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Core Practice 5.  Look for and express regularity in repeated reasoning.

Mathematically proficient students pay attention to repeated calculations as they carry them out, and look both for general algorithms and for shortcuts. For example, by paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, they might abstract the equation (y – 2)/(x – 1) = 3.

 Checking for many points whether they belong to a particular line or not departs from typical exercises with linear functions in a good direction. It’s nice to see this kind of assignment legitimized by the Standards. Once the points are classified, there are so many meaningful questions that can be asked about them. Do any 2 points belonging to the line produce a slope of 3? Can a point belonging to the line and one not belonging to the line produce a slope of 3? Can any two points not belonging to the line produce a slope of 3? If yes, what can we say about such point?

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Intro. Proficient students expect mathematics to make sense. They take an active stance in solving mathematical problems. When faced with a non-routine problem, they have the courage to plunge in and try something, and they have the procedural and conceptual tools to carry through. They are experimenters and inventors, and can adapt known strategies to new problems. They think strategically.

These are practices that expert mathematical thinkers encourage in apprentices.

 I would like to see a more detailed description of an expert mathematical thinker. Right now it sounds more like a mathematician than a highly quantitatively literate person.

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