The Structure of Popovers

We spent a lovely week on Mount Desert Island this summer, playing in Acadia and eating lots of popovers, which are apparently kind of a thing in that part of Maine. When we returned home, the kids asked if we could make some popovers ourselves. Happily, I dug out an old favorite, this Maida Heatter recipe, which says it makes ten popovers. We use a standard twelve-muffin tin, so I asked my five-year old daughter, Daphne, to butter all the way around the outside and leave the middle ones empty. When she was done, I asked, “How many popovers are we making?”

Daphne looked at the tin for a minute, and then started counting from the left-hand side, one column at a time. She looked at the first column and said “Three” without counting, which means she is now able to visually recognize, or “subitize,” three objects as a group. Then she moved her finger to the second column, counted the top cup, skipped the middle one, and counted the bottom one. “Four, five.” She kept going: “Six, seven. Eight, nine, ten.”

I found her count really interesting because I expected her to count all the way around the outside, following her butter tracks. Instead, she looked at the layout of the muffin pan, and used the grid. This is an exciting development! She is looking for some structure, rather than plunging right into counting without a plan. I’ve watched this thinking of hers grow over this summer, starting when a friend asked her, “Do you think you could count all those boats?”

Daphne took one look and answered, “If they were in a line I could.”

Daphne is feeling a need for organization, and for developing strategies to keep track. As numbers get bigger and problems get more complex, she is looking under the numbers for structures like lines, columns, patterns, and groups. These structures are essential for the important mathematics to come, like place value and operations.

Counting doesn’t get the respect it deserves. When I watch kids count lots of things, I see them develop that need for organization. After a child tries to count a pile of action figures, sneezes on figure 31, and has to go back to the beginning because she’s lost track, she starts to think about strategies that might work out better next time. Before long, students are naturally counting out groups of 10, 20, 50, or 100, like the students in these videos: Counting Collections with Kindergarteners and Counting Collections with Third Graders. (This Teaching Children Mathematics article by Julie Schwerdtfeger and Angela Chan has a nice description of the value of counting collections in the classroom.)

In our home, we count all the time. We count anything and everything, from popover tins to ponytail holders to toys on the floor the kids haven’t picked up. After all that real and varied counting, Daphne is starting to count strategically. Once we made the batter, I said, “It looks like we have a lot. Maybe we’ll be able to fill the whole pan. How many popovers would we make if we filled these empty ones also?” I pointed to the two empty cups in the center of the muffin tin.

She thought for a moment, then sub-vocalized, “Ten. Eleven, twelve.” She turned to me and said, “Twelve!”

I asked, “Where did the ten come from?”

“When I counted it before.”

“You didn’t need to start back at the beginning and count them all?”

“No, Mommy. I remembered the ten. I know that’s ten. I just counted from there. Ten…then eleven, twelve.”

Her older sister, Maya, came barreling into the kitchen and said, “Daphne! You’re counting on! I heard you!” She explained the term to Daphne, who was grinning broadly.

I asked, “How did you learn to count on, Daphne? That’s the first time I’ve ever heard you do it!”

She said, “Well, I just counted a lot of times, and then I started remembering the first number. When I counted it again, it hadn’t changed.”

When I coach teachers, I sometimes hear primary teachers instruct kids to count on instead of count all. “I don’t want to hear anybody go back to one. Keep track of where you were and count on.” I hear intermediate teachers say, “I don’t want to see anybody using repeated addition. You need to multiply.” Or I’ll hear, “I don’t want to see any dots. Make gridlines. No more dots.”

I think about this kind of talk a lot. On the one hand, I’m glad when teachers have a sense of the thinking that’s on the horizon for the student, and it’s sometimes appropriate to nudge students toward more efficient techniques. On the other hand, I’ve never seen this kind of nudge work. What if, instead of disparaging kids’ strategies and telling students they have to abandon their trusted techniques, we think about creating the need for more organization, better keeping track, more efficient strategies? For example, if kids insist they can multiply by repeated addition, what happens if we give them some bigger numbers to multiply, and then keep quiet? The children will eventually become frustrated–exasperated even–with the laborious calculations. Now there’s an opening! Now they feel the need for an improved strategy. Now they are looking for a different, more appropriate structure than the one they were overtaxing. And now they own all kinds of good math thinking, which we get to honor as theirs. If we’ve created a classroom climate where students are respected as mathematical thinkers, they’ll be perfectly comfortable saying, “This problem was a total nightmare and I’m sure I made some mistakes doing it this way! I think I need to figure out a new way to do it.”

Daphne believed there were ten buttered muffin cups because she’d counted them herself. She believes in her new strategy of counting on because she’s verified her counts enough times to trust them. She owns her thinking. I see no skipping over these phases. If students enter school without this kind of experience, we need to provide opportunities for them to build this foundation in our classrooms. Otherwise, the student who is counting on because his teacher taught him a procedure, (whether he is convinced it works or not), is the same student who finds an answer of 6,080 instead of 680, and doesn’t bat an eyelash.

The recipe, by the way, made 12 perfectly imperfect eggy popovers, which we ate with butter and jam. Enjoy!

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