Yesterday my calculetes took their first assessment — a algebra boot camp to help us prepare for limits. We focused on rational functions, piecewise functions, and basic function transformations (focusing on exponential and logarithmic functions). I haven’t graded their assessments. You know what? I don’t know why I’m talking about that at all. This post isn’t about that. It’s about…
… what came next … today …
Each year — this is my third year doing this particular rant — I am always surprised that we can go weeks (this year, we went 2 weeks, last year it was 6 weeks) before I ask my kids:
WHAT IS CALCULUS?
We do all this algebraic review, students are settled into class and into their routines, and then POW. I hit ’em with this question.
And now I can say for three years straight, I have been met with totally silence. Followed by a student saying “the thing after pre-calculus.”
Commence rant:
“Why are you here? What is this course you signed up to take? What’s the purpose of this course? Why have you been working your whole time in high school for this? I mean, you took a course last year called pre-calculus. And yet, here we are, already well into our year in a course called calculus, and NO ONE KNOWS WHAT THE HECK IT IS!
Seriously? SERIOUSLY?
Well, good. Let me tell you what it is.
[Insert discussion of the “tangent problem” and the “area problem”]
It’s the study of the very small to learn about the very big — to learn about things you never knew you could know. Like the basis for much of physics. Given just a little information about something moving — a roller coaster, for example — and a knowledge of calculus, what can’t you do? And let’s talk about how little you actually know about space. What figures do you know the areas of?
[Commence students calling out things like “square” and “circle” — of course followed by figures they know the volumes of.]
How sad and pathetic is this? You can only find the areas of silly, putrid little pretty shapes. What about the real world? What about this shape [commence drawing of crazy shape on board] or the volume of this figure [commence drawing of crazy volume on board]. I mean, seriously, think about it. Look at a sphere. What’s the volume? 
[slight pause, building suspence]
WHERE THE HECK DOES THAT COME FROM? I mean, really? You have no idea. It just popped out of nowhere and you never questioned it. In this course, you’ll know, not just accept, that it is true.
Your lives are about to be changed. [1]”
[1] Okay, so I didn’t say this last sentence, but dang, I wish I had. You know kids, I don’t know if it got through to them. But I love doing this each year. It never fails to shock me how it is that these kids work so hard so they can take this vague thing called “calculus” — they’ve even taken a course called “precalculus” which, if the name were accurate, was meant to prepare them for this course — and they come in not knowing anything about it. What other course could you be over 2 weeks in, and ask the students what the course is about, and they won’t be able to answer?
Continuous Everywhere but Differentiable Nowhere 
Sam, this is funny… every year I ask my precalc kids what exactly “precalculus” is. I know there’s no good definition, but they always answer “the class before calculus”. So I guess we’re just going around and around!
(At least you can give them a good vision of calculus! Precalc not so much.)
Great lead in, but you never did really answer your question. What is your answer to them? My answer is that calculus is about infinite summations and quantities called infinitesimals, or hyperreals, that are “alive” entities, dynamic rather than static things that are always in a process of getting arbitrarily smaller. Limits, of course, are a different perspective. They give structure or definition to the “approaching zero” process. It is the process, not destination, that is of primary interest. See here, class, I hold in my hand an infinitesimal dx. Now, it just buzzed out of sight but it is still there, because when I pop it into denominators, nothing blows up! So now the next philosophical question is: can we consider dy/dx to be a fraction given that it follows all the standard operations of division?
Actually, I guess you did sort of hit it with “It’s the study of the very small to learn about the very big .” But that definition also applies to atomic theory, no? :-)
True, but I don’t want to give them a treatise on calculus. I just wanted to get them excited about what’s on the horizon. Pieces here and there.
Honestly, I don’t know what I’d do if I were asked to write 1 page on the definition of calculus — the scope, the structure, the philosophical backbone. Maybe one day this year I’ll be so inspired to see what I come up with.
Is this a genuine rant or a mock rant you give to your kids?
I wouldn’t expect many high schoolers to know what Calculus is before they have taken the course. You say they’ve signed up for it, well sure… its the “next class”. They’re not exactly choosing it.
I understand that you want to explain what they’re about to be studying and that you want to provide motivation, but do you seriously expect them to be able to give you an answer?
How many kids could answer “What is Algebra?” in week 3 of that class?
Oh, TOTAL TOTAL mock rant. I was jumping up and down, and getting all excited about what we are going to be able to do with the course, now that we’re finally getting started on proper calculus topics…