Rote learning in Maths (1 Viewer)

sida1049

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I agree with you. With English imo being the best example here: the act of memorising an essay and regurgitating it for an exam is rote-learning, but the initial conception and construction of that essay itself requires a level of critical thinking and understanding on the part of the student.

In regards to the integral example, we have it committed to memory due to the sheer amount of practice that we do. However, I would think most top students would understand where it comes from and know how to derive it, so I personally wouldn't call it rote learning.
This is actually an interesting point, and I'm going to assert that integration is the quintessence of rote-learning. The only real understanding one has for integration is the whole "area under the curve" thing, which is fine. But to analytically solve integrals, that understanding is useless and it only really comes down to "how many integrals have you seen?", and bashing as many familiar integrals and rules you know against the problem until (a) by some measure-zero chance you solve it, (b) give up, or (c) realise the integral is impossible and then give up.

Perhaps this is just my own views on maths here, but particularly with respect to analytical integration, I doubt that there is a real "understanding", and that the top integrators approach the problem with a mechanical blindness and brute-force that is the quintessence of rote-learning. When a difficult integral is solved analytically, the correct approach is almost always discovered by trial-and-error from a history of practice, not deliberated understanding.
 

Trebla

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I don’t see much difference in trying to figure out a difficult integral and figuring out a difficult Olympiad problem in terms of skills drawn upon.

Both require problem solving skills (i.e. the skill of applying your knowledge) which is completely separate to the method of acquisition of said knowledge (rote learning or understanding the concept).

A good simple analogy is following the road rules in theory vs. practice. You can learn the road rules by rote or by understanding why they exist. However, it is a completely different matter to apply your knowledge of the road rules (whichever way you acquired it) to your driving.
 

sida1049

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I don’t see much difference in trying to figure out a difficult integral and figuring out a difficult Olympiad problem in terms of skills drawn upon.

Both require problem solving skills (i.e. the skill of applying your knowledge) which is completely separate to the method of acquisition of said knowledge (rote learning or understanding the concept).

A good simple analogy is following the road rules in theory vs. practice. You can learn the road rules by rote or by understanding why they exist. However, it is a completely different matter to apply your knowledge of the road rules (whichever way you acquired it) to your driving.
This is a fair point, and I agree. My point was that to be successful at integrating is to be successful at rote-learning by acquiring a library of integrals and techniques through sheer brute-forced practice, and it's rather inaccurate to say that integration success comes from deliberated "critical thinking". I guess the same applies to olympiad questions - they're difficult precisely because intuitive understanding is difficult if not impossible. A lot of research mathematics too, actually. Many a times have I read through 5 pages of proof, where I can see validity, but no understanding nor critical thinking (as we commonly know it).

For this reason, I think using mathematics as an analogy for other things often (read: almost surely) falls short. Where the practice of road rules is straightforward, and the construction of an English essay is very deliberate and intentional, mathematics is none of those.
 

blyatman

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I agree with you. With English imo being the best example here: the act of memorising an essay and regurgitating it for an exam is rote-learning, but the initial conception and construction of that essay itself requires a level of critical thinking and understanding on the part of the student.



This is actually an interesting point, and I'm going to assert that integration is the quintessence of rote-learning. The only real understanding one has for integration is the whole "area under the curve" thing, which is fine. But to analytically solve integrals, that understanding is useless and it only really comes down to "how many integrals have you seen?", and bashing as many familiar integrals and rules you know against the problem until (a) by some measure-zero chance you solve it, (b) give up, or (c) realise the integral is impossible and then give up.

