The
Blocks Task:
Comparative Analyses of the Task With Other Proportion Tasks and Qualitative
Reasoning Skills of SeventhGrade Children in Solving the Task


Tables
and Figures







TABLE
1
Effect on k, the Change between k_{1}
= a_{1}/b_{1} (or k_{1} = a_{1}
x b_{1}) and k_{2} = a_{2}/b_{2}
(or k_{2} = a_{2} x b_{2}). From Change
Between a_{1} and a_{2} (a)
and Between b_{1} and b_{2} (b)


Da

b

+

0



Missing


+

?

(+)



(+)



(?)


()

0

+

(+)

0

(0)



()


()



+

(?)

+

()

?

()


()

Missing


()


()


()


()

Note.
the entries +, and 0 in the table body are qualitative values of k.
The ? means the value of k is indeterminate
with only qualitative reasoning. The
means that value of the k is indeterminate,
because either a or b
is missing in the problem information. The entries in parentheses
refer to invariance of products: the others refer to invariance of
ratio.


(top) 



Da

k

+

0



Missing


+

?

(?)



(+)



(?)


()

0

+

()

0

(0)



(+)


()



+

()

+

()

?

(?)


()

Missing


()


()


()


()

Note.
the entries +, and 0 in the table body are qualitative values of b.
The ? means the value of b is indeterminate
with only qualitative reasoning. The
means that value of the b is indeterminate,
because either a or k
is missing in the problem information. The entries in parentheses
refer to invariance of products: the others refer to invariance of
ratio.


(top) 



Db

k

+

0



Missing


+

+

(?)

+

(+)

?

(+)


()

0

+

()

0

(0)



(+)


()



?

()



()



(?)


()

Missing


()


()


()


()

Note.
the entries +, and 0 in the table body are qualitative values of a.
The ? means the value of a is indeterminate
with only qualitative reasoning. The
means that value of the a is indeterminate,
because either b or k
is missing in the problem information. The entries in parentheses
refer to invariance of products: the others refer to invariance of
ratio.


(top) 



Da

b 
+

0



Missing


+ 
+

+

?


0 
+

0




 
?






Missing 




Note.
the entries +, and 0 in the table body are qualitative values of
k. The ? means the value of k
is indeterminate with only qualitative reasoning. The
means that value of the k is indeterminate,
because either a or b
is missing in the problem information. The entries in parentheses
refer to invariance of products: the others refer to invariance
of ratio.


(top) 



Da

k 
+

0



Missing


+ 
?

+

+


0 


0

+


 




?


Missing 






(top) 

TABLE
6
A Comparison of Proportional Reasoning Tasks on Several Task Variables


Tasks

Variables 
Blocks

Balance
Scale

Rate
and Mixture

Fullness


Numeric
versus nonnumeric^{a} 
Nonnumeric 
Numeric
or nonnumeric 
Numeric
or nonnumeric 
Numeric
or nonnumeric 
Number
of relational propositions 
4 
3 
3 
3 
Types
of quantity involved in: 




The problem information

Extensive
and intensive 
Extensive 
Extensive 
Extensive 
The task question

Extensive 
Product
of measures 
Intensive 
Intensive 
Types
of reasoning 
Multiplicative
and additive 
Multiplicative 
Multiplicative 
Multiplicative 
Type
of invariance 
Invariance
of product 
Invariance
of product 
Invariance
of ratio 
Invariance
of ratio 
Mathematical
principles 
Product
composition
and
Product decomposition
and
Additive decomposition

Product
composition 
Product
composition 
Product
composition 
Relationship
between problem quantities 
Multipliermultiplicand 
Product
of measures 
Partitive
division
Quotitive division
Functional 
Partitive
division
Quotitive division
Functional 
Physical
principles on which the problem is based 
Uniform
density 
Angular
moment conservation 
Uniform
diffusion 
Uniform
pressure of liquid at rest 

^{a}All
tasks can be stated either as numeric or nonnumeric. However, the
blocks task as it is presented in this article is nonnumeric; the
balance scale task as used by Siegler (1976) is stated in two versions,
numeric and nonnumeric; rate and mixture tasks were also used in
the research on proportional reasoning in both versions; and the
fullness task as used by Siegler and Vago (1978) is nonnumeric.


(top) 

TABLE
7
Frequencies of Strategies Used for Problem Representations


Problem
Representation

Strategy 
Structure

Complement

Isolated


Matching 
28

0

0

Imposed matching 
5

0

0





Complete balance 
0

1

0

Incomplete balance 
14

19

0

Deficient balance 
0

7

0





Counting 
0

1

29





Total 
47

28

29



(top) 

TABLE
8
Frequencies of the Solution Processes Across Children's Mathematical
Ability Levels


Ability
Level

Solution
Process 
High
(n = 5)

Middle
(n = 6)

Low
(n = 6)


SM 
20

8

0

SIM 
5

0

0

CCB 
1

0

0

CIB 
6

22^{a}

5

CDB 
2

9

20

IC 
0

1^{b}

6

Uncatagorized 
11

14

23


Notes.
Each subject received 9 items. SM = structurematching; SIM = structureimposed
matching; CCB = complementcomplete balance; CIB = complementincomplete
blalance; CDB = complementdeficient balance; IC = isolatedcounting.
^{a}Includes the 14 exceptions who formed the structure representation
and used the incomplete balance strategy.
^{b}This is the one exception who formed the complement representation
but used the counting strategy. 


(top) 

FIGURE 1
Item 
Pair
(A, Bi)


Given
weight relationship between
A & B

Pair
(C, Di)


Tobefound
weight relationship between
A & B








1. 


=



=


A

B_{0}


C

D_{0}


2. 


=



>


A

B_{0}


C

D_{1}


3. 


=



<


A

B_{1}


C

D_{0}


4. 


>



>


A

B_{0}


C

D_{1}


5. 


>



>


A

B_{1}


C

D_{1}


6. 


>



Indeterminate


A

B_{1}



D_{0}


7. 


<



Indeterminate


A

B_{0}


C

D_{1}


8. 


<



Indeterminate


A

B_{1}


C

D_{0}


9. 


<



<


A

B_{1}


C

D_{1}


FIGURE
1 Diagrams illustrating the relations among the blocks for the
nine blocks tasks



(top) 

FIGURE
2
FIGURE
2 A network diagram for the multiplicative principles structure



(top) 

FIGURE
3
FIGURE
3 A network diagram for the additive principles structure


(top) 

FIGURE
4
FIGURE
4 A network diagram for task variables structure


(top) 

FIGURE
5
FIGURE
5 Diagrams illustrating the decktop structure that children perceived
in making the structure representation. 




(top) 

FIGURE
6
FIGURE
6 A network diagram for the components in the structure representation
of the blocks tasks and relations among them.


(top) 

FIGURE
7


FIGURE
7 Diagrams illustrating two states: One state (describing Blocks C
and D) is viewed as resulting from the other state (describing Blocks
A and B). 


(top) 

FIGURE
8
FIGURE
8 A network diagram for the components in the complement representation
of the blocks task and relations among them.


(top) 

FIGURE
9


FIGURE
9 A network diagram for the components in the isolated representation
of the blocks task and relations among them. 





(top) 