Pre-service secondary mathematics teachers' (PSMTs) understanding and ability of constructing a proof is not only important for their own learning process, but also important for these PSMTs to help their future students learn how to do proofs. Therefore, this study is focused on and explains PSMTs' behaviors that they revealed throughout the proving process of a proposition. In this qualitative case study, the participants were fifteen volunteer PSMTs from a public university in Turkey. The participants were given a proposition which they were asked to think aloud and prove it on the blackboard. The findings of this study show that PSMTs' behaviors and thoughts regarding the given proposition were limited. In detail, PSMTs had difficulties in application of mathematical language and notations, understanding the meaning of the given proposition, knowing where to get started on a proof, using examples efficiently, using appropriate and efficient methods to construct the proof, and defining logical structures of the proposition to construct the proof.

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Augmented reality (AR) helps three dimensional, virtual objects to be viewed, interactively, in a real-world setting. AR technology is used in many fields such as medicine, advertisement, military, industry, and increasingly in education. AR has an important role in concretizing educational platforms and achieving permanent learning. This study aims to evaluate studies on the use of AR in mathematics education through meta-synthesis. 20 research articles, two master's theses, and two PhD theses, published between 2010 and 2021, centered on the use of AR technology, utilizing either qualitative or mixed methodology were analyzed in this study. The databases used for selecting the texts including Google Scholar, EBSCO, Education Resources Information Center, Elsevier Scopus, ProQuest Dissertations and Theses Full Text, Springer Link, Taylor & Francis Online, TUBITAK ULAKBIM Dergipark, YOK Thesis Centre. The studies included in the meta-synthesis have been grouped according to their descriptive properties, research methods, data gathering tools, data analysis, the mathematics subjects they focus on, their validity and reliability, and the conclusions and suggestions they present. The similarities and differences between these studies have been compared, and in accordance with the nature of meta-synthesis they have been categorized into themes for synthesis. Parallel to technological progress, it was observed that the use of AR in mathematics education has tended to increase over the years. In conclusion, most of the studies on AR in mathematics education focus on increasing spatial ability. It was seen that the visualization and concretization aspects of AR were often used. This meta-synthesis offers suggestions for future researchers who are interested in the use of AR in mathematics education.

Proof facilitates conceptual and meaningful learning in mathematics education rather than rote memorization. In this study, incorrect theorems and proofs are used to assess secondary school pre-service mathematics teachers' proof assessing skills. Using the case study method, the study is conducted on preservice mathematics teachers studying at the Department of Mathematics Education. There were eight preservice mathematics teachers selected from each grade, resulting in 32 participants in total. A semistructured proof form containing 13 questions was used to collect data, which was analyzed using content analysis. As the analysis reveals, pre-service mathematics teachers are highly likely to make incorrect decisions regarding theorems and proofs, and the margin of error is unaffected by grade level. Moreover, pre-service mathematics teachers tend to use proving terms incorrectly and, at times, are unable to differentiate between terms that are commonly used in proving. The pre-service mathematics teachers are believed to have learned proofs by rote rather than understanding how proofs work. With the help of interviews and tests created for different proof methods, it has been suggested that pre-service mathematics teachers should be tested on their proof evaluation skills in more detail.