PROSPECTIVE TEACHERS' PART-WHOLE DIVISION PROBLEM SOLUTION STRATEGIES

Walter E. Stone, Jr., University of Massachusetts - Lowell, wesjr@mediaone.net

The purpose of this paper is to report results from a piece of a larger research study (Stone, 1999) that investigated the factors that contribute to prospective teachers' abilities to solve part-whole division problems. The factors investigated were fraction type, that is, how prospective teachers' success was related to the type of fraction (unit fraction or common fraction) presented in part-whole division problems and whether or not a visual image of the data presented in the problem contributes to successful solution strategies. Prospective teachers' part-whole division problem solution strategies are the focus of this paper. Prospective teachers use a combination of proportional reasoning, concept of unit, and fraction as operator strategies when solving part-whole division problems.

Introduction and Rationale

Prior research suggests that preservice and inservice elementary through intermediate level teachers misunderstand many fraction concepts and procedures (Ball, 1990; Post, Harel, Behr, & Lesh, 1991). Ball (1990) found that preservice teachers' knowledge of fraction division is particularly weak. Post and his colleagues (1991) found that many practicing teachers had difficulty ordering fractions, finding equivalent fractions, performing fraction operations, and solving proportional reasoning problems.

Prospective teachers' abilities to solve part-whole multiplication (or fraction as operator) problems has been investigated by Behr, Khoury, Harel, Post and Lesh (1997). Behr and his colleagues found that prospective teachers' abilities to solve part-whole multiplication problems were strong. These researchers paid special attention to strategies used by the subjects and found that they were able to effectively use two related operator subconstruct strategies; duplicator-partition reducer, and stretcher-shrinker.

Part-whole division problems are problems that define part of a quantity and ask for the whole of that quantity (Greer, 1992). For example, "Two-thirds of the children order chocolate milk with their lunch. Six children order chocolate milk. How many children order lunch?" is a part-whole division problem. Post and his colleagues (1991) found that intermediate level teachers were not as successful when solving part-whole division problems. These researchers found that a mean percentage of 69% of their subjects could successfully solve part-whole division problems. Approximately 87% of the teachers were successful at solving an item in which a unit fraction was presented. This percentage dropped to 78% when common fractions were presented in the problems. Post's study focused only on achievement. Teachers' understanding of part-whole division relationships nor the strategies they used when solving part-whole division was not presented. The research reported in this paper presents solution strategies and errors prospective teachers make when solving part-whole division story problems.

Methodology

The sample for this study consisted of 66 students enrolled in a graduate elementary mathematics methods course at a private urban college in the Boston metropolitan area. During November of 1998, data were gathered for the study using two forms of a written instrument called "The Part-Whole Test," an instrument consisting of twelve part-whole division problems. One-half of the problems presented employed a combination of prose and illustration. For example, the problem "One-twelfth of the cans of soda in the refrigerator is diet soda. There are three cans of diet soda in the refrigerator. How many cans of soda are in the refrigerator?" was accompanied by an illustration of three cans of diet soda. One-third of the fractions presented in the problems were unit fractions; two-thirds were common fractions.

Data Analysis

Subjects' solutions to problems from the Part-Whole Test were scored for correctness and coded by solution strategy. Solution strategies included concept of unit and the thinking of a fraction as an operator or a ratio (Behr et al, 1997; Vergnaud, 1983). Subjects who used the fraction presented in the problem in an equation or divided the whole number part by the fraction were classified as using an operator strategy. Subjects who set up a proportion were classified as using a ratio strategy. Subjects were classified as using a concept of unit strategy when the number that corresponds to the unit fractional amount was used to find the whole. Consider the following problem:

Thirty-five gallons of paint is of the paint needed to paint a building. How many gallons of paint are needed to paint the building?

Subjects were classified as using an operator strategy if they set up the equation or performed the operation . Subjects were classified as using a ratio strategy if they set up the proportion .

When employing a concept of unit strategy, subjects used strictly multiplicative or a combination of additive and multiplicative reasoning when solving part-whole division problems. Using multiplicative reasoning, a subject stated that of the paint needed is 7 gallons. Since is one whole, 7 times 7 is 49 gallons; the amount of paint needed to paint the building. Using additive reasoning as well as multiplicative reasoning, a subject stated that since of the paint is 35 gallons, more of the paint is needed since is equivalent to one whole, or the amount needed to paint the whole building. Since of the paint is 7 gallons; of the paint is 7 + 7, or 14 gallons. Thus, the total amount of paint needed is 35 + 14, or 49 gallons.

