The continuous research on mathematical attainment is a part of the International Project on Mathematical Attainment (IPMA) in which such countries as Brazil, Czech Republic, England, Hungary, The Netherlands, Ireland, Japan, Poland, Russia, Singapore and USA are participated. The aim of this project is to monitor the mathematical progress of children from the first year of compulsory schooling throughout primary school and to study the various factors which affect that progress, with the ultimate aim of making recommendations at an international level for good practice in the teaching and learning of mathematics. In Japan, the total of eight different public primary schools have agreed to participate in the project. We asked all two-cohort children and their classroom teachers from these schools to be involved and to take mathematical attainment tests for six years. At the present we have carried out five tests, i.e. Test 1, Test 2, Test 3, Test 4 and Test 5 to about 300 children of first cohort for four years. The purpose of this paper is to analyze the data of three tests, i.e. Test 3, Test 4 and Test 5 to investigate children's progress of mathematical attainment in order to find out some suggestions for improving the teaching and learning of mathematics at these primary schools. In our previous paper (Koyama et.al., 2002), according to the percentage of correct answer to each test item, we made such categories as high [H], medium [M] and low [L] attained items. We defined the fixity of mathematical attainment such that for three tests if a child's changing pattern of correct (1) or incorrect (0) on an item is [1→1→1] or [0→1→1] then the child's mathematical attainment on the item is fixed. As a result of analysis in terms of these categories and the fixity of mathematical attainment, we found out the following: ・There were five different types of [H→H], [M→H], [L→H], [L→M] and [L→L] from Test 3 to Test 4 based on the progress of each some test item in Test 3 which is learned at the third grade. ・There were two different types of [L→M] and [L→L] from Test 4 to Test 5 based on the progress of each some test item in Test 3 which is learned at the fourth grade. ・We found that the only basic item in Test 3 which is learned at third grade was insufficiently fixed among children. These results suggest that the teaching and learning of mathematics at the third grade was more effective than that one at the fourth grade in these schools and that more efforts should be made in the teaching and learning of contents such as decimal, fraction and division with remainder.
