Rational Number Project Home Page 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. Due to the large number of figures, this document is contained in two windows. The main document will open in a browser window behind this window.

Tables and Figures

 Table 1 Table 2 Table 3 Table 4 Table 5 Table 6 Table 7 Table 8

 Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9

 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 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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