This begs the question, though - what is "doing mathematics"? First, it's easy to list a few things that it is not. Doing mathematics is not:
- taking notes
- listening to me talk
- drill & kill assignments
- taking tests
Already, so there is one guide post - minimize the amount of time doing these activities.
As for doing mathematics, I like the way that Keith Devlin describes it:
"Doing math" involves all kinds of mental capacities: numerical reasoning, quantitative reasoning, linguistic reasoning, symbolic reasoning, spatial reasoning, logical reasoning, diagrammatic reasoning, reasoning about causality, the ability to handle abstractions, and maybe some others I have overlooked. And for success, all those need to be topped off with a dose of raw creativity and a desire - for some of us an inner need - to pursue the subject and do well at it.
However, I'm having a difficult time describing exactly what "doing mathematics" means. In my mind, it's more of "I know it when I see it", so I need to work on clarifying what it means. Some activities that I think do fall under this umbrella:
- Solving novel problems
- Student-generated questions
- Cooperative groupwork
- Communicating reasoning and thinking
- Proving conjectures
- Finding patterns
This list could go on and on, but my takeaway has been this. All planning should be focused on maximizing the amount of time that my students are doing mathematics.