The fear of many but always one of my favourite subjects. For me and most of my friends the bogey subject was English because of the stupidity of English as it is taught in New Zealand schools and some of us (but not me) actually did double maths (calculus and statistics). This is how it should be. Maths makes sense and is understandable. It is a fundamental skill and that's part of the problem.
As one progresses through school it becomes readily apparent that the immediate everyday applications of the maths reduce. This is a problem because for students their immediate perception is, "When the hell would I ever need to use this? What is the point?" To be honest, I'm not really sure when we'd ever use imaginary and/or complex (i.e. those with real and imaginary parts) numbers, let alone in everyday life. I'm not sure what to do about this, aside from complex it was never really something that I was troubled by but that doesn't mean I know why that is the case. It's probably connected to liking maths. Perhaps the only thing to do is make sure that students have a clearer idea that school maths is like a pyramid... a clear progression exists. That cannot happen at current because so many students lack/forget the fundamental skills so it's retaught year in year out. Why is this so?
The problem starts in primary. I don't think anyone is ever going to contest that and there are some things that are quite good. Doubling and halving, for example, is a good strategy to use mentally. Explaining through examples. If I have eight people and four apples, how many apples can everyone have? It's good to understand what the signs actually mean. This is, though, probably where the "application obsession" takes root.When everything is expressed in terms of beans (a la Blackadder), counters, apples (as above) or whatever the student becomes used to their being an immediately obvious real world state. It's probably best, therefore, to move past these numerical processes as soon as possible. This means not stopping to consider inequalities in year three (these confused me no end) and leaving them to when they'll be needed again (invariably college). I'm not sure the focus on speed that exists is necessarily helpful either. Certainly, I never did particularly well on the speed tests but it seems to be no trouble whereas people who did could, at times, be stumped by 8+5. But, maybe, that's just me being irritated by struggle of my youth.
There are so many strategies that are taught. So many, in fact, that they're largely forgotten as soon as vertical adding and multiplication is introduced. What is the point of spending years going over these when they're all rendered obsolete by a week's worth of lessons in year seven? Childhood Arthur CD-Roms would suggest that this way of doing things is introduced much earlier in the US. Given they do worse than us in PISA, possibly too early. This may have changed but I don't think so based on the article in the Herald referred to before. Margi Leech's views are somewhat similar to mind. Certainly, reading that article has influenced this post on a number of levels. However, I am a little unclear by what she means when she says "patterns" particularly in terms of what that would actually mean.
By improving the understanding and/or familiarity of students with numerical processes algebra can be introduced earlier. The first time we had to deal with letters was in year eight. The now abandoned entrance exam for the local college was in large parts completely foreign to us. My year nine class had a roughly even split between "knew what factorising was" and those who had never heard of it. These are realities that should not have existed. I wouldn't go as far as introducing logs by year six but maybe that's working for my cousin and my cousin's school. NCEA Maths is all about understanding a concept and figuring out how to apply it to a context/situation (typically this is the excellence part). We'd improve those results by making sure that students get solid foundations in primary. There main difference between school algebra and the maths before it is that there are unknowns everywhere. It is impossible to do well with letters when the skills with just numbers are insufficient.