General thoughts about learning

[Obis]

Let me start with my experience from the past, 3 years ago, when I was still in high school. We had a very strong physics teacher then, who understood physics relatively deeply (at least compared to teachers of other subjects, like mathematics). He was famous for his extremely low marks. On a 10 point scale, it was said, that God gets 10, professor gets 9, student genius gets 8, etc. The majority of students were getting 4 or 5, including myself. This teacher had good teaching abilities, his problems weren't too hard, and the students were actually capable of understanding it, so where was the problem? The problem was in the presentation of other subjets, for example, mathematics. Mathematics was presented as a collection of algorithms and equations, which you apply to solve various problems. The application was blind and mechanical, basically it was symbol manipulation, without understanding their meaning.
The same strategy was used for physics by the majority of students, including me. When you tried to solve a problem, you would write what is given, for example, m and a. Then, you would write what is required, for example, F. Then you would try to remember any relationships, that would relate F, a, and m. You would remember F = ma and solve the problem! However, this was sufficient only for very primitive problems, and the problems the teacher gave us wasn't like that. Even though they weren't hard, they simply needed something more than this strategy : they needed imagining the situation, thinking about the meaning of the symbols, thinking about the meaning of their relationships.
In the last semester, 3 months before the final physics school-leaving exam, my approach to learning changed dramatically. Now I spent quite some time independently reading and trying to understand, tried to imagine everything. My marks in the last semester (from first to last) were 7,8,9,8,10,10 and I got a 100/100 from the exam (which is acquired by the top 1% scorers nation-wide).

My main point here is to accentuate the importance the way you see, the way you approach learning. In fact, I think that one of the main differences between the so called "talented" and "non-talented" students is the way they approach learning, their learning habits. When I see some of my current classmates, their lack of understanding, lack of motivation and low marks, I don't see lazy, non-talented people. I see people, who saw a completely wrong presentation of learning, and, sadly, accepted it, conformed to it. It's the system that I blame, not them.

Now I'm in university, however, I still see (even though, less) the same problem. An algorithm is presented, no time is spent in trying to understand it, what it computes, why it works. Simply a few similar problems are presented, that require direct application of this algorithm, this supposed to "teach" us the algorithm. Yes, after boring and tedious time applying it, it temporarily gets stuck into your memory, however, understanding an algorithm and remembering it is a completely different thing. Human mind is fundamentally bad at remembering meaningless information, computing something blindly and mechanically, doing everything very precisely, without any arithmetic or similar mistakes. However, in a lot of cases, I see that this is exactly what is required by some of the professors. This makes me angry a little bit, I find myself arguing with some professors (calmly enough), I'm trying to convince them, how useless and wrong it is.

Now, one of the reasons I'm writing all this is that I understand, that it is possible, that I'm actually completely wrong. I'm only learning mathematics for a few years, while the professors are doing it for the past 40 years. Maybe I'm just over self-confident, maybe It's I who doesn't have a clue how to approach learning. Maybe I'm just extrapolating my own experience, maybe what works for me, will not work for the majority of others. In fact, when I said, that 90% of my learning is reading and 10% is problem solving, I've felt some pressure.

Hence, I'm deeply interested in what do You think about all this, what do You think is the right approach to learning?

 

[Nano-Passion]

Understanding instead of memorizing.

Plain and simple. The road to understanding the material tends to differ for different people, or so I assume. This is a very complex subject and requires one to choose their words carefully as to not state things too simply. Regardless, I tend to tackle learning something along these manners:

1) Read the material and understand as much as possible. But at this stage I will not fret when I don't understand something and spend hours on it. I will postpone that that in hopes that the holes of my understanding will be filled as I do problem later on.
2) Any assumption in the book that I don't understand I will try as much to confirm the assumption myself mathematically. This is speaking of simple things though, not necessarily anything too rigorous since I'm only up to Calc I & Physics I. I'll try to ration my time rationally, pun intended.
3) I'll do as many problems as possible. Any problem that I don't understand I will stop at nothing till I understand what went wrong intuitively or mathematically. This is the time where I really try to learn and to fill every hole of my understanding.
4) It is recommended that you try to fiddle with equations that you come across and try to play with things, see where they don't work and expand it, etc. But I've found my math is rather premature to do this leisurely.

