The Math Practices: The How Behind the What
You are all familiar with the Standards for Mathematical Practice for grades K–12. They essentially describe the thinking processes and habits of mind students need to develop a deep understanding of math. But I think it’s important to periodically review the math practices because as teachers we need to be intentional about which math practices we are emphasizing with any given problem or lesson. This intentionality allows us to model throughout the year how we want students to engage with mathematics, because the way students approach math is just as important as the mathematics concepts they learn.
I get asked frequently how to incorporate all eight practices into a single problem or lesson. It’s probably not possible to teach all the practices at once and it’s certainly not advisable. The practices are complex and multi-faceted, and it takes time and thoughtful introduction and practice for students to internalize them. Students are more likely to truly learn the practices if we focus only on a few in any given lesson, although students will most likely use all eight practices in a week.
So let’s look at each practice, how you can help students delve deeper into the practices, and how you know when they are successfully using each practice.
Math Practice 1: Make sense of problems and persevere in solving them. Basically, this means that students need to understand the math concepts behind the problem, explore different entry points, consider various strategies, and struggle productively until they find a viable solution. Furthermore, students should be able to apply their current knowledge to new mathematical situations and check their answers to problems using a different method.
How can you support problem-solving and perseverance?
- Provide students with rich math tasks
- Help students develop strategies to solve problems
- Give students adequate time to process and engage with problem
- Prioritize the process rather than the solution
- Encourage rich student discourse about possible strategies and solution paths
- Ask questions to help students struggle productively
How do students demonstrate problem-solving and perseverance?
- See challenges as an opportunity to learn
- Restate a problem in their own words
- Find a valid entry point
- Develop a solution path(s)
- Consider and apply strategies
- Monitor their progress and make course adjustments as needed
- Communicate their ideas with classmates
Math Practice 2: Reason abstractly and quantitatively. Students use multiple representations, make sense of the quantities and their relationships, determine whether their solutions make sense, and decide if further exploration is needed. Students think flexibly about numbers and quantities.
How can you support mathematical reasoning?
- Help students contextualize problems
- Connect symbols and equations to problems
- Provide students with appropriate tools and manipulatives
- Model strategic thinking
- Encourage students to discuss strategies
How do students demonstrate mathematical reasoning?
- Make sense of quantities and their relationships in problems
- Develop coherent representations of problems
- Make conjectures and present supporting evidence
- Think flexibly and efficiently
- Justify and defend their thinking and ideas
- Demonstrate the validity of their answers
Math Practice 3: Construct viable arguments and critique the reasoning of others. Students make conjectures, build a logical presentation—backed by representations and computations—to support their conclusion, and justify their thinking to their classmates. Equally important, students listen to other ideas and can effectively question and, if needed, respectfully counter classmates’ positions.
How can you support mathematical argument and critique?
- Provide a safe environment that supports student discussion
- Model respectful argument and critique
- Facilitate discussion and avoid “rescuing” students
- Pose questions that elicit student clarity
- Provide work samples for students to analyze
How do students demonstrate mathematical argument and critique?
- Make conjectures and develop supporting evidence—numbers, words, visuals, manipulatives
- Defend their solution path and process
- Compare the efficacy of various arguments
- Distinguish between correct and faulty reasoning
- Listen to and ask classmates about their ideas and strategies
Math Practice 4: Model with mathematics. Students use mathematics to model situations, make predictions, and draw conclusions. They reflect on whether their solution makes sense and improve upon their model if it has not performed as expected. Additionally, students can apply mathematics to real-life scenarios.
How can you support mathematical modeling?
- Provide students with a variety of appropriate tools
- Model how particular tools and visuals support an efficient solution path
- Provide students with adequate time to explore a variety of possible tools and manipulatives
How do students demonstrate mathematical modeling?
- Write an equation to describe a situation
- Make predictions using tools and/or visuals to solve problems
- Make connections between representations
- Reflect upon model and make refinements as needed
- Apply prior knowledge to new problems and contexts
- Simplify problem through effective modeling
Math Practice 5: Use appropriate tools strategically. Students consider available tools to help them make sense of and solve a given problem. They understand how the tools work and can select the tool(s) that will produce the most efficient result.
How can you support the strategic use of tools?
- Understand how to use tools and manipulatives yourself
- Provides students with ready access to a variety of appropriate tools
- Discuss with students when they need tools (and when they don’t) to clarify their thinking
- Encourage robust discussion about which tools produced efficient results
How do students demonstrate strategic use of tools?
- Understand how various tools works and when each might be most helpful
- Explain why they chose a particular tool
- Determine if tool(s) produced a reasonable result
Math Practice 6: Attend to precision. Students communicate precisely and use mathematics terms correctly. They calculate accurately and efficiently. Student precision improves with daily opportunities to practice.
How can you support precision?
- Model use of mathematics vocabulary
- Emphasize efficient strategies for accurate computation
- Correct misconceptions
- Spotlight units of measure and symbols
How do students demonstrate precision?
- Communicate with clear mathematical language and precise answers
- Calculate efficiently and accurately
- Use units of measures and symbols correctly
Math Practice 7: Look for and make sense of structure. Students can discern patterns and structure in mathematical concepts. They are able to break apart and explain the structure or pattern.
How can you support making sense of structure?
- Call attention to and discuss patterns and structure
- Provide opportunities and time for students to discover patterns
- Ask questions about underlying patterns and structure
How do students demonstrate that they are making sense of structure?
- Notice generalizations and patterns
- Use generalizations and patterns to help solve problems
- Use patterns to predict results and to determine if answers are reasonable
Math Practice 8: Look for and express regularity in repeated reasoning. Students see and explore relationships between problems and understand how the reasoning behind one problem can be applied to different situations.
How can you support repeated reasoning?
- Present problems that are related
- Point out relationship between problems
- Encourage students to discuss how they might use reasoning on other problems
How do students demonstrate repeated reasoning?
- Apply mathematical rules to specific situations
- Use generalizations and patterns to help solve problems more efficiently
- Develop shortcuts using generalizations