 Volume III, Issue II Heather Clayton, the author of Making the Standards Come Alive!, is the principal of Mendon Center Elementary School in Pittsford Central School District, New York. She is also a co-author of Creating a Culture for Learning published by Just ASK.

## Standards for Mathematical Practice

The habits of a vigorous mind are formed
in contending with difficulties.

 The Common Core State Standards for Mathematics have two sets of standards: the Standards for Mathematical Content and the Standards for Mathematical Practice. The content standards are different for each grade level and outline what students are expected to understand and be able to do at each grade. They are organized by domain or concept. Each domain includes related clusters of standards for each grade. The Standards for Mathematical Practice, however, are the same eight standards across all grade levels K-12. As stated in the Common Core, they represent the “expertise that mathematics educators at all levels should seek to develop in their students.” In other words, they describe what it means to “do” mathematics and apply mathematical content. These standards represent the kind of thinking students do as they are learning the content and how we want them to engage with mathematics. Teachers have the challenge of planning for and teaching both the content standards and the Standards for Mathematical Practice.

### Fourth Grade Content Standards Exemplar

4.OA Operations and Algebraic Thinking (Domain)

Use the four operations with whole numbers to solve problems. (Cluster)

1. Interpret a multiplication equation as a comparison, e.g., interpret 35= 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons and multiplication equations. (Standard)
2. Multiply or divide to solve word problems involving multiplicative comparison, e.g. by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (Standard)
3. Solve multi-step word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. (Standard)

### Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning

### Principles Guiding the Implementation of Standards for Mathematical Practice

The Standards for Mathematical Practice should be embedded in the Content Standards.
One way to assess students’ understanding of the content standards is through the practice standards. For instance, when asking students to apply the content learned

• Is the student able to justify why an answer is correct or a specific rule applies?
• Can the student approach a problem and determine a way to solve it?
• Can the student write equations or expressions relevant to a specific task?
• Does the student notice different patterns and structures embedded in problems?

## Teachers must carefully analyze the tasks they are assigning and be intentional about which practice standards will be emphasized and where they will be emphasized.

 The practice standards should not become lessons that are separate from the content expected at each grade level. Rather, the practice standards should be integrated with the content required in the content standards. The practice standards must be given the same care and attention as the content standards and not be an afterthought or assumed to be “always happening.” Rather, teachers must carefully analyze the tasks they are assigning and be intentional about which practice standards will be emphasized and where they will be emphasized. As suggested in Implementing the Common Core Mathematical Practices, one way for teachers to think about this is to look at each mathematical practice in relation to assigned tasks and consider the potential for students to be engaged in each practice: high, medium, or low.

The Mathematical Practices are standards, and they require explicit teaching.
The Common Core is very clear in saying that these standards “are not intended to be new names for old ways of doing business.” They are designed to move us forward with a sense of urgency, towards more robust and lasting learning in mathematics. It is not enough to simply have them posted on the wall, or to say that they are happening all of the time in all lessons. If that were true, they would be occurring with no intention, purpose, or depth of learning. Instead, these practices need targeted and explicit instruction provided in the context of the content the students are learning.

Not every Standard for Mathematical Practice will be emphasized in every lesson.
In order for students to learn how to engage in the practice standards, teachers need to choose which practice standards will be emphasized during particular tasks. Then, teachers can “turn up the volume” on those practices by explicitly modeling them and sharing examples of how students are applying them when engaging with the content. It is simply not realistic to think that eight practices will be an area of focus in every lesson if we want our students to learn how to engage in the communication, representation, reasoning, and proof required in mathematics.

Relationships exist between the Standards for Mathematical Practice.
While listed as eight separate standards, connections exist between the practice standards. Some are more general than others and, therefore serve as overarching ideas.

