What is the mindset of a mathematician? Is it someone who just follows rules? Or is it someone who has a mad sense of creativity like Mozart or Einstein? Alternatively, is it someone who has a deep sense of clarity and structure? Or is it someone who has a very fast brain? Is it someone who does extremely fast calculations or someone who takes their time with very deep understanding.
There is no clear answer to this. Surely there existed some mathematicians who were excellent because of their inner discipline (e.g. Andrew Wiles who proved Fermat’s Last Theorem - and - funnily, one of my professors at Oxford University) but there are also some mathematicians whose lives resemble those of artists with their unstructured, powerful outbursts of creativity - for example Galois who died in a duel or Erdos who was essentially homeless for most of his life.
Probably what all mathematicians, no matter how excentric, have in common, is their love for mathematics as a subject and their ability to think abstractly about problems.
They also had to have a growth, rather than a fixed mindset. They did not care if they are good at what they are doing (Einstein famously failed his mathematics degree), but instead they tried to gain more understanding despite any boundaries or limitations.
In school, this type of mindset is perhaps more important than the individual character that the students bring into the classroom. That is the reason why I always start the school year with the following picture and ask students to identify themselves with one of the blobs and then identify the blob that they would like to become.
Some interesting blobs/personalities include:
A mathematician who is feeling victorious when he managed to climb the whole tree. He is standing alone.
Someone playing around at the bottom of the tree, not really caring about climbing up.
Someone happily in the middle, clearly content with themselves and not trying to climb any further.
Someone trying to help other people climb the tree. Someone accepting their help and succeeding in going up.
Someone desperately clinging to what they know, clearly not in a good state of mind to climb any further.
Someone who totally gave up and is at the bottom of the tree. They are crying and not even looking up.
Someone who is determined to climb the tree and devotes most of their energy to it. He acts independently, but alone.
Which mathematician would you like to be? And how can you get there? That is the question that each student should answer before starting their learning as it leads to explicit introspection - a highly transferable skill that will be useful no matter what the student decides to do in their future.