Recognizing Is Not The Same As Understanding

My previous blog post, “Sharing can be Rigorous Work!”, I shared that part of the work we have been doing in the last few months investigates how students develop fractional reasoning based on the work of Dr. Les Steffe. When I taught 2nd grade under the previous California Standards it included exposure to fractions where students had to “recognize” fractions as part of a whole or part of a set. With the new Common Core State Standards formal introduction to fractions starts in third grade where the standards state that students understand the concept of a unit fraction.  Usually in 2nd grade, and sometimes in 3rd, students’ work with fractions activities that look like this: 





These two worksheets only focus on recognition of fractions, matching a model (a rectangle partitioned into four parts with one shaded) to a symbol (¼). What is wrong with this? The models are already partitioned for the student.

In our work with 2nd and 3rd-grade students this spring the first task we had students engage in was sharing a candy bar among 3, 4, 5, or more friends. The focus was on having the students exhaust the whole candy bar and to make sure that each piece was the same size. To an outsider, it may have looked like students were just cutting strips of paper, but we observed students staying engaged throughout all our sessions together. You could almost see the wheels turning in their minds. When we began, many students struggled to focus on the goal of using up the whole candy bar, or the goal of making equal pieces…but not both. As we continued to meet together, many of the students were able to focus on both of these goals at the same time. By the end of our sessions, the 2nd and 3rd-grade students were also able to tell us whether the new pieces would be bigger or smaller when changing the number of people with whom they would be sharing the candy bar. For example, if they shared a candy bar with five people and the next candy bar was shared with four people would the pieces be bigger or smaller? Students understood that the pieces would be bigger.  This is huge!

This is a concept that students struggle to understand.  Yet we had students explaining to us how they knew this to be true based on the experiences they had when sharing the candy bar. Think about how much more of an understanding students would have when working with fractions if they engaged in partitioning on their own instead of just recognizing and matching the model to a symbol.  We want our students to understand fractions, not just recognize them!

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