Abstracts of Presentations

Think fast, think slow: algebra learning

Florence Chiang Siu Sia

Why do students make common mistakes even if they do not conspire? Some of the mathematical mistakes students make could be due to common human cognitive biases. One of the project aims is to explain students' mistakes in Algebra learning through the lens of different psychological modes of operations, namely “Think fast” and “Think slow”.

Difficulties and suggestions for learning independent events

Koh Hui Li

This project examines the difficulties in students’ learning of probability concepts related to “Independent Events”. In addition, specific instructional ideas will be offered help students overcome these difficulties.

Use of games in mathematics classroom

Eunice LEONG

In this talk, the rationale and theoretical principles of using games for mathematics teaching and learning, and the underlying principles of assessment will be discussed. Following that, a conceptualization of the use of games for assessing students’ mathematics learning in the Singapore classroom will be discussed.

Enactment project – Qualitative analysis of survey data

Tong Cherng Luen, Dr Quek Khiok Seng & A/P Tay Eng Guan

In this presentation, we will share an exploratory approach we adopted for analysis of data. Our exploration involved 36 Likert scale items that explored teacher actions during mathematics lessons in secondary school. The outcome of which were 4 factors that could possibly further illuminate pedagogy of mathematics teachers in Singapore secondary schools.

Perspectives on mathematics teaching in the US and Singapore schools

Diem H Vuong

The presenter is one of the three recipients of the Fulbright Distinguished Awards in Teaching fellowship from the United States. During her stint in Singapore, she is conducting an inquiry project on Singapore Mathematics Curriculum and the strategic implementation lesson plans that teachers are successfully engaging students in the secondary mathematics education. The purpose of her project is to improve the quality of mathematical instruction in problem-solving and to close the achievement gap in the students’ mathematical curriculum essential knowledge and skills in Dallas Independent School District. In this session, she will draw on her experiences in the U.S., observations in Singapore schools, and share her perspectives about the teaching and learning of mathematics.

Generalising a numerical linear pattern

Kenan Kok

In this presentation, I will share some generalizing strategies that successful students used to find the far term and functional rule in a numerical linear pattern generalizing task. I will also highlight the consistency of generalizing strategies that these successful students employed in establishing both the far term and functional rule. The implications of these findings will be discussed.

Cognitive processes of mathematical investigation

Dr Joseph Yeo Boon Wooi

I will share from my PhD study some findings of how students think when they attempt open investigative tasks. I will begin with a discussion of the similarities and differences between investigation and problem solving, followed by a model of cognitive processes in mathematical investigation. Then I will focus on some of these processes such as problem posing and conjecturing. In fact, there were some surprises in the empirical data that resulted in me refining the theoretical model.

On an amazing identity of Ramanujan

Toh Pee Choon

In this talk, I will introduce and prove an amazing identity that involves the sum of two squares function. Although the identity is attributed to S. Ramanujan, it does not appear in any of his published works. I will also present generalizations of Ramanujan’s identity that were obtained jointly with H. H. Chan.

On k-restricted overpartitions

Uha

In this talk, we introduce k-restricted overpartitions, which are generalizations of overpartitions. In such partitions, among those parts of the same magnitude, one of the first k occurrences may be overlined. We first give the generating function and establish the 5-dissections of k-restricted overpartitions. Then we provide a combinatorial interpretation for certain Ramanujan type congruences modulo 5. Finally, we pose some problems for future work.

Scott's Information system

Li Zhuolun

In his pedagogical effort to make domain theory more palatable to computer scientists and logicians, Dana Scott introduced the notion of information system. An information system is a formal way of setting up a universe A of tokens, an identified collection of subsets called Con, and an entailment relation between Con and A. This entailment relation is required to satisfy some very natural axioms, and it turns out that an information system induces a domain and vice versa. In this talk, this strong link between information system and domains is explained.

A short excursion into computational geometry

Elaine Wong

Computational geometry is a research field concerning both mathematics and computer science, devoted to designing, analysing and implementing algorithms that solve geometric problems. In this project, some fundamental problems arising in different application domains of computational geometry have been explored. The way that these problems can be transformed into purely geometric problems and how algorithms make use of these geometric properties to solve the problems have been studied. These problems include the closest pair problem, the convex hull and more.

Parking functions

Dong Fengming

The notion of a parking function was introduced by Konheim and Weiss in 1966. Suppose that there are n drivers labeled 1,2,⋯,n and n parking spaces arranged in a line numbered 1,2,⋯,n. Assume that driver i has its initial parking preference f(i), where 1≤f(i)≤n. Assume that these n drivers enter the parking area in the order 1,2,⋯,n and driver i park at space j, where j is the minimum number with f(i)≤j≤n such that space j is unoccupied by the previous drivers. If all drivers can park successfully by this rule, then (f(1),f(2),⋯,f(n)) is called a {\it parking function} of length n. Mathematically, a function f:Nn→Nn, where Nn={1,2,⋯,n}, is called {\it a parking function} if the inequality |{1≤i≤n:f(i)≤k}|≥k holds for each integer k:1≤k≤n. For example, for n=2, (f(1),f(2))=(1,1), (f(1),f(2))=(1,2) and (f(1),f(2))=(2,1) are parking functions, but (f(1),f(2))=(2,2) is not. It can be shown easily that f:Nn→Nn is a parking function if and only if there is a permutation π1,π2,⋯,πn of Nn such that f(πj)≤j holds for all j=1,2,⋯,n. Konheim and Weiss proved that the number of parking functions of length n is equal to (n+1)n−1, which is equal to the number of spanning trees of the complete graph Kn+1.

Parking functions are related to many topics in combinatorial theory. In this talk, I will introduce various extensions of parking functions.