This paper is the third repcrt of our research carried out in 1959-1963. Here, our preliminary research programs of experimental education of number concepts are presented. Ss : 26 three year old children of the kindergarten (Yoji Grbup) of Tokyo Woman's Christian College. Ss are divided evenly into two groups, the older (G.II) and the younger (G.I). (This grouping system is not entirely adequate for our purpose, but the groups had already been determined. ) Experimental design and teaching method: For G. I, the dice-pattern symbols (from one to five) are used as the teaching instrument. These patterns are supposed to act as representations of some collections of objects, and, as symbols of cardinal numbers. The Lirst main tasks of G.1 consist of naming(one, two, three, etc. ), associative learni,ng of the names, mutual discrimination, identiLication, reconstruction and transformation of these patterns,and one to one correspondence between these patterns and various concrete objects. Having mastered them, Ss are taught the operation of addition by summing two patterns, for instance, of two and three, then transforming this sum into the regular pattern of five. Similarly, the operation of subtraction can be taught. Then, our original plan was to introduce the numerical figures (1, 2, 3,etc.) and the signs of plus, minus and equals (+, -, =) and to teach the Ss that the equation (e. g. 2+3=5) or subtraction are parallel to the operations of dice-patterns, and therefore to the operations of collecetions of concrete objects. After this, the range of numbrs would be extended to twenty or thirty. For G.II, teaching devices consist of rote learning of numerical orders, counting of the concrete objects, such as marb.les, from one to thirty. The numbers, here, signify the ordinal numbers. Then, calculating operations of addition are taught. This is done by summing the two collections of marbles, for instance, two and three, then counting this sum in order.The operation of subtraction is similarly taught. Then, it was our original plan to introduce numerical figures, signs of plus, minus and equals and the equations, similarly to G.I. This design is to compare the efficiencies of each two teaching plans of number concept mediated by the cardinal or ordinal number.