The Process Standards demonstrate the ways in which students should develop conceptual understanding of mathematical content, and the ways in which students should synthesize and apply mathematical skills.
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. Mathematically proficient students check their answers to
problems using a different method, and they continually ask themselves, “Does this make sense?” and "Is
my answer reasonable?" They understand the approaches of others to solving complex problems and
identify correspondences between different approaches. Mathematically proficient students understand
how mathematical ideas interconnect and build on one another to produce a coherent whole.
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.
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 analyze situations by breaking them into cases
and recognize and use counterexamples. They organize their mathematical thinking, justify their
conclusions and 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. They justify whether a given statement is always true, sometimes, or never.
Mathematically proficient students participate and collaborate in a mathematics community. They listen to
or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or
improve the arguments.
Mathematically proficient students apply the mathematics they know to solve problems arising in everyday
life, society, and the workplace using a variety of appropriate strategies. They create and use a variety of
representations to solve problems and to organize and communicate mathematical ideas. Mathematically
proficient students apply what they know and 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 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.
Mathematically proficient students consider the available tools when solving a mathematical problem.
These tools might include pencil and paper, models, a ruler, a protractor, a calculator, a spreadsheet, a
computer algebra system, a statistical package, or dynamic geometry software. Mathematically 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. Mathematically proficient students identify relevant external mathematical resources, such as
digital content, and use them to pose or solve problems. They use technological tools to explore and
deepen their understanding of concepts and to support the development of learning mathematics. They
use technology to contribute to concept development, simulation, representation, reasoning,
communication, and problem solving.
Mathematically proficient students communicate precisely to others. They use clear definitions, including
correct mathematical language, 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
express solutions clearly and logically by using the appropriate mathematical terms and notation. They
specify units of measure and label axes to clarify the correspondence with quantities in a problem. They
calculate accurately and efficiently and check the validity of their results in the context of the problem. They
express numerical answers with a degree of precision appropriate for the problem context.
Mathematically proficient students look closely to discern a pattern or structure. They step back for an
overview and shift perspective. They recognize and use properties of operations and equality. They
organize and classify geometric shapes based on their attributes. They see expressions, equations, and
geometric figures as single objects or as being composed of several objects.
Mathematically proficient students notice if calculations are repeated and look for general methods and
shortcuts. They notice regularity in mathematical problems and their work to create a rule or formula.
Mathematically proficient students maintain oversight of the process, while attending to the details as they
solve a problem. They continually evaluate the reasonableness of their intermediate results.