Toh, Pee Choon

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.

We de ne a vector crank to provide a combinatorial interpretation for a certain Ramanujan type congruence modulo 7.

Andrews’ singular overpartitions can be enumerated by Ck,i(n), the number of overpartitions of n where only parts congruent to ±i (mod k) may be overlined, and no part is divisible by k. A number of authors have studied congruences satisfied by singular overpartitions. In particular, congruences for C3,1(n) modulo 3, 8, 9, 18, 32, 36, 64, 72 and 144 have been proved. In this article, we prove new congruences modulo 108, 192, 288 and 432 for C3,1(n).

We have applied the ‘practical paradigm’ in teaching problem solving to secondary school students. The key feature of the practical paradigm is the use of a practical worksheet to guide the students’ processes in problem solving. In this paper, we report the diffusion of the practical paradigm to university level courses for prospective and practising teachers. The higher level of mathematics content would demand higher order thinking skills. Learners without a model of problem solving would often revert to solving by referring to many examples of the same ‘type’ of problem. Polya-type problem solving skills framed by the practical worksheet was used as an attempt to elicit more effective problem solving behaviour from them. Preliminary findings show that they were able to use the practical worksheet to model their solution of problems in the courses.

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.