Investigating high achieveing middle grade students' and their mathematics teachers' proving skills and their choices of arguments

Mathematical reasoning and proof is an innate ability. Developing reasoning skills and the ability of proving constitute such an important step in facilitating successful school life and real-life activities. In recent years, the importance of proof in mathematics education has been recognized widely. Thus, reasoning and proving have become one of the important objectives of mathematics education. Current educational reforms movements have emphasized the importance of proofs at all grade levels from pre-kindergarten through high school. Therefore, the purpose of this study was to investigate high achieving middle grade students' and their math teachers' ability to prove and their choices of argument types. This study included six 7th grade and six 8th grade students, who were studying at a public school located in a district of Tokat province, and 4 mathematics teachers during the second semester of the 2018-2019 academic year. Since the aim of this study was to examine the ability of students' and their mathematics teachers' proving and their preferences of argument types, this study was designed as a case study, which is one of the qualitative research methods. In line with the aim of the study, participating students and mathematics teachers were asked to prove the truthiness of mathematical statements and to evaluate the level of convincingness of the provided arguments in the learning area of numbers, which constitutes one of the most important learning areas in mathematics curriculum. The participants of the study were selected based on different criteria such as academic achievement, end-of-year GPAs, gender, volunteering to participate for students; working experiences, gender and volunteering to participate for teachers. Twelve students and four math teachers were selected based on aforementioned criteria by using purposeful sampling method. In order to determine the ability of proving, the Proving Ability Test (PAT) form, designed by the researcher, was administered to the students and mathematics teachers. After a while administering the PAT form, the Argument Assessment Test (AAT) form, designed by the researcher, was administered to the students and mathematics teachers to determine their preferences of argument types. Later, semi-structured individual interviews were conducted in order to gather more detailed data with the students and the teachers. The PAT and AAT forms as well as individual interview records constituted the main data sources for this study. The findings of the study revealed that the participating students constructed experimental (empirical) arguments for validating the truth of the given mathematical statements the most. In addition to the empirical arguments, the students also constructed narrative-verbal arguments. While empirical arguments were mostly constructed by the participating students, algebraic-formal arguments were the least used method. Visual arguments, on the other hand, were constructed by none of the participating students. During evaluating presented arguments, it could not be concluded that the participating students preferred experimental (empirical), narrative-verbal, visual or algebraic-formal arguments to one another. Other findings of the study demonstrated that mathematics teachers constructed algebraic-formal arguments while validating the mathematical statements the most. In addition, most of the teachers stated that they would assign the least points to the experimental (empirical) arguments and the most points to the algebraic-formal arguments. The overall findings of the study concluded that the students constructed experimental proofs while proving, which align well with the results of other studies in the literature. However, they preferred narrative-verbal, visual and algebraic-formal arguments as the most convincing arguments while evaluating the arguments presented to them. Mathematics teachers, on the other hand, constructed algebraic-formal arguments and evaluated the arguments constructed as algebraic arguments with a higher score. In addition, it could be concluded that students and the mathematics teachers who are responsible for their mathematics education do not have much similarity in their proving skills. Their argument evaluation skills, on the other hand, are half-similar. Based on these results, suggestions will be provided for mathematics teachers and researchers.