Abstracts of Presentations
The school mathematics curriculum consists of many different strands. Many of these topics are taught as separate topics in the curriculum. Such compartmentalized learning of the subject becomes challenging for students. In this talk, we present how in Singapore the mathematics curriculum is undergirded by “Big Ideas” which present the entire mathematics syllabus under one coherent whole. Using this paradigm, the emphasis is on the connectedness of the various parts of the mathematics syllabus. Implications for mathematics teaching and learning are discussed.
Examining a student's attempts at solving challenging word problems through the lens of cognitive-metacognitive interplay and student noticing
Challenging word problems often present students with challenges in terms of understanding and formulation of solution strategies. In my EdD study, I examined the complex interplay between cognitive and metacognitive processes when students solve challenging word problems. In this presentation, I will describe how one of them, Karen, attempted to solve two challenging word problems through two sets of lenses. First, I will zoom into the cognitive-metacognitive processes using ideas from Newman (1983) and Wilson & Clarke (2004). Second, I will zoom out and examine what and how Karen selected, interpreted, and worked with mathematical features—or what they noticed—during the problem-solving process. By layering these two perspectives, I aim to provide insights into her struggles as she worked through the two challenging word problems.
Investigating Teacher Noticing in Technology-integrated Mathematics Classrooms
It is widely recognized that the quality of mathematics instruction is consequential for improving learning in mathematics classrooms. A key aspect of high-quality instruction is centered around how teachers orchestrate their interactions with students around mathematics tasks. However, teachers face multiple simultaneous events, ranging from students writing down solutions to their peer discussions, when orchestrating these interactions in classroom practice. This makes it challenging for teachers to attend to and interpret critical classroom events for the purpose of responding to students’ thinking. Teacher noticing has been widely studied as an important component of teaching expertise, needed for such responsive teaching practices. While teacher noticing has been extensively studied in standard mathematics classroom contexts, less is understood about how teacher noticing unfolds in technology-integrated classrooms, against the backdrop of ensuring high-quality instruction in the age of digital transformation. In this study, I seek to explore and investigate what and how in-service mathematics teachers notice in technology-integrated classrooms by drawing on the framework of Pedagogical Technology Knowledge (PTK), which integrates teachers’ mathematical knowledge for teaching, personal orientations, and technology instrumental genesis. In this presentation, I will share more on the study’s theoretical framing, research design, and some preliminary findings.
Making metacognition visible: Enhancing problem solving in a Chinese primary mathematics classroom
We present our investigation of how 5th-grade students in China made visible parts of their metacognitive processes while solving a non-routine mathematics problem. This was done while they wrote their workings on the Problem Solving Metacognition (PSM) Worksheet. The PSM worksheet is a modification of the Practical Worksheet used by the MProSE project. The modifications focused on capturing students’ metacognitive behaviors. There is a dedicated metacognitive engagement section in the Worksheet. In this section, there are specific anticipated metacognitive make-and-break points: understanding difficulty, inadequate progress, key insights, and uncertain answer. These were designed to prompt students to record their monitoring and regulation through checked boxes and brief statements. The analysis involved tracing the trigger for metacognitive behaviour and their consequent responses – we call marker-response. The analysis revealed am interesting range of metacognitive profiles among the students. In the talk, some of the students’ workings that made visible their monitoring and regulation will be shared. The findings suggest that the use of the PSM worksheet offers a feasible pathway for transforming implicit metacognitive activities into visible classroom practices for explicitising metacognition.
A framework for computational thinking in the mathematics classroom: An activity for teaching and research
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The set IR ={[a, b]: a ≤b} of all closed intervals of real numbers is a partially ordered set with respect to the reverse inclusion order. This poset has many pleasant properties: (i) it is a domain; (ii) it is bounded complete; (iii) it is a dcpo model of the real line (the set R of all real numbers with the usual topology). A poset model of a topological space X is a poset P such that the maximal point space Max(P) (the set of all maximal points of P with the relative Scott topology) is homeomorphic to X. In this talk, I will first explain the major properties of IR. Then present some general results on poset models of topological spaces, in particular poset models of metric spaces.
Aldous’ spectral gap conjecture originally arose from a probabilistic background. From the perspective of algebraic graph theory, it asserts that any connected Cayley graph on the symmetric group $S_n$, generated by a set of transpositions, has the Aldous property: namely, the second largest eigenvalue of such a Cayley graph is achieved by the standard representation of $S_n$. In this talk, I will introduce several known families of Cayley graphs on symmetric groups that exhibit the Aldous property.
In this talk, I will give a visual introduction to how homotopy groups, starting with the fundamental group, can be defined for finite spaces. The key observation is that every finite space corresponds to a finite poset, which determines a finite simplicial complex, and hence a Hausdorff space. It happens that this sequence of constructions preserves the homotopy groups, allowing us to calculate the homotopy groups of finite spaces. I will outline this construction, present some examples of finite spaces with non-trivial fundamental groups, and highlight the key results needed in the proofs.
In domain theory, finitely separated (FS) domains occupy a central place as a robust and natural class of continuous domains. It is well known that the bifinite domains are precisely the algebraic FS-domains, and this immediately raises a long-standing open question: Is every FS-domain a retract of a bifinite domain? Domains of this latter form are often called RB-domains. The guiding analogy comes from the classical fact that every continuous domain can be realized as a retract of an algebraic domain, suggesting that a similar correspondence might hold at the level of FS-domains.
This talk provides an accessible introduction to the problem FS = RB. I will review the definitions of FS-domains, bifinite domains, and RB-domains, illustrating each with examples. We will then trace the motivations behind the conjecture, survey the principal known results, and discuss why the problem has remained resistant to solution. The aim is not only to highlight the technical challenges but also to emphasize the structural insights the problem offers into the fabric of domain theory. The talk is pitched for researchers and students interested in the interplay between algebraic and topological properties of domains.