
CCSS.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 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 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 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 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 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, twoway 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 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.
 CCSS.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 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^{2} + 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.
 CCSS.Math.Practice.MP8 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)(x^{2} + x + 1), and (x – 1)(x^{3} + 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 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.