Developing mathematical thinking - Reasoning, convincing and proving

Reasoning, convincing and proving is part of our Developing Mathematical Thinking Primary and Secondary collections. This page accompanies the Reasoning, convincing and proving Primary and Secondary resources.

This follows on from Exploring and noticing, Working systematically, Conjecturing and generalising, and Visualising and representing.

Show Primary webinar recording

During this webinar we had a go at the following tasks:

First connect three

Dicey addition, using the interactivity found in Dicey operations

Show Secondary webinar recording

During this webinar we had a go at the following tasks:

Connect three

Factors and multiples game

Cyclic quadrilaterals

Logical reasoning and proof are at the heart of mathematics. Mathematicians are not satisfied by vague assertions that something seems to be true, they need to be convinced that there is no doubt.

Our classroom needs to be a place where learners are actively involved, constructing reasoned arguments themselves, rather than simply trusting the teacher.

If we are to offer students opportunities to develop their reasoning skills, we may need to think about the following:

Values and ethos in our classrooms

Mathematics has order and structure that can be understood by students
Students can build on current insights and understanding to gain new insights and understanding
We establish what is mathematically true by deductive reasoning rather than empirical evidence or opinion

Structural considerations

Alan Wigley's 'Challenging Model' offers a useful framework for structuring lessons, in which students are required to make significant mathematical choices, and expected to communicate their ideas and justify their thinking
In Building Thinking Classrooms, Peter Liljedahl offers us 14 teaching practices that have been proven to enhance thinking and reasoning
Making use of rich, and accessible low threshold high ceiling (LTHC) tasks requires all students to provide reasoned arguments at whatever level they are working
Giving students time to think on their own, and an opportunity to discuss their thinking with one or two other people, before communicating their ideas to the whole class ('think-pair-share'), can:
- Send out a very clear message that we don't expect refined, convincing arguments from the start
- Raise expectations about the quality of the arguments that are put forward
- Give students time to rehearse ideas in a safe environment before making them public
- Offer opportunities for students to engage with questions that require sophisticated mathematical thinking
'Convince yourself, convince a friend, convince a sceptic' offers a useful framework to support students in refining their arguments
'Only do for students what they cannot do for themselves' offers a useful guiding principle

Emphasising the importance of mathematical reasoning

We'd like our classrooms to provide an environment in which students behave as part of a community of mathematicians, where rigorous mathematical arguments are valued. Articulating ideas and formulating reasoned arguments is a sophisticated skill that needs to be nurtured.

Below are examples of the variety of mathematical arguments that we would like students to be exposed to. They include some proof sorters (PS), which offer scaffolding for students getting to grips with the structure and language of rigorous arguments.