It has been four years now, since the release of the Common Core standards in Mathematics. We have sought to understand, to implement, to adjust to the shift in practice and expectation. But as Grant Wiggins points out in his recent blog, I'm not sure we have really put our collective mind around the true intent and value to our students.
I did a little bit of informal research and asked a wide range of Math teachers "what is the difference between math before and math now?" Overwhelmingly, the answers started and stopped with the Math practices. Certainly, a intentional focus on what it looks like for students to think Mathematically is a, if not the, critical element. Very few teachers I talked to mentioned the other elements of shift: coherence and focus. The result, unfortunately, is that we ask students to make sense of mathematics that doesn't make good sense. Without a critical reading of the shift in fluency and understanding of objects over procedures, we are unfairly asking students to build a house without a hammer.
It isn't the practices alone that signal the difference between Before CCSS and After CCSS mathematics. While problem solving may be the end goal, it is very difficult for students to develop and implement strategies to solve non-rountine problems when their only tools are procedures and steps memorized in isolation. It is in the balance of understanding, fluency, and practices that represent what it means for students to think mathematically, to use mathematics to describe, question, and analyze their world and their place in it.
The importance of the word "understand" as it is emphasized here, to me is the heart of the shift of the standards - the reason it is critically important to students and also the reason why the shift isn't easy for teachers. What does "understand" mean? Look like? How do I asses it? DO I understand the content?
Lets look at this through the lens of something familiar, subtraction.
When I ask most adults, what do students understand about subtraction? They're answer is generally the same refrain, something similar to the stack-and-subtract-borrow-and-carry method that we were all drilled in through out elementary school.
I also ask them to consider each of the four questions below, making note of how they thought about each one. Go ahead, try it for yourself :)
How many times did you use the standard algorithm?
I love these questions because they demonstrate a central truth of the core. To be fluent in subtraction means so much more than knowing the stack-and-borrow method. Part of fluency is understanding that there are many strategies and having a sense of when to use each one. In question one above, you could use the standard algorithm, but this isn't nearly as efficient as comparing the two numbers, asking yourself, what is the difference between them? 98 is two from 1000 and 1003 is three more than 1000, so the total difference is five. Or you might have thought that the difference between 98 and 1003 is the same as the difference between 95 and 1000. There are many more very efficient and elegant solution paths, but the standard algorithm isn't one of them.
In some situations, the standards algorithm helps us organize many steps of thinking more effectively. In others, like above, it is inefficient, and in others, it doesn't work at all. That's right - it doesn't always work. The standard algorithm work because of the base 10 nature of numbers. If we take the situation to a quantity that isn't in base 10 - like time in question two - our understanding of subtraction is really challenged.
You might be thinking - but, Kelly - those examples just make sense. They do. These common questions make sense to us as adults because we have experienced many chances to develop an understanding of what it means to determine the difference or to find what is left after some value has been taken away. We take this understanding for granted, remembering more vividly having to "learn" the algorithm, which takes much more thinking to make sense of.
But our job as teachers is to understand all the thinking, the nuanced and beautiful meaning-making, that students experience as they grow as mathematical thinkers. It is our job to articulate how this understanding develops so that we can partner with students as they explore how subtraction presents itself in the world around them.
It is also our job to know where this understanding might take them. Students use their understanding of subtraction of numbers to develop an understanding of the laws of arithmetic in algebra (what does 2x - 4y mean?) and to make connections to graphing and what subtraction means on a number line or in the coordinate plane (this is another post). This is the Coherence - mathematics is a discipline of related reasoning. And the Focus - spending our efforts on what has meaning for students.
I learn more math from reading the standards every time I open the document. I hope that you'll provide your own examples of the shifts you have experienced in your own deep reading and study of the standards.