One of my earliest memories is that of working on puzzles with my father. I believe that puzzling is at the heart of mathematics. When faced with a problem, you solve its puzzle, generalise it, and then prove this is true for all. Throughout my teaching career I have used puzzles and experiments to enthuse my students, and to encourage them to think in a mathematical way.
Due to the declining number of students taking maths as a subject at A level and beyond, I have spent a great deal of time in recent years publicising the subject I love. My DR Maths columns and maths shows attempt to persuade people that "Maths is Everywhere" and accessible to all.
I believe that the fundamentals of mathematics are not about difficult formulae, but about logical ways of looking at and thinking about things. One of the challenges for maths teachers is to show children that this is true.