Core Content State Standards… for Learning… as Learning

Math in Grades 3-12

Daniel J. Heisey, Ph.D.Mathematics Coordinator

NJ Department of Education

daniel.heisey@doe.state.nj.us

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Fear and Angst

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Presentation Roadmap• NAEP – gatekeeper for NCLB• Conceptual understanding of fractions• Core Content State Standards [CCSS]

– Grade 3 understand fractions as numbers and build fractions from unit fractions

– Grade 4 add and subtract fractions with like denominators and generate decimal equivalents

– Grade 5 multiply fractions (including mixed numbers) and divide fractions in special cases

• Diagnostics and learning progressions

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NAEP DataReading and Math Gap (ETS summary,

2010) African American, Latino, and poor of all races

vs. their wealthier, mostly white peers– By 4th grade, they are 2 years behind– By 8th grade, they are 3 years behind– By 12th grade, they are 4 years behind

High School Dropouts (75% in jail are school dropouts)

68% of six graders score basic (or worse) in math Only 17% of H.S. seniors are proficient in math

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NAEP Released Item

What fraction of the figure is shaded?

Answer: _______________

Did you us a calculator on this question? ______

Grade 4 NAEP 2007

22% incorrectNJDOE 2011 6

NAEP Released Item

Luis is making a game spinner. He wants the chance of landing on red (R) to be twice the chance of landing on blue (B). Show how he could label his spinner.

Number of blues ____Number of reds ____

Grade 4 NAEP 200759% incorrect

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Gene Wilhoit

Gene Wilhoit is the executive director of the Council of Chief State School Officers [CCSSO].

This video was made in November 2010.It is a presentation to the U. S. Congress about the federal role in the state-initiated Common Core initiative … a historical perspective.

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Gene WilHoit in Congress

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Fraction beginnings: Part vs. WholeGrade 2: CCSS 2.G.3Partition circles and rectangles into two or four equal shares and describe shares using words halves, thirds, half of, third of, etc., and describe the whole as two halves, three thirds, four fourths.

Recognize equal shares of identical wholes need not have the same shape.

Numerals for fractions (1/2, 1/3, 1/4, etc.) are not used at grade 2.

Two fractions are compared only if they refer to the same whole.

Grade 3:

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.

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Note: The above is not a complete map (e.g. proportional relationships).

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Critical areas in grades 3-5 -- plus grade 6 ….. HANDOUT #1

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NJCCCS vs. CCSS 4.NF.3 & 4.NF.4 …………... HANDOUTs #2 & #3

In Grade 5: 4.1.5 B. Numerical Operations 2. Construct, use, and explain procedures for performing addition and subtraction with fractions and decimals with:

• Pencil-and-paper • Mental math • Calculator

In Grade 6:4.1.6 B. Numerical Operations 2. Construct, use, and explain procedures for performing calculations with fractions and decimals with:

• Pencil-and-paper • Mental math • Calculator

The first mention of multiplying or dividing fractions in NJCCCS was “implied” in this single standard in grade 6

>>>>>>>>>>>

There is no lead up, and no progression of ideas. The grade 5 NJCCCS is not very helpful.

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Common Core demands we revamp the mile-wide, inch-deep approach in curriculum and textbooks.

Key moments in the curriculum (like 4.NF.4) demand that we slow down and devote more time to allow for reasoning / thinking / discussing as well as the necessary hard work and practice.CCSS gives three years to the division of fractions thread:• In grade 4, we multiply a fraction by a whole number.• In grade 5, we multiply a fraction by a fraction and we divide a

unit fraction by a whole number or a whole number by a unit fraction.

• Finally, in grade 6, we divide a fraction by a fraction.

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(1) Build on prior work of multiplying whole numbers.

