Snapshot 2: Solving
Missing Addend Problems
I took two tiles from a container and
showed them to Rosa, a 2nd grader.
“How many more do I need so I have
10 tiles?” I asked her. Rather than
showing her the problem in written
form ( 2 + ___ = 10) I presented the
problem verbally and with the concrete
material of the tiles. This question
presented Rosa with a missing addend
problem: Instead of giving numbers to
students and asking them to figure out
the sum, students already know the sum
and one of the parts and have to figure
out the missing part.
“You need 8 tiles,” Rosa answered
quickly with confidence.
“How did you figure out the answer?”
I probed.
Rosa replied, “It’s easy. I know that 8
plus 2 makes 10.”
Next, I showed Rosa a picture of a jar
and explained, “This jar can hold 100
marbles when it’s totally filled.” I showed
her another picture of the same size and
shape jar, this one with 30 marbles
written underneath. I hadn’t drawn
actual marbles, but had roughly scrib-
bled to show a jar that was about one-
third filled. “Can you figure out in your
head how many more marbles I need to
put into the jar so there are 100?”
“You need 70 more marbles,” Rosa
answered, again quickly with confi-
dence.
“How did you figure out the
answer?” I asked.
“I know that 7 plus 3 makes 10, so
70 plus 30 makes 100,” she replied.
I’ve found that using the known fact of
7 + 3 to figure out the answer to this
problem is a typical response.
I gave Rosa another problem. This
time, I showed her 5 tiles. “Figure out
in your head how many more tiles I
need so I have 30 tiles in all,” I said.
Rosa was quiet. After a moment, she
counted softly by 5s to 30, putting up
a finger each time. She looked at her
six fingers and thought. After another
moment, she said, “This one is hard. I
don’t think that six is right.”
What I Learned
The numbers we choose for problems
matter. Rosa’s correct responses to the
first two questions indicated that she
understood the structure of the
problem. The known facts of 8 + 2 and
7 + 3 gave her an anchor that helped
her reason out both problems. Children
usually have a good deal of experience
with numbers that add to 10, with their
fingers as a backup support.
The interviews
revealed the fragile
conceptual base of
the students’
understanding.
With 5 and 30, however, Rosa had no
useful anchor, except for counting by
5s, which didn’t help her solve the
problem. Rosa understood the structure
of the problem—that she was to find
the missing addend—but she lacked
facility with the particular numbers.
Students need experience developing
strategies for mentally computing with
numbers that are not as “friendly” as
single-digit numbers or multiples of 10.
The marbles-in-the jar problem can
help students figure out, in their heads,
how many more are needed to make
100. The teacher can start with multiples of 10, then move to numbers that
end in 5, and then move to all numbers.
Having students share their strategies is
valuable. Also useful are 10-by- 10 grids;
students can color in the number they
have and see how many more 1s and
10s they need to fill the grid. The goal is
to help students develop their number
sense so they increase the range of their
numerical comfort.
Snapshot 3: Interpreting Remainders “Here’s a word problem to solve,” I told Randy, a 5th grader. This was at the
beginning of the school year, and I was
assessing students’ understanding and
skills with division. I showed Randy a
card on which I had written “ 30
students, 4 students in a car.” The card
was to help Randy keep track of the
information in the problem. I
continued, “Thirty students are going
on a field trip. Four students fit in a car.
How many cars are needed to fit all the
students?”
“Can I use paper and pencil?” Randy
asked. I nodded and watched Randy
solve the problem as a long division
problem. He wrote the answer as “ 7 R 2.”
“How many cars are needed to fit all
the students?” I asked.
“It’s 7 remainder 2,” he said.
What I Learned
Even though Randy was able to
compute correctly, the computation
alone was not a sufficient indication of
his proficiency with division. His
answer of “ 7 remainder 2” made sense
numerically but not in the context of the
problem. Students often lack experience
solving problems that call for relating
numbers to real-world situations.
Dividing up things in their lives is a
common experience for students, and
it’s valuable to build on this experience
and situate a good deal of their division
work in word problems. Randy, like all
students, needs many experiences
solving division word problems with a
focus on making sense of the answer.
Sometimes pictorial representations
of problems can help. In my experience,
when asking students like Randy who
don’t make sense of their answers to
draw a picture that shows the 30
students getting into cars in groups of 4,
they often self-correct and give the
answer of 8, which does make sense.
But if students do the bulk of their division work on naked numbers—
