Calculus: Is This All There Is?

(An archive question of the week)

I’m going to do something unusual, and post a discussion that was never archived. I ran across it while searching for the original of an archived discussion to check something, and this one stood out as worth posting around the start of the school year. It’s a question from a prospective math major who was discouraged by his calculus class; several of us joined in encouraging him. And this is not only about calculus!

Is calculus boring?

Here is the question, from Rafael in 2015:

I entered college as a math major, since I was good at math in high school and found it interesting. I am in Calculus 1 now but I'm extremely bored by the content. Does this mean I should probably consider changing majors?

The calculus lessons are not that bad, but I hate doing the homework assignments. They are very tedious and feel very pointless. I feel like I could care less about the material. I realized this is only the first class out of probably 20 math courses I'll have to take if I continue as a math major, which made me consider changing my major.

Is calculus a boring subject or is it just me?

Rafael was wondering if the rest of a math major would be as boring to him as he found calculus to be.

Calculus has fascinating applications

Doctor Greenie was the first to respond, focusing on calculus itself:

Hi, Rafael --

My basic calculus was a long time ago (hmmm... about 55 years), so I don't remember a lot about my feelings about the subject when I first started studying it.  But it seems to me I found it somewhat uninteresting at first.

But then as I got deeper into my studies of calculus I found it quite fascinating to see all the applications it had to the real world.

I was a math major, too.  I was on a course I thought was going to lead to graduate studies in math, which meant studying theoretical mathematics.  That's where I lost my interest.  But calculus with all its visible applications I have always found fascinating, and I still do.

I will invite other volunteers here to provide their responses to your question.

At its base, calculus is just “algebra with derivatives”. I recall looking forward to calculus because it sounded like magic; but if you just looked at the work you were doing, it could seem like merely pushing symbols around on paper. That may be what both Rafael and Doctor Greenie saw in it at first. What eventually got Doctor Greenie interested were the many ways it could be used. Anything we learn, from a sport to a language, can become boring if we focus on the current drill and not on the future benefits.

Calculus is not typical of college math

I was the next to respond, paying more attention to the fact that calculus is not typical of a math major:

I wouldn't judge by Calculus 1, which is not primarily aimed at math majors anyway, and so does not represent what the major is like. AFTER calculus, you will start getting into Math Major territory, and everything will start to change. Even then, some of your courses will probably be taken by many other majors.

Real math is far richer than what the typical calculus course deals with. It will no longer be focused on mere techniques, but will dig in much deeper, emphasizing proofs and bringing in entirely new ideas, rather than just adding new little features (like derivatives and integrals) to the algebra you've been learning for years.

Take a look at the titles of the courses you have to look forward to, and see how different some of them are. Then talk to a few professors in the math department and ask about what you will be learning. Maybe they can give you a taste of what it's like so you can see for yourself whether you like it. But making that decision based on Calc 1 would be like deciding whether you like the food of a country you are visiting based on what you eat in the airport on arrival. You're not  really in the country yet!

My guess was that Rafael didn’t find calculus interesting enough to explore for a lifetime, or even to keep him going for a school career; I wanted him to know that more interesting things are coming. He may end up choosing a different major, but he doesn’t yet have a basis for that decision.

Think through what motivates you

Then Rafael wrote back to clarify:

I think you might be confusing that I think the material in Calculus 1 is too easy and thus boring for me. But that's not the case. I think Calc 1 is not hard but not amazingly easy either. In fact, it can be quite hard at times. I still find it boring though. It's so tedious and bland. Now that you know that thinking calculus is too easy is not the reason I find it boring, do your thoughts still stand? Because they seemed like what you were saying is, 'you are bored now but there are more challenging courses after the calculus sequence'

Neither of us had really talked about difficulty; I think his description of “tedious and bland” is probably more or less what both of us had in mind. I responded with more questions about what bores him about it:

No, I don't think either of us assumed it was a matter of being too easy for you. The word you used in contrast to "boring" was "interesting", and both of us took off from that idea.

Dr. Greenie appears to be supposing that you think it's uninteresting because you don't see its applications, so that you "could care less". (Perhaps you therefore are not motivated to deal with the difficult parts.)

I am supposing that you think it's uninteresting because it is too much busy-work or mere rote procedures, and doesn't explore why things work as they do. That is a common complaint about Calculus 1 from students who really like math -- not that it's too easy (it isn't), but that it is too routine. I wrote of higher courses being "richer", not "more difficult" -- though I could well have used the word "challenging", if I had, with either meaning.

