Mathematical Concepts Vs Problems

[abiyo]

This has been bothering me for a long time. I hope someone in this forum will help me clear my confusions. So the way I approach mathematics is by working carefully over the concepts. I reread the concepts till they become crystal clear.( This has been painful for some of my classes like real analysis but it has also been rewarding). I usually focus on questions after making sure I have no gaps in my concepts. However I always find out that when I do the questions, there are things I didn't really understand(I discover gaps in my conceptual framework). This frustrates me a lot. I assume that if one has mastered a concept, one should be able to do any kind of question. But recently I have been thinking whether I should do maths the other way round. That is start with the problems and go back to concepts. Are concepts in mathematics merely systems developed to discuss a class of certain problems? Could people describe me their approach in learning new mathematics;say topology,abstract algebra? (I am planning to go to grad school so I want to orient my thinking and approach the right way.)

Thanks a lot
Abiyo

 

[ZenOne]

I believe both are needed. I usually get a good understanding of a concept--do multiple problems to get an idea for the possible "curveballs" involved with the concept then re-read the concept to reinforce it after understanding the scope of the problems it includes.

 

[twofish-quant]

I assume that if one has mastered a concept, one should be able to do any kind of question.

Perhaps, but you need to do all sorts of questions in order to master the concept.

Could people describe me their approach in learning new mathematics;say topology,abstract algebra?

What works for me is to make be as concrete as possible. I learned to count when Bert and Ernie showed me the number of cookies. When I want to learn what a Banach space is, I look at examples and play with them.

 

[Delta2]

In my opinion the problem is due to our intuitive understanding and due to fact that our intuition might be wrong or incomplete. Usually there are more than the intuition tell us or focuses at. We usually focus on the "core element" of a concept and we don't look at the "peripheral elements". For example for the concept of a function, we usually focus on the function formula and don't give much attention to the domain and codomain.

So you have to both analyze the concepts and solve problems in order to correct and complete the intuition and master the core and the peripheral elements of a concept.

 

[Simfish]

What works for me is to make be as concrete as possible. I learned to count when Bert and Ernie showed me the number of cookies. When I want to learn what a Banach space is, I look at examples and play with them.

That's a really good idea. But where do you find the examples? Many leading textbooks have very few concrete examples of things like Banach spaces.

 

[abiyo]

Thanks all for replying. I still have a lingering question though. Say you have an excellent textbook on a topic you have never seen before. Do you look at the problems first? Do you read the material first? If you read the material, where do you know which part is more important? My teacher once told me in most textbooks what one needs to know is a fraction of the material presented. How do you know which part is important compared to others?

How many of the questions at the end of each chapter do you finish? What is a target percentage?(I am assuming proof based style questions not number crunching or inverting a matrix?

Is it good to find concrete ways to understand abstract ideas? Like drawing pictures, intuitive explanations and so on. Are not they conceptually deadly in the long run(since they shape the way you form conceptual framework)

Thanks a lot once again
Abiy

 

[chiro]

Thanks all for replying. I still have a lingering question though. Say you have an excellent textbook on a topic you have never seen before. Do you look at the problems first? Do you read the material first? If you read the material, where do you know which part is more important? My teacher once told me in most textbooks what one needs to know is a fraction of the material presented. How do you know which part is important compared to others?

How many of the questions at the end of each chapter do you finish? What is a target percentage?(I am assuming proof based style questions not number crunching or inverting a matrix?

Is it good to find concrete ways to understand abstract ideas? Like drawing pictures, intuitive explanations and so on. Are not they conceptually deadly in the long run(since they shape the way you form conceptual framework)

Thanks a lot once again
Abiy

I find that the best teachers have the ability to say what the entire book is about (its purpose) in one or two paragraphs. Another thing to note is the paragraph will be written in simple everyday language and not require a graduate degree to understand.

With regards to the level of abstractness, the above paragraph will explain something in the most abstract way, but still in simple terms.

With regards to representing ideas, there are tonnes of different ways that work for different people. If I was to say explain one of the roles of science to a lay person I would use a Venn diagram with the sub-circle representing knowledge that can be explained by current theory and the entire circle to represent the entirety of a field. From this you could go further in defining sub-fields and so on but you get the idea. By telling the student that you want the small circles (if you have multiple fields and competing theories) to grow into the big circle, you've pretty much explained one perspective relating to all science.

As for things like mathematical proofs, I tend to think that there is a great body of intuitive understanding of which unfortunately sometimes does not go into the papers or the books of the authors, and a lot of this is unfortunately lost or has to be rediscovered by the students.

