linked. This design is significantly different from traditional instruction in rational number, in which topics are taught separately.

Principle #3: Metacognition

In this chapter, I have not made detailed reference to students’ developing metacognition. Yet the fostering of metacognition is in fact central to our curriculum. First, as the reader may have noted, we regularly engaged the students in whole-group discussions in which they were asked to explain their reasoning and share invented procedures with their classmates. We also designed the lessons so that students worked in small groups to collaborate in solving problems and constructing materials; we thereby provided students with a forum to express and refine their developing understandings. There were also many opportunities for students to consider how they would teach rational number to others, either younger students or their own classmates, by designing their own games and producing teaching plans for how these new concepts could be taught. In all these ways, we allowed students to reflect on their own learning and to consider what it meant for them and others to develop an understanding of rational number. Finally, we fostered metacognition in our program through the overall design and goals of the experimental curriculum, with its focus on interconnections and multiple representations. This focus, I believe, provided students with an overview of the number system as a whole and thus allowed them to make informed decisions on how best to operate with rational numbers.

Final Words

I conclude this chapter with an interchange, recorded verbatim, between a fourth-grade student and a researcher. Zach, the fourth grader, was being interviewed by the researcher as part of a posttest assessment. The conversation began when Zach had completed two pages of the six-page posttest and remarked to the interviewer, “I have just done 1/3 of the test;…that is 33.3 percent.” When he finished the third page, he noted, “Now I have finished 1/2 or 50 percent of the test.” On completing the fourth page he remarked, “Okay, so I have now done 2/3 of the test, which is the same as 66 percent.” When he had completed the penultimate page, he wondered out loud what the equivalent percentage was for 5/6: “Okay, let’s see; it has got to be over 66.6 percent and it is also more than 75 percent. I’d say that it is about 80 percent….No, wait; it can’t be 80 percent because that is 4/5 and this [5/6] is more than 4/5. It is 1/2 plus 1/3…so it is 50 percent plus 33.3 percent, 83.3 percent. So I am 83.3 percent finished.”

This exchange illustrates the kind of metacognitive capability that our curriculum is intended to develop. First, Zach posed his own questions,

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