(2)Build fractions from unit fractions:5/4 means 5 x ¼. It is also ¼ + ¼ + ¼ + ¼ + ¼, which builds from unit fractions using additive reasoning. (see 4.NF.3)

(3) We achieve the more complex case 5 x ¾ by saying, 5 times 3/4 equals (5 times 3) fourths equals 15 fourths. This stresses properties of operations, making arithmetic a rehearsal for algebra.Source:

http://www.p12.nysed.gov/ciai/mst/math/standards/revisedg4.html

Fraction progressions …

• Grade 3: Develop an understanding of fractions as numbers.

• Grade 4: Understand fraction equivalence and ordering. Build fractions from unit fractions and apply and extend

previous understandings of operations on whole numbers. Use decimal notation for fractions and compare fractions.

• Grade 5: Use equivalent fractions to add and subtract fractions (including mixed numbers w unlike denominators).

Apply and extend previous understandings of multiplication and division to multiply and divide fractions.

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Thinking about 3/4 …

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Unit Fractions … 1/b … as building blocks

Grade 3: Students name the “numeral” for each unit fraction as 1/2, 1/3, 1/4, etc. Fractions are built from unit fractions.

3/4 is 3 copies of 1/4 … 3/4 = 1/4 + 1/4 + 1/4

4/5 is 4 copies of 1/5 … 4/5 = 1/5 + 1/5 + 1/5 + 1/5

Note: No need to introduce proper or improper fractions. The quantity 5/3 is the sum of 5 parts of a whole divided into 3 equal parts. Students can easily find 5/3 on the number line.

Grade 3: Students identify “equivalent” fractions as having the same size or the same position on the number line.

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Domain: Number – FractionsCluster: Develop understanding of fractions as numbers.

3.NF.1 Understand a fraction 1/b as the quantity formed by 1 part when a whole is partitioned into b equal parts; understand a fraction a/b as the quantity formed by a parts of size 1/b.

3.NF.2 Understand a fraction as a number on the number line; represent fractions on a number line diagram.

a. Represent a fraction 1/b on a number line diagram by defining the interval from 0 to 1 as the whole and partitioning it into b equal parts … each part has size 1/b and the endpoint of that part based at 0 locates the number 1/b on the number line.

b. Represent a fraction a/b on a number line diagram by marking off a lengths 1/b from 0. Recognize that the resulting interval has size a/b and that its endpoint locates the number a/b on the number line.

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Domain: Number – FractionsCluster: Develop understanding of fractions as numbers.

3.NF.3 Explain equivalence of fractions in special cases, and compare fractions by reasoning about their size.

a. Understand two fractions as equivalent (equal) if they are the same size, or at the same point on a number line.

b. Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6, = 2/3 … explain using visual fraction models.

c. Express whole numbers as fractions, & recognize fractions equivalent to whole numbers.Examples: Express 3 as 3/1 & recognize 6/1 = 6; locate 4/4 and 1 at the same point on a number line.

d. Compare two fractions with the same numerator or denominator by reasoning about their size.

Comparisons need to refer to the same whole. Use symbols >, =, < for comparisons … explain using visual fraction models.

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Conceptual Understanding¾

What happens to the value of a fraction if:

-- the numerators is increased by 1 ? -- the denominator is decreased by 1 ? -- the denominator is increased by 1 ? -- the value of the numerator and denominator are doubled ?

Which comparison is true? … Explain WHY using a number line.

½ < ¾ versus ¾ < ½

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Grade 4 Fractions: Equivalent Fractions

Grade 4: CCSS 4.NF.1 Explain why a fraction a/b is equivalent to a fraction (n x a)/(n x b) using fraction models, with attention to how the number and size of the parts differ even though the two fractions are the same size. Use this principle to recognize and generate equivalent fractions.

This fundamental property of equivalent fractions impacts much of the computation and procedural work in grade 4 …

Comparing … adding … subtracting … finite decimals …

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Grade 4 : Fraction QuestionsCreate an area model for 2/3 = (4 x 2)/(4 x 3).

How can 2/5 + 1/3 be represented as a length?

Part of a whole vs. Part of a part.

Convert mixed numbers to improper fractions.

