The aim of the study was to analyze and identify students’ most frequent response types in translating among various representations of fraction concepts. In this manner, the research question was formulated as "What are the most frequent responses that sixth, seventh, and eight grade Turkish public elementary school students made in the Translations among Representations Abilities Test (TRAT)?". Mathematics is a field that is composed of abstract concepts, which are found to be very hard to grasp by many school children. In fact, the effective learning does not require direct learning of abstraction. Rather, external representations and embodiments of the mathematical concepts to be learned can be used by learners (Cramer, 2003). As Cramer states, representations of mathematical ideas can vary in complexity. By this common idea, the NCTM standards (NCTM, 2000) emphasize the importance of using multiple representations in mathematics instruction. Similarly, the reform-based elementary mathematics curriculum in Turkey underlines the importance of developing students’ abilities in problem solving and communication by using multiple representations (MEB, 2005). The concept of fraction is considered as one of the most abstract, complex, and problematic issue children encounter in their pre-elementary years (Aksu, 2001; Behr, Lesh, Post, & Silver, 1983). In order to develop fraction sense, it is claimed that most children need to practice the use of multiple representations (Pape & Tchoshanov, 2001) as it is a convenient concept to be interpreted in many ways. Since this concept is foundational in mathematics education, it is studied starting from the earliest grades of elementary education. Therefore, studying students’ abilities related to fractions could provide important insights for the improvement of mathematics education. Lesh, Post, and Behr (1987) suggested that mathematical ideas can be represented in five different modes: (1) manipulatives – like fraction bars, paper foldings; (2) pictures – like number lines, region, discrete objects models; (3) real-life situations – where knowledge is organized from real life; (4) verbal symbols – specialized sublanguages related to domains like logic, etc. (5) written symbols – specialized sentences and phrases. These different representations in modeling fractions concept can maximize learning as well as will help students to develop deeper understanding (Behr et al., 1983). Lesh et al. (1987) presented a model representing the translations among different representations. The Lesh model stressed that not only are these distinct types of representations are important in their own rights, but translations among them also are important. In this study, a model including five types of representations, partly reflecting the Lesh Translation Model, was identified. Through a more general classification the model mainly consists of symbols, pictures, and real-life situations, which were the main concerns of the study.

The sample for the study consistent of 1456 Turkish middle grade students; 441 sixth grade, 510 seventh grade, and 505 eighth grade students. There were 781 male and 675 female students. Stratified cluster random sampling and convenience sampling were used to obtain a representative sample of the population. In this study, the participants completed the ‘Translations among Representations of Fractions Test’ (TRFT), which was developed by the researchers. For the present study, 12 items of the TRFT were considered. In these items, students were asked to translate basic types of representations into the expected ones. Students’ responses which were coded as incorrect (0) were analyzed based on the representation types. Each item were examined for all subjects and made a frequency analysis. By this frequency analysis, most common incorrect response types were identified and given with their percentages. The Cronbach’s alpha internal consistency coefficient of the TRFT was calculated to be .90 with 1456 students, indicating a high reliability.

The findings were presented according to the representations types which were number line, discrete objects, region, and symbolic models. There were 3 items that involved translations from different types (symbol, discrete objects, region) to number line representations. In general translations tasks that involved the number line representations were among the most difficult ones, which had relatively low percentage of correct responses (39% to 47%). For instance, in one of the items, when students were given the fraction of 10/15 in discrete model representation, students first marked 15 (denominator) on the number line and partitioned it to 15 parts starting from 0. Then they identified the 10th marker line to represent the fraction 10/15. In the second type of response students did not mark a point on the number line, but attempted to show the fraction by arrows. The arrow below the number line was used to represent the denominator (15) and the arrow above indicated the numerator (10). Both types of responses were the most frequent incorrect ones in all questions involving a translation to number line. There were two items that included translations from different types to discrete model. Noticeable responses with errors involved translations from number line model, especially due to the misinterpretation of the fraction given on the number line. Most problematic aspect of the responses regarding the region model was due to the representation of improper fractions. For instance, when the fraction 9/4 were given in the symbolic form, more than half of the incorrect responses (60%) included one whole (drown usually in a rectangular shape) divided into 9 parts and 4 parts were shaded. In another item, when they were given the fraction 2 located on a number line, a common incorrect response of students represent only the fractional part of the expression (10%) There were three items that include translations from different types (number line, region, and discrete objects) to symbolic representations. Typical problems were linked to the misinterpretation of the other representations. For instance in an item, the fraction 1 was located on the number line and students were asked to represent it in symbolic form. There were students who attempted to write only the fractional part (25%). In another item, when the fraction 1 was given in region model and students were asked to write it in symbolic form, about 15% of the incorrect responses were 8/12, where 12 denoted the number of 1/6 pieces in total, and 8 represent the shaded parts in the question. Conclusions: The results of the analysis revealed an overall lack of ability in translating among various types of representations. Students’ most frequent incorrect response types were those including number line models and improper fractions. In the following section, possible reasons of incorrect responses were discussed with comparisons of related studies in four categories. Number Line Model: It was seen that there was a tendency to consider the fraction numeral as two separate whole numbers. In representing a proper fraction, students tend to take the segment between 0 and the denominator as unit and partition it like in region model instead of partitioning the segment between 0 and 1. This finding is consistent with the findings of many other research studies (Behr, et al., 1983; Ni, 2000). They stated that since the majority of students’ experiences had been part-whole interpretation of fraction in a region context, they treat the whole number line as the unit instead of the length between 0 to 1. Region Model: Although, the part-whole interpretation is usually introduced early in the elementary school curriculum, many students had given incorrect responses to those items. Many students, for example, represented as to a region model. The reason most probably stem from the models which represent improper fractions. Teachers and textbooks, in Turkey use visual representations of fractions mostly with proper fractions, in which case only one whole would be needed at all times. This finding with Turkish sample is consistent with the findings of other studies, such as Behr,et al.(1983) and Wu (2002), who also found out that students usually have problems in representing improper fractions to region models. Discrete Objects Model: Although discrete objects model was not a familiar type of representation used in the fractions concept for Turkish students, number of students’ acceptable responses was relatively high in terms of translating from different types of representation to discrete objects model. Most of the students’ errors in the translations involving discrete model usually originated from their lack of interpretation of other representations, such as number line. Symbolic Representation: In schools, since most of the instructional attention has been paid to the symbolic representations and little work is being done about relating it to different representations, students have problems in translating symbolic representations to others. In fact, they tend to write symbolic forms of the fractions even if it was not the required model in the TRFT tasks. Similar patterns of student responses were reported by Behr et al. (1983). Implications: Turkey is in the process of reforming school curricula. The results of the study will enable mathematics educators to assess what necessary skills students acquired and what still need to be mastered. In this sense the findings might helpful in planning curriculum and improving the instructional practice. To the results of the most common incorrect responses, since the most problematic items were those involving number line models, teachers should pay attention to this model, not only in fraction concepts, but earlier in developing the number sense. Teachers should notice more examples representing part-whole interpretation (with distinct type of region models), particularly with improper fractions.