Monday, August 15, 2016

Why Calculus? Specifically, "Why Do I Have to Take Calculus?"

If you're here, you probably are finding yourself taking a calculus class and wondering why oh why did the dreaded calculus monster show up in my curriculum? Maybe you're optimistic and curious about what if any benefits learning this particular material might hold for you and your future? More likely, you view this as an obstacle to your eventual degree and success, after all, machines can do calculus far more rapidly and more accurately than nearly any human could hope to.

And you'd be right.

Calculus is the very last of what my late, favorite professor used to call, "cookbook mathematics". It doesn't require any creativity, every problem can be solved through brute force and rote memorization, and computers churn out answers faster than you can formulate a problem that would require a human pages and pages of written out work.

But it does hold a hint of the essence of true mathematics.

Calculus is possibly the first time where a creative solution will save time and effort. Where you start to understand what is meant by an "elegant" solution. Where, if you have a good teacher and strive to understand the inner workings, you will understand more than is presented to you. Calculus allows you to glimpse the truth of mathematics and it is in calculus (specifically, rigorous calculus courses) where students get their first hint of what modern mathematics entails.

Let me be clear, if you are worried about calculus and you see no value in its completion, you can get through a calculus class in much the same way you've probably gotten through every math class to date: rote memorization of formulas, problem types with prescribed strategies, and effectively turning the course into an advanced version of the baby toy where you are presented with blocks of different shapes and the goal is to push through holes of the same shape. To these students and perhaps to all students when you are despairing: flash cards and drills will be enough to get you through. There is no creativity required, only a memorized set of steps.

That advanced version of identifying blocks and what holes the blocks go through is not why you are taking calculus—nor why you are taking any mathematics course.

If we run the numbers, you probably do not even need calculus in your job. Hell, you can get through life without owning and operating a car, why would you need to own and operate a set of mathematical skills?

Although that car does make things easier for you—as long as you keep it tuned up and know what situations to use it in.

The reason why you're taking calculus is most probably because calculus is damn hard to get through by rote memorization. It's possible and this is usually the route that I see students take, but at the end of the day you're trying to memorize a blizzard's worth of snowflakes. Every snowflake may be unique, but do you remember the differences between each and the path it took to fall?

Calculus (and especially calculus 2 of the calculus series for some reason) is considered the hardest math class because you need to know everything you relied on memorization to get through before. You're expected to know all the arithmetic, algebraic, graphing, and logical rules that you learned throughout every math class you took before. If you can keep those straight at the same time as learning all the new rules of calculus, then you are better at memorization than I am.

You are taking calculus because when memorization fails, it is understanding, logic, reasoning, and problem solving that you fall back on.

You are taking calculus because solving a problem like you've never seen before with techniques that you understand but weren't sure could be used is something you don't see in many math classes before the dreaded calc, but it is something you see everyday out in the "real" world.

You are taking calculus not to learn calculus, but to learn to think like a mathematician, like a problem solver, with the language that mankind has discovered/codified to help them conquer the universe around them and beyond anything we've ever seen.

If you are afraid, you can cheat yourself by getting through via memorization and drills. I've seen many, many, many students do so before. But if you can overcome that fear and approach calculus the way it is designed, with the knowledge of your previous math courses backing you up, then you may start to utilize the skills of calculus in everything you do. You'll take your knowledge and synthesize it in ways you couldn't imagine before.

And, if you are required to take a calculus course for your degree, then know there  are people with your degree that do use calculus and they use it in their jobs and they use it regularly—maybe not in an entry level position and there are definitely routes within your chosen career field that you can get away with reliance on machines or not be aware of the calculus operating behind the scenes even as you advance through your profession—but they do use it.

How much more valuable an employee could you be by understanding calculus? How much further and how much faster can you advance in your career by knowing and understanding calculus that so many of your peers take for granted or dismiss out of hand? If you are going to be taking a calculus class anyway, and if memorization is the harder route for nearly everyone taking it, why not take the less traveled path of attempting to understand it? You will arm yourself with a valuable set of skills that makes you more valuable and prevent headaches down the line of your chosen career path when you actually do need to learn the skills presented now.

Finally, I would like to say that it may not be your fault if you've relied on memorization to get through calculus or past math classes. Note that several times I've required the need for a good teacher. I'm not going to jump on the bandwagon and trash your teacher until I've spoken to both of you and witnessed them in action, and it could be that they are under the same limitations you find yourself under. You may feel the need to rely on memorization and not be given the option of beginning to understand. I will say if you are struggling to understand calculus, then there is a good chance that the missing key is somewhere in past math courses. Examine what the course is expecting you to already understand before you start stressing about understanding something new.

Good luck!