From the Editor’s Desk
14th November 2023

Students in the science stream in A Levels may choose a career in Mathematics or Physics. However, they likely do not know where to start, or even which options are available to them. This article aims to shed light on the myriad of options which students can choose from if they are keen to pursue Mathematics and Physics in future.

The starting point to a career in Mathematics and Physics is the type of university degree students must choose. Students can study a general Mathematics or Physics degree, or they can opt for a more specialised degree which deals with a particular area of Physics, such as nuclear physics, if they aim to work in places which are trying to harness nuclear energy as a viable energy source in the event that fossil fuels are no longer available.

The specificity of the degree is highly important as it will affect students’ progression in university. It is completely fine if students do not know what topics they would like to specialise in, as this often arises in their final year of university when they would have a better understanding of the underlying concepts in Mathematics. However, students should do some research and explore which jobs are not only attainable and feasible, but also are complementary to the kind of lifestyle which they want to live.

Some students enjoy the more theoretical applications of Mathematics, hoping to pursue an academic career, with potential occupations such as being a lecturer in a university. Those who are interested in studying a subject for its own sake could consider taking a degree in Pure Mathematics. However, others enjoy a more practical application of Mathematics, which will grant them a broad range of occupations in many different fields. For the students who enjoy studying the practical applications of Mathematics, Applied Mathematics could serve them well.

For instance, students with a good knowledge of Mathematics could work in aviation or in the military, designing technology and pushing the frontiers of human capabilities and knowledge. The finance industry is also appealing to mathematicians, with markets requiring better models to encapsulate the uncertainty of the financial markets. Some of the best economists are Physics majors as their love for modelling natural phenomena in the real world could translate to making models in hopes of reducing the amount of chaos and spotting trends in finance.

The tech industry would also appeal to people studying mathematics and who possess a working knowledge of programming, as the rise of the use of Artificial Intelligence to track things and people in real life as well as predict decisions means that mathematicians will be in high demand.

Students should think about which careers are in high demand in Singapore if they want to increase their prospects after studying mathematics or physics. As Singapore is a highly-developed society which makes use of many technological advancements to simplify the daily lives of its citizens, there are many different fields to be considered when choosing a career. Students should look around them (they sometimes need to go no further than the MRT station) to see a potential career and use for their degrees. When designing trains and railways, physics and mathematics are essential to determine which materials and how to build efficient transportation systems.

To conclude, when thinking about careers in Mathematics or Physics, students should do ample research as a career is not a small decision to make. Often, for careers in any field, students will invest a high amount of time and effort and they should be sure that they will at least minimally are interested in what they are doing.

From the Editor’s Desk
16th October 2023

“Critical thinking” is a term that is thrown around a great deal in the field of education, but often, people do not stop to question what this term means exactly. When faced with a new set of information, the act of critical thinking involves not blindly accepting a piece of information, but rather questioning its base assumptions, examining whether the new information coheres with existing literature, and evaluating its implications and consequences.

Mathematics as a discipline can be shown to improve critical thinking as it tests assumptions, proves statements, and can be used as a tool to give a statement empirical evidence to back it up. Mathematics should be viewed as a tool to provide answers to questions arising from critically thinking about something – critical thinking does not often involve intense mental gymnastics, but rather questioning assumptions and asking “How do I know what I know?”

1. Testing assumptions using statistics

Firstly, there are many assumptions that are made, as humans are creatures who like to generalise and put everything they observe in neat categories or boxes to make the chaos of life easier to comprehend. For example, if we experience a particular phenomena many times (such as encountering many red cars on the road), we may believe that there are many more red cars on the street than they actually are. Furthermore, if we have never seen something (like perhaps a rainbow holographic car), we may believe that it does not exist – this is called the Black Swan effect. As such, to improve our critical thinking, we should always found our perceptions based on statistics to provide ourselves with a more educated assumption.

2. Extrapolating using Mathematics

Extrapolation is a natural aspect of human life – people want to know how much they have in future. For example, if a farmer buys a new machine, he wants to know how much he could produce with that new technology as opposed to on his own, even though he does not have the machine yet. This is called extrapolation, which is refined using Mathematics. Extrapolation is an important part of critical thinking, as it is important to consider how (if any) the past influences the future to better prepare for it.

