Open up a maths textbook and you’ll find carefully explained ideas, precise definitions and neat diagrams. A textbook *should* be the ideal introduction for a student as they learn new maths concepts.

Except, it’s not.

Even when students dutifully follow what’s provided, they miss out on important features of concepts.

What's going on? And how learners can move beyond surface-level knowledge to meaningfully make sense of mathematical ideas? Let me explain-

**Why mathematical definitions are not enough**

The typical textbook definitions we come across are accurate, but also just *a bit too perfect*. They’re abstracted ideas that hide all the colour of what that concept really is about.

Let me show you just how much more there is to the humble triangle, by taking you through a simple exercise.

**1. What is a triangle?**

Start a list. Write down as many ideas as you can.

Now draw some triangles. How would you describe them? Add to your list.

**2. Some ‘triangle-like’ shapes**

Here are some shapes. Some are triangles, some aren’t. What else would you add to your list?

**3. Compare your ideas**

Here are just some of the features of triangles that teachers brainstormed at a workshop I ran.

Are there similarities to your list? Is there anything else you would add?

**How deep, conceptual understanding is built**

The triangle is one of the first shapes that students learn about in maths. Although it’s so familiar, it’s far from simple.

If students were to only consider the typical definitions in textbooks, there would be so much about this shape that they’d miss out on.

Instead, **to develop deep and enduring conceptual understanding **– whether of a triangle or any other mathematical idea – **learners need to look at what a concept is not and also what it’s similar to.**