# College Math Teaching

## March 30, 2014

### About that “viral” common core meme

Filed under: class room experiment, editorial, pedagogy — Tags: , — collegemathteaching @ 10:09 pm

This is making the rounds on social media:

Now a good explanation as to what is going on can be found here; it is written by an experienced high school math teacher.

I’ll give my take on this; I am NOT writing this for other math professors; they would likely be bored by what I am about to say.

My take
First of all, I am NOT defending the mathematics standards of Common Core. For one: I haven’t read them. Another: I have no experience teaching below the college level. What works in my classroom would probably not work in most high school and grade school classrooms.

But I think that I can give some insight as to what is going on with this example (in the photo).

When one teaches mathematics, one often teaches BOTH how to calculate and the concepts behind the calculation techniques. Of course, one has to learn the calculation technique; no one (that I know) disputes that.

What is going on in the photo
The second “calculation” is an exercise designed to help students learn the concept of subtraction and NOT “this is how you do the calculation”.

Suppose one wants to show the students that subtracting two numbers yields “the distance on the number line between those numbers”. So, “how far away from 12 is 32? Well, one moves 3 units to get to 15, then 5 to get to 20. Now that we are at 20 (a multiple of 10), it is easy to move one unit of 10 to get to 30, then 2 more units to get to 32. So we’ve moved 20 units total.

Think of it this way: in the days prior to google maps and gps systems, imagine you are taking a trip from, say, Morton, IL to Chicago and you wanted to take interstate highways all of the way. You wanted to figure the mileage.

You notice (I am making these numbers up) that the “distance between big cities” map lists 45 miles from Peoria to Bloomington and 150 miles from Bloomington to Chicago. Then you look at the little numbers on the map to see that Morton is between Peoria and Bloomington: 10 miles away from Peoria.

So, to find the distance, you calculate (45-10) + 150 = 185 miles; you used the “known mileages” as guide posts and used the little map numbers as a guide to get from the small town (Morton) to the nearest city for which the “table mileage” was calculated.

That is what is going on in the photo.

Why the concept is important

There are many reasons. The “distance between nodes” concept is heavily used in graph theory and in operations research. But I’ll give a demonstration in numerical methods:

Suppose one needs a numerical approximation of $\int^{48}_0 \sqrt{1 + cos^2(x)} dx$. Now if one just approaches with by a Newton-Coats method (say, Simpson’s rule) or by Romberg, or even by a quadrature method, one runs into problems. The reason: the integrand is oscillatory and the range of integration is very long.

But one notices that the integrand is periodic; there is no need to integrate along the entire range.

Note that there are 7 complete periods of $2 \pi$ between 0 and 48. So one merely needs to calculate $7 \int^{2 \pi}_0 \sqrt{1+cos^2(x)} dx + \int^{48 - 14 \pi}_0 \sqrt{1+ cos^2(x)} dx$ and these two integrals are much more readily approximated.

In fact, why not approximate $30 \int^{\frac{\pi}{2}}_0 \sqrt{1+cos^2(x)} dx + \int^{48 - 15 \pi}_0 \sqrt{1 + cos^2(x)}dx$ which is even better?

The concept of calculating distance in terms of set segment lengths comes in handy.

Or, one can think of it this way
When we teach derivatives, we certainly teach how to calculate using the standard differentiation rules. BUT we also teach the limit definition as well, though one wouldn’t use that definition in the middle of, say, “find the maximum and minimum of $f(x) = x-\frac{1}{x}$ on the interval $[\frac{1}{4}, 3]$” Of course, one uses the rules.

But if you saw some kid’s homework and saw $f'(x)$ being calculated by the limit definition, would you assume that the professor was some idiot who wanted to turn a simple calculation into something more complicated?