However, extrapolation is not a wild guessing game, but it needs to be based on existing data. If we want to predict the population of a country in future, it stands to reason that with the current medical technology and food security, the population of a country will likely go up. How much will it go up by? We then need to look at existing population data to observe how much it has increased in previous years to get a rough estimate of how much it will increase this year.

3. Tethering abstract concepts using Mathematics

Economics is the social science which studies how to use limited resources and allocate them amongst competing people. There are many theories in Economics which govern the allocation of resources, and these theories remain in the abstract until Mathematics is involved. The field of Econometrics involves quantifying these theories so their effects in practice can be better examined. Abstract theories about what happens in the economy hold no water until they can be shown in numerical terms, which shows the weight of Mathematics in tethering these ideas.

The aforementioned techniques are just some ways of improving critical thinking with Mathematics. However, less focus should be placed on the quantitative aspects of improving critical thinking, but the emphasis is instead on asking the right questions. By asking the right questions, such as how the piece of information is derived and how information is portrayed, the quantitative tools can be bent and melded to answer these questions, which is the foundation of how to think critically.

From the Editor’s Desk
11th October 2023

The practice of teaching and learning Mathematics in Singapore has changed over many years, not only in terms of the contents of the syllabus, but the focus on certain courses as per the needs of society. As society develops and technology matures, there will be different needs and skills students will have to fulfil as the job landscape changes. For example, AI was an unthinkable technology before recently, with many people dubbing it science fiction. Now, the use of ChatGPT and other Artificial Intelligence tools is widespread and extremely common.

1. Staying relevant in a changing world

The technological landscape is changing, with more advanced computing tools making it easier to solve rudimentary problems in less time and with less effort than a real human. The real challenge of the modern world is to think in terms of a big picture – how does someone stay relevant amidst the changing times? The answer is by harnessing the technology available in creative ways to solve existing problems, and to prevent the technology from being misused.

An example of this in real life is the application of anti-cheating software in detecting which university students use ChatGPT to commit academic dishonesty. The advent of ChatGPT has an unintended consequence of misuse and abuse by students attempting to get better grades by passing off AI-generated work as their own. As such, this problem must be mitigated by programmers who harness the existing ChatGPT technology to detect when it is being misused by creating software to detect certain patterns in the words in the assignment.

2. Increasing technological involvement

In today’s technological landscape, there are more avenues for technology to be incorporated in Mathematics. An example are Graphic Display Calculator, where students can draw graphs and make complex calculations. These calculators would not have been in use in the past, but nowadays, students are capable of doing more than their predecessors have. Nowadays, computers and various software have made it far easier to undertake larger projects which require more complex calculations. As such, the Mathematics curriculum needs to evolve to make space for these solutions to be implemented as part of learning and problem-solving.

3. Ensuring Singapore’s competitiveness in the global stage

It is important to ensure that Singapore remains competitive in the global stage with regards to performance in Mathematics. Singapore has consistently outperformed other countries in academic performance, as the standards of Mathematics in Singapore are considerably higher as compared to other countries. Singaporean students are consistently made to perform beyond their age level. The Mathematics curriculum should continue to evolve as it should teach students new skills and ideas which put them ahead of others in their grade.

4. Preparing students with the basic skills for university and employment

The Mathematics syllabus should prepare students with the basic skills not only for university, but also for employment. In the real world, there are no “exams” in an academic setting – rather, people will need to use their existing knowledge to solve real world problems. Moreover, Mathematics in theory will not be the same as the application, as in real life, there are many other factors to consider, such as budget and other departments who have different functions.

Teachers will need to keep up with the latest trends and developments in teaching Mathematics. Changing variations of questions are one example of how the Mathematics syllabus in Singapore has changed. Teachers will need to think about the context in which a student is learning a particular topic, moving beyond teaching to prepare students for exams and into teaching for a more practical application – preparing students to handle future novel problems.

From the Editor’s Desk
6th October 2023

For this article, a H1 Math student who has passed A Levels with flying colours was interviewed. She is now in her 3rd year of Computing and Law in Singapore Management University. This article hopes to inspire H1 Math students and let them know that they will overcome A Levels.

1. What are your biggest takeaways in H1 Math?

In the area of probability and statistics, hypothesis testing was the most helpful area as it has helped me make better decisions in my personal life as well as in university, where statistics is compulsory. Such knowledge and exposure to different distributions (albeit limited) has helped me ease myself into statistical thinking for computational topics such as AI or Data Science.

