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Harel, G., Behr, M., Post, T., & Lesh, R. (1987). Qualitative differences among seventh grade children in solving a non numerical proportional reasoning blocks task. In J. Bergeron, N. Herscovics, & C. Kieran (Eds.), Proceedings of the Eleventh International Conference, Psychology of Mathematics Education PMR - XI Volume II (pp. 282-288). Montreal, Canada: PME.

 
QUALITATIVE DIFFERENCES AMONG 7TH GRADE CHILDREN IN SOLVING A NON NUMERICAL PROPORTIONAL REASONING BLOCKS TASK
 

Guershon Harel (Northern Illinois University)
Merlyn Behr (Northern Illinois University)
Richard Lesh (WICAT Systems)
Thomas Post (University of Minnesota)*

 
This study aimed to examine differences in the problem representations strategies employed by low performance and high performance 7th grade children in solving a non numerical proportional reasoning task. The non numerical task, which was developed especially for this aim, involves weight and number relationships between blocks -- a derivative of the density concept. Three categories of representations and four categories of operators were used by the children. The differences between these representations and strategies are like those characterized in research of expert-novice differences among adult solvers in other domains of problem solving, such as physics problems.
 

The process of solving a problem starts with the formation of a problem representation which happens in two stages: (1) converting the problem presentation into internal mental entities and relations among them; (2) selecting operators to produce new slates of knowledge from existing states. The problem solving process proceeds by applying a solution strategy, which means searching the goal state by instantiating the operators previously selected. In certain problem domains, such as physics problems (e.g. Chi, Glaser, and Rees, 1981; Larkin, 1977), the role of problem representation in the solution process is well documented in the research of problem solving. In the domain of proportional reasoning research, on the other hand, the matter of problem representations has been apparently ignored. The efforts have been to investigate the effect of context and structural variables on children's performance in solving proportion problems without considering the mental processes that account for this effect. This is a second study in our attempt to investigate the cognitive aspects of proportional reasoning. In the first study (reported in this volume by Behr, Harel, Post and Lesh) we analyzed missing value proportion problems and suggested a problem solving model which takes into account problem presentation, problem representations and operators.

The study reported here investigates differences between low performance and high performance grade-7 children in forming initial representations and selecting operators in the process of solving a non numerical comparison proportionality task. Data reported suggest differences in children's problem solving stages which are like the novice-expert differences characterized by other studies with adults in physics problems (e.g. Chi, Glaser and Rees, 1981).


Procedure


The research paradigm used was that of a modified teaching experiment which was replicated at the two experimental sites. DeKalb, Ill. and Minneapolis, Minn over a period of about 17 weeks. A total of 18 grade-7 children, nine at each site, participated in the research. Each site involved three children judged to be of low mathematics ability and achievement, three of middle ability and achievement and three of high. The major assessment consisted of four one-on-one interviews.

The tasks in this report were given as nine problems, among more, in the third interview. The task involves two pairs of blocks (A, B) and (C, D). Blocks A, C and B, D were constructed from the same kind of unit-blocks. A1 and B1, respectively; the unit-blocks in A were larger in size than the unit-blocks in B. The number of unit-blocks in A was smaller than the number of unit-blocks in C, and these numbers remained constant across tasks. Three different instances of blocks B and D were used. B-1, B0, and B1, and D-1, D0, and D1 respectively; the number of unit-blocks within each of the pairs (B-1, D-1), (B0, D0), and (B1, D1) was one less, the same, or one more compared to the number of unit-blocks in A and C. respectively. The subjects were asked to judge the weight relationship between C and an instance of D based on one of three given weight relationships, less than, greater than, or equals, between A and an instance of B.

The three pairs (A, B1) reflect 3 different number relationships. This crossed with the three possible weight relationships, <, >, and -, results in nine possible given weight and number relationships. Each of these relationships can be associated with a requirement to find the weight relationship between C and one of the three instances of D. This results in 27 possible problem situations. The nine problems selected and presented to the children are described in detail in following table.

 
Item
Pair (A, B1) presented
Given weight relationship
Pair (C, D1) presented
Correct weight relationship to be found
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
(A, B-1)
>
(C, D0)
Undetermined
7
(A, B0)
<
(C, D-1)
Undetermined
8
(A, B1)
<
(C, D0)
Undetermined
9
(A, B-1)
<
(C, D-1)
<
 
Complexity of the blocks task. Traditional tasks used in research on proportional reasoning involve the requirement to decide which of the relations equal to, less than, or greater than holds between multiplicative relationships a/b and c/d. The nonnumeric proportion task used in this study is more complex than these standard proportion tasks because several relationships must be inferred and coordinated before the final relational judgment can be made in the criterion component of the task.

Qualitative proportional reasoning is involved in two episodes in the solution of the blocks task: one is the coordination of the number and weight relationship between A and B to determine the weight relationship, if possible, between A1 and B1; the other is in the coordination of the weight relationship between A1 and B1 and number relationship between the added amounts, uA1 and vB1, to determine the weight relationship between C and D.

Results

Two processes in the children's responses were examined: one process, reflecting the initial representation of the problem, relates to how they initially viewed the structure of each block and how they interpreted the relationships between these structures; the other process, reflecting the strategies children used to solve the problems, relates to the inferences they made to find the final answer. These representations and strategies were separately classified into 3 and 5 different categories, respectively.

We identified three different initial problem representations of the blocks task, Structure, Complement, and Atomic; listed in order from most to least sophisticated.

Structure representation. If the Structure Representation was used, each block was envisioned as consisting of two parts, deck and top, and the observable number relationships between the tops and between the decks within the pairs (A, B) and (C, D) were identified. This representation also included the property that each of the pairs (A, C) and (B, D) was constructed with the same size unit-blocks, A1 and B1, respectively. The following figure describes elements in this representation including the given and the required relations between the weights of the blocks.

