I used prompts such as these to discuss what to do in math when we are struggling to
learn new concepts. During this particular discussion I wanted these students to
become more aware of when they were not able to move forward in solving prob-
lems. The Comparing and Connecting prompt was particularly applicable. I asked if
they remembered being stuck with anything before. They gave examples, including
a story of actually being stuck in the mud. Using this story, I explained that the way
they were playing the game was like being stuck in the mud without a direction or a
reason for getting out.
I told them we would be playing the game again, but this time I wanted them to
tell their own stories of what they were learning from playing the game. We slowed
the game down, not in the sense of repeating directions or asking students to slow
down, but by methodically reviewing the game to investigate connections to prior
knowledge and taking the time to focus on the important mathematical concepts.
I became an active member of the group, modeling my thinking as I played the
game along with the students. If the array was 4 8, I would say “Maybe 4 8 is
a combination I need to work on, so I am thinking of an easier combination I
already know, such as 4 4.” I would point to a smaller part of the array to show
the 4 4 area, write the equation on my whiteboard, and ask students if I was fin-
ished finding the product of 4 8 (see Figure 19–2). Sometimes students recog-
nized that I was not finished, but they were unable to explain what I needed to do
next. I would then think aloud and say, “I have solved part of the area of the whole
array, now I need to figure out the dimensions of the other part of the array that I
did not use to find the product of 4 8. So, what do I need to do next?” My goal
was to encourage students to think of the next steps. I also explicitly used mathe-
matical words and wrote number sentences to illustrate the distributive property
(4 8 [44][44]), and the commutative property (8 4 or 4 8).
I set the small group off to play the game in pairs and immediately
checked in with Heather and her partner. I began by asking them to explain
the purpose of the game. They answered, “Multiplication and using what we
know.” They were simply reading back the words I had written on the board
earlier to explain what was going to happen in math that day. I asked, “What
makes you say the game is about multiplication and using what we know?”
They said, “Because that is what is written on the board.” I then asked, “How
does using what you know help you with combinations you are working on?”
and they replied, “We count the squares” (in the array). They counted the
squares by 1s. I was not convinced that they understood that they could use
easier combinations to solve for more difficult combinations. So, I drew two
small arrays that fit into one bigger array to demonstrate visually how the dif-
ferent combinations might help them figure out the bigger array combination
(see Figure 19–3).
TAKINGRESPONSIBILITY FORLEARNING