Investigation of pre-service secondary mathematics teachers' reasoning errors reasoning deficiences and reasoning gaps within the context of proof

Mathematical proof is the most important, component that enables the abstract terms in mathematics like definition, axiom, theorem, symbol, sign to have a meaning and the relationships of mathematics' main components between each other to reveal. In this regard, proof, rather than showing the validty of a mathematical result or being a basic mathematical operation, is a process that includes intensive cognitive operations like association, abstraction, anticipation, generalisation and reasoning. Reasoning, is a tool that enables abilities like discovering, developing strategies and making hypotheses to be used during proof process. Therefore proof and reasoning which are in the center of mathematics are handled as a whole by lots of researchers. Mathematical reasoning enables the individuals to make consistent and reasonable decisions on the results they obtained and it is employed in the proof process most. The development of a proper proof understanding occurs only when individuals gain various mental thinking ways. The students develop some strategies through making reasonings in the proof process. But sometimes they generate invalid proofs after falling into a systematic error as a result of these strategies. These errors are called as reasoning errors. Especially the pre-service mathematics teachers at the undergraduate level, because they meet with the abstract and axiomatic side of the mathematics, have difficulties in comprehending the mathematical concepts, the relationships between the concepts, definitions and theorems, in generaing proofs through interpreting those and in making right reasonings. That the students develop right reasoning in proof process is significant in terms of undertstanding mathematical relations. Therefore in this process determining the students's wrong reasonings and conducting studies aimed at investigating how these errors developmentally altered during their university education is important. Moreover detecting the points focusing on reasoning during proof process in the courses where proofs are taught intensively at the undergraduate level and that have critical roles in developing right reasonings and analyzing the reasoning errors made during this process are also important. This study aims to determine the pre-service high school mathematics teachers' reasoning errors, reasoning deficiences and reasoning gaps in proof process, to investigate how these errors alter developmentally during their university education and to analyze deeply to what extent do the some courses attended by the students during this process effect developing mathematical reasoning, determining appropriate strategies and reasoning errors. In line with this purpose, a developmental research was designed to determine how reasonings of the pre-service mathematics teachers at different grade levels (1., 2., 3., 4. and 5. grade) changed during their university education. The data was collected through tests for determining reasoning errors, clinical and semi-structured interviews, course observations, fieldnotes and audio-video recordings obtained during proof process. The data obtained through observations was analyzed with the aid of a observation form. For analyzing the data from the tests for determining reasoning errors and from clininical interviews, the reasoning errors determination scale was employed. In consequence of the study it was found that the pre-service teachers at different grades have similar reasoning errors, reasoning defficiencies and reasoning gaps in proof process. Besides, reasoning errors, deficiencies or gaps showed significant alteration in terms of frequency according to the grade levels. Reasoning errors like considering a result obtained from a special case as a proof and starting with wrong assumptions, reasoning defficiencies like inability in using mathematical language, failure in starting proof or in continuing a already started one, reasoning gaps like conflict between the written proof steps and the obtained result, not stating the justifications for the utilized suggestions were seen to decrease towards higher grades significantly. On the other hand reasoning errors, deficiencies or gaps like starting with the result, ignoring the definition range of the related mathematical statement, not recognizing that the Contrapositive of a theorem is its equivalent were observed to resist. Moreover, at the courses where proofs are taught intensively, components and indicators focused on reasoning and reasoning errors differentiated at the dimensions of mathematical language and systematization, focusing on student reasoning and proofing techniques. In accordance with the results obtained from the study, some recommedations on what the lecturers should pay attention to for enabling the pre-service teachers to develop right reasonings in the courses where proofs are taught intensively at the undergraduate level are given.