ANALYSIS: STRUCTURE AND HIERARCHY,
A missing value proportion (MVP) problem involves three given values and an unknown to be found under the constraint of maintaining the proportion relation. Variables which affect performance on MVP problems have been classified by Tourniaire and Pulos (1985) into student- and task-centered, and the latter sub classified as structural and contextual. This paper presents a theoretical analysis of selected structural variables to give a detailed description of MVP problem structures. This leads to hypotheses for hierarchies of problem complexity, and of preference for instantiating problem operators as solution strategies, and to a partial hierarchy of problem difficulty. This analysis will make a substantial contribution to this research area by providing a theoretical structure to guide the manipulation of these structural variables.
The conceptual process of problem solving is described in terms of a problem presentation, problem representations, problem operators, knowledge of the problem domain, and solution strategies (i.e., instances of the problem operators). Investigations in the problem solving found that the problem representation formed by the solver is dependent on the problem presentation on the solver's knowledge of the problem domain (Greeno, 1978; Chi, Fetovich, Glaser, 1981). Problem difficulty is also a function of the problem representation.
Any verbal MVP problem involves two statements which express a ratio between two quantities, we refer to these as per statements. One of these, the closed per statement, relates two known quantities; the other relates one known and one unknown quantity - the open per statement. The structural variable of order is defined in terms of the location of the unknown (left or right) within a per statement, and in terms of the location of the open per statement (top or bottom) with respect to the closed per statement. This defines four problem categories as BL (bottom left), BR, TL and TR.
A second structural variable is unit of measure. On the sub-variable of measure space, MVP problems very according to whether one or two are involved, and for those with two measure spaces, whether the quantities within, or between, per statements are from the same or different spaces. Here we identify three categories of MVP problems - one-by-one: one measure space; two-by-one: same measure space within per statements, but different between; two-by-one: different measure spaces within per statements, but the same between. The sub-variable dimension distinguishes between different units within a measure space, minutes and hours, for example. Dimension yields four sub categories of the one-by-one and two subcategories of reach of two-by-one and one-by-two. (see Harel and Behr, in preparation.)
A third structural variable, left to a subsequent analysis, is the partionability of the quantity to which the unit of measure refers. Example: Money vs. child. Our observation is that this may affect the choice of operation - multiplication or division - which a child makes in formulating a solution procedure for a MVP problem.
The divisibility variable (or integer ratio), which we have extended to pairs of rational numbers, has been shown in prior research to affect the problem difficulty. When an integer ratio exists the direction of the divisibility will affect problem difficulty. The variable of common divisor is related to divisibility. It appears that children have an order of preference to operate with a pair of numbers which (a) have an integer ratio, (b) have a common divisor other than 1, and (c) are relatively prime (Behr et al., in preparation).
The four levels of the problem variable order, eight of unit of measure, sixteen of divisibility, identify (4x8x16) 512 MVP problem structures. Of these the 256 with dimension 1 form the subclass of ratio problems and the remaining 256, those with dimension 2, the subclass of rate problems.
PROBLEM REPRESENTATIONS AND PROBLEM OPERATORS
Our MVP problem solving model is consistent with that proposed by Simon and Hayes (1974). In order to solve a MVP problem we assume the solver has a set of knowledge structures about the problem. This included knowledge of: the initial and goal states of the problem, problem operators (information processes) that transform a given problem state into another, and constraints under which the operators can be applied. We refer to problem states formed by the solver to determine a solution path from the initial problem state (the problem presentation) to the goal state as problem representations. Our conceptualization of the solution process is that a solver uses one or more problem representations for understanding the problem, and intermediate representations for exploring the relationships among problem components in order to choose the appropriate problem operators, and finally, the solver will arrive at a choice for procedural representation on which an operator will be instantiated as a solution strategy. This procedural representation that a solver forms will reflect both salience of the structural variables and the interaction of these with the solver's preference for how to instantiate a problem operator.
We distinguish three types of problem operators for MVP problems. The first class consists of structural transformations by which the solver, responding to certain structural variables, changes the problem structure. Changing the direction of the divisibility, or the order of the unknown, are possibilities. The second class consists of the unit rate operator. This transforms a closed per statement of the form "a per b" to an equivalent one of the form "1 per b/a" or b/a per 1/" The third class consists of procedural operators. These are applied to a procedural representation to solve the MVP problem. The first class of operators transform intermediate representations to other representations with the ultimate goal of achieving a procedural representation. The unit rate operator transforms a procedural representation into another procedural representation. The procedural operators act on components of the problem structure in a procedural representation to arrive at a value for the unknown.
To illustrate the structural transformations we consider an MVP problem of the form "a per b, c per x." The total of eight structural transformations, which are of three types, form a group of order eight under ordinary composition of transformations. The first type, rate reciprocation, changes the problem structure to "b per a, x per c." This transformation gives the reciprocal of each within-per-statement rate pair. The second, per statement reciprocation, interchanges the order of the two per statements to get the problem structure "c per x, a per b." The third, measure space reciprocation, changes the structure by interchanging b and c or a and x. For a MVP problem with two measure spaces this transformation changes the quantities in the per statements so each per statement is changed from a within measure space comparison to a between space comparison or vice versa. It changes the structure "a per b, c per x" to "a per c, b per x" or to "x per b, c per a."
