The Blocks Task:
Comparative Analyses of the Task With Other Proportion Tasks and Qualitative Reasoning Skills of Seventh-Grade Children in Solving the Task
A nonnumeric task called the blocks task was developed to investigate childrens understanding of the proportion concept. The first part of the article discusses this task through two cognitive task analyses: The first task analysis establishes an isomorphism between the blocks task and several real-world problem types; the second task compares and contrasts the blocks task with three other proportion-concept tasksrate and mixture tasks, the balance scale task, and the fullness task. The second part of the article reports a study of three aspects of adolescents' solution of the blocks task: (a) the basic problem representation that the children constructed in response to the task presentation and the corresponding solution strategy they used, (b) relations between the problem representations and the strategies, and (c) differences among high-, middle-, and low-ability children in terms of problem representation and solution strategy. The investigation resulted in the identification of three categories of problem representations and three categories of solution strategies. A hierarchy of sophistication was evident among the problem representations and among the solution strategies, and a strong relation was observed between the levels of sophistication of the problem representation and solution strategies; furthermore, ability level correlated highly with the level of problem representation and solution strategy.
This article is organized in two major parts. In Part 1, we present and discuss a proportion task that we developed called the blocks task; this task is new to the research on the proportion concept. It has numerous variations and has the attribute of being nonnumeric rather than numeric; thus solutions to variations of the task are based on qualitative rather than quantitative reasoning (à la Chi, Feltovich, & Glaser, 1981). As a qualitative task, the blocks task does not include numeric variables, such as "presence of integer-ratio" and "numerical complexity," which are known to interfere with children's solution of proportion problems (Tourniaire & Pulos, 1985), and thus it provides a clearer view through which to observe children's reasoning about proportionality constraints. In Part 2, we report a study of the qualitative reasoning adolescents use in solving variations of the blocks task.
OVERVIEW OF PART 1
Our presentation and discussion of the blocks task describes features of the blocks task and provides a possible solution of this task, which does not involve numerical computations. Additionally, we involve the blocks task in two comparative analyses.
The first analysis compares the blocks task with more real-world types of tasks by showing an isomorphic structure between these and the blocks task. The purpose of this analysis is to suggest that the type of thinking involved in solving the blocks task likely occurs in real-world or at least school-type problems.
Our second analysis compares and contrasts the blocks task with three traditional tasks used in research on the proportion concept, using six variables identified as influential in problem-solving processes. These tasks are the rate and mixture tasks (Karplus & Peterson, 1970: Noelting, 1980a, 1980b; Tourniaire, 1986; Tourniaire & Pulos, 1985), the balance scale task (Inhelder & Piaget, 1955/1958; Siegler, 1976), and the fullness task (Bruner & Kenney, 1966; Siegler & Vago, 1978). One of the five significant observations or outcomes about the blocks task from this comparative analysis is that it is based on relatively simple physical principles and, therefore, is easily understood by children. The relative simplicity of the physical principles results in less interference of the task on children's reasoning, and, thus, the task provides a clearer window through which to observe children's reasoning about proportionality constraints. A second conclusion of this analysis is that, although the blocks task is easy to administer because the problem data and problem question can be easily stated, it is cognitively more demanding and requires more processing than the other tasks compared." The third outcome of this comparative analysis suggests where the blocks task fits into the research picture on the proportion concept, and the fourth provides researchers with a deeper understanding of the cognitive demands involved in solving variations of both the three traditional tasks and the blocks task in terms of mathematical and physical principles and numeric, semantic, and propositional features. The fifth result of this comparative analysis is identification of several variables of these tasks that could be systematically varied to determine task characteristics that facilitate or debilitate childrens performance on these classes of proportion tasks.
OVERVIEW OF PART 2
Part 2 reports a study investigating the qualitative proportional reasoning of adolescents as indicated by analysis of protocols from one-on-one interviews during and following their solution of nine variations of the blocks task. The study involved Grade 7 students classified as having low, middle, and high mathematical ability. The focus of the data analysis was identification and categorization of the problem representations the children formed and solution strategies they used. Of particular concern was investigating whether a relation existed between the problem representations and the solution strategies, across the three ability' levels. We found four kinds of problem representations that we classified into three categories; these categories had an observable order of sophistication. Similarly, we identified six kinds of solution strategies that we classified into three unequal categories. We give a theoretical argument for an ordering of these in terms of their sophistication, within and among the categories. Based on these orderings of the problem representations and solution strategies, we found a positive relation between the level of representation and the solution strategy used: Children who formed higher levels of representations also used higher levels of solution strategies. Using this positive relation between problem representation and solution strategy, we formed six distinct representation-strategy pairs; we call each of these pairs a solution process. Labeling the three types of problem representations and the three categories of solution strategies as high, middle, and low, based on level of sophistication, led to two high-high representation-strategy pairings (or solution processes), three middle-middle, and one low-low. We found a positive relation between solution process and ability level of the students, with higher levels of solution process being used by higher ability students.
PART 1: THE BLOCKS TASK
We start this section with a description of the blocks task. To highlight the non- numeric attribute of the blocks task, "'e illustrate how it can be solved without arithmetic computation. Then we establish a structural isomorphism between the blocks task and numerous real-world problems, none of which has been used in research on the proportion concept. Finally, we show that the blocks task is different from the traditional tasks used in research on proportional reasoning. Specifically, we show that the blocks task is different in aspects of the knowledge base required t-or solution and problem structure compared with three major tasks used in research on proportional reasoning: rate and mixture tasks (e.g., the orange concentrate task used by Noelting. 1980a, 1980b), the balance scale task (Inhelder & Piaget. 1955/1958; Siegler, 1976), and the fullness task (Bruner & Kenney, 1966; Siegler & Vago, 1978).
Description of the Task
The task involved two pairs of blocks (A, B) and (C, D) like those shown in Figure 1: Blocks A and C were constructed from large (L) building blocks, Blocks B and D from small (S) building blocks. There were fewer building blocks in A than in C. These numbers remained constant across variations of the task. 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 building blocks in B and D within each of the pairs (B-1, D-1), (B0, D0), and (B1, D1) was one less, the same, or one more compared with the number of building blocks in A and C, respectively. Given information about the weight relation between A and B (<, =, or >) in the context of a visually observable number relation between them, the subjects were asked to determine the weight relation between C and D.
The three pairs (A, B-1), (A, B0), and (A, B1) reflect three different observable number relations. This, crossed with the three possible weight relations between A and variations of B (<, >, and =), results in nine possible weight and number relations. Each of these relations can be associated with a requirement to find the weight relation between C and one of the three instances of D (D-1, D0, and D1). This results in 3 x 3 x 3 or 27 possible problem situations. These 27 situations are not entirely distinct from each other, in the sense that they can be organized into three categories, each consisting of nine isomorphic situations (in terms of the problem structure). The nine problems selected for use in this study are representatives from these categories; they are illustrated in detail in Figure 1.
A possible solution path. All instances of the blocks task are solvable without arithmetic computation. The blocks were constructed so that the order relation between the numbers of building blocks could be easily determined by visual observation; neither the actual numbers of building blocks nor the numeric difference between them is necessary to obtain the required weight relation. To illustrate, we present such a solution path for Item 5 of Figure 1. Of course, other solution paths, quantitative and/or qualitative, are possible. This solution path consists of three steps or solutions for three subtasks: The first subtask is to determine the order relation between the weight of the large building block, L, and the small building block, S; the second subtask is to determine the order relation between the total weight of the composite block added to A and B to create C and D, respectively; and the third subtask is the final goal to determine the order relation between the weight of C and D.
In the first subtask, it is observable that:
It is given that:
From Steps 1 and 2 the weight relation between L and S can be inferred:
Thus the goal in the first subtask is achieved. In the second subtask, it is observable that:
Steps 3 and 4 imply:
Thus, the goal in the second subtask is achieved. In the third subtask, from Steps 2 and 5, the required weight relation can be determined, namely:
Thus, the goal in the third subtask is achieved.
Qualitative proportional reasoning is involved in two episodes in this solution path: One is the coordination of the number and weight relations between A and B to determine the weight relation, if possible, between L and S (e.g., deriving Step 3 from Steps 1 and 2); the other is in the coordination of the weight relation between L and S and the number relation between the parts added to A and B to determine the weight relation between these added parts (e.g., deriving Step 5 from Steps 3 and 4).
This solution path also demonstrates the nonnumeric nature of the task, resulting in qualitative as opposed to quantitative solution. Quantitative tasks do not include numeric variables, such as "presence of integer-ratio" and "numerical complexity," which are known to interfere with children's solution of proportion problems (Tourniaire & Pulos, 1985). In this regard, this feature of the blocks task is an advantage over quantitative tasks, because it provides a clearer window through which to observe children's reasoning about proportionality constraints.
The question of whether the blocks task has any relation to the real world may be raised because of concern over whether or not the reasoning displayed in dealing with the blocks task has any relation to thinking on real-world problems. We answer this by showing that the blocks task is structurally isomorphic to numerous real-world situations. Several examples follow.
The isomorphism between the blocks task variations and these problems is derived from a structural mapping among objects and relations included in the problem statement.
Structural mapping among objects. To see this mapping, first notice that each involves a linear function f. In the blocks tasks, weight is a linear function of the number of building-blocks1; in Problem 1, the amount of water is a linear function of the number of pumps. When f is a linear function and n is the number of measure units U (number of blocks, number of pumps), then f(nU) = nf(1U). For example, the weight function has this property: The weight of A-denoted by f(A)-is the weight of n building blocks of size L, or f(nL); that is, f(nL) = nf(L). In addition to the existence of a linear function in the listed problems and the blocks task variations, each also includes (either in the problem statement or in the problem question) the following objects:
1. Two measure units, U1 and U2. For example, in the blocks task, U1 and U2 represent the L and S building blocks, respectively; in Problem 1, they represent the kind of pumps, ALPHA and BETA, respectively.
2. Two pairs of quantities, (a, c) and (b, d), where a and c represent the amount of the measure unit U1 and b and d the amount of the measure unit U2. For example, in the blocks task, a and c represent the number of L building blocks (those that compose Blocks A and C), and b and d represent the number of the S building blocks (those that compose Blocks B and D); in Problem 1, a and c represent the number of ALPHA pumps (2 and 5, respectively), and b and d represent the number of BETA pumps (3 and 7, respectively).
3. Two pairs of values of the function f[f(aU1, f(cU1] and [f(bU2, f(dU2)]. For example, in the blocks task, f(aU1) and f(cU1) represent the weights of Blocks A and C, respectively, and f(bU2) and f(dU2) represent the weights of Blocks B and D, respectively; in Problem 1, f(aU1) and f(cU1) represent the amount of water that can be pumped by 2 or 5 ALPHA pumps, respectively, and f(bU2, and f(dU2) represent the amount of water that can be pumped by 3 or 7 BETA pumps, respectively.
Structural mapping among relations. This structural mapping is derived from the fact that each of the problems just shown and each variation of the blocks task include the four relational information components described next; the first three are stated or derived from the problem statement, and the fourth is asked for in the problem question:
1. The order relations between the numbers a and b. For example, the blocks task provides the order relation between the number of building blocks in A and B by making these relations visually observable. In Problem 1, the numbers a and b are given to be 2 and 3, respectively: thus the order relation between a and b can be derived.
2. The order relations between the numbers c and d. As in the previous component, the blocks task provides the order relation between the number of building blocks in the other pair of blocks, C and D, by making these relations visually observable. Similarly, in Problem l, the numbers c and d are given to be 5 and 7, respectively; thus the order relation between c and d can be derived.
3. The order relation between f(aU1) and f(bU2). For example, the blocks task provides the order relation between the weight of Block A and the weight of Block B; Problem 1 provides the order relation between the amount of water that can be pumped by 2 ALPHA pumps and 3 BETA pumps.
4. The order relation between f(cU1) and f(dU2). For example, the blocks task asks about the order relation between the weight of Block C and the weight of Block D; Problem 1 asks about the order relation between the amount of water that can be pumped by 5 ALPHA pumps and 7 BETA pumps.
The same problem analysis holds if the function f is the distance function in Problem 2 or the volume function in Problem 3.
It should be noted that, although this analysis establishes a structural isomorphism between variations of the blocks task on the one hand and real-world problems on the other, we still do not know if subjects would apply similar solution processes to solve these types of tasks or if, because the problem context varies across tasks, these solution processes would vary accordingly.
Comparison Among Proportion Tasks
The objective of this section is to compare the blocks task with three other major types of tasks mentioned earlier, which have been used in research on the proportion concept. The analysis reveals similarities and differences among these types of tasks. The focus of the comparison is on the following variables: (a) numeric versus nonnumeric; (b) propositional type; (c) types of quantities involved in the problem- extensive., intensive, or product of measure; (d) multiplicative invariance (with two subvariables- invariance of ratio and invariance of product) versus additive invariance; (e) mathematical principles underlying the solution of tasks; (f) semantic relations between the problem quantities; and (g) physical principles underlying the problem situation.
Numeric versus nonnumeric variables. A proportion task can be classified as numeric or nonnumeric according to whether or not arithmetic computations are required in getting its solution. The nonnumeric category includes all variations of the blocks task used in this study. Variations of the blocks task that belong to the numeric category can be created by, for example, changing Item 7 in Figure 1 as follows. In this item as described, it is given that Block A is lighter than Block B w(A) < w(B), and it is observed that Block A has the same number of building blocks as Block B, n(A) = n(B), and Block C has more building blocks than Block D, n(C) > n(D). Under these conditions, the required weight relation between C and D is indeterminate. But, if the given weight relation is numerically quantified-say, B weighs twice as much as A, 2w(A) = w(B)- then by counting the number of building blocks in Blocks C and D-n(C)= 30, n(D) = 29-one can arithmetically compute the required weight relation as follows:
It is given that 2w(A) = w(B) and observed that n(A) = n(B). This implies that:
From 1 and 2,
From 3 and 4, w(D) > w(C).
The numeric category includes some variations of the balance scale task; others fall into the nonnumeric category. The former includes those with more weight on one side but with the weight on the other side farther from the fulcrum. To solve these variations of the balance scale task, the numerical value of the moment for each side of the fulcrum must be computed and compared. Nonnumeric variations of the balance scale task include those with equal amounts of weight equidistant from the fulcrum and tasks with equal amounts of weight in opposite directions and unequal distance from the fulcrum.
The numeric category includes all rate-and-mixture missing-value proportion tasks, because to solve these problems the missing value must be computed from the three rate quantities given in the problem (e.g., find x in 6/5 = 9/x; for an analysis of solution models of these problems, see Harel & Behr. 1989) . The numeric category also includes comparison-type proportion tasks whose solutions cannot be derived solely from the qualitative value of the order relation on corresponding quantities between the rate pairs. For example, in the comparison task., "Which is greater, 3/5 or 7/8?", qualitative values that 3 < 7 and 5 < 8 are insufficient to determine the answer. In addition to knowing the directionality of the difference between 3 and 7 and between 5 and 8, quantification of these differences is necessary.
Included in the nonnumeric category are rate and mixture tasks, such as "Which is greater, 3/8 or 3/9?" or tasks with less obvious solutions such as "Which is greater, 3/8 or 2/9?" No computation is needed to solve these tasks; their solutions can be derived solely from the order relation between the corresponding quantities in the two rate pairs. For example, the solution, 3/8 > 3/9, can be derived from the observation that the same numerator (3) in one fraction is divided by 8, whereas in the other, it is divided by a number greater than 8 (9). The same reasoning applies with any other two fractions in which the numerators are equal and the denominator of one fraction is greater than the other. Problems such as "Which is greater, 7/8 or 10/11?" may also be placed in the nonnumeric category , because one can reason about their order by comparing with reference number .
The fullness task as presented in Siegler and Vago (1978) is also included in the nonnumeric category, because it can be solved without arithmetic computation. A different presentation of the task, however, could make this task numeric. For example, if the quantities in the ratios of the filled-space height (or volume) to the empty-space height (or volume) of the two cups were numerically specified, then knowledge about the order relations between corresponding quantities (i.e., height or volume) between the two ratio pairs would not be sufficient in some cases to determine the order relation between the two ratios, as in the example, "Which is more full, a cup with filled-space height to empty-space height of 3:5 or one with ratio 7:8?"
This discussion indicates that a proportion task can contain numerical quantities and still be considered nonnumeric. The numeric versus nonnumeric classification is made solely on the basis of whether or not the solution can be obtained without numerical computation. (Of course, the problem solution and, therefore, this classification depend on the solver's representation of the problem and the knowledge he or she brings to the problem.) Thus, the nonnumeric tasks can further be classified into two subcategories: those that include numerical quantities (quantified tasks) and those that do not (number-free tasks). as in the following examples:
Thus with respect to the numeric-nonnumeric variable, we find that all variations of the blocks task used in this study were nonnumeric. Some variations of balance scale tasks used by Inhelder and Piaget (1955/1958) and Siegler (1976) are numeric; others are nonnumeric. Traditional variations of missing-value rate and-mixture tasks are numeric; comparison rate-and-mixture tasks split into nonnumeric and numeric depending on whether or not a quantified difference between corresponding between-rate (or ratios) terms is needed and obtainable from the presentation to make the comparison. The fullness task (Siegler & Vago, 1978) is nonnumeric. In some of the nonnumeric tasks, the requested relation is indeterminate, because the quantified difference between corresponding between-rate terms is necessary but not obtainable from the data. The earlier observation that the classification of a task as numeric or nonnumeric is not uniquely determined according to whether the task presentation does or does not include numerical data -leads to consideration of the variable of propositional type for further comparison of these tasks.
Propositional type. In this section, we consider the difference between the four tasks with respect to the type of propositions they contain-assignment or relational (Mayer, 1985). An assignment proposition is a statement in which a measurement is assigned to an attribute of an object (Thompson, in press). For example, in the proposition "Mixture A contains 3 cups of water," the measurement 3 cups is assigned to the volume attribute of water. A relational proposition is a statement expressing a relation between two quantities (Mayer, 1985), as in the statement: Mixtures A and B contain the same amount of water. The difference between problems that include numeric quantities and those that do not, in terms of proposition type, is apparent. Consider, for example, the previous two problems. Problem 1 contains four assignment propositions corresponding to the four problem quantities-and no relational propositions, whereas Problem 2 contains two relational propositions and no assignment propositions.
In terms of propositional type, the blocks task differs from the other three types of tasks, even if all four were presented (or represented) only by relational propositions. The difference is in the number of relational propositions: A blocks task presentation (or representation) would have four relational propositions: the given weight relation between A and B, the observable number relation between A and B, the observable number relation between C and D, and the required weight relation between C and D. The balance scale task would have three relational propositions: the given (or observed) order relation between the weights on the two sides of the fulcrum, the given (or observed) order relation between the distances of the weights from the fulcrum and the required moment relation. Rate and mixture tasks would also have three relational propositions; these are the order relations between corresponding quantities in the two rate quantities and the required order relation between the rate or mixture concentration, as in Problem 2. The fullness task as used by Siegler and Vago ( 1978) cannot be analyzed with respect to type of propositions, because the problem information was presented visually, not verbally. However, to solve a nonnumeric fullness task. interring and encoding three relational propositions is sufficient. These propositions could be, for example, the relation between the heights of the containers, the relation between the heights of the liquids, and the required order relation between the fullness of the two containers.
Quantity type. Further distinctions among the four tasks under investigation can be made by considering the type of quantity involved in the problem information and in the question part of the task presentation. The problem information for rate and mixture tasks, the fullness task, and the balance scale task, involves extensive quantities, such as number of miles and amount of time (rate), number of cups of water and number of cups of orange concentrate (mixture), height of empty space and height of water space (fullness), and amount of weight and distance from the fulcrum (balance scale). The question part of rate and mixture tasks and the fullness tasks involves intensive quantity, such as speed and taste, and fullness, whereas the question part of the balance scale task involves moment which is a product-of-measure quantity. The problem information of the blocks task also involves extensive quantities-number of building blocks and weight-but, because the size and weight of the building blocks can vary directly or inversely with respect to each other, density, which is an intensive quantity, is implicitly involved. The question part in the blocks task and its isomorphs, in contrast to the other tasks, involves an extensive quantity (i.e., the weights of Blocks C and D).
Study on the development of intensive and extensive quantities (e.g., Schwartz, 1988; Strauss & Stavy, 1982) would corroborate the conjecture that this conceptual difference affects the type of knowledge required for task solution.
The type of quantities involved in the problem information or in the problem question corresponds to the type of solutions solvers employ to solve multiplicative problems. An analysis of these solutions provides a classification of multiplicative problems into two main categories in which the four tasks under consideration can be placed. A further analysis of these categories reveals classes of mathematical principles underlying the solutions of multiplicative problems. These analyses are discussed in the sections that follow.
Multiplicative invariance-invariance of ratio versus invariance of product - and additive invariance. The four types of tasks that we are comparing are instances of two general categories of proportion tasks: invariance of ratio and invariance of product. The classification of tasks into these categories is determined according to the solution type - based on ratio comparison or on product comparison-commonly used by subjects. Rate and mixture tasks, which ask for comparison of intensive quantities, belong to the invariance-of-ratio category. The correct solutions children give for these tasks involve comparison of two ratios. In the orange concentrate task (e.g., Noelting, 1980a), the ratios of the amount of orange concentrate to the amount of water for two mixtures are compared to determine which would taste more orangy or if they would taste the same. Also, the reciprocals of these ratios, or the ratios of one pan to the whole, are compared by some children. The balance scale task belongs to the invariance-of-product category. The correct solutions used by children in Siegler's (1976) study involved comparing the products of the distance-weight values for each side of the fulcrum to determine which side would go down.
Essentially, the fullness task belongs to the invariance-of -ratio category. This is because, to solve the fullness task (i.e., to determine which of two glasses is fuller or whether they are equally full), one has to compare the ratio of empty- space volume to filled-space volume between the two glasses. However, the process of determining the order relation between these two ratios, which Siegler and Vago (1978) taught children for a nonnumeric fullness task, also involves comparing products. In the process they taught, a decision was made about how full each glass was (less than half full, half full, more than half full, or all full), and then the fullness of the two was compared. To compare the fullness of a glass to one that is half full, one needs to determine the order relation between the volume of the empty space and the volume of filled space. Because each of these volumes is the product of a respective height (e.g., a and b) by a cross-sectional area of the glass (e.g., c), the order relation between them (i.e., between a x c and b x c) and, thus, the question of their relative fullness depend on reasoning about the variation or invariance of a product.
According to the solution path suggested earlier for the blocks task, this task (and its isomorphs) would belong to the invariance-of-product category. Consider Item 5 in Figure 1. According to the line of thinking suggested for the solution of this item (see earlier discussion of a possible solution path) , it can be seen that it consists of three subtasks. The first subtask is to determine the order relation between the weight of the large building block L and the small building block S; the second subtask is to determine the order relation between the total weights of the composite blocks added to A and B to create C and D, respectively: and the third subtask is the final goal to determine the order relation between the weights of C and D. Solution of the first two subtasks involves reasoning about invariance of product. To achieve the goal in the first subtask, the weights of A and B must be conceptualized and compared as two products: The number of building blocks (L or S) times the weight of each building block (L or S) in each of A and B, respectively, equals the weights of A and B, respectively. The weights of L and S appear as "unknown factors" in these conceptualized products, and the weight relation between them must be derived from the known number and weight relations. In turn, this weight relation is used to achieve the goal in the second subtask by representing the weights of the composite blocks added to A and B to create C and D, respectively, as products of the number of building blocks composing the composite block added and the weight of each building block (see the complete solution suggested earlier).
The third subtask of the blocks task points to an additional difference between this task and the three traditional tasks under consideration: Solutions of the traditional tasks solely involved reasoning about the relation between products or ratios (i.e., multiplicative reasoning), whereas the blocks task involved both multiplicative and additive reasoning. Additive reasoning is required in the third subtask of the blocks task, because its goal is achieved by coordinating additively the weight relation derived in the second subtask with the given weight relation between A and B. From the second subtask, it was derived that the composite block added to A to create C is heavier than the composite block added to B to create D, and it is given that A is heavier than B. Because the weight of C equals the sum of the weight of A and the weight of the composite block added to A to create C, and the weight of D equals the sum of the weight of B and the weight of the composite block added to B to create D, it is concluded that C is heavier than D. We call such tasks (e.g., the third subtask of the blocks tasks) additive invariance tasks.
This analysis shows that tasks labeled under the general category of proportion tasks are of two types: invariance of ratio and in variance of product. In the next section, we present a refinement of this categorization and show that very different reasoning is involved in solving the different types of tasks that resulted. These differences lie in the mathematical principles that must be invoked to solve these tasks correctly.
Mathematical principles. Important goals for mathematics education include identifying the knowledge that constitutes adequate reasoning for additive invariance and multiplicative invariance tasks and offering learning activities that can help children develop such knowledge. In this section we present classes of mathematical principles on which additive reasoning and multiplicative, or proportional, reasoning can be based. These principles are easily derived from the axioms of an ordered field, and, thus, mathematically, they are self-evident, at least for mathematically sophisticated people. For children, however, as research on the concept of proportion implies, these principles are not obvious. The following analysis identifies these principles and distinguishes their roles in solving different tasks, but it does not deal with the question of what experiences are needed to help children acquire these principles, so they can apply them in solutions of different types of tasks, additive and multiplicative.
Earlier, we classified proportion tasks into two broad categories- invariance of ratio and invariance of product - according to the ways subjects commonly solve: these tasks. Under this classification, rate and mixture tasks and the fullness task belong to the invariance-of-ratio category, whereas the blocks task and the balance scale task belong to the invariance-of-product category. Because the blocks task requires additive reasoning, in addition to proportional reasoning, we identified an additional category - the additive invariance category.
The point to be made in this section is that very different mathematical principles and, thus. different reasoning patterns, are involved in solving these tasks. To describe these principles, a refinement of each category is useful. In the following discussion, we identify subcategories of tasks from the invariance-of- product and invariance-of-ratio categories and describe the principles underlying their solutions; then we turn to the subcategories of additive in variance tasks and their solution principles.
There are two subcategories of the invariance-of-product category:
Likewise, there are two subcategories of the invariance-of-ratio category:
Each task from anyone of these four subcategories involves three number pairs - a and b, c and d, and either a pair of ratios a/c and b/d or a pair of products a x c and b x d. The order relation between quantities within two of the three pairs is given, and the problem is to determine, if possible, the order relation within the third pair. Solving such problems includes two steps: Find out whether the third order relation is determinable and, if it is, determine what that order relation is. Accordingly, the knowledge involved in solving such tasks relies on principles that we place in two categories: order determinability principles and order determination principles. The order determinability principles specify the conditions under which order relations between quantities within two pairs can lead to the action of declaring whether the order relation between quantities with- in the third pair is determinate or indeterminate (e.g., if a and b are equal, but c and d are unequal, then the order relation between the products a x c and b x d, or between the ratios a/c and b/d, is determinate). The order determination principles specify the conditions under which order relations between two quantities within two pairs can lead to the action of declaring the order relation between the two quantities within the third pair to be exactly one of: less than, greater than, or equals (e.g. , if a and b are equal but c is greater than d, then a/c < b/d).
We have identified two pairs of classes of multiplicative principles: One pair concerns the order determinability principles, the other the order determination principles (see Figure 2). Two classes, one from each pair, correspond to the invariance-of-product category; one is called the product-order determinability principles class, the other the product-order determination principles class. The other two classes correspond to the invariance-of-ratio category; they are the ratio- order determinability principles class and the ratio-order determination principles class. Each class of product-order (determinability or determination) principles is further divided into two subclasses - product composition (PC) and product decomposition (PD) - depending on whether the principles are used to solve tasks from the find-product-order subcategory or the find-factor-order subcategory, respectively. Similarly, each class of ratio-order (determinability or determination) principles is further divided into two subclasses - ratio composition (RC) and ratio decomposition (RD) - depending on whether the principles are used to solve tasks from the find-rate-order subcategory or find-rate-quantity-order sub- category. respectively. Figure 2 further presents the principles in each subclass. The order determinability principles are denoted by PCi, PDi, RCi, and RDi, where i = 1, 2, or 3. A determination principle that corresponds to each of the determinability principles ascertains that the required order relation is determinate (i.e., the first principle in each of the subclasses). These are denoted by [PC1], [PD1], [RC1], [RD1], respectively. For example, an instantiation of [PD1] is: If c > d and a x c < b x d, then a < b.
Within the order determinability principles class, each of the four subclasses consists of three principles. Let us describe these principles and examples to illustrate how they are used in solution processes of proportion tasks. We start with the product composition subclass; it consists of:
To illustrate how these principles are used to solve multiplicative problems, consider the following three tasks:
In the first task, the order relation between the distances from the fulcrum is the same as the order relation between the weights of the two objects; this, using PC1, leads to the decision that the required order relation between the moments (i.e. , the product of weight and distance) is determinate. In the second task, on the other hand, the order relation between the distances from the fulcrum conflicts with the order relation between the weights; this, using PC2, leads to the conclusion that the order relation between moments is indeterminate. In the third task, although the order relation between the distances is given, the order relation between the weights is missing; thus, using PC3. the order relation between moments is indeterminate.
The second subclass of order determinability principles is the product decomposition subclass; it consists of:
Examples of how these principles are used in problem solutions include the following:
In the first task, the order relation between the distances from the fulcrum conflicts with the order relation between the moments (i.e., the product of weight and distance), which, using PD1, leads to ascertaining that the required order relation between the weights of L and S is determinate. In the second task, on the other hand, the order relation between the distances from the fulcrum is the same as the order relation between the moments, which, using PD2, leads to the conclusion that the order relation between the two weights of L and S is indeterminate. In the third task, although the order relation between the distances is given, the order relation between the moments is missing; thus, using PD3, the order relation between the weights is indeterminate.
The class of ratio-order determinability principles also consists of two subclasses. The first is the ratio composition subclass; it consists of the following principles:
The second subclass of the ratio-order determinability class is the ratio decomposition subclass, which consists of the following principles:
Once a determinability principle is applied and the requested order relation is found to be determinable, a determination principle can be applied to ascertain whether that relation is the less-than, equals, or greater-than relation. A determination principle that corresponds to each of the determinability principles ascertains that the required order relation is determinate (i.e. , the first principle in each of the preceding subclasses). We denote them by [PC1], [PD1], [RC1], [RD1], respectively. For example, an instantiation of [PD1] is: If c ³ d and a x c < b x d, then a < b.
The two classes of determinability and determination principles are summarized in Tables 1, 2, and 3. Table 1 describes the knowledge about the qualitative relationships among problem quantities needed for solving tasks from the two subcategories of tasks, find-rate-order and find-product-order, which, as was shown earlier, correspond to the determinability principles, RCi and PCi, where i = 1, 2, or 3, respectively, and to the determination principles classes, [RC1] and [PC1], respectively. Success on tasks from the find-rate-order (or find-product- order; parentheses hereafter correspond to parentheses in Table 1) subcategories can be achieved by reasoning about how a qualitative change in a1 to get a2 and b1 to get b2 affects the size and, thus, the comparison of k1 and k2, where k1 = a1/ b1 and k2= a2/ b2 (or k1 = a1 X b1 and k2 = a2 x b2). The changes in a1 to a2, b1 to b2, and k1 to k2 can be denoted by a, b, and k, respectively, and the qualitative value (or directionality) of these changes by + , 0, and , according to whether the change is an increase, no change, or decrease, respectively. In Table 1, working in pairs of values from the vertical and horizontal axes to the corresponding value in the table body gives information about how qualitative changes in rate quantities (or factors) affect the qualitative value of the rate ( or product). In the body of the table, "?" indicates that the k cannot be uniquely determined according to the given information about a and b: There are instances for which k is + , instances for which it is 0, and instances for which it is . Therefore, to determine k, more information about a and b is needed. For example, in the case in which it is given that a and b are both increase ( + ; e.g. , the upper left corner of Table 1), k cannot be uniquely determined; but with additional information, such as the multiplicative increase of a1 to a2 is greater than the multiplicative increase of b1 to b2, k is determinate.2 The s in Table 1 indicate that the k is indeterminate, because information on either a or b is missing, not just insufficient as in the "?" cases. Thus the s correspond to principles RC3 (PC3), which state that k is indeterminate, because either a or b is absent.
