A module M is called ss-supplemented if every submodule U of M has a supplement V in M such that U \ V is semisimple. It is shown that a …nitely generated module M is ss-supplemented i¤ it is supplemented and Rad(M) Soc(M). A module M is called strongly local if it is local and Rad(M) is semisimple. Any direct sum of strongly local modules is sssupplemented and coatomic. A ring R is semiperfect and Rad(R) Soc(R R) i¤ every left R-module is (amply) ss-supplemented i¤ R R is a …nite sum of strongly local submodules.

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The purpose of this study was to examine the relationships between metacognitive awareness, mathematical literacy self-efficacy beliefs and problem solving skills, and the effects of metacognitive awareness and mathematical literacy self-efficacy beliefs on problem solving skills. This study used a correlational survey design, and was conducted in the Faculty of Education of Ondokuz Mayıs University in the 2014/15 spring semester. The sample group, selected used convenience sampling, consisted of a total of 171 prospective teachers. Data were collected using Metacognitive Awareness Inventory (Akın, Abacı & Çetin, 2007), the Problem Solving Inventory (Şahin, Şahin & Heppner, 1993) and Math Literacy Self-Efficacy Measurement (Özgen & Bindak, 2008). For the data analysis, descriptive statistics, Pearson correlation, and regression analysis were used. Significant relationships were found to exist between the problem solving scores on the one hand and metacognitive awareness and math literacy selfefficacy scores on the other. In addition, the first stage of the regression analysis showed that metacognitive awareness scores are a significant predictor of problem solving skills, explaining 15% of the variation. In the second stage, it was found that metacognitive awareness and math literacy self-efficacy scores are both significant predictors of problem solving skills, together explaining 18% of the variation. This study shows that significant relationships and interactions exist between metacognitive awareness, mathematical literacy self-efficacy, and problem solving skills.

In this study, the perceptions of prospective classroom teachers on science, technology, engineering and mathematics (STEM) education approaches have been examined through metaphors. The participants in this qualitative research project included 138 prospective teachers studying between the 2016-2017 academic year in the department of elementary education from the faculty of education at a public university in Turkey. In order to determine the perceptions of STEM teachers selected from different class levels, the prospective teachers were first educated about STEM education. Next, a form, on which the statement, "STEM Education (Science-Technology-Engineering-Mathematics) is like ...because ... " was presented to the prospective teachers, and they were asked to write a metaphor for STEM education, as well as to explain the reason. The data obtained from the forms were analyzed through content analysis. From the data analysis, the valid metaphors on STEM education created by the prospective classroom teachers were collected under 9 different conceptual categories, which were developed on the basis of common characteristics. These categories reveal that the prospective teachers regard STEM as a useful, necessary and highly appreciated approach that involved complementary fields. In some categories, it was found that there were significant differences in the rate of participants according to gender and class level.

Ratio-proportion is one of the topics that middle school students have difficulty in comprehending. This study aims to determine the learning difficulties that seventh-grade students face while learning about ratio-proportion. Three commonly used strategies to solve ratio-proportion problems are buildup, unit ratio and cross-multiplication. The study also investigated the effect of the envelope technique, which included these three strategies and was developed to improve comprehending and success of students. The study was designed as an experimental study and 34 seventh-grade students participated in the study. The Ratio-Proportion Achievement Test developed by the researcher used to collect the data. The results illustrated that students frequently have misconceptions while solving ratio-proportion problems by confusing the direction of the operation in reducing and expanding the numbers constituting the proportion. It was concluded that the envelope technique is a method that assists students in reaching a rapid, correct solution in ratio-proportion problems, subsequently increasing their mathematical achievements.