Why textbook definitions don’t help students

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.

What we typically see. Textbook definition: "Triangles can be named using the vertex labels. e.g. triangle BAC, triangle DEF. Triangle: A plane (3D) shape with three straight sides and three angles". This is missing colour.

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?

Some 'triangle-like' shapes. Some are triangles. Those that aren't have: curves or aren't connected. Source: Horne & Watson, 2008

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?

What is a triangle?
A 2-dimensional shape
A 3-sided shape
3 angles
Strong shape (can carry weight)
A polygon
All other polygons can be broken into triangles
Connected sides 
Interior angles add to 180o
Can tesselate (if equilateral…)
Closed shape
Straight edges
Can be drawn on curved surfaces

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.

Key idea: Meaningful understanding comes when we look at what an idea is not and what it’s like.

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