Maths is a science

Maths is a science. What is a science? How's it differ from an art? The best way to compare the two is to consider cookery. Watching Gordon Ramsay at work in his kitchen, it is obviously a place where artists are at work (and in Mr. Ramsay's case a tempramental artist). The results are variable, and if they are not good enough, they are binned and the creative act is performed again. Each time there is the chance that the product will not meet Mr. Ramsay's high standards. It is a matter of the skill of the individual chef whether or not these standards are met.
A science, on the other hand would be the cooking occuring at a cake factory. The owners have measured everything. They mix the ingredients in known quantities, they stir for a known time, with a known force; a known amount of the mixture is poured into a container, which is placed in an oven at a pre-determined temperature, which is closely monitored, for a particular period of time, and the temperature profile is controlled over that time. The factory owners know that if they repeat everything in exactly the same way, every time, they will arrive at the same results every time. There is no swearing of the shop floor (and very possibly swearing there would be a disciplinary offence), the pressure to perform is off, the system does the work.
Mathematics is a system that is used for problem solving. If one is faced with a problem, one can either follow the Gordon Ramsay model, behave like an artist and rely on one's own skill and flair to work out the logic of the solution; alternatively, one can use the science of mathematics, which has already been used to solve many types of problem successfully. All one has to do is recognise the type of problem one is facing and apply the appropriate proven strategy. The choice is yours.

Different exams different approaches

I have recently been looking at the International "O" level exam for chemistry. Yes, maths is involved in chemistry too. This is the exam that children are expected to take at the age of 16, the same age as children take the GCSE exam in the UK. Of particular interest to me were the calculations regarding the number of moles. In the UK's exam the pupils are given a chemical equation. They are told the weights of reagents (or concentrations and volumes of any solutions used) and asked to find the quantity of product. All very straightforward.
For the International "O" level, which is taken in places like Singapore and Hong Kong, the pupil has to know the chemical formulae for the reagents, be able to put them into a chemical equation and balance that equation, do the moles calculation (which is identical to the GCSE calculation), and then draw a conclusion from the numerical result of that calculation. Basically this is 4 separate GCSE questions all concatenated. These 4 questions are not signposted for the pupil as separate parts of the question, the pupils have to know the route through the problem themselves.
The skills needed to do the "O" level question are all maths skills. Maths reasoning is needed to determine the chemical formulae of compounds. Maths skills are needed to balance the chemical equation, and maths skills are needed to do a moles calculation and draw a quantative conclusion. Are we saying that in the west are children are unable to do the "O" level type question because they lack mathematical ability, and have to be fobbed off with the GCSE question?
I would suggest that the "O" level question more closely represents the skills that an employer would want from a laboratory technician, than does the GCSE, which at best demonstrates a partial ability.
Globalisation means that the western nations are competing head on with the likes of Hong Kong and Singapore. But how will our children be able to compete if they are not equipped at school with the knowledge the need? Are we letting our children down?

Maths using images of real things

Young children do not think in an abstract manner. They like to think of things. This is called concrete thinking. When given a maths problem, for example 2 + 3, they don’t think of an abstract 2 and an abstract 3, they want to think of 2 things and add to them 3 things. This is why they count on their fingers. In primary schools the teachers tell the pupils not to count on their fingers. This is an attempt to force them to abandon concrete thinking and to embrace abstract thought. I believe that this is wrong. Children will start to think in an abstract manner as and when they are ready to do so. If they need their fingers to count on I think that they should use them. Certainly, teachers should hold up abstract thought as a target for the child to aim for, but don’t ban them from using concrete examples. Don’t ban children from counting on their fingers. After all teachers don’t ban children from moving their lips when they read, they let them do so, confident that as the child becomes more proficient at reading, they will cease; similarly, teachers should allow children to count on their fingers until such time that they are ready to count without the bother of using them. At Puppet Maths we encourage children to visualise numbers as sets of dots. They can then count the dots in their mind’s eye, then subsequently, by recognising the patterns formed by the dots, learn to do arithmetic without having to count out each time.

Teaching by stimulating the child's mind

When I was at school, there were still 12 pennies in a shilling (and 20 shillings in a £) so from the age of seven I was expected to count in bases 12 and 20 in addition to base 10. How did I do it? I imagined a big pile of pennies, which, when they reached the height of 12 got turned into a shiny silver shilling (not that I often saw a shilling, they didn’t circulate, they were kept aside for use in electricity meters). As I did my calculations I imagined pennies being put on or taken off the pile.

