‘Everyone has an opinion, but not everyone’s opinion is of equal value.’ What is your view?

‘Everyone has an opinion, but not everyone’s opinion is of equal value.’ What is your view?

JC General Paper

‘Everyone has an opinion, but not everyone’s opinion is of equal value.’ What is your view?

Define opinion and its purpose.

An opinion is a view or judgment formed about something, not necessarily based on fact or knowledge. Opinions are predicated on the individual’s current understanding and analysis of perceived phenomena, and its dynamic nature brings value and meaning to the uniqueness of individuality and colorful context to otherwise mundane events.
Thesis statement

While everyone has an opinion and every opinion should be given fair amount consideration in order for society to best function and cater to the interest of every individual as much as possible, the harsh truth remains that not everyone’s opinion is of equal value. Firstly, opinions predicated upon evidence are more reliable to derive inspiration from compared to one which manifests purely from emotions and personal bias. Secondly, psychological bias renders the notion that everyone’s opinion holds equal value and be given the same amount of consideration to be impractical. Hence, everyone has an opinion and is indeed entitled to have one, but not everyone’s opinion is of equal value for society.

(Counter) Argument 1:
Every opinion should be equally considered in order for society to best function and cater to the interest of every individual as much as possible

Elaboration: The conscious societal effort to hold everyone’s opinion with equal regard protects the interest of every individual, especially of those from minority groups. If we were to disregard opinions which do not interest the majority of us, it can lead to immoral and biased behavior, policies and consequences.

Example: Severe racism against blacks in the US in the past was due to the majority group’s absolute lack of consideration of opinion and interests of the perceived minority group.

Link: Every opinion should be carefully regarded in order for society to not neglect the interest of certain groups in the midst of societal progress.

(Counter) Argument 2:
Every individual is unique and as valuable, and hence every opinion carries a perspective that has their own distinct and equal value which should not be ignored.

Elaboration: Based on the premise that “all lives are equal”, all opinions should be equally considered as every opinion represents a distinct and unique individual who is as valuable as another.

Evidence: In a democratic government, every vote from every eligible individual is considered and carries the same weight, acknowledging the notion that every individual citizen is of equal value.

Point: Every opinion matters as every individual is unique.

Argument 3:
Informed opinions hold more meaningful value

Elaboration: “You are not entitled to your opinion. You are entitled to your informed opinion. No one is entitled to be ignorant.” – Harlan Ellison

Opinions predicated upon evidence are more reliable to derive inspiration from compared to one which manifests purely from emotions and personal bias. Opinions direct societal progress, and hence informed opinions lead to more informed decisions which would allow society to function and develop in its best interest.

Example: In the political context, a deep understanding of politics and the local political scene would compel the informed citizen to support the most capable political party. This informed opinion leads to be more logical and rational political vote which brings great positive value to the country, as opposed to having disgruntled citizens angrily voting for political parties which capitalize on emotions without decent consideration for practicality.

Link: Informed opinions is of greater value than uninformed ones.

Argument 4:
Personal bias renders the notion of everyone’s opinion holding equal value impractical

Elaboration: Opinions generate the greatest amount of real tangible value when they resonate with the masses. And the extent of how well an opinion is received by the masses is influenced by psychological bias driven by factors such as age, the perceived credibility and prestige of the opinion-bearer and how well that opinion aligns with our own personal beliefs.

Example: In the campaign for gender equality, it is evident that Emma Watson’s efforts hold greater influence and are held in higher regard than most people who have been essentially advocating and saying the same thing. This is the reason why charities tend to tap onto the prestige of celebrities to amplify their impact.

Link: It is impossible for everyone’s opinion to be genuinely held with equal regard due to personal bias.

Back to the A’levels H1 General Paper 2016 Paper 1 Solutions

To what extent should the arts in your society focus on local rather than foreign talent?

JC General Paper

To what extent should the arts in your society focus on local rather than foreign talent?

