学習機構の解析に関する方法論的研究:V : 学習能力系列の解析(2)

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The purpose of this article is to propose an analytical method for series of abilities which lead to the attainment of the same goal. This kind of analysis is very important for programming a course of study and for many other instructional purposes. Such series may take on various structures. Figures in (1-1) and (1-2) are examples of the structure. [figure] where A_ij is a step ( a particular ability) in the series G is the goal of the instruction A_pq is the ability higher than A_p+1r A_11 directly leads to the attainment of G Series (1-1) have a systematic structure, but series (1-2) have comparatively complex structure. The series which have systematic structure like series (1-1) are called complete-multi-series, and the series which have complex structure like series (1-2) are called incomplete-multi-series. The complete-multi-series (1-1) is analyzed as follows. Suppose that a_ij^* is an item of an achivement test which measures the ability A_ij in series (1-1). the response pattern of a pupil to a battery of items is represented as (a_11, a_21, ……, a_ij, ……) where a=1 or 0 1 designates "pass" and 0 "fail" in item a. If a pattern X is such that a_i1=1 and a_i2=0, Y is a_il=0 and a_i2=1, and Z is a_i1=1 and a_i2=1, we call Z a duplicated pattern of × and Y. We shall consider only those who passed a_11, i. e., those who attained G. There will be difference in response even among them. A pupil may attain G while failing in some of the earlier steps, and another pupil may atain the same G while failing in different set of steps. These patterns will show alternative routs for attaining G. Response patterns which have led to (G) and yet have no duplicated structure between each other, are called Mutually Independent Serial Response Patterns. Let R (a_ij, k) be the quantity of Mutually Independent Serial Response Pattern with a_ij=1, then R (a_ij, k)=2^k-i_i-1C_j-1 where _k is the maximum value of (i) in series (1-1), and it is called the length of series (1-1). _lC_n=l (l-1)……(l-n+1)/1・2・……(n-1)n Σ^^i__j=1R(a_ij, k)=2^k-1 Let S (a_ij, a_pq) be defined as follows S (a_ij, a_pq) =Σa_ija_pq where Σ is sum with respct to Mutually Independent Serial Response Patterns. (i〓p, j〓q) then S (a_ij, a_iq)=0 when J〓q (5) Using (3)〜(5), a complete-multi-series (1-1) is abstracted from the Mutually Independent Serial Response Patterns. Incomplete-multi-series (1-2) may also be abstracted analytically. A tentative method is proposed for determining each terms in these formulae empirically.
1962-12-30

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中嶽治麿
大阪府科学教育センター

学習機構の解析に関する方法論的研究:V : 学習能力系列の解析(2)

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