In a similar manner, Tables 2 and 3 describe the knowledge about the qualitative relations among problem quantities needed for solving tasks from the two subcategories of tasks, find-rate-quantity-order and find-factor-order, which, as was shown earlier, correspond to the determinability principles, RDi and PDi where i = 1, 2, or 3, respectively, and to the determination principles classes, [RD1] and [PD1], respectively.
Recall that solving the blocks task requires, in addition to multiplicative reasoning, additive reasoning. This led us to include in our analysis the additive invariance subcategories and the mathematical principles needed to solve tasks in these subcategories. There are two such subcategories:
The order determinability principles classes for these additive invariance subcategories are the additive composition (AC) principles subclass and the additive decomposition (AD) principles subclass, each consisting of three principles:
As in the multiplicative case, once a determinability principle is applied and the requested order relation is found to be determinable, a determination principle can be applied to ascertain whether that relation is the less-than, equals, or greater-than relation. A determination principle corresponding to each determinability principle ascertains that the required order relation is determinate (i.e., the first principle in the preceding subclasses). We denote them by [AC1] and [AD1]. For example, an instantiation of [AD1] is: If c ³ d and a + c < b + d, then a < b. In a manner similar to that for the multiplicative case, the two classes of determinability and determination principles for additive invariance are summarized in Tables 4 and 5.
We conclude the additive principles presentation with Figure 3, which describes the additive principles structure in a way similar to how Figure 2 describes the multiplicative principles structure. We included in Figure 3 the subtraction component that corresponds to the ratio component in Figure 2, in the sense that, as the ratio structures are the inverse of the multiplicative structures, the subtractive structures are the inverse of the additive structures. We did not specify the principles in the subtractive structure, because none of the tasks in our comparative analyses involves subtractive reasoning. However, following the description of the other classes of principles, one can easily construct the subtractive principles classes.
This analysis classifies the set of tasks (which in the research on proportion concepts were labeled under one category called comparison proportion tasks; see, e.g., Tourniaire & Pulos. 1985) into four subcategories: find-product-order, find-factor-order, find-rate-order, and find-rate-quantity-order. The analysis further describes in detail the mathematical principles needed to solve tasks from these categories. The blocks task (which as shown in the previous section belongs to the invariance-of-product category) involves three subtasks: The first belongs to the find-factor-order subcategory, the second to the find-product-order subcategory, and the third to the additive invariance subcategory. Accordingly, the knowledge involved in the solution of the blocks task (and its isomorphs) must involve more mathematical principles than the traditional tasks; in contrast to the latter tasks, which involve only multiplicative principles, the blocks task involves additive principles as well. Later, in the second segment of this article, we show that children's failure to solve some variations of the blocks task can be accounted for by assuming a lack of familiarity with principles needed to solve these variations, or a lack of knowledge of how to apply these principles in the solution process.
Semantic relations among the problem quantities. In this section, we compare the proportion tasks under investigation based on the semantic relations existing among problem quantities. Separate discussions are given for tasks involving invariance of ratio and those involving invariance of product. We start with the blocks task. The semantic relation between the problem quantities in the blocks task is conceptually different from the relations in the other three tasks. As was indicated earlier, in the blocks task, the weight of each block can be thought of as a product of two quantities: the number of building blocks composing the block and the weight of each building block. Accordingly, the role played in the product by the first quantity (number of building blocks) can be conceived of as an (integral) multiplier and the role of the other (the weight of each building block) as the multiplicand. This multiplier-multiplicand relation - simple proportion relation in Vergnauds (1983, 1988) terms, or mapping-rule relation in Neshers (1988) terms involves two measure spaces: number of building blocks and weight. The use of this relationship is ordinary in everyday activities and in school word problems and is usually expressed as repeated addition. It is based on a set subset relation, the operation union of sets, and the concepts of cardinality and measure. All these are acquired informally through everyday activities.
The balance scale task, in contrast, involves three measure spaces: The quantities multiplied are derived from two independent measure spaces weight and distance and their product creates the measure space of moment. The semantic relation among these three quantities product of measure in Vergnauds (1983) terms is formal, in the sense that is acquired through instruction. Moment is a vector quantity (not a scalar quantity) defined as the cross-product of two vectors: weight (the net gravitational force acting on an object hung on one side of the fulcrum) and (directional) distance from a fixed point (the fulcrum) to the point on which the object is hung. For many balance-scale-task variations, this definition is not necessary because they can be solved based on an intuitive knowledge acquired through inactive experience. These include many nonnumeric variations, such as, "If two boys, Tom and John, sit on opposite ends of a seesaw at an equal distance from the center and Tom is heavier than John, which side would go down, Toms side or Johns side?" This definition, however, is the foundation for the physical principle, w1 x d1 = w2 x d2 (where w1 and w2 represent the weights of two objects, each hung on another side of the fulcrum and d1 and d2 their distances from the fulcrum, respectively), which children do not acquire spontaneously from everyday experience, and it must be used in solving numeric variations of the balance scale task.
The other two types of tasks, rate and mixture tasks and the fullness task, are of invariance-of-ratio type. The semantic relations between the problem quantities within a ratio can have either partitive division meaning, quotitive division meaning, or functional (relational) meaning. Consider, for example, this ratio: 12 cups of water to 4 cups of orange concentrate. One can change 12:4 to the unit-rate 3:1 and think of 12:4 as the number of cups of water per one cup of orange concentrate, or change 12:4 to 1:⅓ and think of the number of cups of orange concentrate per one cup of water; both indicate a partitive division interpretation of the relation between the ratio's quantities. The ratio 12:4 can also be thought of a rate, namely, that the amount of water (orange concentrate) is some number of times as much as the amount of orange concentrate (water): this relation in contrast to the former one, is a quotitive division interpretation. Finally, the ratio 12:4 can have a functional meaning; namely, one of the ratio quantities, 12 or 4, is a dependent variable of the other. For some children, this dependency does not necessarily exhibit a multiplicative relationship, such as x -> 3x (or x -> 1/3 x), but rather an additive relationship, such as x -> x + 8 (or x -> x - 8). These children, when comparing two ratios, say 12:4 and 13:5, map the additive relationship between 12 and 4 (e.g., 4 -> 4 + 8) onto the additive relationship between 13 and 5 (e.g., 5 -> 5 + 8) and would claim that they are the same.
We have identified different types of relations among the problem's quantities in the four types of tasks analyzed. In the blocks task, these relations are of multiplier-multiplicand type; in the balance scale task they are of cross-product type; and in the mixture and rate tasks and in the fullness task, they have either partitive division meaning, quotitive division meaning, or functional meaning. Research in the domain of multiplication problems suggests that these relational differences between problem quantities affect task performance. Vergnaud (1983, 1988), for example, indicates that simple proportion problems (those with multiplier-multiplicand relations) are easier than product-of-measure problems (those with cross-product relations); others (e.g., Fischbein, Deri, Nello, & Marino, 1985) indicate that partitive division problems and quotitive division problems differ in difficulty. Concerning the functional relationships, research on proportion reasoning gives evidence for the use of both multiplicative and additive strategies by children (see, e.g., Noelting, 1980a, 1980b). However, it is not clear from this research what accounts for these strategies and whether the structural mapping between the two ratios as described here corresponds to children's conceptual bases for these strategies.
Physical principles. A fundamental difference among the four tasks in question lies in the principles underlying the interactions between or among the problem quantities. We believe that these principles are the basis for taking account of the proportionality constraints in solving these tasks. The principle involved in the blocks task is about homogeneous density and states that the weight of each block is equally distributed among the separate identical building blocks that compose it. This principle is spontaneous, in the sense that it is acquired in everyday activities such as lifting several objects of the same size and weight. In the analysis of problem isomorphs described earlier, this principle was expressed in terms of linearity of the weight function: w(nU) = nw(1U). Similar principles, though less spontaneous, are involved in the mixture and rate tasks. In the mixture task, the principle involved is about uniform diffusion between liquids: in the rate task, the principle is about uniform rate, such as speed or work. The fullness task involves the principle about liquid and states that, at all points at the same level within a liquid at rest, the pressure is the same; this principle guarantees the uniform level of the liquid and the absolute separation of the water space from the empty space. Knowledge about the balance scale involves the principle of conservation of angular moment, which states the conditions of equilibrium that must be satisfied if a balance scale is to remain balanced or fall toward one of the two sides of the fulcrum. These conditions are nonintuitive and less spontaneous when dealing with the summation of the products of weight and distance on each side of the fulcrum.
As can be seen from this analysis, compared with the other three tasks, the blocks task is based on relatively simple physical principles acquired through everyday experience. It should be noted, however, that we are not claiming that, for children to solve one of these tasks successfully, they must explicitly know the physical principles underlying the task. Our argument - which must be tested experimentally - is that at least an intuitive understanding of the physical principles underlying the situation of the task is necessary for solving the task correctly.
Summary of the task comparison analysis. We have compared and contrasted the blocks task with three major tasks used in the research on the concept of proportion - rate and mixture tasks, the balance scale task, and the fullness task - with respect to several task variables and subvariables. Figure 4 organizes these variables with their subvariables around four structures: the numeric structure, the semantic structure, the propositional structure, and principles structures. This comparative analysis identifies the special attributes of the blocks task with respect to the other tasks, revealing similarities and differences among the four tasks, as summarized in Table 6.
Two attributes make the blocks task relatively free of interference from physical or mathematical characteristics that do not pertain to the concept of proportion. In the blocks task, the relations between the problem quantities are of the simple proportion type (à la Vergnaud, 1983), and the principle underlying its problem situation is the uniform density principle, which is acquired through everyday activity. At the same time, the blocks task is cognitively more demanding and requires more processing. As can be seen from the fact that the blocks task consists of more relational propositions than the other three tasks, it involves more mathematical principles than the other tasks, and to solve this task, proportion reasoning and additive reasoning must be applied.
