Dancing in Time
Walking through the park yesterday evening I was discussing the state of elementary mathematics education (up to the end of secondary school). A rough calculation indicates that, when a student enters the infants’ school at the age of five, about 1400 hours of mathematics education lies ahead of her before she leaves high school. That’s the equivalent of 168 eight hour days or about 6 months full time, at least the first half of which occurs when the child is at her maximum rate of learning.
In that time we have three types of mathematics to teach:
- maths which is useful, indeed essential. That must include the basic operations on the positive and negative integers and fractions.
- maths which, while perhaps not useful per se gives the child a good grounding in logical thinking
- maths which is inspirational and will lead the child to choosing mathematics as a career
The question is how we do, and should, divide the 1400 hours between these three types.
To attack this question, consider something which seems to absorb an inordinate amount of secondary school time: solving quadratic equations firstly by factorisation and then by formula. Into which of my three categories does it fall? Certainly I can’t think that, other than for specialist mathematicians or engineers, it’s particularly useful and specialists could learn the formula in an hour later in life. Solving by factorising could possibly be considered an exercise in logical thinking (“two things multiply together to give zero, so one must be zero” ) but that doesn’t seem to be the way it’s taught (I need to say here that I’m not a maths teacher, I’m extrapolating from the experience of my children and the children of friends who come to me for extra tuition). And I think I would be hard pressed to find solving quadratic equations as being inspirational unless it were extended to the break with formulae between quartics and quintics and a touch of Galois in the night.
Another example of lost time seems to be the teaching of trigonometry from right-angled triangles. I teach aviation groundschool and even those students just leaving high-school have no clue that trigonometry can be applied to calculate the cross-wind component when the wind is blowing from 230 degrees at 20 knots and we are landing on runway 25. As they don’t even associate trigonometry with this calculation then I’m not sure what they’re being taught.
So, what proportions do we need of my three categories? At least initially much will have to be the useful: counting, integer arithmetic, etc. although there seems to be no reason why finite fields could not be introduced through multiplication tables very early on. Remember that children are at their most receptive at this age but, to be conservative I would accept that, for the first 4 years at least 70% of the work will have to be “because it’s necessary”. Thereafter that will probably drop to 40% and tail off to zero by the age of 15. That gives about 600 hours (or 75 full eight hour days) to essential and useful mathematics which intuitively seems to be fine. Of course, a lot of that could be taught in such a way as to extend the child’s thinking abilities but let’s keep things simple.
That means that we have about 800 hours left to teach the inspirational and “thought process” mathematics and the obvious question is what should be taught. Certainly elementary group theory (or groupoids if you prefer) should be accessible to anyone and the basics of calculus taught graphically should extend the thought processes. Number theory could introduce the children to a lot of unsolved problems to break with the traditional view of all mathematics being solved, mostly by ancient Greeks. Trigonometry as series rather than as the ratio of the sides of a right-angled triangle would be interesting. It seems impossible to fill 800 hours with this though.
So what about calculators and computers? Where do they fit into this syllabus? I’ve always thought it a shame that most children have no idea what most of the buttons on their calculators actually do. Looking at my own calculator, I see buttons for nCr, nPr, n!, standard deviation and log which could be easily explained.
Anyway, how can we get quadratic-free schools? I assume that there is, in each country, a centralised curriculum defined by a team of experts who have decided that solving quadratic equations by factorising (and what proportion of quadratic equations actually factorise; without sitting down and doing the work I assume that almost all (in the strict sense) won’t) is more important or more useful than, say, knowing the elements of number theory. How does one reach these experts?
There is, by the way, a fascinating snippet from Ma and Pa Kettle Back on the Farm (1951) on the Mathematics under the Microscope blog at the moment where Pa proves to at least his and his wife’s satisfaction that a fifth of 25 is 14. What’s interesting, apart from the original arithmetic, is that they assume that everyone knows that multiplication is repeated addition and how to do long division. I wonder whether that scene would have the same effect today.