The purpose of this study is to examine pre-service middle school mathematics teachers' (mis)conceptions related to definitions, classifications, and inclusion relations of convex quadrilaterals through a case study research design. The participants of the study were 20 pre-service middle school mathematics teachers who attended a must course, "The Methods of Teaching Mathematics" in a public university in Ankara, Turkey. Purposive sampling strategy was used to select the participants. The data were collected through an achievement test before and after taking the methods of teaching mathematics course. The findings of the study revealed that pre-service teachers had misconceptions in hierarchical classification of quadrilaterals, dual inclusion relations (e.g., rhombus-trapezoid), and difficulty in formulating minimal definitions of quadrilaterals before taking the course. However, the findings also indicated that the course had a positive impact for eliminating those misconceptions and supporting their conceptions. Several suggestions were made based on these findings.

As a part of a long-term design-based research project, this study aimed to investigate the mean differences among the pre-, post-and retention tests of seventh grade students regarding their proportional reasoning skills. The data of this paper were collected from 30 seventh-grade students enrolled in a public school in Ankara, Turkey in the first and second semester of 2017-2018 academic year. Participants of the study were engaged in a teaching experiment aimed to develop their meaningful understanding of ratio and proportion unit through authentic activities. The intervention prolonged 30 lesson hours and Proportional Reasoning Test was applied to the students at the beginning of the instruction as pre-test, at the end of the instruction as post-test, and 5 months later as retention test. A one-way repeated measures of ANOVA was conducted to compare scores on the Proportional Reasoning Achievement Test at Time 1 (pretest), Time 2 (posttest) and Time 3 (retention test). Findings revealed that there was a significant difference among proportional reasoning skills. More specifically, there was a statistically significant increase in Proportional Reasoning Achievement Test scores from Time 1 to Time 2 but the difference between Time 2 and Time 3 was not statistically significant. Findings revealed the significant effect of intervention on students' proportional reasoning skills and retention of those skills. Based on the findings it could be deduced that designing authentic interventions could have an impact on enhancing and enduring students' proportional reasoning skills.