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
- Investigations
- 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.
I have the same goal! I think a good, simple working definition for 'doing math' is the asking and answering of bigger and better questions. This is a pretty solid approach to learning in general, but if the questions are in the least mathematical (pure, applied, or otherwise), then answering them is most likely doing math.
ReplyDeleteI do want to say, however, that taking a test can sometimes be a great math activity. As a student, I used to love a good test. I felt like they were a challenge from my teacher, a little 45 minute session in which I get a handful of good problems to take on. Something I like to do if I test is put on WAY too many things to be done in the period. In a graded environment this is disaster, but if the students understand it's all gravy, them they can take them home, mull them over, and come back later.
In life and in math, you get what you get. The best questions are the ones you carry around with you until you finally get them.
Great post and a great starting mindset for the year!
Hans Freudenthal (the great Dutch math educator) would agree. He defined mathematics as a "human activity." Rethinking mathematics as a verb and understanding what is meant by mathematizing is a powerful and positive way to change students' perceptions of mathematics.
ReplyDeleteGood luck with the rest of your school year!
Paul: I think that's a good big picture definition, but it's still too vague for me - what exactly are bigger and better questions?
ReplyDeleteAs for testing, I absolutely agree that tests can be good math activities *if* we can change the culture around testing. With my semi-SBG policy, I'll be trying to move in that direction, but I think that's a long ways away in public schools.