Any child who didn’t use their imagination in this way that must have been an enormous disadvantage. They must have faced enormous difficulties counting in 3 bases (10, 12 and 20). Later, when I was 8 we were expected to count in bases 14 and 16 (there being 16 ounces in a pound, and 14 pounds in a stone). Then at age 9 we were asked to count in bases 22, and 8, there being 22 yards in a chain and 8 furlongs in a mile (The chain was actually a decimal unit, as there were 100 links in a chain, and 10 chains in a furlong). We were expected to count in all these different bases before we reached the age of ten years! And we did. Why? Because we rose to meet what was expected of us. This is why it is so important not to dumb down maths.
At Puppet Maths we believe in academic excellence. We will not dumb anything down. We believe in making academic ideas and concepts accessible to the young mind, and we have found the use of puppets to be the ideal vehicle to achieve this.
Puppets stimulate children’s imaginations, and once they are imagining then they can see piles of pennies, or lines of numbers, or patterns in their mind’s eye.

Children love animation

Children have vivid imaginations. Maths is not often taught in a manner that allows the child to use their imagination. Maths is considered to be far too serious a subject to let children harness their imagination. What sort of answers would a child come up with if it used its imagination when doing maths?
Q. “What’s 1 + 1?”
A. “Pink bunnies”.
But is this the case? Children want to please the adults in their lives. They know when it is appropriate to be serious and when to be frivolous. They might use frivolity as an excuse to avoid work, but they know exactly what they’re doing, when they do it. The aim of Puppet Maths is to harness children’s imagination to make maths fun. And Maths should be fun, without having to dumb it down.
Children love animation, it simply catches their imagination. The use of puppets immediately propels a child to start using the imaginative part of their mind. When they’re doing this, the child’s mind is open for learning. This is why Puppet Maths is so effective at teaching maths, it engages children’s open minds.

Many ways to solve a maths problem

It is a humbling experience for any maths teacher when a pupil comes up with a better way to solve the problem than that used by the teacher himself. Naturally, the teacher teaches the standard method, and when faced with a maths problem, as a matter of routine, launches into a solution based on applying that standard method. This saves the teacher from having to think about the problem. Then along comes a pupil who has thought about the problem, and using imagination reframes it in terms that are significantly simpler. This allows the pupil to shortcut the route to the answer. This ability to think about a problem and reframe it with simplicity is the mark of a good mathematician. Unfortunately, this is not a skill that is taught in most school maths classes. At Puppet Maths, we think about how maths problems should be approached, and we encourage our pupils to think about how they might be solved. We present our pupils with a number of different ways to view the same data, and a number of different ways to manipulate that data to arrive at an answer. By this means we show our pupils that there is more than one way of considering a maths problem and that there is more than one method reaching a satisfactory answer. We want our pupils to be aware of this diversity, so that they can use their imaginations to arrive at the best path to a solution. We, at Puppet Maths, want to lead children away from the idea that the way that teacher does it is the only good way of doing it, and that it must only be done by that method. Whether at home or at school, we want pupils to develop insight and, yes, cunning into their relationship with numbers.

Don't judge students

An unhappy thing happened today, the connector to my external hard drive got torn from the printed circuit board that it’s mounted on. It needs 9 solder joints to reattach it, but its a surface mount component, so it’ll be hard to get at with a soldering iron. If I had a reflow oven then soldering it would be straightforward, except the disk drive would have to be removed from the circuit board, and no doubt that would require unsoldering. Not so simple. At least maths is easier.
Maths can be solved in your head, no fiddling around with hot tools. The nastiest material you come across with maths is ink (that is unless you think that paper is more offensive). It’s one of the reasons that I gave up semiconductor engineering to teach maths, there’s no hydrofluoric acid or arsene involved in doing sums. When viewed from this perspective, maths really is easy.
If maths is such a good option, why do so many children hate it? My theory is that it’s because, the subject of maths forms a pyramid. Every new thing that a pupil learns depends upon them knowing the preceding stuff. If there is something that they “don’t get”, then from there on, they are at sea. Young children can be very shy of pointing out that they don’t understand something, they’ll pretend that they do, and go to great lengths to hide the fact that they don’t. In this way they act against their own best interests, but, of course, they don’t understand it that way, because they don’t see the bigger picture. This has a lot to do with their status within their peer group, and not being seen as being weaker than the others around them, weaker than those who they are competing with. This is where Puppet Maths helps. Children know that they are superior to puppets, and so are more inclined to admit their inability to do something, because it carries no social stigma. They know they are not going to be picked on by the puppets; they know they are not going to be judged by the puppets.