Maths

[andrewducker]

This annoys me significantly.

A-level maths was apparently the hardest of the A levels, and there was too much information for the amount of teaching time they had, so they cut it back somewhat.

Which is fine.

What annoys me is that they acknowledge the problem (differing levels of ability, limited teaching time, the need to provide an indication to business/university of what level of ability people have) and then come to a conclusion that's clearly yet another bodge.

The answer, as far as I can see, is clear - it's just not politically easy. It's to break the subject down into much smaller pieces, and then rather than saying "Pure Maths: B" have a report which tells you exactly what a student does and doesn't understand. Do they understand basic geometry? Do they understand differentiation? Do they understand integration? Maths _isn't like_ English - you either understand the concepts and can work with them, or you don't. I mean, sure, you might be slower than someone else, so there might be scope for a two grade system of "Understands" and "Is a genius at", but that should show up really well just by seeing whether a student has understood a lot of 'chunks' or just a few.

So what I'd want is a series of small modules, each one of which being a step forward on a variety of different branches of the tree of mathematics. Which gives them clear guidance on their progress, and makes it obvious to others what exactly they can and can't do.

Comments

[allorin.livejournal.com]

Well said.

[azalemeth.livejournal.com]

Speaking as someone who is currently doing AS maths and further maths, I feel very confident to say that there are those who do find mathematics a very challenging option, and then there are sad gits like me who find the whole thing dumbed down beyond recognition. Previously, for maths you sat pure modules 1 to 6, and applied in a mixture of ordinary and further maths (if you were doing it). P1 started off with the 'new basics' (trig rules, polynomial integration and differentiation), and the other pure modules built from that. Applied modules would be mixed in for variation. Now, however, the first maths module is C1, which contains the higher kind of GCSE question - content that was assumed knowledge for P1! How the government can honestly say that the course hasn't been dumbed down when we end up being between a term and a year behind what we might otherwise is beyond me....

Your idea, though a good one, fails for one main reason: Colleges, and schools, don't like more exams. They have to pay for them. They have to provide lots of data, work, and money to the exam boards, who in turn cock things up (I got a "Corrected Figures" sheet for physics and chemistry, correcting loads of misprints), take a long time (over three _months_ to mark a paper), and occasionally just put a big, fat, "Result pending" on your certificate. While having chapter-by-chapter tests is usually a good idea, and (my) teachers try and do it locally, the pressure and college mania that goes into an external examination (a month of revision for everyone, whether they get it or not, setting up exam rooms, cancelling lessons, providing cover teachers...) would really put them off it. Personally, I think that the module system as it stands is about right for A levels - the actual content of the modules however, is way off! It used to be, for example, that you could not do physics without also mathematics, and the reason for this is clear - newton did not say that F=ma, he said that F = m (dv/dt). There are lots of freaky things we can do with calculus that yield beautiful results, but may require a mathematical brain in order to do so. Instead, every _single_ equation is, I quote, "A three letter approximation of the more complicated laws" (from my textbook!). And, the worst thing is that due to the nature and style of the questions there are those who just do not understand it, and do not care.

[azalemeth.livejournal.com]

Eeep. Lot of text. Sorry, rant triggered :P.

[thadrin.livejournal.com]

What's all this about units and AS levels and stuff?
I seem to have missed something.

I also seem to have missed the point of differentiation and integration. I learned them, did the exam and forgot them.

I remember A-level maths consisting of being shown a thing on the board and then being sent away to do it thirty odd times from the book. If I hadn't had a tuor to actually explain it all to me I would never have passed.

[andrewducker]

What's all this about units and AS levels and stuff?
I seem to have missed something.

Well the education system does change on a regular basis :->

I also seem to have missed the point of differentiation and integration. I learned them, did the exam and forgot them.

That was my problem - it was never explained to us what they _meant_ and why you might use them. It was only at university that I discovered that they're incredibly useful for (a) working out the rate of change of curves and (b) working out the area under a curve. When you're dealing with things that accellerate (for instance) this is vital - and we couldn't have landed on the moon without them...

Just call me "old git"

[missedith01.livejournal.com]

Frankly, it's all changed too much since I was there for me to sensibly comment, but I distinctly remember learning differentiation and integration at O Level and being told exhaustively what they were for.

