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12/28/17, 8(39 AMStandards for Mathematical Practice | Common Core State Standards Initiative

Page 1 of 6http://www.corestandards.org/Math/Practice/

Standards for Mathematical Practice

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The Standards for Mathematical Practice describe varieties of expertise that mathematics

educators at all levels should seek to develop in their students. These practices rest on important

“processes and proficiencies” with longstanding importance in mathematics education. The first

of these are the NCTM process standards of problem solving, reasoning and proof,

communication, representation, and connections. The second are the strands of mathematical

proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning,

strategic competence, conceptual understanding (comprehension of mathematical concepts,

operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately,

efficiently and appropriately), and productive disposition (habitual inclination to see

mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own

efficacy).

Standards in this domain:

CCSS.MATH.PRACTICE.MP1 CCSS.MATH.PRACTICE.MP2 CCSS.MATH.PRACTICE.MP3

CCSS.MATH.PRACTICE.MP4 CCSS.MATH.PRACTICE.MP5 CCSS.MATH.PRACTICE.MP6

CCSS.MATH.PRACTICE.MP7 CCSS.MATH.PRACTICE.MP8

CCSS.MATH.PRACTICE.MP1 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP1/) Make sense of

problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem

and looking for entry points to its solution. They analyze givens, constraints, relationships, and

goals. They make conjectures about the form and meaning of the solution and plan a solution

pathway rather than simply jumping into a solution attempt. They consider analogous problems,

and try special cases and simpler forms of the original problem in order to gain insight into its

solution. They monitor and evaluate their progress and change course if necessary. Older

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students might, depending on the context of the problem, transform algebraic expressions or

change the viewing window on their graphing calculator to get the information they need.

Mathematically proficient students can explain correspondences between equations, verbal

descriptions, tables, and graphs or draw diagrams of important features and relationships, graph

data, and search for regularity or trends. Younger students might rely on using concrete objects

or pictures to help conceptualize and solve a problem. 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.

CCSS.MATH.PRACTICE.MP2 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP2/) Reason abstractly and

quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem

situations. They bring two complementary abilities to bear on problems involving quantitative

relationships: the ability to decontextualize—to abstract a given situation and represent it

symbolically and manipulate the representing symbols as if they have a life of their own, without

necessarily attending to their referents—and the ability to contextualize, to pause as needed

during the manipulation process in order to probe into the referents for the symbols involved.

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.

CCSS.MATH.PRACTICE.MP3 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP3/) Construct viable

arguments and critique the reasoning of others.

Mathematically proficient students understand and use stated assumptions, definitions, and

previously established results in constructing arguments. They make conjectures and build a

logical progression of statements to explore the truth of their conjectures. They are able to

analyze situations by breaking them into cases, and can recognize and use counterexamples.

They justify their conclusions, communicate them to others, and respond to the arguments of

others. They reason inductively about data, making plausible arguments that take into account

the context from which the data arose. Mathematically proficient students are also able to

compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning

from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary

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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. Later, students learn to determine domains to

which an argument applies. Students at all grades can listen or read the arguments of others,

decide whether they make sense, and ask useful questions to clarify or improve the arguments.

CCSS.MATH.PRACTICE.MP4 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP4/) 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. In middle grades, a student might apply

proportional reasoning to plan a school event or analyze a problem in the community. By high

school, a student might use geometry to solve a design problem or use a function to describe

how one quantity of interest depends on another. Mathematically proficient students who can

apply what they know are comfortable making assumptions and approximations to simplify a

complicated situation, realizing that these may need revision later. They are able to identify

important quantities in a practical situation and map their relationships using such tools as

diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships

mathematically to draw conclusions. They routinely interpret their mathematical results in the

context of the situation and reflect on whether the results make sense, possibly improving the

model if it has not served its purpose.

CCSS.MATH.PRACTICE.MP5 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP5/) 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. For example, mathematically proficient high

school students analyze graphs of functions and solutions generated using a graphing calculator.

They detect possible errors by strategically using estimation and other mathematical knowledge.

When making mathematical models, they know that technology can enable them to visualize the

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results of varying assumptions, explore consequences, and compare predictions with data.

Mathematically proficient students at various grade levels are able to identify relevant external

mathematical resources, such as digital content located on a website, and use them to pose or

solve problems. They are able to use technological tools to explore and deepen their

understanding of concepts.

CCSS.MATH.PRACTICE.MP6 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP6/) 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, and labeling axes to clarify the correspondence with

quantities in a problem. 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. By the time they reach high school they

have learned to examine claims and make explicit use of definitions.

CCSS.MATH.PRACTICE.MP7 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP7/) 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.

Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for

learning about the distributive property. In the expression x + 9x + 14, older students can see

the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a

geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They

also can step back for an overview and shift perspective. They can see complicated things, such

as some algebraic expressions, as single objects or as being composed of several objects. For

example, they can see 5 - 3(x - y) as 5 minus a positive number times a square and use that to

realize that its value cannot be more than 5 for any real numbers x and y.

CCSS.MATH.PRACTICE.MP8 (HTTP://WWW.CORESTANDARDS.ORG/MATH/PRACTICE/MP8/) Look for and express

regularity in repeated reasoning.

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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. By paying attention to the calculation of slope as they repeatedly check whether points

are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y

- 2)/(x - 1) = 3. Noticing the regularity in the way terms cancel when expanding (x - 1)(x + 1), (x - 1)

(x + x + 1), and (x - 1)(x + x2 + x + 1) might lead them to the general formula for the sum of a

geometric series. 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.

Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content

The Standards for Mathematical Practice describe ways in which developing student

practitioners of the discipline of mathematics increasingly ought to engage with the subject

matter as they grow in mathematical maturity and expertise throughout the elementary, middle

and high school years. Designers of curricula, assessments, and professional development should

all attend to the need to connect the mathematical practices to mathematical content in

mathematics instruction.

The Standards for Mathematical Content are a balanced combination of procedure and

understanding. Expectations that begin with the word "understand" are often especially good

opportunities to connect the practices to the content. Students who lack understanding of a

topic may rely on procedures too heavily. Without a flexible base from which to work, they may

be less likely to consider analogous problems, represent problems coherently, justify conclusions,

apply the mathematics to practical situations, use technology mindfully to work with the

mathematics, explain the mathematics accurately to other students, step back for an overview, or

deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively

prevents a student from engaging in the mathematical practices.

In this respect, those content standards which set an expectation of understanding are potential

"points of intersection" between the Standards for Mathematical Content and the Standards for

Mathematical Practice. These points of intersection are intended to be weighted toward central

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and generative concepts in the school mathematics curriculum that most merit the time,

resources, innovative energies, and focus necessary to qualitatively improve the curriculum,

instruction, assessment, professional development, and student achievement in mathematics.