As you may know from my posts planning out the skills and projects for my fall course on geometry, I recently completed a 2-week project on fractals with my students. I had never taught fractals before, so most of this experience was completely new, and I have a lot that I could have done better.
I plan to follow this post up with discussion of some successes of the fractal project, but today I plan to focus on the negative aspects.
- Poor motivation. After the first two days introducing the Sierpinski Triangle, the Sierpinski Carpet, and connections between the former and patterns in Pascal’s Triangle, I put together a presentation about how fractals actually represent growth and objects in the real world better than many basic geometrical objects. While there was some pushback about this (stop signs are octagons, buildings are rectangular prisms, the Koch snowflake doesn’t look like real snowflakes), probably stemming from my confusion of ‘natural’ with ‘real-world’, the students were mostly intrigued by this presentation. However, most of the actual pieces of the project, like drawing a Sierpinski carpet and computing its area at each iteration, didn’t seem to connect with the real world or my presentation.
- “This is too much work.” N0w, I’ve heard that before, but at times during this project students complained that I was asking them to do too much drawing, too much measuring, too much calculating, too much reading, and too many practice questions. Moreso than other projects so far this year in geometry (volume, tessellations, & similarity/scale)
- Reading = hated with a vengeance. I came across what I felt was an excellent explanation of fractal dimension, at a reading level I believed was appropriate. So after a brief introduction, I asked the students to read, ask questions, and then do some practice with finding dimensions of various objects. This went horribly. I think — to some extent — because they were being asked to read (of all things!) in a math class at all, an idea to which all members of the class were vehemently opposed. Or maybe I misjudged the text’s reading level and accessibility. Perhaps fractal dimension was too difficult a concept to have them read about independently, even if the writing was at the right level.
- Fractal Dimension. Partly due to the point just discussed, but even with revisiting and reteaching, I did not manage to teach this concept clearly to my students. The logarithm part of the formula was a mystery (not taught until Algebra 2), the idea of any of these fractals being infinitely realized (instead of stopping at a particular iteration on the way) was not accepted by my students, and what to count in terms of number of copies and scale factor (two pieces of the formula) was rarely clear to them. I feel that this part of the project was by far the weakest, and I welcome suggestions for how I could improve it.
- SBG Skills Quizzes. It started with me formulating an extremely vague skill#12 back on my original skills list. “Students will demonstrate an understanding of fractals and self-similarity.” I mean, come on. In practice, I decided that meant that students would follow an iterative process to draw unfamiliar fractals. This ended up seemingly testing their understanding of the words used to describe the procedure rather than fractals or self-similarity. While the first two quizzes on this skill went poorly, I think I succeeded with the question (and my students succeeded with their answers) on the third. Still, I welcome any advice on what sort of quiz question would better fit my nominal skill, and/or how to phrase the skill better.
OK. Tomorrow: what went right with the Fractals Project. Stay tuned for the exciting details of computer-based exploration and cross-curricular collaboration!