I recently read "In Search of Understanding: The Case for Constructivist Classrooms" and I was really caught by this quote:
"Piaget suggested that the creation of new cognitive structures springs from the child's need to reach equilibrium when confronted with internally constructed contradictions; that is, when perception and "reality" conflict."
"Nussbaum and Novick suggest a three-part instructional sequence designed to encourage students to make the desired conceptual changes. They propose the use of an exposing event, which encourages students to use and explore their own conceptions in an effort to understand the event. This is followed by a discrepant event, which serves as an anomaly and produces cognitive conflict. It is hoped that this will lead the students to a state of dissatisfaction with current conceptions. A period of resolution follows, in which the alternative conceptions are made plausible and intelligible to students, and in which students are encouraged to make the desired conceptual shift." (p. 298)
We started our unit on exponential growth yesterday and I'm pushing myself really hard to take this approach to my curriculum design. Here's how it has played out so far:
1. I opened with this population counter and asked students to make their best estimate about what the counter would read at an exact date and time. They were just watching the counter so it was pretty difficult to get any data or make any calculated guesses. | 
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2. I asked them how they thought we should break down the problem and what information they would need. The responses were somewhat varied, but most hovered around something like the response at the right. It was obvious that they were assuming a linear/constant growth (their "perception"). | 
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3. Based on their existing perception, I gave students (essentially) what they asked for and prompted them to make some predictions about the next few decades. | 
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4. Then, I revealed this slide. You can imagine their surprise when their constant growth didn't match up with "reality." I think this is what Nussbaum and Novick mean when they say "cognitive conflict" or when Piaget talks about "when perception and reality conflict." | 
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5. I asked them to revisit their predictions based on what they noticed in the new data. There was some variation in their response, but the common theme was something like the image on the right. | 
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This is where we left off for today. In my opinion, students have made the cognitive shift I was hoping they would. They came in thinking population was linear and left knowing that it isn't. They also have some intuitive ideas about how to make predictions based on how it actually is growing. It's sooooo tempting to step in and explain to them about percent increase, exponential growth, and (for some) derivatives. But, if I did that, I would completely rob them of the opportunity to construct that understanding for themselves. I do know one thing…they're about ready for this awesome video tomorrow!
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