Perhaps this is just my own views on maths here, but particularly with respect to analytical integration, I doubt that there is a real "understanding", and that the top integrators approach the problem with a mechanical blindness and brute-force that is the quintessence of rote-learning. When a difficult integral is solved analytically, the correct approach is almost always discovered by trial-and-error from a history of practice, not deliberated understanding.
Hmm fair enough. As an engineer, I tend to value the applications of integrals over doing mindless indefinite integrals since the latter are done computationally in the real world, and analytical attempts serve little purpose apart from being interesting exercises for your brain. Topics like the old 4u volumes required you to formulate integrals, which in turn required students to understand that integrals are essentially infinite sums. Students who understand this concept have little trouble developing their own integral expressions, whether they choose to use the disc methods, shell method, or whatnot. This is why I was so sad to see it cut out from the new syllabus.

When I first teach integration to my students, I really stress the fact that integrals are infinite sums, and I use this concept to derive simple things like the arc length formula. I'm hoping that an understanding of integrals as infinite sums will help them when it comes to the applications of calculus section, which is a topic where many students struggle.

In the real world, integrals such as

which represents mass, require a solid conceptual understanding of integrals. Another example is Gauss' law of magnetism:

This is a mathematical statement which basically states that magnetic monopoles do not exist, and the ability to infer the physical interpretation from of the mathematical result requires an understanding of what integrals physically represent.

Obviously this is all beyond high school math, but I'm just trying to illustrate that a solid understanding of what integrals represent is very useful in the real world, where ones ability to analytically evaluate integrals (ie the "rote learning" part) is rarely utilised.
 
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Trebla

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This is a fair point, and I agree. My point was that to be successful at integrating is to be successful at rote-learning by acquiring a library of integrals and techniques through sheer brute-forced practice, and it's rather inaccurate to say that integration success comes from deliberated "critical thinking". I guess the same applies to olympiad questions - they're difficult precisely because intuitive understanding is difficult if not impossible. A lot of research mathematics too, actually. Many a times have I read through 5 pages of proof, where I can see validity, but no understanding nor critical thinking (as we commonly know it).

For this reason, I think using mathematics as an analogy for other things often (read: almost surely) falls short. Where the practice of road rules is straightforward, and the construction of an English essay is very deliberate and intentional, mathematics is none of those.
Rote learning every single method out there is one way to be successful at integration. You can be good at integration without rote learning, especially if you are good at recognising patterns and have good foresight on what an approach will lead to.

Classic example is integrating say x3ex2. You can memorise the choices you should make to differentiate/integrate in the IBP but another approach is to actually stop, think ahead and realise there is just one choice which makes life easier. The latter approach using an element of foresight to deduce the best approach is how people typically figure out these integral problems, not rote learning.
 

sida1049

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[/QUOTE]
Rote learning every single method out there is one way to be successful at integration. You can be good at integration without rote learning, especially if you are good at recognising patterns and have good foresight on what an approach will lead to.

Classic example is integrating say x3ex2. You can memorise the choices you should make to differentiate/integrate in the IBP but another approach is to actually stop, think ahead and realise there is just one choice which makes life easier. The latter approach using an element of foresight to deduce the best approach is how people typically figure out these integral problems, not rote learning.
I understand what you're saying, that there is a strategy to integration that good students use. My point is simply that this strategy is a form of memorisation, and is directly the result of rote-learning a library of integrals and techniques. The reason why you and I recognise immediately what needs to be done here, is precisely because we poured many hours into practicing this kind of thing, and have seen and used this strategy ad nauseam.

Maybe this is kind of a chicken-and-egg thing; is the strategy essentially distinct from rote-learning, or just another manifestation of it? I lean towards the latter, but I suspect this might just be semantics.

i agree with everything you wrote and wait with bated breath on that latex
 

Trebla

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I understand what you're saying, that there is a strategy to integration that good students use. My point is simply that this strategy is a form of memorisation, and is directly the result of rote-learning a library of integrals and techniques. The reason why you and I recognise immediately what needs to be done here, is precisely because we poured many hours into practicing this kind of thing, and have seen and used this strategy ad nauseam.

