The Standards for Mathematical Practice
The 8 Standards for Mathematical Practice describe the behaviors of mathematically proficient students. Mathematics teachers at
all levels should seek to develop these behaviors. The practices rest on “processes and proficiencies” of longstanding importance in
mathematics education. These process standards and mathematical proficiencies represent the recommendations of the National
Council of Teachers of Mathematics and the National Research Council respectively.
Every lesson will include some of these practices although every practice may not be represented in every lesson. Student
behaviors are connected to teacher behaviors.
Practice #1 Make sense of problems and persevere in solving them
Students:
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Analyze and explain the meaning of
the problem

Analyzing relationships between
various representations

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

Discussing solutions with one
another and explaining or reflecting
on thinking
Teachers:

Provides time for students to
discuss problem situations

Pose rich problems and/or ask
open ended questions
(generalizable tasks)

Provide wait-time for
processing/finding solutions

Circulate to pose probing questions
and monitor student progress

Provide opportunities and time for
discussing problems, cooperative
problem solving and reciprocal
teaching

Allow time for explaining
understanding and reasoning
Question to Develop Mathematical Thinking
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Implementing Standards for Mathematics (SMP) “Look/Listen For” by Melisa Hancock – 2012 (Questions by Learning Services USD 259)
How would you describe the problem in your
own words?
How would you describe what you are trying to
find?
What do you notice about...?
What information is given in the problem?
Describe the relationship between the
quantities.
Describe what you have already tried. What
might you change?
Talk me through the steps you’ve used to this
point.
What steps in the process are you most
confident about?
What are some other strategies you might try?
What are some other problems that are similar
to this one?
How might you use one of your previous
problems to help you begin?
How else might you organize...represent...
show...?
Practice #2 Reason abstractly and quantitatively.
Students:

Represent a problem with
symbols
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Explain their thinking

Use numbers flexibly by
applying properties of
operations and place value

Understand the meaning of
quantities, not just how to
compute

Examine the reasonableness
of their answers/calculations
Teachers:

Ask students to explain their
thinking regardless of accuracy

Highlight flexible use of numbers
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Facilitate discussion through
guided questions and
representations

Accept varied solutions and
representations (manipulative
materials, drawings, online
renderings) of problems
Question to Develop Mathematical Thinking
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What do the numbers used in the problem
represent?
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What is the relationship of the quantities?
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How is _______ related to ________?
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What is the relationship between
______and ______?

What does_______mean to you? (e.g.
symbol, quantity, diagram)
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What properties might we use to find a
solution?
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How did you decide in this task that you
needed to use...?

Could we have used another operation or
property to solve this task? Why or why
not?
Practice #3 Construct viable arguments and critique the reasoning of others
Students:
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Understand and use prior
learning in constructing
arguments

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
Teachers:
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Provide opportunities for
students to listen to or read the
conclusions and arguments of
others

Establish and facilitate a safe
environment for discussion and
risk taking is valued

Ask clarifying and probing
questions

Avoid giving too much
assistance (e.g., providing
answers or procedures)
Question to Develop Mathematical Thinking
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What mathematical evidence would support your
solution?
How can we be sure that...? / How could you
prove that...? Will it still work if...?
What were you considering when...?
How did you decide to try that strategy?
How did you test whether your approach worked?
How did you decide what the problem was asking
you to find? (What was unknown?)
Did you try a method that did not work? Why
didn’t it work? Would it ever work? Why or why
not?
What is the same and what is different about...?
How could you demonstrate a counter-example?
Implementing Standards for Mathematics (SMP) “Look/Listen For” by Melisa Hancock – 2012 (Questions by Learning Services USD 259)
Practice #4 Model with mathematics
Students:

Analyze and model relationships
mathematically (expression or
equation)

Represent mathematics to
describe a situation either with
an equation, diagram and
interpret the results to draw
conclusion

