Maths and the Law

When I was at school the one thing that I really hated having to do were mathematical proofs. At that time I could not see the purpose of them of having to prove the rule, I could not see why I could not assume the rule was true and then show that it was self consistent. Happily since then I have learnt the reason for the need for the rigour of a formal proof. Formal proofs are needed because, unlike disciplines such as Physics and Chemistry whose rules are determined by Mother Nature, mathematics has been devised by man. When a new mathematical rule is developed it has to be shown to be consistent with all the other mathematical rules already in existence. This is the reason why maths involves proofs. A mathematical proof demonstrates that in all cases the new rule will conform with the rest of mathematics. This is what differentiates maths from that other man made construct – the law. Maths is logical, self consistent and produces comparable results, whereas the law is arbitrary, inconsistent and produces many paradoxes which have to adjudicated by human intervention in the person of a judge. This is why Maths is a science, whereas practicing the Law is an art. At Puppet Maths we don’t just teach the rules of maths, we explain the logic of why the rules have been formulated in the way they have. This way pupils get to learn not just what the constituent parts of the science of maths are, but to understand the reasons why the rules of maths have developed in the way they have.

The purpose of maths.

In the US they call it “Math” in England it’s “Maths”… the Americans only have one, the Brits use the word in the plural. In German it is “Mathe” again the plural, so is this a case of the Europeans recognising the plurality of the subject, the fact that it contains many disciplines or is it simply that the Americans are better at abbreviating words? Nevertheless, it raises the question “What is Mathematics”? What is it for? When I was at school, I believed that it was about numerical manipulation. As a result I thought of geometry, for example, as a peripheral activity rather than a central plank of the subject, more a historical overhang that was being studied for completeness rather than for active use. But I was missing the point. What maths is really about is solving problems. Every day people are faced with problems, and the framework provided by maths can help them solve these problems. Mathematics provides proven ways to think about problems, and informs people of the approaches to take to arrive at a solution, approaches that have been shown to be robust, that work each time. Often there are a number of ways to approach the problem, one way might be numerical, another might be graphical, alternatively the solution might be arrived at via an abstract diagram. All these are component parts of the subject that we call mathematics. The more branches of mathematics that a person has knowledge of, the more options that person has to apply to find a solution to their problems. At Puppet Maths we teach the context in which mathematical techniques can be used, to demonstrate to the pupils the real world use that the maths they are learning can be put to and examples of the problems that it can help solve. This shows the pupils that they are studying maths for their own best interests, and not simply because adults like to make their lives difficult.

Visualising maths

Solving maths problems starts with understanding what the problem is. To do this one of the most powerful techniques is to draw the problem. A drawing enables you to visualise the maths problem. In drawing, one is forced to make the abstract maths concepts and numbers concrete... it is impossible to draw an abstract concept, the very act of drawing causes it to take on form (of one sort or another). Puppet Maths is a visual presentation of maths, consequently, by the very nature of the medium we work in, we have to give maths form. The form we give it is that which I was taught as a child (there were always good maths teachers out there). We use dice to get children thinking of numbers as a series of dots. We organise these dots into piles, which, when they reach a height of 10, magically change into a ten. What does that mean? Well to explain we shift our description to one involving coins... each dot becomes a penny piece and ten of these in a pile is equivalent to a 10 penny coin... and in turn ten of these 10p pieces (florins) can turn into a £1 coin. The purpose is to get the pupil thinking about real objects rather than struggle with strange shapes like "3" and "5". [Incidently these two shapes look alike to younger pupils, an appreciation of left and right doesn't necessarily develop until a child is 8 or 9 years old - my own daughter, when she started at school, would write a line of text from left to right, and then write the next line from right to left in mirror lettering - and she would see nothing unusual in it]. When we teach fractions we get the pupils to imagine them as slices of a pizza, or as a position along a line, so that they have a means for visualising what the numbers represent. The Singapore maths course takes a similar approach, making maths visual, so that the pupil can understand the problems they are given. We at Puppet Maths are proud to be working under the same principles as the Singapore maths course.

