After returning from an inspirational week of math learning at NCSM and NCTM in the Bay Area of California, I walked away thinking two things. The first was “Wow! I am really exhausted!” The second was “How do I help teachers implement math tasks in the K-5 world?”
The first step for me was to do some reading. Lucky for me, I met a great gal (@jgough) who invited me to a Twitter slow chat book talk focused on this book:
It is short and sweet and gets right to the heart of task planning and implementation, all while sharing the research that illustrates the importance of the shift to student led learning. All that to say…I was hooked!
In the book, a pizza task regarding whether a half is always equal to half got my creative wheels turning and I immediately jumped in. In the spirit of planning with the 5 Practices in mind, I created a planning tool, which looked like this after I laid out my plan:
Next stop, 3rd grade! During my reading, it became clear to me that an important feature to strong math task implementation is mathematical discourse. Fortunately, Miss Heather Dabney and her students are rock stars in this department and she allowed me to engage in this task with her students.
Because the task has multiple correct answers possible, I began with a Which One Doesn’t Belong? (wodb.ca) activity.
The student responses were so unique from one another and really drove home the fact that more than one person could be correct and that we must actively listen to and consider all answers given. As Devon stated, “At first I thought #4 didn’t belong because it had two big squares, but now I see that #2 also doesn’t belong because #2 only has one skinny column at the bottom.” Perfect! They were ready to think about pizza.
We began with this essential question: Is a half always equal to a half? I shared with them a story of my friends who had dinner (without me!) and said that they each had half of a different pizza. One friend, Jose said he ate more pizza than my other friend, Ella. Ella said they had the same amount. I asked the students, “Who do you think was right?” After some quiet think time, I sent them off to their desks to begin working on this question independently and asked them to record their thinking on their whiteboards in pictures, numbers, and words.
Most students gave answers much like this one. This class had worked for months on the idea that fractions are a number, just like any other whole number they had seen before. It was clear that many were using this reasoning to support their claim that Ella was correct.
Thankfully, there were a few divergent thinkers in the room who could help push our thinking toward the meaning of the numerator in relationship to the denominator.
At this point, I knew students had enough processing time to engage in productive small group discussions. I gave each group of 4 students ONE sheet of nearly blank paper (thank you Tracy Zagar! Ode to a Blank Paper) and asked them to begin discussing their ideas and recording all the possible answers they could justify together.
This is when things got really interesting. A majority of the groups were still focused on the 1/2’s being the same size and many of the group responses looked like this:
As most students were convinced of this answer, I had group 2 share their thinking first. As I scanned the room while Betzaira shared group 2’s thinking, I saw many students nodding their heads in agreement.
The second group I asked to share had made the claim that 1/2 is equal to 1/2, just as group 2 did, but they pushed the reasoning of the class further by thinking about pizzas of the same size that were cut into different quantities of slices. This was an important step because were were beginning to reason about the importance of size in relation to fractions. Yes!
In anticipating student responses during the planning phase, I thought that students would struggle to see an answer other than Ella being correct, so I prepared the following questions to ask groups to provoke deeper thinking:
- Could Jose be correct?
- Could they both be correct?
Group 6 had originally also agreed with the claims of groups 2 and 4, but I challenged them to consider whether Jose could be correct as well. When they recorded the response below, I knew this was going to be just the right thinking to close our whole class discussion with.
Unlike the nodding heads of agreement when the other groups shared previously, I saw many students puzzling over this group’s answer. Enter Talk Moves!
I asked student to consider what group 6 had shared and selected a few students to REPEAT the idea in their own words. I then asked student to REASON by asking whether they agreed or disagreed with group 6’s claim that the size of the pizzas mattered when considering who was correct. At this point some agreed, yet some were still convinced that 1/2 is always equal to 1/2. I then used TURN & TALK to allow students more time to consider this new idea. It was after just a few brief minutes that Christopher, my original skeptic of this new idea, raised his hand. He said to the class, “It’s like when you go to get pizza and you get to have a small one all to yourself. Those pieces are really small. But when you go to get pizza for your birthday party and you order the party size pizza. Those slices are really big. If I looked at 1/2 of the small pizza and 1/2 of the large pizza next to each other, they would not be the same size. Half does not always equal half.”
As Christopher said this, I surveyed the room. Previously skeptical students were now nodding their heads in agreement that 1/2 does not always equal 1/2 half. Have they all been convinced yet? I don’t think so. Are they all on the road to a mathematical discovery? Absolutely.