Mathematicians Take Risks

Too often in math class, mathematically interested students are shut down in class, often by well-meaning teachers who either don't know the underlying mathematical concept, feel publicly challenged,  see math as a set of rules and formulas, or perhaps are on a tight schedule to "cover" the curriculum.  

Most mathematicians feel mathematics is not so much being obedient to a set of fixed rules and regulations, but rather using creativity and risk taking to solve elegant problems. Paul Lockhart, mathematician, says, "Math is not about following directions, it's about making new directions." They are most intrigued and challenged by unsolved or open problems. This requires taking risks, something you might not attribute to learning/teaching math. Mathematician James Tanton says, "Math is being able to engage in joyful intellectual play--and being willing to flail (even fail!)."

Zager talks about the joy children experience when discovering or realizing new mathematical understanding in their everyday lives. It doesn't matter to them if a mathematical law or property already exists in some math textbook.

In a grade two class, Zager observes a math lesson and notices several things. First, the teacher uses wait time, wanting deep thinking and thoughtfulness, not simple ideas and speed. Also, she keeps students focussed on mathematical thinking, not unrelated matters. The teacher makes sure they are using specific and clear mathematical language. Finally, she teaches students to challenge themselves and take risks. 

PROMOTING OBEDIENCE VS. ENCOURAGING RISK

Obedience: Memorizing algorithms is necessary. "What's the rule about adding fractions?"
Risk: Different ways to solve problems exist; try to figure out which work best. "That's an interesting approach. Will it work all the time? Show your work in a way people can follow."

Obedience: Smart and easy are common words. Speed is key. "Wow, that was fast! You must be smart!"
Risk: Challenge and try are common words. "This problem is really interesting. Nice job, you really tried hard! I have a challenge for you today."

Obedience: Students speak up when they know the answer.
Risk: Students speak when they have a question, notice something, have an idea, build on a student's thinking, agree or disagree, or have an answer.

Obedience: Students are passive and do what they're supposed to. 
Risk: Students are encouraged to takes risks and expect them to come up with novel ideas. "You might be inspired by Alvin's example. Is there a volunteer to try? We will help."

 Source: Zager, Tracy Johnston, Becoming the math teacher you wish you'd had, 2017

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