Abstracts of Presentations
“One of the first and foremost duties of the teacher is not to give his students the impression that mathematical problems have little connection with each other, and no connection at all with anything else. We have a natural opportunity to investigate the connections of a problem when looking back at its solution. The students will find looking back at its solution really interesting if they have made an honest effort, and have the consciousness of having done well. Then they are eager to see what else they could accomplish with that effort, and how they could do equally well another time. The teacher should encourage the students to imagine cases in which they utilize again the procedure used, or apply the result obtained. Can you use the result, or the method, for some other problem?” (Pólya, 1973 pp.15-16). Cai and Hwang (2023) give a comprehensive treatment on how to implement problem posing in the classroom. In this talk, we refocus on Pólya’s problem posing trajectory that emanates from the result or the method of a recently solved problem. We propose that making the prompts “Can you use the result for some other problem? Can you use the method for some other problem?” explicit can direct the student into posing meaningful problems as mathematicians would. Examples will be discussed.
References
Cai, J., & Hwang, S. (2023). Making mathematics challenging through problem posing in the classroom. In Mathematical challenges for all (pp. 115–145). Springer International Publishing.
Pólya, G. (1973). How to solve it: A new aspect of mathematical method (2nd ed.). Princeton University Press.
A collaborative experience with poetic inquiry
Grounded on human’s tendency to repeat or rephrase each other, poetic inquiry, as a research method, examines how recurring or parallel lines contribute to the (mathematical) meaning constructed within a (mathematics) discourse. In this sharing, I discuss the collaborative use of poetic inquiry by three researchers (including myself), in presenting and analyzing transcribed classroom dialogue, with particular attention to how repetitions, parallelisms and rhythms of various linguistic aspects, gestures and actions are evident in shaping students’ mathematical sense-making. Using a single transcript from a mathematics lesson, we experimented with different approaches of noticing poetic structures, as influenced by our respective research interests. Rather than seeking a singular interpretation of the data, we illuminate how each poetic form can offer a different way of thinking about students’ mathematical learning processes. Through reflecting on our collaborative process and the interpretive entanglements it brought to the fore, we argue that poetic inquiry provides a generative methodology for deepening our understanding of students’ mathematical engagement and meaning-making processes.
Augmented reality (AR) in Primary Mathematics: Effects on spatial skills
Spatial skills are important for learning geometry, but visualising and mentally transforming geometric objects can be challenging for primary students. This talk examines how augmented reality (AR) can be integrated into mathematics lessons to support spatial learning. It draws on a three-month quasi-experimental study involving 86 Grade 6 students in two intact classes at a public primary school in eastern China. Guided by embodied cognition and inquiry-based learning, five instructional modules combined AR visualisation and manipulation with students’ predictions, exploration, and explanation. Spatial skills were assessed before and after the intervention across four dimensions: recognition, movement, orientation, and measurement of geometric figures. Using selected classroom tasks, the presentation will illustrate the curriculum design, examine the dimension-specific findings, and consider exploratory gender-related patterns and the study’s limitations.
Mathematical Creativity in Schools: A Review of Literature and Theoretical Frameworks
This presentation provides a comprehensive review of the literature on mathematical creativity among school students. It traces the conceptual shift from early psychological models of divergent thinking to domain-specific frameworks, such as relative creativity and the Four-C model. The review synthesizes key cognitive, affective, and pedagogical dimensions, including Multiple Solution Tasks, creative self-efficacy, and problem posing, to examine how creativity is defined, evaluated, and fostered in classroom settings. Emerging trends and gaps in the literature will be highlighted to discuss implications for teaching and curriculum design.
Beyond problem posing: Reimagining mathematical inquiry through noticing, wondering, and extending
This sharing will examine emerging directions in the research on mathematical problem posing and consider what these directions might mean for mathematics teaching and future research. Particular attention will be given to the role of the teacher—not simply as a provider of problem-posing tasks, but as a facilitator and modeller of mathematical inquiry who notices students’ mathematical ideas, asks questions that open possibilities, and helps students refine and extend their thinking. The presentation will highlight several promising research frontiers.
