In Calculus today, I had students work in groups to discuss questions related to the conceptual understanding of first and second derivatives. (If you recall from a previous blog, many of my students did poorly on this section of their last quiz.) Students were given the graph of the second derivative and asked questions like, "Where is the original function concave up?" and "Where does a maximum occur?" Some students understood quickly, but other students still had trouble understanding that what they were viewing was a graph of the derivative and not a graph of the actual function. At one point, I realized I could have written the question at the left better. It should have read "on what interval(s) is the derivative negative". Some students were thinking in terms of ordered pairs or points and not intervals. By the final question in our set of 5 questions, almost all the students were able to explain how the various parts of the graph of the derivative were related to the graph of the original function. Why did students finally get it? I think a main reason was students were discussing the answers. Students who understood the concept worked hard to justify their thinking to their classmates. They wanted them to understand it, too.
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