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December 1, 2013
Vol. 71
No. 4

Mathematical Practices for Deep Understanding

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A steady dose of formative assessment is the key to getting students comfortable with the mathematical practices the Common Core standards require.

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Ask a random sample of adults (excluding any math teachers) what they dislike most about math, and you'll probably hear two words: word problems. But when we ask adults this question in the future, their responses might be radically different—if their teachers helped them get accustomed to using the Common Core Initiative's recommended standards for mathematical practice while they were in school. Maybe they'll even answer, "I loved word problems!"
These eight mathematical practices are the antidote to teaching mathematics as a series of "plug and chug" procedures ("Do this for this kind of problem"). Students who make sense of word problems don't start by asking, "What kind of problem is this?" They start by trying to figure out what the problem means. What does the problem ask? What information is given, what needs to be found, and which mathematical procedures will lead to that information? Students will approach word problems with questions like these once they master the mathematical practices. They'll not only be able to solve mathematically rich problems, but they'll also appreciate math's usefulness.
Students won't master the standards for mathematical practice overnight, even after schools begin implementing the Common Core State Standards. The practices address habits of mind, thinking processes, and dispositions that help students develop "deep, flexible, and enduring understanding of mathematics" (Briars, Mills, & Mitchell, 2011, p. 20). Teachers will need to both give students problems that require them to use the practices and create environments that support student discourse and risk taking. And students will need a steady diet of feedback on their performance.
Math educators will need to change three things to nurture these practices. Besides changing instructional strategies and materials, teachers will need to change their assessments (so items measure mathematical practices as well as computational skills), and make their feedback focus on students' mathematical reasoning, modeling, and other practices—not just on correct answers. Instruction, assessment, and feedback will all need to focus more on higher-order thinking skills, communication, and collaboration.

The Key: Formative Assessment

Formative assessment can be a powerful tool to integrate cognitively demanding tasks into students' math work. Teachers should share with students clear learning targets that include the eight practices. Because a "target" isn't a target unless students are aiming for it, the first step is finding ways to show students what this kind of mathematical thinking looks like and how they'll know whether they are engaging in it. Students must have clear "look-fors" to monitor the quality of their work and their thinking. Most important, lessons should include ungraded opportunities for students to try out strategies, construct mathematical arguments, and receive feedback on their efforts.

Helping Students Think About Their Thinking

To explore how a teacher might leverage the power of formative assessment, let's look at how 3rd grade teacher Renee Parker modified her instruction to help students develop Mathematical Practices 1 (make sense of problems) and 3 (construct viable arguments). Renee and her colleagues developed a student-friendly rubric to improve their learners' ability to communicate about their thinking while solving problems. They devised five problems, each of which required students to draw diagrams, write number sentences, and explain their reasoning and the steps they used to solve that problem. They then focused on one problem each week (Parker & Breyfogle, 2011).
The rubric provided a clear set of things students should look for in their work (for example, "Was all the important information from the problem used?"). The students discussed weekly problem work with their teacher and classmates, so they got a clearer idea of items on the rubric (for example, "I write what I did and why I did it") looked like. Working through these problems helped all Parker's students learn, but her lower-achieving students improved the most. Students especially improved in writing explanations that included mathematical vocabulary and strategies.
This project included changes to all three areas—instruction, assessment, and feedback. Instruction came to include clear targets, which were clarified by the rubric, student work examples, and discussion around both. Assessment changed: The weekly problems required students to demonstrate that they could use mathematical tools, mathematical models, and viable arguments. These assignments were formative; they allowed students to practice and improve, think intentionally about their own reasoning, and talk with others as they continued to reflect and learn. Feedback changed: It included dialogue about rubric items that helped students make sense of the problems.

Teaching with the Mathematical Practices in View

Zeroing in on one problem shows how a teacher might change these three elements of lessons to help learners simultaneously master both that problem and the mathematical practices. The sample word problem in Figure 1 is aimed at 5th graders and maps to the domain of problem solving at Depth of Knowledge Level 3 on the Common Core framework. The problem asks learners to first compute the total number of crayons in four boxes holding 64 crayons each, and then find the answer to several other questions focused on how to divide these crayons among 32 students. Math problems at this level "require reasoning, planning, or use of evidence to solve problems … citing evidence and developing logical arguments for concepts" (Webb, 2002, p. 10).

Figure 1. Sample 5th Grade Word Problem

Mrs. Brown bought 4 boxes of crayons at the store to share with her students. Each box contained a total of 64 crayons.

Part A

What is the total number of crayons Mrs. Brown bought at the store? Explain your answer using diagrams, pictures, mathematical expressions, and/or words.

Part B

Mrs. Brown wants to give each of her students an equal number of the crayons she bought. There are 32 students in Mrs. Brown's class. How many crayons should each student get?

Part C

How many more boxes of crayons does Mrs. Phelps need if she wants each of her students to get 12 crayons? Explain your answer using diagrams, pictures, mathematical expressions, and/or words.


We use this problem because it's a good example of the kind of work students need to do to become skilled with the mathematical practices. Let's look at how teachers might shift their teaching methods to reinforce the practice of constructing viable arguments as students tackle this problem.

Applying These Principles

We've zeroed in on one of the eight mathematical practices and one word problem, but the principles illustrated in this example apply to helping students develop skills in all eight practices and in a wide range of mathematical content.
Note that the changes to instruction, assessment, and feedback we've portrayed here all involve formative assessment strategies: communicating clear learning targets and criteria for success, designing performances of understanding that match the learning targets, providing feedback that feeds students forward, and asking questions that make student thinking visible. Teachers will need a thorough understanding of what each practice looks like and how students typically learn mathematics content to use these strategies effectively.
All these strategies are interdependent. Focusing instruction on the quality of students' mathematical arguments won't work unless you've communicated to students what a mathematical argument is—and made constructing one their learning target. Having a target won't help unless students have multiple opportunities to solve word problems in various ways and talk with others about that problem solving. The formative learning process is a cycle. At its best, it's an upward spiral, getting students to mastery.
References

Briars, D., Mills, V., & Mitchell, S. (2011, February 23). Deeper dive into the Common Core State Standards: Leading with the mathematical practices [Webinar]. Denver, CO: National Council of Supervisors of Mathematics. Retrieved from www.mathedleadership.org/docs/events/webinars/NCSMCCSSWebinar2011-02-23Presentation.pdf

Daro, P., Mosher, F., & Corcoran, T. (2011). Learning progressions in mathematics: A foundation for standards, curriculum, assessment, and instruction (Report #RR-68). Philadelphia, PA: Consortium for Policy Research in Education Research Report. Retrieved from www.cpre.org/learning-trajectories-mathematics-foundation-standards-curriculum-assessment-and-instruction

Parker, R., & Breyfogle, M. L. (2011). Learning to write about mathematics. Teaching Children Mathematics, 18(2), 90–99.

Webb, N. L. (2002). Alignment study in language arts, mathematics, science and social studies of state standards and assessments for four states. Washington, DC: Council of Chief State School Officers.

End Notes

1 The eight mathematical practices are make sense of problems and persevere in solving them, reason abstractly and quantitatively, construct viable arguments and critique the reasoning of others, model with mathematics, use appropriate tools strategically, attend to precision, look for and make use of structure, and look for and express regularity in repeated reasoning. For a detailed description of each practice, see www.corestandards.org/Math/Practice.

Ceri Dean is the vice president of field services at McREL. She is responsible for the development and successful implementation of field services. 

She holds a doctoral degree in curriculum and instruction from the University of Connecticut at Storrs and a master's degree in atmospheric science from Colorado State University.

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