Thank you Math establishment… for designing a curriculum that requires us to keep plowing through material. For connecting teaching salaries to student test results and for keeping so much pressure on our class time that it feels like all we can do is skill and drill. For creating a culture in which students think that math is only a series of formulas that they need to memorize, repeat, use on the test, and promptly forget.
Thanks! No really… Thanks! Because you’ve set the stage for kids’ minds to be blown when they are given real opportunities to study interesting mathematical questions. This is a long post, but I had such a good experience, that I wanted to share details. Skip straight to the bottom if you want to take the materials and run. I promise I won’t judge.
In January, and again in March, my Expeditionary-Learning High School dedicates a full school week to “intensives.” Students choose from a list of course options and engage in in-depth study for five days straight. This January, I led a study of Fractal Geometry. A few kids signed up because they were interested in the subject already; more had math credits to recover. But in spite of themselves, they had a blast.
Although this was far from a skill and drill study, it was rigorous work. I’m all about learning targets. This study had to be differentiated, as I had a group of students from 9th -12th grade all working together. I teach college art as well as middle and high school math (I’m thinking of getting my 7-12 art certification), and as an art teacher, differentiation is often easy and natural. Ideas and techniques become more sophisticated, but a beginner or a master can engage with creating a portrait. The umbrella of Fractal Geometry cast a wide shadow, and allowed for many points of entry.
I’ve been especially interested in the current conversation in math education about what makes a problem “real world.” Dan Meyer has distilled some of the more interesting arguments HERE. I think that Fractals make for “real world” math study within multiple definitions. Self-similarity is everywhere around us and is easy to spot – it’s actually hard NOT to see it once you’ve identified the property, and kids were pointing all over the place “…look a fractal!” In his book, Fractals Everywhere (2000), Michael Barnsley puts it well:
Fractal geometry will make you see everything differently. There is a danger in reading further. You risk the loss of your childhood vision of clouds, forests, flowers, galaxies, leaves, feathers, rocks, mountains, torrents of water, carpet, bricks, and much else besides. Never again will your interpretation of these things be quite the same.
Kids were seeing fractals in nature, and in special effects, but they were also highly engaged with the abstractions that only exist within the math. I had multiple kids emailing me with discoveries late into the evenings.
I’ve included all of the materials and plans below in case they might be useful for you. Here are some (not so brief) highlights from the week:
We’re in Portland, Maine, so I began with a local version of the coastline problem: How long is the coast of Peaks Island? I printed maps of Peaks, and asked kids in pairs to use progressively smaller and smaller rulers and to try to answer the question. We collected our data, recorded it on a chart and examined the results together. “Wait a minute. The coast gets longer and longer depending on the ruler!” I let this sit in the air, and just moved the discussion forward by asking questions without adding my two cents. Showing is so much cooler than telling! It felt poetic to begin our study with the problem that brought Mandelbrot into the spotlight. We visited his famous paper later in the week.
You can see the care put into this visual exploration of iteration!
Next, we began an informal study of the Koch snowflake. Students drew the first few iterations, and we generated a whole bunch of questions. I nudged the kids who were ready to examine what happens to the perimeter and area as this shape is iterated. Again, I was deliberate at this point about not doing any direct teaching. I left their questions unanswered, which helped to build drama and curiosity.
We spent the afternoon of the first day with our anchor “text:” the excellent NOVA special, Hunting the Hidden Dimension. Even for NOVA, this is a really exceptional resource; challenging, but highly understandable. It’s worth purchasing a copy for the better resolution, but you can find it on youtube HERE. I let kids know that their knowledge would be assessed through a piece of writing about one aspect of fractals, and gave them a note-catcher to record observations and questions. I let them know ahead of time that everyone would be expected to share some learning, and I’ve found that a graphic organizer like this helps kids to remember to record their thinking while they watch a video. We stopped the film about halfway through to share our first key learnings, and repeated this process at the end.
I heard some really beautiful comments amongst kids at the end of the day. “I see math totally differently than I did yesterday.” This is why I am really grateful to all those who tried to ruin math for these kids.
On the second day, we delved more deeply into the Koch snowflake. We learned about sums of infinite series, and the paradox of an object with finite area but infinite perimeter. Wait. WHAT? Yup! Minds blown again. I introduced Sierpinski’s triangle, along with some more “mathematical monsters,” (they loved this historical context), and we made some charts to examine some of the properties of these classical fractals. Work ranged from concrete counting of triangles to algebraic generalizations of nth terms to summing infinite series. Differentiation in math can be so tough, but this was a place where it really came together, and I felt like most students were really working toward their personal best. After a really rigorous morning, we spent the afternoon creating original works with two excellent sites: recursivedrawing.com, and Fractal Maker Express.
Day three, we looked through a “fractal library” I’d put together (I had a range of books; Fractals: A Graphic Guide was the one that really resonated in a broad way), and then played the chaos game. HERE is a very nice electronic version, but I think that this is very effective if you print triangles on transparencies, pass out dice, and let kids play manually. The chaos game has a truly unexpected result, and this comes together well when you stack the transparencies. I asked kids to make predictions (in writing) every few minutes, and monitored carefully so that when each one got it, they didn’t spoil it for the others. It was entertaining to see kids noticing the drawings of Sierpinski’s triangle all over the room. “Wait! It’s that triangle!”
Clint Fulkerson’s Division-1
I made a connection with a local artist, Clint Fulkerson, who uses the logic of fractals to create some really cool art. He was generous enough to let a herd of kids into his studio, and he was articulate about how his work uses properties of self-similarity and recursion. He rode a nice line between a casual attitude, and rigor in his work; one thing I hope the kids take away from this week. Check out his work HERE. Clint has a show at the Portland Museum of Art’s Family Spaces, up through this July.
Kids were really inspired by Clint’s work, as well as his work ethic. We spent the fourth day working on some individual and group projects. Some kids created a paper version of the dragon fractal, while others studied strange attractors and the Mandelbrot Set. We watched some excerpts from The Colors of Infinity, Arthur C. Clark’s 1995 fractal documentary. Wow, documentaries have come a long way in 20 years (…and special effects, thanks to fractals!).
Friday was for finishing up loose ends, publishing, individual reflections, group debrief, student write-up for the newsletter, and presenting our work to the school. One of the best moments for me was when one of the students opened up our presentation by saying, “I thought that fractal geometry was going to be dry and boring, but it was actually ridiculously interesting.”
Honestly, it’s hard to say how much of the week we spent doing rigorous math – it depends on how you define this. And I had the advantage that this week did not have to stick to the confines of “Algebra 2.” But I truly believe that any time spent on what some might consider “not mathy,” was more than made up for by their contagious interest and excitement about the topic, which gave me the leverage to dig deep.
Here are some materials and resources I used or created. Send me a note if you’d like a more comprehensive resource list. We use standards based grading, so you’ll see that the language matches “meets” and “exceeds.” I’m hoping that the school runs this intensive every year, so please leave your thoughts or additional resource ideas in the comments. Thanks in advance!
Fractal Entrance Ticket Day 2; Fractal Entrance Ticket Day 3; Fractal Entrance Ticket Day 4; Fractal Exit Ticket Day 1; Fractal Exit Ticket Day 2; Fractal Exit Ticket Day 4; Fractal Intensive Evaluation Form; Fractal Intensive Learning Targets; Fractal Vocabulary Blank; Hunting the Hidden DimensionNotecatcher; Peaks Island Tabloid; Sierpinski’s Triangle; Student_Newsletter_Writeup; The Chaos Game; The Koch Snowflake