Problem Solving Skill vs. Knowledge

[andytoh]

Problem Solving Skill vs. Mathematical Knowledge

Here goes more of my mathematical musings:

Problem Solving Skill vs. Mathematical Knowledge. Which is more important to you? Spending time to improve one will cost you time to improve the other. You must excel at both to become a mathematician, but while you are still developing, which do you place more emphasis on?

By the way, you should not be counting assignments that are forced upon you, unless it is something you know you would want to do anyway. In other words, which do you try to develop more during your "free math time?"

 

[Moridin]

Problem solving can be regarded as mathematical skills in a sense. How did mathematics evolve? It evolved through, finding a new section where the current levels of mathematics did not work, realizing that the mathematical tools available was not sufficient and inventing a way to tackle this new problem. Let's take an example.

Donald has three apples. Doris takes away five of Donald's apples. How many does he have left?

That is a very strange question. You can only go down to 0 apples, because after 0, you have no apples left. With the mathematics today, we can confidently say that Donald has minus two apples left. Not that it makes any sense in a realistic way, but because it can be applied to new problems relating to loans and so on.

If one just have started with calculus, one probably doesn't know as much mathematics in that particular subject than one that has studied calculus for a longer period of time. The more exercises and problems one solves satisfactory, the better one becomes in that particular area and the better one is to addressing new situations that one hasn't been confronted with earlier. The more familiar one becomes with calculus for instance, the better one can address and solve problems that deal with calculus, even as the level of difficulty increases. In other words, a better problem solving skill.

There is a clear distinction between mathematics and physics for example. Mathematics is about solving problems, while physics is not only about solving problems. There isn't really something called conceptual mathematics, because mathematical knowledge is in itself a tool for problem solving.

 

[andytoh]

Well said. I admit I used to be choice 5, but I've changed drastically to choice 2.

 

[Crosson]

I voted for the fourth choice "exercises only occasionally".

I think practicing lots of problems is a lazy way to learn. If you want to get better and solving problems, then concentrate on being better at solving problems.

Think of it his way: when a studio orchestra is given a piece of music for the first time, the first or second attempt is the recording that is eventually packaged and sold.

Imagine instead of this sheer effort of concentration, they wanted to slowly "learn" the piece all afternoon practicing small sections of it, eventually working up towards the hard parts. This method would work also, but it is lazier (because it does not require the brief but extreme effort put forth in the first method).

 

[Gokul43201]

"I only focus on being able to solve problems. What good is knowledge if I can’t apply it ?"

This sounds internally inconsistent. The second sentence indicates that you need knowledge to solve problems. The first one suggests that you can largely ignore knowledge at the expense of learning to solve problems.

 

[Crosson]

This sounds internally inconsistent. The second sentence indicates that you need knowledge to solve problems. The first one suggests that you can largely ignore knowledge at the expense of learning to solve problems.

Knowledge is not just one thing, it comes in many types. Some types of knowledge are applicable, and some states of knowledge are impossible to communicate.

 

[andytoh]

"I only focus on being able to solve problems. What good is knowledge if I can’t apply it ?"

This sounds internally inconsistent. The second sentence indicates that you need knowledge to solve problems. The first one suggests that you can largely ignore knowledge at the expense of learning to solve problems.

The first choice means you only like solving problems with the knowledge you already have and don't care much about expanding your knowledge to vast horizons. I'm only limited to 100 characters to describe each choice. You know what the choices 1-5 mean roughly.

 

[Gokul43201]

Okay, I get it now.

 

[Gib Z]

I voted choice 5, and it bothers me alot. I can not apply it at all. But I am afraid of starting all over again, problem solving elementary questions..I know its stupid...I don't know what to do...

 

[acm]

Choice 3, I try to proove all of the theorems I need to use and try to understand the definitions that enable me to attempt the proofs. Exercises are great when you are clueless and need to grasp the information to proove the formula.

 

[Swapnil]

I think that both problem solving skills and new mathematical knowledge are important. Mathematics without its applications would just be an Art and ...well... personally, I am not real fond of Art. However, without new mathematical knowledge, progress would be really slow and inventions and discoveries would soon come to a standstill.

 

[complexPHILOSOPHY]

I usually read through the vocabulary and concepts, work through some examples and then tackle some problems in that section. If I am working through them really easy, then I try to find harder ones. Once I get stuck, I go back and make sure I understand the vocabulary and concepts and go back and work through the hard ones again.

I also try to figure out how various methods might work together to construct a more unified model of mathematics for myself.

The only thing I don't like doing is memorizing formulas without understanding how to derive or prove it without the formula first -- I like to solve new types of problems completely on my own (if it's a reasonable enough to do so).

Is this an okay way to develop both skills or is there a better method for studying mathematics?

I am curious how others prepare.

 

[andytoh]

Is this an okay way to develop both skills or is there a better method for studying mathematics?

I am curious how others prepare.

That's the purpose of this poll.

 

[AsianSensationK]

Is it weird that everytime I see a new theory or idea in a course, I keep thinking about how familiar it all looks?

Seems to me like expanding your knowledge really boils down to problem solving skills. Of course, I voted for option 3 anyway. It fits best with my personal philosophy about how to study.

 

[Sojourner01]

'Knowledge' of what? Doesn't knowledge - as it were - just boil down to understanding how to solve problems anyway?