 
 
Complement representation. If the Complement Representation was used, the children attended to the fact that the number of units in C (and D) was greater than the number of units in A (and B). They also attended to the property that the corresponding blocks were constructed with the same size unit-blocks. Due to these noticed qualities of the blocks, their representation focused on blocks C and D, where C was viewed as an addition of units on A (and D as an addition of units on B). The following figure describes this representation as a network of two states, 1 and 2. In state 1, blocks A and B and the relation between their weights are given; state 2 is a result of changing state 1 by adding units blocks to A and B and getting C and D, respectively.
 
 
Atom representation. If the Atom Representation was used, the children viewed each block, separately, as consisting of individual unit-blocks, and the number relationships between the tops of the blocks was considered. Elements of this representation are shown in the following figure.
 
 

Five main categories of distinct strategies were identified; listed in order from most sophisticated to least sophisticated, they are as follows: Matching, Balance with three distinct instantiations (Complete, Incomplete, and Deficient), and Counting.

Matching strategy. If the Matching Strategy was used, the child would begin by looking at the relationships between pairs of blocks (A, B) and (C, D). The child would first notice that the number of unit-blocks within the decks of these pairs is equal. Then, he or she would determine the number relationship, µ, between the number of unit-blocks in the tops of A and B, respectively, and the number relationship, µ*, between the number of unit-blocks in the tops of C and D, respectively. The next step would be to acknowledge the given weight relationships, W, between blocks A and B and the required weight relationship, W*, between blocks C and D. Children would then observe one of two relations between relationships: One was that the number and weight relationships between A and B are the same relation ( <, =, or > ), i.e.. µ = W; the other was that the number relationship between A and B and between C and D are the same relation ( <, =, or > ), i.e., µ = µ*. Depending which relationship was determined by a child, one of the following two rules, or operators, was instantiated: (1) µ =W ? W* = µ*; (2) µ = µ* ? W = W*.

Imposed matching strategy. Problems 3, 4, and 7 (see the above table) can not be solved by the Matching Strategy because neither one of the sufficient conditions µ = µ* or µ =W in the above rules holds; this posed a problem to those children who depended on this strategy. After finding they were unable to solve a problem using the Matching Strategy one of two avenues were taken. Either the children would use a fall back strategy (i.e., fall back to a less sophisticated strategy) or would use a derivative of the Matching Strategy, which we call the Imposed Matching Strategy. When using this strategy, the child would suppose an equals number relationship between the tops of blocks C and D, so the sufficient condition µ = µ* would hold and rule (1) could be applied to conclude W = W*. Based on the latter relationship he or she would conclude the required relationship between C and D.

Balance strategy. Within the Balance Strategy category, there are three different instantiations: Complete, incomplete, and Deficient.

Complete balance strategy. If the child solved a task using the Complete Balance Strategy, three relationships were considered. First the children considered the relationship between the weights of A and B. This can be visualized as blocks A and B on a pan balance. The children went on to determine the number of blocks added to A and B which created blocks C and D. At this point blocks C and D are on the pan balance. In order to determine the relationship between the weights of C and D, the children used the weight relationship between the unit-blocks A1 and B1.

Incomplete balance stratagy. The Incomplete Balance Strategy is similar to the Complete Balance Strategy. First the Children considered the relationship between the weights of A and B and then determined the relationship between the number of units added to A and B to solve the problem. Thus, this strategy ignores the relationship between the weights of the unit-blocks.

Deficient balance strategy. In the Deficient Balance Strategy only the rela1ionship between the number of units added to A and B was considered to solve the problem; the other two relationships, the number and weight relationship between A and B, were ignored.

Counting strategy. The most simplistic strategy was the Counting Strategy in which the answer to the task was determined by comparing the number of unit-blocks in C and D.

Relationships between representations and strategies

An analysis of the relative sophistication of the representations and strategies described earlier and the relationships among them will be discussed in Harel and Behr (in preparation). A concise description of these relationships is shown in the diagram below. The diagram shows that the most sophisticated representation - Structure Representation - calls for Matching Strategy, Complete Balance Strategy, or Incomplete Balance Strategy in this order of frequency; the less sophisticated representation - Complement Representation - calls for Incomplete Balance Strategy and Deficient Balance Strategy and Counting Strategy in this order of frequency; and Atom Representation calls only for Counting Strategy. This result is consistent with current theories in problem solving which attribute qualitative differences between a novice and an expert to variability in the quality of their problem representations, especially, in the initial stage of problem analysis. An expert's reasoning about a problem leads to a problem representation that contains structural features of the problem. This representation is superior to that of a novice whose reasoning leads to a representation which incorporates only the surface features of the problem. The sophisticated problem representations of the expert lead to successful solution strategies, while the more primitive representations of the novice lead to unsuccessful solution attempts (see, for example, Chi, Glaser and Rees. 1981).

 
 

 

REFERENCES


Harel, G., Behr, M. J. Representations and strategies of non numerical proportional reasoning task by low performance and high performance children (in preparation).

Chi, M. T. H., Glaser, R. & Rees, E. (1981). Expertise in problem solving. In R. J. Sternberg (Ed.), Advances in the Psychology of Human Intelligence (vol. 1). Hillsdale, NJ: Lawrence Erlbaum.

Larkin. J. H. (1977). Problem solving in physics (Tech. Rep.). Berkeley: University of California, Group in Science and Mathematics Education.

* This research was supported in part by the National Science Foundation under grant No. DPE-8470077. Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of National Science Foundation.

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