developing and testing a scale for measuring students ... - EMIS

DEVELOPING AND TESTING A SCALE FOR MEASURING STUDENTS’ UNDERSTANDING OF FRACTIONS Charalambos Y. Charalambous University of Michigan The study reported in this paper is an attempt to develop and test a scale for measuring students’ understanding of fractions. In developing the scale, several criteria proposed in previous studies for examining students’ construction of the notion of fractions were employed. A test consisting of 44 tasks related to the five subconstructs of fractions was developed and administered to 351 fifth graders and 321 sixth graders. A two-parameter Item Response Model was used and the scale developed was analyzed for reliability and fit to the data. The analysis of the data revealed that the scale had satisfactory psychometric properties. Hierarchical cluster analysis also suggested that the 44 tasks within the scale could be grouped into three clusters, according to their level of difficulty. The findings of the study are discussed with respect to teaching and learning fractions.

INTRODUCTION In the mid 1970s Kieren (1976) proposed that the concept of fractions is multifaceted and that it consists of five interrelated subconstructs: part-whole, ratio, operator, quotient and measure. Since then, several researchers have put forth and tested a number of criteria for examining students’ understanding of the different subconstructs of fractions (e.g., Baturo, 2004; Boulet, 1999; Lamon, 1999; Marshall, 1993; Stafylidou & Vosniadou, 2004). Yet, the extent to which those criteria can form a scale for measuring students’ understanding of fractions remains an open question. The present paper aims to address this research gap by developing and testing such a scale. The summary of the relevant literature that follows provided guidelines for the development of a test assessing students’ construction of the concept of fractions. The part-whole subconstruct of fractions is defined as a situation in which a continuous quantity or a set of discrete objects are partitioned into parts of equal size (Lamon, 1999). To develop the part-whole subconstruct of fractions, students should understand that the parts into which the whole is partitioned must be of equal size; they should also be able to partition a continuous area or a discrete set into equal parts and discern whether the whole has been partitioned into equal parts. In addition, they should develop the idea of inclusion or embeddedness (i.e., the parts of the numerator are also components of the denominator) and understand that as the number of parts into which the whole is divided increases, their size decreases (Boulet, 1999). Finally, a full understanding of the part-whole subconstruct requires that students develop unitizing and reunitizing abilities (Baturo, 2004) which allow them to reconstruct the whole based on its parts and repartition already equipartitioned wholes. The ratio subconstruct considers fractions as a comparison between two quantities. To grasp the notion of fractions as ratios, students need to construct the idea of relative amounts (Lamon, 1999). They should also comprehend the covariance-invariance 2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 2, pp. 105-112. Seoul: PME.

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Charalambous property, according to which the two quantities in the ratio relationship change together so that the relationship between them remains invariant. The “orange juice task” (Noelting, 1980), in which children are asked to specify which mixture of orange juice makes the juice the most “orangey” has been widely employed to examine whether students have developed these ideas. In the operator interpretation of fractions, rational numbers are regarded as functions applied to some number, object, or set (Behr et al., 1993; Marshall, 1993). One could think of the operator as an application of the numerator of the fraction to the given quantity, followed by the denominator quantity applied to this result, or vice-versa. Alternatively, one could consider the operator as a transformer that changes the size – but not the shape – of a figure or changes the number of elements in a set of discrete objects (Lamon, 1999). To master this subconstruct, students should be able to identify a single fraction to describe a composite multiplicative operation (i.e., a multiplication and a division), and relate inputs and outputs (e.g., a 3/4 operator results in transforming an input quantity of 4 into 3) (Behr et al., 1993). According to the quotient subconstruct, fractions are the result of division, in which the numerator defines the quantity to be shared and the denominator defines the partitions of the quantity. To develop an understanding of this subconstruct, students need to be able to relate fractions to division and understand the role of the dividend and the divisor in this operation. Mastering the quotient subconstruct also requires that students develop a sound understanding of partitive and quotitive division (Marshall, 1993). The measure subconstruct conveys the idea that a fraction is a number; this subconstruct is also associated with the measurement of the distance of a certain point on a number line (Marshall, 1993; Stafylidou & Vosniadou, 2004). Several researchers have argued that, despite their relative understanding of the aforementioned subconstructs of fractions, many students appear not to fully understand that fractions are an extension of the number system (Amato, 2005; Hannula, 2003). Hence, Lamon (1999) refers to a qualitative leap that students need to undertake when moving from whole to fractional numbers. A robust understanding of the measure notion requires that students comprehend that between any two fractions there is an infinite number of fractions. They should also be capable of locating a fractional number on a number line and identify a fractional number represented by a point on a number line (Hannula, 2003). In this context, the present study sought answers to two research questions. First, to what extent can the criteria proposed above help develop a scale with good psychometric properties to measure students’ understanding of fractions? And second, provided that such a scale can be developed, do tasks that measure different subconstructs of fractions differ in their level of difficulty and their contribution to the development of the scale? Answers to these questions are important both for teaching and assessing fractions.

