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# Zen and the Art of Mathematics

Posted: 21/06/2012 08:46

I am a child of the 70s and suck at mathematics. I married a maths teacher to get out of doing basic maths at home. Nevertheless, what we share in common is problem-solving. This morning the 'Torygraph' is banging on about standards of mathematics being worse than in the 1970s, and Gove leaks news that he wants to return secondary education to the dark days of O Levels.

Why the dark days? Because, my wife rants this morning, in the seventies kids were drilled with basic arithmetic, which is not mathematics at all. The Kings College study reported in the 'Torygraph' says "15% of modern pupils failed to achieve the most basic standards - showing they can successfully solve problems involving doubling, trebling and halving - compared with just seven per cent in the mid-70s." Multiplying and dividing are basic arithmetical manipulations, not problem solving!!

Yesterday, my wife reports, she had a class that were investigating the internal angles of polygons. Theese are a class that are 'bored with maths' and don't engage, but they got extremely excited when they figured out, for themselves, that the sum of the internal angles of a given shape increase by 180 each time a side is added to the shape. So, a square has 360 degrees of internal angles and a pentagon (five sides) has 540 degrees. But, they surmised, does a hexagon (six sides) have 600 or so degrees? One could make this reasonable assumption, but no- they worked out for themselves that the increase is by 180. And they went on to work out how to mathematically prove this- which is a technical proof, which is beyond my mathematical ability. This wasn't some applied maths problem where Johhny is travelling on a bus with three and six in his pocket: you know what I mean if you grew up in the seventies. No, this was pure, abstract maths, done for the sake of delighting in numbers.

What does Gove want us back to? Back to using our heads for simple arithmetic, the sorts of manipulations that young people do today on the calculators on their mobile phones. They don't need to hold that basic task in their heads, which frees them up to focus on the goal of mathematics which is the appreciation of the beauty of patterns of number in the world and in abstract shapes. He wants young people today to get bogged down in minutiae of detail and take their eye off the goal. He wants worker drones with no sense of their own capabilities and intelligence. He doesn't want people to solve problems, he wants clerks to write down sums in a book, perhaps to help balance the national deficit.

What mathematics should be aiming at is problem solving. My technical area in university is problem analysis and decision-making, and yes, I have used number in helping people make decisions. Only the other day, I was constructing a turnover/investment scenario building tool for a small social enterprise to help them decide by how much they should grow, and how much investment and cashflow money they need. I didn't do the basic sums - I didn't need to - I have a speadsheet for that. But I do have to know how to design the spreadsheet, and understand how to goal seek and scenario build. These are skills my step-father, who was frighteningly quick at mental arithmetic, doesn't have.

Maths in school today is, and should remain, about patterns, problems and decisions. Let young people use the technology to hand to advance them to more important skills, just as slide-rules were eventually allowed to simplify look-up tables. When they practice these higher skills, the basic skills follow. The true function of arithmetic can be found when trying to solve a more complex problem. You don't need to sit there completing sheets of sums to practice mental division, when you don't know how to analyse the problem.

Maths, like topics in universities today, has been broken down into ever small Lego brick type units of knowledge, fragmenting our synthetic understanding of a problem. We are told we have to build up towards a problem with these little lego bricks, with no sense of what the purpose of the brick are. we need to reverse this, and communicate the purpose better, and allow students to work back to find the 'lego bricks' of knowledge or arithemetical manipulation that they need.

Mathematics needs to have a sense of Timothy Gallwey's Inner Game. Maths teachers need to communicate the 'bounce and hit' of maths, the sounds of geometry on the racquet. Teacher of maths shouldn't be teaching pupils with ever smaller and detailed instructions, but to communicate the purpose of maths. They need to model that love of the beauty of number, and show the pupils where they can find the tools to further deepen their understanding of problems. There speaketh the guy who took three attempts to pass both O Level and GCSE maths.

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02:18 PM on 06/21/2012
Arithmetic is, indeed, only a small subset of Mathematics - but you *should* leave school being able to do the latter. Indeed, you shouldn't have any problems with basic arithmetic operations, should be comfortable with addition, subtraction, multiplication and division, and be able to understand fractions and bases before the time you're 13, and arguably much earlier - so that you can then go on and study the more 'Mathematical' concepts.
07:21 PM on 06/21/2012
Half of folk on the planet are not so clever but they need to be able to count.
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Christos Palmer
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12:28 PM on 06/21/2012
Tim, I also find it difficult performing simple mathematical calculations, and I don't think school or the qualifications I took, GCSE's had anything to do with it. Some people are just not able to grasp numbers.

I'm one of them.

Problem solving isn't a, excuse the pun, a problem for me, I'm actually quite good at solving computer glitches. Its all down to the way our brains are wired, and I think my circuit board was wired the wrong way round.
07:34 PM on 06/21/2012
There are systems of teaching arithmetic by beginning with tallying. Then to computation.

But these methods of teaching have died out as obsession with imitation of processes used at higher levels took over.

Teaching methods from thousands of years ago were better. Teaching tallying is the key. Most folk do not even know what it is now. Ancient folk mostly tallied. They did not count.

This book should be studied by every primary school teacher:

http://ebookee.org/From-One-to-Zero-A-Universal-History-of-Numbers-Repost-_875420.html
11:49 AM on 06/21/2012
I agree with much of what you write, but I struggle with this.

"They don't need to hold that basic task in their heads, which frees them up to focus on the goal of mathematics which is the appreciation of the beauty of patterns of number in the world and in abstract shapes."

I regularly read students work and notice an obvious mathematical mistake, for example a student who wrote that there were 24,000 street robbies in Bethnal Green compared to 1,500 in the Greater London area. This is somewhat similar to someone saying that their foot weighed more than their body.

Unless and until you have a grasp of the basic concepts, for example the connections between the abstractions of numbers and the real life, you cannot move on to appreciate the joy and beauty of maths. It is not either/or. I strongly believe that it is only when we get a 'feel' for numbers that we can safely use a calculator. Too many people tap in the number and write out the answer and get no sense that they have tapped in a number that is far too big or small by mistake. Once you grasp the basics you look at the answer on the calculator (oh, OK, mobile phone) and say "Huh, that don't look right."
05:35 PM on 06/21/2012
You are correct...it's all very well to 'free up the mind', but there must be a basic appreciation of numbers, size, scale, and a rough idea of what an answer on a calculator ought to be, even if the precise result, to nth decimal point isn't known. Also, not all school leavers will get work in computerised industries,... many of them could well end up in retail environments, or say engineering, and a total reliance on a calculator would just not be adequate. Oh yes, and what happens when the lights go out... when there is no power ... just a thought.