Throughout my teaching career, I’ve always told students that teaching them to be nimble problem-solvers and good critical thinkers is high on my priority list. However, when it comes time to test the students, I have struggled to find an appropriate way to genuinely and fairly test problem-solving. When I include a novel problem that requires some creativity, my students perceive it to be an unfair, “trick” question. Of course, I emphasize problem solving in collaborative projects, but 30% of my course grades are usually from tests. If problem-solving is really important to me, I ought to find a way incorporate it into my major assessments.
On a recent regular- track Pre-Calculus test, I think I’ve made some good progress. The first 80% of the test was fairly traditional and non-calculator. I think I tapped into some higher-order thinking skills, but not much problem-solving. When the students had completed this section, they turned it in and got the following two problems.
The first problem requires the students to imagine how to complete an even function. In class, we identified even functions and proved that certain functions were even, but I never asked the students to do this. Also, we practiced how to reflect a function over the x-axis, but we had never considered how to reflect over the y-axis. The second problem asks the students to consider the absolute value as a transformation. This is also something that I might have considered in an Honors section of Pre-Calculus, but we hadn’t talked about in this class.
While the students were working, I walked the class and watched them explore. As I had hoped, the students were trying many different things. A few students needed some encouragement to try many varied examples and that is not surprising given how little they have explored mathematics in their high school careers. On the second problem, there were a few students who had used the function from first example and said, “Mr. Hinkley, the transformation doesn’t do anything.” With some encouragement they tried a wider variety of functions.
Overall, I was very impressed with the results. They spent quite a bit of time working on these two problems and were largely successful. I got some very detailed explanations of how the absolute value transformation worked. Rather than following a repetitive process like transforming a simple function, the students were engaged in real mathematics.
My hope is to follow this format for all the rest of the tests this year and see if the students are able to progress in their skills.