Research School Network: Maths teaching: Non-examples Amarbeer Singh Gill explains why showing what something ​‘isn’t’, can help us understand what it is.

On Monday 20 and 27 April, we’re running two online twilights exploring common challenges and potential solutions in maths teaching. You can sign up here. This blog focuses on an approach we’ll explore in the second session: the use of ​‘non-examples’ examples.

Maths includes a wide range of concepts for students to learn and understand. These are varied and often abstract, from parallel and perpendicular in geometry to using letters to represent unknowns in algebra, or relative frequency in probability.

Because of their abstract nature we might try to teach them using multiple examples, but what if there was a more efficient way?

The challenge of abstract concepts

Let’s consider the concept of parallel lines. One option would be to simply give students a definition of this:

Now, this is clearly quite complicated full of potentially unfamiliar language, so we could offer a simpler definition:

This definition is easier to understand than the first, with less complicated vocabulary, but it’s still relatively abstract. So, a common approach here would be to show an example:

This solves the problem of abstraction by giving a concrete example, however it also introduces new possibilities to what parallel could mean:

• Does the definition also mean two lines that are completely horizontal?
• Or two lines that are a specific distance apart?
• Or two lines that are the same length?

This is where non-examples come into play.

The power of non-examples

As the above example demonstrates, the challenge with only providing true or correct examples is that it runs the risk of an infinite number of possibilities.

A non-example helps with this as it provides some boundary conditions for the concept (McCrea, 2019), particularly when the non-example is minimally different to the correct example ie they share a number of common features (Engelmann & Carnine, 1991).

Let’s continue with the example of parallel lines to exemplify this. Instead of just showing one example, or even only showing correct examples, we’re going to show a sequence of options that include both correct and non-examples:

We could either talk through these examples, or ask students to first select which ones do contain examples of parallel lines and then explain why (or why not). So non-examples can clearly be helpful when introducing concepts.

But we could also come at this from another angle to think about the power non-examples can have: verifying understanding. Let’s consider this scenario:

• One option here would be to simply ask students for the definition of parallel lines. This isn’t a bad approach but it runs the risk of students providing the correct definition whilst having an incomplete understanding that we don’t pick up on because we haven’t given them any boundary conditions to contend with.
  
  Even if we asked them to come up with examples, it’s likely that they’d come up with something like examples 1 or 2 in the boxes above.

• Giving them non-examples (like 3 above) or more ambiguous examples (like 4, 5, and 6) to consider achieves both of our initial goals (checking for understanding and activating prior knowledge) but now allows us to make much more valid inferences about students’ actual understanding of the concept.

In summary, non-examples can be incredibly helpful when understanding new/​abstract concepts as they provide boundary conditions, and when compared with correct examples ​“encourages students to attend to the details that define a concept” (McCrea, 2019, p.52).