on spontaneous correspondence€¦ · 2. the techniques of experiments (1)reproduction of figures...

1

On Spontaneous Correspondence

Syuntaro Sate

i)epar tment of Mathematics, Facd ty of Ar ts and Science, F ukushima Unit?ersity

Contents

S1.PrefaceS2. About the date of investigations,examinee and co-workersS3. The preparations and the techniques of experiments

1. The tools of experiment2. The techniques of experiments

(1)Reproduction of figures(i ) Badlystructured figures(ii ) Open series(iii) Closed figures, the shape of which did not depend on

the number of elements used(iv) Closed figures of which the shape depended on the

number of n arbles(v) More complex closed figures, less familiar to the child

(2)Single rows・§4.The way of investigations and its results

(1)Reproduction of figures(i ) Badlystractured figures(ii ) Open series(ill) Closed figures, the shape of which did not depend 01'

the number of elements used(iv) Closed figures of which the shape depended on the

number of marbles(v )More complex closed f igures,less familiar to the child

(2) Single rowsS5.The relation between the results of investigations and I‘Q.S6. The relation between the results of investigations and the time

which are needed

S7.Conclusions

25

§1 Preface

For about 40 years Jean Piaget and his collaborators-Geneva School-have been publishing many theses dealing with the formation of cn cepts in children. He has studied the concepts of number,geometry, space, etc.

Professor Marshall Stone of the University of Chicago has pointed out the importance of his studies and he says as follows.

2

26 Sci Rep. Fukushima Univ .,Not4

‘mumjnatjng psychological investigations,. particularly those of Piaget, pointthe way to hitherto unrecognized pedagogical possibilities, even though we、can perhaps catch no more than a challenging glimpse of their true extent.in the present stage of our knowledge.';1、1:Anyway, it is true that his work stimulated students all over the we「ld・

Many reexaminations were tested, and even though they are not ag「ee with in detail , according to the following theses 〔2〕~〔11 his assertion has been ackno- wledged gradually.

The author stated in :1-,as follows;The exparts who take part in infant education have already Set uP tfie COn-f erence of inf ant edtlcatjort in spring 1962, and they a「e now P「aCtiCing Sy-stematjcal education based on the cognition of child「on・;'Recently,the educational authorities had stated needfu11ness of infant edu-

cation, and at last “the course of study of the infant school''was Published On March 21,1964. This announcement was warmly welcomed by the People-

what process pursues the child's spontaneous correspondence ? This P「o- cess ls an urgent problem to establish the infant education Systematically・

J.p1aget published the noticeable report based on his investigation・n・l Ac- cording to his report,114〕 piaget tried to analyse the mechanism Of cO「「eSPOn- dence, no longer considering its results, but examining its Spontaneous development, I.e. in situation in which the child is compelled to find the CO「「eSPo- ndence of his own accord and to make what use of it he Can・ And afto「 his・ Investigation, piaget mentioned that there are three Stages bete「e the Child gains one_one correspondence.

That is to say;s tage l . The child uses only g1obal comparision, imitating the COnf igu「ati-

on of the model without attempting exact quantification・stage 11 . There ls one_one correspondence, but it ceases to exist When the

figure is distorted.stage Ill . There is exact correspondence and lasting equivalence・ 1:',J.pjaget used swiss children as examinee for his investigation・The authO「

uses children In Fukushjma prefecture as examinee and iS going to examine

whether the stages which Piaget said are admitted o「 not. And if Piaget'S results would be trite lefts consider hereafter the best way Which Child ex-

perience in order to establish spontaneous cO「「eSPOndenCe・

§2 About the date of investigations, examinee and co-workers

(1)Date Feb_ 18~25, (2)ExamineeChildren in a nursery4 yeat・s old children

1964 I_Q.test Feb_14,1964

in Fukushima city10 boys 10 girls total 20 children

3

On Spontaneous Correspondence 27

5 yeasr old children 10 boys 10 girls total 20 children6 years old children 10 boys 10 girls total 20 children

sum-total 60 children

〔Notice) Children are selected at random_(3)Co-workers4 senior students of Educational Department of Fukushima University.

Tomoku Kageyama, Satico Kamiyama,FumikoTakano& Isako Yokozawa_

The stuff of a certain nurcery.

§3 The preparations and the techniques of experiments

1.The tools of experimentO Marbles (They are made of blue glass)100O Match-sticks 50O Drawing papers (19c観x26cm) 4

2.The techniques of experimentsHere our material will not be ef such a kind that the correspondence is su-

ggested by the fact that the objects are qualitatively complementary, and in our questions there is no suggestion of method to be used in making correspondence.

Investigators are four, one of them is operator who operates something to be necessary for questions and experiments, and another one is an assista- nce of the opertor and she prepares for the next experiments and puts away tools which have already used, and the last two record faithfully conversation and reaction between operator and examinee. And investigators pay attention sufficiently not to give shock for the examinee_

(1) Reproduction of figures(i )Badlystruetured figures

,(_A given Model 1 〕Mt M ,1 Mate 1

which are put badlystructured on a drawing paper Mt,- figure transformed from M,

4

28

Mate 1

Sci.Rep. Fukushima Univ Not4

7m-atch_sticks which are put badlystructured on a drawing paper

① we present the child with the model M,made with marbles,and ask himto take the same number of marbles as M, contained_ (Question. There isa number of marbles: pick out the same number.)

② we make him recognize that the number of marbles on the drawing paperis as same as M,. (Question.Is that as same as this (M,)?)

③ we transform M, into M',. Then we ask him the number of marbles onthe drawing paper is as same as M',. (Question. Is this (Mt,) as same asthat ?)

④ We ask him the reason of ③. (Question_ Why ?)⑤ w e put away the marbles on the drawing paper, then ask him to take the

same number of match_sticks as Mt contained. (Question. Pick out match-sticks as same as M,_)

0 we put away the matches on the drawing paper, then ask him to take thesame number of marbles as model Matcl contained.(Question.Pick out ma-rbles as same as Matc1.)

⑦ we put away the marbles on the drawing paper, then ask him to take thesame number of match_sticks as Mate 1 contained. (Question. Pick outmatch-sticks as same as Mate 1.)

(ii) Open series

1:A given Model 2 〕M , M

000 0

0000

Mate2

- 00-- 00 -- 00-

- 00 -

M _ 8 marbles are put in two para11el rows on the drawing paperM':1- figure transformed from M_Mate2 _ 8 match_sticks are put in two parallel rows on the drawing paper

w e shall have the investigation just the same technique as (i) , Substitut- ing the model M_,Mt2 and Mate2 for M,, Mt,and Mate 1 respectively・

5


,iii) Closed figures, the shape of which did not depend on the numberof elements used

〔:A given Model 3 )M 3

0 0 0 0 0 0 0

0 0

M t3

0 0 0 0 0 0 0 00

Mate3

M3- 9 narbles are in a circle on the drawing paperMt3- Model M81s transformed into a ellipseMate3 - 9 match-sticks are in a circle on the drawing paper

We shall have the investigation just the same technique as (i) , substitu- ting the model M_, Mt3 and Mate3 for M,, M'1 and Mate 1 respectively.

(iv) Closed figures of which the shape depended on the numberof 置1arbies

〔:A given Model 4 )M 4 M t4

0 0 0

〇

〇

〇

〇

〇

〇

Mate4

-0 - 0 - 0

一0 一一一0 -0 -

M4- 9 marbles are put in a square on the drawing paperM'4- Model M4 is transformed into a rectangleMate4 - 9 match-sticks are put in a squarしon the drawing paper

We shall have the investigation just the same technique as (i) , substitu- ting the model M,, M'4 and Mate 4 for M,, M', and Mate 1 respectively.

6

Sci.Rep. Fukushima Univ.,Not4

(v)More complex closed figures, less familiar to the child

〔A given Model 5 〕M 3

000 0

00 0000 0 0

0 〇〇

〇

〇

〇

〇

〇

〇

〇

〇

〇〇

〇

Mate5

M.- 13 marbles are put in a rhomb on the drawing paperM;5- Model M5 is transformed into Mt5Mate5 - 13 match_sticks are put in a rhomb on the drawing paper

We shall have the investigation just the same technique as (i) , substitu- ting the model M,, M'5 and Mate 5 for Mt, M', and Mate 1 respectively・

(2) Single rows

〔A given Model 6 -〕 M6

M6- A row of 7 marbles on the drawing paperA - a fundamental figureB - a dense figureC- a extended figure

C

A

B

0 We present the child with the model A made with marbles, and ask himto take the same number of marbles as A contained. (Question. There isa number of marbles :pick out the same number on the drawing paper.)

② We make him recognize that the number of marbles on the drawing paperis as same as A. (Question. Is that as same as this (A)?)

③ We transform A into B. Then we ask him the number of marbles on thedrawing paper is as same as B_ (Question.Is this (B)as same as that ? )

④ We ask him the reason of ③_ (Question.Why?)⑤ We transform B into C_ Th n we ask him the number of marbles on the

7


drawing paper is as same as C. (Question. Is this (C)as same as that ?) 13 We ask him the reason of ⑤.(Question_Why ?)

§4 The way of invettigations and its results

We'11 give representative_ examples for reference. And the author will classify these results in the stage I ~m according to Piaget_

(1) Reproduction of figures

( i ) Badlystrueture(, figures

◆Stage INaoko T.(4:8)

When we ask her to take the same number of marbles as M,, she takes out of a box slowly the marbles one by one seeing the model. She tries to arrange them just like Mt_・''Are they the same ?''

' ' Y e s . ' '

In fact,the collection she has just made does contain 1 extra element.

Next, she tries to correspond matches to model M,,・“Will you pick out the same number of match-sticks as this (M,)?'She picks out 6 match-sticks one by one, and sc one is less than M,, butshe says“There are the same_”

◆Stage IIMinako S. (6;10)

The child is shown Mt, and she is ,、sked to pick out the same number of marbles, she picks out 7 marbles on the drawing paper corresponding one to one to M,.・“Are they the same ?"'Yes.''Then,we transform M, into M',.

・'Now is there the same number of marbles ?''No.''

・“Which is more ?''“Here (pointing to her copy).'

・“Why ? ”''Because, they are too close together.''

・“Then,how can you get the same number -'''She moves the model M', into M,.

・''Is there the same number of marbles ? f'''Yes.''

8

Sci.Rep. Fukushima Univ_, Not4

O Stage IllHumiko Y.(6;9)

The child is shown M,, and she is asked to pick out the same number of marbles, she at first counts the number of M, and she picks out 7 marbles on the drawing paper.・'Is there the same number of marbles ?”

' ' Y e s . ' '

We transform Mt into M',_・“Is there the same number of marbles, too?''

“ Y e s _ ' '

,''Why ?''Because there is the same number of marbles.''Then, we show her Matcl and we ask her to pick out- the same number

of matches, she picks out 7 match-sticks on the drawing paper.・''Are they the same ?''

“Yes_''

Now, though we elected 4~6 years old children as examinee, classifying them in 3 stages we get (Table t ) .

According to this (Tab.1) , we f ind that the older children will be, the smaller the children who belong to stage I .

Many children who are 4 years old belong to stage I , many children who are 5 years old belong te st-

. (Tab age I or Ii , and many chil-dren who are 6 years old be-

agelong to stage m. __ _ _,

As for the develapment,there is no difference between boys and girls.

(ii) Open series

O Stage IHiroyuki Y.(4;3)

The child is shown M,.and he is asked to pick out the same number of marbles, he puts them just as (ii) - 1 on the drawing paper.

1 、)

stage

boy

gi rl

total

boy

i . gir 1

total

boy

gir l

total

total

0

9

10

10

0

0

9 1

16 20

0 1 20

1 ' 10

) 10

' 20

10

10

9

( 11) - 1

〇〇

〇〇

〇

〇

〇

〇

〇


(i i ') - 2

( 11) - 3

〇〇〇

〇

〇

〇

〇

〇〇

〇

〇

〇

〇

〇

〇〇

33

_- '゙Are they the same ?”' Y e s . ' '

He puts match_sticks, marbles and match-sticks according to the models M , Mate 2 and Mate 2 respectively just as (ii) - 2:,(ii) - 3 and (ii) - 4・二Stage 11

Hiroki 0 . (:i ;4)we present the child with the model M2, and ask him to take the Same

number of marbles, he reproduces two para11el rows of marbles, Copying the configuration of the model M2.・''Are they the same ')”

f Y e s . , '

We transform M. into M'_.・''Are they the same,too ?''

' N o '

W h y ' ' '

Where are there more ? ''・Here (pointing to his copy) .'

..・・How can we get the same number ?''He moves the marbles just as the model.

