Abstract:

The National Council of Teachers of Mathematics (2000) states, “effective teaching requires knowing and understanding mathematics, students as learners, and pedagogical strategies” (p.17). Thus, a complete understanding of a mathematical topic is needed to have the ability and knowledge to teach a topic (Ball et al., 2008). Ball (1990) adds that teachers’ knowledge positively relates to the student’s understanding and achievement, but it is necessary to understand the structure of this knowledge as it affects teaching (Ponte, 2012). Many studies examined teacher knowledge regarding mathematics topics such as addition, subtraction or multiplication (Carpenter, 1988; Ball, 1990; Leavitt, 1998; Turner, 2007; Rowland, 2007; Turner & Rowland, 2008), fractions (Borko et al., 1992; Ball, 1990; Isiksal-Bostan & Cakiroglu, 2008; Tekin-Sitrava,2019), algebra (Girit-Yildiz & Akyuz, 2019; Li, 2007) and measurement (Livy et al.,2012; Dogan Coskun et al.,2018). However, it can be said that studies on teacher knowledge of division operation (DO) are limited. DO is first introduced in primary school and is vital due to its strict relationship with other operations. Researchers also emphasized the importance of DO for developing conceptual mathematical knowledge and performing higher cognitive tasks (McCallum & Schmitt, 2011; Ball, 1990; Yorulmaz, 2018). In light of this information, it is critical to investigate pre-service primary school teachers’ (PPST) knowledge of division operation as they are prospective teachers who will teach in primary schools. Hence, this study aims to investigate the nature of PPSTs’ mathematical knowledge regarding how they defined DO and adapted it while writing and solving DO problems based on the Knowledge Quartet (KQ) model (Rowland et al., 2003). The study was conducted with three PPSTs in Turkey. All PPSTs completed all required mathematics method courses and were in the final year of the teacher education program. The data was gathered through a test including five open-ended questions on DO. Written responses of PPSTs describing DO, explaining its meanings, and accordingly writing and solving division problems while adapting their knowledge of the meanings were obtained. The data were examined using the Foundation dimension of the KQ model. The findings showed that the knowledge of PPSTs regarding DO could be organized under the codes of overt display of subject knowledge and concentration on procedure based on the Foundation dimension. According to overt display of subject knowledge, the knowledge of PPSTs regarding DO is limited as they used ill-defined descriptions while explaining DO. Also, their knowledge of DO is limited to only fair sharing meaning, and they do not consider other meanings. For instance, one PPST stated, “DO is to break an object into pieces, e.g., breaking a book into two pieces, dividing, and sharing.” Another PPST said, “DO breaks a whole into equal or non-equal pieces,” and ignored the necessity of equal-sized groups. Further, when PPSTs wrote DO problems, they could write only fair sharing problems such as “A teacher has 12 pencils and wants to share them with her three students. How many pencils will each student get?” Additionally, according to concentration on procedure, their knowledge to represent DO is limited to conventional methods. For example, when they solved the problems they wrote, instead of different representations such as drawing, all PPSTs used the standard algorithm.

Keywords:

Teacher knowledge, pre-service teachers, division operation.