Beyond School Mathematics

Abstract

In this chapter, we unfold a two-sided painting of the tertiary mathematics education landscape in Singapore. One side displays how the education system in Singapore prepares her students for further learning of mathematics at university, while the other portraits how mathematics is taught at the tertiary level in the mathematics department of some Singapore-based universities. Regarding the pre-university mathematics education at ‘A’-level, we examine some of the major syllabus changes for Mathematics, making sense of these changes through the analytical lens of curriculum orientation. In passing, we also looked at the H3 Mathematics curriculum and its implementation, and the niche school NUS High School of Mathematics and Science. For the tertiary mathematics education, we rely on the collective wisdom of seven mathematics professors who have rich experience in teaching undergraduate mathematics from the top four local universities. The story of what goes beyond school mathematics in Singapore brings forth an important message, that is, tertiary mathematics education is responsive to shifts in educational policies occurring at schools—one which is unique of Singapore.

Appendix

An exemplar for creating learning experience in H2 Mathematics Syllabus 9758

Complex Numbers

Lesson Objective

Based on an ‘old’ idea of $C + iS$, this learning experience involves the students to create the imaginary counterpart of a sinusoidal voltage function across a resistor arising from an alternating current source. By so doing, the students reinvent the phasor of the voltage function, which takes advantage of the vector nature of complex numbers, and exploit it to calculate the resultant voltage function that results from adding in series two alternating current sources that are not necessarily in phase. Engineers use this method, called phasor analysis, to think and reason about alternating current voltages, and related quantities.

Problem

An alternating current source has the following voltage function:

$$V_{1} = 3\cos \left( {2t + \frac{\pi }{4}} \right),$$

where $V_{1}$ is the voltage (V) across a given resistor, and t is the time lapsed (s) since the source was turned on.

Another alternating current source whose voltage function is given by

$$V_{2} = 4\cos \left( {2t + \frac{\pi }{6}} \right)$$

is now placed in series with the above-mentioned source so that the resultant voltage is calculated by their sum:

$$V_{1} + V_{2} .$$

What is the amplitude and the period of the resultant voltage?

Mathematical Content Knowledge

For the first voltage $V_{1}$, we create the imaginary sine counterpart of the function $3\sin \left( {2t + \frac{\pi }{4}} \right)$ and construct the complex voltage function:

$$3\left( {\cos \left( {2t + \frac{\pi }{4}} \right) + i\sin \left( {2t + \frac{\pi }{4}} \right)} \right) = 3e^{{i\left( {2t + \frac{\pi }{4}} \right)}} .$$

Similarly, for the second voltage $V_{2}$, we have the complex voltage function:

$$4\left( {\cos \left( {2t + \frac{\pi }{6}} \right) + i\sin \left( {2t + \frac{\pi }{6}} \right)} \right) = 4e^{{i\left( {2t + \frac{\pi }{6}} \right)}} .$$

Now, we sum these two complex voltages together:

$$3e^{{i\left( {2t + \frac{\pi }{4}} \right)}} + 4e^{{i\left( {2t + \frac{\pi }{6}} \right)}} .$$

A preliminary investigation using a GCs reveals that the above sum can be reduced to a single trigonometric function.

From the vector geometry of complex numbers, one can show rigorously that

$3e^{{i\left( {2t + \frac{\pi }{4}} \right)}} + 4e^{{i\left( {2t + \frac{\pi }{6}} \right)}} = {\text{Re}}^{{i\left( {2t + \alpha } \right)}} ,$ where $R = \sqrt {\left( {4\sin \frac{\pi }{6} + 3\sin \frac{\pi }{4}} \right)^{2} + \left( {4\cos \frac{\pi }{6} + 3\cos \frac{\pi }{4}} \right)^{2} }$ and $\tan \alpha = \frac{{4\sin \frac{\pi }{6} + 3\sin \frac{\pi }{4}}}{{4\cos \frac{\pi }{6} + 3\cos \frac{\pi }{4}}}$, which in particular are independent of t.

Further exploration

The phasor addition works because the two voltages are of the same angular frequency. A natural question to ask is how one can tackle the case when the angular frequencies are different. Use a GCs to investigate this situation.