Multiply fractions by whole numbers.

What visual fraction model shows 0.2 > 0.17 ?

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Area models

Each block shown represents 1/6 multiply unit fractions 1/2 x 1/3

1/3 of 1/2 = 1/6

1/2 of 1/3 = 1/6

multiply whole numbers 14 x 6

Multiply Fractions (Area Model)

2/3 x 4/5 = 8/15 (2-dimensional arrays)

2/3 of 15/15 = 10/15

4/5 of 15/15 = 12/15

either 8/15 is 4/5 of 10/15 … or 8/15 is 2/3 of 12/15

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Multiply mixed numbers (area model) A typical grade 5 problem 4 ½ x 7 ¼ = ??

A composite model showing 4 areas

7 ¼

4

½

4 ½ x 7 ¼ = (4 x 7) + (½ x 7) + (4 x ¼) + (½ x ¼) 32 ⅝ = 28 + 3 ½ + 1 + ⅛

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Length model :: Unit Fractions

How can 2/5 + 1/3 be represented as a length?

Draw a model of 2/5 plus 1/3 using the number line.

0 1

Reference: Lamon, Susan. Teaching Fractions and Ratios for Understanding, 2nd Edition, New York, NY: Routledge, 2008.

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Length model :: Unit Fractions

What fraction is located at the point “X”?

0 X 2/3

Reference: Lamon, Susan. Teaching Fractions and Ratios for Understanding, 2nd Edition, New York, NY: Routledge, 2008.

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Part of a whole vs. Part of a partPart of a part: Bill had 2/3 of a cup of juice. He drank

1/2 of his juice. How much juice did Bill have left?

This problem cannot be solved by subtracting 2/3 - 1/2 because the 2/3 refers to a cup … but the 1/2 refers to the amount of juice Bill had and not to a cup of juice.

A similar problem: If ¼ of a garden is planted with daffodils, 1/3 with tulips and the rest with vegetables, what fraction of the garden is planted with flowers?

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Mixed and Improper Fractions

4 2/5 = 4 + 2/5 = 1 + 1 + 1 + 1 + 2/5

Tricks are not helpful to students with memory problems, think about the concept and purpose to the calculations

“Four and two – fifths” … “and” says to add …

4 + 2/5 = 5/5 + 5/5 + 5/5 + 5/5 + 2/5 = 20/5 + 2/5 = 22/5

Trick: The whole number times the fraction’s denominator plus the numerator equals the new numerator. The number in the denominator does not change.

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Multiplication of fractions by a whole number

4 2/5 is not the same as 4 x 2/5 Tricks are not always helpful …

“Four times two fifths” … “times” means add 2/5 four times

4 x 2/5 … 2/5 + 2/5 + 2/5 + 2/5 = 8/5

TRICK: Multiply the whole number by the numerator and that product becomes the new numerator in the answer.

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Visual models for decimal fractions The “tenth” scale vs. the “hundredths” scale …

0.2 < 0.17 is not true

2 is “less than” 17 but …0.2 = 0.20 which is 0.03 greater than 0.17

Compare all fractions by referencing the same part ...

0 0.1 0.2 1/4 0.3 0 0.10 0.20 0.25 0.30

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How do Fraction Models Differ Across Grades

Concrete Manipulatives: Grades K-2

Representational Drawings: Grades 2-3

Abstract Number Procedures: Grades 3-4

1. There were 16 apples. Rhonda ate 1/4 of them. How many are left?

2. Mom used one-third of 12 eggs. How many are left? 3. Tom ate 4 hazelnuts, which was 1/8 of the nuts. How many

were there in the first place? 4. Lisa used $5 to buy a gift which took 1/3 of her savings. How

much did she have in the beginning? How much does she have now?

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The Khan Academy – a resource The web-access to this video library is free and individual

videos can be purchased for 99¢. The videos show only a blackboard and chalk writing. The invisible “teacher” explains the math by audio.

Not the lessons are not aligned to the CCSS skills.

