This report aims to analyze the series of abilities to learn, which arrive at (G), from the point of view of the formative-process of the ability (G).
At first, the following models of the series were set,
(1) Uni-serial model_??_(1) where A
i is ability in uni-series and it is called equal weight element. G is the goal of the series. A
i is higher than A
i+1 (2) Substitutive-serial model_??_(2) where A
l(l=1,2,...,k-1,k+1,...,m) is equal weight element.A
ki,(i=1, 2) is called substitutive element. A
1is equivalent to G.(3) Multi-serial model_??_(3)
Secondly, the substitutive series (2) is analyzed as follows. Suppose that a
i* is the item of achievement test measuring the ability A
i in series (2), then the response patterns of pupils to the achivement test are represented with (a
1,a
2, a
m) where a
i=1 or 0 1 is correct answer for item a
i* 0 is wrong answer for item a
i* Let number of response patterns in (a
1,a
2,...a
m), which had a
i=1, represent with R(a
i), then according to scalability of series (2),_??_(4) Let f(a
1,a
2,...a
m) be degree of achievment, obtained by the pupils who had response pattern (a
1,a
2,...a
m), on the goal (G) and let Q (a
2) define as follows,(5) where Σ is sum of all combination, on the conditioning series (2), in (a
1, a
2,at
i-1,a
i+1,...a
m), a
i=1 or 0,(l=1, 2,...,i-1,i+1,...m) R(a
i) is number of response patterns in (a
1,a
2,...a
m) which had a
2=0. Then (6)
Actually, substitutive elements have character like equal weight element from (a
1=1,a
2=1,...a
m=1) to (a
1=0,a
2=0,...a
k-3=0,a
k-2=1,a
k-1=1...,a
m=1) in scalogram, and additional elements (A
i1,A
i2,...A
ip), which do not belong in the series (2), are mixed in the actual data. From the above point of view, formulas (4); (6) are refined as follows.(8)
Thus, applicating formulas (7); (8) substitutive series of abilities can be approximately abstracted from actual data.
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