

AN
ASSESSMENT INSTRUMENT TO EXAMINE KNOWLEDGE OF MULTIPLICATION AND DIVISION
CONCEPTS AND ITS IMPLEMENTATION WITH INSERVICE TEACHERS


Guershon Harel, Northern Illinois University Thomas Post, University of Minnesota Merlyn Behr, Northern Illinois University 

Fischbein, Deri, Nello, and Marino (1985) argued that conception and performance of multiplication and division problems are unconsciously derived from three intuitive models. The model associated with multiplication problems is repeated addition. Under this model the roles played by the quantities multiplied are asymmetrical (Greer, 1965). One of the quantities multiplied, called multiplier, is conceived of as the number of equivalent collections, while the other quantity, called multiplicand, is conceived of as the size of each collection. These conceptions impose the constraint that multipliers must be whole numbers; thus they reinforce the misconception that products must be larger than multiplicands, or "multiplication makes bigger." For division, Fischbein et al, (1985) suggested two intuitive models, one is associated with equal sharing, or partitive division problems, the other with measurement, or quotitive division. In the partitive division model, an object or collection of objects is divided into a number of equal fragments or subcollection. Associated with this model are the constraints that divisors must be whole numbers and smaller than dividends; these constraints result in the misconception, or constraint, that quotients must be smaller that dividends, or "division makes smaller." The quotitive division model is associated with division problems in which it is required to find how many times a given quantity is contained in a larger quantity. The only constraint imposed by this model is that divisors must be smaller than dividends. Fischbein et al., (1985) addressed the question concerning the conflict between the intuitive models and the formal operations of multiplication and division with 5th, 7th, and 9th graders. Tirosh, Graeber, and Glover (1986) addressed the same question with student teachers, and further investigated the similarity between children’s errors and student teachers’ errors in solving multiplication and division problems. However, in these studies only one variable was controlled, the number type variable (i.e., whole number, decimal > 1, decimal < 1), leaving other variables, such as the textual, contextual, and syntactic variables uncontrolled. Mangan (1986) addressed these questions in a study which controls systematically the textual and the number type variables, with extensive age range of children and adults: primary and secondary school students, further education, university students, and student teachers. The instrument we have developed to investigate questions related to multiplication and division concepts is different from those used in these studies, in the number of confounding variables it controls. The instrument consists of a written, short answer problems followed by oneonone interview assessment, where the problems used were constructed by controlling five variables, the textual, structural, contextual, numerical, and syntactical variable. Textual variable. In analyzing the textual structure of multiplicative problems, Nesher (1988) identified two subcategories within the isomorphism of measures category (Vorgnaud, 1983): mapping rule (e.g., "There are 5 shelves of books in Dan’s room. Dan put 8 books on each shelf. How many books are there in his room?") and multiplicative compare (e.g., "Dan has 12 marbles. Ruth has 6 times as many marbles as Dan has. How many marbles does Ruth have?"). In analyzing these categories we found that that there is a fundamental difference between them (Harel, Post, and Behr, 1988). While in multiplication mapping rule problems the roles of the quantities multiplied  as multiplier or multiplicand  can be directly derived from the problem statement, in multiplication compare problems, these roles are interchangeable, depending on the solver’s interpretation of the phrase "times as many as." Accordingly, a division compare problem can be represented either as partitive or quotitive depending upon the solver’s interpretation of the roles of the quantities in the corresponding multiplication problem. This implies that the theory of Fischbein et al., (1985) cannot be examined on multiplicative compare problems, since their structure cannot be determined prior to testing. Our instrument addresses two aspects concerning the textual variable. First, it tests Fischbein’s theory with problems whose textual structure is that of the mapping rule type. Second, it tests the solver’s representations of problems with the multiplicative compare structure (for more on the latter aspect, see Harel et al., 1988). Problem structure variable. The instrument includes the three problem structures, multiplication, partitive division, and quotitive division. Context. The context of the problems used is that of rate. The only rate problems included are of the following types: weight, price, and consumption. Number type. The instrument tests knowledge of multiplication and division concepts in the two domains of the number system, decimals and fractions. The problem quantities – multipliers and multiplicands and divisor and dividend – were systematically varied across number type, whole number, decimal > l, and decimal < 1, and across the order relation between problem quantities. In addition, the instrument includes problems in which the quantities varied across whole number, fraction > 1, and fraction < 1. Syntactic. This variable is derived from the analysis by Herel and Behr (in press). It involves the subvariables "location of the missing value" and "coordination of measure spaces." (See Conner, Harel, and Behr, this volume). IMPLEMENTATION As has been said earlier, our instrument is distinctive from others’ because it controls a wide range of confounding variables. In addition, the instrument has been used to examine the multiplication and division knowledge of a different population, inservice elementary school teachers. The objective was to assess teachers’ knowledge, with respect to the following questions: (a) do inservice teachers solve correctly formal problems (i.e., problems which conflict with the intuitive models)?; (b) what strategies do teachers use to solve formal problems?; (c) to what extent the teachers’ errors in solving formal problems are similar to those of children?; (d) to what extent the solution processes used by teachers to solve multiplication and division problems are similar to those used by children?; (f) what impact does the textual structure of multiplicative problems have on the problem representation and problem difficulty?; (e) are the intuitive rules equally robust in solving multiplicative problems? The latter question needs more elaboration; it will be given in the following section. Levels of robustness of the intuitive rules. A careful examination of different studies led us to conclude that the intuitive rules are not equally robust in problem solutions. Consider, for example, the table below which shows the percentage distribution of responses to problems 16, 17, 20, 21, and 22 from Fischbein et al.’s (1985. p. 12). All these problems are of partitive division type which violate the some intuitive rule, "divisor must be smaller than the dividend." Despite this uniformity, the results are strikingly different: the percentages of correct responses on Problems 16 and 17 are much lower than of those on Problems 20, 21, and 22. Fischbein et al.’s explanation to this is that in Problems 16 and 17 the students’ tendency was to reverse the roles of the numbers as a divisor and dividend, "had they done that in Problems 20 to 22, however, they would have ended up with a decimal divisor! It appears that, faced with having to cope with a violation of the partitive model’s rules, the pupils chose instead not to reverse the numbers" (p. 13). From this result we concluded that different intuitive rules within the partitive model may not be equally robust in solving partitive division problems: in Problems 20, 21, and 22 the children preferred to cope with the rule, "divisor must be smaller than dividend," than with the rule "divisor must be a whole number." 



