Neatness and accuracy II

When I was at College, there was an American electrical engineering student with whom I was acquainted. She had spacial alignment problems. Whenever she wrote anything she wrote along an arc. She could write along a line for about the first half of the page, but then the text curved and ended up running parallel to the side of the page by the time she got to that side of the page.
Whereas it was possible to read what she'd written, and wasn't difficult to understand if it contained English words, when she wrote maths it was a nightmare to decipher. This is because, in Maths, the position of symbols is important. When adding, subtracting and multiplication, all the units are in the units column, all the tens are in the tens column and so on, in fractions some numbers are at the top and others are at the bottom.
Neatness is important in Maths to the extent that it is possible to know what column the digit is in and where it is located above or below other numbers, and of course to the extent that the numbers are legible. However, beyond that neatness is not a prime consideration.
Think of any film in which a "professor" is depicted. On the blackboard behind them there is almost always some calculation or other, complete with arrows linking one idea to another, and numbers highlighted by cirles. But nevertheless, the numbers are legible and in a clear relationship one to the other. It is a mix of neatness and exuberance, this is what mathematics is all about, a mixture of discipline and immagination.

Neatness and accuracy

I was teaching a class of children fractions. They had got the idea and were doing well with the questions I had set them. There was one girl who had got the first 4 questions correct, but who had made a mistake in the fifth one, before going on to get the sixth correct...
I pointed out her error to her, and suggested that she cross out the wrong calculation and do it again. She was reluctant to do so. The correction would only have required her to cross out a couple of numbers and replace them with others, but she declined to do so. Her problem was that she thought that that would make her work look untidy. She would rather have her sum wrong and her page neat than have the correct answers and minor crossing out on her page.
This is a problem, especially with girls, they have been praised for the neatness of their work early in their school career, and they elevate this criterion above that of getting the correct answer.
Although neatness in layout is a requirement of mathematics, it should not be put above getting the correct answers. A compromise is required.

The Mighty Maths Machine

Introducing to you the "Mighty Maths Machine". Children prefer to think in concrete terms rather than abstractly. Children do not need to know how a mathematical algorithm works to use it to get the right answer (although it is preferable if they do know). Based on these observations, Puppet Maths has created the "Mighty Maths Machine". This is the physical embodiment of a algorithm. It consists of a chamber into which the pupils put the input numbers, a handle which they (metaphorically) turn when they perform the operations involved in the algorithm, and a tray into which the answer is delivered.
This mechanical analogy helps children to understand the purpose of the routines they perform when they're doing arithmetic, and gives them confidence that by just following the rules they've been taught, they will get the correct answer.

Why Maths is easy

Maths is easy. Why is maths easy? Because you don’t have to understand it to do it. All you have to do is learn the routines and apply them. Of course it is better if one understands why one has to do the calculation in a particular way, but is not necessary. When I was at school, subtraction of a larger number from a smaller number was accomplished by “borrowing” a ten, which then later had to be paid back. As a 9 year old I heard another pupil in the class ask the teacher where the ten was being borrowed from. I remember thinking “That’s a good question, I don’t know the answer to that” and listening in carefully. The sums we were doing must have be money calculations, because the teacher replied crossly, “I don’t know… from the bank.” I remember thinking of the teacher “She hasn’t answered the question”, but I also remember my next thought, “It doesn’t matter. As long as I continue to follow the routine I will continue to get the sums right, and that’s all that matters.” So I never questioned where the ten came from again until I started teaching maths. It was then that I realised that my teacher hadn’t evaded the question in quite the way I’d thought. In the method of subtraction I was taught the ten is indeed borrowed from outside the sum. It’s an extra ten that is introduced and corrupts the numbers in the sum until such time as it’s paid back. I didn’t understand at the time, but then I didn’t need to.

