8

The Pieces Get Skinnier and Skinnier

Assessing Students’ Ideas About Fractions

Marta Garcia Johnson

As a fourth-grade teacher of a self-contained inclusion class,* I use a variety of as-
sessments to monitor my students’ progress and drive my instruction. Daily obser-
vations of performance tasks, open-ended pencil-and-paper tasks, unit assessments,
and anecdotal notes are among the ways I capture how my students articulate their
understandings. This picture of individual strengths and weaknesses is especially
useful as I plan to engage my students with learning challenges in meaningful
mathematics. From my observations and informal assessments, I attempt to pin-
point conceptual “breaks,” places where meaning breaks down for students who are
struggling. Even if they have attained a benchmark on a unit assessment, their abil-
ity to retain and apply the ideas in subsequent work can be tentative. The follow-
ing vignette is an example of how I implemented this cycle of assessing, planning,
teaching, and assessment during our curriculum unit on fractions.
My primary goal for the fractions unit was to help students develop strategies
for comparing and ordering fractions using area models, first using physical mod-
els and drawings, then without the benefit of making a drawing or using manipu-
latives. With area models, students divide shapes into the appropriate number of
equal sections (the denominator), then color in the number of sections that cor-
respond to the numerator to create a picture of the fraction. I knew that for many
of my struggling students, it would be critical that they encounter multiple op-
portunities to work with these foundational ideas using manipulatives and draw-
ings. As they became more flexible with these, I hoped they could internalize the
pictures and relationships to form mental images.
During the first few days of the fractions unit, students worked to name frac-
tional parts of an area model. This helped them develop concrete images of frac-
tions and supported their understanding of fractions as representing equal parts of

*Chapter 22 is another essay about this class.