After drawing 8 ladybugs, Heather had to recount the number of ladybugs at
least 3 times and recount the number of legs many more times. As she counted
the legs, she often lost track of her count. She was unable to skip count and was
even having difficulty keeping track of numbers as she counted by 1s. I wondered
how she would ever be able to reason through the process of double-digit multi-
plication problems in the future.
This interview and my other observations were the beginning of my own
understanding of why Heather wasn’t able to function as an independent learner
in my fourth-grade class. I recognized that to be a more independent learner, she
needed many more experiences making sense of numbers as she explored key
mathematical concepts.
Playing Factor Pairs
An early opportunity to help Heather arose during a math workshop one day in
October when I observed students playing a game called Factor Pairs (Russell
et al. 2008c). In this game, students use array cards to find products of any multi-
plication combination up to 12 12. Factor Pairs is engaging for students and
provides opportunities for students to work on multiplication combinations. The
goal is for students to focus on using easier combinations they already know to
determine products of more difficult combinations with which they are not yet
fluent. To play the game, students take turns picking up an array card with the
factor side faceup (see Figure 19–1). They have to name the product either by
“just knowing it” or figuring out an efficient way to count the squares. If their
answer is correct, they keep the card. Students play with a partner and keep track
of combinations they know by dividing a recording sheet into two columns:
“Combinations I Know” and “Combinations I’m Working On.”
A small group of students, including Heather, were having a great time playing
the game and seemed to be working cooperatively, but I suspected they were playing
the game without thinking about the mathematics. I stopped to listen to their con-
versation and observe their actions to determine what strategies they were using. It
quickly became clear that they were simply going through the motions of playing the
game. They would turn over an array card, say they knew the product, and put the
card into the pile of combinations they knew. When I asked if the factor pair belonged
in the pile of combinations they were working on or was a combination they already
knew, they would frequently say they knew it even when it was clear that they did
not. These students were simply guessing, not reasoning about multiplication.
On the other hand, those students who were playing the game using mathe-
matical reasoning would look at the array, talk with each other about how to
break the factors into workable numbers, use the arrays as visual models, ask each
TAKINGRESPONSIBILITY FORLEARNING