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.

From the Editor’s Desk
8th May 2023

Students taking the International Baccalaureate (IB) Programme will encounter reflections in nearly all aspects of their course. For example, students must write reflections during the entire process of writing their Extended Essay, after undertaking their presentation for Theory of Knowledge, and for their compulsory Creativity, Activity, and Service (CAS) portfolio. Writing reflections is an integral skill which IB students need to score the maximum number of points.

The objective of writing a reflection is not to test a student’s knowledge of a particular subject, but rather to prove that the student has earnestly attempted the activity by allowing the student to share some of the learning points he/she has taken away from his/her attempt.

The following guiding questions or format can be used for writing reflections:

I. What was the activity or project I attempted to undertake?
II. What objective did I have in mind when attempting the task?
III. Did I achieve my objective? Why or why not?
IV. What positive experiences did I have from this activity?
V. What negative experiences did I have from this activity?
VI. After the conclusion of the activity, what are the key learning points I took away?

The above format can be applied generally to all the aforementioned projects which require reflections. Students who are lost or stumped as to what to write can pick several (or all) of the guiding questions and craft their reflection in their own words. Students should remember that writing reflections is highly different from writing an academic essay, and they should draw on their own personal experiences instead of citing sources.

Admittedly, it requires some creativity to write reflections as they are more of a personal anecdote rather than a well-researched essay. However, these guiding questions can help students who lack the time to be creative in their approach to quickly and succinctly write reflection essays which will no doubt impress their respective readers.

From the Editor’s Desk
18th Jan 2023

When studying Mathematics, it is not enough to do practice questions from the textbook or understand the concepts in theory. It is imperative that a student drills with past questions before their examinations, as this will help them identify the pattern which is prevalent in these questions to score well for the subject.

Students should perform these practice exam papers and time themselves so that they are sure that they can finish answering the questions in time. It is important that students simulate the actual examination as much as possible, as they will be used to exam conditions. Students who are more used to examination conditions will be less likely to panic when they sit for their actual exams.

Alternatively, if students want to revise a particular topic, they could obtain a book which sorts the past examination questions by topic. If a student is weak in a certain topic, they should focus on revising that topic and understanding the relevant concepts for the topic. For example, if the student needs help with translating graphs, they could do all the past year questions for the topic and check with their teacher or tutor if they are unsure about their work.

Aside from doing the questions as practice, students should mark the questions according to the answer scheme. It is not enough to merely do the questions; students must find out why the answers they have given are right or wrong. Students must take the initiative to find out the reason for their mistakes and how they can improve themselves. At times, the mistakes made by students are due to their carelessness, and at other times, the mistakes made are due to the student’s lack of understanding. By knowing the cause of the mistake, students will be able to improve themselves.

From the Editorial Team
23rd December 2022

When deciding on a Junior College, students and their parents alike must contemplate the difference between A Levels and the International Baccalaureate (IB) curriculums. This article will hopefully shed some light on the key components of the IB curriculum which are not present in A Levels, so parents can make a meaningful comparison.

In the IB curriculum, students can take subjects at either Higher Level (HL) or Standard Level (SL). In Singapore schools which offer the IB diploma, students typically take 3 HL and 3 SL subjects, so as to not overburden them.

The maximum number of points obtained for each subject is 7 points. Students taking 6 subjects will be able to obtain 42 points from their subjects, with another 3 points obtained from the combination of the grades attained for Theory of Knowledge and the Extended Essay. The highest amount of points obtained is 45, which is considered a perfect score.

There is a greater emphasis on independent research and assignment writing, along with preparing for the major exams at the end of 2 years spent studying the content. Students will undertake their various assignments and essays, all the while studying for the exams, which allows them to have a more holistic perspective of education.

Below are 4 components of the IB diploma which students must consider before choosing to undertake it:

Individual Assessments for Higher and Standard Level Subjects

Both HL and SL subjects have Individual Assessments (IAs) which comprise at least 20% of the subject grade. These Individual Assessments can be externally or internally assessed by the school itself. For example, the Mathematics subject for both HL and SL has a Math Exploration IA where students must choose a topic of interest and write about it, demonstrating how their thesis statement has been proven.

Compulsory Theory of Knowledge Component

The Theory of Knowledge (TOK) component is a quasi-philosophy subject which deals with epistemology. In short, it is about how we think about the way we think. There are 2 graded aspects of this subject which count towards the final IB score – a presentation and a 1600-word essay. The presentation can either be done individually or in pairs, and the essay must be written individually. Students will be allowed to choose their own “Knowledge Question” for their presentation. Every year, the questions are released for the essay and students may choose from the options available.

Extended Essay

The Extended Essay is a 4000-word essay written by the student independently with the guidance of an Extended Essay mentor. This essay must be from one of the student’s Higher Level subjects and it must be independently researched. For example, if the student’s chosen subject is Biology, he/she must undertake the relevant experimentation on his own, designing his methodology. There will be opportunities for the student to check in with the Extended Essay mentor and submit reflections about the process so that the student knows he is going on the right track.

Creativity, Activity, and Service (CAS)

Although the CAS component is not graded, it is compulsory for students to submit a portfolio of the other activities they have undertaken during the IB diploma, showing that they have taken part in a holistic educational experience. Co-curricular Activities (CCA) can be included in this portfolio. Creativity is fulfilled by undertaking artistic endeavours such as dancing, music, art, or photography; Activity is fulfilled by taking part in sporting events, and Service is fulfilled by performing community service for others.

To conclude, students must decide for themselves whether they are willing to undertake these additional components instead of devoting their full time and energy to prepare for a major exam at the end of two years. Students more suited for the IB diploma are creative, unafraid to think outside the box, and good at time management.