There are many ways to solve a maths problem. As long as the person solving the problem sticks to the rules of mathematics, then the problem can usually be solved in a variety of ways. There is no single path to the correct solution. This is one of the attributes of maths that makes it EASY. This is also central to the philosophy of Singapore Maths.
By way of example, if we take the multiplication 7 x 9, one way of arriving at the correct answer is to learn the times tables. Learning these is boring, that is why at Puppet Maths we use the Sands-Daniels Musical Times Tables. These uniquely, in my experience, set the times tables to well known tunes that you can hum, and they are without the vast quantities of extraneous verbiage that have nothing to do with multiplying numbers that so many musical times tables are afflicted with. The music triggers the memory of the lyrics and so removes the fear of getting the words wrong thereby making the times tables fun to learn, and fun to recite. We advise that having learnt the songs that the pupil recite them in their head when they require to recall the product of two digits.
However, if one hasn’t been lucky enough to have learnt your times tables using the Sands-Daniels songs, one can still arrive at the correct answer to the calculation by other means. Another way would be to put 7 dots on the page nine times over and then count them up. Alternatively one might write down 7, add 7 to it to get two lots of 7, then add 7 again, and again, and again, until one had added 7 nine times. Both these method would produce the answer.
Another method would be to notice that 9 is almost 10. One could multiply 7 by ten to get 70, and then reason that since we did not want ten lots of 7 only nine lots, so we could then subtract one seven to get our answer.
But there are more approaches… one could look for a pattern in the 9 times table. Whenever 9 is the multiplicand, the product of 9 and some multiplier is such that its 10s digit is one less than the multiplier, and the tens digit and the units digit of the product add up to nine… so in the case of 7 x 9, seven is the multiplier, so the tens digit of the product will be one less, i.e. 6, and the units digit of the product will be whatever added to 6 makes nine, i.e. 3. The product is 63.
Alternatively, one might look for a pattern in the 7 times table. The seven times table follows the pattern on a mobile phone number pad. For this you ignore the 0 button, and use just the other nine buttons. Starting with 7 at the bottom left hand corner, that is one seven. To find two sevens move up the key board. The rule is every time you move up the keypad then you add 10 and the units is given by the number on the key pad. So for two sevens you move one up the keypad, which gives you a ten for moving up the keypad and a four for the units, as that is the number on the keypad button – result 14. For three 7s you move up the keypad again, that adds a ten for moving up, we now have two tens, and the units are given by the number on the button which is now 1 – three sevens are 21. For four sevens we go back down to the bottom key of the middle row of the keypad (that is the button marked 8). Because we have not gone upwards with this move we don’t add another 10, so our tens digit is still 2, but now our units digit, given by the button, is an 8, - four sevens are 28! Five sevens… we move up the keypad so we add a ten giving us 3 tens now and the number on the keypad button is a 5, five sevens are 35, and so on. Doing this we discover that nine sevens are 63.
At Puppet Maths we teach that as long as the pupil sticks to the small number of rules of maths (there aren’t many… and unlike language where there are irregular verbs, there are no irregularities in maths, it always obeys the rules) then they should find the answer by hook or by crook.

Teaching one thing the better to teach another

Significant figures is a confusing concept for most pupils. This is simply because they have been taught to ignore all the leading and trailing zeros in the numbers they write. Because these are not even seen, pupils are not aware of their presence, and ironically, when they are asked to ignore them, they become confused. The solution is to talk about the leading and trailing zeros early on. If one refers the number 100 as "0000100… but we ignore the leading zeros" and talk that talk, then quite quickly pupils will become fed up with you wasting their time by always talking about the leading zeros which we are destined to forget about. Similarly if one refers to 0.21 as "0.210000… but we are going to ignore all the trailing zeros", again pupils will quickly become familiar with the concept of trailing zeros which we ignore. Then, when we start speaking of significant figures, the pupils will be familiar with the concept of insignificant figures (they’ll have had them up to their eyebrows) and they won’t struggle.

Again I am writing about the abstract nature of mathematics. Why would anyone think in the abstract when they can think in terms of things. It's why we draw diagrammes before we set about solving a problem. It's why scientists create analogies when dealing with conceptual ideas... what is an electron? No one has ever seen one, that's for sure. Sometimes it behaves like a little ball, on other occasions it behaves like a wave. Depending on which is the most appropriate way of thinking for the problem in question, the physicist chooses either one description or the other. In maths we should adopt the same strategy. Think of a mechanical analogy. If we're adding up numbers: think of piles of coins, if we're solving equations: think of a balance, if we're performing algebra: think of boxes containing unknown numbers which we move about. Maths is hard when we think of it in an abstract way. So don't think of it that way. At Puppet Maths we teach our pupils to imagine the situations where the maths might be applied. This not only gives relevance to the mathematical routines, but actually makes the maths easier. At Puppet Maths we are dedicated to making maths both easy and fun.

Mechanical analogies allow pupils to imagine whats going on.

The sine function

A pupil was having difficulty with the concept of a sine. What is a sine? Why should the ratio of the length of the line in a triangle opposite an angle, to the length of the line adjacent to the angle matter? Why would anyone bother with it? What was the point? Because of the use of the electronic calculator, to this pupil a sine was just a magical number that appeared when he pressed a button… he was perplexed, where did the calculator get it from? When I learnt about the sine function at school, I was given a book of mathematical tables. When I looked for the sine of an angle I could see what the numbers for other angles were. I got a feeling for the relative values of the function for various sizes of angle. I learnt by observation that the sine function varied from –1 to 1. I realised that calculating the sine function itself was difficult to achieve… that’s why I was using a table containing pre-calculated values to look up the value I needed. But none of these observations are available to the modern pupil, just the magic calculator button.
I explained the sine function as the distance that a shaft connected to rotating wheel moves vertically and that the cosine is the distance that the shaft moves horizontally. As soon as I’d done that, the pupil saw the reason for the function, that it was about knowing where the piston on an engine was relative to the position of the crankshaft, or how the various levers on a loom move as the driving wheel move around. At Puppet Maths we relate maths functions to the real world, so that they become relevant to the pupils.