Toh, Pee Choon

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.

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).

Quek, Tay, Toh, Leong, and Dindyal (2011) proposed that a design-theory-practice troika should always be considered for a designed package to be acceptable to the research users who, in this case, are teachers and schools. This paper describes the fine-tuning to the MProSE problem-solving design made by the teachers in the school after first round of teaching. This process involved teacher input from their experience, and detailed time-consuming discussions and learning between the researcher-designers and the teacher-implementers.

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

Assessing how well students understand proofs is difficult. One way to achieve this is to present an erroneous proof and require students to identify the error. Twelve erroneous proofs on various topics in mathematics at the Secondary school level are presented as examples.

Using known formulas for R(a;b)(n), the number of representations of n as a times a square plus b times a square, we prove 21 conjectured formulas of Melham on the number of representations of n as sums of triangular, pentagonal and heptagonal numbers. We also demonstrate how our technique can be used to prove the other 277 conjectured formulas of Melham concerning representations by other gurate numbers.

This paper considers several generalisations of an interesting problem posed in the 2021 William Lowell Putnam Mathematical Competition. We describe how the authors embarked on a collaborative problem-solving journey which resulted in solving two of these generalisations and how computational thinking guided our approaches.