Maybe this is kind of a chicken-and-egg thing; is the strategy essentially distinct from rote-learning, or just another manifestation of it? I lean towards the latter, but I suspect this might just be semantics.



i agree with everything you wrote and wait with bated breath on that latex
My point was not about the application of a memorised/prescriptive strategy, but rather the construction of a strategy from scratch by analysing an unfamiliar problem and applying deductive reasoning (e.g. to arrive at the best choice of what to differentiate/integrate).

Maybe you were directly taught to use that deductive reasoning approach, but I certainly wasn’t (I was taught to rely on LIATE). When I saw that integral for the first time I had not practiced on or been taught on anything like it. Yet I still managed to figure it out myself using what I knew and coming up with the deductive reasoning that I described earlier. Any good student who has very little experience with IBP can still figure out that integral independently just the same.

I’m sure people who are way smarter than me do not heavily rely on ‘memorised strategies’ or familiarity to solve a problem, but are capable of constructing their own solutions using their own thinking - particularly on unfamiliar problems (there are many ways to solve the same problem).

I also do not consider the notion of constructing a strategy from scratch (separate to memorising a strategy) as rote learning. That would suggest that any logical argument is considered rote learning, simply because the premise/axiom/assumptions are “memorised” even though the logical linkages (which make up the argument) are constructed from scratch.
 

sida1049

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My point was not about the application of a memorised/prescriptive strategy, but rather the construction of a strategy from scratch by analysing an unfamiliar problem and applying deductive reasoning (e.g. to arrive at the best choice of what to differentiate/integrate).
Okay I think this is our primary source of confusion. I was talking about memorised approaches (see below).

Maybe you were directly taught to use that deductive reasoning approach, but I certainly wasn’t (I was taught to rely on LIATE).
I actually have never heard of LIATE before. I solved that integral through trial-and-error ("brute force", as I think of it) in my head since there I could see only two real ways to go about it. I wouldn't call this critical thinking nor deductive reasoning, however.

I also do not consider the notion of constructing a strategy from scratch (separate to memorising a strategy) as rote learning. That would suggest that any logical argument is considered rote learning, simply because the premise/axiom/assumptions are “memorised” even though the logical linkages are constructed from scratch.
Me neither and I don't believe I have implied this; the first person to have integrated something like that couldn't have used rote-learning.

Specifically you said earlier that
  • You can be good at integration without rote learning, especially if you are good at recognising patterns and have good foresight on what an approach will lead to.
This struck me as little paradoxical. To have "good foresight" to see ahead into what an approach will lead to requires an established experiential library. To simplify, this is the scenario of
  1. I've rote-learned Problem A.
  2. Problem B is similar to Problem A.
  3. Therefore, I can solve Problem B.
This, however, is clearly in no way equivalent to the analogy to the analogy of calling logical argument based on memorised axioms as rote-learning. I think the scenario you had in mind might have been different.
 

Trebla

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Okay I think this is our primary source of confusion. I was talking about memorised approaches (see below).



I actually have never heard of LIATE before. I solved that integral through trial-and-error ("brute force", as I think of it) in my head since there I could see only two real ways to go about it. I wouldn't call this critical thinking nor deductive reasoning, however.



Me neither and I don't believe I have implied this; the first person to have integrated something like that couldn't have used rote-learning.

Specifically you said earlier that
This struck me as little paradoxical. To have "good foresight" to see ahead into what an approach will lead to requires an established experiential library. To simplify, this is the scenario of


This, however, is clearly in no way equivalent to the analogy to the analogy of calling logical argument based on memorised axioms as rote-learning. I think the scenario you had in mind might have been different.
Maybe today it looks obvious that you are drawing from your personal experience but I am pointing out that a student could never have experienced that type of ‘tricky’ integral and still be able to independently construct the solution from scratch - using only logical deduction and basic IBP (which you agreed is not rote learning).

