I thought a lot about lesson design today. Our school enrolled in Key Curriculum Press’s service which provides pre-made Geometer’s Sketchpad (GSP) lessons with worksheets. The one we did today allowed the students to change the limits of integration graphically by changing the amount of area under a curve that is calculated. As they drag a button to increase the area under a curve the area is plotted as a function of the upper limit of integration. it’s easy to see that the area under a constant line is linear and the area under a linear function is quadratic. I’d say 30 % of kids saw the connection to derivatives when they saw the area graphed in the same place. it was pretty exciting. I told them that they were as smart as Newton. Which was probably a lie. One who figured out the relationship particularly quickly said it was because they were studying area under the curve in physics. Another pointed out that the timing was perfect because while we learned methods to find area under the curve in Calculus they were finding physical interpretations in physics class.
I did get annoyed a bit at the worksheet which seemed to me resulted in kids getting bogged down in unimportant details. The problem with technology is that they end up figuring out relationships that don’t end up being something that gets study in first year Calculus. It’s really hard to design a program and a list of questions that go with it that focus kids on the key points and not the fun of moving random graphs around. I definitely decided that there was not enough white space of the GSP worksheet and that all of the text made it too many things to really concentrate on in the time given. So I told my kids to use the worksheet as a guide, to skip problems if they felt like it and to focus on finding a formula for the function of the area plot.
We summed up the day with me explaining why it was so crazy that the area under the curve followed formulas we’d developed for slope. Like, change the world amazing. Best thing that happened in the 17th century amazing! As a result of visiting Berkeley this weekend I decided to start asking kids more often what was going on in their heads. I asked them if the relationship between area and slope seemed random or impressive. One said that it seemed obvious. I took them back to the limit definition of the derivative and the integral and asked again if it seemed obvious those things were connected and maybe so they could go to lunch they agreed that it was special.
I remember thinking the Fundamental theorem of Calculus was anti-climatic because it was obvious that integration was the opposite of differentiation because that is how our teacher explained it when we first saw it. I hope my kids see it differently. Tomorrow we need to take all of our conjectures and use some summation notation to look at how we write all these things rigorously. My kids have been involved in a few days of semi-guided exploration and it will be interesting to see how that might change their ability to interact with formal notation. I hope that they actually get the point of it instead of seeing it as a bunch of crazy symbols. I hope by having them problem solve about area using their own reasoning they see that these were conclusions arrived at logically.
One kid asked about the proof of the Fundamental Theorem of Calculus and I had to admit I’d forgotten it. I better look that up for next year!
So Research Questions for the day:
How does having kids figure out their own methods of finding area affect their conceptual understanding of integration?
How does teaching anti-differentiation as a guess and check process affect self-efficacy when faced with an integral problem?
What is the optimal design for GSP activities surrounding the derivative?
How can the design of applets help focus kids on key points?