Abstract thinking

Another big problem children face is that of abstract thinking. I once taught in a primary school on Humberside, where I was asked to tutor the problem child of a class. This little boy desperately wanted to learn and get approval, but wasn’t particularly able. As a consequence, he continually disrupted the class by demanding attention from the teacher, who with 25 other pupils to teach, couldn’t spend time with him.
This boy’s main problem with maths was his stage of mental development. He had not developed abstract thinking. This was evidenced by his inability to multiply 50 by 3 when asked to by his teacher. His attitude was principally “How can anyone do that?” Who learns their 3 times table as far as 50 times? Who learns their 50 times table?
I then asked him to draw a shape. What shape is a 50. He didn’t immediately understand me, so I drew an equilaterally curved heptagon (the shape of a 50p piece).
I then asked him what he would have if he had 3 of these. Without hesitation he replied that he’d have £1.50. I then explained that he’d answered the question. 3 lots of 50 was 150, just like he’d said it was.
Once the boy had converted the problem from an abstract one involving numbers, into a concrete one involving things (in this case three 50p pieces) he had no difficulty with the arithmetic.

Numbers are not unique

One of the big problems that many children face is that they think of all numbers as being unique. This was characterised, in the case of my eldest daughter, by her being able to add 3 and 3 to get 6 with ease, but she couldn’t add 300 to 300 because they were “big numbers”. Children like her do not understand the number system, usually because it has never been explained to them. They do not realise that instead of there being an infinity of numbers, there are really only 10 numbers (0 to 9) and therefore, they will never have to add numbers which have a sum greater than 18.
Because you cannot add different things together, you have to add tens separately from units, and hundreds separately from tens, and so on. This means that all the numbers that you have to deal with are small ones

You can't add different things together

You cannot add different things together. This is true in life. Whereas 2 oranges plus 3 oranges is the same as 5 oranges, 2 oranges plus 3 apples remains 2 oranges and 3 apples. The former is the addition of the same things, and you can add them together, whereas the latter is a collection of different things, you can't add them.

 The same is true of mathematics. Adding a "one" and a "ten". You cannot say "we've got two "one" digits here, we'll add them together and get a 2 digit, because these "ones" are different things. The one in the "units" column signifies 1 unit, the one in the "tens" column signifies 1 ten. They are different things so you can't add them. The answer to the sum "ten plus one" is 11, a one in the tens column and a one in the units column, one ten and one unit, which is identical to the problem we started with. No addition has been done. Why? Because you cannot add different things together.

The English language obscures this fact, for example the number "twenty two" appears to be a unique number rather than a combination of two numbers... this is where the Germans have an advantage, in German it is made plain that 22 is a combination of two separate numbers, zwei und zwanzig, or two AND twenty is obviously a combination of two things and not a unique individual number.

There are only ten numbers, these are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 this means that one doesn't have to add any sum whose result is greater than 18. How hard can that be? The times tables multipling these numbers together results in 100 facts that one has to remember... but 10 of these facts [0 x anything] gives the answer of 0, and half of the remaining 90 facts are the same as the other half [2 x 3 = 3 x 2]. So now we're down to 45 facts that need memorising. Within these 45 facts there are patterns, so they are not random. There is the rule for the 9 times table wherein the digits of any multiple add up to 9. Multiples of any even number is an even number. Multiples within the 5 times table end in either a 0 or a 5. So these 45 facts are not just random. Put them to music, so that the tune prompts recall and learning one's times tables becomes easy.

Dyslexic arithmatic

The problem that some children have with Maths are not obvious. If we take, for example, a dyslexic child. These children sometimes say that they cannot add up a column of numbers because the numbers move about. Those of us who haven't experienced this phenomenon find it hard to understand quite what is happening within their brain, but it appears that their perception of the position of any given digit changes with time. So they perceive a number that is in the units column, to be temporarily in the tens column and vice versa. One solution to this is to use highlighter pens to highlight all the numbers in the units column yellow, all the numbers in the tens column red, all the numbers in the hundreds column blue... then the pupil does not have to add numbers based on their position, but on the basis of their background colours.

