Toh Tin Lam

Studies have shown that teachers do not have good understanding of calculus concepts. This paper reports a study of teachers' content knowledge of calculus, on 27 in-service mathematics teachers. A questionnaire dealing with the concept images and concept definitions of various calculus concepts was administered to the group of participating teachers. The responses to the questionnaire showed that most of the participants had not built up sufficiently rich and comprehensive concept images related to the various differential calculus concepts, and they generally turned to procedures in handling calculus tasks. This study sheds light on the type of calculus content needed by school teachers.

This paper examines the calculus curriculum in the current Singapore secondary and pre-university levels. Two concepts, (1) increasing and decreasing functions and their derivatives, and (2) the second derivative test for the nature of stationary points, are elaborated. An example of an incorrect calculus item in a national examination is brought up in relation to conditional reasoning involving calculus concepts. We reckon that the current emphasis on procedural knowledge in calculus is useful. However, we argue that formal conditional reasoning should not be introduced prematurely for school students.

A case study of calculus instructional material (comprising of lecture notes and tutorial practice worksheets) designed by teachers from a Singapore pre-university is presented in this paper. As textbooks have not been available for mathematics at the pre-university levels, the instructional material was a product of the teachers’ collaborative work based on their interpretation of the school mathematics curriculum. Through a study of the instructional material, the teachers’ instructional goals and their alignment to the curriculum were examined. It was clear that the instructional material was more than a mere compilation of resources for the instruction; the teachers’ effort was also in building the close connection within and across the concepts and sub-topics. The discrete parts of the content were organized under a “big idea” in the lecture notes. Anchor questions were used in the tutorial practice worksheets to facilitate students to recognize the similarity of the underlying structures of seemingly different questions. In addition to the teachers’ articulated instructional goals such as covering all the key concepts, reducing students’ cognitive load, or developing their algorithmic mastery, the study revealed the unarticulated goal as to raise the students’ cognitive growth, which could be explained by the APOS cognitive growth model. In aligning to the school curriculum, it was observed that the teachers made judgement to tap on the advantage of the spiral curriculum to advance their students’ understanding of calculus from the secondary level using a higher perspective, and to better their students’ understanding of calculus concepts in addition to focusing on algorithmic mastery. The effort to engage their students in problem-solving and the use of technology to develop higher order thinking or enhance conceptual understanding remained implicit.