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# Learning effectively: The difference between Understanding and Knowing

You are tasked with teaching a two nine year olds multiplication. At the end of a few weeks, there will be a test to assess their learning. The test format is such that they are required to know the multiplication table for every combination of digits from 1 to 10. You devise two different teaching methods, and you try one on each child.

The first method of teaching is simple, but dull, and perhaps even painful for both teacher and student. You guide them through memorising the possible combinations. And so you begin teaching them that 1x1=1, 1x2=2… up till you’re done with 10x10=100. While this is certainly not the most fun thing to do, it does get the job done. The test comes around, and they score rather well. The child now knows the multiplication table. But that’s it. No further insight was gained, and the child is no more enlightened than before you made them memorise the table.

The second way, unlike the first, takes more effort, time, and patience. This alone is enough to discourage many from taking this approach. However, it is infinitely more rewarding and enriching than the first. For this second process, you start by explaining what multiplication actually means. For instance, you tell the child that 4x3 actually means that you are taking four groups of 3. Therefore, 4x3 is actually 4 times of 3, which is why the “x” sign is also read as “times”. You then ask the child “If there are 4 groups of 3, how many in total are there?”, and let them ponder the answer. You then explain that 4x3 is just a total of 3+3+3+3=12.

Now, instead of telling them the multiplication table, you get them to work it out with this newly gained understanding, one at a time. And so they begin. 1x1=1, 1x2=2, 1x3=3 … After the first few, a realisation hits them: one multiplied by a number, is just that number. And that’s not just a rule, it makes sense conceptually as well. One times any number, no matter how large, is just one group containing that number. And so if you ask them to figure out 1x20, they’d be able to say 20 straightaway.

This however, is not immediately obvious to the first child who simply memorised the table. Unless they figured the idea out by themselves (which is a tall task to ask of a child), they wouldn’t be able to tell you the answer to anything past 1x10, because that’s what they have memorised and that’s all they know.

The second child, after feeling the satisfaction of their discovery, is now motivated to move on to the next number: 2. They start with 2x1 which is 2 times of 1, giving 2. Then another realisation strikes: two times of a group of one is exactly the same as one group of two, as we are just grouping them differently. They then discover that 2x3 is the same as 3x2 (i.e. multiplication is commutative), and this is true for any pair of numbers. Delighted that they have now effectively cut their workload by half, they press on, working out the rest of the numbers diligently until they have completed the entire table.

They are given the same test, and they score extremely well, probably better than the first child. However, if you look beyond the test, I think it’s fair to say the second child has truly learnt much more than the first. The second child, unlike the first, is well equipped to work out, say for example, 13 times 14, or any other pair of numbers for that matter.

I’m sure we all want to be the second child. To discover and gain insights into a subject, feeling satisfied along the way. This is why I strongly believe all students should aim to achieve understanding rather than just superficial knowledge of a subject. However, it is also the responsibility of a good teacher to encourage that.

In my teaching experience, I find that many students struggle simply because they never had the chance to build that understanding. Too often they were told to “just memorise the formulas”, or even worse, to “just memorise how to do each question type” by their teachers. This is a recipe for disaster for many reasons. Without the understanding, there is nothing to build on, and so things snowball until it becomes an unmanageable mess. Studying becomes painful because nothing seems to make sense, which further demotivates the student.

I was never satisfied with just memorising stuff and knowing things superficially. I always made it a point to get to the bottom of things when I was studying. Naturally, that paid off, and I ended up doing well for exams. Given my understanding, I was able to manage whatever questions were thrown at me. This is why I always strive to help my students achieve a true understanding of a subject, rather than just getting them to memorise a bunch of stuff. The next time you attempt to learn something, ask yourself: “Do I really understand it, or do i just know it?”, and the answer will point you in the right direction to learn better.