How Much Math Should You Master Before Moving On?

[robertjford80]

When you study math what % of the material do you try to understand before moving on? I suppose there are big differences between mathematicians and physicists. I would imagine that if you're a mm then you would probably want to understand more than 95% before moving onto the next chapter. As for myself, I'm extremely impatient. I want to know jump from Classical Physics to Quantum Field Theory quick, consequently I skate by with just enough so that I'm not lost. I realize this is a fault but I'm just not patient enough. My plan is to get the basic gist of the material, then plug in the gaps of my understanding later. I get bored when I get bogged down in an intractable problem. It's so much easier for me to just drop the confusion and move on. So I probably understand about 80% of the material before I move to the next chapter.

 

[Whovian]

I tend to try to understand the concepts in a section, and maybe memorize the really important and hard to derive formulae, but write down all the little things stuff's used for, such as triple integrals for Centre of Mass.

 

[Stephen Tashi]

As for myself, I'm extremely impatient.

I am also.

My plan is to get the basic gist of the material, then plug in the gaps of my understanding later.

When it comes to studying for courses, keep in mind that there is a difference between understanding and drill. You can understand how to do material but not be able to do problems quickly and accurately. If your achievements are going to be measured by tests, you must drill as well as understand.

I also tend to open math books in the middle. (Fortunately, I'm not student any longer.) However, there is a danger to this because you can fool yourself into thinking that you understand material. This is likely to happen to people who don't pay close attention to the legalistic aspects of mathematics, such as the precise wording of definitions. If you tend to "put things in your own words", you can skim through a math book building your own little mathematical world and find out later that what you dreamt up is wrong. Get the definitions straight. Aim for 100% on that.

 

[robertjford80]

I put the definition in my own words because I don't like the way mathematicians word their definitions. I'm not a student either so I don't have to do the endless drills so that I can Ace a test. Thank God for that.

 

[Stephen Tashi]

I put the definition in my own words because I don't like the way mathematicians word their definitions. I'm not a student either so I don't have to do the endless drills so that I can Ace a test. Thank God for that.

Then I think you shouldn't skip much material because you have the habits of many impatient people who do self-study. They formulate their own personal interpretation of technical material and their picture of what it says turns out to be full of misconceptions.

 

[oddjobmj]

The way definitions and proofs are written serves an important purpose. Sure, they're sometimes not enjoyable to read or memorize, but it is necessary to build an accurate picture of the model. There really isn't a better way to summarize these details as they're there for a reason.

Often some of the details of a given proof are not useful in the problems that you will do in class so you might never use that little bit directly. However, again, all those little details are very important to building an accurate model of these ideas.

Regarding the %s you're looking for it's hard to answer in the way you worded it for me. I tend to work backwards. I look at the big idea, ponder for a while about what implications there could be, and then go back to fill in the details.

This might contrast with what Stephen Tashi is recommending in the sense that you will start with incomplete models but it comes with a major benefit; you will build a more comprehensive and resilient mental model this way than if you focus on only the smaller picture in order of progression.

 

[camillio]

However, there is a danger to this because you can fool yourself into thinking that you understand material.

I double this. When it comes to some more difficult subjects (e.g. martingales) I sometimes have several "true" notions of what this or that means. This notions develop and change, as "my ideas" don't fit the theory, but it is quite easy to accept false idea of understanding.

This is likely to happen to people who don't pay close attention to the legalistic aspects of mathematics, such as the precise wording of definitions. If you tend to "put things in your own words", you can skim through a math book building your own little mathematical world and find out later that what you dreamt up is wrong. Get the definitions straight. Aim for 100% on that.

Again my case. With CS background, I tended to understand math in my own way, rather than to accept the only precise wording. It was very fine for someone studying CS. But I've started with pure math a year ago (both fortunately and unfortunately not as a student) and still struggle very much with that relaxed approach.


To answer the original question - I move to next chapter only when I feel that I understand the actual one. That means, I can write definitions/lemmas/theorems with only minor or none faults and the same applies to proofs. Being impatient, I still have the terrible bad habit to examine only few problems.

 

[20Tauri]

Are you talking about work for classes or independent study? I don't really think of it in terms of what percent of the material I understand. I take notes in class, learn a bit more by bludgeoning my way through the homework (and going to office hours and/or working with a friend), and then study for the exams. I suppose since I aim for As and Bs, I am hoping to understand >85% of the material? I'm a math student, for what it's worth.

