Abstracts of Presentations

Use of comics in teaching mathematics

Toh Tin Lam

This talk presents a research carried out by the speaker and his colleagues on using an alternative approach of teaching mathematics through the use of comics. Research has shown that school-going students have particular attraction for comics and educators have begun the use of comics in schools to capture. Educators have reported that the use of comics does not necessarily simplifies the mathematical content to be taught in school; on the other hand, it includes equipping students with the important skills of combining both words and visual cues in interpreting given messages – an important aspect of the twenty first century skills. The speaker will discuss the process of designing the teaching package, the implementation in the Singapore mathematics classrooms and some preliminary findings resulting from the study.

A study of classroom assessment techniques in mathematics

Wang Kangcheng & Jeremy Ibrahim bin Abdul Gafar

This session will report the experiences of two undergraduate students’ participation in the Undergraduate Research Experience on CAmpus (URECA) research project. During the presentation, the students will also report the findings of their research on mathematics classroom assessment practices and preferences. Through a series of teacher interviews and lesson observations, the project seeks to identify and understand assessment strategies used by teachers in the mathematics classrooms in Singapore.

A study of mathematics homework in Singapore secondary two classrooms

Alina Khaw Han Ron

Mathematics homework is an integral part of students' academic routine in Singapore. The instructional purpose of homework is multi-faceted and the approach to assess students' understanding through homework is multi-dimensional. It is imperative for teachers to select meaningful tasks to foster and develop students' robustness in understanding. In this presentation, I will discuss the nature and source of mathematics homework as well as teachers’ and students’ perspectives about the role of mathematics homework in two separate Singapore Secondary Two mathematics classrooms.
A video-based professional development programme for Thai primary 6 teachers

Muanfun Yaowiwat

A video-based professional development (PD) programme was developed and conducted for a group of Primary 6 mathematics teachers in Thailand. This PD aimed to enhance teachers’ ability to interact to the students. A mediation strategy framework was introduced to the teachers to engage them in critical reflection of the video lessons. During the presentation, the conceptual framework of the PD, the details of PD and some preliminary findings of the study will be discussed.

The seven hierarchical levels of understanding of vectors of some Singapore students

NG Swee Fong*, SEOH Bee Hua**, LEE Yee Tyng**, Judy NG Suan Khee**
*National Institute of Education
**Hougang Seondary School, Singapore

Vector is an entity in Euclidean space that has both magnitude and direction. Although this topic is only a small section of the secondary school mathematics curriculum the notations and the underlying concepts are crucial to learning higher mathematics and to the study of physics in particular. Piecemeal reports in official reports provided by ‘O’ level examination boards (e.g. University of Cambridge, 2011, 2013 and 2014) showed that Singapore students’ performance in this important topic was less than satisfactory. In particular they had difficulties working with vector notations and failed to understand that two or more vectors are equal only if they have the same magnitude AND the same direction. Such reports although useful do not provide a coherent picture of students’ knowledge of vectors and what are their gaps of knowledge. The purpose of this report is to provide a systematic breakdown of the knowledge held by166 Secondary 4 Express (16+) and 68 Secondary 5 (17+) Normal Academic students from one secondary school. This study used the Ruddock Instrument, which was constructed bound by two important threads - abstraction and complexity and differentiated into 7 hierarchical levels of understanding, to assess these students’ understanding of vectors. Students who could operate at Level 4 and above were described as capable of operating at an abstract level without the aid of any concrete representations to assist them. Students in this study were proficient with vector items at levels 1, 2, and 3. This meant that these students were able to add vectors, carry out simple scalar multiplications, and operate with vectors involving negative integers. Such items have a visual aspect, diagrams or grids are provided and solutions usually involve one step. However they had difficulties identifying vectors of the type a▁x, where the scalar a can be negative integers and/or rationals and where diagrams are constructed using oblique lines or when it was necessary to ‘imagine’ the solutions. Furthermore students were unable to explain their solutions. Some students have difficulties differentiating between the notations associated with vectors and translation and those used to represent points on the Cartesian plane. Some suggestions to improve performance will be discussed at the presentation.

Optimal orientations of G vertex-multiplications

Tay Eng Guan

For a graph $G$, let $\cal D(G)$ be the family of strong orientations of $G$, and define $\stackrel {\rightarrow} d \hspace{-0.1 cm}(G) = \min \{d(D) | D \in \cal D(G)\}$, where $d(G)$ (resp., $d(D)$) is the diameter of the graph $G$ (resp., digraph $D$). Let $G$ be a given connected graph of order $n$ with vertex set $V(G) = \{v_1, v_2, \cdots, v_n\}$. For any sequence of $n$ positive integers $(s_1, s_2, \cdots, s_n)$, let $G(s_1, s_2, \cdots, s_n)$ denote the graph with vertex set $V^*$ and edge set $E^*$ such that $V^* = \cup_{i=1}^n V_i^*$, where $V_i^*$ are pairwise disjoint sets with $|V_i| = s_i$, $i = 1, 2, \cdots, n$; and for any two distinct vertices $x, y$ in $V^*$, $xy \in E^*$ iff $x \in V_i$ and $y \in V_j$ for some $i, j \in \{1, 2, \cdots, n\}$ with $i \ne j$ such that $v_iv_j \in E(G)$. We call the graph $G(s_1, s_2, \cdots, s_n)$ a $G$ vertex-multiplication.

