Mathematical Practices

The Standards for Mathematical Practice describe the

behaviors of a proficient mathematician.  
The "practices" describe varieties of expertise that we seek to develop in students.

1. Make sense of problems and persevere in solving them.                Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

2. Reason abstractly and quantitatively.
   Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.
   
   
3. Construct viable arguments and critique the reasoning of others.
    Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.
   
   
4. Model with mathematics.
   Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation.
   
   
5. Use appropriate tools strategically.
   Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.
   
6. Attend to precision.
   Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other.
   
   
7. Look for and make use of structure.
   Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.  
   
   
8. Look for and express regularity in repeated reasoning.
   Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Practice #1: Make Sense of Problems and Persevere In Solving Them

Students:

Analyze and explain the meaning of the problem
Actively engage in problem solving (Develop, carry out, and refine a plan)
Show patience and positive attitudes
Ask if their answers make sense
Check their answers with a different method

Because Teachers:

Pose rich problems and/or ask open ended questions
Provide wait-time for processing/finding solutions
Circulate to pose probing questions and monitor student progress
Provide opportunities and time for cooperative problem solving and reciprocal teaching

Practice #2: Reason Abstractly and Quantitatively

Students:

Represent a problem with symbols
Explain their thinking
Use numbers flexibly by applying properties of operations and place value
Examine the reasonableness of their answers/calculations

Because Teachers:

Ask students to explain their thinking regardless of accuracy
Highlight flexible use of numbers
Facilitate discussion through guided questions and representations
Accept varied solutions/representations

Practice #3: Construct Viable Arguments and Critique the Reasoning of Others

Students:

Make reasonable guesses to explore their ideas
Justify solutions and approaches
Listen to the reasoning of others, compare arguments, and decide if the arguments of others makes sense
Ask clarifying and probing questions

Because Teachers:

Provide opportunities for students to listen to or read the conclusions and arguments of others
Establish and facilitate a safe environment for discussion
Ask clarifying and probing questions
Avoid giving too much assistance (e.g., providing answers or procedures)

Practice #4: Model with Mathematics

Students:

Apply prior knowledge to new problems and reflect
Use representations to solve real life problems
Apply formulas and equations where appropriate

Because Teachers:

Pose problems connected to previous concepts
Provide a variety of real world contexts
Use intentional representations

Practice #5: Use Appropriate Tools Strategically

Students:

Select and use tools strategically (and flexibly) to visualize, explore, and compare information
Use technological tools and resources to solve problems and deepen understanding

Because Teachers:

Make appropriate tools available for learning (calculators, concrete models, digital resources, pencil/paper, compass, protractor, etc.)
Use tools with their instruction

Practice #6: Attend to Precision

Students:

Calculate accurately and efficiently
Explain their thinking using mathematics vocabulary
Use appropriate symbols and specify units of measure

Because Teachers:

Recognize and model efficient strategies for computation
Use (and challenge students to use) mathematics vocabulary precisely and consistently

Practice #7: Look For and Make Use of Structure

Students:

Look for, develop, and generalize relationships and patterns
Apply reasonable thoughts about patterns and properties to new situations

Because Teachers:

Provide time for applying and discussing properties
Ask questions about the application of patterns
Highlight different approaches for solving problems

Practice #8: Look For and Express Regularity in Repeated Reasoning

Students:

Look for methods and shortcuts in patterns and repeated calculations
Evaluate the reasonableness of results and solutions

Because Teachers:

Provide tasks and problems with patterns
Ask about answers before and reasonableness after computations

                                                                    (Lightbulbs and Laughter, 2014)