What exactly do I want to know? That question has been rattling around in my head a lot since reading Mark’s post on exit cards. In his post, he talks about different kinds of exit cards – those intending to engage a student’s metacognition, those examining procedural knowledge, those targeting concepts, and those intended to clarify misconceptions. While I try to incorporate all of those things into my lessons and my assessments, I really haven’t been as intentional in thinking about the kind of thinking my exit cards elicit as I might be.
As I have been teaching my students about the GCF and LCM, I have tried to incorporate a wider array of exit card types.
Following lessons on how to find the GCF and LCM, I gave my students two numbers and asked them to find both the GCF and LCM. This was intended to assess whether they could correctly find the GCF and LCM I think knowing where students are with regard to a skill is important. Knowing a skill isn’t enough, though.
My primary curriculum is Connected Math. They have several great investigations in which students explore problems utilizing the GCF and LCM. They pose real questions and students solve the problems. The questions explore how the GCF and LCM are used in real contexts but don’t specifically direct them to them as solution paths. Students determine when two ferris wheels of different sizes will align, when two different species of cicadas will simultaneously emerge, how many equal-sized snack packs can be made from a given set of apples and packages of trail mix, and how many students a set of juice boxes and crackers will feed.
Following this exploration I asked students what it was about a problem that would indicate that finding the GCF would be a good solution path and what it was about a problem that would indicate that finding the LCM would be a good solution path. (This is my version of one of the CMP reflection questions at the end of the investigation.) I wanted to know if they had grasped the fundamental understanding that the LCM is used to solve problems of alignment and that the GCF is used to solve problems of grouping.
Each time that I posed this question (I teach four sections of this class), I left plenty of time for students to answer it and also for us to debrief the question. I wanted the individual information, but I also wanted to de-privatize student thinking. After about five minutes of thinking and writing, we discussed the question. It was eye-opening for me. Every student had been able to work through the investigations pretty successfully, but only about a third of them were able to articulate how they knew what approach to take. This was surprising because of the discussions that we had during each investigation. When we explored the question of ferris wheel alignment, I knew which students had jumped to the LCM and intentionally did not have them talk about their thinking until the end. Instead, I drew out students who had struggled a little, who had to draw a picture or a model to figure out what was happening. Only after we talked through their thinking, did I call upon students who had seen finding the LCM as an appropriate strategy. I used a similar strategy for the other investigations. I expected that the act of modeling the situation would help the students who struggled a little bit with the problem to see that they were looking for alignment. Yet, when the time came, it was difficult for them to articulate it. I’m not sure what this says about their depth of understanding vs their ability to translate their understanding into words. I am certain, however, that posing the question and asking them to respond to it pushed them to think more deeply and that the ensuing discussion was more important for them than it was for the ones who were able to readily answer the question.
Having taught these concepts for several years now, it has become clear that students confuse the GCF and the LCM fairly frequently. They don’t seem to deeply grasp the fact that the GCF is a factor and must be less than or equal to the numbers and that the LCM is a multiple that must be greater than or equal to the numbers. They don’t always make sense of the problems they face and don’t have those alarm bells screaming that something doesn’t make sense when they try to find/use the wrong one. Over the last few years, I have tried to make this thinking a lot more explicit as we discuss the concepts in class and on the quizzes/tests that I give. Last week, I posed this question on a quiz:
- Choose all that apply. When finding the GCF of two numbers,
- the GCF is always less than the two numbers
- the GCF may be less than the two numbers
- the GCF may be equal to the two numbers
- the GCF is never less than the two numbers
- the GCF is always greater than the two numbers
- the GCF is never equal to the two numbers
Despite my efforts to draw this out in our previous discussions, the response was only fair. I decided I needed to keep working with this with my students so I incorporated the same sort of question into an exit card following a lesson dealing with fraction subtraction. This is the exit card.
As I have explored using different kinds of exit cards, I have learned a lot about what my students know and what they don’t know. I think looking deeper gave me a better understanding of my students’ thinking and gave my students a better understanding of the math.