.・・Is there the same number of marbles,. then ?''''Yes.'This child reproduces correctly the f igure after some trials between like

objects and unlike objects, but the correspondence does not mean lasting equivalence, and equivalence ceases to exist when the figure is distorted.': Stage IE

Yumi 0.(6;2)when she is asked to reproduce M, she takes marbles one by one out of

10

34 Sci_ Rep. Fukushima Univ Not4

a box, and puts them on the drawing paper at random. ・“Are they the same ?”“Yes.''We transform M・ into M'2.

・“Are they the same, now ?;'' Y e s . ' f

Why ? '“None takes out of the marbles''

The results obtained with this test can be classified into 3 stages, and we

(Tab. 2)

boy

4 gi rl

9 1 1

total

01_ _ 10

total

boy

gir l

total

18

6一・9

2 1 0

5 1

2.7

一

一

一

一

2.2

4

10

10

20

get (Table 2) .According to (Tab2) ,

we find that the older children will be the smaller the children who belong to stage

l .

Many children who are 4years old belong to stage I ,many of 5 years old belong to stage l or II , and many of 6 years old belong to stage III .

As for the development,there is no dif ference between boys and girls.

boy

g-「l

to-

o.,一,

1・0

1

9 1

18

10

(iii) Closed figures,the shape of which did not depend on the number ofelemerts used

1二. Stage 1Keiko S.(4;7)

we present the child with M3 made with marbles, and ask her to take the same number of marbles. She makes a circle as large as M,, but she puts 12 marbles on lt. (see (iii) - 1)・“Are they the same ?''

' ' Y e s ' '

Where is this (indicating a marble on the copy)?';Here (pointing at random to those of the model) ."Next, she corresponds match-sticks to M3, and she tries and tries again

and she makes a f igure as (iii)- 2,(iii)- 3, but she fails to make a circle of 9.

11


〇〇〇

〇

〇

〇

〇

〇

〇

〇〇〇

35

'I don't know.・'She says, and gives up.Then we present the child with Mate 3 made with matches,and ask her to

take the same number of match-sticks, she makes a circle of 12 match-sticks. ・'Are they the same ?”

' Y e s . ' '

(: Stage jlYasuaki T. (、4 ;9 )

when he is asked to take the same number of marbles as M3, he puts 12 n arbles at random.・‘・Is there the same number of marbles ?'''No.'I want them to be the same.”

He counts 9 corresponding to Mg, and removes the 3 marbles.Next,we transform M3 into M'3.

-・''Is there the same number of marbles, too ? 'He counts them both,and he says

'Yes.・':l Stage litYoiti K. (i ;_i i

when he is asked to take the same number of marbles as Mu, he makes a circle of s, but he counts again and he reproduces a circle of 9.・'Is there the same number of marbles ?''

' Y e s . '

We transform M3 into M'3..・'Are they the same ?'''Yes. Even if you stretch them, the number of marbles remains the same.f'

The rest,Its obtained with this test can be classified into 3 stages, and we get (Table :,.

According to this (Tab.3) , we f ind that many of 4 or 5 years old chi1- aren belong to stage l , and we find only one child who belongs to stage II .

As for the development, there is no difference between boys and girls.

12

36 Sci. Rep. Fukushima Univ.,Not4

(Tab. 3)

- -一二二:: stage -1 1 、、: 、_? 1

a g e s e x 、、、. 1

1-boyl girl

1 total1

t hey 1-girl

total

g,r,

E

19

°1 20

1I II ' III , totall

_ _ - -- --- 1- -- ・- - 一一 9 1 1 0 ' 10_ _ __ __-- --1-- - -

10 0 1 0 1 Io

151

1.3

一

0

0

0

10

-10.一20

一

一

---一

-

l・l

9

7.6

一

(lv) closed fjsures of which the shape depended on the number of ma「bios

O Stage ITomoko H.(4;0)

we present the child witt, M4 made with marbles, and ask he「 to take the same number of marbles, she puts Only two...‘,Is there the same number of marbles ?''“Yes.''

.“where Is this?this? _ this (indicating some marbles On the model M,)?'she points to two marbles alternatively. As for the model Mate4,She

gives the same behaviour.O Stage II

Seizi S.(5;9)He can correspond correctly the marbles to M4.