PART 2: THE STUDY
This study was conducted as one part in an ongoing teaching experiment concerned with investigating the development of the proportion concept in children. The instruction dealt with concepts associated with rational numbers, ratios, rates, and proportion problems. The nine variations of the blocks task that provided the data for this study were included in a more extended one-on-one interview assessment in the context of this teaching experiment. During the course of instruction and in the prior assessment interviews, the children had many opportunities to work on, solve, and discuss proportion problems. They had no prior exposure to the blocks tasks.
Subjects. The teaching experiment was replicated at two sites: DeKalb, Illinois, and Minneapolis, Minnesota. A total of 18 Grade 7 children, 9 at each site, participated in the experiment. They were chosen based on teacher recommendation and pretest scores so that each site involved children believed to be of comparable ability levels. At both sites 3 children were selected at each of high, middle, and low mathematical ability levels (children at either extreme in mathematical ability were not chosen).
Administration of the task. The two invariant Blocks A and C were permanently attached to a small board on which there was space to add different instances of Blocks B and D, respectively. In this way the student's attention could be easily drawn to the respective pairs of blocks, (A. B) and (C. D). The interviewer and the child were seated with the child to the interviewer's right or left in such a way that both were able to look directly at the four blocks involved in a given task variation but were at a distance so that the child could not easily touch or lift them. The weight relation between Blocks A and B was shown on a small sign (e.g., "Block A is heavier than Block B"). This sign, placed directly behind the pair at the same time that the interviewer read the information about this relationship, stayed in view throughout the time the child worked on that particular variation. The nine variations of the blocks task were given in the order in which they are shown in Figure 1, in a one-on-one interview that took about 60 min and involved about 40 items concerning the concept of proportion. Children were given adequate time to complete each variation of the blocks task.
At the beginning of the blocks-task sequence, the interviewer read the following information to the child:
I will give you information about the weights of two blocks [pointing to Blocks A and B]. The information will be about which is heavier or lighter or if they are the same weight. You are to use this information about these blocks [pointing to A and B] to compare the weights of these two blocks [pointing to C and D]. In some I might say A is heavier than B, in another that A is lighter than B, and in another that A and B have the same weight. So you see, sometimes I will say that the smaller block is heavier, sometimes lighter, and sometimes the same weight as the larger block.
Then after placing the appropriate instance of Blocks B and C on their respective boards to define the given task, the interviewer would say, "If Block A weighs (more, less, or the same) than/as Block B, what about the weights of Block C and Block D?" In every case, the child appeared to understand the directions and the task requirements.
Results and Discussion
The data obtained were in the form of verbal protocols of one-on-one interviews with the children as they attempted to solve the blocks tasks. Data analysis was conducted by two investigators (Harel and Behr) with help from graduate-student research assistants. On a first reading, at least half the protocols across subjects within each task were read by the two investigators. This first reading suggested that it was possible to identify the problem representation that children had formed during comprehension of the problem, as well as the structure of the solution strategy used. This seemed possible, because many of the children would refer to the components of the blocks - the tops of Blocks A and B, for example, and state in their own words some observation about a relationship between the number of building blocks in these problem components. In some protocols, clearly this observation was made by the child before an attempt was made to make deductions and inferences about the requested weight relationship between Blocks C and D; in other instances, there were intermittent observations about problem components and deductions and inferences. Overall, it was possible, in a majority of the protocols, to distinguish clearly between relational statements about the given components in the task and deductive and inferential statements. We attributed the relational statements about problem givens as a reference to the child's problem representations and the deductive and inferential statements as references to the solution strategy (Chaiklin, 1989; Larkin, 1989). In this way we were able to make conjectures about the form of the children's problem representations and about the structure of their solution strategy. The statements of these conjectures were refined and confirmed during the second reading of the protocols. In this reading the two investigators read each protocol independently and coded it for the type of problem representation and solution strategy represented. The ratio of the number of like codes (representation and strategy) to the total number of protocols coded was .95. Disagreements were resolved by discussion between the two coders.
We identified three kinds of problem representations - structure, complement, and isolated - and three categories of strategies - matching (with two variations, matching and imposed matching), balance (with three variations, complete, incomplete and deficient), and counting. We observed a high degree of systematic correspondence between the task representation the children formed and the solution strategy they used; this led us to consider the confluence or synthesis of a representation-strategy pair and to call it a solution process. The analysis presented in this section suggests that there are six such representation-strategy pairs for which there was a high frequency of correspondence between representation formed and strategy used to call a solution process.
In the rest of this section, we (a) present data to establish the existence of these solution processes and investigate their conceptual basis; (b) present data on and discuss the issue of systematicity in the use of a particular solution strategy for a particular problem representation, and the correspondence between the solution processes and the three levels of mathematics ability (low, middle, and high) from which subjects for this study were chosen.
When a child formed a structure representation, the strategy usually chosen was the matching strategy (or the imposed matching strategy). We call this confluence of problem representation and solution strategy the structure-matching (SM) solution process (or the structure-imposed matching, SIM, solution process). In general terms, the structure representation includes the block structures, tops and decks, within each block (see Figure 5), and the comparison of these structures across blocks, with respect to the number of the building blocks and the weight of the blocks. The matching strategy can be described in general terms as an attempt to map the observable and given relations within one pair of blocks onto those within the other pair of blocks to determine the required weight relation; the imposed matching strategy is a derivative of the matching strategy, in the sense that some children use hypothetical reasoning to change the problem situation so that the matching strategy would apply. Detailed descriptions of the structure representation, the matching strategy, and the imposed matching strategy follow.
Structure representation. If the structure representation was used, each block was envisioned as consisting of two structures: the deck and the top (see Figure 5). In all items presented, the deck was always a rectangular solid that contained the same number of building blocks (4 x 3 x 2; see Figure 1), whereas the order relations among the number of building blocks in the tops varied. The observable order relations on the number of building blocks L (those that compose Blocks A and C) and S (those that compose Blocks B and D) in the tops of the decks of the pairs (A, B) and (C, D) were identified. Usually, this representation also included the observation that each of the pairs (A, C) and (B, D) was constructed with the same size building blocks, L and S, respectively. Figure 6 describes elements in this representation, including the given and the required relations between the weights of the blocks.
The following two protocol excerpts indicate what children who formed the structure representation typically said. Pointing actions are described in parentheses; our interpretations of their remarks in more formal terms are offered in square brackets.
There's the same amount of squares on the bottom [The deck structures contain the same number of building blocks], and there's the same amount of squares on top [The top structures contain the same number of building blocks] (pointing to C and D) ...in both of them [A and B]. And these ones (pointing to the building blocks of A and the building blocks of B) are the same kind of these blocks (pointing to the building blocks of C and the building blocks of D) [Blocks A and C are made of the same building blocks. Ls; Blocks B and D are made of the same building blocks. Ss].
Both (pointing to the decks of C and D) have equal [C and D have the same deck structure]; okay, I look at these two (pointing to A and B) have the two decks like that [A and B have the same deck structure, and that is the same as the deck structures of C and D] and the three on top [A and B have the same Top structure]. And the same with this (points 10 C and D) [C and D have the same Top structure]. Those two are equal (pointing to A and B) ...umm this (pointing to C and D) has the two decks like that one, and they have the equal top piece.
Matching strategy. If this strategy was used, the child would begin by looking at the relation within pairs of blocks (A, B) and (C, D). The child would first notice that the decks of these pairs are equal in number of building blocks and then would determine the number relation between Blocks A and B (or between their tops) and the number relation between Blocks C and D (or between their tops). The observable number relationship between Blocks A and B, or between their tops, is referred to as N(A, B), and the observable number relationship between Blocks C and D (or between their tops) as N(C, D). In the next step, the child would acknowledge the given weight relations between Blocks A and B and would attend to the question of the required weight relation between Blocks C and D. The given weight relation between Blocks A and B is referred to as W(A, B), and the required weight relation between Blocks C and D as W(C, D) .The child would then observe one of two relations holding between the relations already described: One was that the number relation ( < , = , or > ) between A and B and between C and D are the same relation, that is N(A, B) = N(C, D); the other was that the number and weight relations (<, +, or >) between A and B are the same relation, that is N(A, B) = W(A, B). Depending on which relationship was determined by a child, one of two rules was used.
We present two examples to illustrate application of these rules. For the first rule, suppose that the top of Block A has fewer building blocks than the top of Block B; the top of Block C has fewer building blocks than the top of Block D; and the weight of Block A is greater than the weight of Block B (e.g., Item 5 in Figure 1); that is, N(A, B) and N(C, D) would both be the less-than relation and W(A, B) the greater-than relation. In applying the first rule, the child would say something to the effect that, because N(A, B) = N(C, D), then W(C, D) = W(A, B), which means that the weight of C is greater than the weight of D. As an illustration of the second rule, suppose that Blocks A and B have the same number of building blocks; Block C has more building blocks than Block D; and Blocks A and B weigh the same (e.g., Item 2 in Figure 1). In this case the child would say something to the effect that, because N(A, B) and W(A. B) are the same relation (i.e., both are the equality relation), N(C, D) and W(C, D) must also be the same relation; that is, because N(C, D) is the greater-than relation, W(C, D) will also be the greater-than relation; thus C is heavier than D.
The existence of the matching strategy, as exemplified by the use of these two rules, is best illustrated by presenting some children's protocols. An application of the first rule can be found in the following responses. In Item 9 (Figure 1), it is given that A is lighter than B, and it can be observed that A and C have more building blocks than B and D, respectively. Sarah uses this rule saying:
Well, here you have three (on top of A), and here you have two (on top of B) [N(A, B) is the > relation]; this one (A) is lighter than this one (B) [W(A, B) is the < relation]; And here you have six (on top of C), and here you have five (D) [N(C, D) is the > relation), so this one (C) is lighter [W(C, D) is the < relation).
Sean uses the same rule for the same item, saying "Over here, A has more than B [N(A, B) is the > relation], and it's (A) lighter [W(A, B) is the < relation], so over here C has more than D [N(C, D) is the > relation], so it's lighter [W(C, D) is the < relation].
An application of the second rule can be found in the following response. In Item 6, it is given that A weighs more than B, and it can he observed that A has more building blocks than B, whereas C and D have the same number of building blocks. Ann uses the second rule. saying:
A is heavier than B [W(A, B) is the > relation], and it's got an extra block on it [N(A, B) is the > relation], and over here (C and D) ...this helped me because they have the same number of them [building blocks; N(C, D) is the = relation], so they (C and D) weigh the same [W(C, D) is the = relation too].