And my main point was that you haven't yet seen what a math major is like, so, regardless of what it is that bores you, you shouldn't assume that it will be true of the rest of the program.

Can you explain more about what bores you? I think you do mean what both of us assumed, namely that it doesn't interest you. But what WOULD interest you? What did you find interesting about previous courses? What do you wish it were like? How is it different from what you expected? We can give better answers if we have a clearer picture of what bores you, and what wouldn't bore you.

I can easily understand that it be hard to say exactly how you feel about a math course; I’ve seen students with all different sorts of negative attitudes, and know not to make assumptions about what lies behind a complaint.

Rafael made a good attempt:

I remember finding Precalculus interesting. I can't recall why exactly but I had more fun learning that than Calculus. 

I used to find the ideas of calculus really interesting. Like I was amazed at being able to find the slope of curves and really liked the idea of an integral. I used to think "how is it possible to find the exact area under any curve!?". But then when I got to actually doing the calculus and actually learning it in class it's not nearly as exciting as I had hyped it up to be.

A problem with me is I get very excited at new and interesting concepts but then when I get there and learn them they are not interesting anymore. Like right now I like the idea of 3-D calculus (multivariable calculus) but I'm sure when I get to that class, I will not find it as fascinating as I imagine it will be now. I'm always looking ahead of things because I'm bored by the current topic. Like a few weeks ago I was reading ahead in the book on integrals and now weeks later integrals bore me! It makes no sense.

It would be hard to pinpoint exactly what bores me about my calculus class. I just know it's a pain to do my homework. So much that right now I'm just focused on passing the class with an A and stopped trying to think about the "why" behind things. I just want to get it done with. I do remember really disliking curve sketching. So tedious and not at all fun to do.

I know you asked several questions but I have difficulty pinpointing the exact answers.

One thing we can see here is that it is newness that excites him. (That reminds me of my own expectations of calculus.) On the other hand, he does like to think about “the why”, which is fundamental to being a good mathematician; that has just been stomped out of him by the nature of the present class. That’s actually a good sign.

Be patient through the dull parts

Now Doctor Floor (in the Netherlands) stepped in:

Let me drop in with some thoughts.

The thoughts you describe I think are quite common among mathematicians and mathematics lovers. I myself didn't study maths in the United States, in my home country things are somewhat different, but I do remember the disappointment about "calculus" - this being like 30 years ago. 

What I disliked was that concepts used were basic, homework often repetitive and tedious. Homework focused a lot on calculating techniques, while I wanted to understand better and dive deep. I wasn't really thrilled by the intermediate value theorem, but later I realized how valuable the theorem is.

To get better in maths you need to do the dull work as well in order to create a firm base. Deeper concepts appeared later in my math study. Sometimes very difficult or even too difficult to understand. It was more satisfying to me. But I had to be able to rely very firmly on what I had learned: perhaps not so much the contents of beginning calculus, but the needed precision. Being precise turns out to be very important. Combined with being creative.

I don't think you should drop maths because of this "Calculus 1". Perhaps you should talk to an older student to ask him what he (she) is learning and how he judges his Calculus 1 now.

So calculus students aiming for a math major need, on one hand, just to be patient because the best is yet to come, but also to learn this material thoroughly, because parts of it will be foundational to later, deeper studies. (Compare those drills in sports or music, compared to thrilling games or concerts to come later.)

Find the excitement

I now replied, initially talking to Rafael as if he were just a child expecting everything to be fun, but really knowing he was better than that:

My first reaction is something you're probably not going to like: Suck it up! This is life. Not everything you will do in college, or in life, will be scintillating every moment. Part of the purpose of college is to mature you -- to teach you to persevere in things that are not always pleasant, in order to accomplish a goal.

No matter what you do, there will be things that are fun, and things that are just hard work. You do them both, because you want to excel, and you know that the hard work is necessary. Of course, you want to choose a major, and a career, that you find interesting and rewarding; but that will be the big picture, not a constant throughout the process. You can expect parts that will not be exciting.

If you approach all of life by quitting anything as soon as it is no longer new and shiny, please don't get married and have children! These are very rewarding things (at least as much as math), and are a lot of fun (almost as much as math), but also involve slogging along and carrying out commitments that are not fun, just because you know they have to get done in order to get to the parts that are fun (and sometimes not even for that). Marriage and kids get old, just like calculus does -- but old and familiar does not mean they are not worth your best efforts. The same is true of your major (whatever it turns out to be) or your job (whatever you end up doing).