・''Are they the same ?''' ' Y e s . '

We transform M4 into M'4.・“Are they the same ?;'He counts the number of marbles of M'4 and his copy, but he makes no

reply .・''Is this different ?''''Yes.”

・“Where are there more ?''He keeps silent.

・''Don't you know ?''“I don't know.''

He can make one_one correspondence, but he loses equivalence When the figure is distorted.

13

The results obtained with this get (Table 4) .

According to this (Tab.1) , we find that the older

childrea will be, the smallerthe children who belong to st-age I .

Almost of 6 years old ch-ildren belong to stage m.

As for the development,there is no difference betweenboys and girls.

On Spontaneous Correspondence 3r

◆Stage IIIMasahiro K.(5;8)

seeing M4 he reproduces the model by marbles. We transform M4 into M , 4 ・

・'Is there the same number of marbles ? ''“ Y e s . ' '

・''Are you sure ?''“Yes.''

we remove one from the model and we keep his copy as it was.・cAre they the same ? '“No.”

・'Why ? ''“Because,you take away one.';

Then,we add one and we construct M'4 again.・“Are they the same,now ?'''Yes.''Why ?''

‘Because,you make the number of marbles as same as the first one (M4).''

test can be classified into

(Tab

stages, and we

l 4

boy

girl

total 15

2

3

total

10

10

20

boy 10

10

20

boy

'g-「1一to-

(v )More complex closed figures, less familiarto the child

O Stage INaoko T.(4;8)

Looking at M., she makes a rhomb of 14 elements just as (v)- 1.

10

10

18

(V) - l

〇〇〇〇

〇〇〇〇〇〇

〇〇〇

〇

20

14

・8 Sci.Rep.Fukushima Univ_,Not4

・・“Is there the same number of marbles ?''Yes.''

・“Where are those (indicating sori,e marbles on the M.)? ''There (pointing at random to those of her copy ).''

O Stage EYasuaki T.(4;9)

Looking at M,., he puts marbles correctly. ・・“Are they the same? ''“Yes.'

We transform M5 into Mt。.・''Are they the same ?''''I don't know.'He also succeeds in making the correspondence between marbles and ma-

tch-sticks, and he can recite numeral to )19.O Stage IE

Hiromi K. (5;9)We present her with M5.

・''How many marbles do you need to make the same one ?''' 1 3 ' '

We transform M5 into M'5.・“How many marbles do you need to make the same one ? '

' ' 1 3 '

Is the number of the later as same as th.e former ? "''Yes.''

The results obtained with we get (Table 5) _

Most of 4 and i years

old children belong to stage l , and most of f; belong to stage Ill

The reason of this will be the very fact that the figure is less familiar to them and the figure is more complicated.

As for the development,there is no difference between boys and girls.

this test can be classified into

(Tab ) )

boy

girl

total

boy

g-「l

to-

一

6

-

stages, and

- - ----- - - l

E IE 1 total

9

10

19

一

l

一

一

0一0.0

0

0

0

1一1

2

18

一

一

1

1

2

0.0

20

10l

10

20

9

8

17

15

(2) Single rows

On Spontaneous Cor respondence 3g

◆Stage ITakao T_(5;7)

We present the child with model A.・“There are marbles, aren't they ?Pick out the same number of marbles andputs them.''

Looking at the model,he takes 10 marbles and he makes a dense row with them so that his row is as long as A.・'?Is there the same number of marbles?''

Yes.''We transform A into B.

・“Now,are they the same ?''No. '

・'Why ? 'There are more here (pointing to his row).''

・''What can we do to make them the same ?''He closes up his row .

・'Are they the same,now ?”u Yes.''

We transform B into C. ,・''Are they the same,now ?';

There are more here (pointing to the model C.) ''・ Stage Il

Seizi S.(5;9、1Concerning model A,he makes a correct correspondence. But as for B

・'' Is there the same number of marbles ?''' ' N o . ' '

・ “ W h y ? ' '

''Becauseyouc1osetogether the model B.''・'Then,where are there more ?''

Me (pointing to his copy) .''We transform B into C.

・“Are they the same ?''“No.'Why ? ''

“Because you spread here (C) .”・“Where are there more ?''“Here (pointing to the model C) .”

◆Stage IllMasaharu K.(6;1)

Concerning model A, he makes a correct correspondence, and as for B, C he insists the equality saying “You only moved them”or “You only closed

16

Sci.Rep. Fukushima Univ., Not4

together.''