 

[mathwonk]

This is hard to answer. But an interesting argument has been made by the Harvard physicist Eric Mazur, that students who really understand concepts can do problems, but not the other way around. This suggests that one should do the reading and then wrestle with well chosen questions that reinforce the concepts, before working traditional problems. But it takes an expert to devise the right questions. He has a wonderful lecture on youtube you can probably find.

 

[discrete*]

I find that there needs to be a tender balance between the two, and sometimes that's easier said than done. There are times where I would much rather just keep reading through the texts, and other times where I just want to sit down and put pen to paper. But there's got to be a balance between the two, in my opinion.

Thanks all for replying. I still have a lingering question though. Say you have an excellent textbook on a topic you have never seen before. Do you look at the problems first? Do you read the material first? If you read the material, where do you know which part is more important? My teacher once told me in most textbooks what one needs to know is a fraction of the material presented. How do you know which part is important compared to others?

I like to read cover-to-cover, if possible and beneficial. I find that if I skip chapters or even sections of chapters, I end up feeling malnourished and having to constantly looks backwards. That said, there are times where reading a book cover-to-cover is just out of the question for me; like when things are present in a book that I've already covered and am comfortable with, or when things are just out of my reach at the time being. This happens more often than not. I'm currently working through Stoll's Set Theory and Logic, however I've previously studied Logic and am comfortable enough with it that I'm only going through the set theoretic aspects of the book.

How many of the questions at the end of each chapter do you finish? What is a target percentage?(I am assuming proof based style questions not number crunching or inverting a matrix?

This is a tough question. My answer is that it varies greatly. Again, I'll use Stoll's Set Theory and Logic for the simple reason that I'm currently studying it. If I look back in my notes, it seems like I complete between 40 and 90 percent of the exercises. The ones that are on the low end are usually chapters with either trivial exercises, true of false questions etc. However, even in such a case, I make sure to complete at least 1 or 2 exercises from every section of exercises (i.e.: maybe 3 or 4 true or false, maybe some computational ones etc.). I go heavy on the proofs, always. I am never satisfied with my proof techniques, so I seek them out. A lot of time I will try and rework proofs that are presented in the examples but from a different perspective -- this helps a lot, not just mechanically, but conceptually.

Is it good to find concrete ways to understand abstract ideas? Like drawing pictures, intuitive explanations and so on. Are not they conceptually deadly in the long run(since they shape the way you form conceptual framework)

I find that I like to learn the idea in all its abstractedness. I never draw diagrams and I loathe being taught in an "intuitive manner". If I'm looking for real-world connections it's only because I find it interesting to do so. And this is very rare, I do this maybe 10 percent of the time.

I was in a class where we had an open discussion board, and many students seemed to be having trouble with some of the more abstract concepts. I was able to give them real world examples (which they all said helped them very much), but only because I had a good understanding of the abstract concept. However, if the Professor had presented the real-world examples first, I'd have been the one in need of help. I'd much rather live and operate at the most abstract level that I can -- I personally don't like many fields of applied maths for this reason. I just can't overcome the feeling that using a pure subject to describe something else is sort of watering down the purity of the subject.

These are just my opinions, and my methods of study. I've found that most people differ from me in these regards, but I've learned a lot from many different styles of study and found what works best for me. That's what I'd encourage you to do. If you feel like you'd better understand the material conceptually, than go ahead and emphasize that. If you feel like you need to do droves of exercises, by all means, do it. What's important is that when you close the book, you are comfortable in what you've learned and could convey it to others if need be. After all, isn't that the real test of knowledge -- our ability to pass it along? Good luck.

 

[DrummingAtom]

This is hard to answer. But an interesting argument has been made by the Harvard physicist Eric Mazur, that students who really understand concepts can do problems, but not the other way around. This suggests that one should do the reading and then wrestle with well chosen questions that reinforce the concepts, before working traditional problems. But it takes an expert to devise the right questions. He has a wonderful lecture on youtube you can probably find.

Wow. That video blew me away. My last physics class was watching my professor do endless problems in class. We would touch on some concepts and occasionally do peer instruction but the concepts never sunk in. Then our professor would give a section of 10 multiple choice questions on the exam. That multiple choice section was always the worst part of the exam on average for the class. My professor would then admit that we should spend more time on the concepts in lecture. We would do concepts for one class and then go right back to only doing problems until the next exam.

 

[abiyo]

Thanks Chiro, discrete*, mathwonk and DrummingAtom for your replies.