An investigation into misconceptions of primary school mathematics pre-service teachers' on the derivative of multi variable functions subject

If a function has more than one variable, this type of function is called a multivariate function. Multivariate functions are used in all areas of life. For example, the income from each of the products sold in a stationery (revenue from book sales x, income from book sales y, income from photocopy printing z, etc.) individual independent variable, the total revenue dependent variable, refer to multivariate function in the total revenue function. In order for this concept, which is closely related to daily life, to be used in implementation, the concept of derivatives must be learned in multivariate functions. However, the misconceptions that may occur about these concepts negatively affect the realization of meaningful learning. It is of great importance to identify the misconceptions about this subject in order to prevent the negativity that may be encountered in learning. In this research, it is aimed to determine the misconceptions of elementary mathematics teaching undergraduate students about the derivative of multivariate functions. For this purpose, the misconceptions of mathematics preservice teachers regarding some definitions and concepts about derivative of multivariate functions, the concept of derivatives by partial derivative and directional derivative and the geometric applications of derivatives in multivariate functions were examined. The research is based on cross-sectional survey designs from quantitative research patterns. In the study, the sample selected from the population was selected from nonprobability sampling methods through convenience sampling. The study was carried out with 81 preservice teachers who were studying in the Department of Elementary Mathematics Teaching in the spring semester of the 2020-2021 academic years. In the research, a data collection tool called "Test for Determining the Misconceptions about the Derivative of Multivariate Functions" was used, consisting of 18 open-ended questions developed by the researcher. The questions in the test used are aimed at exposing the misconceptions of preservice teachers regarding the concepts involved in this subject, rather than measuring their knowledge of the derivative of multivariate functions. The data collected from the sample in the study were first analyzed by descriptive analysis, then inferential analysis and analyzed with the IBM SPSS Statistics 23.0 package program. As a result of the research, it was determined that preservice teachers had various misconceptions about the derivative of multivariate functions and the possible causes of this situation were examined. Also, the research emphasized the misconceptions that cause students to reach the correct conclusion. Accordingly, it was determined that 40.7% of preservice teachers had misconceptions regarding definitions and concepts related to derivatives of multivariate functions, 45.7% related to partial derivative concept, 29.6% related to directional derivative concept and 19.8% had misconceptions regarding the geometric applications of derivatives in multivariate functions. It was seen that many preservice teachers defined the concept definition of derivative by partial derivative and directional derivative as intuitive and noted their deficiencies in the field language. Moreover, it was found that the preservice teachers tried to apply their conception of a particular question in similar questions encountered. It has been observed that the lack of knowledge or misconceptions in the preliminary knowledge of some preservice teachers leads to the formation of new misconceptions. It is thought by the researcher that the reason for the misconceptions of preservice teachers is that they contain many abstract concepts in the content of derivatives in multivariate functions, lack of knowledge or misconceptions in the preliminary knowledge of students in new learning situations, questions that will reveal the misconceptions of the students during the teaching phase and the use of mathematical language are not taken care of. Finally, some recommendations have been made in line with the results obtained. Keywords: Analysis III, misconceptions, partial derivative, directional derivative, multivariate function.