The results obtained with this test can be classified into

3 stages,and we get(Table 61.The most important thing

about this test is brilliant the change from the stage I to the stage IE.

(Tab. 6)

lE l tota1

boy 10

16

、1 0

4 0

10

'20

boy

girl

total

boy

17

§5

-

.-2

43

4」・4一o

46

41

48

49

4-0

4-

42

43

44

b b b b b b b b b b

g g g g

I

I I[

l

I I I

I

I Il I I I Ii

一I

I

I I

(2)

I I I II I I I I I E一I I I I

I.Q. minu teslll

20 11

6 0 6 5 5 3 5 3 0

2 2 2 2 3 1 2 2 3

4

9

5

6

1

1

2

2

10

10

.20

一10

girl

total

10

20

In order tomake clear the stage he be1ongs to, time he needed, lefts make a (Table 7 ) .

(Tab_ 7 ) 4 years old children. experimentl Child age 1 (l 、) (1j ) (jj j) 1_(jv) ・ (v)

The relation between the results of investigations and I .Q

his I.Q.(Tanaka B)and

17

41

I I I I

I I I I I I I I I I

5

6

7

8

9

10

4

4

.-

4

4

4


4546474849410

.-。、一- I.Q. minut experiment

(iii? (iv) - (v)

years old children

(ii、) (i) age chi ld

16

6

6

4

0

3

7

5

5

9

1

2

3

3

2

1

2

1

1 23

18

l8

12

4

2

9

2

7

6

1

3

1

4

1

2

118

0

3

,o

5

5

7

3

8

9

9

9

9

0

0

0

9 1 109

1- - - - 117

9

2

9

6

9

6

0

0

7

1

10

0

8

7

8

9

1

8

1

1

1

El

l

II II ]l I I I l]I II

II

EI

I

I

I

I

Ef

I

]]f

lI

Ill I I

l

- I I I

0

1

2

4

)

6

7

8

9

10

b s1

hi

bib 54

b 55

b 56

b 57

b 言b 5

9

b 510

g

g 5 g 3

g 4

.. 5 g 55

g 65 g 7 5 1

g 85

g 9

5g 1o

22

18

42

child age

Sci.Rep. Fukushima Univ.,Not4

years old children- - experiment (i) (ii) (iii) (iv (v) (2)

I.Q

6-

62

63

f4

65

66

67

68

69

6a.

61

62

63

64

65

66

67

68

b b b b b b b b b b

一g g g g g g g g

69

60

g g

0

0

1

4

-o

6

7

8

9

10

0

0

1

2

3

4

5

6

9

10

]ff

EI

Il-

-]f

EI

III

III

Ef

EI

Il

EI

Il I

Ef

lI

IIf

EI

III

III

III

]I[

]Ef

m 1]I

IE・

Ef

mM一

-fl

mM一

T,

El

f[l

m一

EI

I1

111

mm一

m

Ei

m一田一田一-I[

l[j

ill m m一 m一

111 11 130

1II 1241

of f 501

jif f 164

El l 24

I置 1 184

]E

lfI

IIl

I m--IIIIIEIm一-I[m一m

I II

EI

If

Ifl

EI

EI

II[

EI

If. m

8

0

0

9

4

3

1

1 96

127

119

135

6

9

4

5

3

6

0

5

0

6

3

4

4

5

1

1

1

1

1

1

1

minutes

15

9

1

6

6

3

0

0

0

5

2

2

1

2

1

2

c・一2

2 28

15

15

13

24

18

8

7

7

1

1

2 17

20

w here hi (i=4,5,6) means i years old boybj (j=1,2,- ,10) means examinee boyg is an abbreviation for girl

Now, we rearrange 5,6 years old children according to thei「 ages and sexes and let's calculate the mean value of their I。Q。. Then We get a(Table8) .