In this talk, we shall discuss the relation between $\stackrel {\rightarrow} d \hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n))$ and $d(G)$. While it is trivial that $d(G) \le \stackrel {\rightarrow} d \hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n))$, we shall show that if $s_i \ge 3$ for each $i = 1, 2, \cdots, n$, where $n \ge 3$, then $\stackrel {\rightarrow} d \hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n)) \le d(G)+2$. This result naturally gives rise to a 3-classification of graphs of the form $G(s_1, s_2, \cdots, s_n)$ according to $\stackrel {\rightarrow} d \hspace{-0.1 cm} (G(s_1, s_2, \cdots, s_n)) = d(G)+i$, $i = 0, 1, 2$.

A ‘rough’ introduction to semicontinuous lattices

Beertino Romerow Woe

Since its introduction by D. S. Scott in late 1960s, continuous ordered structures (e.g., continuous lattices, domains and continuous posets) have been extensively studied. Continuous ordered structures provide the semantic model for functional programming languages. Central to domain theory is its affordance to describe the phenomenon of approximation and convergence. Earlier this year, Zou, Li & Ho explained how domain theory can be perceived as a theory of approximation by giving it a rough set-theoretic foundation. Importantly, they gave a characterization of continuous posets based on certain rough set approximation operators that were specifically manufactured to cope with the irreflexive nature of the famous way-below relation that is prominently featured in domain theory.

Closely related to the notion of continuous lattices is that of semicontinuous lattices – first invented by Zhao in 1997. Semicontinuous lattices are a generalization of continuous lattices and have very pleasing properties similar to their continuous counterpart. By contrasting semicontinuity with continuity, we hope to derive sufficient background knowledge to create a rough set-theoretic foundation for the theory of semicontinuous lattices.
Group lasso method with application on the biomedical spectroscopic data

Zou Lin

The spectroscopic data have been widely used in biomedical area for classification and calibration. High-dimensional spectroscopic data, having as many as 1000 wavelengths, consist of many overlapping absorption bands sensitive to the physical and chemical states of the compounds. How to extract most useful structure feature from the spectra data to identify the compositional differences so as to classify different types of the samples has been a very attractive but challenging area of research in recent years. Nowadays, feature selection methods such as Lasso, Sparse Partial Least Square (SPLS) are quite often used but has the limitation of wavelengths interpretation due to the complicated dependent structure in the biomedical spectroscopic data. Sparse Group lasso method, characteristic with grouping effect, has shown great promise in terms of physical interpretation. This method is illustrated on biomedical spectroscopic data, and provides great evidence regarding the active ingredients with high classification rate.

 

Properties on chromatic polynomials of hypergraphs

Ruixue Zhang

In this talk, I will introduce some results on chromatic polynomials of hypergraphs. A hypergraph $H$ consists of two sets $V$ and $E$, where $V$ is a finite set and $E$ is a subset of $\{e \subseteq V : |e| \ge 2\}$. A proper $k$-coloring of $H$ is a mapping $\psi : V \rightarrow \{1, 2, \cdots, k\}$ such that $|\psi(e)| \ge 2$ holds for each $e \in E$. Let $P(H, \lambda)$ be the number of proper $\lambda$-colorings of $H$ when $\lambda$ is a positive integer. The function $P(H, \lambda)$ is a monic polynomial of degree $|V|$.

The main result I will introduce is that the real zeros of chromatic polynomials of hypergraphs are dense in the whole set of real numbers and the complex zeros of chromatic polynomials of hypergraphs are dense in the whole complex plane. We also prove that for any graph $G = (V, E)$, the number of totally cyclic orientations of $G$ can be determined by the value of the chromatic polynomial of some hypergraph at $\lambda = -1$. In addition, some results on the multiplicity of zeros 0 and 1 for chromatic polynomials of hypergraphs which are different from chromatic polynomials of graphs also will be presented.

Penalized classification and feature selection with its application to detection of Chinese herbal medicine

ZHU Ying

Fourier transform infrared (FTIR) spectra of herbal medicine consist of many overlapping absorption bands sensitive to the physical and chemical states of compounds. Often, only a small subset of spectral features is found essential. Direct implementation of linear discriminant methods on high-dimensional spectroscopic data provides poor classification results due to singularity problem and highly correlated spectral features. In this study a penalized classification model using FTIR spectroscopy was developed to discriminate between two species of Ganoderma (a traditional Chinese herbal medicine) by incorporating the high correlation structure of the spectral variables into the model. The well-performed selection of informative spectral features leads to reduction in model complexity, improvement of classification accuracy, and is particularly helpful for the model interpretation of the major chemical compounds of Ganoderma in relation to its medicinal efficacy.