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NCTM standards state that "In effective teaching, worthwhile mathematical tasks are used to introduce important mathematical
ideas and to engage and challenge students intellectually."
An ideal atmosphere for learning is one in which:
 | student discussion and collaboration are encouraged
| | students are expected to justify their thinking
| | students feel comfortable questioning the reasoning of the teacher and other students
| | students respect each other and the teacher
| | students feel comfortable making conjectures and suggesting alternative problem solving strategies
| | students feel comfortable presenting results.
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This atmosphere requires the teacher to be a facilitator of learning rather than a dispenser of knowledge. For example, instead of stating the definition of a convex polygon, a teacher who actively engages her students would give examples of convex and non-convex polygons and ask the students to create a definition and justify their choice.
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