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In solving mathematics problems there’s a certain flow engaged and at different stage of the process different skill sets are required. In this article I wish to share my opinion of this flow, and to highlight this is generally how I pinpoint where students’ weaknesses are.
Disclaimer: All are just my opinion and it’s a huge simplification of the actual process.
Step 1: Know your definitions
Generally, this is pure memory work. It is paramount to know your definitions; without it you can’t even engage the questions at all. It’s like trying to play a game without knowing the rules or goals.
Step 2: Techniques and tricks
For most people this involves algebra manipulation, sketching graphs. You need to be able to exhibit the necessary skills involved to get to the answer. Practice makes perfect here in improving your speed, reducing chances of carelessness etc. Some people may get it faster than others (we all learn and absorb at different paces), but you will get there eventually with enough practice. But of course, you can get there faster by engaging a tutor to point out and correct your work properly, or by simply reflecting on your mistakes to minimise the odds of you repeating them.
In a football context it’s like knowing how to dribble and pass the ball, the more you practice the smother you get. And people film themselves at practice in order to review (with their coaches/tutors) and work on their weaknesses.
Step 3: Linking across concepts
This is what we call higher order questions, where students are required to connect across concepts. Technique from one chapter may be used to solve questions in another, or what feels like a question in chapter A is a question from chapter B. In these questions, it’s important to understand the setter’s intent in order to comprehend and solve the question thoroughly.
Only students who have exposure to such challenging questions by practice, discussions with friends or self-thought can conquer these. Some students seem to get it more naturally due to a sharp Math instinct, focus.
Continuing my football analogy, it’s like attempting a challenging free kick or scoring under challenging conditions. Only players who have trained adequately under similar conditions can deliver it. Again, some players make it look so easy due to talent (and countless hours of practice).
I will show an example on quadratic functions:
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