Math/CS Majors: Best Books to Learn Theory & Concepts Easily

[Sduibek]

Math/CS majors and grads: Which math books do you recommend for theory and concepts?

I am retaking college pre-calc Trig and still struggling (C on the first exam). I have yet to ever receive from an instructor or assigned book the following:

1. Why I should care about math / how not to hate it
2. What math actually is (in other words what is the purpose and essence of the language of mathematics)
3. High-level concepts and theory (versus a barrage of exercises and proofs)
4. Words instead of just numbers and diagrams
5. How to make it not dry and boring and really hard
6. How to approach a problem that is baffling you and feels to be above your skills / knowledge.

I realize many of the above are similar but I separated them to (hopefully) make replies easier.

I love programming, and I like understanding how things work, creative design [I draw, make music, write comics and make video game mods], and learning, but nothing has ever motivated me to not think of math as a horrible black hole that sucks away my emotional and mental well-being while flipping me off.

Thank you very much in advance.


EDIT: For the record I'm asking my instructors directly if they have ideas for such books, just in case you think this post is 100% inaction whining :P

 

[mathwonk]

you know that's pretty much how i feel about questions like this.

 

[berkeman]

I am retaking college pre-calc Trig and still struggling (C on the first exam). I have yet to ever receive from an instructor or assigned book the following:

1. Why I should care about math / how not to hate it
2. What math actually is (in other words what is the purpose and essence of the language of mathematics)
3. High-level concepts and theory (versus a barrage of exercises and proofs)
4. Words instead of just numbers and diagrams
5. How to make it not dry and boring and really hard
6. How to approach a problem that is baffling you and feels to be above your skills / knowledge.

I realize many of the above are similar but I separated them to (hopefully) make replies easier.

I love programming, and I like understanding how things work, creative design [I draw, make music, write comics and make video game mods], and learning, but nothing has ever motivated me to not think of math as a horrible black hole that sucks away my emotional and mental well-being while flipping me off.

Thank you very much in advance.


EDIT: For the record I'm asking my instructors directly if they have ideas for such books, just in case you think this post is 100% inaction whining :P

Are you really sincere in your first question? If you don't know why math is important, or refuse to believe it, that is going to be a pretty hard barrier to get past.

Are there any free tutoring resources available to you at your college? Often the honor societies will have some free tutoring sessions where you can attend and ask some questions. It would probably help you to talk with some mentor type person who can help you with some of the problems you are struggling with -- they can often start to show you the cohesivness and utility of the math you are learning. They can also often help you learn tricks that make it easier to remember and apply the math concepts you are learning.

One thing that I find helps a lot is to do as many word problems as you can. Word problems force you to think "big picture" more in setting up math problems, and it won't seem as dry as it sounds like it is for you now.

 

[Sduibek]

I don't necessarily mean I don't know why math is important, more that that's my emotional reaction -- for example when I'm at the pounding-head-on-desk or brain-turning-to-pudding point, the unwilling mantra "don't care, don't care, don't care" starts running through my head. Not sure how else to explain it.

I think I will look into tutoring for sure.. there's some free tutoring resources at my school that have been marginally helpful, but to be honest if I have to pay someone to get my brain into the groove that's not really a bad deal.

Re: tricks, that's actually the main thing I'm looking for. Things like, on crazy-looking XYZ equation asking to verify a proof, do you first combine like terms, factor, rationalize, etc. There are cases when one of these in certain order is MUCH easier but I really would like to learn a system for more efficiently knowing which ones to try first. Otherwise it's really just a more complicated version of doing guess-&-check while trying to factor something like 2x^2 -8x +20 which of course is ridiculously inefficient. I know how to do all that algebra stuff but it's never clear to me which ones are preferable systematically in which scenarios.

 

[berkeman]

There are definately tips and visual tricks that help when doing different kinds of math. In trig, I usually show folks who I'm helping how to draw the sin, cos and tan graphs for the first period (zero to 2*Pi) above each other on a 3-horizontal-axis graph. I use that graph to help with lots of problems involving relationships between the various functions.

I also try to bring most concepts back to physical situations if possible, since that helps folks to internalize and visualize what is going on. As you get farther along, like in differential calculus, you can do lots of things with word problems and diagrams to show what is going on.

Hang in there!

 

[micromass]

Re: tricks, that's actually the main thing I'm looking for. Things like, on crazy-looking XYZ equation asking to verify a proof, do you first combine like terms, factor, rationalize, etc. There are cases when one of these in certain order is MUCH easier but I really would like to learn a system for more efficiently knowing which ones to try first. Otherwise it's really just a more complicated version of doing guess-&-check while trying to factor something like 2x^2 -8x +20 which of course is ridiculously inefficient. I know how to do all that algebra stuff but it's never clear to me which ones are preferable systematically in which scenarios.

The problem is that most things in mathematics are essentially guess-and-check things. There are some situations where a nice algorithm can be given: do this first and that second and it always works, but this is rarely the case.

