Abstracts of Presentations

Quality of Singapore mathematics teachers' written justifications

Chua Boon Liang

Constructing mathematical justification is an important activity in the mathematics classroom. However the justifying skill is well known to be rather elusive to many students. So how do adults like mathematics teachers fare in the construction of justification? Drawing on findings from the Justification in Mathematics (JiM) research study that aimed to examine the quality of justifications produced by secondary school students and mathematics teachers, I present and discuss the performance of 55 mathematics teachers in justification tasks that span across the different content strands in the Singapore secondary mathematics curriculum. In particular, I focus on the quality of justifications produced by the teachers to offer some insights into the way they dealt with justification tasks.

Generalising the functional rule for a figural quadratic pattern

Tan Zhi Lie (Sean) & Chua Boon Liang

Students worldwide often face difficulty in generating the functional rule underpinning a linear pattern generalising task. The difficulty can get even more pronounced when the figural pattern depicts a quadratic relationship and is not presented as a sequence of successive configurations. This paper draws on a research study conducted in Singapore that investigated the generalisation of four figural quadratic patterns by 32 Year 8 students (15 boys, 17 girls) from one secondary school. The figural quadratic patterns were presented in two different formats: (i) a sequence of three sucessive configurations, or (ii) a single diagram or a sequence of two or three non-successive configurations. The students were distributed into two groups: one assigned to the former format and the other to the latter format. Of the 32 students, 15 of them (7 boys, 8 girls) were given the patterns with successive configurations whilst the remaining 17 (8 boys, 9 girls) worked on patterns with non-successive configurations.

This paper focuses on the students’ performance in just one of the four figural quadratic patterns. Specifically, the success rates, the kind of generalising strategies adopted and the kind of rules generated by the two groups of students will be described in greater detail using examples of students’ generalisations. Overall, about 72% of the students in the two groups combined succeeded in generating a correct functional rule, producing a variety of mathematically equivalent rules using a wide spectrum of generalising strategies.

A study of Indonesian grade 10 mathematics textbook

Teresa Wijaya

Textbooks form the bedrock of activities teachers enact during their lessons. They also often represent the intended mathematics curriculum of education systems. As noted by research the role of textbooks sometimes is even more evident and direct in the teachers’ instruction than the curriculum itself. In this presentation, I will explain my study about three sets of grade 10 mathematics textbooks that are used by schools in Jakarta. These schools have consistently produced good outcomes for students in mathematics, based on the result of the National Exam. The analysis is carried out in a qualitative method with two dimensions involved: physical characteristics (features) and knowledge development (sequence of topics, methods of delivery, purpose of the mathematical tasks, and level of cognitive demand). The framework created to analyse these textbooks draws from the works of several renowned researchers.

Encouraging construction of knowledge by stuednts

Pearlyn Lim Li Gek

The 21st century brings about challenges that require students to be critical and inventive thinkers. They also need to be self-directed learners who question, reflect, persevere and take responsibility for their own learning (MOE). The aim of my study is to encourage students to be self-directed learners who can construct knowledge with minimal explicit teaching. I studied the open-ended approach, the post-teahouse approach (which originated from China) and the idea of productive failure and have incorporated the strengths of the three pedagogies into an approach of teaching that may be suitable for use in Singapore. The gaps in the existing approaches are also addressed in this adapted approach. In this presentation, I will discuss what I observed when lesson packages based on the adapted approach were carried out in two primary mathematics classrooms.

Relational understanding triumphs over instrumental understanding:

The case of Singapore primary four children’s understandings of odd and even number

Teo Kwee Huang, Evergreen Primary School

Activities with odd and even numbers are especially rich for helping young children to focus on the structure of numbers and the relation between one number and its neighbouring number. Children need to make reference to such structural properties underpinning odd and even numbers to make meaningful algebraic generalisations and justifications when operating with odd and even numbers. This study explored the types of justifications provided by 18 primary four children from one Singapore primary school as they described the relationships between pairs of odd numbers, even numbers and a combination of odd and even numbers. The findings suggest that children of all abilities had the capability to explain odd and even numbers relationally and applied them in making analytical justifications about sums of odd and even numbers. Children with relational understanding of odd and even numbers offered better and more convincing analytical justifications about sums of odd and even numbers than children with instrumental understanding or with vague understanding of odd and even numbers who could only provide empirical justifications about sums of odd and even numbers. Children with incorrect understanding of odd and even numbers could not abstract, generalise and justify about sums of odd and even numbers. These findings offer evidence that teachers need to teach for relational understanding of odd and even numbers. Also, the mathematical work of teachers in encouraging students, provoking, supporting, pointing, and attending with care, is critical to the development of young children’s awareness of structure.

