Squares within Triangles - A New Perspective
[S2 Maths - Congruence and Similarity]
When it comes to Maths, most of the things actually performed seem to be useless in today's world. When were we ever going to use the rate of water flow in real life, when we simply needed to turn on the tap and fill the kettle with water for boiling? Depending on one's future career, your time in school could be next to useless when framed that way.
And it's a difficult argument to convince otherwise. Primarily, Maths is segregated into two different categories - Pure Mathematics, and Applied Mathematics. The latter is something that is used in work-life most of the time, be it designing machinery or working in high voltage environments. But, before one can understand the applied side of Mathematics well, it is important to learn about the pure side of Mathematics.
Today I'll share with you a video on how to change your perspective in tackling a question that is otherwise difficult.
As shown in the video, it always helps to start with a simple example (relaxing the restriction, as he says), and use the simple example to solve the problem. If you have ever taken the PSLE exam paper, you would quickly realise that this particular element of Maths which requires creative thinking is often removed from your curriculum in school.
It is a sad reality that school teachers only need you to follow the formula present within the textbook's topics. They are, by extension, not required to teach you about the more complex inner workings behind the formulae they teach if it's not expected of the students to learn about it. Overworked and underpaid, the teachers have it rough in regards to their job and having to meet the expectations of not just the government, but the parents who feedback to the school without prior knowledge of their dilemma.
But, there is a lasting euphoria in learning something intrinsically true. It is worth learning the pure aspect of Mathematics, to open our eyes to new perspectives. It helps us learn to be tolerant toward set restrictions, flexible in facing hard problems, creative in ways we approach the trials and tribulations later on in life by giving hands-on experience in overcoming strife and difficulty.
You wouldn't have naturally come up with ways to isolate squares within triangles unless you were given a lot of time to think through how it is possible. But with problems like that presented to us, encouraging us to think deeply and critically on what is required, one could say that solving Maths problems helps you become a better problem solver as a whole. So the next time you see a difficult Maths problem, try not to glaze over it and leave the problem to someone else - Give it an honest go instead.
(Interesting trivia: Italian mathematicians in the 16th century conducted maths duels, where they gave each other difficult questions they could solve, and they would win if the opposing party could not. It was a way of showing not just their capacity at maths, but at problem-solving; and those who excelled were offered positions as teachers, engineers, councillors and so on. Those who did not do well would risk losing their teaching jobs and fame.
The most famous example was a notable duel between Niccoló Fontana Tartaglia and Gerolamo Cardano over the cubic formula. You can Google up their feud and history to learn more, I believe one of the universities published an official historical paper titled the Cardano-Tartaglia Dispute which chronicles the details well.)