The question of how different rule violators effect differ entry the solution of multiplicative problems needs further investigation. As a first step, we classified multiplicative problems according to the types of rule violations. Consider, for example, the partitive model, which was characterized by the three rules: (a) divisors must be whole numbers; (b) divisors must be smaller than dividends; and (c) quotients must be smaller than dividends. Based on the type of rule violation, partitive division problems can be classified into six categories: PV(O), PV(a), PV(b), PV(a, b), PV(a, c), and PV(a, b, c), where P stands for partitive, V for violation, and a, b, and c for the rule being violated (0, when no rule is violated). Thus, PV(a), for example, is the set of partitive division problems which violate Rule (a). Using this classification, we were able to identify a complexity hierarchy among multiplication and division problems, and to suggest models for the cognitive processes involved in solving these problems. RECOMMENDATIONS Fischbein et al. (1985) believe that the intuitive models correspond to features of human mental behavior that are "primary, natural, and basic" (p. 15). Acknowledging the fact that these models are behaviorally and intuitively meaningful, teachers use them to introduce the operations of multiplication and division; thus they reinforce the conflict with the formal concepts of these operations (p. 1516). To resolve this didactical dilemma, several recommendations were offered. Fischbein et al., recommended that teachers should "provide learners with efficient mental strategies that would enable them to control the impact of these primitive models" (p. 16). Greer (1985) chose a different direction: he recommended that teachers should "aim to widen the range of models available to the pupils" (p. 74). However, neither Fischbein nor Greer specified his recommendation. Our recent work with the multiplicative field and results from the assessment of inservice teachers’ knowledge suggest that the mental processes involved in the comprehension and solution of the intuitive problems (i.e., problems which conform to the intuitive models) are fundamentally different from those involved in the formal problems (i.e., problems which conflict with the intuitive models). As such, transitional stages from the domain of intuitive problems to the domain of formal problems are required. Our conclusion is that these transitional stages must include the concept of proportion, which can enable learners to represent formal problems as missing value proportion problems. Intuitive solution strategies are available to this representation. These strategies involve determining the multiplicative relationship between two given quantities and extending that relationship to the other two quantities to find the unknown quantity (see Harel and Behr, in press; Vergnaud, 1983). Primary data with inservice and preservice elementary school teachers and results from work with 7th graders (see Sellke, Behr and Voelker, this volume) support this approach. However, still more work is needed to find how to integrate the two domains of problems into a one coherent domain within the multiplicative field. REFERENCES Fischbein, E. Deri, M., Nello, M. & Marino, M. (1985). The rule of implicit models in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education, 16, 317. Greer, B. (1985). Understanding of arithmetical operations as models of situations. In J. Sloboda and D. Rogers (Eds.) Cognitive Processes in Mathematics. London, Oxford University Press. Harel, G., & Behr, M. (in press). Structure and hierarchy of missing value proportion problems and their representations. Journal of Mathematical Behavior. Harel, O., Post, T., & Behr, M (1988). On the textual and semantic structure of mapping rule and multiplicative compare problems. Proceedings of the Tenth International Conference of PNE. Budapest. Mangan, C. (1986). Choice of operation in multiplication and division word problems. Unpublished doctoral dissertation, Queen’s University, Belfast. Nesher, P. (1988). Multiplicative school word problems: Theoretical approaches and empirical findings. In M. Behr and Hiebert (Eds.) Number Concepts and Operations in the Middle Grades. Reston, VA, National Council of Teachers of Mathematics. Tirosh, D., Greaber, A., & Glover, P. (1986). Preservice teachers choice of operation for multiplication and division word problems. Proceedings of the Tenth international Conference of PNE. London. Vergnaud, G. (1983). Multiplicative structures. In R. Lesh and M. Landau (Eds.) Acquisition of Mathematics Concepts and Processes. NY, Academic Press. This research was supported in part by the National Science Foundation under grant No.TEI8652341. Any opinions, findings, and conclusions expressed are those of the authors and do not necessarily reflect the views of National Science Foundation. 
