Common Core State Standards for Mathematics

William McCallumThe University of Arizona

University of Connecticut, 1 April 2010

Composite of high achieving countries

matics standards from 21 states (Figure 2) and 50 districts(Figure 3, page 11) were also developed and compared tothe A+ composite. (For more details on the methodology,please see page 47.)

While examining the A+ composite, it is important tokeep in mind that this figure represents a “core” curriculum,not a complete curriculum. Our goal in developing thecomposite was to find out which topics at least two-thirds ofA+ countries believed to be essential. Not surprisingly, thesecountries’ points of agreement resulted in a smaller set of

topics in our composite than any one of these countries in-cludes in its national curriculum.4

To represent the full scope of a complete mathematicscurriculum in a typical A+ country, roughly three topicswould have to be added at each grade level in addition tothose listed in Figure 1. As noted in the last line of Figure 1,the average number of topics that would have to be addedrange from one (in grades four and five) to as many as six (ingrades two and seven). This is important information forAmericans who understand that there is a need for a com-

4AMERICAN EDUCATOR SUMMER 2002

FIGURE 1A+ Composite: Mathematics topics intended at each grade by at least two-thirds of A+ countries.Note that topics are introduced and sustained in a coherent fashion, producing a clear upper-triangular structure.

TOPIC GRADE: 1 2 3 4 5 6 7 8 Whole Number Meaning ! ! ! ! !

Whole Number Operations ! ! ! ! !

Measurement Units ! ! ! ! ! ! !

Common Fractions ! ! ! !

Equations & Formulas ! ! ! ! ! !

Data Representation & Analysis ! ! ! ! !2-D Geometry: Basics ! ! ! ! ! !

Polygons & Circles ! ! ! ! !

Perimeter, Area & Volume ! ! ! ! !Rounding & Significant Figures ! !

Estimating Computations ! ! !

Properties of Whole Number Operations ! !

Estimating Quantity & Size ! !Decimal Fractions ! ! !

Relationship of Common & Decimal Fractions ! ! !

Properties of Common & Decimal Fractions ! !

Percentages ! !

Proportionality Concepts ! ! ! !Proportionality Problems ! ! ! !

2-D Coordinate Geometry ! ! ! !

Geometry: Transformations ! ! !

Negative Numbers, Integers & Their Properties ! !

Number Theory ! !Exponents, Roots & Radicals ! !

Exponents & Orders of Magnitude ! !Measurement Estimation & Errors !Constructions w/ Straightedge & Compass ! !3-D Geometry ! !

Congruence & Similarity !

Rational Numbers & Their Properties !Patterns, Relations & Functions !Slope & Trigonometry !Number of topics covered by at least 67% of the A+ countries 3 3 7 15 20 17 16 18 Number of additional topics intended by A+ countriesto complete a typical curriculum at each grade level 2 6 5 1 1 3 6 3 ! – intended by 67% of the A+ countries ! – intended by 83% of the A+ countries ! – intended by 100% of the A+ countries

I Mathematics topicsintended at eachgrade by at least twothirds of A+countries.

I A+ countriesdetermined by lookingat statisticallysignificant differencesin 8th grade scores on1995 TIMMS

I On average an A+country would have1–6 more topics pergrade level in itscomplete curriculum.

Composite of U.S. State Curricula

SUMMER 2002 AMERICAN FEDERATION OF TEACHERS 5

mon, prescribed curricular core, but also believe some localdiscretion must be accommodated. The A+ composite showsthat, at least in math, it is eminently sensible and doable tothink of some math topics as part of a required core taughtin particular grades and others as topics that can float ac-cording to, say, state or district discretion.

The A+ CompositeFigure 1 presents the A+ composite for mathematics bytopic and grade. The 32 topics listed are those that are in the

national curricula at a given grade in at least two-thirds ofthe A+ countries. As evidenced by the “upper-triangular”shape of the data, the A+ composite reflects an evolutionfrom an early emphasis on arithmetic in grades one throughfour to more advanced algebra and geometry beginning ingrades seven and eight. Grades five and six serve as a transi-tional stage in which topics such as proportionality and co-ordinate geometry are taught, providing a bridge to the for-mal study of algebra and geometry.

More specifically, these data suggest a three-tier pattern of

FIGURE 2State Composite: Mathematics topics intended at each grade by at least two-thirds of 21 U.S. states.Note that topics are introduced and sustained in a way that produces no visible structure.

TOPIC GRADE: 1 2 3 4 5 6 7 8 Whole Number Meaning ! ! ! ! ! !Whole Number Operations ! ! ! ! ! !Measurement Units ! ! ! ! ! ! ! !

Common Fractions ! ! ! ! ! ! ! !Equations & Formulas ! ! ! ! ! ! ! !

