Riled Up: Here’s the Steaming Pile of “Nitty-Gritty” You Ordered

If you haven’t been interpreting most of these posts as me yelling and slobbering, you’re wrong. I have an extreme obsession with teaching that bleeds into every millisecond of my day. It causes me to find the seeds for investigations and lessons in things that I should just be enjoying. This is how I know that I am supposed to be a teacher: it consumes my life, I have no control over it, and I don’t mind a bit.

Which is why it makes me sad when the grind-stone pushers come out. I don’t really take part in online discussion boards, or commenting in general, mostly because I fear being taken the wrong way. These blog posts take me hours to write and edit, and whenever I write a comment, it always feels deflated and unsubstantial (thank you all for the well written and useful comments, though; mine just seem to suck)

However, I have been receiving a few comments (on this site and others) that I’d love to address. Mostly, the accusation that what I’m doing here is nothing more than the Splenda™ of math and science ed: it might taste good (debatable), but in the end there’s no nutritional content. This is obviously offensive, but I do have to admit that I rarely write about what specific 500 problems a night my little cherubs have to do (Teehee, they don’t). I’ve always wondered, are they supposed to drink a protein shake after that many problems? When will their noses be sharp enough? So, in short, I’ve been accused of running a snake oil business by some folks: I guess I’m not doing enough “nitty-gritty.”1

I have no doubt that these are the same people who pulled Dan Meyer into their backwards arguments. In no way am I saying that I have the readership, clout, sweater vests, or California-cool of Señor Meyer, but I will say that neither of us are looking for kids to “just fill up a water tank” and call it a day. It’s about the thinking. It’s about seeing the natural need for mathematics, and then laying it down like Frusciante on a guitar track.

You can keep giving kids the sterile garbage that book publishers give you, and they will keep right on code switching. They’ll have one set of mathematics they use in their real life, and one set of math they regurgitate for you. Do you want that? Can you sleep at night with that on your shoulders? I don’t even know which set would be more anemic. The one that the student makes up on their own without guidance, or the one that’s so hamstrung that it has to be pulled on a tether by a teacher to find food.

Here’s a fantastic example worked out in all of its horrifying detail. I believe that this example would be useful at some level in  geometry, pre-calc, and/or calculus:

The Method of Exhaustion

Yes, that’s tongue-in-cheek. Maybe it’s the end of the year, but I’ve been Cornally-Hulk more often than not.

The challenge is this: Use an increasingly complex inscribed polygon to find the area of the general circle of radius r.

but, like, with a million more triangles

When would I use this? On an SBG quiz after spending a long time discussing areas, limits, and sums. If you start with things like this, which many of us do, you’ll lose kids. They don’t care about circles. They don’t have this “math is pretty” mentality. If you ask them to derive Πr2 in any number of goofy ways, they’ll just look at you and say, “But, we already know that formula.” To most students this is the end of math: get the formula, use it for some esoteric application and then move on to the next. That’s why this example should only be used at the end, to help motivate an understanding of the beautiful near self-consistency of math (holla’ atcha Göedel, more on him later).

Where to begin? Well, we need to sum a bunch of triangular areas. In fact, an infinite number of infinitesimally small areas. (<- That’s calculus in a nutshell, btw) So let’s start from the very beginning:

This lengthy sum can be rewritten more succinctly as:

And now we want to allow for an infinity of these little areas:

This looks suspiciously like calculus. We should now endeavor to find a formula for the area of our triangle friends:

The hypotenuse of our right triangle is r, and the angle of each is related to a whole circle. This means that the legs of this triangle can be related using r and some trig:

Rearrange and simplify to get the base and the height:

Plug those in to the area of a triangle formula:

Use a trig identity to clean that up:

Clean that up further:

Now let’s combine the dry ingredients with the wet for our area of a circle:

This part’s the trickiest: only things that have i‘s in them participate in the sum. Nothing has an i in it, so all things can be moved out of the sum:

The summation of the number 1 just yields n (or in other words, one added up n times = n):
Here’s where the magic happens. If you try the limit, you get n going to infinity, and the sine term going to zero. Zero multiplied by infinity is indeterminate. We then endeavor to create a form that can be handled by L’Hopital’s rule (i.e. play games with the n):

Remember that the inverse of n is also n to the negative one. After performing L’Hopital’s rule, I get something beastly but not unusable:

This simplifies to:

And further:

Finally, the limit of the cosine term goes to one, leaving:

And that’s why we don’t post example board work on blogs. Here’s a statement of how I spend my time in class and why.

1. The real reason is that most of us are fantastic at board work. We write out our notes, and present them well. What’s the point of blogging about it, when it’s not new or experimental? Your book is full of examples.