[0olong.livejournal.com]

It's amazing how nearly-universal the experience is of being taught things like calculus and trigonometry by teachers who somehow never think to explain what they are or why anybody should care. The basic ideas of differentiation (finding a rate of change, or the slope of a graph) and integration (the reverse, finding the area under a graph) are so simple, and important in all sorts of ways (understanding economics, statistics, physics-related problems) that it's just embarrassing that so many maths teachers seem to have such trouble conveying them. These are ideas which primary-school kids can grasp without trouble, much as they might have some trouble putting them into practice.

[0olong.livejournal.com]

God knows how universities are going to cope with students trying to do science subjects, knowing even less maths when they arrive. When I started my physics (+ philosophy) degree they were already constantly exasperated by how much mathematical knowledge they could take for granted a decade or two before was entirely new to most of the students coming in.

For my part, I reckon the whole approach to teaching the subject is at fault. We should be teaching more basic-yet-'advanced' concepts much younger, and making good use of computers to encourage kids to explore them in a visual, playful way.

It's interesting that students are coming to university knowing less and less maths, at a time when in principle it's getting easier and easier to teach.

I wonder how much tweaking assessment is likely to help. It would be helpful for universities to get a breakdown like this, I'm sure, but it would be a lot of added overhead... and as things stand, they can always give people 'how much maths do you know so far' tests when they arrive, which is what Bristol did with us.

[andrewducker]

What I didn't make entirely clear is that I think that doing things in small chunks will help a lot _because_ most people get in trouble in maths when they fail to grasp a concept in the timespan allotted and then the class moves on to something else that depends on that concept.

So saying that you can't move on to module 3.6.7 until you've fully grasped module 3.6.6 would mean that you couldn't find yourself out of yuor depth and lost, which seems to happen to a lot of people.

[0olong.livejournal.com]

Hmm... so what would you do with people who failed a particular module?! Magic up some extra teaching time for someone to take them through where they went wrong? Leave them to get on with it on their own, maybe with some cards, or books or something, to work from?

[andrewducker]

Self-directed learning is the answer. I think teachers should be there to clear up misunderstandings and help people along, and that this is far more productive than standing at the front of 30 people explaining what they think the average person would want to know, boring 1/3 and confusing 1/3 of them.

[0olong.livejournal.com]

I get where you're coming from, but see, I came from a maths system (up to GCSE level) based on an approach something like this, and it was hopeless... quite possibly because it was very badly implemented, but it was off-putting, to say the least.

The way it worked was that every single person in the class would be working on something different, and essentially left to figure everything out for themselves - except for the one person at any given time (usually someone who was especially struggling) who the teacher could actually talk to - I guess there were probably about 6-8 of these in the average session, out of a class of almost 30.

There were no actual lessons, where the teacher tried to explain things to the class, after some time around the second year.

So... the way it didn't work was that a good three quarters of the class had no input at all from teachers in the average lesson; only a small minority of concepts were ever actually explained by a human; and when you got stuck - which naturally happened to almost everyone on a very regular basis - you could either hope that one of your peers could help you out, or hope (usually in vain) that the teacher would have the time to get to you this lesson.

Having said this, I suspect that a broadly similar approach might actually work with judicious use of computers... so that kids could look things up when they didn't understand them, and ask for (and give) help online, so that they're not left timewastingly floundering when they get stuck.

[andrewducker]

Aaah, you had the opposite experience to me - I was taught in that way and found it worked very well - and most of my class didn't need much teacher input and could work at their own pace fairly well. I suspect you're right about teaching aids - the better the quality of the teaching materials the more students can do without the teacher.

[channelpenguin.livejournal.com]

Given that I am currently teaching (albeit teaching adults navigational theory in an evening class), I should have some useful comments to make on the teaching side of this - but probably not yet...

It is pretty difficult to deal effectively with differing levels of ability/knowledge and people's differing reactions to not knowing, or not knowing what to do.

I am trying to do as little waffle as I can get away with, and as much time for the student to practice the techniques (with me supporting them and answering questions as required). Even with as obviously-applied subject as this, it is still hard to get over how it relates to the real world of sailing a boat...

I think, as people have said that that might be the real trick with teaching (and understanding) - "What can I use this FOR?"