The role of elementary and secondary school students' understanding of concepts and their usage of concepts-procedures in mathematical problem solving process

The aim of this study is to determine the role of elementary and secondary school students' understanding of the concepts and their usage of concepts-procedures in mathematical problem solving process. To fulfill this aim, we investigate the relationships among students' "understanding of the central concepts relevant to the problem", "usage of concepts-procedures" and "reaching a conclusion" during the mathematical problem solving process.Phenomenology, as one of the qualitative research designs, was used in our research. The participants of the study were selected from Edirne and Tekirdağ (Turkey); three were 8-grade students studying for SBS, and three were 12-grade students studying for YGS and LYS.The data for the study were collected through six interviews with each of the students between November 2009 and October 2010. However, most of the data which were appropriate for the aim of the study were gathered through three main interviews in mathematical problem solving environments, conducted in average 3-5 week periods between March 2010 and June 2010. The tools used in data gathering process were "interview questions forms" included some mathematical problems from SBS and ÖSS in 2008, 2009, and the papers that the students produced to solve problems during the interviews.In data analysis process, we used "descriptive analysis" and "content analysis", which are certain approaches in qualitative data analysis. In content analysis, we used qualitative research techniques used in grounded theory; such as, open, axial and selective coding.With some additional results, our main result of the study is that though the participant students either from an elementary school or a secondary school have wrong or deficient understanding of the central concepts in the problem or are not able to make explanations regarding the definitions of some central concepts relevant to the problem, these facts do not seem obstacles for them to use concepts-procedures and reach the right answer, in many of the problems. When "usage of concepts-procedures" of the participants who reached the wrong answer, or could not decide between two answers, or could not find any answers at the end of problem solving process were examined, it was established that sometime during the problem solving process, these participants either used the inappropriate concept/procedure, or used the appropriate concept/procedure in a deficient or wrong way, or did not use any concepts at all.The findings also demonstrated that the main reason for a participant who solved a problem correctly, though s/he does not know some of the central concepts in the problem well is the lack of the need for thinking about the meanings of the concepts, knowing their definitions and using the definitions directly while solving the problem. In addition, it was observed that in both elementary and secondary school participants' "reaching the answer" in problem solving process, generally their "usage of concepts/procedures" played a more decisive role rather than their "understanding of the concepts", and that in most of the problems they used -not predominantly conceptual solutions- but mostly predominantly procedural solutions. In the light of the findings of the research, some suggestions were made for elementary and secondary school mathematics teachers, the specialists preparing the mathematical exam problems and authors of mathematics textbooks, and mathematics educators.Keywords: Mathematical concepts, concept definition, problem solving, understanding of the concepts, concept-procedure usage, conceptual knowledge, procedural knowledge.