The Right Equation for Math Teaching

The Common Core State Standards for Mathematical Practice require a new method of teaching. Know what to look for in your classrooms.
By Deborah Schifter and Burt Granofsky
Principal, November/December 2012
Web Resources

Full implementation of the Common Core State Standards for mathematics is still a few years away for many states. But district and school leaders are faced with many decisions now—from curriculum adoption to teacher professional development—that will influence the long-term effec­tiveness of this bold initiative. What will the Common Core State Standards actually look like when implemented in real classrooms with real students?

School leaders have a significant task in front of them, as they are key agents in ensuring that the Common Core fulfills its promise of transforming mathematics teaching and learning across the coun­try. To lead an effective adoption of the standards, they need to know why a Common Core approach is different from what existed before, what mathemat­ics instruction should look like, and how to support teachers throughout the implementation phase.

Why the Common Core Is Different

A fundamental critique of American education is that it is “a mile wide and an inch deep.” The authors of the Common Core responded to this criticism by creating a set of standards that approach mathemat­ics instruction in quite a different way than most of the sets of state standards that it replaces.

First, the authors emphasize that the Common Core is both focused and mathematically coherent. It is focused because there are fewer Common Core State Standards than existing state standards. It is coherent because it supports large conceptual issues at the heart of K-12 mathematics, and considers how those concepts develop from grade to grade.

Precisely because of this coherence, the content standards of the Common Core cannot be read as discrete items. Content addressed in different standards is connected and sometimes overlaps. It would be a mistake to think that even a veteran teacher could fully cover all of the mathematical ideas in any one standard in a single lesson.

Second, the Common Core promotes eight Standards of Mathematical Practice that identify mathematical “habits of mind” educators should seek to develop in their students at all levels. These practices—such as constructing viable arguments, critiquing the reasoning of others,


and communicating with precision— often take years to develop, but are essential for success in mathematics.

Common Core State Standards are different from many state standards in that they require teachers to give each practice explicit, focused attention to inculcate students in these math­ematical ways of thinking. Once a class begins to enact the practices, however, they become a seamless part of math­ematical discussions.

Principals need to know that it is the interplay of the Standards of Mathematical Practice and the con­tent standards that make the Com­mon Core such a robust set of guide­lines. The Common Core embraces the idea that teaching can be nonlin­ear, with various types of classroom experiences all supporting the same instructional standard.

When principals observe a lesson, they should expect to see students working through partially formed (or even incorrect) ideas about mathemat­ics. They also should expect to see teachers who are pushing their stu­dents to think like mathematicians— to justify their ideas, communicate with peers, and construct arguments, even if those arguments lead to some interesting tangents. These are all necessary steps for students to take as they develop enduring understandings about mathematics.

What Common Core Math Instruction Looks Like

Diving directly into mathematics itself can illustrate exactly what instruc­tion that aligns with the Common Core State Standards can look like, and can help principals understand what to look for when conducting walkthroughs. Let’s examine the idea of “properties of operations,” which appears in grade 2. The individual standards detail what students need to know and be able to do. The cluster heading, in bold, puts the group of standards (or cluster) in a larger context. (Three of the five standards within the cluster are excerpted below; emphasis added.)

Use place value understanding and properties of operations to add and subtract.

2.NBT.5. Fluently add and subtract within 100, using strategies based on place value, properties of operations, and/ or the relationship between addition and subtraction.

2.NBT.7. Add and subtract within 1,000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method.

2.NBT.9. Explain why addition and subtraction strategies work, using place value and the properties of operations.

Some educators may interpret the understanding of “properties of operations” to mean that students must memorize definitions for the properties. This interpretation misses the point of the Common Core. Rath­er, teachers need to provide experi­ences that allow students to make sense of how these properties interact with the calculation strategies they use every day.

Consider the expression 46-18=? and what it can show us about how to teach properties of operations. Some students might incorrectly subtract the smaller digit from the larger in each column, regardless of the order of the digits, and say the difference is 32. Students tend to persist with this error even after they have been cor­rected or have had opportunities to see that it is wrong.

