# MATHEMATICAL MUSINGS by mathteacher24

These are some of my thoughts about teaching high school mathematics. My blog focuses specifically on lessons and activities that I have done in my classroom with my students.

## Friday, December 15, 2017

### Teach 180: Conceptual Understanding in Calculus Revisited (Day 71)

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.

## Thursday, December 14, 2017

### Teach 180: The Two Hour Delay (Day 70)

Teachers are in a caring profession and we frequently look out for others, often at the neglect of our own health and wellness. Today we had a 2-hour delay and rather than getting extra work done for school or getting caught up with housework, I chose to take care of myself. I slept for an extra hour and then did a cardio workout in my basement. (A graph of my steps is at the right.)

That helped me to be ready to take on the day, reschedule a missed meeting, meet with four students, give a test, attend a faculty meeting and attend an optional meeting called "Faculty Fellows". This month's topic is facilitating class discussions and faculty read excerpts from "A Classroom Revolution: Reflections on Harkness Learning and Teaching".

In case you are wondering how my Calculus class did with the activity I described in my previous blog, the first two classes were dropped today due to the two hour delay. There was no Calculus class today, but I will be doing the Plicker activity with them tomorrow. Tune into tomorrow's entry to see how we did.

That helped me to be ready to take on the day, reschedule a missed meeting, meet with four students, give a test, attend a faculty meeting and attend an optional meeting called "Faculty Fellows". This month's topic is facilitating class discussions and faculty read excerpts from "A Classroom Revolution: Reflections on Harkness Learning and Teaching".

In case you are wondering how my Calculus class did with the activity I described in my previous blog, the first two classes were dropped today due to the two hour delay. There was no Calculus class today, but I will be doing the Plicker activity with them tomorrow. Tune into tomorrow's entry to see how we did.

## Wednesday, December 13, 2017

### Teach 180: Assessing for Conceptual Understanding Fail (Day 69)

What do the following numbers represent?

Yesterday I gave my students a quiz to test conceptual understanding of first and second derivatives. The maximum possible score on the quiz was 16. You can see that a few students had a strong conceptual understanding of these ideas, but many didn't. What type of conceptual difficulties did they have?

Let's look at the back of the quiz first. (Note: The front of the quiz was low level vocabulary questions and calculating the second derivative of f(x) = 1/x to determine concavity.)

Yesterday I gave my students a quiz to test conceptual understanding of first and second derivatives. The maximum possible score on the quiz was 16. You can see that a few students had a strong conceptual understanding of these ideas, but many didn't. What type of conceptual difficulties did they have?

Let's look at the back of the quiz first. (Note: The front of the quiz was low level vocabulary questions and calculating the second derivative of f(x) = 1/x to determine concavity.)

The other questions I asked were:

- Based on the first derivative, when does the original function have tangent lines with negative slopes? Explain how you know from the graph.
- Based on the second derivative, does the original function have an inflection point? Explain how you know from the graph.
- Based on the second derivative, when is the original function concave up? Explain how you know from the graph.

For some reason students thought that the graph on the left was the original function and spoke about derivatives of the parabola. When some analyzed the second derivative, they said that there was no change in concavity, because the graph had a constant slope and since the slope was positive, the function was always concave up. Others talked about the function having a minimum at (0, -3). Although the first derivative has a minimum at that point, the original function does not have a minimum at that point.

Based on what students wrote, you can see that they have been in a calculus classroom. However, they had a hard time making the connection between the calculus ideas (concavity, inflection points, extrema) and the graphs of the derivatives.

Tomorrow, I will be starting class by assigning students to small groups of 2-3 students with one of the top scoring students in each group. Then, I'll have them do some similar questions using Plickers. Having students discuss the solutions with their peers should help some of them to get a better conceptual understanding of what is being shown on the graphs for the derivatives. I also have a derivative matching activity and I'll probably use that on Friday or Monday.

I called this blog post "Assessing Conceptual Understanding Fail", but actually it helped me to uncover some of the incomplete understanding students have. Learning about what my students know/don't know and understand/don't understand and modifying my teaching based on that is a win.

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