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C'est la Z

Math isn't always fun but you can do fun things with math

Garth Flint recently wrote another post talking about some of the PD going on in his neck of the woods. Garth talks about the disconnect between the professors putting on the PD and what goes on in the K12 classrooms of the attending teachers. Here's the money quote:

“Why do all college CS profs think everyone loves math? Want to turn kids off to programming? Throw math at them.”

To a certain extend it makes sense - professors are content specialists. Getting a doctorate and doing research are deep dives into specific subject area specialties. What's more, once you get beyond teaching early for-major classes, you're also teaching a population that's just as passionate about the subject as the professor is. While it's true that there are plenty of non-major courses and non-majors who take higher level courses, college educators live much more in a subject specific echo chamber than K12 educators.

At the high school level, it's very different. Years ago my principal sat in on a lesson. I have no recollection of the lesson but afterwards she commented that at one point I mentioned something that I found cool. "So what" she said. How does that make it cool for the kids. While it's true that our passion can be infectious she was right. It has to be cool for them.

Since so many CS Profs and even HS CS teachers have strong math backgrounds and really like math, that becomes a frequent fall back. Truth be told, although I'm not a math guy, I've fallen back on that myself. I've done the quadratic formula, correctly derided in Garth's post and the traditional mathy recursion problems.

I'm guilty of this but not apologetic. It's ok to do some mathy stuff becuase there isn't a single magic bullet. Some kids will indeed love math, some games, some text processing, some art, some music and some other. There are a lot of domains in which we can apply programming and CS solutions and it make sense for us to use as few or many as necessary to best reach our kids.

Math, though, is a little different than domains like games or music. We don't need games or music to program up solutions to problems but we sometimes do need math.

So, what do we do?

At least we should try to use math when there's a point to it. Maybe it won't be loved but at least there's a chance at it being appreciated.

Let's look at the quadratic formula:

Given a quadratic equation in the form: \(ax^2+bx+c=0\)

You can find the roots using the quadratic formula:

\begin{equation} x = \frac{-b\pm \sqrt{b^2-4ac}}{2a} \end{equation}

What's so cool about that? How about the discriminant? That's the \(b^2-4ac\) part. What's the point?

I didn't really appreciate either until much later when I studied computer graphics. In computer graphics, there's a technique called Ray Tracing. It's typically results in very shiny surfaces and historically, ray traced images have included lots of spheres. When you see an add with really smooth shiny stuff, particularly ones with bright specular highlights, it's probably ray traced. I believe the movie "Cars" was the first movie that was entirely ray traced.

I wouldn't do ray tracing with a class until it had at least pre-calc level math maturity but for an advanced group, it's very approachable.

The process involves tracing rays form the viewers eye through each point on the screen and seeing what object it hits first. Once it figures that out, it spawns rays from that point to each light source and sees what light sources it hits. It also spawns rays for reflection and refraction.

Let's say you have a 1000x1000 screen. That would be 1,000,000 rays but since you might actually "supersample" - that is, subdivide each pixel on the screen into 16 subpixels and trace each of those rays and average the result. That's a lot of rays to trace and it can be crazy slow. I wrote my first ray tracer in the mid 90s. It didn't super sample and was pretty simple. I started a render the night I finished it. By the next morning, it had only completed three lines.

Since for each ray you have to figure out which object it hits first, you've got to do a lot of math to figure out ray/object intersections. It turns out that the easiet is probably the ray/sphere intersection. It might seem that intersecting a box might be easier but intersecting a box is actually a lot of work. You have to:

  1. Note that a box is really 6 sides each being on an independent plane

  2. Figure out which plane the ray hit first

  3. Figure out if that intersection point is bounded by the four sides of that face of the box.

For a sphere, you have to go through a bunch of substitutions and simplifications but you end up with a single simple equation to solve. A quadratic equation and for this you can use the quadratic formula.

You're still left with a huge number of rays to compare with a huge number of spheres.

This is where the discriminant comes into play. The discriminant tells you the nature of the roots of the equation. For example, if the discriminant is negative, you have complex roots. Geometrically, if the discriminant is negative, it means the ray you're testing misses the sphere entirely. Here are the possibilities;

Discriminant Means for a sphere / ray intersection
< 0 Ray misses the sphere
= 0 Ray is tangental to the sphere
> = Ray intersects the sphere at two points

Since most rays will miss most spheres it means that instead of solving a bagillion quadratic equations with the square root and the division, you just do the discriminant and throw most of them away.

Then, your solutions also tell you something useful:

Roots Means
double root Ray hits sphere at one point
both negative Ray is behind the viewers eye
both positive Ray is in line of sight - choose the smaller root for intersection point
one negative one positive Eye is inside the sphere

This was the first time I really "got" how useful something like both the quadratic formula and the discriminant in particular could be.

While this isn't a lesson for newbies, there are a number of areas where we can do something neat that uses math rather than doing math for it's own sake.

I've written about using the distance formula for recommendation systems and an intro to decryption. Also using statistics for question answering. Other approachable programming projects that can make math more interesting would be clustering or a Bayes based classifier.

If we look, we can find lots of places where teaching and using math makes sense. This way, the math and the CS both support and reinforce each other. Much better of a way to do it than merely because I thought it was "cool."

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