Impact of the Number Type on the
Fischbein. Deri, Nello, and Marino (1985) argue that students' conceptions of and performance on multiplication and division application word problems (hereafter multiplicative problems) are unconsciously derived from primitive intuitive models that "correspond to features of human mental behavior that are primary natural and basic" (p. 15). They suggest their theory to account for conceptions, such as multiplication makes bigger and division makes smaller, identified in previous studies (e.g., Bell, Fischbein, and Greer, 1984; Vergnaud, 1983; Bell, Swan, and Taylor, 1981). These conceptions are in accord with the operations of multiplication and division in the whole number domain but incongruent to these operations in the rational number domain; thus, they block the way to correctly solve many multiplicative problems whose quantities are decimals or fractions.
In our recent effort to better understand the multiplicative conceptual field, and in particular the transition phase from the additive structure to the multiplicative structure, we probed Into this question of incongruity We found that many questions and concerns are still open and need further investigation. Among these we addressed the following: (1) the impact of the number type on the solution of multiplicative problems; (2) the impact of the textual structure on problem interpretation; and (3) solution models used by teachers to solve multiplicative problems. To address some of these questions empirically, we developed an instrument that controls a wide range of confounding variables known to be influential on subjects' performance on multiplicative problems and used it with inservice and preservice teachers. In this chapter we report on an investigation of aspect (1); the investigations on the other two aspects will be reportedseparately.
In this chapter, we dealt with several aspects of the impact of the number type on the relative difficulty of multiplicative problems. We reexamined the findings from other studies concerning this impact, investigated the "absorption effect" notion suggested by Fischbein et al. to account for differences in subjects' performance on multiplicative problems with different non-whole-number operators and probed into the level of robustness of the intuitive rules derived from Fischbein et al.'s models. The observations and findings reported in this chapter are summarized as follows:
1. The instruments used in Fischbein et al. and the studies that followed it do not control for many variables known to be influential in problem solution. We offered a framework for an instrument that controls for a wide range of confounding variables: number type, text, structure, context, syntax, and rule violation.
2. Our data is consistent with the finding that subjects' model for multiplication is the repeated addition model, and for division, subjects' models are partitive division and quotitive division.
3. Our data do not support Fischbein et al.'s notion of the "absorption effect": No significant difference in performance was found between multiplication problems with multipliers whose whole part is relatively large and those with multipliers whose whole part is relatively small. Moreover, the absorption effect does not apply to division problems.
4. Evidence was shown for differential robustness of the intuitive rules associated with Fischbein et al.'s models.
5. The type of multiplicand seems to have an impact on problem solution when the multiplier is smaller than 1.
This research has raised several questions for further investigations. First, this and other studies focused on one type of problem quantities: decimal numbers. The question of whether subjects encounter similar difficulties with multiplication and division problems that involve fractions has never been directly addressed. There is a reason to believe, however, that fractions and decimals do not have the same effect on the solution of multiplication and division problems. More specifically, it seems easier to solve multiplication and division problems in which the operator (i.e., the multiplier or divisor) is a fraction than in those in which the operator is a decimal. A rationale for this is based on the fact the naming rule of fractions is different from the naming rule of decimals: Under these naming rules, it is easier to identify the role of a problem quantity as an operator or operand if the quantity is A fraction than if it is a decimal; therefore it is easier to recognize its relation to other problem quantities. For example, the two propositions in the statement, "John had 5 ounces of ice cream and he ate x of the amount he had" are easier to connect if (the operator) x is a fraction, say 2/5, than if x is a decimal, say 0.40.
Second, a further distinction among the intuitive rules derived from Fischbein et al.'s model is that some of the rules are associated with the problem information, others with the problem solution. In multiplication, the rule that the multiplier must be a whole number impose 3 a constraint on the type of multiplier provided in the problem information; in contrast, the rule that the multiplication makes bigger restricts the problem solution to be a number greater than the multiplicand. Similarly, in partitive division, the rules that the divisor must be a whole number and the divisor must be smaller than dividend are problem information rules, whereas the rule that the quotient must be greater than dividend is a problem solution rule. Finally, in quotitive division, the rule that the divisor must be smaller than dividend is a problem information rule; no problem solution rule is involved. This raises the question of whether problem information rules are equally robust as the problem solution rules.
