Abstracts of Presentations

Mathematics Education
Types of talk and mathematical knowledge for teaching problem-solving: Dialogic analysis of a lesson study

Professor Stéphane Clivaz, Lausanne University of Teacher Education, Switzerland

The presentation will focus on a research aiming to investigate mathematical knowledge for teaching problem-solving in lesson study among primary school teachers (Clivaz, Batteau, et al., 2023; Clivaz, Daina, et al., 2023) . Eight meetings were analysed, revealing mainly three types of talk, cumulative, disputational, and exploratory, evolving across the lesson study cycle. We will consider some theoretical and methodological issues in linking the analysis's micro, meso and macro levels. We will also present some of the results, showing that knowledge levels differed among the types of talk, offering insights into teacher knowledge development.

Clivaz, S., Batteau, V., Pellet, J.-P., Bünzli, L.-O., Daina, A., & Presutti, S. (2023). Teachers’ mathematical problem-solving knowledge: In what way is it constructed during teachers’ collaborative work? The Journal of Mathematical Behavior, 69, 101051. https://doi.org/https://doi.org/10.1016/j.jmathb.2023.101051

Clivaz, S., Daina, A., Batteau, V., Presutti, S., & Bünzli, L.-O. (2023). How do dialogic interactions contribute to the construction of teachers' mathematical problem-solving knowledge? Construction of a conceptual framework. International Journal for Lesson and Learning Studies, 12(1), 21-37. https://doi.org/10.1108/IJLLS-03-2022-0031

Contradiction in Andy’s Mathematical Knowledge for Teaching MK(f)T and Mathematical Knowledge in Teaching MK(i)T

Chia Su Ngin, MME, NIE

Many researchers have examined how teachers transform their knowledge in theory into practice. This transformation is often difficult to capture and can be problematic. In this presentation, I would first briefly introduce how the design of SSA (Sort Sequence Act) methodology can be used to capture the complexity of teachers’ knowledge based on Ball’s Mathematical Knowledge for teaching (MKT) framework.

Using the case of Andy, I will discuss how a teacher who may be rich in knowledge for teaching may not transform his knowledge effectively into practice.

Exploring teachers’ metacognition and its support in the development of students’ metacognition

Lee Mei Ling Sharon. MME, NIE

Teachers play an important role in promoting metacognition in the classroom. This study explores teachers' metacognition in the context of its support in the development of students' metacognition, and aims to understand how teachers draw on their metacognitive practices to impact their instructional practices and support students' metacognition in the classroom. The qualitative study focuses on two mathematics teachers who taught Normal Academic level mathematics, examining their orientations, resources, and goals through an online survey, an interview using stimulus text, a classroom lesson observation, and a video-stimulated recall interview. The findings emphasise the importance of teachers developing their pedagogical understanding of metacognition and mathematical knowledge for teaching. Additionally, the findings highlight the significance of high-quality professional development experiences and instructional resources in facilitating students' metacognition. This case study also provides valuable insights for the design of effective teacher professional development programmes for metacognition.

Investigating secondary students’ procedural skills, conceptual understanding, and application of measures of central tendency

Teo Lin Lin. MME, NIE

This study investigated secondary students’ procedural skills and conceptual knowledge of mean, median, and mode, and their ability to select and apply appropriate measures of central tendency across various contexts. Eighty-five participants completed a paper-and-pencil test designed to assess these competencies. A mixed-methods approach was employed, combining quantitative analysis of test scores with qualitative examination of students’ written work to synthesize and triangulate findings. Findings suggested discrepancies between students’ procedural proficiency, conceptual understanding, and practical applications, highlighting the need to enhance students’ ability to connect procedural knowledge with conceptual understanding in real-world contexts.

Mathematics Education in Asia

Professor Berinderjeet Kaur, MME, NIE

In the 4th International Handbook of Mathematics Education, one of the five sections is devoted to Mathematics Education in Asia. But ‘Why a Focus on Mathematics Education in Asia?’ This presentation by the editor of this section, will share with participants a response to the above question. It will also share insights beyond perceptions through a cursory tour of the five chapters in the section, namely:
  • Mathematics Education in Asia – Curriculum and its Origins by David Lindsay Roberts, Rakhi Banerjee, and Catherine Vistro-Yu;
  • Myths and Realities: Teaching of Mathematics in High Achieving East Asian Countries by Yew Hoong Leong, Oh Nam Kwon, Keiko Hino;
  • East Asian Students’ Mathematics Performance: A Values-based Macroeducation Perspective by Wee Tiong Seah and Ting Ying Wang;
  • by Xinrong Yang and Gabriele Kaiser; and
  • Beyond Asian and Western Traditions of Mathematics Education by Jinfa Cai, Anne Watson, Binyan Xu

Mathematics
Distances in Pythagorean Graphs

A/P Tay Eng Guan, MME, NIE.

