Suppose that the pupils understood the subject matters (α
1=(α
1, α
2,..., α
k)) and didn't underst and (α
0=(α
k+1, α
k+2,..., α
m)), when the new subject matters (α=(α
1, α
2,..., α
m)) had been given by an educational action (A) to the pupils who had ha d fundamental abilities (S
1, S
2,..., S
n)(where S
i=1 or 0, 1 is understanding S
i, 0 is not understanding).
Then subject matters (α
1=(α
1, α
2,..., α
k)), which the pupils understood, are called region of readiness to learn of fundamental abilities (S
1, S
2,...S
n) and it is represented by R
s(α).
From the above point, R
s(α) is represented as follows, _??_ (1) where α
i=1 is understanding α
i, α
j=0 is not understanding α
j.
In this report, readiness region is analyzed from the following points.
(1) To consider the characters of readiness region.
(2) To abstruct the elements of readiness region.
(3) To consider the characters of boundary elements
The results obtained (1)-(3) are as follows.
(1) Let fα
i (S
1, S
2,..., S
n) be degree of achievement, obtained by the pupils who had fundamental abilities (S
1, S
2,..., S
n), on subject matter (α
i). Then, readiness region R
S(α), which is represented by formula (1), is defined by as following formulas (2),(3).
_??_ (2)
where i=1, 2,..., k
??_ (3)
where i=k+1, k+2..., m Let φ(S
1, S
2,..., S
n)(α
1=1, α
2=1,..., α
k=1, α
k+1=0, α
m=0) be ratio of the pupils, who have understood subject matters (α
1, α
2,..., α
k) and had fundamental abilities (S
1, S
2,..., S
n), to the all pupils who had fundamental abilities (S
1, S
2,..., S
n).
Then, from formulas (2),(3),
_??_ (4)
and in any other combination of (α
1, α
2,..., α
m)(where α
i=1 or 0, i=1, 2,..., m).
_??_ (5)
(2) According to the above consideration, readiness region will be abstructed by formula (6) in real material.
_??_ (6)when (α
1*, α
2*,..., α
m*) satisfied formula (6), R
S(α)=(α
1*, α
2*,..., α
w*)
(3) Let R
S(α) be readiness region and let R
S(α) be outside of readiness region, then the boundary elements(α
l) have following characters.
_??_
_??_where _??_ represents intersection of A and B.
and so
1>fα
l(S
1, S
2,..., S
n)>0
1>ψ(S
1,S
2,..., S
n)(α
1=1, α
2=1,..., α
l-1=1, α
l=1,
α
l+1=0,..., α
m=0)>0
1>ψ(S
1, S
2,..., S
n)(α
1=1, α
2=1,..., α
l-1=1, α
l=0, α
l+1=0,..., α
m=0).
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