Successful mathematicians understand curriculum concepts, are fluent in mathematical procedures, can solve problems, explain and justify their thinking, and have a positive attitude towards learning mathematics.
Exploring, questioning, working systematically, visualising, conjecturing, explaining, generalising, convincing, proving... are all at the heart of mathematical thinking. The activities below are designed to give students the opportunity to think and work as mathematicians.
For problems arranged by curriculum topics, see our Secondary Curriculum page.
For problems arranged by mathematical mindsets, see our Mathematical Mindsets page.
Exploring and noticing - Secondary teachers
Age: 11 to 16
These problems will offer your students opportunities to explore and reflect on what they notice.
Working systematically - Secondary teachers
Age: 11 to 16
These problems will offer your students opportunities to appreciate the value of working systematically in a variety of contexts.
Conjecturing and generalising - Secondary teachers
Age: 11 to 16
These problems will offer opportunities for your students to use specific examples as a springboard to generalising.
Visualising and representing - Secondary teachers
Age: 11 to 16
These problems will encourage your students to draw on their visualising skills, and make use of multiple representations.
Reasoning, convincing and proving - Secondary teachers
Age: 11 to 16
These problems will challenge students to develop their reasoning skills and create convincing arguments.
Thinking mathematically - Short problems
Age: 11 to 16
A collection of short problems which require students to think mathematically.
In this film (also available outside the UK) the mathematician Andrew Wiles talks about his personal experience of seeking a proof of Fermat's Last Theorem. He describes what it is like to do mathematics, to be creative, to have difficulties, to make mistakes, to persevere, to make progress, to have a dream and love what you are doing so much that you are willing to devote yourself to it for a long time. Of course, each mathematician's experience is different, and most mathematicians do not work alone for such prolonged periods without discussing their work with others, but much of Andrew Wiles' experience is shared amongst mathematicians, and reminds us of the rewards of perseverance in the face of difficulty.
You may like to read the following books to find out more about what it means to think mathematically:
Learning and Doing Mathematics by John Mason
Thinking Mathematically by John Mason, Leone Burton and Kaye Stacey