When I first saw the Common Core State Standards' (CCSS) Standards for Mathematical Practice (SMPs), I was so incredibly excited. Simply brilliant. Now, I thought they were brilliant because they were kinda exactly what I wanted them to be. They were the most important parts of math, in my view. Back in the early 1990's, when I decided to teach the lessons of math without numbers or Bunsen burners, this was what I was looking to teach to students who got freaked out in math class.
Math teaches us how to think. Well, every content area should teach us how to think. Math is not special in that regard. As Ted Sizer said, schooling is about learning to use one's mind well. Math is one way to teach children how to use their minds well.
The thing is, it's not the more traditional math content standards that teach us how to think, how to strengthen our minds. They are just tools. And we can fill the toolbox with tools, but that does not teach us when to use them, or why. Tools are not thinking skills, and tools are not habits of mind. But in my view, math provides incredible opportunities to teach students these so valuable lessons. Math can be incredibly useful for providing lessons worth learning for a lifetime.
The hardest math problems are so often word problems. Real world content problems. Not the naked math problems whose solution paths are simply the algorithms that we were taught to apply. Not the problems that make obvious what tool to use. No, the challenge is figuring out how to get from the word problem to the naked math—and that's with grossly simplified problems without any of the actual messiness and complexity of the real world.
And that's where the SMPs come in. They are habits of mind that enable us to make sense of messy and complex situations in the real world, find the actual problem, and recognize the tools we might apply. And then to figure out which parts of the mess and complexity are the information that our tools need as input, or the gaps that we need to fill with outputs from our tools—often from our mathematical toolbox.
SMP 1: Make sense of problems and persevere in solving them.
SMP 2: Reason abstractly and quantitatively.
SMP 3: Construct viable arguments and critique the reasoning of others.
SMP 4: Model with mathematics.
SMP 5: Use appropriate tools strategically.
SMP 6: Attend to precision.
SMP 7: Look for and make use of structure.
SMP 8: Look for and express regularity in repeated reasoning.
Even on the surface, few of these appear to be truly focused on math or the procedural tools of the CCSS's content standards. Sure, from a mathematical perspective, the links are obvious. But links to other content are also pretty obvious from other content perspectives. The SMPs are a set of incredibly important thinking habits that allow us to better make sense of the world. And then, once we have done that, we might know what we are trying to accomplish and how our math toolbox might help us to do so.
Ironically, the only SMP that does not fit this view—that the SMPs are about bridging the gap from the complexity and messiness of the real world to our (math) toolbox—is SMP 5. Oddly, CCSS's official explanation of SMP 5 is not about the math toolbox of the Standards for Mathematical Content. The examples given are "pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software." These are not the internal skills and procedures math students develop mastery of through their math education. No, instead they are the external tools that they might buy (or download). SMP 5 has always struck me as odd. The title suggests that it should be my favorite, perhaps the one that focuses all the others. But whoever wrote that explanation does not think like I think and does not understand the other math practices…perhaps at all.
The others, however, are clearly all about making sense of what we don't yet understand. As that same page says—in its closing section on connecting the practices to the content standards—these practices are what students draw on to "consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations."
The tragedy is that the norms of math instruction make it impossible to teach students to grow their command of these important habits. Rather than embracing the messy complexity of the world and our responsibility to help students make sense of it, math instruction and assessment remove as much complexity as possible. They want a clear sightline to each content standard and an uncluttered workspace in which to apply it. Disconnected tools, without real purpose. Which leaves students unable to transfer their use from the obvious, simplified problems of worksheets and math tests to the challenges and realities of their actual lives outside of math class.
This is no minor tragedy. It is the fundamental flaw of math education in America. And unfortunately, although large scale assessment has the ability to push instruction to pay more attention to the important parts of our state learning standards, math assessment has also bailed on the most important parts of the Common Core State Standards for Mathematics.