Meeting Alan Schoenfeld and hearing about his work spending three years analyzing one hour of video trying to really understand what math teachers do might be inspiring me to excessive reflection. I keep wondering about what kids are thinking, what I am thinking and what types of mental processes are they actually using to solve problems.
We were finding the volumes of various objects this week in Calculus. Students brought in a variety of fruits and muffins so that they could take measurements and calculate volumes. The “stop bluffin with your muffins” group wanted to know if mini muffins or large muffins are a better value when considering volume. We found out that mini muffins are a better value when you consider volume but not when you consider weight. I asked which one tastes better, since that is probably a lot more important than price. This question led to the realization that the higher the density of a muffin the better it tastes. This turned into a graph of taste versus density and some fantastical math related to our imaginary taste scale. It was awesome to be able to actually hold a muffin and point out the radius and try to decide which part we wanted to be the x-axis and what the cross sections would look like. The upcoming math ed researcher in me wonders what benefits using a manipulative like this has on understanding of the volume process. I know that kids can check to see if answers are reasonable by making an estimation-or as I suggested to the group trying to find the volume of an iphone-just drop your object in some water and measure! They iphone group wanted to know about processing power versus volume but gave up because they couldn’t figure out how to write functions to model the curved edges.
Another group who wanted to find the volume of an ellipsoid(see Wikipedia) did not give up even when they discovered that solving the problems was going to involve double integrals and that they’d need to learn a bit of Calculus III to finish their project. I ended up invested in the problem and had to relearn double integrals so that we could figure it out. I cheated a bit with google as well. There are forums online answering almost any question you might expect in a traditional Calculus class and I was reminded of my future professor who studies what type of interaction happens in online forums. After we’d set up the integral I realized we needed to know trig substitutions from Calc II and thought that it was crazy to expect them to learn another new chapter so we defaulted to using the super simple formula for an ellipsoid which is as easy an simple as finding the volume of a sphere using 4/3 pi r^3. It felt a little ridiculous to do all that work just to find the final answer with a one-step formula but I had plenty of fun trying to explain that a double integral represents volume and a triple integral hypervolume! They persisted through all this math because they really wanted to know the volume of a genie lamp to see how much space a genie had to live. They were adamant about pointing out that it was NOT a tea pot and that it is called a lamp because you can light it to provide illumination.
I definitely noticed that the project was a learning experience for the students who were still really struggling to relate the mathematical notation of integrals to a real-world volume. Since I’m counting this project as the assessment I never know what to do with the conversations where I realize that a student is super confused about the fundamental concept of adding infinitely many infinitely small cylinders to create a volume.
Now we are on to talking about infinite sequences and series in Calculus. The concepts of convergence and divergence are crazy and kids have all sorts of interesting ideas surrounding these topics. They always want to know what infinity minus 1 is equal to. We’ll see how discussion of convergence and divergece goes. So far we talked about the Koch snowflake and I think(hope) that they can comprehend how adding additional small pieces sometimes creates infinity and sometimes creates a number. This snowflake is great because it has a finite area and infinite perimeter so kids are immediately curious. I honestly need to do some work to figure out what types of ideas they are gathering from all my random infinity factoids and problems.