Using only their knowledge of basic IBP and deductive reasoning the student could:
- Analyse the integral and identify a list of choices to differentiate/integrate
- Realise that some choices like integrating dx and differentiating x3ex2 lead to a dead end; either by hand or by observing the pattern in the algebra to foresee that ahead
- Narrowing the list of options allows them to deduce an appropriate way forward

The above approach doesn’t require any new knowledge outside of IBP, nor does it require any previous experience of similarly ‘tricky’ integrals. It is simply a logical thought process of analysis and deductive reasoning - something that good maths students already have without needing to be taught.

You are basically suggesting that to you can only answer the question with a strategy acquired only by rote/experience, not one that is independently derived. This is not true at all and contradicts with my (and possibly a lot of others’) personal experiences when I saw this type of question for the first time and similarly for a lot of other math problems.

Basically my point is that you do not become good at integration or maths purely by familiarity or rote. A big complementary part of it is knowing how to construct a logical solution from scratch to solve an unfamiliar problem.
 

sida1049

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I largely agree with you, but I've noticed we commonly work with different understandings of the terms we're using. I only really disagree with your representation of what I'm saying.

As previously mentioned, I not at all believe that a problem can only be solved by rote learning. Proper mathematical exploration is definitely something that good maths students do. I believe that this is what you're trying to explain and believe that I disagree with. We've been talking over eachother here, so I hope to have resolved that.

My initial point is that integration is a perfect example of a skill that requires an experiential library to be good at - not passable, but good. Can a student who has never seen a particular kind of problem work out how to solve it? Of course. But to be a "good" analytic integrator? Someone who can tackle difficult integrals successfully? Do as many integrals with as much variety as you possible can, so you have an array of tools and develop the necessary foresight. This is the only serious method.
 

Trebla

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I largely agree with you, but I've noticed we commonly work with different understandings of the terms we're using. I only really disagree with your representation of what I'm saying.

As previously mentioned, I not at all believe that a problem can only be solved by rote learning. Proper mathematical exploration is definitely something that good maths students do. I believe that this is what you're trying to explain and believe that I disagree with. We've been talking over eachother here, so I hope to have resolved that.

My initial point is that integration is a perfect example of a skill that requires an experiential library to be good at - not passable, but good. Can a student who has never seen a particular kind of problem work out how to solve it? Of course. But to be a "good" analytic integrator? Someone who can tackle difficult integrals successfully? Do as many integrals with as much variety as you possible can, so you have an array of tools and develop the necessary foresight. This is the only serious method.
So we both agree that a difficult integral can be solved either by:
- calling upon prior experience; or
- using problem solving skills to derive a solution from scratch

How is that consistent with your primary assertion later on suggesting that having prior experience is the only way to be able to solve difficult integration problems?

I am not disputing the claim of existence (lots of practice/experience is a way to be good at integration). What I am disputing is the claim of uniqueness (that it is the only way to get there). Sure, there are people who get good at integration purely by building a big “experiential library” to call upon. But there are also people who are good at integration with nowhere near as big of an “experiential library”, mainly because they have a strong general aptitude for problem solving - which translates to the ability to independently derive a solution from scratch.

People who are good at integration are usually characterised by a mix of both an experiential library (for the familiar) and a strong aptitude for general problem solving (for the unfamiliar) - it is not only the former.
 

sida1049

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So we both agree that a difficult integral can be solved either by:
- calling upon prior experience; or
- using problem solving skills to derive a solution from scratch
Yes, definitely. Or a mix of both of course, where the student can recognise familiarities with previous problems as well as new, unfamiliar aspects.

How is that consistent with your primary assertion suggesting that having prior experience is the only way to be able to solve difficult integration problems?

