Continuing the discussion of Proof
The work of establishing an environment in which students feel safe to do academic work with one another is a daily business requiring constant attention.
As a teacher, it is appealing to think that once a correct mathematical idea has been stated, then everyone who has heard it has understood it, but this is generally wishful thinking.
One of my colleagues wondered if spending a full class period on a skill that could be taught in five minutes was a good use of time.
It is the task of teachers to prepare students not only for their future courses and student but for engagement in the world.
The three acts that are critical to the development of quantitative literacy as well as higher level of mathematical fluency are: questioning, reasoning, and representing.
Questioning
Student questions are exceptionally rare in schools and they become more rare as students move up through school. Questions are rarely considered as a valuable end product in their own right.
The teacher encourages student questioning by asking students to be skeptical and to form probing questions for their peers, whether or not they understand the work.
Reasoning
The practice of reasoning is a means through which students learn to connect the different places that they visit in the mathematical environment.
Representing
Representation has four important roles:
- the opportunity to stabilize ideas
- the opportunity to edit, refine and produce clarity and elegance
- the opportunity to communicate with others
- the opportunity to discover
These opportunities are minimized if students are given repeated experience with only one type of representation such as numbers.