# Mathematics

Common Core State StandardS for

mathematics

Common Core State StandardS for matHematICS

Standards for mathematical Practice 6

Standards for mathematical Content

High School — Number and Quantity 58 High School — Algebra 62 High School — Functions 67 High School — Modeling 72 High School — Geometry 74 High School — Statistics and Probability 79

Glossary 85 Sample of Works Consulted 91

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Introduction Toward greater focus and coherence

Mathematics experiences in early childhood settings should concentrate on (1) number (which includes whole number, operations, and relations) and (2) geometry, spatial relations, and measurement, with more mathematics learning time devoted to number than to other topics. Mathematical process goals should be integrated in these content areas.

— Mathematics Learning in Early Childhood, National Research Council, 2009

The composite standards [of Hong Kong, Korea and Singapore] have a number of features that can inform an international benchmarking process for the development of K–6 mathematics standards in the U.S. First, the composite standards concentrate the early learning of mathematics on the number, measurement, and geometry strands with less emphasis on data analysis and little exposure to algebra. The Hong Kong standards for grades 1–3 devote approximately half the targeted time to numbers and almost all the time remaining to geometry and measurement.

— Ginsburg, Leinwand and Decker, 2009

Because the mathematics concepts in [U.S.] textbooks are often weak, the presentation becomes more mechanical than is ideal. We looked at both traditional and non-traditional textbooks used in the US and found this conceptual weakness in both.

— Ginsburg et al., 2005

There are many ways to organize curricula. The challenge, now rarely met, is to avoid those that distort mathematics and turn off students.

— Steen, 2007

For over a decade, research studies of mathematics education in high-performing

countries have pointed to the conclusion that the mathematics curriculum in the

United States must become substantially more focused and coherent in order to

improve mathematics achievement in this country. To deliver on the promise of

common standards, the standards must address the problem of a curriculum that

is “a mile wide and an inch deep.” These Standards are a substantial answer to that

challenge.

It is important to recognize that “fewer standards” are no substitute for focused

standards. Achieving “fewer standards” would be easy to do by resorting to broad,

general statements. Instead, these Standards aim for clarity and specificity.

Assessing the coherence of a set of standards is more difficult than assessing

their focus. William Schmidt and Richard Houang (2002) have said that content

standards and curricula are coherent if they are:

articulated over time as a sequence of topics and performances that are logical and reflect, where appropriate, the sequential or hierarchical nature of the disciplinary content from which the subject matter derives. That is, what and how students are taught should reflect not only the topics that fall within a certain academic discipline, but also the key ideas that determine how knowledge is organized and generated within that discipline. This implies

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that to be coherent, a set of content standards must evolve from particulars (e.g., the meaning and operations of whole numbers, including simple math facts and routine computational procedures associated with whole numbers and fractions) to deeper structures inherent in the discipline. These deeper structures then serve as a means for connecting the particulars (such as an understanding of the rational number system and its properties). (emphasis added)

These Standards endeavor to follow such a design, not only by stressing conceptual

understanding of key ideas, but also by continually returning to organizing

principles such as place value or the properties of operations to structure those

ideas.

In addition, the “sequence of topics and performances” that is outlined in a body of

mathematics standards must also respect what is known about how students learn.

As Confrey (2007) points out, developing “sequenced obstacles and challenges

for students…absent the insights about meaning that derive from careful study of

learning, would be unfortunate and unwise.” In recognition of this, the development

of these Standards began with research-based learning progressions detailing

what is known today about how students’ mathematical knowledge, skill, and

understanding develop over time.

Understanding mathematics

These Standards define what students should understand and be able to do in

their study of mathematics. Asking a student to understand something means

asking a teacher to assess whether the student has understood it. But what does

mathematical understanding look like? One hallmark of mathematical understanding

is the ability to justify, in a way appropriate to the student’s mathematical maturity,

why a particular mathematical statement is true or where a mathematical rule comes from. There is a world of difference between a student who can summon a

mnemonic device to expand a product such as (a + b)(x + y) and a student who can explain where the mnemonic comes from. The student who can explain the rule

understands the mathematics, and may have a better chance to succeed at a less

familiar task such as expanding (a + b + c)(x + y). Mathematical understanding and procedural skill are equally important, and both are assessable using mathematical

The Standards set grade-specific standards but do not define the intervention

methods or materials necessary to support students who are well below or well

above grade-level expectations. It is also beyond the scope of the Standards to

define the full range of supports appropriate for English language learners and

for students with special needs. At the same time, all students must have the

opportunity to learn and meet the same high standards if they are to access the

knowledge and skills necessary in their post-school lives. The Standards should

be read as allowing for the widest possible range of students to participate fully

from the outset, along with appropriate accommodations to ensure maximum

participaton of students with special education needs. For example, for students

with disabilities reading should allow for use of Braille, screen reader technology, or

other assistive devices, while writing should include the use of a scribe, computer,

or speech-to-text technology. In a similar vein, speaking and listening should be

can fully reflect the great variety in abilities, needs, learning rates, and achievement

levels of students in any given classroom. However, the Standards do provide clear

signposts along the way to the goal of college and career readiness for all students.

The Standards begin on page 6 with eight Standards for Mathematical Practice.

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Standards define what students should understand and be able to do.

Clusters are groups of related standards. Note that standards from different clusters may sometimes be closely related, because mathematics

is a connected subject.

domains are larger groups of related standards. Standards from different domains may sometimes be closely related.

number and operations in Base ten 3.nBt Use place value understanding and properties of operations to perform multi-digit arithmetic.

1. Use place value understanding to round whole numbers to the nearest 10 or 100.

2. Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

3. Multiply one-digit whole numbers by multiples of 10 in the range 10-90 (e.g., 9 × 80, 5 × 60) using strategies based on place value and properties of operations.

These Standards do not dictate curriculum or teaching methods. For example, just

because topic A appears before topic B in the standards for a given grade, it does

not necessarily mean that topic A must be taught before topic B. A teacher might

prefer to teach topic B before topic A, or might choose to highlight connections by

teaching topic A and topic B at the same time. Or, a teacher might prefer to teach a

topic of his or her own choosing that leads, as a byproduct, to students reaching the

standards for topics A and B.

What students can learn at any particular grade level depends upon what they

have learned before. Ideally then, each standard in this document might have been

phrased in the form, “Students who already know … should next come to learn ….”

But at present this approach is unrealistic—not least because existing education

research cannot specify all such learning pathways. Of necessity therefore,

grade placements for specific topics have been made on the basis of state and

international comparisons and the collective experience and collective professional

judgment of educators, researchers and mathematicians. One promise of common

state standards is that over time they will allow research on learning progressions

to inform and improve the design of standards to a much greater extent than is

possible today. Learning opportunities will continue to vary across schools and

school systems, and educators should make every effort to meet the needs of

individual students based on their current understanding.

These Standards are not intended to be new names for old ways of doing business.

They are a call to take the next step. It is time for states to work together to build

on lessons learned from two decades of standards based reforms. It is time to

recognize that standards are not just promises to our children, but promises we

intend to keep.

domain

ClusterStandard

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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.

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

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).

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,

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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

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