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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.
Being curious, resourceful, resilient and collaborative are all valuable mathematical mindsets. We hope that the activities below will give students opportunities to develop these desirable characteristics.
These problems will exploit students' natural curiosity and provoke them to ask good mathematical questions.
These problems require careful consideration. Allow your students time to become absorbed in them.
These problems require resilience. Encourage your students to persevere - there's often a great sense of achievement when we've had to struggle.
These problems are ideal to work on with others. Encourage your students to share ideas, and recognise that two heads can be better than one.
In this film (available here if you live 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 also be interested in this paper, Habits of Mind: An Organizing Principle for Mathematics Curricula, in which Cuoco, Goldenberg and Mark call for mathematics curricula to have ways of thinking about mathematics at their core.