19

(Tab

ex p

8)l

l

1

43

I .Q num. I.Q I .Q

boy gi r1 total

1029296

97 118 103

11899

105


(i)

(ii)

(iii)

(iv)

(v)

(2)

vi

al一yl

al

yl

al

yl

a1一vi

al

o「一t.o.1「t

o.-「t

o「t・一o「t

b

gto・b

gto

b

gto

b

'glto

一b'glto

10697

101

976

861一一841

344.011

111一111

boy

girl total

boy

girl total

7 8

一-,

9

98

9

-0

7

99

9

i i6

124 117

999797

96 1)7 111

136 141 138

109 109

998

138 142 140

3 1072 1145 110

9 1387 145

16 141

3 1073 1106 109

9 138g 140

18 139

1 1181 1102 114

yi

al

yl

a1

o「t

o.-「-

b

gto

b

gto

96 131 119

1019397

829一0-,/.6

9

.0

g

1

.--

1

111

9

00

7

135 140 139

2 1083 10:;5 106

8 140

9 140 17 ' 14(1

According to this (Tab.8) , we can conclude that comparing the

20

44 Sci .Rep_Fukushima Univ., Not4

ages children, the higher their I.Q. will be, the closer to the stage lit they will be.

§6 The relation between the results of investigationsand the time which are needed

According to (Tab.7) the time which are needed to complete the tests by 4,5 and 6 years old children are 24,22 and 20 minutes respectively.

Therefore we can conclude that the older they will be, the shorter the time will be.

§7 Conclusions

As you know, we want to know what might be called quantifying operations, i.e. the elementary operations of correspondence, equating, etc. which constitute the logic of number. In a word, we would like to concentrate on the problem of the genesis of operations as such, but not the problem of perception.

And now we have investigated (1) the reproduction of figures and (2) single rows according to the same way as Piaget had done lt. The results of our investigation are classif ied into 3 stages as Piaget.

The children at the ist stage do not feel the need for a quantitative evalu- tjon, and they have no precise notion of cardinal number. They therefore confine themselves in evahlating the given sets, to global qualitative comparision (such as long, short,wide, narrow, dense, spread out,etc.), without co_ordination of the qualities that are compared_

These results are coincide with Piaget's. But we have the following exce- ptional examples.

Tosiya D.(4;0), seeing the face of experimentalist, has pick out marbles unlimitedly. As for the experiment (iv) such as closed figures of which the shape depended on the number of marbles, e.g. a square, he can't count the marbles, he can't correspond the marbles neither reproduce the figure.In other words, he can't clo the very g1obal comparision, and he continues to pick out marbles until we ask him “Are they the same?''_We shoud say this child pre-stage l , but we dare to make him belong to stage I .

The children at the 2nd stage get out of the global comparision, and thei「 analysis of fjgtlre and quality becomes more correct, and they begin to feel the need for a precise quantitative evalution. But, according to our exPe「i- ments, the number of the children at the 2nd stage are less than we have expected. And this is remarkable in experiments (iii) and (v) (from Tab.3 and Tab.5) .

21


The children at the 3rd stage can do exact operational correspondence,and even if figures are changed, they keep lasting equivalence.

Now, let's consider each figure. The children who are 4 or 5 years old can't correspond correctly the closed figures (iii) , the shape of which did not depend on the number of elements used.And, in the case (v) ,. too, more complex closed figures, less familiar to the child, it is the same as above mentioned.

According to the results of our investigation, we get series (2)or (iv) ,(ii) , (i) , (iii) , (v) ;where (2)or (iv) is more easy to the children, and(v) is more difficult to the children.

If we consider these results from the point of view of materials, the chi_ 1dren can correspond correctly between like objects. On the contrary,between non-likely objects, for instance in the case where one corresponds match-sticks to marbles, one puts marbles as long as match-sticks by disturbing match-. sticks.

So, we can establish the order of introduction as follows.① We must exercise between like objects, and the order of introduction is (2)

or (iv) , (ii) , (i) , (iii) , (v) .② Then let's exercise between non-likely objects, and the order of introduc_

tion is as same as ①.

References

〔1)Stone, Marshall H.:Reform in School MathematicsNeu1 Thi'liking m School Mathematics. (Organization for European Eco_nomic Cooperation,May, 1961)p.23

〔2)Dodwe11, P.o. :Children's Understanding of Number and Related Con_coptsCailad. J. Psych. XIII, 3 (1960)pp.191~203

(3)Elkind, D.:The Development of Quantitative Thinking; a systematicreplication of Piaget's studiesJ_Genet.Psych_XCVIII (1961)pp.37~46

〔4) Elkind, D.:The Development of the Additive Composition of Classes inthe Child (Piaget Replication Study m)J Geptet.Psych.XCIX (1961) pp.51~57

