Singapore Maths Tuition - University and JC

From the Editor’s Desk
1st August 2023

When we use technology today, it is often taken for granted its development over the years. By tracing the development of technology, we need to think about the definition of technology in the first place. Technology is not limited to computers – rather, the wheel is also a form of technology. Mathematics has an invaluable relationship to technology – it allows there to be a systematic way of reproducing technology and measuring its effectiveness.

1. Edison’s light bulb

Although Edison took credit for the invention of the light bulb, he played more of a conceptual role and firm principles for its success. His staff approached problem solving systematically, using Mathematics to conduct a cost analysis for the success of the lightbulb. Edison rightly predicted that electric lights would replace gas burners. To build a high-resistance lamp filament, Edison calculated the cost of the copper coils by using Ohm’s Law, as he realised that by increasing the resistance of the lamp filament, he raised the voltage in relationship to the current, making his light bulb more economically competitive than the gas burners.

2. Transitioning to industrial society

The first industrial revolution happened in England, which caused a shift from a feudal system to a disciplined, industrial capitalism. Discipline was enforced upon workers by the use of timesheets and clocks to regulate workers’ behaviour. Mathematics helped workers and factories divide the day into a certain number of hours, with designated activities according to time. The day was compressed into 10 hours, with bell ringing to signal a change in time.

3. Ergonomics – The Science of Work

As society became more industrial, Ergonomics (the science of work) was invented based on laws of Physics to help workers become more efficient. The Science of Work imagined the human body as a motor, and sought to get rid of fatigue and maximise the energy of human bodies. Although this may sound nebulous in our current climate, the Science of Work was an attempt to meld the natural world with the concepts of Physics.

From the Editor’s Desk
25th July 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 generalize 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 phenomenon 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. To improve our critical thinking, we should always base our assumptions on statistics in order to make sure we are more informed.

2. Extrapolating using Mathematics.

Extrapolation is a natural aspect of human life – people want to know how much they have in the 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
22nd July 2023

Game Theory models the strategic interaction between at least two players in a situation with set rules and outcomes which helps predict, understand, and explain activities. The players in Game Theory do not know what the other party could do, but they are aware of what actions they can take. Moreover, one party’s choices will influence the outcome for the other, which means that the parties are interdependent.

This article explores the use of Mathematics in an area of law called mediation. Mediation is not like adjudication before a judge in a court of law, but parties have the opportunity to present their ideas and views to an objective mediator who will guide the parties to achieve a common solution in line with their goals. The process of mediation could be likened to a game, where the outcome of the mediation depends on the actions of the parties, with a payoff received by each party after a certain outcome, and parties receive a set of common information.

A key concept in Game Theory is the Nash equilibrium, when both parties cannot independently change their strategy without ending up in a less desirable position. To illustrate this concept, consider the Nash Equilibrium within the Prisoner’s Dilemma, where both the prisoners confess to reduce the risk of the worst possible outcome, when maintaining their innocence would have landed them both in better consequences.

In the Prisoner’s Dilemma, there are two criminals who are being interrogated separately. The state lacks the evidence to convict them of the crime they have committed – only the evidence to convict both on a lesser charge, resulting in each spending 1 year in jail. The prosecutor wants to convict each criminal on the more serious charge and incentivises each criminal to confess and implicate the other. He tells each criminal that if he confesses but his partner does not, he will go free, and his partner will be locked up for 10 years. If both confess, they will be put away for 3 years, but if neither confess, both will be let go. The dilemma lies in whether both prisoners will refuse to confess without communicating with each other, or whether they will cheat.

Game Theory can be applied to the mediation process as with proper communication facilitated by an observant mediator, parties can avoid Nash equilibriums and strive for better outcomes. The Nash Equilibrium is a valuable strategy for mediators as they can gauge the parties’ understanding of the mediation process and their plans to walk away from it with a desirable outcome. The Nash Equilibrium is highly relevant during reframing as the mediators must avoid a situation where the parties “trap” themselves within a Nash Equilibrium by refusing to compromise and communicate.

In the Prisoner’s Dilemma, both criminals want to reduce the risk of the worst possible outcome and confess although staying silent would have been best for both. The Nash Equilibrium arose due to the prisoners’ inability to communicate. By acting out of self-interest while trying to guess at the other party’s decision, the parties in mediation could similarly trap themselves in a Nash Equilibrium as their actions are based on what they think the other party would do.

In mediation, law students who are aspiring to be mediators could view mediation in the Game Theory lens – that each party is an agent who makes certain decisions based on certain cues. Good mediators will then facilitate communication and attempt to defuse the tension between parties so that parties do not act rashly, trapping them in a cycle of decisions which would leave them worse off than before.