This question is indeed interesting considering how Singapore has stated her explicit aims of being a global arts hub on one hand, and the recent online Youtube controversy on the other hand about Stephanie Koh proclaiming that she is not proud of being a Singaporean due to the lack of an arts scene here. She claims the lack of this art scene makes the environment very stifling.  This “complain” of Stephanie Koh is just one of the many laments that we hear from the local art talents here. So how exactly should we bridge the gap between the nation’s aim versus the grudges of local talents? Should we focus on the local art talent or the foreign talent in order to elevate and revitalize our arts scene first?

Foreign talents definitely!

  1. Focusing on foreign talents who have the necessary expertise and connections will create a platform for future local arts talents to leverage on and develop their own artistic potentials. Many global cities have already established their own niche area in the arts scene and it will be extremely difficult for Singapore to break through and crave out a name in this sector without some foreign expertise and help. International collaboration will go a long way in ensuring that Singapore makes waves in the art scene- small but impactful waves should be the main objective of Singapore in order to attract some attention which can go a long way for local art talents.
  1. Having world-renowned foreign talents in the local arts scene will serve as a “pull” factor in getting tourists into Singapore and help to boost our economic performance, considering that Singapore is heavily dependent on the service industry. The arts industry is a huge untapped potential area for Singapore to develop and attain tourist dollar.  Singapore has implemented the Foreign Artistic Talent scheme to encourage international arts professional to become PR here in order to improve the arts and culture landscape.
  1. Singaporeans are likely to trust foreign productions that have been successful overseas than a local production that they have not even heard much about. Singaporeans in general are very skeptical and conservative and may not like to try anything new, and would rather get “their money worth” on a foreign production. The whole aim is not on whether Singapore is focusing on local or foreign production, rather using these foreign productions to get Singaporeans even interested in the arts scene, so that local arts will have more recognition in time to come.

E.g. Productions like “Lion King and “Shakespeare in the Park” have high recognition overseas which translate to high interest by the locals- goes a long way in getting Singaporeans to start enjoying arts production.

Locals definitely!

  1. Locals are talented and should be given a voice and platform to express their works, otherwise it will lead to a brain drain eventually. Local talents will be able to add a domestic angle to productions and cater better to the local audience due to their understanding of local social issues.  Prominent people who have left include Stephanie Koh.
  1. Governments should spend more money in grooming our local talents and providing them the platform to tap into the local market before venturing overseas. It is a sector that Singapore can be proud of and even develop their nationalistic pride surrounding it. For instance, we have the Young Artist Award to encourage the development of young artistic talents. SOTA has also been created to better cater and keep our young local talent pool.

Students should note that by the nature of the subject, there is several other possible pointers too, so feel free to discuss freely below!

Back to: 2015 A’levels H1 General Paper (8807) suggested solutions

Mathematical Computing #1

University Mathematics

With regards to a recent post, I shall share further on Mathematical Computing and how we can start on it.

So to assist me in not over-simplifying the journey, I’ve personally signed up for an online course with Coursera to learn Python. I have prior experience with Python, C++, Matlab, R & Stata. So yes, I might not struggle in the course much but I can refresh on the basics and hopefully recall my struggles and share here.

Not the Snake! Credits: python.org
Not the Snake! Credits: python.org

Firstly, I recommend Python as a beginner mathematical computing language. It is also the language that is being taught to Year 1’s in NUS (NTU teaches Matlab instead). I think Python is more straight forward and easy to understand. Furthermore, it sets the foundations straight for future endeavours into other language too. I also recommended the Python Course to a fellow educator whose son wants to learn a bit.

Does your math need to be strong to do mathematical computing? Not really since this is altogether a different thing. Math focuses on problem solving, so the implantation of a algorithm simply hasten the process.

Understanding A-level differentiation questions

JC Mathematics

Many students seem to know how to solve a Min/ Max problem easily but do not understand. I hope this gives them some insights into Min/ Max problem.

Firstly, what is a Min/ Max problem? It is those differentiation questions that ask us for the largest area, smallest volume, minimum cost, etc. All of this falls under a simple idea of minimum or maximum problem.

Credits: A-level H2 Mathematics (9740) Paper 1 2010 Question 9
Credits: A-level H2 Mathematics (9740) Paper 1 2010 Question 9

Take the above question as an example. Here, we are trying to obtain the minimum external surface area. So most students should be able to say they will find an equation of area and then differentiate it, equate to zero and solve.

