• Mathematics | Standards

    for Mathematical Practice

    The Standards for Mathematical Practice describe varieties of expertise that

    mathematics educators at all levels should seek to develop in their students.

    1 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 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.

    2 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.

    3 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,

    Common Core State Standards for MAT HEMAT ICS

    standards for mathematical practice,

    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 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.

    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. 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.

    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. 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 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.

    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,

    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.

    Common Core State Standards for MAT HEMAT ICS

    standards for mathematical practice |

    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. Later, students will see 7 x 8equals the

    well remembered  7 × 5 + 7 × 3, in preparation for learning about the distributive

    property. In the expression x2 + 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)2 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.

    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. 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)(x2 + x + 1), and (x – 1)(x3 + 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.