We define a c-Pythagorean graph, where c is the largest integer in a primitive Pythagorean triple, PGc as an infinite graph with V(PGc) = {(x, y) | x, y are integers} and E(PGc) = {(x1; y1)(x2, y2) | (x2 - x1)2 + (y2 - y1)2 = c2}. We consider the problem of finding the distances between vertices. A closed-form expression for the smallest non-trivial case of c = 5 is given and generalisations to higher values of c are considered. The problem can be seen as an extension of the Knight’s Tour Problem. Computational approaches seem necessary to explore the problem.

The distribution of Laplacian eigenvalues of graphs

Xu Leyou, SPMS, NTU

The Laplacian matrix of a graph is the difference of the diagonal matrix of vertex degrees and the 0-1 adjacency matrix. The eigenvalues of the Laplacian matrix of a graph G are called the Laplacian eigenvalues of G.

Any Laplacian eigenvalue of an n-vertex graph lies in the interval [0,n]. The distribution of Laplacian eigenvalues of graphs is relevant to many applications related to Laplacian matrices. Little is known about how the Laplacian eigenvalues are distributed in the interval [0,n]. In this talk, we discuss results relating the number of Laplacian eigenvalues in specific intervals and graph invariants.

Double Itô-Wiener integral

Tay Yong Khin, MME, NIE

My project is motivated by the Itô-Wiener integral in the stochastic case. By drawing parallels from the Itô-Wiener integral, we define our own version of this integral in the non-stochastic case. I modified the Henstock division to the Itô-Wiener division on the interval T2=[0,1]×[0,1]. Following the stochastic case, this new division considered the diagonal Gc={(x1,x2 )∈ T2:x1=x2 } and we use the function f0 (x) for the Riemann sum where we set the value to be zero if x lies on the diagonal. Additionally, the tags can lie either outside or within their respective subintervals. Hence, multiple rectangular intervals can share the same tags. In my presentation, I will share the motivation and development of this theory, along with key results such as the Henstock Lemmas, some convergence theorems and lastly the iterated integral.
Formal Concept Analysis and Its Applications

Alicia Goh, MME, NIE

This study explores the application of Formal Concept Analysis (FCA) to analyse geometric relationships among quadrilaterals. FCA systematically organises relationships between objects and attributes using a concept lattice, a structure that encodes meaningful classifications. The study begins by introducing fundamental FCA concepts, including ordered sets and complete lattices. It then examines formal contexts, formal concepts (extent and intent), and the Basic Theorem on Concept Lattices. Additionally, different representations of concept lattices, such as line diagrams and lectic order, are discussed. A key component of this research involves constructing a formal context where quadrilaterals serve as objects and their geometric properties, such as those related to sides, angles, and diagonals, serve as attributes. From this context, a concept lattice of quadrilaterals is generated by manually finding all extents of the context. The concept lattice visually represents the hierarchical relationships between different quadrilateral types. The study further explores attribute implications and pseudo-intents, which discovers logical dependencies among geometric properties. This research contributes by clarifying certain aspects of FCA theory through filled-in proof details and remarks. Furthermore, the creation and analysis of the quadrilateral concept lattice offers a fresh perspective on geometric classification and the relationships between different quadrilateral types. Future research directions include exploring many-valued contexts and leveraging computational tools to extend FCA applications.

Statistical inference for multivariate functional data

Asst/Prof Zhu Tianming, MME, NIE

As technology continues to advance rapidly, multivariate functional data (MFD) have become increasingly prevalent across diverse disciplines, including biology, climatology, finance, and many other fields. Although MFD are encountered in various domains, the development of methods for hypothesis testing on mean functions, especially for the general linear hypothesis testing (GLHT) problem, has been limited. Current methods primarily rely on permutation-based or asymptotic tests. In this talk, I will give a brief overview of the GLHT problem for MFD.