Finally, fractions season is upon us. I’m sure there are people who dread it, but I’m not one of them. I love the complexity of teaching fractions, the way you have to choose your words so carefully around them, the way they act like other numbers even when you don’t think they will, the fact that they allow you to talk about the number line, the spaces in between and how they are the thing … the spaces are the THING! The very appearance of a fraction is just so intense. It looks like two numbers, but it’s one number! You just can’t beat all of that for teaching fun.
For the month of March I’m going to write a series of posts on my classroom experiences with the teaching and learning of fractions. As the math coach at my school I have the opportunity to work in 3rd, 4th and 5th grade classes simultaneously, so I get to really dig into the CCLS fractions progression with kids and teachers, and see firsthand the ways in which students’ ideas about fractions build, slip backwards, and leap forwards.
In January, I had the chance to see Phil Daro (one of the authors of the CCLS) speak at Teachers College in New York City. Phil said a number of things that stuck with me, but the one that I’ve been thinking about the most this past month came from a story he told about the authors’ debate over whether or not to include long division in the CCLS (“it’s surprisingly unuseful,” Phil said, “…these days, with calculators…”). When they consulted with a group of Japanese teachers on why they included long division in their curriculum, the Japanese teachers said something like, “Long division is such a wonderful context for teaching the unfinished business of place value.”
The idea of unfinished learning is a powerful reframing of “learning gaps,” a term I hear often (along with “strugglers,” “kids who can’t access this math,” and a plethora of other names for, well, regular kids who are learning math at a pace that doesn’t seem to match with the pace of the curriculum or the state testing schedule). It brought me back to a related sentence from Bill McCallum’s blog post of long ago that I still can’t stop thinking about: “Much unfinished learning from earlier grades can be managed best inside grade level work when the progressions are used to understand student thinking.”
I’ve been carrying the idea of unfinished learning as an opportunity, not a hindrance, and the idea of using the progressions to understand student thinking about fractions, into a whole bunch of 5th grade classrooms this past month. One thing I’ve seen is that when you welcome unfinished learning, instead of feeling threatened by it, you can get pretty excited about how much of it there is to explore.
Last week in one of my 5th grade classrooms, the teacher posed this question: Is 7 x ⅘, and ⅘ x 7, the same as (7 x 4)/5? Exploring these three related expressions, students drew tape diagrams to match each one and talked animatedly with their partners about their ideas. When they came together as a whole class to share their thinking, one student seemed particularly unsettled. “I know that 7 x ⅕ is 7/5, so 4 of those is 28/5,” Allison started. (Me: marveling at her application of the unit fraction work she did in 3rd and 4th grade to this new concept for her.) “But the tape diagram would look like this, and I don’t understand what that has to do with (7 x 4)/5.” She showed her tape diagram to the class and other students raised hands to try to answer her question.
“7 x ⅘ is the same as ⅘ x 7 because in multiplication you can switch the numbers around,” a boy said (Me: marveling at his application of the commutative property of multiplication, an idea introduced in 3rd grade, to this new situation.) (Also, this didn’t exactly speak to Allison’s question – have you noticed this is how kids often respond to each other? With something only sort of related?)
Allison still looked unconvinced. “What do you mean?” she said.
“You know, like 6 groups of 2 is the same as 2 groups of 6,” another girl chimed in. A little side discussion ensued in which several kids made the motion of turning a rectangular array one way and then the other to illustrate the commutative property. “And so if you were going to make a tape diagram for this one,” the girl said, pointing to (7 x 4)/5, “first you would make 7 4s, or I guess you could make 4 7s. And this other one is like 7 four-fifths. So it’s like you have 7 copies of 4 but now the 4 is four-fifths.”
Allison paused. “So 7 x 4 means 7 copies of 4?” she asked.
At this moment her teacher and I exchanged a look of genuine surprise. Here was a bit of unfinished learning, not about fractions, but about the meaning of multiplication (one meaning of multiplication, anyway), bubbling up in this fractions/multiplication/division discussion. Without this conversation, we never would have known that Allison had unfinished learning around the meaning of 7 x 4. I should also say that Allison is not a “struggler” in math class, and she generally appears to be making sense of grade-level work. So whatever unfinished learning she had around single-digit multiplication has not been holding her back. What I loved about this moment was the way that Allison was really working to make the math make sense to her, bringing everything that she understood from before, along with all of her unfinished learning, into her math community. And the other kids were bringing all of that too. And they were just figuring stuff out together, like putting together pieces of a puzzle, a puzzle that doesn’t get finished in one school year.
There are a lot of different joys that come from teaching, and one of them is when you’re really wanting kids to understand something and then they do. But another kind of joy, that’s newer to me, is the joy that comes from just watching kids wrestle with ideas and NOT reach that moment of understanding. Watching kids wrestle with ideas and letting the conversation end unsettled, letting the learning be unfinished, and thinking that true honest learning mostly goes that way.