Before attending the workshop, participating teachers ask their students to find the number that they could put in the box to make the following open-number sentence a true number sentence: 8+4=□+5. At the workshop, the teachers share their findings with the other participants. Fewer than 10% of the students in any teacher’s class solved the problem correctly.* The majority of the incorrect responses were 12, with a number of responses of 17. These findings, which surprised most teachers, have led them to begin to listen to their students, and a number of teachers have engaged their students in a discussion of the reasons for their responses. The teachers’ experiences have precipitated a discussion in the workshop of how students are thinking about equality and how these misconceptions might have been acquired. The discussion generates insights about how children are thinking and what teachers can learn by listening to their students. Although the teachers recognize the students’ errors on this problem, however, they do not have a good idea of how they would address the misconception.

The workshop leader introduces several true and false number sentences as a context to challenge children’s incorrect notions of equality. Examples include 8=3+5, 17+9=36, 23=23, 17+26=27+16, and 76+7=76. The task is to decide whether the sentence is true or false. Sometimes the decision requires calculation (e.g., 74–57=17), and sometimes it does not (e.g., 67+96=96+67). The teachers work in small groups to construct true and false number sentences they might use to elicit various views of equality. Using these sentences, their students could engage in explorations that might lead to understanding equality as a relation. The sentences could also provide opportunities for discussions about how to resolve disagreement and develop a mathematical argument. The teachers work together to consider how their students might respond to different number sentences and which number sentences might produce the most fruitful discussion.

SOURCE: Falkner, Levi, and Carpenter, 1999. Used by permission of the authors.