DAVEL: We have just solved 9 8, which is 72. So 9 7 is... 1 group less.
LUCÍA: Yes! Yes! 72 7?
JULIO: No wait! First we had 8 plates with 9 cookies, now we have 7 plates
with 9 cookies, so we have to take away 1 plate of 9 cookies.
ALL: Yeah!
ALEJANDRO: 72 – 9 63. 9 7 63.In this conversation, Davel brought to the other students’ attention what was
already familiar to them and how they could use it to solve a nonfamiliar fact. This
time, he didn’t need my prompting. Then Julio transformed the problem into a fa-
miliar context, one that could help them reason whether they had to subtract
7or9.
The process of using prior knowledge in new situations is automatic for math-
ematicians and successful students but needs to be taught to students who strug-
gle in mathematics. In this lesson, I explicitly helped the students link prior
knowledge to the new situation. Our previous discussion about the cookie trays
served to anchor the conversation. If I hadn’t done so before sending them off to
work, they would have treated 9 8 and 9 7 as problems isolated both from
each other and from any previous knowledge they had.
Making Connections Among Strategies
As mentioned previously, one goal was to help these students understand the re-
lationship between skip counting and multiplication. To do this, I gave them the
following problem: I will pack 23 party favor bags for my son’s birthday. Each bag
will have 4 toys. How many toys do I have to buy? Some students used multipli-
cation and some used skip counting. I wanted to focus on the connection between
those two strategies, so I purposefully chose to share the work of three students
who used skip counting and multiplication.
Lucía
23 4 92
4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, 48, 52, 56, 60, 64, 68, 72, 76, 80, 84, 88, 92Fleurette
23 4 92
10 4 40
10 4 40
3 4 12
40 40 12 92Are We Multiplying or Dividing?