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Abstracts of Presentations
Mathematical Modelling in Schools - an applied mathematician's lament
Ang Keng Cheng
For mathematical modelling to succeed in the classroom, teachers need to be properly prepared, suitably supported and well resourced. In reality, however, not every mathematics teacher has had formal training or experience in mathematical modelling or applied mathematics as a student. Moreover, the unpredictable nature of a mathematical modelling lesson has further discouraged teachers from planning and carrying out mathematical modelling lessons. To address these issues, the “Mathematical Modelling in Singapore: From Research to Practice” project, aimed at building relevant resources and support structures for teachers, based on past research work done and material collected over a decade, was launched. This talk discusses the motivation for the project, and the process which led to its outcomes.
A Study of Malaysian Teacher Characteristics and Grade 8 Student Engagement in Mathematics Classrooms: Evidence from TIMSS 2015
Mok Yuen Teng
This presentation, draws on the TIMSS 2015 data, and examines Malaysian teacher characteristics that relate to their quality of instruction as perceived by their students. It also explores student attitudes toward learning mathematics and investigates if the quality of instruction is significantly related to student attitudes.
Orchestrating discussions around typical problems: noticing as pedagogical reasoning?
Choy Ban Heng & Jaguthsing Dindyal
In this presentation-discussion, we will share some of our insights from an OER project looking at teacher noticing in the context of orchestrating learning experiences (OER 03/16 CBH). In particular, we will highlight some surprising findings, and propose a connection between noticing and pedagogical reasoning for discussion.
A study of the school mathematics curriculum (secondary)
Prof Berinderjeet Kaur et al.
In this presentation-discussion we will share some of our observations about the instructional core and models of instruction that guides teaching and learning of mathematics in the classes of competent secondary school mathematics teachers.
On the skewness of Cartesian products with trees
Tay Eng Guan
The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. In this talk, we show that skewness is additive for the Zip product under certain conditions. We then present results on lower bounds and exact values of Cartesian products of graphs with trees. This is work done by Ouyang Zhangdong, Dong Fengming and Tay Eng Guan during the former’s visit to NIE in 2017/18.
On the existence of real roots of chromatic polynomials of hypergraphs
Zhang Ruixue
A mixed hypergraph H is a triple (V,C,D), where V$ is a finite set, C and D are subsets of {eV: |e|> 1}. For any positive integer k, a proper k-coloring of H is a mapping φ: V{1,2,…,k} such that |{φ(v): ve}|< |e| holds for each eC and 1<|{φ(v): ve}| holds for each eD.
Let Pmix(H,x) denote the function that counts the number of proper k-colorings of H whenever x=k is a positive integer. This function is a polynomial and is called the chromatic polynomial of H. In this talk, I will introduce our latest work on the study of zero-free intervals (-,0) and (0,1) of Pmix(H,x) for under certain conditions on H. This result is an extension of known results on the zero-free intervals of chromatic polynomials of hypergraphs and graphs.
An Open Problem in Domain Theory
Ng Kok Min
Every RB-domain is an FS-domain. However, whether the converse is true is an open problem in Domain Theory. Lawson (2008) showed that the pointed domain of closed balls, B(Rn), of the familiar Euclidean space Rn is an FS-domain. While much have been known of the Euclidean space, it is surprising that it remains unknown if B(Rn) is an RB-domain for n greater than 1. This then naturally leads to the suspicion that B(Rn) can serve as the counterexample to the open problem. In this talk, we discuss the problem and some of its related progress.
On the Evenly Spaced Topology on Z
Jeremy Ibrahim Bin Abdul Gafar
In number theory, the infinitude of primes is a well-known result that was first established and recorded by Euclid around 300 B.C. By all standards of rigour in modern mathematics, Euclid's style of reasoning still stands as an excellent model for the proof method by contradiction. Another less known but nonetheless curiously interesting proof was due to Hillel Furstenberg. Hillel Furstenberg’s proof of the infinitude of primes is a celebrated topological proof that the set of integers contains infinitely many prime numbers. The proof was published in 1955 in the American Mathematical Monthly while Furtsenberg was still an undergraduate student at Yeshiva University. Central in Furstenberg's proof is a type of topology defined on the setof natural numbers which is called evenly spaced topologies on Z. Later, S. Golomb continued along this line and look at another evenly spaced topology defined on Z and its properties in connection with its number-theoretic implications expounded.
In my Final Year Project, I studied Furstenberg's proof of the infinitude of primes, noting which properties of the evenly spaced topology are important in the proof. I also carried out a further study into the topological properties of the Furstenberg's evenly spaced topology, other than those involved in the proof.
Hamilton Path Decompositions of Complete Multipartite Graphs
Hang Hao Chuien
There has been interest in problems concerning decomposition of graphs into Hamilton cycles and into Hamilton paths, for many years. In 1976, Laskar and Auerbach showed that a complete multipartite graph can be decomposed into Hamilton cycles if and only if it is regular of even degree. We prove a corresponding result for decompositions of complete multipartite graphs into Hamilton paths. In particular, we prove that a complete multipartite graph K with n>1 vertices and m edges can be decomposed into edge-disjoint Hamilton paths if and only if m/(n-1) is an integer and the maximum degree of K is at most 2m/(n-1). This is joint work with Darryn Bryant and Sara Herke