Previous studies of integral calculus have mainly explored students' conceptual and procedural knowledge; only a few have focused on students' metacognition in relation to integral calculus. The study reported here explored students' metacognitive knowledge of integral calculus by interviewing nine first-year university and eight Year 13 students. The design of the interview questions was based on the structure of metacognitive knowledge in Revised Bloom's Taxonomy. The findings suggest there are differences between students' metacognitive knowledge at Year 13 and the first year of university. In particular, the importance of knowing the rationale behind the theorems and formulas was not obvious for Year 13 students. Moreover, students' metacognitive knowledge could be developed further at both levels, particularly in terms of developing strategies to identify how integral calculus questions could be solved and to check the outcome of problem-solving.Educ. Sci. 2020, 10, 55 2 of 20 interesting to know if there are any corresponding differences in the metacognitive knowledge each group developed. Moreover, several studies highlighted the issues students face during the transition from secondary to tertiary level (e.g., [12,13]). For instance, many students may face difficulties in university since mathematical concepts are presented in more depth and problems require more technical and conceptual understanding in comparison to secondary level teaching [14]. Exploring students' metacognitive knowledge between these two levels could shed light on which aspect of Year 13 students' metacognitive knowledge needs further development to be able to fit with the academic rigor of studying mathematics at the tertiary level. These research questions will be explored in relation to the Revised Bloom's Taxonomy (RBT) [1] structure of metacognitive knowledge, as it is a useful tool for exploring the teaching, learning, and assessment of mathematics [15].
Literature ReviewIn this section, metacognition, metacognitive knowledge, and previous studies in relation to integral calculus are described to frame the study.
Metacognition and Metacognitive KnowledgeSince the introduction of the term 'metacognition' by Flavell in 1976, different operational definitions of metacognition have been introduced (see, [16]). A group of scholars believed cognition and metacognition are separate entities and hierarchical in nature (e.g., [17]), while another group believed cognition and metacognition are integrated (e.g., [18]). For instance, Shoenfeld [17] believed metacognition works as an executive control, helping to solve mathematical problems by self-regulating, monitoring, and controlling the allocation of resources. On the other hand, Kramarski et al. [18] highlighted to teach metacognition effectively, its teaching should be embedded in the teaching of the content.Despite these differences, many scholars agree that metacognition is a model of cognition that works at a meta-level to cognition by monitoring and controlling cognitive tasks [1...