 

[robertjford80]

Are you talking about work for classes or independent study?

Independent study

I don't really think of it in terms of what percent of the material I understand. I take notes in class, learn a bit more by bludgeoning my way through the homework (and going to office hours and/or working with a friend), and then study for the exams. I suppose since I aim for As and Bs, I am hoping to understand >85% of the material? I'm a math student, for what it's worth.

You see I don't get grades so the temptation to just skip over something that I don't understand is simply unbearable. Plus I hate moving at a snail's pace. I like to know that in about 3 months I will be at level X, that helps keep me excited.

 

[20Tauri]

I don't know how to advise you for purely independent study--the only successful independent work I've done in math was under the guidance of a professor. If you have access to a professor or a friend who's taken the class, maybe you could get ahold of a syllabus and/or some old exams so that you'd have a way of judging your progress?

 

[robertjford80]

I don't know how to advise you for purely independent study--the only successful independent work I've done in math was under the guidance of a professor. If you have access to a professor or a friend who's taken the class, maybe you could get ahold of a syllabus and/or some old exams so that you'd have a way of judging your progress?

Well, the best way for me to judge my progress is to find out if I understand the steps in arriving at a mathematical answer. As of now I have a pretty good grasp of what's going on but sooner or later my bad habits will catch up with me i think

 

[dh363]

I think for math, you really need to make sure you understand everything, in terms of the rigorous language. Thinking about it in terms of easier, more relaxed definitions I find helps conceptually, but does not give one the tools to solve any proof problems. One needs to be able to think abstractly and let go of the crutches of attaching general, abstract concepts to intuitive ones. I self study a lot, and I generally have to understand fully all the definitions, all of the concepts, and all of the proofs before moving on. I also like to go back and look over every theorem and see if I can reproduce a quick proof sketch of the theorem. If I can't, I read over the theorems again and try to really gain an intuition for why it's proven the way it is.

 

[robertjford80]

I also like to go back and look over every theorem and see if I can reproduce a quick proof sketch of the theorem. If I can't, I read over the theorems again and try to really gain an intuition for why it's proven the way it is.

yea, i could see how if i were wanting to be a mathematician or a theoretical physicist i would do that, but i simply don't have the time for that level of depth

 

[Harrisonized]

If you're studying math just for physics, you definitely don't need the rigor.

For example, what physicist ever uses the definition of continuity and smoothness to use calculus to solve physics problems? Have you heard of any physicist that uses the delta-epsilon limit to solve physics problems? What physicist cares about the details of the various integral transforms, or even a rigorous proof of integration? Solving integrals in the form ∫f(x)/[exp(x)-1] dx? Forget complex analysis. Just ask a mathematician. Or go to your computer and Google up Wolfram|Alpha.

Forget about proofs. Just make sure the material makes sense to you. If you need a concept, look it up, get the important results, and learn it on the spot. Physics is about applying math to real situations, so your solution is tested through experiment rather than rigor.

However, if you're like me, you'll want to actually learn the math and maybe some easy proofs, but nothing that requires too abstract of thought. Although it pissed me off, the TA I had last quarter, a second year grad student, had no idea what the Fourier transform is. All he did was quote results and tell us that the rest of it isn't "too complicated". It was entertaining to attend his section only because he'd make a fool of himself with his illegal mathematics every time.

 

[robertjford80]

If you're studying math just for physics, you definitely don't need the rigor.

For example, what physicist ever uses the definition of continuity and smoothness to use calculus to solve physics problems? Have you heard of any physicist that uses the delta-epsilon limit to solve physics problems? What physicist cares about the details of the various integral transforms, or even a rigorous proof of integration? Solving integrals in the form ∫f(x)/[exp(x)-1] dx? Forget complex analysis. Just ask a mathematician. Or go to your computer and Google up Wolfram|Alpha.

Forget about proofs. Just make sure the material makes sense to you. If you need a concept, look it up, get the important results, and learn it on the spot. Physics is about applying math to real situations, so your solution is tested through experiment rather than rigor.

However, if you're like me, you'll want to actually learn the math and maybe some easy proofs, but nothing that requires too abstract of thought. .

I mostly agree with you. Like everyone else I have limited resources and I have to pick and choose what to study very carefully. I used to like proofs because as a philosopher I find them interesting, but proofs are just too hard for me, they're written in a highly opaque language so now I mostly skip over them but a few proofs I understand.