strategies for discussions makes more and more students actively participate in
discussions as the year goes on.
There is evidence in the video that the way Dee structures and facilitates dis-
cussions already draws the active participation of students with a wide range of
understanding of division and numbers in general. For example, one student is
comfortable enough to count on his fingers to check the answer to a small sub-
traction problem, and another student talks about negative numbers in his
response. Also, some students show they have some clear ideas about how to deal
with remainders, although others share that they think there is no answer to the
problem. Through this discussion, Dee tries to help individuals move forward
from where they are in their thinking about division as well as helping the whole
group move forward in their understanding of division.
Communicating Ideas
In another interview, Dee says that one of her goals is for her students “to be very
specific in their answers.” Her interaction with one student, Louise, is an exam-
ple of how she tries to help students clearly communicate their thinking. She
repeats what Louise says (that there is no answer to 36 ÷ 8) but turns what she
says into a question: “So there’s no answer because it went over 32?” This makes
Louise think again about her response and whether it is true and whether it is
exactly what she wants to say. Dee says a few times, “So there is no answer?” and
then expects Louise to explain why she thinks there is no answer. When Louise
says something about what is left over that is not quite correct (half of a bus, half
people), Dee takes her literally, which forces Louise to rethink how to phrase her
idea to exactly reflect what she means. Dee uses humor to illustrate how Louise’s
answer is not precise, but also to help make the students comfortable.
As Dee helps Louise communicate her ideas, she helps her think through
some ideas about division and remainders: Is there an answer to a division prob-
lem where the amount in the groups doesn’t fit evenly into the total? Can you
have leftovers in an answer to a division problem? How do you answer a division
problem with remainders? How does the answer in the context of a story problem
relate to an answer to a bare numbers problem? What do the numbers represent?
Listening to Dee’s interaction with Louise may help other students strengthen
their understanding of division because they might have similar ideas and confu-
sions and it might be helpful to hear Louise’s responses.
By the end of the discussion, there are some things about division that many
of the students seem to understand and there are some things they are clearly still
trying to figure out. Most of the students seem to understand that in division you
are dividing a quantity into equal groups. There is some evidence that students
What Do We Do with the Remainder?