Investigation of the instrumental and relational understanding of elementary school mathematics teachers in terms of the meaning of fractions

The process of constructing a mathematical concept in a meaningful way has an important place in both the permanence of knowledge and making sense of other mathematical concepts. The instrumental and relational understanding introduced into the literature by Richard R. Skemp focuses on "mathematical understanding". Relational understanding may be defined as "Mathematically knowing what to do and how to do it" where instrumental understanding may be defined as "unreasonable rules" (Skemp, 1972; 1994; 2002). In order to determine what "mathematical understanding" is in the sub-learning area of fractions, which contain many misconceptions as in every learning and sub-learning areas, the problem sentence of this study was determined as "How are elementary school mathematics teachers' relational and instrumental understanding about the meanings of fraction in terms of instruction?" Since this study aims to explain and analyze relational and instrumental understandings in depth, a case study, one of the qualitative research methods, was adopted. The participants of the study consist of 6 elementary school mathematics teachers working for the Ministry of National Education. The data collection tool was prepared by the researchers and includes 10 scenarios consisting of solutions that contain misconceptions about the five meanings of the fraction. The data obtained from the interviews conducted online with the participants were divided into themes and sub-themes in the light of the content analysis method and the framework created by Skemp. The answers of the two scenarios created for each sub-meaning of the fraction were examined by taking into account the professional experiences of the teachers. According to the findings of the study, regarding the relational understanding; teachers used subthemes of questioning the logic of operations, having the subject applied on a different problem, relating / narrating with daily life, associating with a different topic / discipline, using concrete material, a different way of representation / notation, and the definition of the concept. In the context of instrumental understanding, they used the subthemes of saying the rule, using a rule-based memorization strategy, telling step-by-step operations, using the solution of a known problem / using a prototype, asking questions to lead to the correct solution, writing an equation, and using a confirmation sentence. It was observed that teachers' understanding types also changed according to the meanings of fractions. For the scenarios presented regarding the meanings of part-whole and measurement, teachers generally use relational and instrumental understanding together; In terms of ratio, division and operator, it was determined that sub-themes representing instrumental understanding in general are used more than sub-themes representing relational understanding. Another result obtained within the scope of the research is that the teachers cannot explain some of the misconceptions for the part-whole and measurement meanings of the fraction, in addition to this, they can explain the reasons of the misconceptions in the scenario situations presented regarding the meanings of ratio, division and operator. As the professional experience increased in terms of division meaning, the situation of solving the questions given in the scenarios decreased.