Too often, this is what maths looks like to outsiders:

Yet, work with students and you notice how deep, enduring mathematical understanding comes from far more than just examples on a board or endless exercises.

We can’t just teach students to copy processes or lines of working out. We can’t just show them new definitions. That’s not enough.

So, what’s missing?

## What holds maths learning together?

Students need opportunities to:

Analyse, argue, compare, contrast, convince, critique, deduce, elaborate, explain, infer, justify, persuade, reflect and weigh up.

When they do these things, students can then:

- make observations and sort through the ideas in their head,
- link and understand the logic behind steps,
- actually
*trust*that the maths they’re doing isn’t just some process to be memorised, but that it makes purposeful, meaningful sense.

You might think of this collection of opportunities as the glue that holds all mathematical learning together. Helpfully, there’s a name for this: **mathematical reasoning**.

**What does it mean to get stronger in mathematical reasoning?**

Now, students can be engaging in these ‘glueing’ opportunities (i.e. via those verbs listed above), but how might you support them to become stronger and more sophisticated in their reasoning?

Here are 5 stages of development in mathematical reasoning, adapted from NRICH:

STAGE 1: A student can **describe **what they did.

STAGE 2: A student can **explain **what they did, by offering reasons – whether or not these reasons are correct. At this stage, they may not have a coherent argument.

STAGE 3: A student can provide a coherent or complete argument to **convince** someone of what they did. The student uses a ‘chain of reasoning’ to produce their argument, even if the mathematics is not all accurate.

STAGE 4: A student can **justify **what they did using a complete argument that is logically correct.

STAGE 5: A student can **prove **what they did using a watertight and mathematically sound argument. The argument may be based on generalisations and the underlying structure of mathematical ideas.

Just as the mathematical proficiencies can be applied across different ages and any topic, so can these stages. You might also find these stages useful as a formative assessment tool.

## Summary

Mathematical reasoning helps students to better understand and see the purpose behind what they learn.

Over time, students can get stronger and more sophisticated in their reasoning.