Data Representation & Analysis ! ! ! ! ! ! ! !

2-D Geometry: Basics ! ! ! ! ! ! ! !

Polygons & Circles ! ! ! ! ! ! ! !

Perimeter, Area & Volume ! ! ! ! ! ! !

Rounding & Significant Figures Estimating Computations ! ! ! ! ! ! ! !

Properties of Whole Number Operations ! ! ! !Estimating Quantity & Size !Decimal Fractions ! ! ! ! ! !Relationship of Common & Decimal Fractions ! ! !Properties of Common & Decimal Fractions Percentages ! ! ! !Proportionality Concepts ! !Proportionality Problems ! ! !

2-D Coordinate Geometry ! ! ! ! ! !

Geometry: Transformations ! ! ! ! ! ! ! !

Negative Numbers, Integers & Their Properties ! ! !Number Theory ! ! ! !Exponents, Roots & Radicals ! ! !

Exponents & Orders of Magnitude ! !Measurement Estimation & Errors ! ! ! ! ! ! ! !Constructions w/ Straightedge & Compass3-D Geometry ! ! ! ! ! ! ! !

Congruence & Similarity ! ! ! !Rational Numbers & Their Properties ! ! !Patterns, Relations & Functions ! ! ! ! ! ! ! !

Slope & Trigonometry Number of topics covered by at least 67% of the states 14 15 18 18 20 25 23 22Number of additional topics intended by states to complete a typical curriculum at each grade level 8 8 7 8 8 5 6 6! – intended by 67% of the states ! – intended by 83% of the states ! – intended by 100% of the states

I Mathematics topicsintended at eachgrade by at least twothirds of 21 U.S.States.

I On average a statewould have 6–8 moretopics per grade levelin its completecurriculum.

I From Schmidt, Houang,and Cogan, AmericanEducator, 2005.

Focus

The composite standards [of Hong Kong, Korea and Singapore]have a number of features that can inform an internationalbenchmarking process for the development of K–6 mathematicsstandards in the US. First, the composite standards concentratethe early learning of mathematics on the number, measurement,and geometry strands with less emphasis on data analysis andlittle exposure to algebra. The Hong Kong standards for grades1–3 devote approximately half the targeted time to numbers andalmost all the time remaining to geometry and measurement.

—Ginsburg, Leinwand and Decker, 2009

Mathematics experiences in early childhood settings shouldconcentrate on (1) number (which includes whole number,operations, and relations) and (2) geometry, spatial relations,and measurement, with more mathematics learning timedevoted to number than to other topics.

— National Research Council, 2009

Coherence

Schmidt, Houang, and Cogan (2002) say that content standards andcurricula are coherent if they are:

articulated over time as a sequence of topics and performancesthat are logical and reflect, where appropriate, the sequential orhierarchical nature of the disciplinary content from which thesubject matter derives. That is, what and how students aretaught should reflect not only the topics that fall within acertain academic discipline, but also the key ideas thatdetermine how knowledge is organized and generated withinthat discipline. This implies that to be coherent, a set ofcontent standards must evolve from particulars (e.g., themeaning and operations of whole numbers, including simplemath facts and routine computational procedures associatedwith whole numbers and fractions) to deeper structuresinherent in the discipline. This deeper structure then serves as ameans for connecting the particulars (such as an understandingof the rational number system and its properties).

Concepts

The mathematical process goals should be integrated in thesecontent areas. Children should understand the concepts andlearn the skills exemplified in the teaching-learning pathsdescribed in this report.

— National Research Council, 2009

Because the mathematics concepts in these textbooks are oftenweak, the presentation becomes more mechanical than is ideal.We looked at both traditional and non-traditional textbooksused in the US and found this conceptual weakness in both.

—Ginsburg et al., 2005

There are many ways to organize curricula. The challenge, nowrarely met, is to avoid those that distort mathematics and turnoff students.

—Steen, 2007

Standards for Mathematical Practice

I Make sense of problems and perservere in solving them

I Reason abstractly and quantitatively

I Construct viable arguments and critique the reasoning of others

I Model with mathematics

I Use appropriate tools strategically

I Attend to precision

I Look for and make use of structure

I Look for and express regularity in repeated reasoning

Reason abstractly and quantitatively

Mathematically proficient students make sense of the quantities and theirrelationships in problem situations. Students bring two complementaryabilities to bear on problems involving quantitative relationships: theability to decontextualize—to abstract a given situation and represent itsymbolically and manipulate the representing symbols as if they have alife of their own, without necessarily attending to their referents—and theability to contextualize, to pause as needed during the manipulationprocess in order to probe into the referents for the symbols involved.Quantitative reasoning entails habits of creating a coherent representationof the problem at hand; considering the units involved; attending to themeaning of quantities, not just how to compute them; and knowing andflexibly using different properties of operations and objects.