Rewording and anomalous information in statistics word problem solving

Professor Queena Lee Chua, Ateneo de Manila University, Philippines

Numerous studies have shown that a number of concepts in statistics lend themselves to fallacies. Students often tend to rely more on fuzzy ideas rather than on mathematically rigorous interpretations of these concepts. This study investigates whether anomalous information causes an increase in questions generated by college students while they solve statistics word problems. 40 science undergraduates are presented with different versions of each of ten statistics problems: (a) original, (b) deletion of critical information, (c) addition of contradictory information, (d) addition of salient irrelevancies, and (e) addition of subtle irrelevancies. Results show that most of the transformed versions trigger more questions than the original (p < 0.01), with the deletion versions triggering more questions than do others. The role of rewording in statistics word problems is discussed in light of these findings, and cognitive models (such as the obstacle hypothesis) are used to explain pertinent differences. Implications of these findings for pedagogy and research are also offered.

The Catalan conundrum

Toh Pee Choon

The Catalan numbers are ubiquitous in mathematics. We will briefly survey the history of these numbers, describe several combinatorial objects that are enumerated by them, and finally consider a higher dimensional generalization.

The travelling salesman problem (TSP)

Goh Han Sin

Using enumeration to solve the TSP will eventually cause the programme to grind to a halt. This paper presents our attempt to write a programme called “2-opt plus” with inspiration from improving Dynamic Programming’s (DP) run-time and memory storage, by incorporating features of Branch-and-Bound algorithm and the structure of the enumeration tree. The DP set-up is replaced with the enumeration tree structure to alleviate the problem with memory storage. Simultaneously, we introduce the present as well as the future 2-opt improvement checking as our bounding criteria to prune the enumeration tree, eliminating tours which are not optimal. This results in a significant reduction in the number of nodes visited to solve the problem compared with only using the present 2-opt improvement check, improving the run-time of the programme as compared to enumeration.  
Class of functions defined by means of neighbourhood assignments

Dewi Kartika Sari

A neighbourhood assignment for a topological space (X,τ) is a mapping δ:X→τ such that x∈δ(x) holds for each x. A mapping f:X→Y from a topological space X to a topological space Yis called AO-separated if for any neighbourhood assignment ε on Y, there is a neighbourhood assignmentδ on X, such that for any x1,x2∈X, x1∈δ(x2) and x2∈δ(x1) implies f(x1)∈ε(f(x2)) or f(x2)∈ε(f(x1)). In this talk, we shall give a brief overview of the characterizations of AO-separated function. By considering many conditions in the codomain, we will show that if the codomain is a metrizable space then the class of AO-separated functions is equivalent to some existing class of functions such as the class of continuous functions and the class of weakly separated functions.

Irreducible Completion of Spaces

Ho Weng Kin and Hadrian Andradi

In recent years, the usefulness of irreducible sets in domain theory and non-Hausdorff topology has expanded. Notably, Zhao and Ho (2015) developed the core of domain theory directly in the context of T0 spaces by choosing the irreducible sets as the topological substitute for directed sets. Just as the existence of suprema of directed subsets is featured prominently in domain theory (and hence the notion of a dcpo – a poset in which all directed suprema exist), so too is that of irreducible subsets in the topological domain theory developed by Zhao and Ho. The topological counterpart of a dcpo is thus this: A T0 space is said to be irreducible complete if the suprema of all irreducible subsets exist. In this talk, we show that the category, icTop+, of strongly complete T0 spaces forms a reflective subcategory of a certain lluf subcategory, Top+, of T0 spaces.

Topological data analysis

Zhao Dongsheng

Data analysis (or analysis of data or data analytics) is a process of inspecting, cleansing, transforming, and modeling data with the aims of identifying useful information, suggesting conclusions, and providing support for decision-making. It is one of the most useful and active areas of applied mathematics.

Data analysis has got multiple facets and various different approaches. One of the relatively new and rapidly growing branches of data analysis is the topological data analysis (TDA). The objective of TDA is to use the topological tools to discover / identify the geometric shapes in data. In this talk, we shall present an introduction to TDA.

The topics will include basic topological concepts, some geometric shapes of data, persistent homology of a space and practical examples of TDA applications.