When teachers know what students know: integrating mathematics assessment

Theory Into Practice, Autumn, 2004 by Maryl Gearhart, Geoffrey B. Saxe

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A second theme concerned the challenges children face in their efforts to coordinate representations (Siegal & Smith, 1997). Many Seeing Fractions tasks require children to draw or cut an area or a fractions strip, and then label with a numeric solution. Children's responses often differed in the two media, suggesting that each medium affords certain kinds of insights, but constrains others.

In one example we discussed, a child solved this 3-step problem:

1. I invited 8 people to my party (including me) and I had 6 cookies. How much did each person get if everyone got a fair share?

2. Later my mother got home with 2 more cookies. We can always eat more cookies, so we shared these out equally too. This time how much cookie did each person get?

3. How much cookie did each person eat altogether?

The child drew 8 stick figures, and solved all steps of the problem by cutting circles and pasting fractional pieces of cookies next to each figure. For step 1, he gave 1/2 cookie to each person, and then 1/4 cookie, explaining that "I knew I couldn't give one to each person, so first I gave everybody a half. There were cookies left over, so I cut those into fourths. That worked!" He labeled his solution '1/2 1/4.' For step 2, he dealt another 1/4 cookie to each person, saying, "I'll see if a fourth works. It does!" He labeled his solution '1/4.' Now each stick figure had one half, one fourth, and another fourth. But he did not use the graphics to solve the final step, instead solving step 3 with the column addition shown in Figure 4. In a series of three additions, he added the whole number in the denominators, ignoring the numerators as well as the fraction value represented by the relationships between numerators and denominators. He explained that "'2' and '4' and '6' means 'how many pieces,' so I added those." His approach is one of many different ways that children may treat numeric fractions representations as whole numbers (Gelman, 1993; Mack, 1995), without coordinating their drawings with numeric notation. In our discussions, we considered how developmental progress with fractions requires children to construct new understandings of both concepts and written notation, and these may not develop in tandem (Saxe, Taylor, McIntosh, & Gearhart, 2003).

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Implementation: Teacher as professional educator

Implementation activities emphasized the importance of ongoing assessment of children's thinking. Sometimes we suggested assessment approaches for particular lessons, and other times teachers shared and critiqued methods of their own design. We viewed assessment as a potential component of all practices--open-ended questions during a class discussion, brief queries during small-group activities, open-ended tasks for journals and quizzes. Over many meetings, we considered the challenges of each strategy: observation and inquiry during student activities (how to focus the purpose of an observation), whole class discussions (how to interpret and integrate "wrong" answers), assessment of students' written work (how to design a good task and evaluate students' responses), peer assessment (how to help students pay attention to important mathematics), portfolio assessment (how to organize and assess a collection of work, and engage students productively in the process). Teachers role-played assessment methods in our meetings and piloted assessments in their classrooms.