Further support for the existence of the matching strategy is shown in the following two protocol excerpts. The first gives clear evidence of the fact that Sarah realized that, if the condition of the second rule is false, then no conclusion is possible (i.e., if N(A, B) W(A, B), then no conclusion can be made about the required weight relation between C and D). Jean went a bit too far and apparently claimed that N(A, B) N(C, D) implies that W(C, D) W(A, B). In Item 3, it is given that A weighs the same as B, and it can be observed that A has more building blocks than B, whereas C and D have the same number of building blocks. Sarah responded to this item saying:
Well, here you have two (on top of A), and here you have three (on top of B) [N(A, B) is the < relation], and you said they weigh the same [W(A, B) is the = relation]. Here you have six (on top of C), and here you have six (on top of D) [N(C, D) is the = relation], so you can't know which one is heavier.
In the same item, Sean observed that N(A, B) ≠ N(C, D) and that W(A, B) is the equals relation, and then concluded that W(C, D) cannot be the equals relation. He says:
Well, they (C and D) look the same [N(C, D) is the = relation], but they're not as equal as this (A and B) ...this one (A) has three on the top, this one (B) only two [N(A, B) is not the = relation], so if they're equal [A and B in weight; W(A, B) is the = relation], then these two (C and D) can't be equal [in weight; W(C, D) is not the = relation].
Imposed matching strategy. Items 3, 4. and 7 (see Figure 1) cannot be solved by the matching strategy, because neither of the sufficient conditions N(A, B) = N(C, D) or N(A, B) = W(A, B) in the previous two 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 was 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 mentally substitute another block for one of the given blocks, so that the condition N(A, B) = N(C, D) would hold, and then apply the first rule to conclude W(C, D) = W(A, B). Based on this conclusion and using a transitive reasoning, the weight relation between the given Blocks C and D could be determined. However, children did not exp1icitly use the transitive reasoning but did give evidence of using it in the solution process. As an example of the imposed matching strategy consider the response of Jon (who consistently used the matching strategy to solve other items) to Item 7, in which it is given that A is lighter than B, and it is observable that A and B have the same number of building blocks, whereas D has one building block less than C:
Okay, umh, this one D is missing one to be equal to C, to have the same amount of parts. But C would be lighter than that (pointing to D) [W(C, D) would be the < relation, i.e., W(A, B) = W(C, D)], if it had this (pointing to the "missing" part in D), if it had that part (continues to point at missing part in D)[if N(A, B) = N(C, D)], so I think C would be lighter.
Jon first used the rule N(A, B) = N(C, D) implies W(C, D) = W(A, B) to derive that Block C is heavier than the hypothetical Block D. Then, it seems that he used transitive reasoning to argue that, because the hypothetical Block D must be heavier than the presented Block D (because it has one more building block), Block C must be heavier than Block D.
In the complement representation, the problem situation consists of two states; one state (describing Blocks C and D) is viewed as resulting from the other state (describing Blocks A and B) that undergoes a change in number of building blocks (see Figure 7). When this representation was formed, the strategy usually used was the balance strategy. The reasoning in this strategy is analogous to the reasoning involved in determining the change in the tilt of a balance scale when the sets of objects on each pan undergo a change in number. To correctly determine this change, one must take into account the initial tilt of the pan balance, the number of blocks added to each pan, and the weight relation between individual objects added to respective pans. Children who used this strategy usually did not take into account all these constraints. Depending on whether a child took into account all three, two, or only one of these constraints, his or her response was classified into one of three instantiations of the balance strategy complete balance, incomplete balance, and deficient respectively. Details of the complement representation and its confluence with each of these strategy instantiations are discussed later.
Complement representation. If the complement representation was used, the children attended to the fact that the number of building blocks in C was greater than in A, and that the number of units in D was greater than in B. Due to these noticed qualities of the blocks, their representation focused on Blocks C and D, where C was viewed as resulting from adding units to A, and D as resulting from adding units to B. Figre 8 describes this representation as a network of two states. 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 unit blocks to A and B to get C and D, respectively.
Evidence for this representation can be identified in the following protocol. Ann described the blocks in Item 4 by looking at a qualitative relation among them, saying: "This D has more [building blocks] than B, and C has more [building] blocks than A." Skip, on the other hand, described the blocks in the same item by looking at a quantitative relation among them, saying: "This one D got two more than that one B, and this one C has got three more than that one A."
Complete balance strategy. In the complete balance strategy, three relations were considered: (a) the relation between the weights of A and B (which can be visualized as having Block A on one pan of a balance and B on the other pan), (b) the number of building blocks added to A and B to create Blocks C and D (at this point, Blocks C and D are on the pan balance), and (c) the weight relation between the building blocks L and S. Only one child, Ann, exhibited application of the complete balance strategy, saying, in response to Item 7,
They're the same [weight] (C and D) ...because here (A and B) have the same number of [building] blocks, except A is lighter than B ...and here there's six [building] blocks (top of C) and here only five (top of D) (observing that more is added to A to make C than is added to B to make D) ...so I think they (C and D) weigh about the same. These (building blocks in D) are lighter than these (building blocks in C), and these (building blocks in B) are lighter than these (building blocks in A), but still C and D weigh the same.
We refer to the solution process consisting of the complement representation and the complete balance strategy as the complement-complete balance (CCB) solution process.
Incomplete balance strategy. The incomplete balance strategy is similar to the complete balance strategy. First, the children considered the relation between the weights of A and B and then determined the relation between the number of units added to A and B to solve the problem. In contrast to the complete balance strategy, this strategy ignores the relation between the weights of the building blocks. An application of this strategy is seen in Saul's response to Item 4: "This one's (A) heavier than that one (B) ...then, this one (C) ...would be heavier than D. ...Well, because, this one (C) still has three more than that (A), and ...this one (D) has two more than that (B)." A second example of this strategy is seen in Dave's answer to Item 2: "They're the same weight (C and D). ...You got the same amount of this (A and B). ...Ya only added three more on here (C) ...and only three more on there (D)." We refer to the solution process consisting of the complement representation and the incomplete balance strategy as the complement-incomplete balance (CIB) solution process.
Deficient balance strategy. In the deficient balance strategy, only the relation between the number of units added to A and B was considered to solve the problem; the other two relations, the number and weight relations between A and B, were ignored. The following protocols illustrate this strategy. In answer to Item 4, Dave says:
This one will be more [in weight] (points to C) ... (C) you added three more, so it will weigh more two added over here (D). Here (C) you added three more, so it will weigh more.
To Item 5, Daves response was:
C is heavier than D, and here (C) you added twice the amount (the top of C is twice the amount compared with the top of A), and here (D) you didnt add twice the amount (the top of D is less than twice the top of C).
We refer to the solution process consisting of the complement representation and the deficient balance strategy as the complement deficient balance (CDB) solution process.
The most simplistic solution process used by children to solve the blocks task consists of the isolated representation and the counting strategy, which we refer to as the isolated-counting (IC) solution process. In these representations and strategies, children ignored some problem components and did not consider relations among others.
Isolated representation. If the isolated representation was used, the children considered the number relations between Blocks C and D or between their tops and the number relations between Blocks A and B or between their tops. They did not, however, relate Blocks A and B to Blocks C and D. Moreover, in some cases Blocks A and B were completely ignored. Elements of this representation are shown in Figure 9.
The evidence for this simplistic representation can be found in the following protocols. When describing the blocks in Item 3, Lisa says: "Well, A and B, A has one [building] block less. There is six here (C) and six there (D)." In response to the same item, Chris said: "There is the same amount of squares (L, S) in each cube (C, D)," ignoring Blocks A and B.
Counting strategy. In the counting strategy, the answer to the task was determined by comparing the number of building blocks in C and D, disregarding the size of the blocks and the given weight relation between A and B. A very short response to Item 4 given by Lisa illustrates this simplistic strategy: "D is lighter than C because it (D) has one less thing; it has five on top, and C has six on top."
Conceptual Bases for the Solution Processes
The key question about the solution processes that we have identified concerns their conceptual bases. In this section, we try to answer the question of what constitutes children's thinking in producing these solution processes and what accounts for their failure to solve the blocks task, in terms of the mathematical principles we have identified in the first part of this article.
Structure-matching solution process. Each of the two rules associated with this solution process can be viewed as an analogical mapping between two systems. For example, the first rule N(A, B) = N(C, D) implies W(A, B) = W(C, D) is a mapping between the system consisting of Blocks A and B and the system consisting of Blocks C and D, under which the direction of the number relation that is, N(A, B) and N(C, D) and the direction of weight relation that is, W(C, D) and W(A, B) are preserved. If this interpretation is true, the matching strategy ref1ects a simplistic solution process in which only the surface features of the problem are considered. This is rather surprising because, as we show later, the matching strategy was consistently used by the highest ability students, who applied proportional reasoning in solving other traditional ratio tasks, such as the mixture tasks. We offer an alternative viewpoint about the conceptual base of this strategy: Our hypothesis is that the rules associated with the matching strategy reflect reasoning about ratios, in which children violated or failed to apply some of the mathematical principles determinability and determination identified earlier.
Because our analysis of the conceptual base for each rule associated with the matching strategy seems to be the same, we continue our illustration based on use of the first rule: N(A, B) = N(C, D) implies W(C, D) = W(A, B). Our hypothesis is that, between the supposition part and the conclusion part of this rule, children made additional derivations about ratios; but, due to their limited ability to express their reasoning about the problem information and the problem question in terms of ratios among the number and weight of building blocks, these derivations were not observed in the protocols. In the hypothesized solution process, the supposition N(A, B) = N(C, D) reflects an attempt to determine the order relation between the number ratios, n(A)/n(C) and n(B)/n(D), where n(A), n(B), n(C) and n(D) represent the number of building blocks in Blocks A, B, C, and D, respectively; and the conclusion, W(A, B) = W(C, D), reflects an attempt to derive the required weight relation from the order relations between the weight ratios, w(A)/w(C) and w(B)/w(D), where w(A), w(B), w(C) and w(D) represent the weight of Blocks A, B, C, and D, respectively. More specifically, we hypothesize that children used the following rule, which is a three-derivation rule rather than a one-derivation rule as suggested earlier:
There is nothing in the protocols that we could interpret as supporting the hypothesis that children who applied the matching strategy thought about ratios or made the two intermediate derivations suggested here. However, assuming that children did use the number ratios and the weight ratios indicated in this rule, we have data from other sets of ratio tasks that indirectly suggest that the three derivations are likely to be made by these children.
Children's observation N(A, B) = N(C, D) is about the order relations between two pairs of numbers. It means that the order relation between n(A) and n(B) is the same as the order relation between n(C) and n(D); or, putting it in number-transformation terms, n(B) and n(D) are results of the same type of transformation "value increases," "value decreases," or "value stays the same" - applied to n(A) and n(C), respectively. We hypothesize that children made this observation in the context of the number ratios, n(A)/ n(C) and n(B)/n(D). That is to say, they observed that the numerator and denominator of the ratio n(B)/n(D) are results of the same number-transformation - value increases, value decreases, or value stays the same - applied to the numerator and denominator, respectively, of the ratio n(A)/n(C). Their goal was to determine the order relation between these number ratios based on the observation N(A, B) = N(C, D).