There lies many fine details between what I wrote that is left out. I'll leave others to fill the holes of my strategy. You can easily write a 1000 page book about how to solve problems. Solving problems is a marvelous feat of the human mind, completely detached from our original programming to survive in the wilderness. A scheme or way of thinking is often needed to know how to solve problems, something that isn't rigorously lain down but is left for the students to properly attain for themselves. It comes easier for some more than others, depending on their scheme or approach. One who paid no attention to their previous studies throughout their life would notice that it is something largely taken for granted, for the reason that certain schemes or way of thinking has been carefully raised with the student throughout their education. Others not fortunate enough to adapt to their studying environment must learn to adapt or perish. Luckily your brain is a very malleable piece of machinery.

 

[Highway]

you need to understand what's going on before you can expect to solve problems...

are you working on the section wrt angular momentum? do you know the corresponding equations and how they are applied? the same goes for the quotient rule, etc.

once that's done, you have to practice as many problems as you can from either your textbook, or an older edition of different authors book that is on cramster or something that you have fully worked out solutions to, so that you can check your work for each step of the process to see wtf you are doing wrong.

im not saying use cramster like most kids do to cheat on your hw, but if you need worked out solutions as a means to practice, try to find an older edition of a book that they have on there (you can probably get it for $10 or less online, as a really beat up used copy), and practice the odd problems. cramster has free access to the odd problems of a lot of books, but the even solutions you have to pay for since they are usually assigned for homework by most teachers. use these odd solutions to practice against.

 

[Nano-Passion]

you need to understand what's going on before you can expect to solve problems...

are you working on the section wrt angular momentum? do you know the corresponding equations and how they are applied? the same goes for the quotient rule, etc.

once that's done, you have to practice as many problems as you can from either your textbook, or an older edition of different authors book that is on cramster or something that you have fully worked out solutions to, so that you can check your work for each step of the process to see wtf you are doing wrong.

im not saying use cramster like most kids do to cheat on your hw, but if you need worked out solutions as a means to practice, try to find an older edition of a book that they have on there (you can probably get it for $10 or less online, as a really beat up used copy), and practice the odd problems. cramster has free access to the odd problems of a lot of books, but the even solutions you have to pay for since they are usually assigned for homework by most teachers. use these odd solutions to practice against.

OP, a membership in cramster is the best money that you can spend! Get it!

 

[victor.raum]

Spot on Obis, very well worded :-)

 

[turbo]

Test-taking and knowledge don't always line up all that well, especially if you are expected to apply that knowledge to real-world problems. One of the best trouble-shooters that I have ever known in the field of papermaking was an "engineer" with no degree, just lots of experience, and Beloit promoted him as a "field engineer".

I ended up being paired with him on a couple of tough jobs, in part because he was a "field engineer" on the start-up of my company's first paper machine, and we got to know each other well. One mill that we went to was having persistent wet streaks in the sheet, and couldn't make any salable paper on that machine. When we got there, he asked me if I wanted to troubleshoot the wet end or the dry end. I chose the wet end. After a couple of hours, I looked him up and told him what the problem was (mis-aligned breast roll due to improper use of the cable-type tensioning system). He came back to the wet end with me, looked at the problem (as I saw it) and called the production manager, mill manager and their minions into a meeting. He was a heavy hitter, but he let me present my results. The engineers and paper mill superintendents and assistant superintendents disagreed with my findings, so my mentor said "Let's go. They can lose tens of thousands of dollars a day while they ignore you." The mill manager and production manager chased us out of the room to ask what their problem was. They fixed the problem and were right back in production. Very old paper machine that had been taken over by clueless foremen and managers.

 

[homeomorphic]

My main point here is to accentuate the importance the way you see, the way you approach learning. In fact, I think that one of the main differences between the so called "talented" and "non-talented" students is the way they approach learning, their learning habits. When I see some of my current classmates, their lack of understanding, lack of motivation and low marks, I don't see lazy, non-talented people. I see people, who saw a completely wrong presentation of learning, and, sadly, accepted it, conformed to it. It's the system that I blame, not them.