So how does this relate in real life?

Most products you see in supermarkets, are related to such a problem. We have a constraint usually, in this case volume. And we have factors we can vary, in this case, the dimensions which affects the surface area. We want to minimise it simply to cut costs. Other words, given this single constraint of volume, we find the dimensions of the box that gives the smallest surface area.

The equation of area we found is a graph and when we equate the first order derivative to zero, we are attempting to find stationary points which can be either maximum or minimum points here, that is, the minimum area or maximum area. Then we check with the second order derivative to show it is indeed a minimum area.

Some students do ask, but in reality, we would’ve more constraints like weight, etc. To learn more on this, you can read more on lagrangian function which allows us to consider even more constraints. This will require knowledge beyond A-levels.

Solutions to this questions can be found here.

A different perspective to transformation

JC Mathematics

Okay, many students get confused over when to do what for transformation. I thought I should try to clarify a bit. First of all, it doesn’t matter whether you translate or scale first! x and y transformations are independent so do them in whichever order you like. Most importantly, we need to understand that every single transformation gives a NEW function.

I’ll discuss a bit on how I help my students to do it safely.

So you can see it as a replacing “x” by “x+a” for example. The function f(x) becomes f(x+a) & if f(x) = (x+1)(x-2) then it becomes (x+a+1)(x+a-2). We literally replace the x by x+a. This is a translation in -a units along x axis.

If we extend this and consider replace x by 2x then we have f(2x+1). We apply the same method and it gives (2x+a+1)(2x+a-2). This is a scaling by 1/2 units parallel to x axis.

Switch this around.

Consider instead we replace x by 2x first then x by x+a.

The functions will be f(x) becomes f(2x) and then f[2(x+a)] = f(2x+2a). Here we scale by 1/2 units parallel to x axis than translate by -a units along the x axis. The resultant curve will be different from previous! Of course, we can also translate by -2a units along to x axis then scale by 1/2 units parallel to x axis, that is f(x) -> f(x+2a) -> f(2x+2a).

The act of replacing x by ___ is always consistent. But please don’t write it in this manner during exam; I’m merely suggesting an alternative way to do it correctly.

Consider something else, say f(-|x|).
For this we will replace x by -x first then replace x by |x|. Other words, we reflect about y-axis first then do |x|..
You should realise that replacing x by |x| then x by -x is pretty meaningless since we have f(x) \rightarrow f(|x|) \rightarrow f(|-x|) = f(|x|). So the last transformation of -x is redundant.

Using this idea of replace x by ___, we in general do something I call like “outside in” if you want an easier brainless method.

Example: f(-2|x|-4)

Starting we f(x), we go for a translation of 4units in along x axis, then a reflection about y axis then scale by 1/2 parallel to x axis then |x|. So really doing from furthest of x towards x, like outside in. You can try writing it out in function terms.

Students who need further clarification, feel free to discuss. I think it is a lot easier to understand this way. 🙂

How to craft a good essay introduction?

How to craft a good essay introduction?

JC General Paper

The GP Essay – Writing the introduction

Like every other piece of writing, a GP essay begins with an introduction. Personally, I find the introduction the hardest part to write. Even as I type this little advice column, you would never have guessed how long I took to draft this one, tiny paragraph.

Now why do I say the introduction is the hardest part? Because most experienced GP markers can tell if a student’s essay is yay or nay based on the introduction alone. To understand this, one needs to know the functions of a GP essay introduction, which are:

  1. To set the context of your essay (background information)
  2. To define the question (definition of terms)
  3. To state your stand and address the task (thesis statement)

Thus, if you mess up or fail to do any of the above, the marker roughly knows that the rest of your essay would be –to put it bluntly— crap.

To help you avoid such a situation, here is a simple guide to writing a decent introduction.