That's what physics is. Nobody's interested in how many different wavefunctions you can derive, they want you to do something useful for them, like invent something using those theories to engineer new technologies.

In short - physics IS problem-solving, but you need knowledge to be able to do so. Technique is only one part; accumulated knowledge is another.

 

[JeffKoch]

I don't know how you can develop mathematical knowledge except by working problems, each one of which trains you in how to look at the next mathematical problem and makes the next one easier to solve. Isn't mathematics entirely about solving problems? Even pure mathematics research, say trying to prove Fermat's last theorem, is about solving problems using knowledge gained by solving other problems.

 

[andytoh]

I don't know how you can develop mathematical knowledge except by working problems, each one of which trains you in how to look at the next mathematical problem and makes the next one easier to solve. Isn't mathematics entirely about solving problems? Even pure mathematics research, say trying to prove Fermat's last theorem, is about solving problems using knowledge gained by solving other problems.

Some people read as fast as possible, just memorizing new definitions and theorems, and simply assuming that they can do the exercises in the textbook if they had to.

For example, I had a relativity professor who said he had read an entire algebraic topology textbook in less than one week (on top of his other duties). I doubt he did many (if any at all) exercises from the book. He was brilliant and distinguished, so I assume that he could have solved most (if not all) of the problems if he wanted to.

 

[JeffKoch]

He was brilliant and distinguished, so I assume that he could have solved most (if not all) of the problems if he wanted to.

I doubt it. And even if he could, he'd be handicapped trying to solve harder ones because he didn't learn anything by solving the easier ones. It's kinda like skimming a textbook on Navajo and assuming you could speak the language any time if you had to.

 

[andytoh]

I doubt it. And even if he could, he'd be handicapped trying to solve harder ones because he didn't learn anything by solving the easier ones. It's kinda like skimming a textbook on Navajo and assuming you could speak the language any time if you had to.

It depends what your level of math ability already is. Let's say that the algebraic topology textbook was math level 4 (4th year university). The professor who read the algebraic topology textbook in one week probably had math level at least 6. Since 6 > 4, then yes he could have solved the problems in the textbook if he wanted to. So why was a level 6 person reading a level 4 book? Because he never learned algebraic topology formally before.

For example, some high schools teach foci, directrices, and eccentricities of conic sections, while some don't. A graduate math student who never learned it but realizes he now needs it could skim through that chapter to learn the definitions and theorems (and their proofs) and skip the exercises. He knows that he can do this grade 12 level math if he wanted to. He just needed to learn the topic; he doesn't need to waste time practising in it.

However, if a level 3 student is reading a level 4 textbook and skipping the exercises, then I think he will run into trouble.

 

[JeffKoch]

He just needed to learn the topic; he doesn't need to waste time practising in it.

I guess that's the crux of it, in my view practising = learning when it comes to math.

 

[Swapnil]

Well, I thought about it some more and I think that #4 is the better choice. Problem solving skills are very imporatant and all (and makes math more interesting) but I wouldn't spend a lot of time on them. I would spend just enough time so that I know how, in theory, I would arrive at the answer. Afterall, not all the topics you study in math have applications. Even amongst the one that do have an application, you are not going to be using every one of those applications.

 

[sphoenixee]

I chose choice 2.

New knowledge is great, but unless you know how to apply it and practice with it, you're going to forget it. My favorite encounters with math are through high school math contests, and in this arena at least, practice>>>>knowledge.

I'd say that 2 years ago I knew most of the formulas I know today, and all of the more used ones, but my ability to cleverly use them was certainly not as proficient. These last 2 years, I've greatly improved my problem solving ability, but have not learned much new knowledge, yet my mathematical contest performance has gone up dramatically. (Of course, all not-too-difficult HS contests, i.e. AIME and below can be solved using pretty basic maths, though very nice applications of them)

On the other hand, 1 year ago, I went on a stint to learn Markov chains and Leontief Input-Output Model, but I did not apply this. And what's the result? I can't remember Markov chains for my life right now XD

I think that if you want to use math to do something, practice>knowledge. Reading is for discussing and sounding smart, practicing and doing is for real.

Just my two cents,

sphoenixee

 

[andytoh]

I chose choice 2.

New knowledge is great, but unless you know how to apply it and practice with it, you're going to forget it. My favorite encounters with math are through high school math contests, and in this arena at least, practice>>>>knowledge.

I'd say that 2 years ago I knew most of the formulas I know today, and all of the more used ones, but my ability to cleverly use them was certainly not as proficient. These last 2 years, I've greatly improved my problem solving ability, but have not learned much new knowledge, yet my mathematical contest performance has gone up dramatically. (Of course, all not-too-difficult HS contests, i.e. AIME and below can be solved using pretty basic maths, though very nice applications of them)

On the other hand, 1 year ago, I went on a stint to learn Markov chains and Leontief Input-Output Model, but I did not apply this. And what's the result? I can't remember Markov chains for my life right now XD

I think that if you want to use math to do something, practice>knowledge. Reading is for discussing and sounding smart, practicing and doing is for real.

Just my two cents,

sphoenixee

If your goal is to do well in math contests, then for sure practising with problems will overshadow any new knowledge. However, once you are on your way to earning a PhD, the amount of knowledge required is just so great, that you will not have the time to prepare for contests anymore.