This post is different than most of my other What Do You Think posts. It's more of a big picture view focused on what it means to do mathematics.
I wanted to write a post on the Common Core, but that's a huge topic to address all at once, so instead I'm going to talk about one (unfortunately tiny) part: the eight Standards for Mathematical Practice (or SMPs). You can read the SMPs here if you want and you can read a more detailed breakdown of them here.
What's Great about the SMPs
I love the SMPs. They tell us what it means to do mathematics. Usually when we think of mathematics, and of standards in particular, we think of content. Content standards are things like:
Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.
Content standards are all the stuff you're supposed to be able to do in math. They're the fractions, the adding numbers, the identifying shapes and their properties.
The SMPs are the practices one engages in when doing math. They're things like:
Students make sense of problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. (slightly modified verbiage)
This isn't specific to arithmetic or geometry. It's a general practice that cuts across mathematics. It's what mathematicians do whether they're first learning to add numbers or proving new, cutting edge theorems. To me the SMPs are the heart of mathematics and their inclusion in the Common Core means we expect all students to be mathematicians. They're also the skills that are more likely to be relevant outside the classroom.
So what are they?! The list below is far from comprehensive, but I think it's an accessible overview to some of the big ideas. This book unpacks them in greater detail.
SMP 1. Problem Solving (i.e., Make sense of problems and persevere in solving them)
Most school mathematics involves exercises. The teacher shows kids how to solve a problem and then kids practice on a bunch of similar exercises. But mathematicians don't do exercises, they solve problems. They don't have a strategy for solving them (if they had a strategy, they wouldn't be problems!).
An extensive body of research shows that children learn mathematics best by solving problems—questions they do not have a predetermined strategy for. Virtually everything I've written about on this blog involves problems, because I let the person doing the math figure out their own way of solving it. A problem for a young child might be "You have 10 toys. You want to give 2 to each person. How many friends can you give toys to?" Many adults would see that as a division problem and have a predetermined strategy for how to solve it (many would recall a memorized fact), but young children can invent their own strategies for solving it using blocks, their fingers, drawings, skip counting, etc.
SMP 2. Quantitative Reasoning (i.e., Reason abstractly and quantitatively)
This is all about how we move back and forth between real-world contexts and more abstract, symbolic mathematics. An important part of mathematics is recognizing how we can quantify the world. The world does not come with numbers attached to it, so we have to invent ways to measure the world with number. For instance, creating units (like miles) for measuring distance and using these to measure lengths, or creating more complex units, like speed as miles per hour. When doing mathematics we can do stuff to the numbers (for instance, adding up a bunch of areas) and it will tell us something about the world (like how many square feet of carpet to buy to cover multiple rooms).
SMP 3. Arguments and Reasoning (i.e., Construct viable arguments and critique the reasoning of others)
Doing mathematics requires justifying your thinking. You have to be able to prove that you know what you know. Many would describe proof as the core of mathematical practice. Young children do not engage in formal proof, but they can begin to justify their thinking and explain how they know what they know. They can also create general arguments even if they don't have the formal language of mathematical proof.
It's important to note that proof actually plays a variety of roles in mathematics. On one level it's a way to be sure that we know what we know. However, it also plays an important role in understanding why things are true—this is especially important in school mathematics where there ought to be a bigger emphasis on "proofs that explain."
SMP 4. Model with mathematics
This is a little like SMP 2 on overdrive. Using mathematics in real life is messy. Real life doesn't come in neat little story problems with a clear answer. Instead, people generally have to analyze a real-world problem, do research to understand the problem, think carefully about how they might use math to better understand or solve the problem, work with the mathematics, check back in with the real world and see if they're really making progress, modify their approach (often multiple times), and eventually (hopefully) get some kind of solution in which the mathematics helps provide insight into the problem they are exploring. This is a messy, iterative process.
