Prealgebra students learned about functions in a unit that emphasized (a) representing problems in multiple formats, (b) anchoring learning in a meaningful thematic context, and (c) problem-solving processes in cooperative groups. In posttest results, treatment students were more successful in representing and solving a function word problem and were better at problem representation tasks such as translating word problems into tables and graphs than were comparison students. Similar results were found for students who spoke English as a second language.

This study examined changes in 26 fourth-grade students' early conceptions of rational number representations as a function of receiving one of two curricular interventions. The first group of 12 students received a curriculum that emphasized constructing knowledge through extended problem solving with a single perspective of the rational number domain based on part-whole relations. A second group of 14 students received a curriculum that emphasized a more conceptually diverse multiple perspective view of the domain through problem solving with operator and ratio relations. Analyses of the students' rational number knowledge before and after the interventions indicated that students in the single perspective group produced organizations of knowledge that more frequently diverged from a formal domain analysis than those produced by students in the multiple perspective group. Further, students in the single perspective group increased their focus on superficial surface features. Alternatively, students in the multiple perspective group demonstrated an increased focus on operations that more frequently reflected the underlying mathematical relation conveyed by the representation. The findings indicate that an early exposure to more diverse perspectives of rational numbers assists students in developing more interconnected and viable representation knowledge for rational numbers.KEY WORDS: constructivist curricula, early rational number knowledge, elementary students, mathematical representation, problem solving, rational number perspectives Educational Studies in Mathematics (2005) 60: 37-69

Using cognitive ethnography as a guiding framework, we investigated US and Japanese fourth-grade teachers' domain knowledge of key fraction representations in individual interviews. The framework focused on revealing cultural trends in participants' organization of knowledge and their interpretations of that organization. Our analyses of the interviews, which included a representation sorting task, indicated three major differences that defined US and Japanese teachers' approaches to rational number representation: (1) Japanese teachers interpreted all rational number representations as conveying primarily mathematical information, whereas US teachers interpreted only some representations as conveying primarily mathematical information;(2) the US teachers focused more intently on part-whole relations than Japanese in their interpretations; and (3) Japanese teachers more easily linked rational number representations to more advanced upcoming content in the curriculum. A review of US textbooks used by the teachers reflected their consistency with US teachers' interpretations of the representations. These findings imply that strong cultural differences underlay the approaches that teachers in both nations take to rational number representation and that these differences may help explain established crossnational differences in student reasoning.It has been demonstrated that US teachers are less successful than their Asian counterparts in computing correct values as well as providing indepth explanations when given tasks that involve rational number relations (Ma, 1999). A logical extension of Ma's work is to examine the possibility that differences between US and Asian teachers may, in part, be due to the ways they conceive of the various meanings of rational number representations (e.g., part-whole and ratios). We believe that these differences are more accurately interpreted as representative of cultural approaches to rational numbers in the two nations than simply reflecting individual differences in teachers' cognitive capabilities. In this study we adopted a cognitive ethnography framework to study the knowledge that US and Japanese teachers access when working to organize representations for rational numbers.

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This study argues that maximizing early childhood educators' abilities to create social opportunities for co-construction of knowledge rests on two understudied assumptions, one theoretical and one empirical. Theoretically this study rejects the notion of language as an impartial conveyor of knowledge in favor of one in which math and language interact. This alternative framework is termed MathMediated Language (MML) and argues that the perception of common terms that adults possess is an important part of the knowledge that practitioners possess about linking conceptually related linguistic and mathematical knowledge. Empirical findings from a survey recording participants' reactions to seven categories of terms with mathematical meanings and three categories of distracter terms were analyzed. The data indicated that when asked to think about math, practitioners more readily accessed words for operation terms than relational terms. Additionally, participants demonstrated stronger tendencies toward additive terms conveying addition or subtraction concepts over multiplicative ones conveying multiplication or division concepts. The findings point to patterns in the ways that participants view mathematical language demonstrating that language interacts with even simple interpretations of basic mathematical terminology. The implications of this are that practitioners interpretations of everyday language may influence their ability to see opportunities for teaching mathematical concepts not only in the context of an explicit math lesson but throughout the broader early childhood curriculum.