@article{oai:miyazaki-u.repo.nii.ac.jp:00000193,
 author = {Uda, Hirohumi and 宇田, 廣文 and Uda, Hirofumi},
 journal = {数学教育学研究紀要},
 month = {Mar},
 note = {Introductions of mathematical concepts and introductions of equality of these concepts are really important to learn and understand those concepts. In particular, equality of concepts with different expressions of symbols, like fractional numbers and ratio, is difficult. In the preceding papers, investigated on understanding equality and inequality of fractional numbers for university students and discussed levels of understanding these in three viewpoints一一properties of fractional numbers, functions of fractional numbers and calculations of fractional numbers一一.Many of them interpreted equality of fractional numbers by means of reduced representatives. Moreover, I showed the difference between mathematical definition of equality of fractional numbers and introductions of ones in a primary school. 
In this paper, I investigate and discuss the following matters about equality of ratio.
(1)I investigate levels of understanding equality ratio for university students who belong to an elementary education teachers training course.
(2)I discuss the meaning of ratio made quality of ratio from historical viewpoints, mathematical viewpoints and educational viewpoints. The outline of conclusions is as follows.
(i)Main method of interpretations by university students on equality of ratio is use of properties of ratio rather than use of quotient. This fact is strange because they have learned equality of ratio by quotient.
(ii)Ratio first was introduced as a relation within two quantities of the same kind and gradually was extended as general ratio containing as realtion between two quantities of the different kind. Moreover, Starting from misunderstanding about equality of ratio, the meaning of ratio has changed from the meaning as a relation to the meaning as quantity.
(iii)Understanding equality of ratio from the viewpoint of mathematics is not same to understand that from the viewpoint of mathematical education. There is a great difference between interpretations of ratio as an relation within two quantities of the same kind and that as an relation between two quantities of the different kind. Moreover, there is a great difference between interpretations about equality of ratio within two quantities of the same kind and that between two quantities of the different kind.
(iv)Japanese method defining equality of ratio relative to two quantities of the same kind by mean of quotient is containing not a little matter.},
 pages = {61--69},
 title = {比の相等性},
 volume = {18},
 year = {1992},
 yomi = {ウダ, ヒロフミ}
}