Grade 3 Fractions

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DRAFT

Fractions as representations of numbers

1. Understand that a unit fraction corresponds to a point on a numberline. For example, 1

3represents the point obtained by decomposing the interval

from 0 to 1 into three equal parts and taking the right-hand endpoint of the first

part. In Grade 3, all number lines begin with zero.

2. Understand that fractions are built from unit fractions. For example, 54

represents the point on a number line obtained by marking off five lengths of 14

to the right of 0.

3. Understand that two fractions are equivalent (represent the samenumber) when both fractions correspond to the same point on anumber line. Recognize and generate equivalent fractions withdenominators 2, 3, 4, and 6 (e.g., 1

2 = 24 , 4

6 = 23 ), and explain the

reasoning.

4. Understand that whole numbers can be expressed as fractions. Three

important cases are illustrated by the examples 1 = 44

, 6 = 61

, and 7 = 4×74

.

Expressing whole numbers as fractions can be useful for solving problems or

making calculations.

Fractional quantities

1. Understand that fractions apply to situations where a whole isdecomposed into equal parts; use fractions to describe parts ofwholes. For example, to show 1

3of a length, decompose the length into 3 equal

parts and show one of the parts.

2. Compare and order fractional quantities with equal numerators orequal denominators, using the fractions themselves, tape diagrams,number line representations, and area models. Use > and < symbolsto record the results of comparisons.

Grade 6 Statistics and Probability

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DRAFT

Grade 6 Statistics and Probability

1. Understand that a statistical question is one that anticipatesvariability in the data related to the question and accounts for it inthe answers. For example, “How old am I?” is not a statistical question , but

“How old are the students in my school?” is a statistical question because one

anticipates variability in students’ ages.

. . .

5. Relate the choice of the median or mean as a measure of center tothe shape of the data distribution being described and the context inwhich it is being used. Do the same for the choice of interquartilerange or mean average deviation as a measure of variation. For

example, why are housing prices often summarized by reporting the median

selling price, while students assigned grades are often based on mean homework

scores?

High School

I Number and Quantity

I AlgebraI Seeing Structure in ExpressionsI Arithmetic with Polynomials and Rational ExpressionsI Creating Equations that Describe Numbers or RelationshipsI Reasoning with Equations and Inequalities

I Functions

I Geometry

I Statistics and Probability

I Modeling

Seeing Structure in Expressions

1. Understand that different forms of an expression may reveal differentproperties of the quantity in question; a purpose in transformingexpressions is to find those properties. Examples: factoring aquadratic expression reveals the zeros of the function it defines, andputting the expression in vertex form reveals its maximum orminimum value; the expression 1.15t can be rewritten as(1.151/12)12t ≈ 1.01212t to reveal the approximate equivalentmonthly interest rate if the annual rate is 15%.

2. Understand that complicated expressions can be interpreted byviewing one or more of their parts as single entities.

3. Interpret an expression that represents a quantity in terms of thecontext. Include interpreting parts of an expression, such as terms,factors and coefficients.

4. ...

Comparison of CCSS with A+ composite Topic 1 2 3 4 5 6 7 8

Whole Number: Meaning ! ! ! ! !

Whole Number: Operations ! ! ! ! !

Measurement Units ! ! ! ! ! ! !

Common Fractions ! ! ! ! ! !

Equations & Formulas ! ! ! ! !

Data Representation & Analysis ! ! ! ! ! ! ! !

2-D Geometry: Basics ! ! ! ! ! ! !

2-D Geometry: Polygons & Circles ! ! ! ! ! ! ! !

Measurement: Perimeter, Area & Volume ! ! ! ! ! ! !

Rounding & Significant Figures ! !

Estimating Computations ! ! !

Whole Numbers: Properties of Operations ! ! ! ! !

Estimating Quantity & SizeDecimal Fractions ! ! !

Relation of Common & Decimal Fractions ! ! ! !

Properties of Common & Decimal Fractions ! !

Percentages !

Proportionality Concepts ! ! !

Proportionality Problems ! ! !

2-D Geometry: Coordinate Geometry ! ! ! !

Geometry: Transformations ! !

Negative Numbers, Integers, & Their Properties ! !

Number Theory !

Exponents, Roots & Radicals !

Exponents & Orders of Magnitude Measurement: Estimation & Errors Constructions Using Straightedge & Compass ! !

3-D Geometry ! ! ! ! ! !

Geometry: Congruence & Similarity !

Rational Numbers & Their Properties ! ! !

Patterns, Relations & Functions !

Proportionality: Slope & Trigonometry !

Topic Intended in Common Core Standards !

Intended in More Than Half of Top Achieving Countries

I The number of extratopics per grade levelin CCSS is comparablewith A+ countries.