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Harel, G., Behr, M., Post, T., & Lesh, R. (1992). The Blocks Task: Comparative Analyses of the Task With Other Proportion Tasks and Qualitative Reasoning Skills of Seventh Grade Children in Solving the Task. Cognition and Instruction, 9(1), 45-96.

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The Blocks Task:
Comparative Analyses of the Task With Other Proportion Tasks and Qualitative Reasoning Skills of Seventh-Grade Children in Solving the Task
 
Tables and Figures
 
 
 

TABLE 1
Effect on k, the Change between k1 = a1/b1 (or k1 = a1 x b1) and k2 = a2/b2
(or k2 = a2 x b2). From Change Between a1 and a2 (a) and Between b1 and b2 (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.
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TABLE 2
Effect on b, the Change between b1 and b2
. From a, the Change Between a1 and a2 and k, the Change Between k1 = a1/b1 (or k1 = a1 x b1) and k2 = a2/b2 (or k2 = a2 x b2).


 
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.
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TABLE 3
Effect on a, the Change between a1 and a2
. From b, the Change Between b1 and b2 and k, the Change Between k1 = a1/b1 (or k1 = a1 x b1) and k2 = a2/b2 (or k2 = a2 x b2).


 
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.
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TABLE 4
Effect on k, the Change
Between k1 = a1 + b1 and k2 = a2 + b2.
From Change Between a1 and a2 (a) and Between b1 and b2 (b)


 
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.
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TABLE 5
Effect on b, the Change Between b1 and b2. From a, the Change Between a1 and a2,
and k, the Change Between k1 = a1 + b1 and k2 = a2 + b2.


 
Da
k
+
0
-
Missing

+
?
+
+
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.
insert lc phi here
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TABLE 6
A Comparison of Proportional Reasoning Tasks on Several Task Variables
 
Tasks
Variables
Blocks
Balance Scale
Rate and Mixture
Fullness

Numeric versus non-numerica 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 Multiplier-multiplicand 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

aAll 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.
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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

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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
22a
5
CDB
2
9
20
IC
0
1b
6
Uncatagorized
11
14
23

Notes. Each subject received 9 items. SM = structure-matching; SIM = structure-imposed matching; CCB = complement-complete balance; CIB = complement-incomplete blalance; CDB = complement-deficient balance; IC = isolated-counting.
aIncludes the 14 exceptions who formed the structure representation and used the incomplete balance strategy.
bThis is the one exception who formed the complement representation but used the counting strategy.

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FIGURE 1
Item
Pair (A, Bi)
 
Given weight relationship between
A & B
Pair (C, Di)
 
To-be-found weight relationship between
A & B
             
1.
=
=
 
A
B0
 
C
D0
 
2.
=
>
 
A
B0
 
C
D-1
 
3.
=
<
 
A
B-1
 
C
D0
 
4.
>
>
 
A
B0
 
C
D-1
 
5.
>
>
 
A
B1
 
C
D1
 
6.
>
Indeterminate
 
A
B-1
   
D0
 
7.
<
Indeterminate
 
A
B0
 
C
D-1
 
8.
<
Indeterminate
 
A
B1
 
C
D0
 
9.
<
<
 
A
B-1
 
C
D-1
 

 

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

 
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FIGURE 2


 

FIGURE 2 A network diagram for the multiplicative principles structure

 
 
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FIGURE 3

 

FIGURE 3 A network diagram for the additive principles structure

 
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FIGURE 4

 

FIGURE 4 A network diagram for task variables structure

 
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FIGURE 5

FIGURE 5 Diagrams illustrating the deck-top structure that children perceived in making the structure representation.

 

 
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FIGURE 6

 

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

 
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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).

 

 
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FIGURE 8

 

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

 
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FIGURE 9

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