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# Toh, Tin Lam

#### Using comics to contextualise the teaching of percentages: An adaptation of a comics-based teaching package for primary school mathematics classrooms

2022, Toh, Tin Lam, Cheng, Lu Pien

In this article, an adaptation of a secondary school mathematics comics-based instructional package for primary school mathematics classroom, and the teachers' and students' perceptions about the use of comics in the classroom are discussed. Further suggestions by the teachers on fine-tuning the package are also discussed.

#### Enacting a problem-solving lesson using scaffolding to emphasize extending a problem

2021, Chor, Emily Wai Si, Toh, Tin Lam

In this paper, we describe our conceptualization of teaching mathematical problem-solving at the upper primary level, emphasizing Polya’s Stage Four in extending a problem. Geometry is used as a context of the presentation. The objective is to engage pupils more metacognitively in their problem-solving process. By reviewing existing education literature, features that will support authentic problem solving were identified. The frameworks explored in this study include Polya’s 4-step problem-solving model, Schoenfeld’s framework, and the synthesis of the two frameworks through “Making Mathematics Practical” which utilize an extensive use of teacher scaffolding. The proposed scaffolding stresses pupils to problem solve beyond finding a solution as well as independently check and expand the given mathematics problem.

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

2023, Toh, Pee Choon, Toh, Tin Lam

#### 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”.

#### School calculus curriculum and the Singapore mathematics curriculum framework

2021, Toh, Tin Lam

In this paper, the Singapore school calculus curriculum at the upper secondary and the pre-university levels is examined in the light of the Singapore mathematics curriculum framework. Three key features of the calculus content are discerned: (1) an intuitive approach to calculus supported by the use of technology; (2) an emphasis on techniques; and (3) an emphasis on procedural over conceptual knowledge. Following that analysis, a review of the performance of a group of pre-university students on selected calculus tasks in a calculus survey prior to and after their learning of pre-university calculus is discussed. The students’ performance in the survey shows that many students did not visually identify calculus concepts that were studied procedurally. They demonstrated a lack of conceptual understanding of the calculus procedures. This study suggests that the partial calculus knowledge acquired in the early upper secondary levels might not necessarily facilitate the acquisition of a more complete concept at the pre-university level. Furthermore, the students’ procedural knowledge of calculus did not seem to develop their procedural fluency or flexibility.

#### Use of dynamic geometry software in the teaching of matrix and transformation: An exemplar of a classroom enactment

2023, Toh, Tin Lam, Chen, Kexin, Zhu, Tianming

Purpose and Research Question - In this paper, we propose a SCCG framework for using a dynamic geometry software (DGS) to enact a lesson on “Matrix and Transformation” based on intuitive-experimental approach. Methodology - A systematic literature review was conducted focusing on the impact of DGS on students’ learning, drawing on various learning theories, including Skemp’s relational understanding, social dimensional constructivism, and discovery learning. Findings - We demonstrate with an exemplar the use of SC2G framework in designing one lesson on “Matrix and Reflection” for senior high school students.

#### 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.

#### 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.

#### Teaching and learning complex numbers through problem solving

2022, Toh, Tin Lam

With reference to complex numbers, it is argued in this paper that attention should not only be focused on the practical usefulness or the aesthetics of mathematics to make mathematics attractive to students. Teachers could ride on the affordance of the problem solving mathematics curriculum framework in engaging students in activities that reveal the “power” of mathematics in solving mathematics problems and generalizing the results. The paper illustrates how portions of complex numbers, a pre-university mathematics topic, could be introduced through the various stages of mathematical problem solving. The use of complex numbers is a natural progression from basic algebraic manipulation at the secondary level and could be introduced through expanding a problem in the problem solving process; students could be introduced to the power of mathematics in providing an alternative solution or proof to mathematics problems from geometry and calculus. The roots of a complex numbers can be introduced by teaching through problem solving and re-enacting the (simplified) process of how mathematicians discovered complex numbers.

#### Teachers' instructional goals and their alignment to the school mathematics curriculum: A case study of the calculus instructional material from a Singapore pre-university institution

2022, Toh, Tin Lam

A case study of calculus instructional material (comprising of lecture notes and tutorial practice worksheets) designed by teachers from a Singapore pre-university is presented in this paper. As textbooks have not been available for mathematics at the pre-university levels, the instructional material was a product of the teachers’ collaborative work based on their interpretation of the school mathematics curriculum. Through a study of the instructional material, the teachers’ instructional goals and their alignment to the curriculum were examined. It was clear that the instructional material was more than a mere compilation of resources for the instruction; the teachers’ effort was also in building the close connection within and across the concepts and sub-topics. The discrete parts of the content were organized under a “big idea” in the lecture notes. Anchor questions were used in the tutorial practice worksheets to facilitate students to recognize the similarity of the underlying structures of seemingly different questions. In addition to the teachers’ articulated instructional goals such as covering all the key concepts, reducing students’ cognitive load, or developing their algorithmic mastery, the study revealed the unarticulated goal as to raise the students’ cognitive growth, which could be explained by the APOS cognitive growth model. In aligning to the school curriculum, it was observed that the teachers made judgement to tap on the advantage of the spiral curriculum to advance their students’ understanding of calculus from the secondary level using a higher perspective, and to better their students’ understanding of calculus concepts in addition to focusing on algorithmic mastery. The effort to engage their students in problem-solving and the use of technology to develop higher order thinking or enhance conceptual understanding remained implicit.

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