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Charalambous THE DEVELOPMENT OF THE TEST To address the research questions, a test on fractions (available on request) was developed taking into consideration the preceding literature review and the curriculum used in Cyprus where the study was conducted. Table 1 presents the specification table that guided the construction of the test and the tasks employed to examine students’ performance on each of the five subconstructs of fractions. Subconstruct Part-whole Ratio Operator Quotient Measure

Tasks 1-6, 20, 22-28, 39 9, 10, 13-15, 25, 30, 31, 38 8, 11, 12, 18, 44 7, 29, 40, 41, 43 16, 17, 19, 21, 32-37, 42

Table 1: Specification table of the test used in the study The first six tasks of the part-whole subconstruct asked students to identify the fractions depicted in discrete or continuous representations. Tasks 22, 27, and 39 examined students’ unitizing and reunitizing abilities (Baturo, 2004), whereas the remaining tasks addressed common misconceptions related to the notion of embeddedness, the requirement that the parts be of equal size and the inverse proportional relationship of the size and the number of parts into which a unit is partitioned (Boulet, 1999). Nine tasks were used to examine the development of the notion of fractions as ratios. Those included expressing the relative size of two quantities using a fraction (tasks 9-10), identifying fractions as ratios (task 25) and comparing ratios, based either on quantitative (13-15) or qualitative information (30-31) (the latter five tasks were related to the “orange juice problem”). Task 38 referred to the widely cited problem of boys and girls sharing different numbers of pizzas (Marshall, 1993). The tasks used to examine the operator notion of fractions asked students to specify the output quantity of an operator machine given the input quantity and a fraction operator (tasks 11-12). The remaining three tasks required students to decide the factor by which the number 9 should be multiplied to become equal to 15 (task 8), use a fraction to describe a composite operation (task 18), and specify the factor by which a picture reduced by 3/4 should be enlarged to restore it to its original size (task, 44 Lamon, 1999). Three of the tasks used to measure the quotient subconstruct (tasks 29, 40, and 41) examined students’ ability to link a fraction to the division of two numbers and identify the role of the dividend and the divisor; the remaining two tasks of this category were related to the partitive and quotitive interpretation of division (tasks 7 and 43, respectively). Consistent with previous studies (Hannula, 2003; Lamon, 1999; Stafylidou & Vosniadou, 2004), the tasks of the measure subconstruct examined students’ performance in identifying fractions as numbers (tasks 21, and 32-34) and locating them on number lines (tasks 16-17, and 35-37). Task 19 required students to find a fraction that would be located between two given fractions; in task 42 students were required to identify among a number of PME31―2007

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Charalambous fractions the one that is closer to the number one. It is also important to note that in Cyprus there is a national curriculum used in all elementary schools. Like curricula in other educational settings (Amato, 2005; Lamon, 1999), the Cypriot curriculum places more emphasis on the part-whole interpretation of fractions. The remaining subconstructs are mainly taught in fifth and sixth grades.

METHODS The tasks included in the tests were content validated by three experienced elementary teachers and two university tutors of Mathematics Education. Based on their comments, minor revisions were made to the test. The final version of the test was administered to 351 fifth graders and 321 sixth graders (316 boys and 356 girls). To avoid a single test period of undue length, the test tasks were split into two sub-tests, which were administered to students over two consecutive schooldays. Students had eighty minutes to work on each sub-test. To examine whether the tasks used in the tests could form a scale for measuring students’ understanding of fractions an Item Response Theory (IRT) model was fit to the data. In the scales developed using IRT models, the task parameters ( β : difficulty of the task and α : item discrimination) do not depend on the ability distribution of the examinees and the parameter that characterizes the examinees ( θ : ability) does not depend on the set of the test tasks (Hambleton, Swaminathan, & Rogers, 1991). The data were analyzed by using BILOG-MG (Zimowski, et al., 1996). First, the fitting of the data to a single-parameter model was compared to that of a two-parameter model. Whereas in a single-parameter model all tasks contribute equally to the development of the scale, a two-parameter model allows the tasks to differ in their discrimination parameter. Tasks with higher values of discrimination are more useful for developing a measurement scale, since they are better at separating examinees into different ability levels (Embretson & Reise, 2000). Finally, hierarchical cluster analysis was used to cluster the tasks into different groups, according to their level of difficulty.

FINDINGS The analysis of the data revealed that both a single-parameter and a two-parameter model fit the data well (reliability indices=.88, and .91, for the one- and two-parameter models, respectively). However, since the difference of the -2loglikelihood index of the two models was statistically significant (x2=641.35, df=43, p