· Solving a simplified version

· Testing a particular case/ special case/ extreme case

· Breaking the problem into smaller pieces

· Trying to find a counterexample

· Using all available information

· Making a model or diagram

· Making conjectures (and then testing them)

· Eliminating possibilities

· Working backwards

· Adding helpful conditions

· Innovating

· Returning to the question

And also:

· Being able to stay with a problem over a period of time (an hour, a week, a month)

· Knowing when to take a break

· Knowing when to abandon a strategy

· Being used to struggling

· Being unafraid of mistakes and false starts

· Collaboration

· Flexibility

· Creativity

· Patience

· Persistence

Analytical thinking skills

· Knowing when something has been established

· Being clear about definitions

· Knowing that if we can’t do something, that doesn’t mean it can’t be done

· Understanding that examples are not proof

· Understanding why one counterexample suffices to show falsity

· Being able to articulate a clear argument

· Asking (and answering): how do we know that?

· Critiquing our own and others’ reasoning