Building Intuition for Math Problem Solving

[JoeTheKid]

Hello brand new to the forums and I just have one question.
So I decided to follow mechanical engineering as my degree of choice and I stand firm with my decision, I'm good at building things and finding unique solutions to problems, but the one thing I'm bad at is math... I'm a little above average when it comes to math, but I was wondering if anyone on these forums has any hints or tricks to building strong math intuition or any tips on how to study for math, I would greatly appreciate it.

 

[berkeman]

Hello brand new to the forums and I just have one question.
So I decided to follow mechanical engineering as my degree of choice and I stand firm with my decision, I'm good at building things and finding unique solutions to problems, but the one thing I'm bad at is math... I'm a little above average when it comes to math, but I was wondering if anyone on these forums has any hints or tricks to building strong math intuition or any tips on how to study for math, I would greatly appreciate it.

Welcome to the PF.

I was not very good at math in high school, but I also did not study very much and got good grades. In college I found I had to study much more to get good grades, especially in math. I typically would try to do all of the problems in each chapter that had answers in the back of the book, in addition to the homework assigned problems.

What math courses have you had so far? How much did you study in each?

 

[occh]

The problem with most mathematics courses is that they are taught theoretically with elementary or no applications. In order to get an intuitive understanding of the underlying concepts, I usually do research on the physical applications of the topic to have a mental picture of what it represents. (Obviously once you get beyond the Calc sequence this will be more difficult) When it comes to studying for math, don't bother reading the textbook... DO THE EXAMPLES and refer to the textbook as you need to. Actually going through the procedure as opposed to memorizing the formula is the most important thing. Furthermore, learning how a formula is derived rather than just the end product helps understand the purpose and meaning of what you are doing, and makes the formula easier to recall in the case that you forget the it on an exam or quiz.

 

[JoeTheKid]

Welcome to the PF.

I was not very good at math in high school, but I also did not study very much and got good grades. In college I found I had to study much more to get good grades, especially in math. I typically would try to do all of the problems in each chapter that had answers in the back of the book, in addition to the homework assigned problems.

What math courses have you had so far? How much did you study in each?

Regarding just math, I'm taking at the moment trigonometry and physics 201, Trigonometry comes clear to me during lectures and homework, but I find I have to study and practice physics for hours just to get a shaky understanding of the chapters content, and I read and reread the chapter material from the book but I can never wrap my head around the concepts.

 

[462chevelle]

Regarding just math, I'm taking at the moment trigonometry and physics 201, Trigonometry comes clear to me during lectures and homework, but I find I have to study and practice physics for hours just to get a shaky understanding of the chapters content, and I read and reread the chapter material from the book but I can never wrap my head around the concepts.

For me reading the book is useless, the best thing for me is to watch demonstrations of what is actually happening. Walter Lewin on Mit OCW is a good resource.

 

[Stephen Tashi]

Having intuition is fine, but you shouldn't ignore "formalism". Work on all the skills needed to negotiate math and science. Intuition is only one of them.

 

[chandrasekaran]

building math intuition is based on understanding the things in mathematical way... it need more practice and patient..

 

[axmls]

The best phrase to sum this up, in my opinion, is that math is absolutely not a spectator sport. Watching someone solve a problem and saying "Ah yes, I understand what they did" and actually struggling through a problem until it finally clicks are two entirely different things. It's the latter that builds your intuition. With enough practice, you begin to notice all of the recurring ideas and techniques.

 

[mathwonk]

@462: when you say reading a book is useless, do you realize you may be saying you do not yet know how to read a book? this is definitely a skill worth learning.

 

[homeomorphic]

Read Visual Complex Analysis and do some of the exercises. And practice imagining the arguments and recalling them without looking. Best thing you could ever do for your intuition.

Having intuition is fine, but you shouldn't ignore "formalism". Work on all the skills needed to negotiate math and science. Intuition is only one of them.

