions tutor that he was a student who was capable of learning about arthropods simply because he already had taught himself. Peter was given an unconditional offer. All he had to do was get the 3 “A” levels which the university required of all students and he was in… no competition for grades at all. On the other hand I taught a student at Andover College who had an enormous need to achieve. He was incredibly hard working, but his work was worthy rather than enthusiastic. He was diligent with his mathematics, but he was working at it, he was working hard at it, it was not a passion. That spark of inventiveness, of insight, of exploration was missing. He was not Oxford maths material not least because he worked too hard at it. If he could have achieved the academic results he was achieving, but with less effort then he would have better fitted the profile of an Oxford mathematician. Why was he working so hard at maths, when there are others out there who do not need to do so? Because, he’d learned maths as a series of routines that he had to apply to solve the questions that were posed to him, and he approached the problems from this perspective. He didn’t approach them from the perspective of a fun adventure, in which he could apply his imagination. It is that sort of person that a premier university looks for when selecting its students. Puppet Maths intends to inculcate that adventurous approach to maths from the get go, so that all pupils no matter what their final level of achievement might be (GCSE, "A" level, matriculation) find that mathematics is a fun adventure, is easy, and isn't the toil that so many have experienced of maths in the classroom.

Enthusiasm for a subject

If a parent wants their child to be the next Carol Vorderman, or a similar whizz at maths, then they need to make their child enthusiastic for the subject. People’s minds differ, and their interests differ. Because of the way their minds work, some find one subject easy and another difficult, whereas a different person will have the opposite experience. Much of this is not inherent in the workings of the individual’s brain, but comes from the way they happen to interpret the subject that they are learning. My eldest daughter was daunted by the subject because she thought that to add, she had to remember all the combinations of numbers and what their sum was, and she we intelligent enough to realise that there were an infinite number of numbers and that this was impossible. So she gave up… after all, why try and attempt the impossible? (I have sympathy with this outlook, because, when at the age of 11, I started learning French, I gave up before I started due to what I perceived as the impossibly huge task of learning and remembering all the words of the vocabulary that I knew in English). I was able to help my daughter through this difficulty, but it still put her off maths, and she has underachieved in this field ever since as a consequence. Had she not be put off the subject, then she could have achieved well at it. From these experiencesI learnt that creating enthusiasm for a subject is the main attribute to success in it. When you’re enthusiastic and interested in something, doing that thing is not work.

Over testing in schools

There is a great concern regarding the hyper-accountability of the school system. Pupils are tested almost to destruction, and schools are held accountable for their results. The upshot of this is that teachers teach to the exam, rather than provide the pupils with an education. This failing permeates all school subjects and is just one reason why so many parents have chosen to teach their children at home. Home education / home schooling allow parents to educate their children rather than cram them for an exam. Cramming in schools has reduced mathematics from an imaginative subject, which teaches pupils how to solve problems, to one in which pupils just follow routines for no particular purpose. Success at maths has become the result on a good memory. Pupils who can remember the routines and how to use them are good at maths, those that cannot, aren’t. Many teachers recognise this problem, but they are part of a system which they are powerless to change, and upon which their livelihood depends, so they are reluctant to rock the boat, they simply go along in the direction which their line manager directs them, and they do their best. Puppet Maths sprang out of my disillusion with maths teaching in schools. I wanted to reintroduce imagination into the subject, and make it fun for the pupil. I want to make maths education something that children will enjoy and a subject that they will succeed at.

Home education, a sense of community.

We teach children in the classroom rather than give them books to read and leave them to get on with it. Why? Children are usually not motivated to spend a long time studying, especially when the learning gets challenging, when it is so much more congenial to give up and do something else, so social pressure to keep on task is one reason. Another reason is the provision of feedback. Pupil’s learning needs to be assessed and their strengths and weaknesses identified, so that they don’t remain in their comfort zone repeatedly doing things that they know they can do well and ignoring those things at which they need more practice. But also there is the element of human interaction. The human animal is designed to learn from other humans by social interaction. This is where the internet tends to fall down. There is plenty of information on the internet, some of it text, some audio, some video, but outside of chat rooms and social media sites, where individuals online communicate in real time, there is little social interaction. Puppet Maths intends to address this aspect of learning when it teaches maths. Children often have imaginary friends, or they often relate to a comic character or a fictional character from film or TV. Reality is not a barrier to them in this respect… but it is a character that they relate to. When we at Puppet Maths teach maths we put a lot of effort into characterisation. Each puppet has its own character, so that the child can imagine that they are a participant in the classroom along with the puppets, they can learn from the puppets and even feel superior that they could solve a problem that a puppet struggled with, giving them a sense of achievement. This is especially valuable to children that are being home educated / home schooled, who, however content they may be learning in the family environment, can still feel isolated.

Home Education

Parents turn to home education / home schooling and teach their children at home for a variety of reasons. These range from religious beliefs, to problems with their child’s health or problems of bullying within the school, to a lack of confidence in the ability of the school system to teach effectively. Parents who teach their children at home cannot be expert in all the subjects that are covered by the curriculum. The internet provides a lifeline allowing them to access the information and teaching materials that they need. Puppet Maths is a resource that home education / home schooling parents can access to help them teach maths to their children. As a maths teacher of many years standing, I have learnt a lot about teaching maths on the job. Time and again I have revised my method in the light of experience. Pupils experience problems and misunderstandings that, until I witnessed them, I could not have guessed at. It has been my job to help them overcome these problems, and in my teaching, I try to address these problems in passing to ensure that my pupils do not develop a mental block regarding some subject or other. Home education / home schooling parents do not generally have this experience, and they don’t have the time to learn from their mistakes in finding the best ways of explaining a topic… they want their children to succeed academically. Puppet Maths brings them the benefit of my experience in teaching maths and in avoiding many of the mental blocks that can be introduced in a child’s mind when they don’t understand some aspect of mathematics.