My kids can : making math accessible to all learners, K–5

WORKINGCOLLABORATIVELY

Although Jhali needed this additional time with Diane, it was also important
that she continue to be exposed to the mathematical ideas that we were working
with in class and to participate in that larger classroom community. I use flexible
grouping in my classroom so that my struggling students have opportunities to
collaborate with their peers who are working with grade-level ideas successfully.
So, while Diane narrowed her work with Jhali to a specific focus (the meaning of
numeratoranddenominator), I assessed Jhali’s understanding of the concepts being
investigated by the rest of the class. These students were creating representations
for a variety of fractions including fourths, sixths, eighths, tenths, and fractions
representing more than a whole. They then made all kinds of fraction compar-
isons, including fractions with like and unlike numerators and denominators.
I needed to find out how Jhali was making sense of this mathematics. Was she
able to use the strategies she learned with Diane and apply them to these new
fractions? I checked in with her, asking her to explain the models she had drawn
and how the numerator and denominator represented the fraction. Jhali stated:
“The denominator is how many pieces [sometimes she would say lines] and the
numerator is how many I colored.” Although she used this description consis-
tently, I wasn’t sure if she would be able to draw upon these images when she had
to compare fractions without a visual model. A few days later, I spent some time
assessing the development of Jhali’s understanding.

TEACHER: Which of these is the larger fraction: or?

JHALI: because if I cut the rectangle I have only 3 pieces instead of 5.

TEACHER: So you said you have 3 pieces. Why do you think is more

than?

JHALI: See, I mean the pieces would be bigger because you only have 3. So

you take 1 piece and I take 1 piece but my pieces will be fat and bigger than

yours.

This was progress! Jhali based her second comment on her first idea. Next, I
wanted to see if she would stay with her strategy:

TEACHER: What about and?

JHALI: Maybe is bigger.

TEACHER: What is different about these two fractions from the others you

just compared? Let’s look at these.

JHALI: Oh, the because the pieces are fatter than the pieces. See.

Jhali then drew two circles (one divided into fourths and one into fifths) and
pointed out how the fourth pieces were bigger. (Although her pieces were not of
equal size, she had attempted to make the fourths larger than the fifths.) I went on
to see if she could compare two fractions when the numerators were not equal to 1.

1

5

1

4

1

5

1

4

1

5

1

5

1

3

1

3

1

3

1

5