Well, formalism is needed, too, but less so for an engineer. And you shouldn't take it for granted that people know what they are doing when they present things very formally. On the contrary, I've found it very productive to take it for granted that people usually present things too formally, and that there is a better way to understand things, which can be found by browsing around for the right book and thinking for yourself or talking to the right person, rather than taking what the books say at face value. It is usually not necessary to be so formal (actually, you can be as formal as you want, without raising my objections, as long as you ALSO explain informally what you are doing, if it's not immediately obvious). My experience has always proven that, up until I got to research level math, where I ran into trouble, but I don't think the OP is headed for math research. One of the big difficulties you come across if you are more intuitive is that you have to "swim against the stream", to quote V.I. Arnold's point of view on this issue, because most mathematicians these days are more formal/algebraic. So that artificially makes things more difficult for a more intuitive thinker. I'm not saying intuition is the only thing there is, but it's a kind of glue that holds things together, without which, you just end up forgetting things as soon as you learn them. That's why I think intuition is the most important thing, by far. Having pictures or gut feelings of how things work is what makes it stick in your mind (and for me is the main thing that gets me interested in the subject in the first place).

I think it's partly a matter of style, too. There's some leeway in how formal a mathematician you choose to be. In the past, very intuitive, non-rigorous people like Witten have had a big impact, and on the other hand, more formal people like Weierstrass have also made their mark (albeit more boringly).

If you are going to be a mathematician, it is true that you have to come to terms with the logic of doing proofs and writing things down formally at some point. It would be a shame to let the paranoia of being wrong overtake everything that's fun about math. You need the paranoia if--and only if--you are going to be proving serious theorems. If you're just an engineer, perhaps caution is enough, rather than paranoia.

 

[462chevelle]

@462: when you say reading a book is useless, do you realize you may be saying you do not yet know how to read a book? this is definitely a skill worth learning.

I say this because for me, I can read through a chapter of a physics or math book and get lost in all the notation and symbols. Watching or interacting in a lecture makes it crystal clear. It doesn't seem to make a difference whether I read the chapter in the book or not. I still comprehend everything through lecture. Most books leave steps out or write to much junk in them to even get what you're trying to learn out of them. It could be that I usually don't get to sit down with my textbook until real late at night, but I have no way to tell if that is the case. That's not to say everyone has this issue, but I do. I just don't pick up on enough things just reading the book to do the problems in the book.

 

[axmls]

I say this because for me, I can read through a chapter of a physics or math book and get lost in all the notation and symbols. Watching or interacting in a lecture makes it crystal clear. It doesn't seem to make a difference whether I read the chapter in the book or not. I still comprehend everything through lecture. Most books leave steps out or write to much junk in them to even get what you're trying to learn out of them. It could be that I usually don't get to sit down with my textbook until real late at night, but I have no way to tell if that is the case. That's not to say everyone has this issue, but I do. I just don't pick up on enough things just reading the book to do the problems in the book.

I used to have the same problem. Basically, I had to sit down over a summer and force myself to follow the book very slowly. My book of choice was Spivak's Calculus. I did this after taking Calc I and II. Don't underestimate the ability to learn well from reading the textbook. It definitely is a skill that you can learn.

Now, maybe that doesn't count, because it is a very well-written book. But I have noticed an improvement in my ability to learn from textbooks since then.

 

[JoeTheKid]

Sorry I've been quiet on this thread but, narrowing down the area i want to really improve is translating the understanding i have of the problem into math. For example i can read a physics problem and i will understand what is happening in the problem conceptually but i can't translate my understanding of what is happening into the math, and as for a range of where i start to have this problem is the intermediate problems in my courses book. Hope that makes enough sense to everyone.

 

[axmls]

Have you taken integral calculus yet? Some physics concepts become much more clear after learning new mathematical concepts. An example for me was the displacement function for a constant acceleration, $$x(t) = \frac{1}{2}at^2 + v_0 t + x_0$$

a formula that seemed completely arbitrary when I took algebra-based physics. After learning calculus, this seemed like almost an obvious formula, as the position is simply the integral of the velocity function, which is the integral of the acceleration function.

What I mean to say is that perhaps you'll develop a better intuition on how to go about using the math in these problems as you learn more math. You will recognize when you need to use certain techniques, etc. Of course, the best way to ensure this is to keep practicing physics problems.

