Gravity is quite an easy concept to understand. If two bodies have mass they attract each other. They move together as a system. We understand very clearly how this works and have some simple equations to enable us to work out the forces involved. It’s simple enough for school children to understand.
So, two bodies, very simple. So how much harder can it be if three bodies are involved? It turns out that the answer is “very”. Working out the movements of three bodies (eg the Sun, the Earth and the Moon) as a gravitational system relies on approximations and numerical methods. Four body problems (which arise when adding a spaceship to the mix, say) rely on ignoring the presence of one or more of the bodies in order to create a set of solutions.
Solutions to multi-body problems rely on breaking them down into groups of parameters within which sets of equations can apply.
Education is a bit like this. Typically we have three interacting elements – student, teacher and content. The relationship between the teacher and the content (how much of a subject expert are they) affects the relationship between the teacher and the student and so on. It is iterative and cannot be easily predicted. There is a fourth element as well – the school and how it is organised. The largest divide we have here is between primary and secondary. How learning is organised in the two are very different and so we would expect the interactions between teacher, student and content to be different as well. We also have a fifth body to consider, government. They are like the black hole at the galactic centre in this analogy – distant, impacting little on a day-to-day basis, but ultimately destructive if they get too close to the action. I give you Universal Free School Meals, everyone.
Which brings me to Chinese maths teachers.
I’m a massive fan of teachers learning from each other. Generally speaking it is the best way for teacher learning to occur. So I welcome any input into the teaching of mathematics that any teacher can bring. There will be techniques that are used in China that are new to UK maths teachers (and vice versa) which can be used to enhance practice in this country. At the individual lesson level there may well be some impact. I would suggest that the impact could have also been achieved by empowering the best local (ie UK) teachers to do the same. But that would be churlish and is not the point of this post.
What I pick up from reading about the ‘fact-finding’ visit to China is not predominantly about the brilliance with which any one teacher teaches. What I hear is that the big differences are due to how the school, and the students within are organised. Many of these elements are not in the control of the individual teacher and many of them will not happen.
A good example is the ‘teach maths every day and teach it in the morning” approach. Ok, that can be done at primary (says he from his secondary experience base). But it is not solely the decision of the individual teacher. At secondary it won’t happen. Why? Simply because a school would not have sufficient numbers of maths teachers and could not allow them to be idle for the afternoon. Unless they used the maths teachers to teach other subjects in the afternoon. Put non-specialists onto another subject. So there is an impact on the other subjects. There’s that three body problem rearing its head again.
Another is the “don’t let the child go on until they understand the current topic”. I have a lot of sympathy with this idea. Maths requires this approach perhaps more than any other subject, particularly for the basics. So, again, perhaps more implementable in primary than secondary. But, unless you do the additional work outside of normal lesson time, disruptive to other subjects. Ok then, do it after normal lesson time, 3:30 to 5:00. Again, I’m not completely averse to this idea, but then you have to ensure that the school structure supports it and it is supported by parents. Probably into four body problems with this one.
I have seen many people visit other, successful schools, both in England and abroad. They see the success being achieved and then they bring back one or two elements and try to implement them at home. More often than not they are disappointed by the outcome. They are doing the things right, they just don’t recognise that those things were only successful because they were part of a larger, holistic, whole.
Any attempt to improve the level of maths in the country is to be applauded. Any attempt that concentrates solely on the individual classroom and ignores the wider supporting structural issues will not succeed. There is too much interaction between the different parts of the system for it to do so.
In celestial mechanics there are places called Lagrange Points, named after the great Italian mathematician Joseph-Louis Lagrange. In 1772 he published a paper on the three-body problem and demonstrated* the existence of areas where, given certain conditions, the gravitational effects of the three bodies was in balance. A fourth body could rest there unperturbed by gravitational fields of the existing three bodies.
We desperately need to find some Lagrange Points in education. And put our classrooms there.
*Strictly speaking, Lagrange did not notice this effect in his own work, it was spotted by others.