The Five Big Ideas at Secondary – Mathematical Thinking

Continuing our series on The Five Big Ideas at Secondary, we chat to Jane Hawkins, Assistant Director for Secondary, about mathematical thinking. She explains why it should be part of every maths lesson and shares some ideas of what this might look like.

What do we mean by mathematical thinking?

Mathematical thinking is central to how students learn maths. It includes looking for patterns and relationships, making connections, conjecturing, reasoning and generalising. Students should have opportunities in all lessons to engage with others, to communicate their own ideas and to process others’ ideas. As teachers, we have a responsibility to design lessons that facilitate thinking and create opportunities for students to think deeply about the maths that they are learning.

Why is mathematical thinking important, and how does it fit into the Five Big Ideas?

At the NCETM, we believe all students can learn and enjoy maths. One of the ways that students can succeed is by sharing their mathematical thinking, which is why it is one of the Five Big Ideas and is woven throughout The Essence of Mathematics Teaching for Mastery.

Variation or representation and structure might be strategies that teachers use to encourage thinking and expose mathematical structures, and coherence should be integral to the ordering and sequencing of lessons, whilst mathematical thinking and fluency are states of being that we want students to embody. Thinking mathematically is an important element of what it means to be a good mathematician, and we want students to understand and experience that in all of their lessons.

• READ: Our Ofsted explainer on Reasoning through Talk focuses on how talk enables students to explore and articulate their understanding of the maths they are learning.

What are the benefits of mathematical thinking?

If teachers can change students’ perception of ‘success’ in maths from trying to quickly attain a single correct answer, to instead focus on reasoning, explaining, evaluating and justifying their thoughts, the benefits to teaching and learning are huge. Students will develop learning behaviours that will serve them well within and beyond their maths lessons.

In a typical lesson, we would suggest that the teacher leads interactions with students, including questioning, short tasks, explanations, demonstrations and discussions, which enable students to think, reason, justify and apply their knowledge to solve problems. Equipping students with precise mathematical language will support them to communicate their thinking effectively.

Activities which facilitate mathematical thinking and talk enable teachers to better gauge student understanding. Misconceptions can be addressed promptly, and teachers can adapt lessons to build on what students currently understand.

• LISTEN: Our podcast episode with NRICH on problem-solving explores how rich tasks can support mathematical thinking and classroom talk.

How should teachers incorporate mathematical thinking into their lessons?

Teachers should explicitly plan for mathematical thinking to be a routine part of the curriculum experience for every student.

If students are to think mathematically, they need to be given something mathematically worthwhile to think about. By including task types such as withholding information or comparing solutions, students are encouraged to think more broadly and deeply.

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Even by asking, ‘How do you know?’ after a student's response can draw students’ attention to explaining and reasoning as an important part of mathematics.

• WATCH: Our Five Big Ideas video on fluency explains how focusing on the method rather than the solution can support students' understanding:

What are some of the challenges teachers should be aware of?

Developing students’ facility for thinking mathematically may, in some cases, require addressing students’ preconceptions about what it means to be good at maths. If students believe ‘good mathematicians’ to be those who can compute calculations very quickly, then frequent experiences where mathematical thinking is recognised, praised and valued will be necessary to shift students’ perceptions.

What are your top tips for teachers?

Teachers should plan opportunities for students to think mathematically in every lesson. Allocate time in your lesson design for students to be thinking deeply about maths. Start with what it is you want students to be thinking mathematically about, and then focus on designing and constructing the experiences that will enable them to do so.

Use rich tasks, which give students something mathematically worthwhile to think about. With thoughtful task design, teachers can shift students’ attention to thinking mathematically, rather than attaining a single correct answer.

Listen authentically to your students. Not only does that help to facilitate quality discussion and show that students’ ideas are valued, it is also an invaluable assessment for learning source.

To find out more about the Five Big Ideas, and to discover how your school could start using mathematical thinking across the maths curriculum, contact your local Maths Hub or explore some of the professional development available.

Look out for our explainer video coming soon, Why is mathematical thinking one of the Five Big Ideas in Teaching for Mastery?

Did you enjoy this?

Read our other Five Big Ideas features (more coming soon):

• Coherence
• Fluency.