 

[Nikitin]

Read every proof and understand the reasoning, try to look at the big picture, find the connections between the theorems and formulas and, BY FAR most importantly:

work work work work work work work work work work work work work work work work work work work work work work work work work work work
work work work work work work work work work work work work work work work work work work work work work work work work work work work

 

[MidgetDwarf]

Rule of thumb. Have at least one other textbook of the same subject matter to use as a reference. Sometimes explanations of certain ideas are presented more clearly in other books. Research good books and have them early before the semester starts. Get your feet wet by doing at least the first 2 chapters of a math/physics/what ever before you have to take said class.

Maybe you are reading to fast or extremely impatient? What works for me:
1. Do a power read on a section thinking of what sounds important and how the ideas of that section tie together.
2. Read a bit slower and actually do the examples and question why the math steps look like they do.
3.Now I grab a notebook (I prefer journals such as Moleskine, Gallery Leather, or even Clairefontain (cheapest of these journals)) and analyze the text by writing some key ideas in the journal slowly. The writing slows your reading down so you can absorb more information. Then after you feel ready, tackle some problems and repeat if necessary. When you have time condense the notes you made in your journal.

I choose these types of journals because they are sturdy, will last me a longtime, I can write on the spine for easy reference, and because they cost more than a 99 cent notebook I am careful what I put intomthem.

It may seem tedious at first but overtime it becomes second nature. The faster you practice learning how to read a math/science book, the faster it will be to learn new things on your own. After college how else are you going to learn new things?

 

[jaskamiin]

Solve solve solve solve the problems. And then solve more. As others have stated - it's not enough to just watch someone solve a mathematics problem. Do not fall into the trap of looking at a problem, thinking to yourself - "Oh, I know how to do this." - and then skipping it.

Mathematical intuition is something that for most people needs to be trained. And it sounds like your mind is also like most (read: mine) in that you probably want immediate gratification by waking up tomorrow and "knowing how to math". Simply take it easy, and do problems. Start with easy basic ones until you understand how to algorithmically (or heuristically) solve any given problem of the type, and then move on to the 'logical' next step (for example, knowing trigonometric identities backwards and forwards and then moving on to solving derivatives).

 

[ShayanJ]

Let's hear from a great mathematician John von Neumann:

• Young man, in mathematics you don't understand things. You just get used to them.
  • Reply to Dr. Felix T. Smith at Stanford Research Institute who had said "I'm afraid I don't understand the method of characteristics." —as quoted in footnote of pg 208, in The Dancing Wu Li Masters: An Overview of the New Physics (1984) by Gary Zukav.

So practice and practice and practice, until you get used to it!
By practice, I don't mean just solving problems which lead to a number. You should sometimes practice deriving some formulas and proving some theorems.

 

[austinuni]

Sorry I've been quiet on this thread but, narrowing down the area i want to really improve is translating the understanding i have of the problem into math. For example i can read a physics problem and i will understand what is happening in the problem conceptually but i can't translate my understanding of what is happening into the math, and as for a range of where i start to have this problem is the intermediate problems in my courses book. Hope that makes enough sense to everyone.

Are you talking about being given a problem description that is written in plain English, and then translating that problem description into a set of mathematical equations that could be used to solve some desired numerical quantity, or some other function of the equations? Sort of like those problems you were given in HS algebra, where Train A leaves station B at such-and-such time going speed C, and when will it meet another Train leaving a different station at so-so time (using a very simple example for illustration)? If so, then I can see how that can be non-intuitive to translate these kinds of problems into the appropriate math.

I think it is a multi-step process that needs to be methodically broken into parts. You can't just expect to come up with the equations after one read of the problem. Start with identifying the pieces of the problem description -- isolate them into separate parts. Don't try to put them all together yet. Identify each part of the problem and try to describe mathematically how it behaves. Assign basic units next to each of these parts, such as those for speed, acceleration, pressure, temperature, etc. Does that make sense? Or were you trying to solve a different issue?