Do you understand it or just know it?

In English there is no word for ceasing to understand something (“ununderstanding”) but there is of course a word for ceasing to know something i.e. ‘forgetting’.  This points to something vital about the process of learning that is very often lost by both students and teachers – that the foundation of deep learning is understanding, not knowledge.  This is because understanding is a usually permanent gain in insight, whereas knowledge is often temporary and fragmentary.

But what’s the difference?  And why does it matter?   Here’s an example that may help to illustrate the point.  In A level maths you may be taught that there is something called the discriminant (b² – 4ac) which can be used to determine the number of roots of a quadratic equation.  (If the discriminant is positive it is has 2, if equal to 0 it is 1, and if negative it has no real roots).  I once had some students that knew this but did not know why.  I explained to them that the formula for solving a quadratic contains the discriminant (highlighted in red):

x = [-b ±√(b² – 4 ac) ] ÷ 2a

And then I asked them:  Can you square root a negative number?  To which the answer was no.  So then I asked them:  Well if the discriminant was negative, would that formula give you any answers?  And then I  saw the PENNY DROP moment!  And then I said:  And what if the discriminant was equal to 0, what would happen then?  And one answered:  “Well then it would just be √0 so x would just equal -b ÷ 2a!  So just one answer.”  And then of course they could answer for themselves what happens when the discriminant is positive.

This illustrates what I believe is central to effective learning:  that students not only knowing what is true, but vitally, WHY it is true.  It’s clear to me that those students will now never forget the meaning of the discriminant, because they can’t ‘ununderstand it’.  What’s more the understanding they’ve gained would lend itself to being able to handle other more complex situations.  For instance they would be able to use this insight to answer questions of the form:  “Prove that quadratics of the form kx² – 2kx + k  always have exactly one root.” Whereas students who only ‘know’ the discriminant might well struggle.

So my job as a tutor is always to inculcate understanding in my students, not simply knowledge.  And for students, your aim should always to be to understand what you are being taught, not simply to remember it.