Applying the procedural operators requires additional knowledge, called procedural knowledge, of how to instantiate these operators for a particular MVP problem. Involved is a sequence of finding the relationship between the quantities in the closed per statement (the RCQ operator) and transposing that to an operation on quantities of the open per statement (the ROQ operator) to find the unknown. Consistent instantiations of the RCQ and ROQ operators results in an identifiable strategy. A strategy is valid if it observes appropriate constraints and invalid if any problem constraint is violated. The frequently observed addition strategy is invalid because it violates the constraint of maintaining the proportion relation. Among the valid strategies we consider several instances of a multiplicative strategy.
A multiplicative strategy begins by instantiating the RCQ operator by determining the relationship between the two known quantities in the closed per statement and expressing this relation in the form of a multiplication or division equation. This equation has an unknown value u. Let v denote the computed true value for u. Next the ROQ operator is instantiated by using v and the known quantity of the open per statement in a multiplication or division equation to find the value of the unknown. Thus a multiplication strategy involves sequential instantiations of the RCQ and ROQ operators with some combination of multiplication and division equations. We classify a multiplicative strategy as: a division strategy (DS) or a multiplication strategy (MS), when the RCQ operator is instantiated with a division or multiplication equation, respectively. We thus have the strategies PMS (positive division), and NDS (negative division). Moreover, the equations which are used to instantiate these operators can be formed so that the value needed to be found in either case appears in a missing value equation or as the answer to be found in a direct computation equation. In the first case the equation is indirect (I), and in the second direct (D). Further analysis leads to the following list of 14 strategies for solving MVP problems: PMS-ID, PMS=II, PDS-DD, PDS-DI, PDS-DI*, PDS-II, PDS-II*, NMS-ID, NMS-II, NMS-II*, NDS-DI, NDS-II, and NDS-ID, where the * means that v appears as the result, rather than operator or operand, in the equation of ROQ.
PREFERENCES FOR INSTANTIATING THE RCQ AND ROQ OPERATORS
Our next objective is to hypothesize a hierarchy of children's preference for these strategies. To instantiate the RCQ operator children must consider the direction of the operation (left or right) and also its type (multiplication or division). Instantiation of the RCQ operator involves two given quantities a and b and an unknown quantity u. The equation that is formed to give the relation between a and b can involve a, b, and u in one of two types of operations an din one of three roles of operator, operand or answer. Considerations, too lengthy to discuss in this brief report, lead us to six ways to instantiate the RCQ operator (see Harel and Behr, in preparation). Under the constraint of an integer ratio (non-integer ratio involves other considerations) these six ways, listed in the assumed order of children's preference, are as follows:
Once the RCQ operator has been instantiated, to instantiate the ROQ operator judgments need to be made about four variables; preservation of operation direction, preservation of operation type, the role (operator, operand, or answer) in which to use f, the computed value of u, in the known-to-unknown relationship and the level of structural equivalence between the RCQ equation and the ROQ equation. (See Harel and Behr, in preparation, for information on structural equivalence.) All possible equation pairs by which RCQ and ROQ operators can be instantiated for a MVP problem of the form "a per b, c per x" are given in the table. Similar information for a problem of the form "a per b, x per d" is given in Harel and Behr (in preparation).
Analysis of Strategies
and Preference Hierarchy for Instantiating the RCQ Operators for the Procedural
Representation: a per b, c per x
An analysis for which we refer the reader to Harel and Behr (in Preparation) leads to the identification of exactly 18 procedural representation structures (Pi, i =1,2, ,18). Further analysis based on the preference hierarchy for instantiating the RCQ and ROQ operators and further assumptions (see Harel and Behr, in preparation) allows us to order these from most complex to least complex; we assume the notation to be such that this order is P1 , P2 , , P16 . The total set S, of the 512 problem representations is partitioned by the group of eight transformations into subsets, Si,j. The subset Si,j is the set of all problems pre-presentations which are mapped by a transformation Tj unto the procedural representation Pi. For m ? n, the intersection of Sm,j and Sn,j is empty; that is, there are no two problems representations which get mapped by different transformations unto the same procedural representation. We next consider the sets S1,j, S2,j, S3,j, ,S18,j, the sets of problem representations which are mapped by Tj unto P1, P2, P3, ,P18, respectively. We take the previously established order of complexity on the procedural representations P1 to P18 as an imposed order on the set of pre-images, S1,j to S18,h. We define this imposed order to be the order of problem difficulty on this collection of problem representations. This leads to the following partial (in some sense of partial order) difficulty hierarchy on the set of 512 MVP problem structures (Let > denote greater in difficulty).
Finally our analysis shows that it is not possible to order the levels (Level 1 through Level 8) according to difficulty.
Chi, M.T.H., Fetovich,
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Harel, G. & Behr, M.J. Structure and Hierarchy of missing value proportion problems and their representations. In preparation.
Greeno, J.G. (1978). A study of problem solving. In R. Glaser (Ed.), Advances in Instruction Psychology (Vol. 1). Hillsdale, NJ.: Lawrence Erlbaum Associates.
Tourniaire, F., & Pulos, S. (1985). Proportional reasoning: a review of the literature. Education Studies in Mathematics, 16, 181-204.
* The research was supported in part by the National Science Foundation under grant No. DPE-8470077. Any opinions, findings, and conclusions expressed are those of the author and do not necessarily reflect the views of the National Science Foundation.