An Error Analysis in Division Problems in Fractions Posed by Pre-Service Elementary Mathematics Teachers

AbstractThe present study aimed to make an error analysis in the problems posed by pre-service elementary mathematics teachers about fractional division operation. It was carried out with 64 pre-service teachers studying in their final year in the Department of Mathematics Teaching in an eastern university during the spring semester of academic year 2010-2011. The data collection instrument used in the study was the four-item Problem Posing Test (PPT) developed about fractional division operation. Semi-structured interviews were conducted with 16 pre-service teachers so as to confirm the errors observed in the responses to the PPT and to reveal the reasons behind these errors in detail. Seven types of errors were identified in the problems posed by the pre-service teachers about fractional division. In addition, it was also found that the pre-service teachers overlooked the conceptual aspect of division in the problems they posed about fractional division.Key WordsDivision in Fractions, Problem Posing, Pre-Service Elementary Mathematics Teachers.Fraction concept and mathematical operations with fractions are the main mathematical subjects hard to be understood by students (Alacaci, 2009; Dorgan, 1994; Isiksal, 2006; Kocaoglu & Yenilmez, 2010; Ma, 1999; Smith, 2002; Tirosh, 2000; Yim, 2010; Zembat, 2007). Various methods are used to teach fractions; however, it is very common among teachers to have students start making calculations without setting up an adequate substructure to be used in mathematical operations with fractions (Isiksal; Mack, 1990; Mok, Cai, & Fung, 2008; Rule & Hallagan, 2006; Utley & Redmond, 2008). Students develop limited understanding on fractions as teachers do not include adequate activities in their curriculums, which are likely to ensure students to understand the fractions and mathematical operations with fractions conceptually (Borko et al., 1992; Redmond, 2009; Sharp & Adams, 2002). Researches show that teachers and pre-service teachers, exactly like students, have some difficulties on fraction concept and division in fractions (Ball, 1990; Borko et al., 1992; Carraher, 1996; Ma, 1999; Mok et al., 2008; Redmond, 2009; Sharp & Adams, 2002; Tirosh, 2000; Toluk-Ucar, 2009; Yim, 2010; Zembat, 2007). In their study conducted with 14 mathematics teacher; Post, Harel, Behr, and Lesh (1991) detected that teachers are not able to make any logical pedagogical explanations on mathematical operations with fractions, and that they have superficial knowledge on fraction-related concepts. Isiksal (2006) stated that even though the preservice teachers are able to solve division problems related to fractions, their levels of reasoning for explaining the meanings of such operations are low. Several researchers (Fishbein, Deri, Nello, & Marino, 1985; Graeber, Tirosh, & Glover, 1989) stated that equal division dominant model causes limitations in solving verbal problems bearing fractionrelated points.In the studies conducted to avoid difficulties in fractions and division in fractions, importance of associating fractions and fraction-related operations with tangible situations is emphasized (Arcavi, 2003; Bezuk & Armstrong, 1993; Hasemann, 1981; Keijzer & Terwel, 2003; Li, 2008; Rule & Hallagan, 2006; Steffe, 2002). Linking fractions and fractionrelated operations with the situations faced in real world, makes contributions to the development of reasoning and problem solving skills of students (Cianca, 2006). Problem posing has a very important place in linking fractions and fraction-related operations with the situations faced in real world. Abu-Elwan (2002) states that problem posing makes contributions to establish a relationship between the content of mathematics and daily circumstances, and this is an effective method for developing the students' mathematical thinking.Researches indicate that when teachers and preservice teachers are asked to pose a problem covering division operation in fractions; they, in general, tend to pose multiplication problems or are not able to pose a correct problem (Tirosh, 2000; Utley & Redmond, 2008). …