Teaching and learning complex numbers through problem solving
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.
The use of mathematics competition questions for mainstream primary mathematics classroom instruction
Mathematics competition questions can have a wider use in the mainstream mathematics classroom instructions beyond the rather restrictive mathematics competition environment. Efforts have been taken by the education community to popularize mathematics by utilizing these questions for the general student population. However, most of the work cited were mainly adapting or modifying competition questions for mathematics instruction at the secondary level. In this paper, we propose with two exemplars how selected mathematics competition questions can provide the affordance to engage the general primary school student population in the problem solving processes.
How formal should calculus in the school mathematics curriculum be: Reflections arising from an error in a calculus examination question
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.
Use of dynamic geometry software in the teaching of matrix and transformation: An exemplar of a classroom enactment
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.
A framework for designing comics-based mathematics instructional materials
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.
Scaling up the education research: Magical (SUPERMAGICAL): use of comics in teaching mathematics.
Stratonovich-Henstock integral for the operator-valued stochastic process
In this paper, we introduce the Stratonovich-Henstock integral of an operator-valued stochastic process with respect to a Q-Wiener process. We also formulate a version of Ito's formula for this integral.
Fallacies about the derivative of the trigonometric sine function
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”.
From the past to the future of technology in mathematics education in Singapore
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.
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