To know what trick to apply when requires exercise. Lots of exercise. After a while, you can kind of "see" what you need to do. But this is a lot of hard work, and you got to be willing to give it a lot of effort.

You are not interested in math (and science, I assume), so that makes things harder. I don't know what to say to make you interested. You probably know that math is useful somewhere later in life, even though it is not easy to see now.

Let me tell you why I find math exciting, maybe it is useful to you.
I like to read books and make up stories myself. The author of a book invents a whole new world with awesome characters and wondeful creatures. He is free to invent whatever he wants. But after the invention, he is bound to rules. He can't just change a male character to a female character, for example.

The same is true for math. In math, you get to make up your own wonderful world. You are free to make up whatever you want. If you want to invent a world where xy-yx=-1, then you are free to do so! But after you set the rules, the world takes on his own personality and his own shape. You made the rules, but afterwards you have essentially no control over what happens. You need to discover the laws of the created world. So you have to set up a journey to discover everything about your universe that you can. Sometimes this is easy, sometimes this is hard. In this sense, math is a form of art.

This interpretation of math is not taught in high schools. They reduce it to mindless computations and tricks. They are basically ruining math.
I don't blame you for hating math, I would probably hate it too if I were in high school now. But you need to persist. Once, the day will come that you will need that math somewhere. And then you're going to be glad you know it!

 

[Sduibek]

You are not interested in math (and science, I assume), so that makes things harder. I don't know what to say to make you interested. You probably know that math is useful somewhere later in life, even though it is not easy to see now.

I actually enjoy it when I'm getting the answers. Sometimes I look kind of autistic or something because when I'm chugging through things in my brain I tend to mumble to myself, rock back-&-forth and stare at the ceiling, but when I'm getting the right answers I feel smart and like I'm accomplishing something. It tends to be "heck yeah! right answer, booyah!" for each problem several times until I hit a problem that makes my brain explode. Then back to the booyah problems. It's a pretty weird experience.I'm looking forward to getting those books from Amazon. I've got Student (Prime) so I'll get them all by Thursday :) If it's any interest to anyone who reads this thread (present or future) here's what I got:Master Math: Trigonometry
Ross, Debra Anne

Schaum's Outline of Trigonometry, 4th Ed.
Moyer, Robert

Trigonometry Demystified, 2nd Ed.
Gibilisco, Stan

Functions and Graphs (Dover Books on Mathematics)
I. M. Gelfand

Easy Outline of Trigonometry
Ayres, Frank

Trigonometry
Gelfand, I.M.

Trigonometry For Dummies
Sterling, Mary Jane

The Complete Idiot's Guide to Precalculus
Kelley, W. Michael

2500 Solved Problems in College Algebra and Trigonometry (Schaum's Solved Problems Series)
Schmidt, Philip A.

To know what trick to apply when requires exercise. Lots of exercise. After a while, you can kind of "see" what you need to do. But this is a lot of hard work, and you got to be willing to give it a lot of effort.

This actually was very helpful to read. If one part of it (or perhaps a major part of it) is just doing a ton more problems/exercises, hey at least I know what my game plan is now.

One thing I did two quarters ago (Precalc I) studying for the final was to go over all my tests and copy them onto normal paper, then do every test until I could get 100% on them. I got an A- in that class, which I think is part of why I was so crushed and .. almost ashamed I think, emotionally due to failing Precalc II (Trig) so badly. I got 20% or 50% on every test in that class the first time around.

This interpretation of math is not taught in high schools. They reduce it to mindless computations and tricks. They are basically ruining math. I don't blame you for hating math, I would probably hate it too if I were in high school now.

I actually read a blog post about this while searching for answers to my dilemma.. something about the CS curriculum being all wrong because it makes math boring, something like that. I'm not in high school but I'm going back to school after 10 years of absence, which is pretty much the same thing ;) Going to college for the first time at 26 (now 27 since it's my 2nd year technically) is rough. But, I'm Bipolar so what are yougonnado. While most people were in college I was partying and trying to kill myself, then getting "psychiatrically sober" and reconstructing my psyche. Things like this just remind me that I need to keep going and do the best I can, even if that means retaking a bunch of classes because I fail them the first time during a depressive cycle.

Thanks everyone for your replies!

 

[Sduibek]

For those who may stumble upon this thread in the future, an update for what it's worth:

I'm changing my major. Originally I was thinking some type of Engineering (probably EE or CE) then was more leaning towards CS instead. It made sense because a lot of my friends from high school got majors like that, and I enjoy computers and programming and-blah-blah-blah-whatever.

But, it boils down to the fact that I can spend ten hours on Trigonometry and complete maybe four problems in that time. This level and beyond of math just ain't for me right now. And, considering that I tend to hate it (when it's that drawn-out and gory anyway) it's silly to go into a field hating and struggling immensely with what is basically the basis of the field.

So! Dear reader, if you feel like I did in the OP, consider switching majors! The weight off your shoulders will feel like you've just discovered a new Trig identity.
Wait, bad analogy.