〔5)Hood, H.B.:An Experimental Study of Piaget's Theory of the Deve1o-pment of Number in ChildrenBrt't.J.Psycho1. LIII, 3(1962)pp.273~286

〔6) Wohlwi11, J.F., and Lowe, R.C.:Experimental Analysis of the Devel-opment and Conservation of NumberChild Deue1opm. XXXm ,1(1962)pp.153~167

〔7〕Lovel1, K., Healey, D., and Rowland, A.:Growth of Some Geom-

22

46 S ci Rep.Fukushima Univ.,Not4

etrical ConceptsChtld Detle1opm_ XXXm (1962) pp_751~767

(8〕 syuntaro Sato :The Child's Cognition of Geometrical Figures - especi-ally, on the representation of figures (No.1)-science Reports of the Fac・ulty of Arts a11d Set'once, Fttkushima Uni11. No.12

(1963)〔9) syuntaro Sato :The Childfs CognMon of Geometrical Figures - especia-

lly, on the ability of facsimile of figures and, of selection of figu「oS,and on the grown-up percentage of the ability of facsimile (No.2)-Sa'ertce Ret1orts of the F,acltlty of Arts artd Sctence, Fukttshima UnilJ No.12(1960)

〔:10) Arthur F_Coxford, Jr. :Piaget :number and measurementThe Arithmetic Teacher vol. 10, num.7 (Nov.1963) pp.419~427

〔:11) Arthur F.Coxford, l r. :The ef fects of instruction on the stage placem-ent of children in Piaget's seriation experimentsThe Arithmetic Teacher vol. 11, num. 1 (Jan.1961)pp.4~9

〔12-1syuntaroSato :Child's Cognition of the Conservation of Continuous Qua-ntities

c'cjence Reports of the Faculty of Arts and S tonic, Fu,kushima Univ_ No.13(1964)

(13) J. piaget :The Child's Conception of Number 1952 Routledge & KeganPaul

(14) p. 65 in (13〕〔:15) p. 66 in 〔13)(16-1w .A.Lay :Fuhrer durch Rechnen Unterricht gegriindet auf didaktische

Experimente Leipzig (Nemnich), 1907 (2nd edition)〔17) A. Descoeudres :Le Deve1oppemente do t'Enfant do 2 a 7 ans

Delachaux & Niestle, S.A., 1920〔18〕 0.Decroly :Etudes do Psychogenese Lamertin, 1932

Abstracted

w .A. Lay、16: made a detailed study of the way in which various figures made with 3,4,5, etc., objects arranged as triangles, squares, etc., a「e distinguished by the child, f rom the poll:it of view of perception Of numbe「. The number 4, for instance, is more easily recognized when the objects a「e placed at the 4 corners of a square than when they are placed at random. A. Descoeudres、17、and 0.Decroly、'8.made use of these investigations in their int- eresting research into the development of number. Our point of view here will be different, for while these authors examined what has come to be Called perception of number, i.e. the application of already existing numerical sche- mata to discrete objects perceived in the same field, we shall examine what might be called quantifying operations, i.e. the elementary operations of co-

23


rrespondence, equating,. etc. wh.ich constitute the logic of number. In other words, we shall ignore the problems of perception and shall concentrate on the problem of the genisis of operations as such.

Now,we prepare 6 types of figures according to J.Piaget.;113、(i )Badlystructured figures, e.g., a collection of marbles distributed atrandom, but neither touching nor overlapping, (ii)Open series, e.g., twoparallel rows of marbles, (iii)Closed figures, the shape of which did notdepend on the number of elements used, e.g., a circle, (iv)Closed figuresof which the shape depended.on the mlmber of marbles, e.g., a square,(v)More complex closed f igures, less familiar to the child, e.g., a rhomb,(2)Single rows.According to our results of investigations, the children are classif ied into

3 stages as Piaget.Stage l :G1obal comparisionStage E : Intuitive correspondenceStage If :Operational correspondence

And we can establish the order of in.troduction as follows.① We must exercise between like objects, and the order of introduction will

be (2)or (iv) , (ii) , (i) , (In) , (v) .② Then let's exercise between non-likely objects, and the order of introduc-

tion will be as same as ①.

on spontaneous correspondence€¦ · 2. the techniques of experiments (1)reproduction of figures...

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