How to develop a way of thinking mathematically?

Question

I'm a Computer Science student. I'm struggling in Calculus I at my school. The problem I think is that I don't know how to think mathematically.

I can write a paper on some random topic in English or History really well, but I break out into a cold sweat doing quotient rule ( I would trade all that to be able to have excellent mastery in math). I know I can think analytically about how different social stratification problems intersect both socioeconomically and historically , but I'm at a loss when I try to do the same thing with a Calculus problem. I want to be better at Calculus ( and math in general), but I don't know how. I don't know how to approach a math problem without panicking because I have no idea where to start.

Approaching almost any Calculus ( or ever precal or trig) problem I have zero confidence unless my hand is held through the problem, and even then, when I try to remember how to do the problem, my mind is at a blank.

Also, I have to learn how to think mathematically for my Computer Science major, but again, I don't know how. it affects my performance in my Computer Science classes ( which are horrendous, since I treat Computer Science like Cal and panic when I try to start off a problem ). It feels like at this point, I should switch to a non-STEM major ( which I can do reasonably well in, but will hate myself and every second in the major ) and cut my losses. I failed pre-cal twice ( managed to pass the entrance exam to Cal, somehow, with a lot of difficulty) and I feel that that was a sign to rest the dream of Computer Science/Cal and suck it up and do some other major I'll regret and pay thousands of dollars for.

But, at the end of the day, I want to stay dedicated to Computer Science, and with that, master Math, especially Calculus.

Essentially, how can I bring up my mastery of Math, and preferably fast?

Answer 1

How comfortable are you with pre-calculus mathematics? Let me suggest a road map before getting into calculus:

Solving two equations in two unknowns

Solving quadratic equations

Solving simple problems involving logarithms.

Manipulating algebraic identities such as $(a^2-b^2)=(a-b)(a+b)$

Problems finding the 3 lengths and 3 sides of a triangle when some partial info is given.(this is trigonometry and plane geometry).

Learn how to multiply polynomials; especially the theorem that a polynomial $f(x)$ can be factored as $(x-a) g(x)$ if and only if $a$ is a root of the equation $f(x)=0$.

Understand the geometry when the book says "Plot the curve $y=g(x)$"

If you learn to solve these kinds of problems then calculus will not be difficult.

Answer 2

Start by mastering the basics of trigonometry. You will be surprised how ubiquitous that right triangle is in Calculus. Then like another user just said, get comfortable doing algebraic manipulation. Logarithmic and exponential functions should be next on the list. Don't give up. Don't be ashamed to flip open your algebra I book and do a review. And most of all, don't be afraid to make mistakes!!!!

Answer 3

Here's my advice.

(1) Don't be afraid to be wrong. It's better to do the problem wrong than just stare at it.

(2) Understand things, don't just remember them. Math isn't a random collection of formulas; it's basically problem-solving. If you just memorize the steps to solving a problem - for example, if you just memorize the quotient rule - you might do well on the exam, but you'll never feel confident about it. If you can get to the point where those steps "feel" like the right ones, even without memorization, then you understand it. My personal rule of thumb is that I don't really understand something until I can make a joke out of it - for example, $\frac{d5}{dx}$ is "the rate of change of $5$ with respect to $x$". To me, that feels a bit silly.

(3) Revisit the basics. All the time. It's tempting to just push past a difficult topic, to "just get through it". But math is extremely cumulative, and the hard things aren't going to go away. If you're not comfortable with algebra, calculus is going to be awful. If you're not good at arithmetic, algebra will be hard. There's no shame in going back and re-learning what you've missed. Practice arithmetic and algebra a lot, with everyday examples where you can check your answer - for example, when deciding how to divide eight slices of pizza between four friends, set up the equation $4x = 8$ and solve.

(4) Use your resources. Most professors and teaching assistants are happy to help during office hours. You don't even need to have a good question. Just go talk to them.

(5) Take things one step at a time. Don't try to solve an entire problem at once. Instead, ask yourself what piece of information you could find that would get you a little closer to the answer, and try to get that instead. When a problem seems overwhelming, break it into the tiniest fragments you can think of.

(6) Don't be afraid to be wrong. Instead, try to get a feel for when an answer's not right - for example, if the question was how long it takes to drive from Chicago to New York and your answer was $2000$ years, you probably did something wrong. If you can spot that sort of thing, then it doesn't hurt you to do the problem wrong - you'll spot the error, then go back and try something else.

Answer 4

Building upon P Vanchinathan's great road map, I want to stress out the importance of examples, simplifications and pictures.

Simpler examples give you some insight on the thought processes. For example, when dealing with polynomials, sometimes you can consider a quadratic polynomial (eg., $y=ax^2+bx+c$, or "those nice parabolas") to help visualization.

Pictures are really important too, especially in geometry, trigonometry and graphs of functions. Noting down variables in a picture (such as angles, sides or areas) often leads to some good ideas.

Since you seem to be good at writing, outlining your reasoning before doing the "dirty work" with algebra and such may help.

When you get stuck in an exercise, note the difficulty down and look up if there's no formula or theorem that may help. It can be invaluable especially when dealing with Trigonometry and Algebra, which have so many identities.