A METHODOLOGICAL STUDY CONCERNING AN ANALYSIS OF THE LEARNINGMECHANISM: IV

Analysis of the Series of Abilities to Learn (1)

Abstract

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. A₁is 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 (a₁,a₂, 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 (a₁,a₂,...a_m), which had a_i=1, represent with R(a_i), then according to scalability of series (2),_??_(4) Let f(a₁,a₂,...a_m) be degree of achievment, obtained by the pupils who had response pattern (a₁,a₂,...a_m), on the goal (G) and let Q (a₂) define as follows,(5) where Σ is sum of all combination, on the conditioning series (2), in (a₁, a₂,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 (a₁,a₂,...a_m) which had a₂=0. Then (6)
Actually, substitutive elements have character like equal weight element from (a₁=1,a₂=1,...a_m=1) to (a₁=0,a₂=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.