Abstracts of Presentations

Relevance of Early Numeracy Skills for School-level Mathematics

Dr Jose David Munez Mendez (National Institute of Education, Singapore)

Contrary to common assumptions, research reveals that infants exhibit rudimentary mathematical knowledge, suggesting an innate capacity for numerical understanding from the earliest stages of development. In this talk, I delve into the critical importance of nurturing young children's early numeracy development as a cornerstone for their later success in school-level mathematics. Far beyond simple counting, early numeracy encompasses a diverse array of foundational abilities that lay the groundwork for mastering more complex mathematical concepts introduced later in formal schooling. For instance, 5-year-olds can process non-symbolic ratios. When presented with two gum machines containing a mix of blue and red gums, they select the gum machine with the best proportion of the target color. This capacity serves as a precursor to the acquisition of more sophisticated mathematical competencies such as understanding rational numbers and fractions. Indeed, research consistently demonstrates the predictive capacity of young children's numeracy skills for a multitude of later-life outcomes, including academic achievement, earnings, and employment prospects. Understanding the nuanced interplay between these foundational skills and subsequent mathematical achievements is pivotal for educators and policymakers in designing effective early math interventions and curricular frameworks.

Teacher Noticing and Use of Language as a Resource in the Teaching and Learning of Mathematics

Ms Pauline Tiong (Simon Fraser University, Canada)

While the notion of language as a resource is not new and of increasing interest in mathematics education research, not many researchers focus on understanding the notion from the perspectives of experienced mathematics teachers. In this presentation, I share my Ph.D. research which seeks to understand the existing state of how teachers are noticing and using language (particularly the mathematics register) as a resource in the mathematics classroom. In my research, I interviewed eleven teachers and asked them to reflect upon what they noticed and how they would respond to a series of tasks, designed to illustrate situations which might lead to language-related dilemmas and challenges in using the mathematics register. By accounting for their responses through the lens of language-related dilemmas and orientations, I observed two main categories in which language has been noticed and used in the mathematics classrooms, namely as a resource for developing mathematical understanding, and as a resource for mathematics talk. Additionally, I analysed the teachers’ responses to three tasks through the Mathematics Register Knowledge Quartet to provide an exemplification of their knowledge (or lack of knowledge) of the mathematics register in relation to the teaching and learning of related concepts.

Teaching Matrix & Transformation in Chinese Classroom Using Dynamic Geometry Software: A Framework for Geometry Instruction

Chen Kexin (National Institute of Education, Singapore)

As it is increasingly emphasized worldwide on preparing post-secondary students adequately for their study of undergraduate mathematics, basic elements of advanced mathematics have been incorporated into senior high school mathematics curriculum. Matrix and Transformation is one such topic that provides students with opportunities to engage in deductive reasoning.

In teaching Matrix and Transformation, we modelled after the intuitive-experimental approach with the underlying problem-solving models of Mason (2010) and proposed an instructional framework, the Specializing-Conjecturing-Convincing-Generalizing (SC2G) framework, for using GeoGebra to provide students with learning experiences in problem solving. We believe that this approach facilitates the development of students’ relational understanding, and their construction of knowledge through discovery learning.

We aim to study the impact of GeoGebra on students’ learning through the use of the SC2G framework. Our goal is to develop a design of teaching and learning based on the SC2G framework using design-based research. Through several iterative cycles, we expect to develop a local theory of instruction for using GeoGebra in teaching geometry. We tap on the affordance of design-based research to collect multiple types of data to study the various impacts on student learning and teacher teaching.

Visualizing Mathematics Teacher's Specialized Knowledge Using Commognitive Lens during Online One-to-one Tutoring

Dr. Lu Jijian (Hangzhou Normal University, China)

Digital society's transformation has led us to adopt an increasing number of online one-on-one tutoring models, signifying a significant shift in educational practices. This research is a sequel to our previous exploration of the cognitive processes during one-to-one tutoring, aiming to visualize the mathematical teachers' specialized knowledge (MTSK) through a commognitive lens during online one-on-one tutoring. This study adopted a case study approach, utilizing 270 minutes of computer-supported one-on-one tutoring videos. We have visualized the mathematical teachers' specialized knowledge across the domains of Mathematical Knowledge (MK) and Pedagogical Content Knowledge (PCK). Additionally, we conducted a lag sequential analysis of the teachers' professional knowledge. In conclusion, we have formulated a model of specialized knowledge possessed by mathematics teachers grounded in commognitive principles. Additionally, we have presented recommendations aimed at improving the online tutoring process.

