I am going to start this article off with a story. As I write this article, PSLE exam results just came out and a good percentage of our students got the elusive A star. And even more so, we would like to highlight just one humble story, a story that may resonate with you and your child, that, there is in fact a way out. A way, a method, that if used with discipline can and will help your child. And with every inch of conviction and a congratulatory message from the parent of the child, we would like to share one such success story with you, amidst plenty that we have experienced as teachers. And may I add that this is what teachers live for.

A backdrop of the story is necessary. While this girl did not get the coveted A star, she received an A nonetheless. Let us call her Belle.( Her real name is not Belle) But let us not forget where she started from. In primary 5, she scored less than 50 marks for SA 1. But for PSLE, she got an A.

Surely that is a remarkable improvement. And we applaud the student for it.

So why is that?

What did this student do that others didn’t?

What is it that we do that helps students do well?

Can this be replicated?

Can this system be taught?

Can thinking be inculcated?

Polya’s Problem Solving Strategies

At this point, we want to introduce George Polya’s framework to solving mathematical problems. In 1945, Polya, a mathematician wrote a seminal classic on How To Solve It which sold a million copies and was translated into 17 languages. Polya broke problem solving down to:

• Understand the problem
• Devise a plan
• Carry out the plan
• Reflection on the problem

This set of steps to better problem solving, if you will, has long been introduced in schools in Singapore. And the effectiveness of Polya’s methods have already been proven, recognised and even endorsed by other mathematicians.

The question is : Is there a way to improve on Polya’s problem solving processes?

General Weaknesses Of  Students

Some of the problems that we encountered with students is that they often try to see the end of the question without actually working on what needs to be worked on first. And at other times, students are careless, make wrong assumptions often and also have a first conclusion bias when it came to problem solving. The first conclusion bias is actually a term borrowed from psychology where a thinker tends to think that the first conclusion is correct. In that sense,there is a tendency to attribute correctness to the first set of drawn conclusions. Another problem when trying to solve a mathematical question hinges on the decision making ability of the student. We would classify this issue as incorrect decisions.

So in summary, the other weaknesses that students have while trying to solve questions are :

• Poor decision making ability
• Incorrect assumptions
• Too much guessing involved
• Latent biases such as the first conclusion bias
• Errors in calculation

Any educator who has been in the profession and has had an interaction with students will at some point come upon these issues. Parents too will also face similar issues when tutoring their own kids. That is also the reason why we at www.singaporemathguru.com started our own video learning company to help parents and other educators.

Our solution to the problems above

Our solutions to these weaknesses above are laden, explained away, taught, dissected and grilled into our students with our 330 hour video library to over 10,000 free examination questions from primary 1 to 6. And this is how we solve the problems above.

We incorporate first principles thinking to problem solving. That is, we ask our students to ask themselves what is it they know, from reading the question, that has a state of certainty. Then, work from there to solve the question. That state of certainty is often a statement within the question that contains a very clear message.

Let me list some examples here.

John has 30 beads and Mary has 15 more than John.

So from this statement alone, a student can work a couple of things out.

After reading the statement, the student should start to underline/highlight the key information in the question. Key information is information needed such that the problem can be solved. So in this case, the student should underline or highlight :

John has 30 beads and Mary has 15 more than John.

The next is to react to the statement.

30 + 15 = 45

45 + 30 = 75

They have a total of 75 beads.

So by doing so, the student is incorporating elements of first principles thinking to the question and working from certainty.

So it is not that Mary has 15 less or John has 15 more.

It is that Mary has 45 beads and they have a total of 75 beads.

3 is the magic number

From our experience as educators, 3 seems to be the magic number. Can you examine your thought processes above. Are you working with clarity? Is there a flippant or careless attitude? Do you know the question deeply?

If you repeat to yourself once, you know a little. But if you repeat to yourself thrice and then acting on it, you know deeply. Hence we call our approach:

• Underline
• React immediately

The reaction can be some workings, a model or something visual. This also incorporates elements of first principles thinking and Polya’s advocacy of using a visual diagram or a picture.

A side note though. Elon Musk, a famous entrepreneur and founder of Tesla and SpaceX, uses first principles thinking on a very deep level when trying to solve real world problems. And if it can be used for solving real world problems, it can be used to solve mathematical problems. Elon Musk did say that it is good to “take a position that you are some degree wrong and your goal is to be less wrong over time”. For a man who has solved so many of the world’s complex business problems, perhaps we should borrow some of his wisdom and help our children develop that ability to think.

What is first principles thinking?

First principles thinking is a way of thinking that does not allow for incorrect assumptions and poor decision making to creep into the mind while making a decision. It is likened to a basic assumption that cannot be further deduced.  It is a way to thinking deeper and deeper into the problem until one is left with foundational truths.

And why is this even important?

Because it helps our children solve math questions. No incorrect assumptions. No wild guessing. Just really good foundational truths to build upon. And if you think about it, that really makes sense. The only way that you can get full marks for solving a question is to be right from the start to the end. Build foundational truths, one at a time, till the correct answer is found. So the emphasis should not be placed on the answer but rather, the immediate steps to take to get to these foundational truths.

How does SingaporeMathGuru do it?

With the strategies that Polya has laid out, first principles thinking and the Read,Underline,React Now approach, we have created more than 300 hours of worked video explanations to questions to guide students to become better thinkers and to get that elusive A star.

Our questions are free to peruse and attempt with progress tracked on an exercise level. As these videos are a self learning tool, we encourage parents who are interested to also learn with their kids watching the videos together with them. The videos will contain a whiteboard with a teacher explaining in an exacting amount of detail, what to do, when to do, why to do, where, who and how, to the student, incorporating elements, thought processes and habits that help a student towards the A and the elusive A star.