Preliminary results of data from other sets of tasks can be interpreted with the hypothesis that children overgeneralize the RC1 and [RC1] principles into new principles that violate the RC2 and [RC2] principles. We believe that the order relation between the number ratios, n(A)/n(C) and n(B)/n(D), was determined by children by applying these overgeneralized principles.
Based on the order determinability principle RC1, the order relation between two ratios is determinate if the number transformation applied to the numerator and denominator of one ratio to get the numerator and denominator, respectively, of the second ratio is of the third type - value stays the same - and, if so, based on the order determination principle [RC1], the order relation between these two ratios is the equals relation. If, on the other hand, the transformation applied is of the first two types - value increases or value decreases - based on the order determinability principle RC2, the order relation between these ratios is indeterminate. Children's overgeneralizations of the RC1 and [RC1] principles are that, under any of the three transformation types, the order relation between the two ratios is determinate and the application of anyone of these transformation types results in equal ratios. This result was accumulated from data obtained from other sets of qualitative proportion tasks (see Larson, Behr, Harel, Post, & Lesh, 1989). Children's conceptual bases for these conclusions might be that, because the change between the numerators is the same as the change between the denominators, no change would occur between the two ratios. Thus, children, when observing that N(A, B) = N(C, D), applied these overgeneralized principles by reasoning that because the change in n(A) to get n(B) is the same as the change in n(C) to get n(D), the two ratios, n(A)/n(C) and n(B)/n(D) are equal.
The second derivation, n(A)/n(C) = n(B)/n(D) implies w(A)/w(C) = w(B)/w(D), is believed to be based on the uniform-density property of the blocks, which means that the weight of a block equals the number of building blocks composing it times the weight of one building block: that is, w(A) = n(A) x w(L), w(B) = n(B) x w(S), w(C) = n(C) x w(L), w(D) = n(D) x w(S) and w(A)/w(C) = w(B)/w(D). Thus, the order relation between w(A)/w(C) and w(B)/w(D) is the same as the order relation between [n(A) x w(L)]/[n(C) x w(L)] and n(B) x w(S)/[n(D) x w(S)] which, in turn, is the same as the order relation between n(A)/n(C) and n(B)/n(D); because the conclusion in the first derivation is that the latter ratios are equal, the ratios w(A)/w(C) and w(B)/w(D) would also be equal.
The third and last derivation is: w(A)/w(C) = w(B)/w(D) implies W(A, B) = W(C, D). This can be interpreted as saying that, if two ratios are equal, then the type of change - value increases, value decreases, or value stays the same - between their numerators is the same as the type of change between their denominators. The reasoning involved in this derivation is similar to that involved in the first derivation - N(A, B) = N(C, D) implies n(A)/n(C) = n(B)/n(D) - except that in the first derivation the principles involved are ratio composition type; here they are ratio decomposition types; RD1, [RD1], RD2, and [RD2]. Although our preliminary data mentioned before do not include tasks associated with these principles, it is believed that children overgeneralize these principles similarly to what they do to the order composition principles. Thus the information, w(A)/w(C) = w(B)/w(D), was obtained to derive the weight relationship W(C, D) - the order relation between w(C) and w(D) - which is thought of as a change between the denominators of these two ratios; children might think that, if the ratios are equal, the type of the latter change must be the same as the type of change between the numerators of these ratios; that is, W(A, B) = W(C, D).
Complement-balance solution processes. The interesting aspect of the data reported here is that the three variations of the balance strategy (complete balance, incomplete balance, and deficient balance) were called up by the same problem representation, the complement representation, to form the three solution processes: complement-complete balance (CCB), complement-incomplete balance (CIB), and complement-deficient balance (CDB). The main conceptual difference between these solution processes is in the difference among the representation components taken into account in the strategy used: The CCB solution process takes three components into account: (a) the weight relation between Blocks A and B; (b) the number relation between building blocks added to A and B to create C and D, respectively; and, acknowledging that these two components are not sufficient to determine the weight relation required in the problem, it also involves a derivation of (c) the weight relation between the building blocks L and S- W(L, S). The CIB solution process ignores the third component, whereas the COB solution process considers only the second component. The third component, which involves proportional reasoning, is missing from the latter two solution processes. The following analyses attempt to understand the conceptual base for each of these three solution processes and the mathematical principles children likely used in these solutions.
The CCB solution process takes components (a) to (c) into account and also combines additive principles with multiplicative principles to solve the blocks task. The (b) and (c) components are needed to answer the questions:
When the second question is answered, the first component (a) is needed to answer two additional questions concerning the required weight relation between Blocks C and D:
Using the uniform-density property of the blocks, the weight of the two composite blocks added to A and B to create C and D, respectively, can be represented as w(A + ) = aw(L) and w(B + ) = bw(S), where a is the number of building blocks L composing the composite block A+, and b is the number of building blocks S composing composite block B+. To answer Question 1, the order determinability principles PC1 and PC2 must be used, and to answer Question 2, the order determination principle [PC1] must be used: If the order relation between a and b is the same as the order relation between w(L) and w(S) or if one is the equals relation, then, by PC1 , the order relation between w(A+) and w(B+) is determinate; on the other hand, if the order relation between a and b conflicts with the order relation between w(L) and w(S), then, by PC2, the order relation between w(A+) and w(B+) is indeterminate. If the order relation between w(A+) and w(B+) is found to be determinate, then using [PC1], Question 2 is answered; namely, the order relation between w(A+) and w(B+) is exactly one of the three relations: greater than, less than, or equals.
To answer Questions 3 and 4, this information about the order determinability relation and the order determination relation between w(A+) and w(B+) is combined with the given weight relation between w(A) and w(B). If the order relation between w(A + ) and w(B + ) is found to be indeterminate, then, using AD3, Question 3 is answered; namely, the required order relation between w(C) and w(D) is indeterminate. If the order relation between w(A+) and w(B+) is found to be determinate and its type - greater than, less than, or equals - is determined, then depending on the type of this relation and the type of the given order relation between w(A) and w(B). Questions 3 and 4 are answered by using the AD1, AD2, and [AD1] principles: If the order relation between w(A+) and w(B+) is the same as the order relation between w(A) and w(B), or if one is the equals relation, then, by AD1, the required order relation between w(C) and w(D) is determinate, and, by [AD1], this relation is determined. On the other hand, if the order relation between w(A+) and w(B+) conflicts with the order relation between w(A) and w(B), then, by AD2, the required order relation between w(C) and w(D) is indeterminate. Thus, this analysis shows that the CCB solution process, by taking into account the three components (a) to (c) mentioned earlier, applies determinability and determination principles of two kinds: additive and multiplicative.
The CIB solution process takes components (a) and (b) - the given weight relation between Blocks A and B and the order relation between the number of building blocks added to A and B to create C and D, respectively - into account to solve the blocks task, but ignores component (c) - the weight relation between the building blocks L and S. In taking into account components (a) and (b), this solution process applies the additive principle AD1 and [AD1], and, at the same time, violates another fundamental principle, the "sameness" principle, which states that, in adding two quantities, they must be in units of the same type. By ignoring component (c), it does not apply any multiplicative or proportional reasoning and, thus, does not need any multiplicative or ratio principles. For example, if it is given that Block A is heavier than Block B [component (a)], and it is observable that the number of building blocks added to A to create C - n(A+) - is greater than the number of building blocks added to B to create D - n(B+) - [component (b)], then children who applied the CIB solution process would claim that Block C is heavier than Block D. It seems that the conceptual base for this reasoning is that children additively combine weight with number of building blocks: They add w(A) to n(A+) to get w(C), and w(B) to n(B+) to get w(D). When observing that w(A) > w(B) and n(A+) > n(B+), children apply the AD1 and [AD1] principles to conclude that w(C) > w(D).
The CDB and the isolated-counting (IC) solution processes based their solutions for the blocks tasks solely on one unit of information, ignoring other information necessary for a correct solution: The CDB solution process derives the required order relation between Blocks C and D from component (b) - the order relation between the number of building blocks added to A and B to get C and D, respectively; the IC solution process derives this relation from the observed number relation between C and D, or their tops. In doing so, these solution processes do not seem to apply any of the additive or multiplicative principles identified earlier. An apparent difference between these two solution processes, however, is that, whereas the CDB solution process takes account of all four sets of blocks involved in the problem, by considering the order relation between the number of building blocks added to A and B to create C and D, respectively, the IC solution process takes account only of Blocks C and D, ignoring the other two Blocks A and B.
The three solution processes - CCB, CIB, CDB - share important elements. First, all use the same type of representation - the complement representation. Second, the three sets of problem components used by these three solution processes are in sub-set relation: CCB used components (a) to (c); CIB, only components (a) and (b); and CDB, only component (b). This enables us to compare the sophistication hierarchy of these three solution processes based on the number and quality of unit information used. The hierarchy from most to least sophisticated is CCB, CIB, CDB: The CCB solution process uses three problem components and applies both multiplicative principles and additive principles; the CIB solution process uses two components and applies additive principles but no multiplicative principles; and the CDB solution process uses only one component and applies neither multiplicative nor additive principles.
Based on the fact that the IC solution process takes into account information on one pair of blocks (C and D), ignoring the other two (A and B), whereas CDB takes both pairs into account, we assume that CDB is more sophisticated than IC. Thus, this extends the solution-process hierarchy, from most to least sophisticated, to CCB, CIB, CDB and IC. A sophistication comparison of the SM and SIM solution processes with these four is not possible, because the conceptual bases of the structure-matching processes are very different from those of the other four processes.
In the next section, we present data bearing on two issues: (a) systematicity in the use of a particular solution strategy for a particular problem representation, and (b) correspondence between the solution processes and the three levels of mathematics ability (low, middle, and high) from which subjects for this study were chosen. This suggests a sophistication hierarchy among the solution processes that corroborates the one already presented.
Table 7 indicates the frequency for which a given strategy was used with a particular problem representation to form the solution process we have defined. This table clearly suggests that the correspondence between representation and strategy is very high but not perfect; exceptions exist. The matching strategy or imposed matching strategy was used with the structure representation 33 of the 47 times this representation was made. In the 14 exceptions, the structure representation invoked the incomplete balance strategy. The complement representation called up the balance strategy (complete, incomplete, and deficient) with one exception; then it called up the counting strategy. The isolated representation was used with the counting strategy with no exceptions. Our explanation for the 14 exceptions to the structure representation-matching strategy correspondence is that the child formed the structure representation and attempted the matching strategy or the imposed matching strategy, but difficulty in applying these strategies resulted in applying a "fallback" strategy, the incomplete balance strategy.
