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).
Figure 1. Common Core Standards for Mathematical Practices
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.
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 1. Common Core Standards for Mathematical Practices
- 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.
- Look for and express regularity in repeated reasoning.
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.
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.
Practice Standards by Grade- by Inside Mathematics
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