Subjects' errors were coded as multiplication or addition errors. Multiplication errors occurred when a subject directly multiplied the fraction and the whole number presented in the problem and addition errors occurred when subjects added the fraction and the whole number presented in the problem.

Results

Strategies: The number of subjects and the percentages of subjects using particular combinations of strategies are listed in Table 1.

Table 1

Strategies Subjects Used to Solve Part-Whole Division Problems (Number, Percentage of Subjects)

One Method	Two Methods	Three Methods
Operator (17; 25.7%)	Operator/Ratio (6; 9.1%)	Operator/Ratio/Concept of Unit (5; 7.6%)
Ratio (9; 13.6%) Concept of Unit (16; 24.2%)	Operator/Concept of Unit (9; 13.6%) Ratio/Concept of Unit (4; 6.1%)

Results from Table 1 show that 36% of the subjects did not consistently use the same strategy to solve part-whole division problems. Strategies used by each subject varied from problem to problem. The concept of unit strategy was used by approximately 50% of the subjects. Thirty-six (36) percent of the subjects treated the part-whole division problems like proportional reasoning-missing value problems.

In addition, many of the solution methods involved the use of visual images. In 36% of the successful solution strategies, subjects drew pictures, charts, or diagrams. Subjects appeared to use images to model the data and the relationships present in the problems. Only 12% of subjects used the drawings provided on the Part-Whole Test to model successful solution strategies.

Errors: Of the errors subjects made on part-whole division problems, 25% were multiplicative errors. Twenty (20) percent of the errors subjects made were additive. Non-completion of solution procedures, non-attempts, or calculation errors while applying correct solution procedures accounted for 55% of errors subjects made when solving part-whole division problems.

Conclusions

As can be seen in Table 1, it may be inferred that prospective teachers that participated in this study have flexible conceptions of fraction (the conception of fraction as a multiplicative operator as well as a sum) as proposed by Kieren (1993). Having a flexible conception of fraction involves understanding fraction as both a multiplicative and an additive entity. Sowder and her colleagues (1998) suggest that teachers need this broad, flexible knowledge of fractions in order to help students solve problems that have a "multiplicative structure," or belong to the multiplicative conceptual field.

It is surprising that additive solution methods such as the concept of unit strategy were used to solve many of the part-whole division problems. The persistence of additive solution methods when solving problems that belong to multiplicative structures has been investigated by many researchers (e. g., Kaput & West, 1994). Resnick and Singer (1993) believe that the preference of additive solution strategies finds its genesis in the early development in school of additive properties of number in contrast to the later development of multiplicative properties. Subjects in this study may have known more about the additive than the multiplicative composition of numbers when relating fractional parts to wholes. Additional research is needed on the reasons why some prospective teachers prefer the use of additive solution strategies.

When making errors to part-whole division problems, subjects may have chosen multiplication to solve part-whole division problems because they had difficulty conceiving the whole, or answer to the problem as a fraction of the part (Vergnaud, 1983). The findings in this study lend credence to Vergnaud's (1983) premise that most of the errors made by subjects making multiplicative errors involve multiplying the whole number by the fraction presented in the problem.

Subjects may have based their choice of solution method on the misconceptions that "multiplication makes bigger" and "division makes smaller" (Fischbein, Deri, Sainati Nello, Sciolis Marino, 1985). These are misconceptions based on the ideas that multiplication is "repeated addition:" and division is "repeated subtraction." Additional research is needed on how the misconceptions of "multiplication makes bigger" and "division makes smaller" are related to prospective teachers' abilities to solve part-whole division problems.

References

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Sowder, J. Armstrong, B., Lamon, S., Simon, M., Sowder, L., & Thompson, A. (1998). Educating teachers to teach multiplicative structures in the middle grades. Journal of Mathematics Teacher Education, 1, 127-155.

Stone, W. E. (1999). Factors that contribute to Teachers' Abilities to Solve Part-Whole Division Problems. Unpublished doctoral dissertation, Boston University, Boston, MA.

Vergnaud, G. (1983). Multiplicative Structures. In R. Lesh, and M. Landau (Eds.), The Acquisition of Mathematics Concepts and Processes (pp. 127-174). New York: Academic Press.