Abstracts of Presentations

Proving with proofs

Leong Yew Hoong

Most mathematicians cannot conceive of mathematics without proofs; proofs are the essential contents – perhaps even the ostensible defining trait – of mathematics. However, proofs in its rigorous forms are often considered not easily accessible for many students who have weak foundations in mathematics. What does a maths teacher do when caught in this situation of conflicting instructional goals: keep true to the disciplinarity of mathematics in proving while not presenting proofs in a formal way? We present a way to cope with this dilemma as used in the project schools.

Professional development for teachers of mathematical modelling in Singapore

Tan Liang Soon

A school-based professional development programme (SBPD) aimed at developing secondary school mathematics teachers’ competencies to teach mathematical modelling in Singapore is presented and evaluated in this thesis. The SBPD is characterized by three key dimensions of learning–content dimension to develop teachers’ knowledge and skills, process dimension where teachers are engaged in transformative learning cycles to elicit, enact, and reorganize their orientations in a mathematical modelling classroom, and context dimension to adjust the SBPD programme implementation based on school contextual factors. A multiple-case study approach was adopted to examine teachers’ developmental trajectories in their practice of teaching mathematical modelling. Goal-based decision making analysis of teachers’ practice suggests that this SBPD positively influences teachers’ knowledge and resources, goals and orientations in planning, designing and enacting modelling learning experiences. Implications for the SBPD programme in terms of its contributions to educational theory and practice are also discussed.

Using activity theory to study the development of an (aspiring) professional learning community

Romina Yap Ann Soon

Professional learning communities (PLCs) have been endorsed as social infrastructures that can effectively support teachers’ collective and individual learning. However, fostering effective PLCs can be a challenging task because it needs to consider personal, interpersonal, and institutional orientations, relations, and capacities. Studying PLC building initiatives can provide insight into how these social infrastructures can proceed with more success. In this presentation, I will present how the lens of activity theory can be useful for studying the development of PLC initiatives while taking into consideration their multifaceted nature. To demonstrate, I use the case of a group of secondary mathematics teachers in the Philippines who were collaborating for mathematical problem solving instruction.
Informing mathematics pedagogy: Singapore practices and US reform

Katie Bueker-Sibbit

This session will report the findings of one Fulbright teacher’s research in mathematics teaching practices and beliefs. The study compared Singapore teachers’ practices to recommendations by US reform movements through the United States’ Common Core State Standards. In addition, both US and Singapore teachers and students’ beliefs about math teaching and learning were studied in comparison to the National Council of Teachers of Mathematics (NCTM) recommendations about productive and unproductive beliefs. Through a series of student and teacher interviews and observations, comparisons were made with the hope of informing US reform through Singapore’s success. The researcher will share pragmatic insight into student and teacher beliefs about math teaching and learning, as well as strategies and resources for international best practice in all contexts in the field of mathematics.

Teaching for Metacognition - Teachers working and learning collaboratively in two-tiered communities of practice

Berinderjeet Kaur & Wong Lai Fong

Kaur developed a hybrid model of professional development for teachers in Singapore. The model draws on significant findings of effective professional development programmes. During the presentation she will discuss the enhancement of the model that has led to the development of two-tiered communities of practice in a cluster of schools that were developing their classroom practices to teach for metacognition.
What is the crank of a partition?

Toh Pee Choon

The partition function p(n) counts the number of ways an integer n can be written as a non-increasing sum of positive integers. The Indian mathematician S. Ramanujan discovered many fascinating properties of p(n), chief among which are the three congruences which are now known as Ramanujan's congruences. About 25 years later, the famous physicist Freeman Dyson, then an undergraduate at Cambridge, discovered a remarkable property concerning partitions which he called the rank of a partition. The ranks of partitions provided a combinatorial explanation of two of Ramanujan's congruences but not the third (and most difficult to prove) congruence. Dyson then conjectured the existence of what he called the "crank" of a partition which would explain the third congruence. This was one of the rare instances in the history of mathematics where a mathematical object was named before anyone had an example of what the object looked like. It took another 40 years before Andrews and Garvan discovered what is now known as the crank of a partition. In this talk, I will explain, with the help of many pictures, what are ranks and cranks of partitions.
Quasi-metric spaces and their corresponding poset of formal balls

K.M. Ng, W.K. Ho

First introduced by Weihrauch and Schreiber in 1981, the poset of formal balls can be used to represent metric spaces as a computational model. Many connections and characterizations of the metric spaces using the order structure of the formal balls were discovered over the years. For instance, in a classical result due to Edalat and Heckmann (1998), a metric space is complete if and only if its corresponding poset of formal balls is directed-complete. Analogous to the metric spaces, there are also interesting links between a quasi-metric space and its poset of formal balls. In this talk, we share some of these links and how we attempt to anchor on them and existing results in general order structures to revisit some of the traditional concepts such as the various completions on quasi-metric spaces.
A new topology on the Denjoy space

Dewi Kartika Sari, Peng-Yee Lee, Dongsheng Zhao

We define a new topology on the Denjoy space in terms of a neighbourhood system. The neighbourhood system is constructed from the complete metrizable space C[0,1] with additional properties. During the presentation we will illustrate the space and discuss it.

Chromatic equivalence classes of complete tripartite graphs

Ng Boon Leong

The chromatic equivalence class of a graph G is the set of graphs that have the same chromatic polynomial as G. We review previous research of chromatic equivalence classes of some families of graphs, in particular, complete tripartite graphs. We also find the chromatic equivalence class of the complete tripartite graphs K(1,n,n+2) for all n≥2, a result obtained by A/P Dong and me in 2015.

Rough set theoretic approach to domain theory

Ho Weng Kin

Rough set theory was introduced by Z. Pawlak in 1991 to formalize the notion of approximation in the context of data and information. The upper and lower approximations in rough set theory were recently modified to characterize the notion of continuity in posets by Li, Zhou & Zhou 2015. In this talk, we look at this new approach and investigate how domain theory can be studied through the lens of rough set theory.