Most kids get almost no opportunities to do this in school and it shows. They tend to get worse at using real-world knowledge the longer they are in school, because instead of actually connecting math to the real world they have learned to play the "word problem game." In the word problem game you learn that the real world doesn't matter and instead you just go along for the sake of going along. Note, this isn't to say that all story problems are bad—in fact they have a very important role to play in learning mathematics. But never doing modeling is bad, since modeling is how math is used in the real world. Weather forecasting, understanding the spread of diseases, predicting when social security will run out of money, and projecting population growth all involve mathematical modeling.
SMP 5. Use appropriate tools strategically
Students should know how to use tools, like "pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software." They should know how the tools can help them and know their limits.
This is kind of like the "calculators are not evil" standard.
SMP 6. Attend to precision
Precision covers a lot of ground in SMP 6. It includes precise language (which does not mean you start out a new math unit by going over 8 million vocab words). It also includes specifying units (like "inches"), labeling graphs appropriately, calculating accurately, and making appropriate estimations.
This is an all around, "be as clear and accurate as appropriate" standard.
SMPs 7 & 8. Using Generalizations & Generalizing (?) (i.e., 7. Look for and make use of structure and 8. Look for and express regularity in repeated reasoning)
I'm still not sure I understand SMPs 7 & 8. It was once explained to me that one way to think about it was to imagine SMP 8 coming first. SMP 8 is about noticing patterns and regularities in how mathematics works. Some young-child examples might be noticing that it doesn't matter what order you add numbers in (i.e., 8 + 3 = 3 + 8 no matter what the two numbers are). Another example might be noticing that when you add 10 to a number the tens place goes up by one, but the ones stay the same (e.g., 27 + 10 = 37 ... the "2" changed to a "3" but the "7" stayed the same). Some of these generalizations are big (like these examples). Others may be specific to a particular problem.
Then comes SMP 7, which involves using generalizations to make math easier. For example, knowing you can add numbers in any order could help on this problem: "8 + 5 + 2", because you could switch it around to "8 + 2 + 5" and doing "8 + 2" first is easier for a lot of people.
These two standards can tie in with SMP 3 (the reasoning one) because you ought to figure out why these things are true.
What's Terrible about the SMPs
I hate the SMPs. They're a sad, anemic representation of what it means to do mathematics.
First, they are far too short. They are a paragraph each, yet all teachers from kindergarten to 12th grade are expected to address them in their teaching. They just don't provide enough detail to really be useful in classroom practice. These can be contrasted with the five process standards from the National Council of Teachers of Mathematics' Principles and Standards (which the SMPs are partially based on). In Principles and Standards the five process standards get almost as much attention as the content standards—each standard gets a few pages of description for each grade band (K-2, 3-5, 6-8, 9-12)... Remember the SMPs are one paragraph each for all of K-12!
There are too many of them. This may seem silly, but eight is a bad number of practices to have. The SMPs are based on the five process standards from the National Council of Teachers of Mathematics (NCTM) and the five strands of mathematical proficiency from Adding It Up (free to read online). Five is a good number. It's easy to remember. Plus the NCTM process standards had nice names (Problem Solving, Reasoning & Proof, Communication, Connections, and Representation). These are things you can remember. Contrast that with the list of SMPs above.
They are not consistent in scope or character. Depending on how you understand them, SMP 2 can be seen almost as a subset of SMP 4. Using tools and attending to precision (SMPs 5 and 6) seem to be of a different character than many of the others.
No one can understand SMPs 7 & 8. I'm not sure if I could write a more confusing mess if I tried.
What happened to communication?! One of the NCTM standards was communication. Mathematicians and people who use math in the real world need to communicate their thinking, but that's largely absent from SMPs. There's SMP 3 about constructing arguments, but that has more in common with NCTM's reasoning and proof standard than it does with the communication standard.
They're rarely tested, and when they are tested it's in a simplified way that undermines their inclusion in the classroom. This isn't exactly the fault of the Common Core, but rather the way in which tests are used to assess the Common Core, but these are pretty closely related issues, especially since a new testing regime was part of the Common Core from the beginning.
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How much of the doing side of mathematics was emphasized when you learned math? How was it emphasized?