Designing and Orchestrating productive discussion to build on mathematically significant pedagogical opportunities using typical problems

Dr Choy Ban Heng (National Institute of Education, Singapore)

Orchestrating productive discussions and building on mathematically significant pedagogical opportunities during lessons are high-leverage classroom practices that position student thinking at the heart of adaptive teaching. Doing this usually requires teachers to select or adapt mathematically rich tasks for use in their classrooms, which can be pedagogically challenging. This raises an important question: How can teachers design and carry out adaptive teaching given the limited curriculum time? In this paper, I will present the case of Mary, who is part of a lesson study team, to illustrate how the affordances of typical problems can be harnessed to create mathematically significant pedagogical opportunities for building student thinking. Implications for practice and research will be briefly discussed.
Erdos-Ko-Rado Combinatorics

Dr. Ku Cheng Yeaw (SPMS - NTU, Singapore)

The Erdos-Ko-Rado theorem is a cornerstone result in extremal combinatorics that gives the size and characterization of the largest intersecting families of k-subsets of a finite set. The theorem has been extended to many other combinatorial and algebraic objects. In this talk, I will provide a glimpse into this fascinating area, and some problems I find interesting, particularly for permutations and perfect matchings.

Multivariate Statistical Methods for Discriminating Geographical Origins of Red Wines using Spectroscopic Data

B Viveka (National Institute of Education, Singapore)

This research delves into the utilization of multivariate statistical methods, specifically Principal Component Analysis (PCA), Linear Discriminant Analysis (LDA) to distinguish the geographical origins of red wines using high-dimensional spectroscopic data. Focusing on Merlot, Cabernet Sauvignon, and Pinot Noir varietals sourced from Australia and France, this study addresses the critical need for accurate wine authentication and provenance determination in the global market. By harnessing the inherent chemical signatures captured through spectroscopic analysis, coupled with advanced statistical techniques, this research aims to develop interpretable classification models capable of effectively discriminating between wines of different origins. The integration of PCA and LDA, along with the rigorous model evaluation and selection afforded by cross-validation, facilitates the creation of reliable predictive models that enhance the authenticity and traceability of red wines. Through empirical validation and analysis, this study contributes to the advancement of methods for ensuring quality assurance within the wine industry.

Pointless Topology

Tan Zheng Han, Hans (National Institute of Education, Singapore)

The word for “topology” comprises two Greek root words “place” or “location” (τόπος) and “study” (λόγος). Classically, point-set topology is concerned with notions that articulate “getting near to a location/point” or “neighbourhood of a location/point” using the concept of open sets. More formally, a topological space X is a set endowed with a distinguished collection t of subsets of X – called open sets – that satisfies three basic axioms that are distilled from those arising from the usual Euclidean topology on the real line, i.e., closure under the formation of finite intersections and arbitrary unions. My talk shows how one can still talk about topological notions without referring to the elements or points of the space, hence the title “pointless topology”. Ironically, we do so by defining “generic points” of the lattice WX of opens of the underlying space X, which are precisely the completely prime filters of WX. This pointless approach is crucial if one pursues the line of constructive mathematics.
Topology and the Infinitude of the Primes

Leong Chong Ming (Nanyang Junior College, Singapore)

The infinitude of the primes is one of the most proved results in mathematics. It is interesting to note that there are several different proofs of this famous result based on entirely different domains of mathematics. Euclid probably gave the first proof by a clever construction which is still relished by mathematics enthusiasts today. One ingenious topological proof was given by Furstenberg in 1955 while he was still an undergraduate. What is surprising about this beautiful proof is the ‘unreasonable’ connection between topology and number theory. In this talk, I shall attempt to re-enact the topological proof supplied by Furstenberg which according to the famous Hungarian mathematician Paul Erdos, was “a proof from the book”.

The Scott Topology

Ho Weng Kin (National Institute of Education, Singapore)

Point-set topology is a mathematical theory that formalizes convergence and approximation by capturing what it takes for points to get near to a location – reminding us of its Greek etymology: “Topology” in Greek means the study (logoz) of location (tottoz). First courses in topology mostly assume the strongest degree of separation, focusing only on Hausdorff spaces, i.e., spaces in which every two distinct points can be told apart using disjoint open neighborhoods containing each. For a long time, non-Hausdorff topologies are misconstrued as unnatural and hence unimportant. In this talk, we introduce the Scott topology which stands as the most important non-Hausdorff topology, possessing many interesting and fundamental mathematical properties; thus debunking the myth that non-Hausdorff topologies are esoteric.