• School Mathematics
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• v.18 no.3
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• pp.457-478
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• 2016

The purpose of this study is to investigate middle school students' understanding and development of function graphs. We collected the data from the teaching experiment with two middle school students who had not yet received instruction on linear function in school. The students participated in a 15-day teaching experiment(Steffe, & Thompson, 2000). Each teaching episode lasted one or two hours. The students initially focused on numerical values rather than the overall relationship between the variables in functional situations. This study described meaning, role of and students' responses for the given tasks, which revealed the students' understanding and development of function graphs. Especially we analyzed students' responses based on their methods to solve the tasks, reasoning that derived from those methods, and their solutions. The results indicate that their continuous reasoning played a significant role in their understanding of function graphs.

• Education of Primary School Mathematics
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• v.21 no.1
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• pp.55-73
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• 2018

The purpose of this study is to investigate the effects of cognitive activity on cognitive activities that students imagine and cope with continuously changing quantitative changes in functional tasks represented by linguistic expressions, table of value, and geometric patterns, We identified covariational reasoning levels and investigated the characteristics of students' reasoning process according to the levels of covariational reasoning in the elementary quantitative problem situations. Participants were seven 4th grade elementary students using the questionnaires. The selected students were given study materials. We observed the students' activity sheets and conducted in-depth interviews. As a result of the study, the students' covariational reasoning level for two quantities that are continuously covaried was found to be five, and different reasoning process was shown in quantitative problem situations according to students' covariational reasoning levels. In particular, students with low covariational level had difficulty in grasping the two variables and solved the problem mainly by using the table of value, while the students with the level of chunky and smooth continuous covariation were different from those who considered the flow of time variables. Based on the results of the study, we suggested that various problems related with continuous covariation should be provided and the meanings of the tasks should be analyzed by the teachers.