- William L Culbertson
Calculus in Middle School
Calculus for Middle School? Why Not?
Right at the start, I’d like to stipulate that there is a difference between explaining Calculus and teaching Calculus to students. A twelve-year-old would have to be quite precocious indeed to have mastered the skills of algebra, as well as learned about the various other functional relations, needed to be able to learn how to do Calculus.
That being said, there is no reason why a middle schooler could not be challenged with the ideas of Calculus. Take the idea of a limit, a core concept that undergirds all of Calculus.
Ask the students, “How close can you come to something without touching it?”
This is the basic idea behind an infinitesimal and one of the fundamental ideas used to introduce the Calculus of limits. In a science class it could also be the springboard to talking about very small things like living cells, viruses, molecules, and atoms.
Tell the students to imagine walking across a room in a strange way. Go half way across and stop. Now go half of the remaining distance. And again half way? Getting close to the wall yet? If you go exactly halfway each time—exactly half, not just close—will you ever actually get to the other wall? How close is close enough?
This explores the idea of limit a little more rigorously and is the way the idea is introduced in a Calculus class. A question of this type can lead to some interesting philosophical discussions with students. Don’t shoot over their heads, but curiosity is curiosity, and middle schoolers have boat loads—especially if they think they're getting away with not doing regular math.
Here’s another problem that looks at limits from another direction. “Suppose a farmer’s field is bordered by a meandering stream. How could you figure out the area of the field?” This is a typical way Calculus books approach integral Calculus.
Break the field down into narrow strips, say a foot wide, that run all the way to the stream. Measure how far it is to the stream in feet. The product of the one foot width with however many feet it is to the stream is the area of the strip. Do each strip and add them all together. Yes, it would be a bit tedious, but it would be pretty accurate. More importantly, it is a familiar, tangible process that demonstrates the process.
Wouldn’t it be more accurate if we made the strips narrower—say only an inch wide? Yes, it would be twelve times the work, but it would more precisely account for the irregularities in the stream
But why stop there? By repeatedly summing smaller and smaller strips of the field, you could calculate the area to any desired accuracy. But again, how accurate is accurate enough? Oops! I guess you could invite a little philosophical discussion of error her as well.
In the unit on geometrical shapes, compare polygons: a triangle to a rectangle to a hexagon to an octagon to a . . . whatever. Go as far as you have pictures, then note, "Gee. The more sides you have, the more it looks like a circle. I wonder how many sides it would take before you can't tell the difference?"
You could play in the arithmetic domain. Challenge the students by saying, “Take out your calculators. Divide 23 by 5. Now divide 23 by 2. Then by 1”
Then ask, “How do the the answers compare? When we divide 23 by smaller numbers, what happens to the answers?”
Now have them divide by numbers less than 1. “Divide 23 by .5. How does that answer compare to what you get when you divide 23 by .25?”
Keep asking, “Why?” In no time they can leave their calculators behind to answer, “What happens if you divide it by .005? By .00005?”
(Now the key questions:) “Describe happens to the answer if you divide 23 by a number that keeps getting closer to zero.” This is another way to introduce the idea about things being infinitely large.
And since you’re going to be infinitesimally close, you’d better be ready to answer the question, “So what happens if you divide by zero?”
[Okay, dividing by zero gets into a whole new territory, but it’s so natural a question that one of those curious middle schoolers will be sure to ask. But please, don’t say, “You can’t divide by zero because it’s not allowed.” The instant reply will be, “But WHY?” I have another blog post that addresses this issue. Be prepared!]
My point here is that you do not have to have a great deal of arithmetical, geometrical, or algebraic experience to talk about the fundamental ideas of Calculus. As a Calculus teacher for many years, I would have been thrilled if more of my students had come to their first class having played around with the ideas of limits, infinitesimals, infinity, and so on.
Brains can (and should!) be boggled as soon as students can understand the basic concepts. If more students understood that mathematics is a science where we ask very thoughtful questions, more students would be intrigued by math rather than repelled.