Every graph tells a story, but can you “read” that story? Do you see the beginning, middle and end (if there is an end)? Do you see how the story builds and changes? These questions are at the heart of a unit I teach my 6th grade students. (The unit is a Connected Math unit called Variables and Patterns). They were also at the heart of a lesson I taught recently in which student were asked to interpret a graph, to read its story.
To begin, I wanted students to zero in on the key elements in a graph that reveal the story but I didn’t want to tell them too much. I decided to try using a chalk talk, which is a technique in a book I have been reading with my PLC, Making Thinking Visible . In the chalk talk, students respond to a prompt on a large sheet of paper. Students “talk” about the prompt by writing on the paper. This allows them to respond to the prompt and also to the responses that other students have written, hence making thinking visible. I decided that my prompt would be each of the six graphs that my students would need to interpret.
I drew each of the six graphs and posted them around the room, each hanging above a blank piece of poster board. As students entered, I assigned them to a specific graph station. I asked them to look at the graph and write what they noticed about the graph. After everyone had completed the task, I had them rotate to another graph station. This time, they could either write about the graph or respond to what someone else had written.
The six graphs were large scale versions of the graphs shown below.
In this phase of the lesson, I was hoping that students would look for markers that tell the story of the graph: what is the initial value, how does the graph change (e.g., is it a constant rate of change or a variable rate of change), is the graph increasing or decreasing in value, does the pattern in the graph repeat, is there an “end”. I wanted to draw out student thinking on these ideas as a launch into the lesson. Since this was the first time that I was using a chalk talk, I didn’t really know what I would get, though. It was a new process for my students, so they wouldn’t have the benefit of experience with the process to guide them and I wasn’t sure if I was presenting the task well since it was my first attempt with it. As I looked at their work, I could see the evidence of our inexperience. The responses varied widely Some students gave responses along the lines that I expected. Some students made up a story for the graph. A few students responded to other students’ thinking, but most used the thinking of other students to create their own elaborate stories. Since I got results I hadn’t anticipated, the debrief of the task was a little different than I expected and didn’t really focus on some of the things I had hoped it would. I didn’t try to force it though because I wanted to give students the chance to make meaning rather than me giving them meaning. I liked the technique as a launch, I just need to pose the prompt a little better next time.
After the debrief of the chalk talk, I had students work in table groups to match the same set of six graphs to a set of seven different stories. I gave them graphs and stories on a set of cards, so it was essentially a card sort. As students worked in their groups, I told them that they must work silently. One person would match a story to a graph. The next person could either match another story to a graph or could change one of the matches on the floor. I did this to ensure equity of voice in the group, the silence and turn taking ensured that no one could dominate the group and that each must contribute. I also did this to encourage students to analyze the choices that other students were making. The card sort used the graphs and stories shown below.
After each group had “completed” the matches, I had the groups do a gallery walk. During the gallery walk, they went around to each of the other groups and considered the choices that the other groups had made as they matched the stories to the graphs. During the gallery walk, they could discuss what they saw with their group members but could not make changes to the work that they saw. At the conclusion of the gallery walk, each group was given time to discuss the choices they had made with their own card sort and make any changes they wished to their group’s card sort.
Finally, we debriefed the card sort. We returned to the large scale graphs on the walls that we had used for the chalk talk. I read one of the stories and asked one of the groups to share which graph they believed it matched. They had to identify the independent and dependent variables and explain how the changes in the dependent variable reflected the story. They also had to give the graph a title. Other groups then had the opportunity to comment or question the group’s response.
One of the things that I really like about this lesson is that students grapple with hard questions. There is the potential for them to wrestle with misconceptions. As they discuss the graphs and stories, their misconceptions are exposed. For example, someone always suggests that one of the parabolas is the amount of daylight over the course of time. Someone always then raises the question of what it means for the amount of daylight to be zero (either initially or in the middle of the graph). At that point, there is always this lovely “oh!” moment in which students realign their thinking.
Next year, I do want to make a couple of changes to the way I did this. First, I will re-work my directions for the chalk talk. In addition to the graph, I will write a prompt asking them to respond to specific components of the graph (initial values, how the dependent variable changes as the independent value increases, etc). Second, I will change the way that I do the card sort a little bit. I think I will give each group a set of dry erase markers and require them to add labels and a title to each of the graph card as they match it to the story. I am curious whether seeing those things written on the graph will help them see their misconceptions before the whole class debrief.