Abstracts of Presentations

Productive Teacher Noticing: An Untapped Resource in Mathematics Education

Dindyal Jaguthsing

During the past few years interest in Teacher Noticing has considerably increased. The idea that teachers attend to, interpret and respond to significant events in the classroom ¬¬(Jacobs, Lamb, & Philipp, 2002; Van Es & Sherin, 2008) is a prominent aspect that has been highlighted in the literature. In this presentation, we highlight the related idea of Productive Teacher Noticing (PTN) that has the potential to enhance classroom instruction even further. We describe the case of Alice, a teacher from our project, who through her practice that involved the use of typical problems, was influential in our re-conceptualisation of PTN in the teaching of mathematics.

Students' Errors and Misconceptions in Quadratic Inequalities

Hng Choon Keong

This qualitative study set out to identify the errors which secondary students commonly make when studying quadratic inequalities; understand what misconceptions students have; and investigate why students make these errors and end up having these misconceptions. Data were collected from 58 students coming from five different classes. The students had to complete a test with 11 items and twelve of them from different ability ranges were selected for one-to-one interviews. The five teachers who taught these students were also interviewed. It was found in this study that students made four main categories of errors aligned with the framework proposed in a study by Godden et al. (2004) – careless errors, procedural errors, calculation errors and application errors. Results also showed that students routinely applied their prior knowledge about equations incorrectly when solving quadratic inequalities.

The title of my talk is Development and Field-Testing of an Instrument for Rating Cognitive Demands of Mathematical Assessment Items

Ng Wee Leng

Teachers’ judgements of the cognitive demand of assessment items have implications for the nature of students’ learning experiences. However, existing taxonomies for classifying cognitive demand are often not customised for A-level mathematics teachers to use. This talk reports on the development and field-testing of a cognitive demand instrument specifically for helping A-level mathematics teachers sharpen their judgements of the cognitive demand of A-level mathematical assessment items.

Making a Teacher’s Thinking Visible through the lenses of Commognition and Teacher Noticing

Chia Su Ngin

Learning Mathematics in Primary Schools is often mediated through the use of multiple representations. However, teachers may not pay enough attention to the way they use these representations. Given that the translations between representations and language may not always be smooth, it may be insightful to examine how teachers mediate learning through the use of multiple representations and language. In this presentation, I will share key ideas in commognition and teacher noticing before I present a case study of how Hannah, a teacher, mediate learning of percentages in her class. I will also introduce the idea of a ‘Mediation Flowchart’ and demonstrate how these flowcharts can be used to describe and analyse a teacher’s use of language and multiple representations.

The Changes of Four Thai Primary 6 Mathematics Teachers’ Reflections on Their Teaching

Muanfun Yaowiwat

This session will report the experiences of conducting a 4-month video-based professional development programme to a group of Primary 6 mathematics teachers in Thailand and some preliminary findings related to the teachers’ changes of their reflections on teaching. Four primary 6 mathematics teachers participated in the study. They attended a 3-day workshop to become familiar with the mediation strategies framework, after which each of the teacher was video-recorded during mathematics lessons where mediation strategies were used. Stimulated recall interviews were conducted for the teachers to reflect on their teaching. During the presentation, the research methods will be also discussed.

Henstock’s Stochastic Integral

Toh Tin Lam

It has been shown over the past decades that Henstock’s approach has been able to give an equivalent definition of the Ito stochastic integral with respect to Brownian motion and even continuous semimartingales. This talk presents how Henstock’s approach could be used to give an alternative definition to the Stratonovich integral with respect to continuous semimartingales, and the possible ways to expand research in this direction.
Open Set Lattice of the Minimal Prime Spectrum of a Multiplicative Lattice

Nai Yuan Ting

The natural abstraction of the set of all ideals of a commutative ring is the multiplicative lattice. The main research task on abstract ideal theory, initiated by R.P. Dilworth and M. Ward, is to extend the results on ideals of commutative rings to multiplicative lattices. We shall address the following problem: for a given multiplicative lattice L and a subspace S of the prime spectrum (all prime elements equipped with the hull-kernel topology) of L, can we find a subset of L that is order isomorphic to the open set lattice of S? In this talk, we consider the case where S is the set of all minimal prime elements of L. The main theorem is that for a reduced coherent multiplicative lattice L satisfying certain conditions, the open set lattice of the subspace of all minimal prime elements of L is isomorphic to the lattice of all normal elements of L. The corresponding result for this theorem in the case of lattice of all ideals of a commutative ring with identity is also obtained.
A Brief Survey of Zero-Divisor Graphs of Commutative Rings

Teo Kok Ming

Let R be a nonzero commutative ring with identity. The {\it zero-divisor graph} Γ(R) of R is the (undirected) graph with vertices that are zero-divisors of R, and distinct vertices r and s are adjacent if and only if rs=0. We define a relation ∼ on R by r∼s if and only if annR(r)=annR(s). Then ∼ is an equivalence relation on R. The compressed zero-divisor graph ΓE(R) of R is the (undirected) graph with vertices the equivalence classes induced by ∼ other than [0] and [1], and distinct vertices [r] and [s] are adjacent if and only if rs=0.

In this talk, we shall give a brief overview of some results on zero-divisor graphs and compressed zero-divisor graphs obtained by various people working in this area.

 

Spanning Trees of Graphs

Lim Xiao Kai

A tree is a connected graph with no cycles, while a spanning tree T of a connected graph G is a subgraph of G such that T is a tree that includes all vertices of G. We explore two ways to calculate the total number of spanning trees of any connected graph G, one via its Laplacian matrix L(G), and another via the degrees of its vertices.

On Congruences for Andrews' Singular Overpartitions

Uha Isnaini

An overpartition of a positive integer n is a non-increasing sequence of natural numbers whose sum is n in which the first occurrence of a number may be overlined. Recently, Andrews introduced singular overpartitions which can be enumerated by ¯¯¯¯Ck,i(n), the number of overpartitions of n where only parts congruent to ±i(modk) may be overlined, and no part is divisible by k. In this talk, we first give a brief introduction for integer partitions and overpartitions, and then proceed to present a brief survey on some results for singular overpartitions. Finally, we present our work on singular overpartitions.