How to Teach Element 1 of 4: Categoricals

Using non-examples is not enough. You have to use them well.

Podcast is AI generated, and will make mistakes. Interactive transcript available in the podcast post.

In the last post we saw that nearly everything, the hundreds and thousands of ideas we’ll come to teach in maths, can be atomised into one of just four elements.

Now:

1. how do you identify whether one of those atoms is categorical

2. and once you do, how exactly do you teach it?

Identifying Categoricals

Spotting a categorical is pretty easy. If you can meaningfully ask the question:

“Is this an X?”

Then X is a categorical.

Is it a triangle? Is it a pentagon? Is it a polygon? All valid questions; all categorical concepts.

These are also categoricals: expression, equation, identity, parabola, prism, histogram, quadratic, acute angle, parallel lines and so on.

But you couldn’t make sense of this question for a concept like gradient:

“Is this a gradient?”

“Which of these are gradients?”

…doesn’t make sense. That’s because gradient is not categorical.

Teaching Categoricals

In short, teach categoricals using non-examples.

But there’s an effective way to do this, and an ineffective to do it.

It’s not enough to just grab any old non-example and compare the two.

We’ve been running training on this for over a decade now. Back in 2013, if you were to explain to someone that we understand something not only in terms of what it is, but also what it is not, that was a revelation. Today, it seems like everyone has been introduced to the idea of non-examples.

This is a wonderful position to be in, and means we can now level up to make sure we use them effectively, deliberately deciding:

1. What non-examples should you select?

2. How should you sequence them?

 Compare these two example sequences:

Before reading on, try to answer the question: which do you think will be more effective, and why?

Before we get to our answer, a few things that are the same in each case:

• We are presenting a sequence of examples, one after the other. We are not presenting many examples all at once and asking students to sort them. We’re not even presenting many sorted examples all at once; they come in sequence, one after the other. The idea of sequence is critically important, and will become a recurring theme in future posts

• Each sequence has only five examples. This is not a hard rule, but a categorical sequence should usually have only 4-8 examples in it. Fewer than that will rarely be enough to communicate the concept. More risks dragging it out: reducing pace, and risking transient information effect

• The sequences are ordered NPPPN. As a rule of thumb, NPPPN is a rock solid structure to get started with. It’s not the only structure that will work, and sometimes it might not be the best structure, but if you’re diving in for the very first time, NPPPN is a solid rule of thumb. Try to stick to it in the beginning

Thanks to these design choices, each sequence is already likely to be more effective than some of the alternative presentations of non-examples out there.

But, one of them also carefully applies three of the four principles of instruction.

So for that reason, here’s our answer to which is likely to be more effective: