It is well known that many schoolchildren fail to represent quantities by fractions. For example, when we ask them to paint 2/3m on a paper streamer of two meters, many of them paint its 2/3, and it results 4/3m. They believe that not only 2/3 of one meter but 2/3 of two meters and even 2/3 of ten meters are also 2/3m. This kind of misunderstanding seems to be derived from the emphasis on the operational aspect of fractions. At the same time, figures adopted in a math textbook strengthen the misunderstanding. That is, the total length of a paper streamer, for example, used to show a length represented by a fraction is restricted to one meter, which curtails of their opportunity to consider the total length of the paper streamer. In this paper, I reported the introductory instruction to fractions, which I taught the third graders in three classes. In one class I used figures of which total amounts were restricted to unit magnitudes, 1m, 1l, and 1 for a number line (a control class). In other two classes I employed figures of which total amounts exceeded the units, 2m, 3m, 2l, and 2, and so forth (experimental classes). Posttest on the representation of quantities by fractions implemented just after the instruction showed significant difference between the control class and the experimental classes. However, research conducted after the posttest revealed that effects of the figures employed in the experimental classes were not so strong as expected. From these results, I discussed how teaching materials adopted in the introductory instruction to fractions should be, and that the instruction had to include a kind of remedial aspect in the case that there were schoolchildren who had some knowledge of fractions with the misunderstanding at the time they learned them in school.
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