Among those in the reform math area, there has been a push to interpret the SMPs (Standards of Mathematical Practice) along reform math ideologies that push certain mathematical “habits of mind” outside of the context in which such habits are learned, as well as a predominate use of collaborative group work and inquiry-based learning. This article provides the description of each SMP as written in the Common Core math standards. (http://www.corestandards.org/Math/Practice) It discusses aspects of each SMP that can be interpreted along conventional or traditional approaches to math teaching and contrasts this with how each one may be implemented under the math reform interpretation.Here's some of his commentary on SMP 1: Make sense of problems and persevere in solving them:
The SMP writeup describes a problem solving mind-set as well as a variety of problem solving strategies that students should have. It is important to realize that the goal of this SMP comes about after years of experience and practice. The ability to solve problems and think mathematically develops over time. Problem solving cannot be taught directly; rather, it is based on mastery of many basic skills. (See (http://www.ams.org/notices/201010/rtx101001303p.pdf )It's a great series, I highly recommend it.
Requisite for learning how to solve problems is an explanation of how specific types of problems are solved using worked examples and practice with routine problems. A set of problems can then escalate in difficulty through careful scaffolding: i.e., by changing aspects of the problem so that students must apply their knowledge of the basic procedure to new forms of the problem. In this way homework is not just a set of repetitive “exercises”. Students progress from simple routine problems to those which increase in complexity and are non-routine. The non-routine problems can then be extended into even more challenging problems. Such challenging problems should definitely be given but students must be able to use prior knowledge of skills and procedures in solving them. The goal of math teaching is to provide sufficient opportunities to apply skills and knowledge so that students know how to turn “problems” into routine exercises.
While the approach described above is a sensible and effective interpretation of this SMP, the reform math ideology that is dominating Common Core implementation is likely to reject it. That philosophy is to regard math as some sort of magical thinking process. It holds that “understanding” the problem and seeing the big picture is math, while the mechanics of problem solving are just a rote afterthought. Worked examples and routine problems are generally disparaged as “non-thinking” and “routine achievement”. The reform approach usually manifests itself as giving students a steady diet of “challenging problems” in an effort to build up a problem solving habit of mind that is sometimes referred to as “sense-making”. Such approach does not accomplish this, however. Instead, the constant pursuit of “challenging problems” stands in the way of developing fluency with certain classes of problems and building on what one already knows.