The effect of explicit and implicit learning of addition and multiplication principles in algebra by students with different mathematical metacognition perceptions on procedural, representational fluency and mathematical explanation

The question of how students intuitively or consciously process the regular and patterned structure of mathematics is an important research topic. Learning processes can be determinative of how fluently an individual performs mathematical operations, how quickly and accurately he/she can use representations, and his/her mathematical explanations. In this study, the effects of explicit and implicit learning of addition and multiplication principles in algebra on procedural, representational fluency and mathematical explanation of students with different perceptions of mathematical metacognition were examined. This study was conducted according to a nested mixed design. The study was conducted in the fall semester of the 2024-2025 academic year with 85 7th grade students enrolled in two middle schools in Mersin. Data were collected using procedural fluency tests, representational fluency test, mathematical explanation test, confidence rating scale, semi-structured interview form and researcher diaries. Descriptive statistical techniques, Robust ANOVA analysis and Chi-Square test were used to analyze the quantitative data. Qualitative data were analyzed using inductive content analysis. As a result of this study, students' mean scores of procedural fluency in addition in algebra differed significantly in terms of learning process (implicit, explicit and learning activities recommended in the mathematics curriculum), mathematical metacognition perception (high and low), time (pre-test, post-test and retention test) and the binary and ternary interactions of these variables; learning process and time effects. The mean scores of students' procedural fluency in multiplication in algebra show a significant difference in the effects of learning process, mathematical metacognition perception and time, and in the binary interactions of learning process and mathematical metacognition perception and learning process and time. Students' mean scores of representational fluency in addition and multiplication in algebra show a significant difference in learning process and time effects, learning process and mathematical metacognition perception and learning process and time binary interactions. The mean scores of students' mathematical explanation in addition and multiplication in algebra show a significant difference in learning process, mathematical metacognition perception and time effects, learning process and mathematical metacognition perception and learning process and time binary interactions. Students' confidence levels of correct answers regarding the principles of addition and multiplication in algebra show a significant difference in terms of the learning process applied. As a result of this study, it was concluded that the students in the experimental groups had positive perceptions of the explicit and implicit learning processes and the learning materials used. It was found that the learning materials used in the explicit and implicit learning processes supported the learning process; the steps that provided opportunities for various representations, creating and solving questions and explaining the solutions stimulated students' interest. Based on the results of this research, it can be suggested to develop learning activities and materials for explicit and implicit learning processes in mathematics education. Keywords: Explicit learning, implicit learning, metacognition perception, implicit-explicit knowledge, algebra learning.