talking through different solutions and
methods broadens students’ understanding and forces them to consider
their answers in relation to their peers
and to make revisions if necessary.
When it appeared that most students were finished, I called the class
together. First, I asked for solutions
to the problem. As each group called
out their answers, I recorded them
on chart paper: 21, 22, 23, and 22
r3. Most students arrived at 22 with
3 remaining but then adjusted their
answer to fit the scenario because r3
dollars didn’t make sense. Some confusion arose around whether the extra
three meant another dollar was needed
( 23), if a dollar needed to be removed
( 21), or if they could just disregard the
extra 3 because they were part of the
last dollar ( 22). A few groups kept 22
r3 because they didn’t know what to
do with the remainder.
I then began the discussion by
asking students to justify their solutions to the class. The first group I
chose was Alli, Josie, and Lila because
when I observed them working during
Act 2, I could see they had a clear justification for why they added an extra
dollar (a good example of a thought-out mathematical argument).
ALLI: We got 23 . . . well, 22 with
some extra so we made it 23.
TEACHER: Can you walk us through
what you did?
ALLI: We didn’t want to divide, so
we multiplied. We did 10 x 8 to get 80
[ 8 being the number of cubes on the
dollar I had shown]. Then we doubled
it to get 160.
LILA: That’s like 20 x 8.
ALLI: Yeah, so then we were 19
away. So that’s two more 8s, which is
22 x 8 = 176.
LILA: But then we had 3 left over, so
we figured that was another dollar.
ALLI: Well, a piece of a dollar, but
you can’t really have a piece of a dollar
so we added one more to get 23.
TEACHER: I think just about everyone
got 22 with a remainder of 3. I’m
curious what others did with that
EFRAN: We got 22 with 3 extra too,
but we thought that each time you
have eight cubes you are losing that
little part of a dollar [that isn’t covered
by the cubes]. And 22 of those little
pieces is like another dollar so we took
off one dollar instead of adding it like
TEACHER: So how many dollars does
your group think are in the roll?
TEACHER: Other thoughts?
KIARA: Jessie and I said 22. It’s like
everyone else. We got 22 with 3 extra
cubes. But we thought those extra
cubes were on the last dollar.
JESSIE: We thought that when you
stick another cube on that first eight,
part of that cube is going to be on
that first dollar. Then the next dollar
will have some of that cube and seven
more of whatever color it is and so on.
[They drew part of their thinking, as
shown in Figure 5.]
KIARA: Each time you add a dollar,
The conversation continued with
other justifications and revisions of
ideas. In the end, the bulk of the stu-
dents agreed that 22 dollars made the
most sense, in large part because of
the argument and diagram from Kiara
and Jessie. At this point, I shared the
Act 3 video that showed me counting
out each dollar in the roll. When I got
to 22, there were cheers and high fives
from the students.
As you read this vignette, you likely
noticed that the bulk of the workload
was on the students. They were in
charge of observing the situation,
asking questions, analyzing information, and deciding how to approach
the problem. The justifications for
solutions and the debate that ensued
involved mostly student voices. The
students were active participants and
had ownership of the task.
Imagine if I gave them a worksheet
with 179 ÷ 8 instead? Would we have
For a discussion of additional math
teaching strategies, see the
online article “Five Practices to
Unleash Problem Solvers”
by Lisa Watts Lawton at
FIGURE 5. In Act 3, Kiara and Jessie
showed their thinking as they explained
In real-life problem solving, we are not given
all the information we need in a nice, neat
package—we must gather and sort it ourselves.