2. Why did you choose H1 Math, and what is the difference compared to other levels?

There are different categories: H1 Math, H2 Math, Further Math, and H3 Math, in the order of difficulty. I chose H1 Math as H1 Math is less in-depth compared to H2 Math. For example, H1 Math only covers basic distribution models, but there is a larger variety of models for H2 Math. H3 Math has the widest range of topics.

3. How did you best prepare for H1 Math?

Doing the entire 10-years series for Math helped a lot as it made me practice with questions which would simulate the questions asked in the exam. Furthermore, I made it a point to go through past mistakes I had made in tests and internal exams, as sometimes the trial exam set internally would be harder than the actual paper. If I did not understand anything, I would consult my tutor in school.

4. Did you have tuition and did it help?

I did not take tuition, but I needed a lot of motivation to practice a lot. Good teachers at school helped to fill in some gaps in my knowledge, as the explanation of essential concepts is key to your understanding. Math was not a hard subject at the H1 level as long as I did my due diligence and practiced.

5. If you could go back in time and tell yourself something, what would it be?

I would tell myself that keeping consistent and investing a fraction of my time for every subject will get you very far. Studying with friends can be helpful, as a good support system you can go for subject consults with and use as a sounding board for clarifying concepts and pushing through the subject together is important for your mental state as well.

From the Editor’s Desk
8th August 2023

A tort is a civil wrong. To understand a tort, we should differentiate it from a crime. A crime is a wrong committed against the state, and when someone commits a crime, the Public Prosecutor will prosecute that person. However, when someone commits a tort, they could be sued by a private party – the state is not involved. It is possible for one action to be both a tort and a crime.

This article will explore the relationship between Mathematics and Tort Law, to show that even lawyers require an understanding of Mathematics during their work. One area where Mathematics is relevant is the computation of damages – the payment the aggrieved party is owed for the wrong committed against him.

To give you, the reader, a better understanding of a tort, let’s suppose that A has a wild, fierce dog chained up to the gate of his house. However, A fails to take reasonable care to restrain the dog, and the unsuspecting mailman comes to deliver the mail and gets mauled, sustaining injuries which prevent him from working as a mailman ever again. The mailman sues A for failing to exercise reasonable care in preventing his wild dog from escaping and hurting him for damages – and A must pay money to the mailman as compensation for his injuries.

Mathematics comes into play in quantifying the damages involved. How should the court go about calculating the damages owed to the mailman by A? One way is to look at the position the mailman would have been in, had the injury not happened. A would thus have to pay for the mailman’s medical bills and all expenses incurred since the dog bit him.

The mailman could also argue that he would lose his future earnings due to the tort. As such, the mailman’s streams of income are evaluated, and an estimate is given as to how long the mailman would have been working those jobs. The concept of extrapolation comes into play, as the court would have to evaluate and make an educated guess as to how much the mailman would suffer in future to calculate how much money A should pay him as compensation.

Another area of tort law which involves mathematics is Loss of Chance, which revolves around the tort of negligence. If the claimant can show that he has suffered a loss, he is able to receive damages. Such a loss could be the loss of physical objects, or even loss of seemingly intangible items such as economic loss or the loss of genetic affinity (such as in a case where the baby born as a result of fertility treatments was not the one having the same genes as the person who donated sperm). However, in Singapore, for medical negligence cases, the courts have refused to recognize a loss of chance. For example, such a case would include a loss chance arising because of the doctor’s negligence, which prohibited a patient’s recovery.

There has been much debate in the law due to this loss of chance area, which relates to Mathematics as it involves quantifying the damages awarded to a patient. This blends the law with science and Mathematics as it requires the law to evaluate the severity of the patient’s illness, the likelihood of recovery, and the extent to which medical negligence prohibited recovery for the patient. A pure legal background will not be sufficient to determine these, and the law must rely on experts such as doctors and researchers to determine these factors. In fact, courts have relied on statistics provided by expert witnesses, and the debate revolves around how reliable these statistics are in proving their case – should they be held as gospel, or should they serve no more than a rough guideline?

As such, students should hopefully see that Mathematics and Science can be applied across many disciplines, including in a court of law.