Abstracts of Presentations

Effectiveness of Teachers’ Participation in a Mathematics Networked Learning Community in Terms of Students’ Classroom Environments and Attitudes to Mathematics in Singapore

Dr Cynthia Seto

Networked learning community aims to promote professional collaboration among teachers from different schools to enhance teaching expertise. As the perspectives of students can provide a valuable source of data for assessing the effectiveness of professional development, a learning environment framework was used to provide process criteria of effectiveness. A new instrument called the Mathematics Classroom Environment and Attitude (MCEA) questionnaire was developed and validated to assess students’ perceptions of classroom learning environment and attitudes to mathematics. The MCEA questionnaire has a total of 40 items that assess the five scales of Cooperation, Teacher Support, Involvement, Problem Solving and Enjoyment.

The MCEA questionnaire was used in a pretest–posttest quasi-experimental design to compare the changes in classroom environment and attitudes of those classes whose teachers’ participated in networked learning community (experimental group) with those classes whose teachers were not in the networked learning community (comparison group).

Data analyses supported the factor structure, internal consistency reliability and discriminant validity of the MCEA questionnaire when used with 375 Primary 5 students from 10 mathematics classes in five schools in Singapore. ANOVA showed that each scale of the MCEA was able to differentiate between the perceptions of students in the different classes. Overall, pretest–posttest changes were larger in magnitude for the experimental group (teachers who participated in networked learning) than for the comparison group (teachers who did not participate in networked learning) for every learning environment and attitude scale.

Simple correlation and multiple regression analyses revealed positive associations between the learning environment and students’ attitudes towards mathematics. All of the four learning environment scales were statistically significantly correlated with attitudes to mathematics. Teacher support and problem solving were significant independent predictors of students’ attitudes towards mathematics for both the pretest and posttest data.

Download presentation

Teachers’ exemplification considerations in the teaching of mathematics

Ng Lay Keow

Examples are intimately intertwined with the teaching and learning of mathematics, yet the routine use of examples and their omnipresence in the mathematics classroom might have undermined perceptions of their significance leading to a lack of relevant research. While researchers have attended to the roles of sub-categories of examples, research into how teachers integrate examples into their teaching, or exemplification in short, remains scarce (Zodik & Zaslasky, 2008). Research has also shown that teachers’ use of examples involves the informed deliberation of a multitude of factors, where the decision is often made instantaneously during the course of a lesson. Given the pervasiveness of examples in mathematics instruction, the examination of mathematics teachers’ instructional examples remains a pressing issue to be uncovered.

Recent developments at the turn of this century have begun to put exemplification in the foreground. Literature also point to a strong link between practitioners’ knowledge and the nature of their instructional examples. Hence, the main study aims to examine the example choice and use by experienced teachers using the multiple case studies approach, in view of its potential to understand complex social phenomena.

To locate four teachers as the key research participants, a questionnaire was constructed, where 121 teachers from 24 secondary schools in Singapore responded. The intent of this sharing is to present part of the questionnaire findings that focus on teachers’ guiding principles in example selection. The results showed that students’ mathematical abilities and the difficulty level of the examples were among the topmost considerations teachers have when introducing mathematical ideas or when selecting homework tasks. There were noticeable variations in teachers’ exemplification considerations for different instructional purposes too. In addition, teachers gave written descriptions of their conception of good examples. The findings are indicative of the interconnectedness between teacher knowledge and their use of examples.

The FOCUS Framework: snapshots of mathematics teacher noticing

Ast/P Choy Ban Heng

In my PhD study, I investigated mathematics teacher noticing—what mathematics teachers see, and how they interpret these observations to make instructional decisions that enhance student reasoning during the planning, delivery and review of lessons. A key result from my study is the FOCUS Framework, which describes the notion of productive noticing. The framework uses a photography metaphor to highlight an explicit focus for noticing, and the key role of pedagogical reasoning (focusing), as essential characteristics of productive noticing. There are two main contributions of the FOCUS Framework: First, it extends existing research on noticing, which focuses mainly on in-the-moment teaching and post-lesson discussions, to emphasise the importance of planning to notice. Next, a theoretical model is developed from the constructs of the framework to pinpoint the specific actions that a teacher can take to attend to, and make sense of, and respond to students’ confusion when they are learning a concept.

In this presentation, I will give an overview of my research on teacher noticing and illustrate how the FOCUS framework can be used to describe and analyse teacher noticing. Snapshots of a few teachers’ mathematical noticing will be shared to demonstrate how the model developed from the framework can provide a picture of teachers’ developing expertise in noticing. Finally, I will highlight some theoretical and practical implications of my work on teacher noticing, and suggest possible areas for further research.