1. Provide background information

Most students struggle with two problems in this area:

  1. What to write?
  2. How much to write?

To address the first problem, providing background information can be done in several ways:

  • Facts and figures
  • Quotes or idioms
  • Anecdotes a.k.a. experiences and short stories

Do note that your introduction also serves to attract your reader to read on so if possible, think of a good hook i.e. interesting background information. If you really cannot think of one, abandon the idea and deliver the cold, mundane facts to avoid wasting any more precious time. For example:

A picture is always more powerful than mere words. What is your view? (2006)

Introduction with a hook:

Approximately 500 years ago, explorers in France stumbled on some cave paintings depicting life scenes in the Palaeolithic era. Fast forward to today, one can find graffiti on city walls and brightly-coloured advertisements advocating how one should live one’s life. Thus, it is without question that for over centuries, pictures have been the mode of choice for conveying messages rather than words…

Introduction without a hook:

“A picture is worth a thousand words”. In the world of advertising, this expression is as good as law. How often do we see books, brands and even election campaigns that are void of illustrations? Indeed, many people …

Regarding how much background information to write, a good length would be between 3-4 lines. This is ENOUGH. Any longer and you are spamming.

2. Define the question

This part is easy because it only requires you to paraphrase the question. However be warned that a marker can tell how well a student understands the question based on his/her paraphrasing so choose your words carefully. Only paraphrase key words in the question.

Now here comes the problem. When GP teachers tell you to define the key words, some students end up doing this:

Can space research be justified nowadays? (2011)

Space research is the development of rockets, satellites or probes for space exploration. Some people think space research is unnecessary in today’s world.

Students make the mistake of mechanically defining key terms. Not only is this boring, but it disrupts the flow of your introduction. Only include this type of definition if it is absolutely necessary in helping the reader understand the context. If not, save it for your essay body.

Instead, you should be subtle when defining the key terms, for example:

Some people would argue that probing the universe for extraterrestrial life or sending rockets into space is a waste of time and resources in the present era.

Look at how much better the language flows in the second example.

3. Write your thesis statement

A thesis statement states your stand (if the question demands it) and tells the reader what he/she is going to read about in the next few paragraphs. A simple thesis statement can look something like this:

sports

The key to good health is lifestyle rather than medicine. How far do you agree? (2010)

I agree to a large extent that lifestyle plays a more significant role than medicine in determining good health. I will now present reasons to justify my stand in the following paragraphs.

To sum up, your whole introduction should be about 7-8 lines. The goal is to keep it short and sweet; like an appetiser to tantalise your reader. Remember that a bad introduction is like a bad first impression and as objective as markers try to be, their outlook towards your essay is inevitably affected by your introduction. Thus if I were you, I would start working on perfecting that first impression.

What is a good estimate?

JC Mathematics

Some of my student asked me how good is an estimate and how do we measure it up.

Let us first consider the difference between an estimate and estimator. We are all familiar with the following formula

s^{2} = \frac{1}{n-1}[\sum{x^{2}} -\frac{(\sum{x})^{2}}{n}]

The above formula is an estimator. It allows us to input our collected data and values to %latex s^{2}which is the estimate. So to put things in perceptive, we first collect samples and obtain a sample variance (<a href="http://theculture.sg/2015/07/what-does-unbiased-in-math-mean/">biased</a>), then input it into the estimator formula to obtain the estimatelatex s^{2}which is unbiased. And let us not forget that the unbiased estimate is still not the same as the true value, that is,latex s^{2} \ne {\sigma}^{2}$

So there are definitely more factors that attributes to a good estimate, unbiased is one. The next in line will be consistent. We question if the estimate is consistent, this means to consider if the estimate tends to the true value as n tends to infinity.

Credits: https://www.google.com.sg/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAYQjB1qFQoTCNv566vBmMcCFVNtjgod7mIJaA&url=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AConsistentEstimationFunction.png
Credits: https://www.google.com.sg/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAYQjB1qFQoTCNv566vBmMcCFVNtjgod7mIJaA&url=https%3A%2F%2Fcommons.wikimedia.org%2Fwiki%2FFile%3AConsistentEstimationFunction.png

This is really intuitive as I explained in class just yesterday.Imagine you are capable of obtaining the weight of EVERY SINGLE person in Singapore, then you will definitely be able to find the TRUE mean weight (granted ladies do not underestimate their weight). Thus, the value found will be consistent with the true value.

