The first way teachers and students modeled mathematics was through the use of tools such as manipulative and representational models. When groups of students were presented with the Sharing Brownie task, they were provided with physical manipula- tives and asked to brainstorm the possible tools they could use to tackle this problem. Some students chose to use the physical manipulatives, some chose to draw their work in order to visualize their fractions, and others started by trying to solve the task numerically. One of the teachers, Mary, noted an incident where “I thought a student was playing with the manipulatives, but when I sat down and talked them through it I realized that they had discovered how to break up the whole into pieces.”

The variety of approaches evidenced through students’ representational models demonstrated the different ways in which they constructed their understandings of the mathematical ideas. The teachers used student models as an opportunity to make sense of their students’ thinking and identify future learning opportunities. Alice commented that “It is clear that this student needs support with naming fractions and labeling” also noting possible extensions such as “a next step might be to show her that 4/4 + V give a total of 5/4.” Cindy learned that her students had a hard time seeing that V five times is 5/4 and can also be expressed as 1 V. She realized that her students had difficulty expressing themselves precisely. Tina realized that her students needed to learn how to define a numerator and denominator, how wholes can be divided into equal parts (e.g., 4/4) but still be wholes, the importance of labeling, and the many ways to physically divide a whole (vertically, diagonally, horizontally).

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