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Multiplication of fractions is a deceptively easy skill for students to learn. In a large school district in Minnesota, fifth-grade students identified as being in the lower quartile on the district's mathematics competency test had great difficulty with all the fraction items except multiplication of fractions. Although only 18 percent of these students could find the sum of 1/2 and 1/8, 75 percent could find the product of 2/3 and 4/7. Students can successfully multiply, for example, 2/3 and 4/7, using their whole-number ideas. The answer. 8/21, can be calculated without considering the meaning of either of the fractions or of the answer.

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Designing didactic objects involves imagining how students can conceive of specific mathematical topics and then imagining what types of classroom discussions could support these mental constructions. This study investigated whether it was possible to design Java applets that might serve as didactic objects to support online learning where 'discussions' are broadly defined as the conversations students have with themselves as they interact with the dynamic mathematical representations on the screen. Eighty-four pre-service elementary teachers enrolled in hybrid mathematics courses were asked to interact with a series of applets designed to support their understanding of qualitative graphing. The results of the surveys indicate that various design features of the applets did in fact cause perturbations and opportunities for resolutions that enabled the users to 'discuss' their learning by reflecting on their in-class discussions and online activities. The discussion includes four design features for guiding future applet creation.

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