Implementing the Common Core Mathematical Practices
The Common Core mathematics standards, like the Principles and Standards for School Mathematics (2000) from the National Council of Teachers of Mathematics (NCTM), include both content and process standards. Content standards include the mathematical knowledge and skills students should learn; process standards specify the mathematical ways of thinking students should develop while learning mathematics content. In the Common Core standards, the process standards are described as eight Common Core Mathematical Practices (see Figure 1) that build on the NCTM process standards (communication, representation, reasoning and proof, connections, and problem solving) and the National Research Council's five strands of mathematical proficiency (procedural fluency, conceptual understanding, strategic competence, adaptive reasoning, and productive disposition).
To emphasize the link between the content standards and the Mathematical Practices, the Common Core standards document indicates that content standards—beginning with the word "understand"—are especially good places for making connections between the content standards and Mathematical Practices.
Given the importance of the Mathematical Practices in understanding and implementing the Common Core mathematics standards, it is essential that teachers, teacher leaders, and administrators have a firm knowledge of these practices. Here are a variety of ways to manage the Mathematical Practices, resources and ideas for designing professional development around the Mathematical Practices, and suggestions for finding and creating math tasks that elicit the Mathematical Practices.
Managing the Mathematical Practices
The Mathematical Practices can seem overwhelming to weave into the curriculum, but once you understand the relationships among them and their potential use in mathematical tasks, the task becomes more manageable.
The Mathematical Practices are articulated as eight separate items, but in theory and practice they are interconnected. The Common Core authors have published a graphic depicting the higher-order relationships among the practices (see Figure 2). Practices 1 and 6 serve as overarching habits of mind in mathematical thinking and are pertinent to all mathematical problem solving. Practices 2 and 3 focus on reasoning and justifying for oneself as well as for others and are essential for establishing the validity of mathematical work. Practices 4 and 5 are particularly relevant for preparing students to use mathematics in their work. Practices 7 and 8 involve identifying and generalizing patterns and structure in calculations and mathematical objects. These practices are the primary means by which we separate abstract, big mathematical ideas from specific examples.
Figure 2. Higher-Order Structure of the Mathematical Practices
Because of their interrelated nature, the Mathematical Practices are rarely used in isolation from one another. Consequently, we can expect students to learn the practices concurrently when they are engaged in mathematical problem solving. However, teachers can highlight specific practices during a lesson to provide students with explicit knowledge of individual practices. In addition to analyzing tasks for their relationship to the content standards, we can analyze them for their relationship to the Mathematical Practices. One way to do this is to rate a math task as having high, medium, or low potential for students to engage in each mathematical practice. As with all higher-order learning, students will need repeated engagement with the practices and feedback on their use in order to develop deep understanding of when and how to use the practices. Over time, teachers can track how often each practice or practice-pair is emphasized during instruction.
Professional Development on the Mathematical Practices
We cannot expect math teachers to automatically begin incorporating the Mathematical Practices in their instruction, but professional development can help. In her MSPnet Academy webinar "Learning to Teach the Common Core," Deborah Loewenberg Ball suggests that teachers need to understand three essential things to teach the Mathematical Practices. First, teachers need an appreciation of how fundamental the practices are to learning the content. Second, teachers need to develop a conviction that all students can—and must—develop proficiency with the Mathematical Practices. Third, teachers need mathematical knowledge for teaching (MKT) with respect to the Mathematical Practices. (For more information on MKT, see Hill, Ball, & Schilling, 2008.)
Ball recommends several activities for each of these steps. One way to help teachers understand the essential nature of the Mathematical Practices to doing mathematics is to explore rich mathematical tasks and analyze the practices involved. A strategy we've used is to give each teacher a set of eight different colored index cards with one Mathematical Practice on each card. As teachers worked on a task, we would stop and ask them what practices they had been engaged with in the last 10–15 minutes. We then asked them to write on their practice cards what it looked like when they were engaged in that practice. In this reflective practice, the teachers identified the Mathematical Practices they were using and documented an example of what they looked like in action. This process can help teachers develop student-friendly language for describing the Mathematical Practices.
The Inside Mathematics website has videos of elementary, middle, and high school students that are appropriate for analyzing students' use of the Mathematical Practices. Many of the lessons in these videos draw on tasks developed by the Mathematics Assessment Resource Service (MARS), which is developing well-engineered assessment tools to support U.S. schools in implementing the Common Core State Standards for mathematics.
Some other activities that Ball recommends to help teachers develop MKT for teaching the Mathematical Practices include the following:
- Engage in solving problems and discussing the practices used in solving them.
- Unpack practices in detail: explain, represent, and consider what might be involved in making them learnable.
- Study cases of students and develop explicit knowledge of how they approach mathematical practices.
- Develop and compare ways to teach specific practices.
To study students' development, use, and learning of the Mathematical Practices, teachers can reflect on the following questions:
- What do they find difficult?