• Making sense of problems and persevering in solving them (1) while attending to precision (6) is relevant in all problem solving situations and illustrates the overarching level of thinking our students need to do whenever they are engaging in mathematic al work. This involves:
• Finding a valid entry point
• Planning a pathway to a solution
• Monitoring thinking
• Relating situations to prior knowledge
• Communicating with others using clear mathematical language and precise answers that have been calculated efficiently and accurately
• Reasoning abstractly and quantitatively (2) and constructing viable arguments and critiquing the reasoning of others (3) focus on students’ skillfulness in
• Making sense of quantities and their relationships in problems
• Creating logical representations of problems
• Justifying thinking using mathematical ideas
• Proving the validity and reasonableness of answers
• Modeling with mathematics (4) and using appropriate tools strategically (5) encourage students in
• Simplifying problems
• Applying prior knowledge
• Representing the mathematics
• Using tools to help them visualize and solve problems
• Looking for and making use of structure (7) and looking for and expressing regularity in repeated reasoning (8) expect students to engage in
• Applying mathematical rules to specific situations
• Noticing generalizations and patterns.

### Middle School Exemplar Problem and Solution from NCTM

Problem
Each tire for a 4-wheel competition monster truck can be used for exactly 14,000 miles. What is the fewest number of spare tires that will be needed for a total of 21,000 miles? Explain how you arrived at your answer.

Exemplary Student Explanation
“I know that two spare tires are needed. A total of 21,000 miles are needed for each of the 4 wheels, so 84,000 miles worth of wear are needed. Tires can be used 14,000 miles each, so 84,000 ÷ 14,000 results in 6 tires being needed. To accomplish this, drive 7,000 miles and swap the 2 front tires with the 2 spare tires. Then, drive 7,000 more miles. After that, take the 2 tires that used to be on the front and put them on the back. (The original back tires have 14,000 miles of wear on them.) Now go the remaining 7,000 miles. The truck has now traveled 21,000 miles and each of the 6 tires (4 original and 2 spares) has had 14,000 miles of wear.”

Practices Embedded in the Above Explanation

• Make sense of problems and persevere in solving them
• Construct viable arguments and critique the reasoning of others
• Model with mathematics

In summary, the Standards of Mathematical Practice demand our attention and commitment. In order for our students to learn mathematical content at the depth expected in the Common Core, students need instruction in the practice standards as well. It is through these standards that students will develop the dispositions and ways of thinking that are the foundation of math learning.

In order to do this significant work, we need experience unpacking each Standard for Mathematical Practice and opportunities to explore each of them in the context of the content we are required to teach. In subsequent issues, we will dig deeply into some of these standards and illustrate what they look like in authentic learning situations.

### Resources and References

“Common Core State Standards for Mathematical Practice.” Washington, D.C.: National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010. Accessed at: www.corestandards.org/Math/Practice.

“Common Core Standards for Mathematical Practice.” Los Altos, CA: Inside Mathematics. Accessed at: www.insidemathematics.org/index.php/commmon-core-math-intro.
These video resources illustrate the Standards for Mathematical Practice.

Hiebert, James and Douglas Grouws. “The Effects of Classroom Mathematics Teaching on Students’ Learning.” Second Handbook of Research on Mathematics Teaching and Learning. Charlotte, NC: Information Age Publishing, 2007.

“Implementing Standards for Mathematical Practices.” Ed. Melisa Hancock. Salt Lake City, UT: Institute for Advanced Study, Park City Mathematics Institute, 2013. Accessed at: www.louisianabelieves.com/docs/common-core-state-standards-resources/guide–teacher-planning-for-math-practice-implementation.pdf?sfvrsn=2.
This PDF provides a summary of each practice, relevant questions to ask students to develop their mathematical thinking, and characteristics of what each looks like when implemented.

Kilpatrick, Jeremy, Jane Swafford, Bradford Findel. Adding it Up: Helping Children Learn Mathematics. Washington, D.C.: The National Academies Press, 2001.

Problem Database. National Council of Teachers of Mathematics. Accessed at: www.nctm.org/problems.

Parker, Frieda and Jodie Novak. “Implementing the Common Core Mathematical Practices.” ASCD Express. Alexandria, VA: ASCD, December 2012. Accessed at: www.ascd.org/ascd-express/vol8/805-parker.aspx.

Van De Walle, John, et. al. Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Pearson, 2012.

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