Misunderstandings
applying a different rule he had been taught about fractions.
I now had information about the instructional help that I
could provide him.
example, 1/8 and 1/5. One piece of a whole that is cut into
five equal pieces is obviously greater than one piece of the
same whole cut into eight equal pieces.
Gerald would most likely have given the incorrect answer
on a written assignment if asked to circle the greater fraction. However, without talking to Gerald and having the
opportunity to probe his thinking, I would not have known
that his misconception was a result of inappropriately
Surprised by Student Thinking
We teachers often rely on students’ written assignments to
assess their skills and understanding. However, as with Alicia
and Gerald, I’ve learned that one-on-one interviews reveal
valuable information that is not available when I rely solely on
students’ written work. This information is essential for
guiding appropriate instructional decisions.
PHOTOS BY SUSAN BARCLAY ©2009 MATH SOLUTIONS
Some history on how I came to this conclusion. In 1993, I
was working on a series of videotapes Mathematics: Assessing
Understanding for ETA/Cuisenaire. The videos included individual student interviews. In preparation, I practiced by
teaching many lessons and conducting dozens of one-on-one
interviews with students at different grade levels. During the
interviews, I probed mathematical strengths and weaknesses
so I could construct a mathematical profile of the student. The
practice experiences were always revealing and sometimes
astonishing, uncovering students’ misconceptions and gaps in
their understanding that I hadn’t recognized before.
I was even more stunned during the actual videotaping.
The lesson called for the students to time the teacher, Carol
Brooks, for one minute while she drew stars on the board.
Carol then talked with the students about how they might
figure out how many stars she had drawn. The discussion led
them to circle groups of 10 stars and count how many 10s
and extras there were. During the lesson, the responses of two
of the 2nd graders—Cena and Jonathan—indicated a firm
foundation of understanding place value. However, when I
interviewed each of them the next day to probe their understanding one-on-one, I was shocked. The interviews revealed
the fragile conceptual base of their understanding in ways that
their teacher had no way of knowing from the context of the
classroom lesson. (The videotaped interviews are available at
www.mathsolutions.com/placevalue/Cena and www
. mathsolutions.com/placevalue/Jonathan.) As a result of this
experience, I began to incorporate more and more individual