There's definitely truth to that, but you have to consider whether some people just have a hidden talent that is dormant until they start doing things the right way. So, I'm somewhat agnostic, but hopeful about that issue. My teaching experience leads me to be more pessimistic, but it could be that the students have had their brains fried so badly in high school that their mathematical talent is very well-hidden. I don't think it's a completely level playing field. Nature doesn't really care about fairness, and if there wasn't some variation in different mental abilities, we wouldn't have evolved the way we did. That said, I think it is undoubtedly true that most students are not performing anywhere near the level they could be. As a teacher, I felt like if I could take over the students' brains and study for them, they would all become A students. The problem is that I could not observe they way they were trying to learn and provide any guidance, except maybe a few studying tips that they might not take seriously. I was confined to facial expressions, a few questions, and answers from a few students who participated in answering my questions. So, when I taught trigonometry and precalc for a year, it was like I was in a kind of jail. Actually, my attempts to try to explain concepts, I think, provoked a lot of hostility towards me. They were happier with a more familiar plug and chug approach. So, in some ways, I do blame the students. But it's not entirely their fault.

Now I'm in university, however, I still see (even though, less) the same problem. An algorithm is presented, no time is spent in trying to understand it, what it computes, why it works. Simply a few similar problems are presented, that require direct application of this algorithm, this supposed to "teach" us the algorithm. Yes, after boring and tedious time applying it, it temporarily gets stuck into your memory, however, understanding an algorithm and remembering it is a completely different thing. Human mind is fundamentally bad at remembering meaningless information, computing something blindly and mechanically, doing everything very precisely, without any arithmetic or similar mistakes. However, in a lot of cases, I see that this is exactly what is required by some of the professors. This makes me angry a little bit, I find myself arguing with some professors (calmly enough), I'm trying to convince them, how useless and wrong it is.

I never had the courage to argue with professors. I suppose this may the reason shyness evolved. I'd be interested to know how these conversations go.

Now, one of the reasons I'm writing all this is that I understand, that it is possible, that I'm actually completely wrong. I'm only learning mathematics for a few years, while the professors are doing it for the past 40 years. Maybe I'm just over self-confident, maybe It's I who doesn't have a clue how to approach learning. Maybe I'm just extrapolating my own experience, maybe what works for me, will not work for the majority of others. In fact, when I said, that 90% of my learning is reading and 10% is problem solving, I've felt some pressure.

I almost have a PhD in math. I think, my approach to try to understand things more deeply has slowed me down a lot. People who just take things on faith can get ahead in system that doesn't reward a deep understanding. So, yes, trying to understand too much might actually put you at a disadvantage, but it's partly because the generally accepted way of doing things is a little messed up. There are times when I think it's appropriate to just take theorems as given to save time, though.

Hence, I'm deeply interested in what do You think about all this, what do You think is the right approach to learning?

I don't know, but it's not plugging and chugging.

Here are some interesting things to read (long).

http://usf.usfca.edu/vca//vca-preface.html

http://pauli.uni-muenster.de/~munsteg/arnold.html

http://www.maa.org/devlin/LockhartsLament.pdf

http://arxiv.org/PS_cache/math/pdf/9404/9404236v1.pdf

 

[homeomorphic]

In fact, when I said, that 90% of my learning is reading and 10% is problem solving, I've felt some pressure.

I should add that I think this is a somewhat separate issue from the rest of the discussion. There is more than one way to do problems and there are also different kinds of problems. Doing the problems can be more integrated into the process of understanding. However, I do spend a lot of my time, just mentally rehearsing arguments of why this is true or that is true in my head to supplement doing the problems. Without this practice of constantly going through arguments in my mind, I would be like some kind of Forgetful Jones of mathematics, which sadly, a lot of mathematicians are (of course, you may emphasize remembering how to derive it, over remembering it directly). They just move from one subject to the next, and only the stuff that gets used again gets remembered. But you have to budget your time according to your finite life span, and reviewing stuff this way does take time.