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Maths behaviourism

Sorry, but alot of people on here think that maths is about practice. I don't believe maths is about practice. Yes practice can make you faster at say algebraic manipulation. However, to understand a topic takes more then just practice.

Sadly, I don't know what that more is but its not practice.

I think people practice but then develope schemas so they don't need to practice it anymore. Also, I think there is stages of thinking or abstraction that a person needs to grasp. Certainly pre school algebra is such a abstraction, then another abstraction is noticing that algebra can be represent by graphs, then abstract algebra is another abstration.

I'm pretty sure I see equations differently to what I did when two years ago. The manipulation part was alot of rote learning, but the shift in how I view equations is not. And by ignoring this and saying maths is just practice is ignoring abstractions that take place.

P.S. Certainly, I'm pretty sure my view of algebra is going to change drastically when I start uni and then even more drastically if I become a professional mathematician.
P.P.S. Behaviourism view learning as just being shaped, that is you get reinforced to produce a behaviour. Taking the example of algebra being reinforced with praise if you spot difference of squares would then mean you develope skill of using difference of squares.
P.P.P.S. I think its largely to do with the subconscious mind. Look up Poincare he wrote stuff on it. I tend to notice that I try to understand something for hours then come back in a couple of days and then suddenly I can understand it. Sadly, I would rather have the eureka moment which I haven't had in about a seven months. A book had a crappy explanation of differential equations but in the book it said look this is implicit differentiation, it dawn on in class that they wanted you to go backwards.

Reply 1

MrShifty

Simplicity

I'd agree up to a point. However, I'd say that practice in the form of playing around with concrete examples of abstract structures and applying theorems does actually help deepen a person's understanding. Again, it's in the same spirit as the whole idea of drawing pictures: an idea can be hard to understand or fully grasp unless its illustrated in some way that can be appreciated, and one of the best ways to do this is to practice and look around some familiar example or realisation of that idea. In research seminars and the like, you'll frequently hear mathematicians advising the audience to think of an object in terms of its most familiar example: for differential manifolds you're encouraged to think of R^n, for Hecke algebras it's the group algebra, and so on. The idea being that you can get a better and quicker appreciation of what's really going on without being encumbered with the need to process all the information in terms of abstract structures.

I think its largely to do with the subconscious mind. Look up Poincare he wrote stuff on it. I tend to notice that I try to understand something for hours then come back in a couple of days and then suddenly I can understand it.

This is an all too familiar story. I've spent hours, sometimes days staring blankly at a few pages of maths, only to find it all suddenly comes together during a second reading at a later date. Some days maths just doesn't want to happen!

Reply 2

DFranklin

MrShifty

This is an all too familiar story. I've spent hours, sometimes days staring blankly at a few pages of maths, only to find it all suddenly comes together during a second reading at a later date.

When I was at Cambridge, it was pretty widely recognized that very few people would be able to answer tough questions on a course lectured in exam term. Funnily enough, by the time the follow on course started the next year, the 'subconscious assimilation' would have occured and they'd be able to cope.

Since my time there, I think they've actually rearranged some of the first 2 years to account for this. There are courses that are lectured in exam term in the first year, but only examined in the second year. It makes a fair amount of sense.

Reply 3

MrShifty

DFranklin

Funnily enough, by the time the follow on course started the next year, the 'subconscious assimilation' would have occured and they'd be able to cope.

I remember often looking back at exam papers I'd sat from previous years and thinking "why in god's name did I find such and such a question so rough at the time". Despite having not touched the material since, it would suddenly be so much clearer, almost embarassingly so at times (the same still applies to this day).

Since my time there, I think they've actually rearranged some of the first 2 years to account for this. There are courses that are lectured in exam term in the first year, but only examined in the second year. It makes a fair amount of sense.

That really is an excellent idea. Whilst I can understand it working in Cambridge, I do wonder if a similar thing would work at other universities. The danger as I see it being that since the basic model of pre-university education is near continual assessment, some students may begin to lose a little direction (my polite euphemism for 'slack off') during the summer vacation. Though I'm sure I'm just being pessimistic!

Reply 4

generalebriety

MrShifty

some students may begin to lose a little direction (my polite euphemism for 'slack off') during the summer vacation. Though I'm sure I'm just being pessimistic!

God, no.

Reply 5

Simplicity

MrShifty

However, I'd say that practice in the form of playing around with concrete examples of abstract structures and applying theorems does actually help deepen a person's understanding.

Maybe, I have been reading too much about Grothendieck. But, yeah I read some studies that say having examples can actually hinder abillity. I think they got school children to learn basic property of commutativity, and they found if they gave children it more abstractly instead of referring to examples of everday life they understood it better.

If you read some things that Grothendieck has wrote he basically says you shouldn't use examples. Although, yeah I admit that I'm abit of a Grothendieck fanboy.

MrShifty

The idea being that you can get a better and quicker appreciation of what's really going on without being encumbered with the need to process all the information in terms of abstract structures.

You got a point. I still need to use the box and ball analogy when thinking about functions. However, I really should get rid off it.

I don't know its like visualisation. Cauchy argued against it because it
will trick you sometimes. If I can remeber correctly, Cauchy was trying to prove something in topology and was trying to visualise it but he said he was tricked as going through it algebraically showed it didn't work if it had a hole in it.

generalebriety

God, no.

So how much work are you doing now?

P.S. I thought Cambridge only did exams in the last year?

MrShifty

The danger as I see it being that since the basic model of pre-university education is near continual assessment, some students may begin to lose a little direction (my polite euphemism for 'slack off') during the summer vacation. Though I'm sure I'm just being pessimistic!

Most unis that do maths the first year of the uni is worth little to the overal grade. Most people just study to pass exams.

Reply 6

My Alt

I don't think it's ever going to be clear cut. Practise is important, as it's the most simple way to go about understanding concepts, in various different scenarios. Siklos always goes on about how doing work yourself is so much more beneficial than having someone else show it to you, and no-one disagrees with him..

Then again there are other factors such as natural talent, that you have already pointed out, which are valid also.

Reply 7

Simplicity

My Alt

Siklos always goes on about how doing work yourself is so much more beneficial than having someone else show it to you, and no-one disagrees with him..

Yeah, but I don't mean computations. Anyway, I don't want to go into the issue off STEP. Certainly, the quality of teachers has an effect on how good you are.

My Alt

Then again there are other factors such as natural talent, that you have already pointed out, which are valid also.

I don't know about that. I think a huge factor that overrides natural talent is how hard you work. Its stupid to reduce mathematical abillity to biologically level, and I think it can't. Also, I think the philosophy of how a person learn has a great effect.

Reply 8

My Alt

Simplicity

Yeah, but I don't mean computations. Anyway, I don't want to go into the issue off STEP. Certainly, the quality of teachers has an effect on how good you are.

I don't think he, nor I, meant specifically STEP, I was thinking more generally. I do agree that quality of teaching helps, quality of teaching also helps someone's work ethic and philosophy etc... It's all very interlinked.

I'd say a reasonable part of a person's philosophy is biological, and part of it is environmental. I'm not here to say that it is entirley biological, but I don't think you should count it out entirley. In fact, I don't think you should count anything out entirley, it's just too hard to consider all the factors. What you end up with is a function in waaay too many interlinked variables, and I think too many of them have similar weight to truly analyse.