The title “Low Floor and High Ceiling” seems to be the gold standard with regards to the type of problems we should all be presenting to our math students. This label is given to a task that is easily accessible to everyone and can be taken to high levels.
My students will often get caught up in the problem wording or are overwhelmed by multiple steps and don’t really know where to begin. These students just need wider (not necessarily lower) access to the problem. Additionally, many students see multiple-step problems as boxes to check off as they complete each step rather than exploring the mathematics of the problem. When students finish the assigned problem parts, they leave the “mathematical” room no matter how much was left to explore.
I am not necessarily looking for low floor, high ceiling problems, but rather how to widen the threshold and keep them in the room.
CPM is a problem-based curriculum, so the rich problems I needed are already written. I set out to explore how to open these problems further to make them more accessible, increase the exploration, and still meet the lesson goals.
Two ways I tried opening up problems:
- I removed the subparts (parts a-d) and asked students to explore the mathematics in the problem. I used the a-d parts as pocket questions to ask as needed during circulation. This helped me to be sure all teams were hitting the mathematical goal. Removing the subparts seemed to be most successful when I gave students some time as a whole class to notice and wonder about the problem before sending them off to their teams. I found that I also needed to spend some time really exploring the problem before giving it to my classes. I was so used to the curriculum questions, I didn’t always explore the many directions students would go.
- I explained a situation (usually orally and/or with pictures) and had students make up their own questions. Students were very engaged in both creating their own and solving other teams’ questions. The only tricky part for me was how to quickly assess their learning. Since the teams all had different questions, they all had different answers. I did find I was really listening to their thinking, though it was tough to hear from all the groups in the allotted time for the lesson. I found this strategy most successful if students recorded their responses through Flipgrid, so there was time for peer assessment as well as my own in video form. I usually had to wait until the next day to close a lesson, but it was very powerful to reference student responses that explained the lesson goal.
Observations of opening up problems:
- Open problems worked best when teams did their work on a common, non-permanent surface (whiteboards) to encourage the exploration and allow for easy revision and addition to their thinking. Having students make their work visible for all was also really helpful for the ensuing class discussion; we quickly arrived at a common understanding of the main ideas for the day.
- Students stayed on core problems for the same amount of time (the time needed for my last team to hit the learning goal) and no one “finished” first, because there was not a set number of problems to finish. This kept the class together and at the same time allowed me to differentiate my teaching with varied questions for each team.
- This benefit led to another equally important result; student misconceptions about speed being connected with mathematical thinking has changed. With a finite number of problems or subproblems, students always knew who finished first and measured themselves accordingly. In this case of open problems, if a team says they are finished, the perception is the opposite; they didn’t investigate the problem deeply.
- Students engaged in and explored mathematics more. Not confined to a set of questions, my students valued exploring a concept rather than completing the work.
- Communication and collaboration increased within and between groups. A student couldn’t just read a problem, write the answer and move on.
- Ownership increased in both learning and mathematics. When students wrote questions or found mathematics in a situation, they made a personal connection, rather than reading and answering a pre-written question.
- My personal lesson planning changed to include more differentiation in my lessons. This has been especially true for my students with a great deal of math background. I think and plan what questions I can give to them to push their thinking further.
In exploring how to open problems up for students, I feel like I have widened my own threshold for teaching mathematics, and will definitely be staying in the room to explore more ways that this approach will benefit my students!