(This post is part of the Math Munch Exploration writing assignment for Justin Lanier’s smOOC. Read more here. If you are here and somehow do not know of Math Munch, you should probably cancel all plans for the next few hours and go here.)
This post is about a piece of mathematics I really love, how I shared it in my classroom, and how it went wrong.
One of my favorite mathematical objects is the Koch Snowflake. It is one of the first fractals discovered and actually predates the word “fractal” significantly. Start with an equilateral triangle and remove the middle third of each side. Stick an equilateral triangle one third as long onto the big triangle. Now you have a six-pointed star made out of 12 line segments, each a third as long as a side of the original triangle. Now repeat this process (remove middle third, add smaller triangle) for each segment.
Now imagine the perimeter of the original triangle is three (side length one). This means that the perimeter of the six-pointed star is four, and each remove-and-add-triangle iteration multiplies the perimeter by 4/3.
So here’s the weird thing. The Koch Snowflake is not any one particular step in this process. It’s the limit of this process, which means that, since the perimeter is being multiplied by 4/3 at each step, the Koch Snowflake actually has infinite perimeter, even though it has finite area! (Fun fact: no step in the Koch snowflake process can tile the plane except the equilateral triangle, but the snowflake itself can, using two different sizes.)
But wait! There’s more! Let’s think about what it means to be one-dimensional; if an object is one dimensional, say, a line segment of length one, it means that a scale factor of three increases the object’s “size” by a factor of three. If an object is two-dimensional, say, a square of side length one, it means that scaling the object by a factor of three increases the object’s “size” by a factor of 9, since 9 is three to the second power. If an object, say, a cube of side length one, is three-dimensional, it means that scaling the object by a factor of three increases the object’s “size” by a factor of 27, since 27 is three the third power.
Now, let’s think about one “side” of the snowflake, sometimes called the Koch curve, starting with a single side of the original equilateral triangle. Let’s call that single segment Stage 0. Now think about the four pieces of the Koch curve (which start life as mere segments in Stage 1). Each segment is not similar to the Stage 1 curve overall, since each segment is straight and Stage 1 is composed of four non-collinear segments. However, the Koch curve itself is “infinitely detailed,” so each of the four pieces is actually a 1/3-scale copy of the whole curve. In other words, scaling up a copy of the Koch curve by a factor of three increases its “size” by a factor of FOUR. This means that, since four is greater than three but less than nine, the Koch curve is more than one-dimensional, but less than two-dimensional! Weird. This was one of the first examples of what is called fractional dimension.
(This is an abstract version of the “Coastline Problem,” which you can read more about on Math Munch here, the gist being that the length of a coastline, due to its fractal nature, depends heavily on the scale at which it is measured. )
I actually got the chance to bring this up in one my classes this past school year. We were studying exponential growth and decay, having gotten to the idea that a simple exponential function f(x) with power x is increasing if the base is greater than one and decreasing if the base is less than one. I had some extra time and decided to try out looking at the Koch curve with one of my classes.
Everything was going well. Students whiteboarded the first few stages in groups and the whiteboards were great, since they allowed easy erasing of the middle of each segment. Many students were quick to figure out that the Koch curve has 4^n segments at the nth stage, and that each segment is of length (1/3)^n at the nth stage, resulting in a total perimeter of (4/3)^n at the nth stage. Everything was awesome; we were doing some real mathematics and hitting up some necessary skills at the same time. Students got into making their drawing as precise as possible and a few groups even made it to stage 5 of the curve.
But then the questions and comments started coming:
“Why are we doing this?”
“Is this a separate standard or part of the exponential functions standard?”
“I hate drawing. I wish we were doing regular math.” (No kidding)
I slowly started to lose my audience, but I dragged them through the discussion, made the mistake of trying to share the infinite perimeter idea by just telling them, and we never really came back to it.
What went wrong? I know the content cold, definitely brought my own enthusiasm for the subject, and I think that playing with the Koch curve has a lot of promise as a classroom activity that builds basic skills while also exploring something weird and cool. There were definitely some small rookie mistakes I made in terms of execution, being a first year teacher, but there are some bigger issues with my first-year classroom that this episode illustrated for me.
1. No matter how much I like something, that doesn’t make kids like it. Enthusiasm is a necessary but not a sufficient condition.
2. I love and value recreational/weird/off-the-beaten-track mathematics, but one of my biggest obstacles at introducing more of that sort of math into the classroom might very well be my own students’ opinions about what math is and what it is not.
3. What sort of message does it send to students when we only do “fun math” when there’s some extra time left over in one period due to a fire drill in the other class? Students are not going to shift gears suddenly, so exploring ideas like the Koch curve really has to be a regular part of class if it’s going to be included at all.
These all seem pretty obvious even as I write them now , but I was just so surprised by how the whole activity fell apart at the end.
One of my many resolutions for my second year of teaching is to work at introducing topics like the Koch curve, but without the assumptions I had the first time. My kids didn’t like it in some cases and didn’t see it as “real math” in many cases; this is not an excuse for dropping it from my teaching, but rather the reason why I should give them more experiences that are confusing, artistic, and weird.