Finally, when we looked at the other studies' data on multiplication and division, we raised the question, Why are problems with a multiplier greater than 1 relatively easy for the subjects despite the fact that they are in conflict with the model of multiplication as a repeated addition? If indeed this model governs subjects solution of multiplication problems, it is not at all clear why the intuitive rule derived from it that the multiplier must be a whole number-is substantially less robust in the case of a non-whole-number multiplier greater than 1 than in the case of a multiplier smaller than 1. Further, it is not all clear what is the conceptual basis for the multiplier I being an index for the relative difficulty of multiplication problems. In fact, Fischbein et al. in their explanation to the observation that the intuitive rule that the multiplier must be a whole number does not equally affect multiplication problems with decimal multipliers, did not differentiate between multipliers greater than 1 and those less than 1. Rather, they suggested the "absorption effect" notion, which differentiates between multiplicative problems according to the size relationship between the whole number part and the fractional part of their multipliers. In this chapter we reported data that are not consistent with this explanation; therefore, further theoretical and empirical investigations are needed to answer these questions.
Bell, A., E. Fischbein and B. Greer. 1984. Choice of operation in verbal arithmetic problems: The effect of number size, problem structure and context. Educational Studies in Mathematics 15: 129-147.
Bell, A. M. Swan, and G. Taylor. 1981. Choice of operations in verbal problems with decimal numbers. Educational Studies in Mathematics 12: 399-420.
Conner, G., G. Harel, and M. Behr. 1988. The effect of structural variables on the level of difficulty of missing value proportion problems. Proceedings of the tenth PME-NA annual meeting, DeKalb, IL.
Fischbein, E., M. Deri, M. Nello, and M. Marino. 1985. The role of implicit models in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education 16:3-17.
De Corte, E., L. Verschaffel, and V. Van Coillie. 198& Influence of number size, problem structure and response mode on children's solutions of multiplication word problems. Journal of Mathematical Behavior, 7: 197-216.
Goldin, G. A. 1986. Review of A Bell, E. Fischbein, and B. Greer: Choice of operation in verbal arithmetic problems: The effects of number size, problem structure and context. Investigations in Mathematics Education 19 (1): 1-7.
Greaber, A., D. Tirosh. and R. Glover. 1989. Preservice teachers' misconception in solving verbal problems in multiplication and division. Journal of Research in Mathematics Education 20: 95-102.
Greer, B. 1985. Understanding of arithmetical operations as models of situations. In Cognitive processes in mathematics, ed. J. Sloboda and D. Rogers. London. Oxford University Press.
_. 1988. Nonconservation of multiplication and division: Analysis of a symptom. Journal of Mathematical Behavior 7: 281-298.
Greer, B., and C. Mangan. 1984. Understanding multiplication and division. In Proceedings of the Wisconsin meeting of the PMENA, ed. T. Carpenter and J. Moser, 27-32. Madison: University of Wisconsin Press.
Harel, G., and M. Behr. 1989. Structure and hierarchy of missing value proportion problems and their representations. Journal of Mathematical Behavior.
Harel, G., T. Post, and M. Behr. 1988. On the textual and semantic structure of mapping rule and multiplicative compare problems. Proceedings of the tenth international conference of PME, Budapest
Harel, G., M. Behr, T Post, and R. Lesh. in press. Teachers' solution strategies of multiplicative problems. Journal of Mathematical Behavior.
Hart, F. M. 1984. Ratio: Children's strategies and errors in secondary mathematics project. Windsor, Berkshire: NFER-NELSON.
Lester, F. FL, and P. Kloosterinan. 1985. Review of E. Fischbein, M. Deri, M. Nello, and M. Marino: The role of implicit models in solving verbal problems in multiplication and division. Investigations in Mathematics Education 18: 10-13.
Luke, C. 1988. The repeated addition model of multiplication and children's performance on mathematical word problems. Journal of Mathematical Behavior 7: 217-226.
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 Number concepts and operations in the middle grades. ed. M. Behr and J. Hiebert. Reston, W National Council of Teachers of Mathematics.
Vergnaud, G. 1983. Multiplicative structures. In Acquisition of Mathematics Concepts and Processes, ed. R. Lesh and M. Landau. New York: Academic Press.
This research was funded in part by grant #CRG-890977 from NATO.