The central limit theorem is a cornerstone of statistical inference, but its classical fixed-dimensional formulation does not directly address settings in which the dimension grows with the sample size. This talk introduces high-dimensional central limit theorems for sums of independent random vectors, emphasizing how the accuracy of Gaussian approximation depends on dimensionality, moment conditions, covariance structure, and the geometry of the sets over which probabilities are compared. We focus on two important classes of sets—hyperrectangles and Euclidean balls—and explain their connections to maximum-type and quadratic statistics, respectively. Selected results and proof ideas will illustrate both the possibilities and the limitations of Gaussian approximation in high dimensions.
Building on this framework, we discuss Gaussian multiplier bootstrap methods and their role in obtaining critical values for high-dimensional inference. We also clarify the distinction between Gaussian approximation for a random vector and normal approximation for a scalar test statistic, with particular attention to quadratic forms. Applications to simultaneous inference and two-sample covariance testing will illustrate how these approximation results can inform the construction and calibration of statistical procedures.
Many statistical procedures work in two stages: they first estimate a nuisance quantity and then make a discrete choice from the adjusted data. Change points are located after the serial dependence of the noise has been estimated or a smooth baseline has been removed; genes are screened after the samples have been normalized. How accurate must the first stage be for the plug-in choice to agree with the choice an oracle would make if it knew the nuisance? The usual answer is a convergence rate, but a selector responds only to how its candidate models are ordered: an error can be large without changing that order, or small and reverse it. For penalized least-squares selectors, an exact identity shows that first-stage error enters each comparison only through a low-rank projection, so that much of it is invisible to the decision. The plug-in choice reaches the oracle's decision once the visible part of the error is of smaller order than the square root of the decision margin, and this order cannot be improved. The theory returns three different verdicts for three common first stages: after autoregressive filtering, any consistent estimator recovers the number of change points; after smoothing, undersmoothing is harmless but oversmoothing creates spurious breaks; after feature-average normalization, a systematic shift persists however precise the first stage. I will illustrate these verdicts with simulations and with air-quality, array-CGH and spike-in microarray data, and end with a glimpse of ongoing work on recovering urban boundaries from nighttime-light images.
This is joint work with Kaiming Cheng (Zhejiang Gongshang University) and Keming Yu (Brunel University of London).
A spanning tree of a graph is a connected, acyclic spanning subgraph, and the resistance distance between two vertices is the elective resistance between them when the graph is viewed as an electrical network. The enumeration of spanning trees and the computation of resistance distances are two classical and closely related topics in graph theory and electrical network theory, with connections to theoretical physics, computer science, and other elds.
In this talk, I will introduce some methods for enumerating spanning trees and computing resistance distances, together with several classical results. I will also discuss some related developments and extremal problems, and conclude by presenting some of our work on these topics.
Let G be a simple graph with a vertex ordering v1, …, vn. Its adjacency matrix is the n × n matrix A = (aij), where aij = 1 if vi and vj are adjacent, and aij = 0 otherwise. Since A is real and symmetric, all its eigenvalues are real. The largest eigenvalue of A, denoted by ρ(G), is called the spectral radius of G. Adjacency matrices arising from different vertex orderings are similar and therefore have the same eigenvalues. Thus, ρ(G) is independent of the chosen vertex ordering.
For a positive integer r and a positive real number α, let Wr(α) be the smallest subset of [0, ∞) containing α and satisfying the following closure rule: for any integer s with 1 ≤ s ≤ r and any positive elements q1, …, qs ∈ Wr(α), not necessarily distinct, set
q = α − ∑i=1s 1/qi.
Then q ∈ Wr(α) whenever either s < r and q ≥ 0, or s = r and q = 0.
A tree is a connected graph which has no cycles. We prove that 0 ∈ Wr(α) if and only if there exists a tree T with spectral radius ρ(T) = α and maximum degree Δ(T) ≤ r. As an application, we characterize the pairs of integers k, r ≥ 2 for which there exists a tree T satisfying ρ(T) = √k and Δ(T) = r. We show that, for any integers k, r ≥ 2, there exists a tree T with Δ(T) = r and ρ(T) = √k if and only if k/4 + 1 < r ≤ k.