Like many people, algebra is a slightly painful word. Rows and rows, indeed columns of columns of x's and y's attacked me at secondary school. I didn't really get what they meant, even though I was actually quite good at solving equations.
Now as a primary school teacher I still have a blind spot when it comes to algebra, there's something about it that I don't quite get.
But I've had a revelation today. I think I know what I've not been quite getting all this time.
I've just read a chapter in a wonderful book by Derek Haylock: "Mathematics Explained for Primary Teachers" (4th Edition). I've been able to access the book through the MaST programme I'm on at Edge Hill University - but it was so good that I bought the whole book from Amazon. It starts with a question that illustrates why I don't get question. I don't want to steal Haylock's thunder, so here's a different version of the same concept:
On a school visit, 6 students are can go for every 1 teacher. There are t teachers, s students can make the visit. Describe the relationship between s and t.
The temptation is to say 6s=t. That is exactly what I did in the equivalent problem that Haylock set me. But then, say 30 students make the trip, then according to the equation I just wrote, I need 6*30 teachers. 180 teachers for 30 students? Slightly over-powering! The answer is s=6t
Haylock makes the point that I'm getting confused between 'things' or 'objects' and variables.
In arithmetic, which dominates primary teaching, I use letters as abbreviations - hence 't' for teachers. There's also m for metres, kg, mm, l, and many more. In algebra, letters never represent abbreviations for measurements, they represent variables - they stand for whatever the number you've chosen. An amount that can be changed. It is precisely for this reason that it is unhelpful to use 't' for teachers and 's' for students, because it provides the illusion that you are representing the actual teachers as a tangible thing., rather than the number of them.
I think many of us in teaching younger children think of algebra as a nice extension to do when the children have really got their arithmetic sorted. But I'm seeing now that if we only ever train children to think arithmetically, than we are doing them a disservice. Algebra is a branch off the same mathematical tree that Arithmetic grows on, it is not a branch that nicely extends from Arithmetic. Algebra develops from recognising and playing with patterns, investigating sequences and seeing how things can be represented as bigger or smaller. Many of us teachers, especially in schools were standards are low, look at these lessons and wonder 'how will this help the children's maths?' And by maths we are thinking of arithmetic and doing well in tests (which for 11 year olds are about 50% arithmetic). We are not thinking of developing the children's brains so they can generalise patterns and represent problems.
I can hear the question being posed. So what? Why should children have to generalise patterns and represent problems?
Well the answer comes down to being able to solve problems with much bigger numbers and larger degree of complexity. I might be able to solve a problem with my arithmetic skills, but if I can represent it I can use a spreadsheet or a scientific calculator to solve it for any number. Likewise I might be able to work out the 15th term of the triangular number sequence, but working out the 77th is a rather harder challenge - I can save loads of time by generalising the pattern, representing it with algebra and calculating from there.
I wonder how many software developers, games designers, app creators and the like can get away with only thinking arithmetically? I don't know anything about how those kinds of jobs work, but I'm sure that some level of algebraic thinking is required for those jobs.
So. An answer to my question: as young as possible. In my next post I'll start to explain how...