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Self-efficacy in mathematics is related to engagement, persistence, and academic performance. Prior research focused mostly on examining changes to students' self-efficacy across large time intervals (months or years), and paid less attention to changes at the level of lesson sequences. Knowledge of how self-efficacy changes during a sequence of lessons is important as it can help teachers better support students' self-efficacy in their everyday work. In this paper, we expanded previous studies by investigating changes in students' self-efficacy across a sequence of 3–4 lessons when students were learning a new topic in mathematics (nStudents = 170, nTime-points = 596). Nine classes of Norwegian grade 6 (n = 77) and grade 10 students (n = 93) reported their self-efficacy for easy, medium difficulty, and hard tasks. Using multilevel models for change, we found (a) change of students' self-efficacy across lesson sequences, (b) differences in the starting point and change of students' self-efficacy according to perceived task difficulty and grade, (c) more individual variation of self-efficacy starting point and change in association with harder tasks, and (d) students in classes who were taught a new topic in geometry had stronger self-efficacy at the beginning of the first lesson as compared to those who were taught a new topic in algebra (grade 10), and students in classes who were taught a new topic in fractions had steeper growth across the lesson sequence as compared to those who were taught a new topic in measurement (grade 6). Implications for both research and practice on how new mathematics topics are introduced to students are discussed. [ABSTRACT FROM AUTHOR]

This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment. [ABSTRACT FROM AUTHOR]