The Common Core State Standards feature eight standards for mathematical practice that are intended to be integrated into every math lesson and help students think mathematically.
These are intentionally called “standards for mathematical practice” because they are ways of thinking about and interacting with math that must be taught and practiced in a strategic and deliberate way. Many educators are intimidated by these practices because some of them sound like they overlap and teachers are unsure of what, exactly, is meant by each one. There are lots of resources out there that provide grade- or subject-specific examples of each practice, which can be extremely helpful. This article is meant to provide you with an overview of each of the practices so that you can begin integrating them into your classroom in a way that makes sense to you.
1. Make sense of problems and persevere in solving them.
This first standard focuses on adaptability. Students are encouraged not to just memorize one way of solving a problem, but having a persevering mindset and attitude that leads to developing new ways to solve the problem. The Common Core website says students should constantly be asking, “Does this make sense?” If it doesn’t, they should have the tools to figure out why and make changes. This mindset can’t be taught in one day and shouldn’t be limited to math. Over time, students will gain confidence as they practice persevering through difficult problems. Using more authentic problems in your class that have real-world implications and connections can help students feel like they are working on problems worth solving.
2. Reason abstractly and quantitatively.
Students should have a strong grasp on abstract and quantitative reasoning. When solving a problem, students can understand the role of symbols and how word problems relate to the actual mathematics involved. For example, Inside Mathematics suggests that students reflect on fractions, communicating what each part actually represents. Giving students opportunities to explain their reasoning will reinforce the importance of number sense and mathematical thinking over rote memorization while preparing them for the following standards for mathematical practice.
3. Construct viable arguments and critique the reasoning of others.
Logic should not be isolated to English papers. Encourage your students to reason with others about mathematical principles. This once again emphasizes the importance of independent thinking and reasoning rather than blind memorization. Deconstruct a theory with your class and have students explain why it is correct, or why it always leads to the same outcome. Have students debate a theory or problem, offering them ownership of their own learning. Challenge students to solve the same problem in different ways, then have a class discussion about which method was the most efficient or effective and why.
4. Model with mathematics.
Students who are truly proficient in math understand that mathematics is a beautiful language that helps us see relationships and understand how things relate to each other in the world around us. They are able to take a complex problem in the real world, model it with graphs, functions, shapes, or equations, and draw meaning from those models. They also know that a solution out of context is useless, so they put their answer back into the context of the problem to ensure it makes sense and make adjustments to their process if it doesn’t.
5. Use appropriate tools strategically.
Students should be able to analyze the tools available to them and then use that knowledge to solve a problem more efficiently. The implications of this and other standards of mathematical practice vary greatly across the grade levels. Obviously the tools that are available to high school students are not always appropriate to be used by elementary students. The goal of this practice is for students to be exposed to as many appropriate mathematical tools, such as manipulatives, protractors, graphs, rulers, compasses, calculators, and computer software, as possible within the classroom, and be given guidance on when and how to use them effectively.
6. Attend to precision.
Accuracy and precision are often confused, and while both are important, precision matters immensely in math. Have your students practice being clear, even when it seems unnecessary. Make sure your students practice legibly writing each step as they solve a problem rather than just scribbling the final result. Often the process of reasoning and communicating reveal more about what a student understands than simply knowing the right answer. Have students use units in conjunction with their numeric answers to aid clarity and ensure the answer makes sense in the context of the problem. Also discuss with students the level of precision that is appropriate in different situations (such as money, measurement, and statistical analysis).
7. Look for and make use of structure.
Patterns and structure are crucial to having a working understanding of math. Once again, a student who can see patterns will do better than one who can memorize one way to solve a problem. Students should be able to apply concepts to a variety of problems rather than only know how to solve one problem. While identifying and extending visual patterns is considered an elementary task, those are not the only types of patterns that exist within mathematics. Students also need to be able to see patterns in formulas, equations, graphs, and more to simplify steps in future problems.
8. Look for and express regularity in repeated reasoning.
While this sounds a lot like one of the other standards for mathematical practice, mathematically proficient students should be able to move beyond recognizing patterns and begin to generalize those patterns, using their observations to check and guide their work. These students will notice that multiply by two always results in an even number, for example, and will realize that they have done something wrong when they multiply by two and get a product that is an odd number. This ability to recognize and apply repeated reasoning can provide shortcuts for students when tackling problems that are familiar, and help them maintain accuracy when encountering new problems.
The standards for mathematical practice aren’t meant to be intimidating. They are meant to articulate what we already know as educators—conceptual understanding is more important than fluency, mathematical thinking is something that needs to be nurtured, and that all students are capable of success in mathematics when given the right tools, support, and practice.
Want more standards help? Discover ways to integrate the engineering practices from the Next Generation Science Standards.
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