To find out what students really
understand about that math lesson,
try one-on-one interviews—
and be ready for some surprises.
How much is 100 minus 3?” I asked Alicia, a 3rd grader. I gave an additional direction, asking her to try solving the problem in her head. “But you can use paper and pencil if you need to,” I assured her.
After a moment, Alicia responded confidently and correctly.
“Ninety-seven,” she said.
I then gave Alicia another problem. “How much is 100
Alicia frowned and said, “I’ll have to use paper and pencil. I
can’t count back that far.” She wrote the problem on her
paper and subtracted, regrouping as she had learned and
getting the correct answer of 2.
Alicia’s response indicated that she had not learned about
the relationship between addition and subtraction. She did
not think, “I know that 98 plus 2 equals 100, so 100 minus
98 has to be 2.” On numerous assignments, Alicia had
demonstrated proficiency with paper-and-pencil skills for
both addition and subtraction. Her written work, however,
hid this important gap in her under-
standing. I now had information
about the instructional help
that I could provide Alicia.
I showed Gerald, a 6th
grader, two fractions—
6/10 and 7/10. “Which
fraction is greater?” I
asked. As with Alicia, I
asked him to try and
decide in his head but assured him that it was OK to use
paper and pencil.
Gerald thought for a moment and then answered incor-
“How do you know that?” I probed.
“The smaller the number, the bigger the fraction,” he
replied. Gerald incorrectly applied what he had learned about
comparing fractions with different denominators—for