The team I work with at AIMS has begun working with students on fractional understanding and it has been an interesting couple of weeks. We are seeing the students have opportunities to learn things we never intended but still fit right in with expectations in elementary classrooms. Personally, I have discovered that when we develop tasks designed to engage the students with mathematics rather than telling them about mathematics, they learn so much more than we intended.
The task we have been working with uses strips of paper that we pretend are candy bars to be shared among a certain number of people. The students are asked to find the size of one person’s share. This will then lead to counting using fractional language and begin developing the concept of parts out of a whole. Sharing tasks help students begin thinking about fractional meaning because they are familiar with sharing in their day-to-day activities. When they are asked to share, it initiates a need for the pieces to be equal. The next step is to use up the whole candy bar with the pieces. We didn’t anticipate this would be a struggle for children, but the students in the initial stage of this development did struggle to think about same size pieces while also trying to “exhaust the whole” candy bar. I will be writing more in future blog posts about the task, and what we are doing in particular, as well as what the students end up learning regarding fractions. For now, though, I want to talk about what they are learning that we didn’t intend.
The first struggle I had with this task as a teacher was the terminology. Phrases like too big, too small, or just right. The students would make a guess at the correct size of one person’s piece and then have to prove if it was the right size or not. They would get three tries in all, which required them to think about whether the guesses in their first and second try were too small or too big. Another challenge involved thinking about what to do with the next piece if they had made their piece too small initially. After only two visits with the students, they are now able to think about the relationships between the pieces, the results, and next steps with relative ease. We didn’t intend for this to be an area of growth with the task but the students struggled their way to understanding these things as they approached the task.
The students also were highly engaged in attending to the precision of their pieces and proving their piece was appropriate. This allowed them to learn about estimation and measurement, as well as Common Core Mathematical Practices. What started as a fraction task turned into a rich experience from which the students were able to develop multiple new understandings.