How formal should calculus in the school mathematics curriculum be: Reflections arising from an error in a calculus examination question
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.
Infusing problem solving into mathematics content course for pre-service secondary school mathematics teachers
This paper presents a re-design of an undergraduate mathematics content course on Introductory Differential Equations for pre-service secondary school mathematics teachers. Based on the science practical paradigm, mathematics practical lessons emphasizing problem-solving processes via the undergraduate content knowledge were embedded within the curriculum delivered through the traditional lecture-tutorial system. The pre-service teachers' performance in six mathematics practical lessons and the mathematics practical test was examined. They were able to respond to the requirements of the mathematics practical to go through the entire process of problem solving and to carry out "Look Back" at their solution: checking the correctness of their solution, offering alternative solutions, and expanding on the given problem. The use of Mathematics Practical has altered the pre-service teachers’ approach in tackling mathematics problems in a positive direction.
On pre-service teachers' content knowledge of school calculus: An exploratory study
This paper reports an exploratory study on the pre-service teachers’ content knowledge on school calculus. A calculus instrument assessing the pre-service teachers’ iconic thinking, algorithmic thinking and formal thinking related to various concepts in school calculus was administered to a group of pre-service mathematics teachers. Their performance on five of the items is reported in this paper. Other than their good performance in the iconic recognition of stationary points, their recognition on points of inflexion, differentiability and notion of minimum points was relatively poor. In addition, they appeared to lack the algorithmic flexibility in testing the nature of stationary points and the formal thinking about definition of an extremum point. The implications of the findings are discussed.
Teaching undergraduate mathematics: A problem solving course for first year
In this paper we describe a problem solving course for first year undergraduate mathematics students who would be future school teachers.
The Mathematician Educator special issue: Mathematics instruction for the future
A study of pre-service teachers' performance on two calculus tasks on differentiation and limit
The purpose of this paper is to report a part of a calculus research project, about the performance of a group of pre-service mathematics teachers on two tasks on limit and differentiation of the trigonometric sine function in which the unit of angle measurement was in degrees. Most of the pre-service teachers were not cognizant of the unit of angle measurement in the typical differentiation formula, and a number of participants recognized the condition on the unit of angle measurement but did not translate this to the correct procedure for performing differentiation. The result also shows that most of the participants were not able to associate the derivative formula with the process of deriving it from the first principle. Consequently, they did not associate it with finding . In the process of evaluating this limit, the pre-service teachers exhibited further misconceptions about division of a number by zero.