Future of Mathematics in Junior Colleges

JC Mathematics

I’ve been teaching H2/ H1 Mathematics for 6 years and the past year papers clearly indicates that they are interested in finding out if students do understand their learning. In my opinion, Mathematics can be studied in several ways, mainly by technical means or academic means.

Technical means will be a students sitting down in Starbucks and do questions from the same topic for the entire day. Some extreme cases will be those that asks me directly for answers keys and regurgitate the model answers. To me, this is unhealthy as students are very limited to the type of questions they are capable of handling. And judging from the past year papers, they will end up with a B only.

Academic means will be a students understanding and appreciating the concepts of how to handle this topic. As I previously shared, H2 math classes that I conduct focuses a lot on handling the concepts, i.e., learning the topic by understanding. And then applying this learnt concepts to solving the questions. This enables students to stretch their abilities. Not to mention, students must also understand that the questions don’t test topically. We see from this question that A-level can mash and test a few topic in one question. So students must be able to look at H2 Mathematics as a subject, instead of topics.

I feel that in 2-3 years time, the syllabus might have an overhaul again. Also the old syllabus of F Mathematics and C Mathematics is very good, and more can be done for H2/ H1 Mathematics by bringing some of the removed topics. Similarly, the lack of focus given to Statistics in H2/ H1 Mathematics is simply sad. And I do think that it will not be long before Ministry of Education begins to introduce Statistics as a subject and Pure Mathematics as another. These two subjects might be stand alone subjects too, allowing students more focus and variety. We already see this being done with H2 Computing, albeit little success.

When this changes take place, students who religiously mug their Ten Year Series, read them will definitely suffer a lot. Thus, I always advise students to keep questioning what they are learning and to ask sparingly.

P.S. I recently gave a few students an assignment: to at least ask me 7 questions weekly. Be it conceptual questions or a practice question.

Euler’s Identity

JC Mathematics
Credits: www.bbc.com
Credits: www.bbc.com

This is one of my favourite formula as it combines many things. We already saw how i and e was linked in the polar form of a complex number. But here, we related the number zero and one. It is quite beautiful, how Mathematics can be. 🙂

Difference between H1 Math and H2 Math

JC Mathematics

Prior to going Junior College, we always hold JC talks at various location to discuss with post O-levels students what to expect in JC. So many were confused with H1 Mathematics and H2 Mathematics. Let’s compare and contrast H1 and H2 Math in A-levels. And is H1 truly easier?

Content difference between H1 and H2 Math

Firstly, H1 Mathematics is 50% content of H2 Mathematics. Topics such as Vectors, Complex Numbers, Differential Equations, etc are not taught to H1 Mathematics students.

Is H1 Math easier?

In short, yes.

But will questions will be easier? No! I think there is a common misconception that H1 Mathematics is easier, only the studying part since students have lesser things to study. However, difficulty of the questions are not being halved or scaled down.

Exam difference between H1 and H2 Math

H2 Mathematics involves two 3-hr papers where paper 1 tests on pure mathematics while paper two comprise of statistics and pure mathematics. The overall composition for pure mathematics and statistics is 70$ and 30% respectively. H1 Mathematics, on the other hand, has one 3-hr paper which consists of 60 marks on statistics and 35 marks on pure mathematics.

What about the prospects?

Now many students are also concerned about the prospects of taking either subject. Personally, I strongly encourage students to take H2 Mathematics as it does prepares them for University well. Believe me, unless you are going into Medicine, Law, Architecture or Dentistry FOR SURE, you probably cannot escape from Math entirely. So the know-how will definitely help.

Pre-requisites wise, I know some students that do not have Additional Mathematics background from secondary schools who managed to get an A for H2 Mathematics. So if they can, why not you? But these students put in double the effort as they know they are already lagging behind. My advice in general will be that, if you don’t have additional mathematics background, think carefully about H2 Mathematics. No doubt it is beneficial and crucial to meeting many university entry requirements, but JC is less than 2 years and you must ask yourself if you are ready and disciplined enough. 🙂