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# Toh, Tin Lam

#### A framework for designing comics-based mathematics instructional materials

2022, Cher, Zheng Jie, Toh, Tin Lam

Purpose and Research Question - In this paper we propose a PATH framework for designing comicsbased
instructional material for classroom lesson enactment through conducting a literature review in mathematics education.

Methodology – Systematic review was made with a focus on the potential benefits of comics for education, in particular, on developing students’ motivation for learning and facilitating their knowledge retention.

Findings – We further demonstrate with an exemplar the use of the framework in designing one set of comics-based instructional material for lower secondary mathematics lessons on mensuration.

Significance and Contribution in Line with Philosophy of LSM Journal – An exemplar of a comics-based instructional material designed according to the PATH-CoHANa framework.

#### Calculus for teaching and learning (CASTLE): An exploratory study

2022, Toh, Tin Lam, Toh, Pee Choon, Teo, Kok Ming, Zhu, Ying, Tan, Victor, Tang, Wee Kee

#### From the past to the future of technology in mathematics education in Singapore

2023, Toh, Tin Lam

This paper discusses the changes in classroom instructions due to technology over the years in mathematics education, and how these changes have impacted mathematics learning and teaching. The impact on learning can be seen over a few phases in Singapore: The use of scientific and graphing calculators has allowed the focus on the developing of higher order thinking skills, while at the same time de-emphasizing the routine computation. With the introduction of various computer softwares such as spreadsheets, mathematics teaching and learning has moved towards the next level of emphasis on coding and computational thinking. Technology can and has been harnassed by teachers to enhance student learning. These will be discussed in details in the talk, with particular reference to the Singapore education context.

#### The Mathematician Educator special issue: Mathematics instruction for the future

2023, Toh, Pee Choon, Toh, Tin Lam

#### Mathematical Problem Solving for Everyone (MProSE)

2020, Toh, Tin Lam, Quek, Khiok Seng, Tay, Eng Guan, Leong, Yew Hoong, Dindyal, Jaguthsing

This project involves the development and implementation of a problem solving package (M-ProSE) in the secondary school mathematics curriculum. It aims to induct secondary school mathematics students into the discipline of mathematics via a programme that turns well established theories of mathematical problem solving into praxis. In contrast with conventional training for mathematics competitions which tend to be restricted to a small number, M-ProSE is designed for all mathematics students Development of the project: In a pilot study conducted over two years in an Integrated Programme of a junior college, the research team observed that students were generally resistant to following the stages of Polya's model. In an attempt to 'make' the students follow the Polya model, especially when they were clearly struggling with the problem, we decided to construct a worksheet like that used in science practical lessons and told the students to treat the problem solving class as a mathematics 'practical' lesson. In this way, we hoped to achieve a paradigm shift in the way students looked at these 'difficult, unrelated' problems which had to be done in this 'special' class. Practical work to achieve the learning of the scientific processes has a long history of at least a hundred years. It is certainly conceivable that similar specialised lessons and materials for mathematics may be necessary to teach the mathematical processes, including and via problem solving. Implementation of the project: M-ProSE is an attempt to teach problem solving in 'practical' setup. Students will be taught Polya's model and problem solving in general in two or three dedicated lectures. The main mode of learning is then through a series of 'mathematics practical' lessons. Students work on usually one or at most two problems which have to be worked out on a special worksheet which requires the student to systematically and metacognitively go through the Polya model. M-ProSe is to be implemented as part of the mathematics curriculum and will be assessed. In order to implement M-ProSE, we need to build the teachers' capacity first to solve non-routine mathematics problems and thereafter to teach problem solving to their students. This involves the researchers conducting a series of workshops for the school teachers to widen their repertoire of problem solving resources. Next, we will develop with the teachers the instructional strategies to teach problem solving to their students, by means of a lesson study approach. Some of the researchers will initially teach some student classes as a model for the teachers before they take over entirely. To contribute to the understanding of teaching mathematical problem solving in general, the researchers will collect data over some cohorts which will enable them to further improve the package and make the package useful to other schools. The evidence collected will provide the basis for pedagogical practices in the mathematics classrooms.

#### How formal should calculus in the school mathematics curriculum be: Reflections arising from an error in a calculus examination question

2023, Toh, Tin Lam, Toh, Pee Choon, Tay, Eng Guan, Teo, Kok Ming, Lee, Henry

This paper examines the calculus curriculum in the current Singapore secondary and pre-university levels. Two concepts, (1) increasing and decreasing functions and their derivatives, and (2) the second derivative test for the nature of stationary points, are elaborated. An example of an incorrect calculus item in a national examination is brought up in relation to conditional reasoning involving calculus concepts. We reckon that the current emphasis on procedural knowledge in calculus is useful. However, we argue that formal conditional reasoning should not be introduced prematurely for school students.

#### On pre-service teachers' content knowledge of school calculus: An exploratory study

2022, Toh, Tin Lam, Toh, Pee Choon, Teo, Kok Ming, Zhu, Ying

This paper reports an exploratory study on the pre-service teachers’ content knowledge on school calculus. A calculus instrument assessing the pre-service teachers’ iconic thinking, algorithmic thinking and formal thinking related to various concepts in school calculus was administered to a group of pre-service mathematics teachers. Their performance on five of the items is reported in this paper. Other than their good performance in the iconic recognition of stationary points, their recognition on points of inflexion, differentiability and notion of minimum points was relatively poor. In addition, they appeared to lack the algorithmic flexibility in testing the nature of stationary points and the formal thinking about definition of an extremum point. The implications of the findings are discussed.

#### Fallacies about the derivative of the trigonometric sine function

2021, Toh, Tin Lam, Tay, Eng Guan, Tong, Cherng Luen

In this paper, several fallacies about the extension of the formula \frac{d}{dx} (\sin x) = \cos x to the erroneous formula \frac{d}{dx} (\sin x^\circ) = \cos x^\circ are discussed. In a Commognitive Theory Framework, misconceptions by ‘newcomers’ can be traced to the use of the word “unit”.

#### Principles in using comics for mathematics classroom instruction

2024, Toh, Tin Lam

Mathematics has always been a difficult subject for many students. Consequently, many students fear the subject and refuse to engage in this subject. There are many educational theories which attempt to address students’ motivational and cognitive issues in learning mathematics. In this lecture, I attempt to propose an alternative approach of teaching mathematics through the use of comics based on my experience of infusing comics into the teaching of mathematics for the low progress learners. I further present a framework of infusing comics for mathematics instructions, and take reference from not only educational theories but also from the perspective of communication theory. In designing lessons using comics, I make reference to how the various elements of the communication model are taken into consideration in the design process. I illustrate the application of this framework with the use of exemplars, making reference from both primary and secondary levels. With the advent of Generative Artificial Intelligence, what appears to be a rather expensive process of developing comics package could potentially become relatively inexpensive, so that educators and designers could invest more time in conceptualizing the content of the comics package. Thus, the theoretical framework for the design process deserves greater attention to researchers and educators.

#### Comics for mathematics instruction for future-ready learners

2023, Toh, Tin Lam, Cheng, Lu Pien, Lim, Lee Hean, Lim, Kam Ming

In this paper we present the views that the roles of comics for mathematics instruction extend beyond the role of addressing the affective needs of students, in particular the lower achieving students. We argue that within the broader framework of contextualization, comics have the potential to reach out to the entire spectrum of students to develop their higher order thinking skills and even raise their cognizance to environmental issues. Two exemplars based on the research carried out by us are presented.