- What is typical at different ages?
- What do they do naturally?
- How do they use language or represent mathematically?
The book The Common Core Mathematics Standards: Transforming Practice through Team Leadership (Hull, Miles, & Balka, 2012) describes strategies for leadership teams to help teachers implement the Common Core standards in mathematics. One tool they describe is the Standards of Student Practice in Mathematics Proficiency Matrix (PDF), which teachers can use to assess students' proficiency with each practice. In addition, several state departments of education are developing tools and resources to help teachers implement the Mathematical Practices. Visit the Core Challenge website for links to these resources.
Tasks and the Mathematical Practices
Because the Mathematical Practices involve higher-order thinking, teachers need to provide students with math tasks that allow for this. The Illustrative Mathematics project is collecting math tasks that are appropriate for the Common Core standards. The site links each task to a standard and includes solutions and a place for comments. The MARS website includes lessons for formative assessment and summative assessment tasks. Two other websites that contain a variety of rich tasks are dy/dan (teacher Dan Meyer's blog) and NRICH.
Although traditional textbooks sometimes lack higher-cognitive-demand tasks that elicit the Mathematical Practices, teachers can modify lower-cognitive-demand tasks to increase their Mathematical Practices potential. In his TED talk, Meyer shows how removing the substeps from a textbook task can transform it from a relatively straightforward procedure to a genuine problem. The example he shows is a problem in which a graphic of a ski lift with four sections is overlaid on a coordinate system. Students use a series of questions to walk through the process of finding out the slope (or steepness) of each section. The final question asks, "Which section is the steepest?" Meyer suggests that the problem is more mathematically engaging if the coordinate system is removed from the ski lift graphic and there is only the final question: "Which section is the steepest?" This revised problem gives students a greater opportunity to think mathematically. In particular, it lets students experience Mathematical Practices 4 and 5.
Several other modification strategies exist. Jennifer Piggott provides four strategies on NRICH. One is to give students the answer to a question and ask them what the question is. For example, rather than ask students to calculate the area of different triangles, ask them to find out which triangles can have an area of six square units. This problem has the potential for students to engage in Mathematical Practices 7 and 8.
Another strategy Piggott suggests is for students to make up problems that meet some criteria. For instance, a set of textbook problems might have students evaluate expressions that involve parentheses (e.g., 2(5+3) ). We could ask students to generate expressions with parentheses and determine the expressions for which the parentheses can and cannot be removed without changing the value of the expressions. This problem lets students engage in Mathematical Practices 2 and 3.
Piggott's third and fourth modification strategies can serve as nice problem extensions. With the "What if?" strategy, a teacher poses a question about what might happen if we change a quantity or other aspect of a problem. For example, we might ask students to find all possible combinations of ice cream sundaes given three types of ice cream, two kinds of fruit, and two types of nuts. Then we could ask, "What if the number of types of ice cream is changed? What if we add a new category, such as whipped cream?"
With the "All answers" strategy, teachers ask a question to determine if all possible answers have been found. For example, in the previous sundae-making problem, we might ask students to justify how they know they have found all possible combinations. Asking "What if?" and "All answers" questions requires students to find patterns; make generalizations; and justify assertions, which are present in several Mathematical Practices.
Intertwining mathematics content and the Mathematical Practices will help turn mathematical learning from what many perceive as memorizing a set of disconnected facts and procedures into an engaging and meaningful way of seeing and exploring the world around us. The Mathematical Practices represent the thinking skills necessary for developing and applying mathematics, and they align well with 21st century skills and other work-readiness skills. As we prepare teachers to engage students in the Mathematical Practices in authentic ways, so do we prepare students for our ever-changing world.
Common Core State Standards Initiative (2012). "Mathematics: Standards for Mathematical Practice." Retrieved on 11/16/2012 from http://www.corestandards.org/Math/Practice.
Hill, H. C., Ball, D. L., & Schilling, S. G. (2008). Unpacking pedagogical content knowledge: Conceptualizing and measuring teachers' topic-specific knowledge of students. Journal for Research in Mathematics Education, 39(4), 372–400.
Hull, T. H., Miles, R. H. & Balka, D. S. (2012). The Common Core Mathematics Standards: Transforming Practice Through Team Leadership. Thousand Oaks, California: Corwin Press.
National Council of Teachers of Mathematics (2000). Principles and standards for school mathematics. Reston, VA: Author.
National Research Council (2001). Adding it up: Helping children learn mathematics. Washington, DC: National Academy Press.
Frieda Parker is doing postdoctoral work at the University of Northern Colorado. Jodie Novak is a professor in the School of Mathematical Sciences at the University of Northern Colorado.
ASCD Express, Vol. 8, No. 5. Copyright 2012 by ASCD. All rights reserved. Visit www.ascd.org/ascdexpress.