One way to make calculus, or whatever, feel worth doing is to keep reminding yourself of your bigger goals (which implies that you need to *have* bigger goals beyond momentary pleasure). It is also a good idea to use the dull parts as occasions to strengthen your broader skills. If the problem is that it is no longer new, MAKE it new by deliberately seeking out different perspectives. (Don't expect your professor to make every technique you learn look amazing; he's having a hard enough time getting those less capable students to barely understand what he's teaching.) Find the excitement in it; or else just keep in mind that there will be more new things you will be able to discover *because* you master what you are currently learning.

If you were someone who loved mountain climbing, you would keep walking through the valleys because they were getting you closer to the peak you are heading for -- and you'd probably also find pleasure in the streams you pass on the way. (You wouldn't find pleasure in the blisters -- but you'd be able to bear them because of the beauty and challenge you know is coming.)

Now, that was my FIRST thought. It's probably far too severe; I doubt that you are really expecting everything in life to be fun. But it's worth considering whether your expectations are reasonable.

My second thought is that you ought to think a little more about why you want to study math in the first place, in order to make your decision. There are a lot of fascinating things about it; but what do you plan to do with it? It may be that you'd be more interested in some field that *uses* math (engineering or science, perhaps). But then, they have to struggle through calculus, too.

Gather information from faculty in math and in related fields, and from others who have gone further than you, and consider whether what is yet to come will be worth the struggle. If you change your mind, there's nothing wrong with that. Just don't jump ship hastily.

You can imagine that I was anxious, wondering whether I had been too severe there! Did I judge his real needs accurately?

Will it all become boring?

He wrote back, having seen the right things in it:

Thank you for your eye opening advice on things not always being fun. But I think that actually gives me too much credit. I was worried that thinking Calculus 1 was boring meant that I didn't really like math. I was  thinking "you are in over your head doing a math major when you're already bored in your first course". It  wasn't a complaint. It was more of a worry.

Any major I would do I'll need the calculus sequence for so I have to take those courses so it wasn't like I was whining and was going to quit on Calculus. I was just saying that 'if this bores you then do you even like math?' Because I was looking at linear algebra and it kind of seemed even more bland. Matrices?? So I had this idea that like all the math major would be like that. Dry, uninspired, dull, tedious, routine, and pointless material. I had thoughts that were telling me, "that is math, you just don't like it and so you need to stop pretending to like it. Liking math in high school doesn't mean you're math major material."

I do love the advice about making something old new by looking at it differently. I'll give that a try.

Yes, he got it. You can like math without liking an introductory calculus course; and you can make the most of even a class that doesn’t make it interesting.

It will get better

I replied,

As I said, those remarks were just my immediate reaction to your statement that things stopped being fun when they stopped being new; I didn't think you would really have that big an issue.

And I sent the reply when I had to go to bed, though I felt I hadn't quite answered your real question. Doctor Floor had done that, I think: being bored in calculus is common among math-lovers. It may even be a necessary sign that you really like math, because that course is not the best that math has to offer, especially as taught to a general audience. (Some schools have a separate calculus course aimed at potential math majors who want a bigger challenge -- not in the sense of a harder course, necessarily, but of going beyond the mechanics.)

Linear algebra can seem dull on the surface -- it's just linear, after all, and what's more boring than straight lines? And some linear algebra courses will focus on the mechanics of working with matrices, which might not excite you. But a good course will get into some amazing stuff. It can be your first introduction to the ideas that are further developed in abstract algebra: that you can invent a new kind of entity (in this case, the matrix) that can be added and multiplied but is not a number, and study how it behaves under various conditions. But it's also very useful -- computer graphics are built around linear algebra, for one thing.

Again, ask a faculty member or two to give you examples of what you will learn in these higher courses, and what makes them interesting. If what interests them doesn't sound interesting to you, then maybe you do need to consider another major. (By the way, getting to know faculty is one of the best ways to make the most of your college education.)

He closed with this:

I just want to end with a sincere thank you for taking time out of your life to address my questions. Advice from someone with so much experience is much better than advice coming from my peers or classmates. Thank you for showing me that I was looking at things the wrong way.

Some of the advice we shared is just for potential math majors; but some of it is for everyone. Whatever class you take, make the most of it, and don’t depend on the teacher to make it fun.

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