Abstracts of Presentations

Big ideas in Mathematics: Exploring the dimensionality of big ideas in school mathematics

Jahangeer Bin Mohamad Jahabar (National Institute of Education, Singapore)

Big Ideas in school mathematics can be seen as overarching concepts that occur in various mathematical topics in a syllabus. Although there has been much interest recently in the understanding of Big Ideas, there is little research done in the assessment of Big Ideas thinking. I will share our development of an instrument to measure the Big Ideas of equivalence and proportionality. We analysed the data collected of some pilot items. Our analysis suggests that Big Idea thinking is a multidimensional construct within most school environments.

Opportunities to implement problem solving in Chinese mathematics classrooms within the context of the new mathematics curriculum standards

Li Xin (Changshu Institute of Technology, China)

Mathematics problem solving has been emphasised for a long time in China as an effective way to achieve the objectives of the mathematics curriculum standards. However, due to various reasons, problem solving methods have not been fully implemented in Chinese mathematics classrooms. In the past two years, a small “earthquake” occurred in China's education sector. The implementation of the "Double Reduction" policy (July 2021) and the introduction of new curriculum standards (April 2022) have changed the learning environment of most primary and secondary students. The changes may provide yet an opportunity for problem solving to take root at scale in the system.

Chinese teacher belief in project-based learning: Focusing on mathematical problem solving

Liu Yixuan (NGS from Beijing Normal University)
Yiming Cao (Beijing Normal University)

In the latest reform in the Chinese mathematics curriculum for compulsory education, project-based learning (PjBL) is recommended in the secondary stage (Grade 7-9). In the curriculum documents, PjBL emphasises open-ended problem posing and problem solving in the multidisciplinary real-life context. Teacher belief is systematised, contextualized and always seen as one of the key concepts in professional development as well as curriculum reform. In this research, I investigate teacher belief in mathematical problem solving (MPS) within PjBL, how teachers design and enact PjBL, and the potential affordances and constraints of MPS within PjBL. In this presentation, I will introduce my theoretical work based on three arenas (design, construction, and curriculum mapping) by Remillard (1999) and an observing grid on teachers’ behaviour in MPS lessons by Rott (2020). The main hypotheses of this study are: (a) teacher belief about MPS within PjBL and the daily classroom would have similarities and differences, (b) teacher belief about MPS would be different before and after designing and enacting PjBL, (c) there would be possible links between teachers’ belief and practice in MPS within PjBL, (d) there would be potential affordances and constraints of MPS within PjBL.

Use of card sorting methodology to characterise a primary teacher's mathematical knowledge for teaching

Chia Su Ngin (National Institute of Education, Singapore)

Many studies attempt to study teachers’ mathematical knowledge for teaching (MKT) through instruments such as multiple-choice questions that evaluate how teachers interpret student thinking, how they select materials for instruction, and how they explain concepts and procedures, amongst others. Many of these instruments require considerable context information and often provide a static view of a teacher’s knowledge. In this presentation, I will describe the design and use of a card-sorting instrument that can elicit a teacher’s mathematical knowledge in teaching. By illustrating how the card-sorting instrument was used to capture a primary teacher’s knowledge about teaching division, I argue that card sorting, as part of a suite of other methods, can be a powerful approach to elicit teachers’ knowledge in teaching. I conclude by highlighting possible directions in refining and developing card sorting instruments for future studies.

Unpacking the 'M' in integrated STEM tasks: A systematic review

Choy Ban Heng (National Institute of Education, Singapore)
Gabi Cooper (Griffith University, Australia)

Despite the promise of integrated STEM for authentic learning, it remains unclear whether such integrated approaches can result in significant learning in STEM disciplines. This is particularly true for mathematics, which is often seen as an “accessory” discipline to Science in many integrated STEM tasks. In this systematic review paper, we investigate the issue of integrating Mathematics by unpacking the centricity of STEM tasks, analysing the connections between Mathematics and other disciplines, and highlighting the different faces of Mathematics presented in these tasks. Our findings suggest that although current STEM tasks anchored in mathematics are problem-centric, they have relatively weak inter-disciplinary connections. Analyses also revealed less emphases on mathematics as a way of knowing. We discussed possible implications and suggestions for strengthening the M in STEM from this review.
Orders - their roles and relationships with other structures

Zhao Dongsheng (National Institute of Education, Singapore)

Order structure is one of the three fundamental mathematics structures. In this talk, I shall first consider some familiar orders in sets of numbers and their relationships with the basic algebraic operations, such as addition and product. Some basic properties of orders and their proofs are considered. The order characterization of limits of sequences, the generalized order convergence, lim-inf convergences and the Scott topology will be discussed.

The Dinitz problem

Michelle Kwan (National Institute of Education, Singapore)

Consider n2 cells arranged in an n×n square and let (i,j) denote the cell in row i and column j. Suppose that for every cell (i,j), we are given a set C(i,j) of n colours. Is it then always possible to colour the whole array by picking for each cell (i,j) a colour from its set C(i,j) such that the colours in each row and each column are distinct?

An introduction to Gröbner bases

Alex Lim Tze Ming (National Institute of Education, Singapore)

This presentation will focus on the problem of solving polynomial equations and how Gröbner bases can be used to find all common solutions of a system of polynomial equations. We will first introduce the basics of monomial orderings, division algorithm, and monomial ideals. We will then delve into the main theory of Gröbner bases, exploring their properties and how to construct them using the Buchberger Algorithm. Finally, we will discuss the applications of Gröbner bases and how they can be used to solve the problem of polynomial equations. This presentation will be of interest to anyone working in the field of computational algebra and algebraic geometry.

Applying Particle Swarm Optimisation (PSO) to solve the Fermat Point Problem

Lim Chu Wei

In talk, I will introduce the Fermat Point Problem and its generalization, and then propose to use a stochastic simulation method called Particle Swarm Optimisation (PSO) to provide a way to locate the Fermat Point in the generalized Fermat Point Problem. We also suggest some further areas where we may explore with regard to PSO.

Probabilistic powerdomains over finite posets: Distribution of weights

Ho Weng Kin (National Institute of Education, Singapore)

In this talk, we look only at those probabilistic powerdomains arising from finite posets. Even in the case of finite posets, things can get interestingly messy.