Download presentation

Teaching and Learning through Concrete-Pictorial-Abstract Sequence

Chang Suo Hui

The key instructional strategy for the development of primary mathematics concepts in Singapore Primary Mathematics is the Concrete-Pictorial-Abstract (C-P-A) sequence (MOE, 2007, 2012). This sequence is evident in the National Syllabus and textbooks adopted in schools. Local teachers design their mathematics lessons using the sequence to facilitate learning among our pupils. However, the process from initial exposure of concrete, pictorial and abstract representations to eventual acquisition of the mathematical concept is not explicated. The facilitation by a teacher in pupil’s learning through the C-P-A sequence is unclear. Without an understanding of the learning process a pupil undergoes, teachers are at risk of underutilizing the sequence (Flevares & Perry, 2001). Herein is a proposed theoretical model to explain the learning process of the C-P-A sequence and the accompanying teaching facilitations to guide pupils to achieve mathematical conceptual understanding. The presentation first covers how the theoretical model is derived from examining works of various key learning theorists such as Bruner, Piaget, Dienes and Skemp. Significant of this theoretical model is its intentionality to examine teaching and learning as an integrated unit (Shuell, 1993). Therefore, the model comprised of a teaching and a learning component. Second part of the presentation covers a comparison of the theoretical model against findings obtained from its implementation with two groups of low-progressed pupils in a primary school. A series of six intervention lessons on the Primary Three concept of equivalent fractions are crafted using the theoretical model as the guiding framework for lesson development. The learning progression of the sampled pupils was tracked using various data sources such as task-based interviews and class worksheets. The similarities and differences in the learning progression of pupils proposed by the theoretical model and its actualization are discussed. The discussion includes implications to the original teaching facilitations proposed. The presentation then concludes by highlighting some issues surfaced from its implementations and reflections of the researcher cum practitioner.

Download presentation

Some recent works on Baire class one functions and problem posing strategies involved

Zhao Dongsheng

Good problems play the central important role in both mathematical research and learning. Being able to find and pose new problems is one of the key skills for researchers in the discipline of mathematics. In this talk, by reviewing recent work on the definition of Baire class one functions, I will try to illustrate some strategies of finding and posing research problems in mathematics.

Download presentation

Mathematical research - Problem posing, problem solving and making connections

Nai Yuan Ting

Throughout my five years of study as a part-time PhD research student, I have learnt that there are three areas that are very important in mathematical research: problem posing, problem solving and making connections.

When posing problems, it is important that the statement of the problem must be phrased in such a way that it does not restrict the scope of the problem, but yet precise enough so as to avoid ambiguity. Through reading journal articles and attending oversea conferences, I have learnt from my supervisor and other mathematicians as to how problems can be posed and how problems are being posed. At conferences, mathematicians are interested in the “big ideas” and “intent” of the paper presentation so that they can transfer such “big ideas” and “intent” to other areas of research as well. Some examples will be given to illustrate the above.

As a teacher currently teaching Mathematics at a junior college, I have always taught my students Polya’s problem solving framework as well as the heuristics used. As a research student trying to come up with conjectures for the problems posed, I find myself sometimes guilty of not using some of the heuristics such as observing patterns and working with simpler cases. Besides the use of heuristics in problem solving to come up with the conjectures, I also realise the importance of using certain tools and techniques when trying to prove certain results. I will give examples of some of these tools and techniques.

Most importantly, I have learnt that there are inter-relation between the various areas of Mathematics; this is especially so for the areas of topology, algebra and order theory. It is necessary to make connections between the area of my research with other areas of Mathematics so as to make the results more meaningful. I will share examples of such mathematical results, and will conclude the presentation with some discussion on “point-free” Mathematics.

Download presentation

An identity via Arbitrary Polynomials

F.M. Dong, W.K. Ho, T.Y. Lee

In this talk, we present a family of combinatorial identities. Applying these identities, we proved two conjectures posed in [Thomas P. Dence, Some Half-Row Sums from Pascal's Triangle via Laplace Transforms, The College Mathematics Journal 38(2007), 205-209].

Download presentation

On a new convergence class in k-bounded sober spaces

Hadrian Andradi

Abstract. Recently, J. D. Lawson encouraged the domain theory community to consider the scientific program of developing domain theory in the wider context of T_0 spaces instead of restricting to posets. We respond to this calling by proving a topological parallel of a 2005 result due to B. Zhao and D. Zhao, i.e., an order-theoretic characterisation of those posets for which the lim-inf convergence is topological. We do this by adopting a recent approach due to D. Zhao and W. K. Ho by replacing directed subsets with irreducible sets. As a result, we formulate a new convergence class on T_0 spaces called Irr-convergence and established that this convergence class I on a k-bounded sober space X is topological if and only if X is Irr-continuous.

Download presentation