An elementary teacher in Massachu­setts, Ms. Stern, was faced with this very error of subtraction, and took an approach that was quintessentially aligned with the Common Core: She addressed the misconceptions behind the error.

First, Ms. Stern asked her students to think about the effect of changing the order of numbers in an addition expression. After some investigation, students were quite sure that chang­ing the order of addends does not affect the sum: 6+3 was the same as 3+6, for example. Then Ms. Stern asked about changing the order of the numbers in a subtraction equa­tion. Since order did not matter in addition, did they think that 17-9 was the same as 9-17? Students responded with a range of ideas, but all con­cluded that order does matter when subtracting.

Ms. Stern’s approach hinged on a number of important elements: her knowledge of the math content and of why students sometimes struggle with subtraction, and a plan for helping stu­dents work through it. She was able to take a commonly held misconception about subtraction and lead her students on a journey where they began to rec­ognize addition and subtraction as dis­tinct operations that behave differently.

So, what came of the lesson?

After two class sessions spent on the order of the numbers in subtraction, Ms. Stern returned to the original equation. When asked to solve 46-18, students were able to successfully work through the problem. Through this activity, Ms. Stern engaged her students in several of the Standards of Mathematical Practices. Her students analyzed mathematical structure concerning addition and subtraction, and after they articulated what they noticed, they created arguments to justify their conclusions and explain their thinking to oth­ers. In the context of teaching a skill (double-digit subtraction), the teacher promot­ed understanding of the properties of operations, and helped her students think like mathematicians.

Ms. Stern’s instruction illustrates how approaching the Common Core as con­stellations of items—specific standards, cluster headings, grade-level critical areas, and mathematical practices— can allow teachers to act on their understanding of how students learn, how they make connections, and how they can develop mathematical power.

The Principal’s Role

School leaders can take two con­crete steps to support teachers and mathematics support specialists as they think about how to implement the Common Core State Standards thoughtfully and faithfully.

Pick a good curriculum. Choosing a mathematics curriculum that sup­ports Common Core ways of teaching and learning about mathematics is an essential first step. There are many strong, Common Core-aligned curri­cula available for elementary, middle, and high school adoptions. However, there also are some curricula that pay lip service to the Common Core without actually embracing its cen­tral tenets. Where curriculum deci­sions are made by committee, school leaders should make every effort to inform the decision-makers about what the Common Core is, what it is not, and what curricula best support this new way of thinking about math­ematics education.

Teachers need time and space to teach. If the authors of a curriculum have made sure their lessons cover the standards, then teachers will be able to put their energy into other important issues—such as prepar­ing lessons, analyzing their students’ work, and collaborating with col­leagues. It is the job of the curricu­lum, not the teacher, to ensure that every content standard has been met within a certain grade.

Invest in teacher pro­fessional development. If Common Core instruction is to transform classrooms, then school leaders must prioritize teacher pro­fessional development in two main areas. First, teachers will need to understand both the mathematical content and the conceptual chal­lenges students deal with when they encounter that content. Using the example from earlier, teachers not only need to know that 46-18=32 is incorrect, but also be able to delve into the reasons why a student would make this mistake.

Second, teachers will require sup­port with the Standards for Math­ematical Practices—both in thinking about how to teach with them, and in learning how to identify evidence of these practices in student work.

Teachers must know that imple­menting the Common Core requires their effort. They cannot deliver the standards directly into students’ minds; there is extensive mathemati­cal thought, practice, and peer col­laboration that needs to happen.

As for principals, they too should study the ideas and approach of the Common Core. The Common Core State Standards are intended to be a new direction for mathematics educa­tion in the United States. Principals have the power to support the new standards’ implementation, helping the standards live up to their promise of improving student learning.

In particular, principals should become familiar with the Standards of Mathematical Practices and learn what it means to enact these practices in K-12 classrooms. They also must remember that learning is messy, and that there are few paved roads from learning to understanding.

Deborah Schifter is a principal research scientist at Education Development Center (EDC) in Waltham, Massachusetts.

Burt Granofsky is a senior writer at EDC.


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