Mathematics is an inherently social activity, in which community of trained practitioners (mathematical scientists) engages in the science of patterns – systematic attempts, based on observation, study, and experimentation, to determine the nature or principle s of regularities in system defined axiomatically or theoretically (“pure mathematics”) or models of systems abstracted from real world objects (“applied mathematics”). The tools of mathematics are abstraction, symbolic representation, and symbolic manipulation. However being trained in use of this tools no more means that one thinks mathematically than how to use shop tools make one a craftsman. Learning to think mathematically means
- Developing mathematical point of view – valuing the processes of mathematization and abstraction and having predilection to apply them
- Developing competence with the tools of the trade, and using in the service of the goal of understanding structure – mathematical sense – making. (Schoenfeld)
Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world.