Learning math-style vs substance (rationalizing Surds)

It’s always important to learn about substances rather than just the style. Unfortunately, we can’t distinguish between them under most situations, and the real test only comes when we get “out of the norm situations” or put the student under test on first principles. That is whether the student can explain the logic, apply the same logic and principles into another situation, extend or even generalize the idea.

Such “superficial learning” can sometimes arise from:

1.      Erroneous teaching stemming from want the lesson to be efficient, or

2.      The instruction was give to solve a specific problem and not to be extrapolated;

3.      Student tried to self-learn by observing worked examples but unfortunately didn’t observe enough

In this post I will try to explain using the rationalization of surds as an example.

Most students will know how to rationalize surds like this:

basic surds question

But they may have learnt that the method is to multiply top and bottom by “flipping the sign of the radical”, which is

basic surds rationalization

This is of course correct, but when faced with another question:

twisted surds question

They do this:

twisted surds question - less pleasant way

and cannot understand why they can do this:

twisted surds question - more pleasant way

Whenever I see students doing the former, I will always question why that action and make them answer if the latter move is ok too. Fundamentally, those who really understand the concept of rationalizing surds understand that it is all about creating a “difference of two squares” moment across the denominator.

I then conclude by asking the student how we should rationalize something this:

rationalizing surds with double radicals

Alas, there are always some who still has the idea of “flipping the sign of the radicals” and say:

rationalizing surds with double radicals (wrong method)

and then it's back to the whiteboard.

Alvin