Independently Studying a Math

[yUNeeC]

Greetings,

Over this summer I plan on independently studying Calculus II and III because I am behind in my math major (just switched from biology)

As a result, I plan to double up on maths with 2 per semester. My other courses are physics and chemistry related, so this is no cake-walk schedule, and I feel learning these two will help immensely. I plan on taking Calc II and Transition to Advanced math next semester, and Calc III and linear algebra the following semester...already knowing half of my maths for each semester should prove beneficial in me still having free-time.

What I am wandering is, what is a good method to go about independently studying these two subjects? Should I try to cover EVERYTHING, or just get the essentials down? For instance, in Calc II, should I just try to master the different methods of integration? Or should I do a broad overview of the entire subject? I am going to make an effort to "master" the entire course, but I have a job over the summer and don't know how realistic this is. I also want to make progress in Calc III because I don't know how much free time I will have over Christmas break.

Any tips?

Thanks.

 

[Skins]

It depends. In addition to formal college classroom learning (I got my degree in Math) I also did a lot of self-learning. Since I am very interested in math in the pure sense I wanted to know everything... for example not just how to Integrate but how it all works and why, So I tend to delve into the most abstract and definative text paying close attention to the theorems, proofs, etc. Of course it helped me to learn to "think mathematically". For example, the mechanisms and motivations behind many of the proofs seemed strange to me until I learned how to do proofs, i.e. the underlying logic that the vast majority of proofs follow.

My approach to learning however was quite time consuming and quite thorough, perhaps at times too thorough for my own good.

The way a lot of teachers explained it to me goes like... the first time around focus on getting the basic ideas down. If you're learning Calculus for example, learn the basic ideas behind how to find limits, derivatives, integrals, etc and what they mean in both a geometric and analytic interpretive sense. Then, you can go back later on and fill in the finer and more abstract details later on. Although this was not the approach that I took it seems to be the method favored and recommended by most instructors whom I have spoken.

 

[zhentil]

I've taught calculus several times, and I would say that your chances of learning calculus on your own without spending too much time on it are very low. Some people are especially gifted at math and can just pick up calculus by seeing a couple examples, but most people find calculus very difficult even in a classroom environment. Here's an example: with integration by parts, some people can look at it and say, "This is cake! I already know the product rule for differentiation." Most people, on the other hand, fumble around and look for mnemonics for which thing to let the u and v be. So it depends which type you are. If math comes really naturally, learning calculus on your own is a breeze.

 

[yUNeeC]

I've learned partial fractions pretty well thus far, and have made progress on the integration by parts technique. Just some of the more outside the box questions elude me. I see the explanation and I learn...but am kind of discouraged by rarely being able to think them through completely (I usually get part of the way through them, and then butcher things). Are the outside the box problems something someone who is good at math should naturally be able to answer, or do they just serve as learning experiences? Is it possible to understand ~90% of these?

 

[ImAnEngineer]

Are the outside the box problems something someone who is good at math should naturally be able to answer, or do they just serve as learning experiences? Is it possible to understand ~90% of these?

I think that's different for everybody. When I study a new subject in math I tend to pick some problems and then look up the answers. Then I study the answers quite thoroughly, and try to understand every single step that is taken and I'll try to get a grip on it. I then ask myself: 'how could I have come up with this myself? What are the characteristics of this particular problem? What would the answer have been if this or that were different? Would I have been able to solve it using the same method? etc..'. Then I write the solution to the problem down without taking another look at the answers to make sure I get it. This way you can teach yourself how to deal with that kind of problem, instead of having to mess around with it, make wrong assumptions, continue to do false steps and getting confused and frustrated. If you've done some problems like this as an 'introduction' it's much easier to do other 'out of the box' problems because it's easier to see the bigger picture.

But hey, that's just me...

 

[Wellesley]

Over this summer I plan on independently studying Calculus II and III because I am behind in my math major (just switched from biology)

As a result, I plan to double up on maths with 2 per semester. My other courses are physics and chemistry related, so this is no cake-walk schedule, and I feel learning these two will help immensely. I plan on taking Calc II and Transition to Advanced math next semester, and Calc III and linear algebra the following semester...already knowing half of my maths for each semester should prove beneficial in me still having free-time.

What I am wandering is, what is a good method to go about independently studying these two subjects? Should I try to cover EVERYTHING, or just get the essentials down? For instance, in Calc II, should I just try to master the different methods of integration? Or should I do a broad overview of the entire subject? I am going to make an effort to "master" the entire course, but I have a job over the summer and don't know how realistic this is. I also want to make progress in Calc III because I don't know how much free time I will have over Christmas break.

Any tips?

Thanks.

If you are planning to major in math, I would think it would be a good idea to understand as much as possible. It will only build in later semesters.

 

[yUNeeC]

I think that's different for everybody. When I study a new subject in math I tend to pick some problems and then look up the answers. Then I study the answers quite thoroughly, and try to understand every single step that is taken and I'll try to get a grip on it. I then ask myself: 'how could I have come up with this myself? What are the characteristics of this particular problem? What would the answer have been if this or that were different? Would I have been able to solve it using the same method? etc..'. Then I write the solution to the problem down without taking another look at the answers to make sure I get it. This way you can teach yourself how to deal with that kind of problem, instead of having to mess around with it, make wrong assumptions, continue to do false steps and getting confused and frustrated. If you've done some problems like this as an 'introduction' it's much easier to do other 'out of the box' problems because it's easier to see the bigger picture.

But hey, that's just me...

Good idea, thanks.