Apply the mathematics they
know to solve everyday
problems

Can determine if a model makes
sense and/or needs modified
Teachers:
Question to Develop Mathematical Thinking
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Allow time for the process to
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What number model could you construct to
take place (model, make
represent the problem?
graphs, write equations to
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What are some ways to represent the
describe a situation, etc.)
quantities?
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Provide meaningful, real
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What’s an equation or expression that
world, authentic, performancematches the diagram..., number line..,
based tasks (non traditional
chart..., table..?
problems)
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Where did you see one of the quantities in
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Provides contexts for students
the task in your equation or expression?
to apply mathematics learned
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Would it help to create a diagram, graph,
table...?
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What are some ways to visually
represent...?
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What formula might apply in this situation?
Practice #5 Use appropriate tools strategically
Students:
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Select and use tools strategically
(and flexibly) to visualize, explore,
and compare information
Select and use instructional tools
and resources to solve problems and
deepen understanding
(manipulatives, drawings,
technological tools, graphs, mental
estimation)
Use external resources to solve
problems
Teachers:
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Make appropriate tools
available for learning
(calculators, concrete models,
digital resources, pencil/paper,
compass, protractor, etc.)
Use appropriate tools
instructionally
Provide opportunities for
students to use estimation and
other mathematical knowledge
to detect possible errors
Make available external
mathematics resources for
students to use in solving
problems
Question to Develop Mathematical Thinking
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What mathematical tools could we use to
visualize and represent the situation?
What information do you have?
What do you know that is not stated in the
problem?
What approach are you considering trying first?
What estimate did you make for the solution?
In this situation would it be helpful to use...a
graph...,number line..., ruler..., diagram...,
calculator..., manipulative?
Why was it helpful to use...?
What can using a ______ show us that
_____may not?
In what situations might it be more informative or
helpful to use...?
Implementing Standards for Mathematics (SMP) “Look/Listen For” by Melisa Hancock – 2012 (Questions by Learning Services USD 259)
Practice #6 Attend to precision
Students:

Recognize the need for precision
in response to a problem; clear
and concise and uses appropriate
mathematics vocabulary

Calculate accurately and
efficiently

Use and explain meaning of
symbols and specify units of
measure
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Specify units, label answers to
clarify correspondence with
quantities in the problem
Teachers:
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Use (and challenging
students to use) mathematics
vocabulary precisely and
consistently
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Emphasizes the importance
of precision in
communication, uses clear
definitions in discussions
Question to Develop Mathematical Thinking
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What mathematical terms apply in this
situation?
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How did you know your solution was
reasonable?
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Explain how you might show that your
solution answers the problem.
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Is there a more efficient strategy?
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How are you showing the meaning of the
quantities?
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What symbols or mathematical notations are
important in this problem?
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What mathematical language...,definitions...,
properties can you use to explain...?
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How could you test your solution to see if it
answers the problem?
Practice #7 Look for and make use of structure
Students:
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Look for, develop, and generalize
relationships patterns and
structure (properties, compose
and decompose numbers)
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Understands WHY rules or
properties “work”

Apply reasonable thoughts about
patterns and properties to new
situations
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See complicated things as single
objects or as being composed of
several objects
Teachers:
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Provide time for applying and
discussing patterns, and
structures that emerge in a
problem solution

Ask questions about the
application of patterns and
relationships

Provide rich tasks that
associate patterns with
properties of operations and
their relationships
Question to Develop Mathematical Thinking
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What observations do you make about...?
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What do you notice when...?
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What parts of the problem might you
eliminate..., simplify...?
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What patterns do you find in...?
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How do you know if something is a pattern?
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What ideas that we have learned before were
useful in solving this problem?
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What are some other problems that are similar
to this one?
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How does this relate to...?
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In what ways does this problem connect to
other mathematical concepts?
Implementing Standards for Mathematics (SMP) “Look/Listen For” by Melisa Hancock – 2012 (Questions by Learning Services USD 259)
Practice #8 Look for and express regularity in repeated reasoning
Students:

Reason about varied strategies
and methods for solving
problems

Look for methods and
shortcuts in patterns and
repeated calculations

Understand the broader
application of patterns and
structure in similar situations

Evaluate the reasonableness
of results & solutions
Teachers:

Provide rich tasks and
problems with patterns that
require reasoning and sense
making

Encourage students to look
for and discuss regularity in
their reasoning

Guide students in evaluating
regularity and use their
thinking to lead to general
formulas
Question to Develop Mathematical Thinking
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Will the same strategy work in other situations?
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Is this always true, sometimes true or never
true? Use evidence to explain.
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How would we prove that...?
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What do you notice about...?
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What is the relationship between . . . . . and
……..? What does this tell us?
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What is happening in this situation?
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What would happen if...?
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Is there a mathematical rule for...? Explain the
rule and where it came from
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What predictions or generalizations can this
pattern support?
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What mathematical consistencies do you
notice ?
Implementing Standards for Mathematics (SMP) “Look/Listen For” by Melisa Hancock – 2012 (Questions by Learning Services USD 259)
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Look/Listen For - Mathematical Practices