Abstracts of Presentations

Problem Solving at Primary Level: Students’ Strategies for Solving Challenging Geometry Problems

Goh Song Eng

The purpose of this study was to explore the various strategies used by Singapore primary students in solving challenging geometry problems. Specifically, the two research questions explored were:
  • What are the strategies used by Singapore primary four and five students to solve challenging geometry problems?
  • To what extent do the strategies lead to productive solutions?
An instrument with four challenging geometry problems covering four main topics of geometry at primary level: length, angle, perimeter and area, was designed and developed. A total of thirty-six primary four and five students took part in the study. All the answer scripts were studied in detail to understand how the students were trying to solve the problems and to identify all the strategies used by them. All the strategies identified were examined to determine if they were used in a useful manner leading to a productive solution. Findings from the data collected revealed a rich list of ten strategies used by students to solve challenging geometry problems: “draw auxiliary lines”, “draw diagrams”, “look for equal parts”, “look for known values”, “solve part of the problem”, “trace the changes”, “transfer unknowns”, “transform figures”, “use geometrical properties” and “use visual estimation”. Except for “use visual estimation”, students could use all the other nine strategies in a useful manner to solve the problems. There were about 40% of all strategies used in a useful manner leading to a productive solution.

A Framework to infuse Financial literacy into Mathematics Curriculum

Yeo Kai Kow Joseph

The prevailing public image of mathematics is an objective, abstract, and inhuman subject. Applications of mathematics to everyday life are often limited to basic number operations, percentages, volumes, and time measurements. In the primary levels, pupils as young as seven-years-old have already began to learn the basic elements of money and finance which are central to developing financial literacy. Since its inception, this has been achieved mainly through out-classroom lessons or enrichment activities, but the infusion of financial literacy (FL) into Mathematics instruction is rare. This presentation reviews the concept of financial literacy and the importance of financial literacy in mathematics education. The author has developed a matrix of mathematics-based financial literacy to help teachers conceptualise this infusion. The mathematics-based financial literacy framework also provides a more holistic conception of FL.

Promoting the Mastery of Problem-Solving Skills in Students via Animation

Han Hui Xuan Dilys

This study presents the use of an animated video to impart mathematical problem-solving skills to lower secondary learners. An animated video on a specific challenging concept of Percentage was developed to scaffold the problem-solving process for learners. In the first segment of the animated video, a real-life scenario is used to contextualise the mathematics problem. Elements of humour were incorporated into the story. A jargon similar to the secondary school learners is used to allow learners to identify with the story. Worked examples are explained with the aid of animation, followed by a series of practice questions of suitable difficulty level and in a variety of contexts to aid the transfer of knowledge to novel mathematics situations. The video ends with a summary to help learners consolidate their learning. This paper also discusses the theoretical underpinnings of our approach. This study also seeks to contribute to the existing repertoire of pedagogical strategies by providing a concrete teaching idea developed from robust pedagogical principles for a specific lower secondary concept.

Translating Productive Failure in the Singapore A-level Statistics Curriculum

Lee Ngan Hoe

In the recommendations made to the revised General Cambridge Advanced Level Examinations (A-level) Mathematics curriculum (Ministry of Education (MOE): Curriculum Planning and Development Division (CPDD), 2012), an emphasis was placed on the use of constructivist pedagogy to deepen students’ understanding of concepts and appreciation of the disciplinarily of the subject, and the development of students’ critical and inventive thinking capacities that are relevant to the 21st century. Current practices in the Junior Colleges (JCs), with its lecture and tutorial system, remain largely didactic with direct instruction being the main pedagogical approach. This presentation examines the efforts undertaken by the research project DEV 03/14 MK to support MOE’s shift in pedagogical approach and updated learning experiences for the JC curriculum, through the translation of the use of Productive Failure (PF) - an empirically-tested and tractable learning design that embodies constructivist principles (e.g., Kapur, 2008, 2010, 2012) - across the A-level Statistical curriculum. Through MOE’s existing processes and structures, the research team embarked on a three-year project, working with the JCs to (i) develop, implement, and refine curricular units targeted six key statistical topics using PF principles; (ii) build teachers’ capacity in implementing PF; and (iii) impact student learning. Results revealed the viability of the PF learning design in the JC mathematics classrooms, the impact of teachers’ consolidation quality on students’ learning, and the importance of effective professional development, in-site and collegial support, and professional learning communities in changing practice. Implications of the findings will be discussed.

Model-based Clustering and its Applications

Zhu Ying

Traditional clustering methods, such as hierarchical clustering, k-means clustering, are heuristic and are not based on formal model. Formal inference is thus not possible. Clustering methods based on probability models have great advantage over heuristic-based algorithms. Model-based clustering assumes that the data are generated by a finite mixture of underlying probability distributions such as multivariate normal distributions. The model then defines clusters and assigns each object to a single cluster. The Gaussian mixture model has been shown to be a powerful tool for its probabilistic foundations and its flexibility with clustering applications in various scientific fields. However, challenges arise when the data dimension becomes large. Model-based clustering of high dimensional data will be illustrated on real-world data sets.

Optimal Orientations of Graphs

Willie Wong

A connected graph G has a strong orientation if and only if no edge of G is a bridge. Founded on Robbins’ one-way street theorem, optimal orientations minimising the diameter of a bridgeless graph have practical value in real-life applications. One example is the conversion of a two-way street system into a one-way system in times of need. Optimal orientations of some special classes such as the complete graphs and complete n-partite graphs were investigated by many. In this talk, we survey the history and some existing results. Alongside, we will share some new results, with emphasis on complete n-partite graphs.

SF-topology

Sheng Chong

By using irreducible sets from a given topology, Zhao and Ho constructed the irreducibly-derived topology, called SI-topology. In this paper, we prove that the directedness condition of SI-continuity can be replaced by the irreducibility. Moreover, we define a new topology, called SF-topology, which is more relax than SI-topology. Some results related to this topology are presented, a new kind of continuity of spaces, called SF-continuity, and a new kind of sobriety, called _-sobriety. In addition, we prove that SF-topology can be induced by a certain convergence structure and provide a sufficient condition for the structure being topological.

A Game on Formal Balls

Ng Kok Min

The notion of formal balls induced by a metric space was first introduced by Weihrauch and Schreiber in 1981. Since then, this notion has proved to be an important tool in linking domain theory and classical analysis, such as providing models for classical topological spaces and characterizing completeness of metric spaces. In this talk, we share how we can use the existence of a winning strategy in a topological game on formal balls to characterize metric spaces.

E-injectivity and E-projectivity

Ho Weng Kin

Projective (dually injective) objects are important objects of investigation in algebra, tracing the origins back to ring and module theories. In this talk, I will speak about projective objects relative to specific classes of epimorphisms; dually, injectives with respect to certain classes of monomorphisms