Table 8 shows the distribution of the six solution processes for the nine items given to the 17 Grade 7 children across the three ability levels. A sophistication hierarchy of these solution processes used by the children can be conjectured from their use distribution among the children. High-, middle-, and low-ability students likely use solution processes ordered from most to least sophisticated.
In general, the data indicate that the SM, SIM, and CCB solution processes were used predominantly by high-ability children (26 of 34 cases); solution process CIB was used predominantly by middle-ability children (22 of 33 cases); and solution processes CDB and IC were used predominantly by low-ability children (26 of 38 cases). This sophistication hierarchy corroborates the analysis of these solution processes provided in the previous section.
SUMMARY AND CONCLUSIONS
One contribution of this article is the comparison among four categories of tasks used for studying the proportion concept - the blocks tasks, rate and mixture tasks, balance scale tasks, and the fullness tasks. This analysis exposed similarities and differences among the tasks in terms of seven variables:
The analysis suggests some potential differences in understandability and difficulty of problems that differ according to these variables. The analysis offers a systematic approach to investigating questions about aspects of problem situations embedding the proportion concept that might be manipulated in an instructional sequence to facilitate children's learning of the proportion concept. Researchers could manipulate these variables in proportion tasks and in experimental instruction to investigate the effect on children's performance.
Another component of our analysis of the blocks task led to identifying a set of basic mathematical principles that underlie proportion tasks. These principles are hypothesized to be necessary to be able to reason proportionally in general, both qualitatively and quantitatively. Investigations to determine to what extent and at what age children have implicit or explicit awareness of these principles are suggested by this analysis. In addition, teaching experiments need to be conducted to determine whether children can gain implicit or explicit knowledge of these principles and whether and how this awareness contributes to the development of proportional reasoning ability and to improved performance on solving proportion problems.
The observation made in the analysis -that some proportion problems concern the invariance of a product. whereas others concern the invariance of a ratio (quotient) - clearly suggests that the development of the ability to reason proportionally requires experience with situations involving such invariance. This has strong implications for research and instruction. Current curricula treat multiplication and division (also addition and subtraction) from the static perspective of finding the answer when problem components are given (e.g., factors and addends are given with a requirement to find the product and sum) but neglect instruction involving the dynamic perspective of questioning whether a given product or quotient (or sum or difference) changes or is invariant when changes (transformations) are made to the problem components. Other related questions concern the direction of change, the magnitude of the change, and what will compensate for a change so that invariance can be achieved. Work on early number (e.g., Carpenter & Moser, 1983) has found that many children use invented strategies for finding sums and differences, frequently called derived-facts strategies. These derived-facts strategies depend on at least implicit knowledge of the invariance or compensation for the variation when a transformation is made on the problem components. Only a small amount of work seems to have been done on investigating instructional situations to determine whether children can learn these derived-addition-and-subtraction-fact strategies (e.g., Steinberg, 1985). We know of no similar efforts in multiplication and division. The circumstances under which a mathematical result is invariant are of fundamental concern to much of mathematics, and this issue is the fundamental question in any proportion problem. Implicit knowledge of this needs to be developed in children in the early elementary grades. How to develop this knowledge is a question for instructional research.
This article provides some rather conclusive evidence that a relation exists between the representation that a problem solver forms for a problem and the level of sophistication of the strategy that the solver uses in attempting a solution. Moreover, study of the student protocols indicates that it is the problem representation that drives the selection of a strategy and not the other way around. Evidence for this is the fact that, in their explanations, children refer to the problem representation before referring to the solution strategy. Before evidence was given of using the matching strategy, for example, the structure representation was indicated by explicating the necessary conditions for the matching strategy to apply.
This important finding leads to some further questions for research. If students are aided through instructional intervention to form problem representations that are associated with higher level solution strategies, will this result in their being able to use the more sophisticated strategy? If students demonstrate the ability to use more sophisticated representations as a result of instruction, can a commensurate increase in understanding of proportionality be detected? Are any instructional effects short term or do they result in some permanent restructuring of the students knowledge of proportions?
This article also suggests a new set or problem types that have not been considered in proportional reasoning research. Moreover, we established a mathematical isomorphism among problems of this class; investigations that compare and contrast children's problem-solving behavior among these problems are still needed.
In the blocks tasks and their isomorphs, more sophisticated strategies require the manipulation and coordination of a larger number of variables than in other problems used to investigate the proportion concept. One only has to look again at Figures 6, 8, and 9 to appreciate the differences in complexity among the structure, complement, and isolated representations, respectively, and among the strategies associated with them. We determined that strategy sophistication was highly correlated with students' mathematical ability. That is, the more able students (ability level was assessed prior to and independently of project activities) used more sophisticated strategies requiring the manipulation and coordination of a larger number of variables. One is reminded of earlier Piagetian definitions of intellectual stages that were inextricably related to the number of variables students could simultaneously manipulate. Recall early explanations of the lack of number conservation as the inability to coordinate both the amount of space between the counters and the amount of space (usually length) required to display the set. In like fashion, the onset of formal operational thinking was believed to be accompanied by the ability to coordinate mentally a large number of variables.
Proportional reasoning, one index of formal thought, shares many of those same perspectives. Higher ability students probably used more sophisticated strategies precisely because there was a better match between the processing capabilities and the demands of the problem condition than there was with the less able students. Thus, this class of problems might be a useful metric in providing insight into students' levels of proportional reasoning development.
function is linear in this case, because the blocks material is of homogenious
multiplicative increase of a1 to a2 is c (i.e.,
a2 = c x a1) and that of b1 to b2
is f (i.e., b2 = f x b1), then a2/b2
= (c x a1)/(f x b1), or a2/b2
= (c/f) x (a1/b1). Because it is given that c is
greater than f, c/f must be greater than 1; thus a2/b2
is greater than a1/b of k
is also an increase.
This work was supported in part by National Science Foundation Grant No. DPE-8470077. Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of the National Science Foundation.
Requests for reprints should be sent to Guershon Harel, Department of Mathematics, Purdue University, West Lafayette, IN 47907.
Carpenter, T. P., & Moser, J. M. (1983). The acquisition of addition and subtraction concepts. In R. Lesh & M. Landau (Eds.). The acquisition of mathematical concepts and processes (pp 7-14). New York: Academic.
Bruner, J S. & Kenny, H (1966) On relational concepts. In J. S. Bruner, R. R. Oliver, & P M. Greenfield (Eds.). Studies in cognitive growth (pp. 168-182). New York : Wiley.
Chaiklin, J (1989). Cognitive studies of algebra problem solving and learning. In S. Wagner & C. Kieran (Eds.). Research agenda in mathematics education: Research issues in the learning of algebra (pp. 93-114). Reston, VA: National Council of Teachers of Mathematics.
Chi, M. T. H., Feltovich, P. J. & Glaser, R. (1981). Categorization and representation of physics problems by experts and novices. Cognitive Science. 5. 121-152.
Fischbein, E., Deri, M., Nello, M., & Marino, M. (1985). The role of implicit models in solving verbal problems in multiplication and division. Journal for Research in Mathematics Education. 16, 3-17.
Harel, G. & Behr, M. (1989). Structure and hierarchy of missing value proportion problems and their representation. Journal of Mathematical Behavior, 8, 77-119.
Inhelder, B., & Piaget, J. (1958). The growth of logical thinking from childhood to adolescence (A. Parsons & S. Seagrin, Trans.). New York: Basic. (Original work published 1955).
Karplus, R.. & Peterson, R. W. (1970). Intellectual development beyond elementary school: 1I. Ratio: a survey. School Science and Mathematics, 70, 813-820.
Larkin, J. H. (1989). Robust performance in algebra: The role of the problem representation. In S. Wagner & C. Kieran (Eds.), Research agenda in mathematics education: Research issues in the learning of algebra (pp. 120-134). Reston, VA: National Council of Teachers of Mathematics.
Larson, S., Behr, M.. Harel, G., Post, T., & Lesh, R. (1989). Proportional reasoning in young adolescents: An analysis of strategies. In C. Maher, G. Golding, & R. Davis (Eds.), Proceedings of the Eleventh Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (pp. 181-186). New Brunswick, NJ: Rutgers University Press.
Mayer, R. E. (1985) Implications of cognitive psychology for instruction in mathematical problem solving. In E. A. Silver (Ed.), Teaching and learning mathematical problem solving: Multiple research perspectives (pp. 123-138). Hillsdale, NJ: Lawrence Erlbaum Associates, Inc.
Nesher, P. (1988). Multiplicative school word problems: Theoretical approaches and empirical findings. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp. 19-40). Reston, VA: National Council of Teachers of Mathematics.
Noelting, G. (1980a). The development of proportional reasoning and the ratio concept: Part I. Differentiation of stages. Educational Studies in Mathematics, II, 217-253.
Noelting, G. (1980b). The development of proportional reasoning and the ratio concept: Part II. Problem structure at successive stages: Problem solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathematics, II, 331-363.
Schwartz, J. L. (1988). Intensive quantity and referent transforming arithmetic operations. In J. Hiebert & M Behr (Eds.). Number concepts and operations in the middle grades (pp. 141-161). Reston, VA: National Council of Teachers of Mathematics.
Siegler, R. S. (1976). Three aspects of cognitive development. Cognitive Psychology, 8, 481-520.
Siegler, R. S., & Vago, S. (1978). The development of a proportionality concept: Judging relative fullness. Journal of Experimental Child Psychology, 25, 371-395.
Steinberg, R. M. (1985, April). Keeping track of processes in addition and subtraction. Paper presented at the meeting of the American Educational Research Association, Chicago.
Strauss, S., & Stavy, R. (1982). U-shaped behavioral growth: Implications for theories of development. In W. W. Hartup (Ed.), Review of child development research (Vol. 6, pp. 547-599). Chicago: University of Chicago Press.
Thompson, P. (in press). The development of the concept of speed and its relationship to concepts of rate. In G. Harel & J. Confrey (Eds.), The development of multiplicative reasoning. New York: SUNY Press
Tourniaire, F. ( 1986). Proportions in elementary school. Educational Studies in Mathematics, 17, 401-412.
Tourniaire, F. & Pulos, S. (1985). Proportional reasoning: A review of the literature. Educational Studies in Mathematics, 16, 181-204.
Vergnaud, G. (1988). Multiplicative structures. In J. Hiebert & M. Behr (Eds.), Number concepts and operations in the middle grades (pp 141-161). Reston, VA: National Council of Teachers of Mathematics.