Well, after a week off doing other things we're back in harness again, back to the writing and editing of scripts. We're still working on persuading children to think of arithmatic operators as things that manupulate equations rather than buttons on a calculator which magically produce an answer.
When I was teaching Physics, I compiled a mechanics question whose answer had a numerical value of "5". All the pupils produced an answer of "4.99". For the weaker pupils I marked this as correct, but for the most able pupils I marked it wrong. The most able pupils, naturally, complained and were not greatly mollified by the explanation that they got it wrong, when others with the same answer got it right, because I expected better from them. Actually, I expected better from all the pupils, but when one has a class of 22, which one sees infrequently, and a tight schedule of work to be completed, one has to be realistic about what one can achieve. My job then was to teach them Physics, not to remedy the shortcomings in their Mathematics. I tried for those whom I thought I could remedy in passing, and accepting the status quo for the others, but it has always rankled with me that this class did not manipulate mathematical expressions properly.
When I moved on to teaching Maths, I made sure that my pupils learnt the algorithms and techniques to solve the problems without continual recourse to their calculators. This is what I intend Puppet Maths to achieve as well. Maths isn't difficult, all it requires is that pupils learn a few routines, and then apply them to the problems they have. The puppets are there to make learning the routines fun, and more memorable.

Writing scripts

Taken a day off from the scripts today. Today has been spent working on the puppets. We've got some new outfits for some of them and we have been customising some of the others. A particular problem we have had is in working the mouths of a couple of them. This proved hard because the puppeteer's fingers were slipping and couldn't gain traction to manipulate the mouths properly. We solved this by equiping the puppeteer with those rubber thimbles which bank tellers use when counting banknotes, only instead of using only one, the puppeteer had them on all of the fingers and also on their thumb.

We have been considering some interesting questions today... such as "How many numbers are there?" and "What does the symbol '=' actually mean?" These are the questions that pupils learning maths ask. They are surprisingly difficult to answer, simply because we adults are so used to using numbers that we take these things for granted. It's as if we have been asked to define the meaning of the word "the". We know what it means, and how to use it, but can we explain it?
Well Puppet Maths sets out to explain the answers to maths questions such as these. Why? Because children get hung up on these types of issues; it stops them from progressing in the subject; and it undermines their confidence with using numbers. So at Puppet Maths, we're going to explain away these awkward questions in a manner that is understandable and interesting to the child. This way we can build confidence, and remove the fear that so many children develop when they're faced with arithmatic.

After a break for the weekend, we're back at work again today editing the scripts for our new videos of puppet maths. We're still busy working our way through the Key Stage 3 (KS3) Maths specification (what we used to call a syllabus). We're drawing on the team's combined experience of teaching maths to pupils of all abilities. One of our central motivations is to introduce immagination to maths. For many pupils maths consists of nothing more than a series of numbers that have to be manipulated by a series of arbitrary and incomprehensible rules. Our aim is to go beyond this by putting meaning to these rules, and doing so in an entertaining manner. Our aim is not only to show that maths is easy, but also to try and make it fun, by appealing to children's immagination.

Well, we've looked at Equations, and rearranging equations, Fractions, and Algebra... and we're still pushing forward through the KS3 specification polishing our scripts. This is a really time consuming activity, but if the scripts aren't right, then the whole learning experience won't be right, so it's right that we take our time. So back to business...

More script reviews today. Myself and my team are slowly working our way through the KS3 specification discussing the best ways to present the Maths methodology using puppets to make for an entertaining and adsorbing learning experience.

Script review

I've spent the last day in reviewing the scripts. This is necessary as we transition from face to face use of puppets in teaching maths to the use of video on line. As we will lack the ability to read each individual situation as it occurs and adjust our teaching to accomodate any failure to grasp a point by the pupils, we must make sure that the video content contains all the explanations and caters for all the forseeable misunderstandings that pupils might exhibit.
Happilly, my extensive experience as a Maths teacher allows me to predict what the most likely misconceptions will be, but it is always necessary to be careful as children are adept at inventing new ones, if they're given half a chance.

Well a lot has been done since the last blog. We've acquired some new puppets that we're going to use teaching maths. We hope that these will allow us to make even more entertaining videos that will grip the immagination of young people and allow us to teach them maths without them realising that they're doing "hard" stuff, so as a consequence they will find maths EASY.