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Misconceptions on how to do well in Math (AP, IB, IGCSE, "O", "A" levels)

Hello, in this post I will address some common misconceptions amongst students in the way they approach Math.

From my experience these are some common viewpoints I encounter the most:

  1. Math is all about understanding, there's no memorisation;

  2. Math is all about practice, so I can just cramp all in right before the exam - and cramping right before the exam will prevent me from forgetting the info;

  3. I have never seen student X doing any practice, yet he always gets the top scores, I'm going to follow his style from now on.

My opinion:

(1): The structure of Math is that its foundation is based on a set of definitions that one has to know very well (memorise), and from those definitions we derive logical results, theorems (connecting the dots) using techniques are outright hand-waving or mesmerising. Everyone ought to know definitions, conditions well in order to approach questions. You can't sit for an exam without knowing what is the condition for the existence of an inverse function (1-1) or what does 1-1 function really mean.

In fact, a common approach to solve questions is to write down/think about the math definitions/conditions mentioned in the question and try to connect the dots.

(2): Last minute cramping is never recommended. The knowledge retention is low after the exam and does not serve anyone well. As mentioned in my previous post the more consistent student always win. The real work is always done and the war won before the first battle even starts. I do not believe Stephen Curry starts practising his shooting only on the night before the finals.

There's a good time for everything, but for sure revising everything only on the night before is bad. For myself, I only refresh all the definitions, conditions, theorems the night before any exams - at best go through some difficult questions that I have bookmarked too.

(3): Practice can take many different forms, not restricted to solving questions on paper. There are students who can actually process all their workings and have a rough idea of the overall solution in their head and they move on. Putting it on a more relatable manner, in our daily lives some of us weigh the pros and cons of a decision fully in our heads, while some people divide a piece of paper into two columns and populate accordingly. It's all about engaging an interface that we are comfortable with and sufficient to resolve the question/issue on hand. I'm not advocating a 100% mental approach, writing your solution out is also training you on speed/time management/minimisation of carelessness.

The most important thing to me is to practice efficiently. After solving a question you should ponder other possible cases/extensions/variations of parameters of the question - we call this extension to other cases or generalisation. It will help you internalise the whole concept better and you might not have to spend the time solving other related questions.

For example, if I solve a quadratic function question requiring the function to have two real roots. I will then ask myself how will I approach the question if the questions ask for:

  • no real roots

  • one real root of order 2

  • real roots

I try to do this for all my classes, after solving a question I ask the kids what other ways can the question be set. It trains them to think like an exam-setter and is also a good check on whether they have missed out any small concepts in the chapter itself.

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