I am not disputing the claim of existence (lots of practice/experience is a way to be good at integration). What I am disputing is the claim of uniqueness (that it is the only way to get there). Sure, there are people who get good at integration purely by building a big “experiential library” to call upon. But there are also people who are good at integration with nowhere near as big of an “experiential library”, mainly because they have a strong general aptitude for problem solving - which translates to the ability to independently derive a solution from scratch.
Theoretically, it is possible someone with no background in mathematics, initiate an exploration of maths from the most elementary skills all the way into solving a hugely difficult open problem. So in the case of even the most difficult integral with an elementary solution, this is possible. But this is not representative. The likelihood of a person achieving this with no prior practice in the topic decreases drastically with the difficulty of the problem. And I'm imagining a really difficult integral, where not having a tour through the different kinds of elementary integrals and techniques makes the problem hopeless for someone with very little prior experience for these kind of problems.

The reason why I don't think there is a conflict between the points we're trying to make, is precisely that the point you're making is more of a strongly theoretical one - for elementary integral problems, it is much more realistic, definitely. But I'm making a statement on the practice. I have no doubt that there are amazing maths students who can turn in a solution for problems they have almost zero relevant prior experience in. This could make them a god at integration, but this is not even remotely representative.

I've made the scope of my statement clear in my first post on this: "When a difficult integral is solved analytically, the correct approach is almost always discovered by trial-and-error from a history of practice, not deliberated understanding." I acknowledge that the rarer possibility is still a possibility. I do somewhat regret my wording the follow in "to be successful at integrating is to be successful at rote-learning" because it sounds very strong, but this statement holds very accurately in practice.

People who are good at integration are usually characterised by a mix of both an experiential library (for the familiar) and a strong aptitude for general problem solving (for the unfamiliar) - it is not only the former.
This is true, but this is just a pointless semantic chicken-and-egg problem, since one can say that an aptitude for something these kind of problems is generally developed as a result of practicing over and over and over.
 

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There is no “chicken-egg” thing here. Aptitude (for problem solving in this context) by definition means an individual’s innate (not learnt) abilities or natural instincts - which is clearly separate from any abilities gained from practice. It is something you and I (and anyone naturally mathematically inclined) draw upon when encountering unfamiliar problems where we can’t solely rely on past experience. It is no different for people who are good at integration when they face an unfamiliar integral (no amount of practice will help much on that).

I agree that it is rare to get by relying on aptitude alone but for the same reason it is also rare to get by relying on experience alone (as you won’t be able to handle unfamiliar questions). This is why I said at the end that typically people who are good at integration have a strong mix of both.
 

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But it's so weird right. If you look at the top students and even some of the top students I've met at Unsw for 4u maths say the same thing when I ask them how did you get out question 16, which is "I've done similar questions before." Is this a form of rote learning by just hammering out hard problems constantly or is it ones ability to problem solve? Obviously they're amazing at Maths to get out question 16 but does this make 4u 'rote learn-able'?
 

sida1049

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Obviously they're amazing at Maths to get out question 16 but does this make 4u 'rote learn-able'?
I've always felt that the thing I disliked the most about 4U was precisely that it was very rote-learny.

There is no “chicken-egg” thing here. Aptitude (for problem solving in this context) by definition means an individual’s innate (not learnt) abilities or natural instincts - which is clearly separate from any abilities gained from practice. It is something you and I (and anyone naturally mathematically inclined) draw upon when encountering unfamiliar problems where we can’t solely rely on past experience. It is no different for people who are good at integration when they face an unfamiliar integral (no amount of practice will help much on that).

I agree that it is rare to get by relying on aptitude alone but for the same reason it is also rare to get by relying on experience alone (as you won’t be able to handle unfamiliar questions). This is why I said at the end that typically people who are good at integration have a strong mix of both.
My mistake.

If by aptitude you really do mean one's a priori ability to do something, then I'm going to have to strongly disagree (oh boy) with this essentialist view. I will firmly assert that had I not trained in mathematics at all, I will have no chance to successfully solve an advanced maths problem. Even if you explain to me what an integral is, what the symbols represent, mathematical logic and so forth, if I had no prior training in mathematics I will not be able to solve it. Mathematics requires a gradual build-up of ability, experience and knowledge. Some people may pick up knowledge quicker than others, sure, but a priori black magic has no role here.
 

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