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.