Here’s a very dispiriting new blog post by Tim Gowers about the dire state of school maths teaching in the UK. I’m a bit surprised, though, that he’s surprised by what he discovered (but then being in the happy situation of teaching Trinity mathmos is likely to colour your view of the world!). Certainly, my experience teaching Cambridge philosophy students — a very bright lot — about half of whom had done A-level maths and got top grades, was that very few of them really understood any maths. They might have picked up a bag of tricks to apply in response to some formulaic questions. But as for some conceptual understanding of classical analysis or an appreciation of the idea of rigorous proof ‘in the wild’ … Sigh. (No wonder philosophy of mathematics can leave them cold!)
And judging from the many comments to Tim Gowers’s post, the decline in school maths teaching is not confined to this declining off-shore island.
Well, there’s no point in moaning about this, though I hope my local friendly mathematicians are exercising what influences they can to help stop the rot. But it does mean that there’s a real problem for teachers of logic and writers of logic books aimed at students. We tend to be mathematically ept ourselves, and students are good at dissembling — or at least, at remaining silent — so we can so easily fail to realize just how mathematically inept so many of them are (through no fault of theirs, let it be emphasized). What to do? I don’t have the teaching problem any more: but I still do have the writing problem. I guess I’ve fallen over the years into working with the principle if in doubt, go slower and explain even more. But it does make for long books to write and long books to read.
In my baby logic course, it isn’t a problem (just makes for slower going). In the 2nd course it means I spend a while teaching functions, relations, induction, number systems, etc. I have to do this even with students who have (nominally) taken “finite math”. Don’t even ask about the advanced (=metamath) course.
I read Tim Gowers’s post after reading yours (thanks). I have taught high school maths in the U. S. for 5 years and have tutored students for decades (while in another career). His experience is one I’ve had many times and his strategy with this student is one I’ve employed, and remain inclined to employ, almost every time.
Your comment about going slow and explaining more is apt. The reality is that we need to do exactly the opposite. We need to ignore the clock, explain less, offer more space to discover and offer better opportunities for exploration at every level in school.
Okay, primary school children are not “ready” developmentally (according to current theory and data) to generalize systematically from specific cases. But, they do recognize patterns and generalize in non-linguistic/non-symbolic ways (e. g., look both ways before crossing the street; the alphabet consists in those 26 characters and no more; the word, “letter” is used in two ways and is not to be confused with “let her”, despite their nearly identical sounds). If they didn’t, they would not be able to learn the multiplication tables.
This topic has been explored in academia, by consultants and is the subject of the expenditure of much political capital in the U. S. I don’t for a minute believe that I have anything new to add to this battle or the research in the field. But, I do want to add my voice to the chorus of parents, children and teachers who want their curricula to encourage exploration, learning self-sufficiency and to promote and reward strategies and outcomes that encourage these outcomes.
I don’t think we really disagree. When I talked about “going slow and explaining more” I was thinking about what we want from university-level logic textbooks, which provide only one ingredient to the learning process. What is best practice in a high-school classroom situation is indeed a different matter, and “curricula to encourage exploration, learning self-sufficiency” rather than the mere rote learning of a basket of half-understood tricks is of course (in some measure) part of what we need.
Such exploration etc approaches seem to be controversial, though, with some of the comments on Tim Gower’s post linking to articles such as these in the Atlantic:
http://www.theatlantic.com/national/archive/2012/11/a-new-kind-of-problem-the-common-core-math-standards/265444/
http://www.theatlantic.com/national/archive/2012/10/its-not-just-writing-math-needs-a-revolution-too/263545/
The second includes a video that ridicules the way a girl has been taught to add multi-digit numbers. I thought the girl showed an admirable grasp of what she was doing and of the ideas involved; but it turned out that she also knew how to add numbers in the traditional way — which seems to be called using “stacking” and is supposedly something they’re not allowed to do in school. (She was taught “stacking” at home.) I suspect she’s the sort of student who’d understand how addition worked almost regardless of how she was taught.