Analyzing pre-service teachers? strategies and representations regarding their solutions to mathematical problems

In mathematics education, it is emphasized that problem solving needs to be extensively considered and class environments where the students share their different problem solving strategies and processes with their peers and teachers need to be created. The purpose of this study is to investigate different solutions, knowledge categories used in collective solution spaces, problem solving strategies and representations used by prospective mathematics teachers during problem solving.This study is a qualitative research where qualitatively analysed data is collected through document analysis. In this research, case study research design is used. Participants of this research consist of fifty senior prospective secondary mathematics teachers in a public university. Research data is collected from written documents. These documents contain participants? solutions to five different mathematical problems. The so-called problems are ?Right Triangle Problem?, ?Handshake Problem?, ?Map Problem?, ?Table Problem? and ?Square Problem?.Each problem is given to participants in different sessions and solutions to these problems are collected in writing. Within the scope of this research the solutions of prospective teachers for each problem are weekly analyzed and in the case of contradiction (incoherent solutions, unreadable writing etc.) the so called participants are interviewed informally by the researcher in order to clarify the solutions. In these informal interviews in order to solve the contradiction, the so called participants are asked to explain their solutions verbally. However these interviews are not recorded and not included within the scope of this resecarh.The solution spaces generated by collecting participants? solutions to these problems, knowledge categories, problem solving strategies and representations are coded in theoritical framework. Data analysis is performed by using thematic coding and content analysis. In the findings of this research, participants identities are protected by assigning nicknames to the participants such as ÖA1, ÖA2 etc.The data analysis reveals that especially for right triangle and handshake problems, knowledge categories in the collective solution spaces created by using participants? solutions are quite comprehensive. The results of this study indicates that prospective teachers have difficulty in solving geometry problems by using algebra.Contrary to the rest four problems, for the table problem, although each prospective teacher produced a limited variety of solutions, within these solutions a large percentage of participants are able to use large variety of knowledge categories.This study reveals that when solving problems most frequently used strategies for right triangle problem, handshake problem, table problem, map problem and square problem are as follows in consecutive order; make a drawing and make a supplemental drawing, using formula, make a drawing, using formula and look for a pattern. Besides, special problem solving strategies which have not been seen in problem solving literature are revealed for some problems.When the representations are examined, symbolic representation is found to be the most frequently used single representation in prospective teachers? solutions for all problems in general. Besides, pictorial-verbal and pictorial-symbolic representation pairs are found to be the most commonly used representation pair among representation pair translations.According to the research findings, in mathematics teacher education process it is suggested for math teachers to be trained in producing different solutions and their consciousness towards their own knowledge base to be expanded. In addition to this, it is suggested for mathematics teachers to be trained in using different solutions approaches, solving problems by relating to different knowledge categories, knowing and using problem solving strategies, and expressing their opinions by using different representations.