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The purpose of this paper is to show the similarities as well as the differences of fundamental fraction knowledge owned by preservice elementary teachers from the United States (N= 89) and Taiwan (N= 85). To this end, we examined and compared their performance on an instrument including 15 multiple‐choice test items. The items were categorized into four different types of fundamental fraction constructs, including part–whole relationship, quotient, equivalence, and meanings of operations. Each item was embedded in the area, linear, or set model except for the items constructed out of the meaning of operations. Several items were featured with a pictorial illustration. Quantitative analysis showed that U.S. preservice teachers were significantly outperformed by their Taiwanese counterparts overall. The difference between the two groups was statistically significant on 12 of 15 items. Findings suggest that preservice elementary teachers from both countries need to be better prepared in their understanding of the meaning of fraction multiplication or division operations. Findings also suggest that U.S. preservice elementary teachers need to be more knowledgeable in dealing with fraction problems embedded in a linear model. Further research is suggested to study the issues raised from the findings.

The purpose of this case study was to investigate the values demonstrated by an elementary school teacher in her mathematics teaching and what values her students perceived. This research adopted the valuing theory (Raths, Harmin & Simon, 1987) and used classroom observations and interviews to document the teacher's mathematics pedagogical values and a questionnaire, interviews, and instructional artifacts to document the students' perceptions of these values. The results of this research identified two educational values and three mathematics pedagogical values that were influenced by the teacher's personal beliefs about Buddhism, Confucianism, and curriculum. Her goals for education were to reinstate the students' original enlightenment and the students' respect for ethics and experts when dealing with people and life; and her values about mathematics learning were that it depends on individual efforts and personal understanding, her central purposes for teaching were to make students understand the mathematics content and to cultivate their problem-solving methods, and her purposes for evaluation were to understand students' learning and to encourage students to correct their errors. The students' awareness of these goals and values did not parallel the teacher's personal priorities. The implications of teachers' pedagogical values on teacher professional development and curriculum reform are discussed.

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