For example, the video is multiplying fractions (a grade 5 CCSS skill) with integer math included (a grade 6 CCSS skill).

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COMMON CORE MATH STANDARDS

Clearer, Fewer, Higher

No more …

… content a mile wide and an inch deep… standards that require “un-packing”… skills re-taught (spiraled) in the next grades

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SC GA

NC

WV VT LA

NH

/RI

DoD

EA IN MO NV

OH

OK

OR SD WA NJ

AZ CA

DC FL MI

NM NY TX MD

MN UT VA CO

ME

WY

TN AL KS MS

ND AR ID AK HI

0

1

2

3

4

5

6

7

Culminating Learning ExpectationIntermediate ExpectationsInitial Learning ExpectationRepeated Expectation

States

Gra

de L

evel

American Institute for ResearchSteve Leinwand

Addition of Whole Numbers

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2 3 4 5 6 7 8 120%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

2 3 4 5 6 7 8 12

Data, Prob, & Stat

Algebra

Geometry

Number

NJCCCS CCSS

Grade Levels Grade Levels

% o

f Sta

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PI’s

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The CCSS Format of StandardsGrade Level

Domain (Topic)

Standard

ClusterAlgebra Symbol

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CCSS Developer Commentary

http://www.americaschoice.org/uploads/Common_Core_Standards_Resources/PhilDaro_MathStandards/PhilDaro_MathStandards.html

In this video Phil Daro applauds Common Core developers as being mindful of common sense skill levels selection “less is more” idea.

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http://successatthecore.com/teacher_development_featured_video.aspx?v=44

CCSS Assessments coming 2014-15 In 2014-15, assessments for the Common Core will be

administered via the internet.

The Partnership for Assessment of Readiness for College and Careers [PARCC] is an alliance of states working together to develop common assessments serving nearly 25 million students. PARCC’s work is funded through a four-year 185 million dollar grant from the U.S. DOE.

PARCC’s partners include 200 higher education institutions and is led by its member states and managed by Achieve.

PARCC’s goal is to ensure that all students graduate from high school college and career ready. (See www.parcconline.org )

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Kinds of Assessments

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• Readiness

• Benchmarks

• Diagnostics

• Common Core is a new paradigm. Teachers will teach to mastery so that all students can demonstrate mastery.

• Diagnostic assessments will be needed to manage the intervention events for struggling math students.

ASSESSMENT NOTE RE: Student Progress

“It is impossible for a norm-referenced test to align with standards. Norm-reference tests tell you how well students are doing compared to each other …

Standards mean that student progress must be compared to the standard, not to how well or poorly others do.”

• Quoted from … Workshop: Teaching to Academic Standards

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1. A batch of muffins requires 2/3 cups of sugar. A loaf of zucchini bread requires 3/4 cups of sugar. How many cups of sugar are needed to make one batch of each recipe?

a. 5/7 b. 1 5/12 c. 1 1/2 d. 1 7/12

2. What is 4 divided by one-half? a. 1/2 b. 1/8 c. 2 d. 8

3. If AX = B and A, B, and X are not equal to zero, then ______ ? a. X = A divided by B b. X = A minus B c. X = B divided by A d. X = B times A

4. Which number is largest ? a. 0.72 b. 0.080 c. 8/9 d. 6/7

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Readiness Test Items

Diagnostic Test ItemsFind the equivalent common fraction.

1. 1/4 a) 11/14 b) 3/5 c) 2/8 d) 4/1

2. 5/6 a) 6/5 b) 50/60 c) 15/16 d) 5/11

3. 1 3/4 a) 7/4 b) 13/4 c) 7/14 d) 3/14

4. 3 2/5 a) 32/5 b) 17/5 c) 6/5 d) 2/32

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Diagnostic Testing Diagnostic tests assess fewer skills – “less is more”Skills are ordered as a learning progression

-- Four test items assess each skill-- Each Item has similar difficulty-- 3 or 4 items correct demonstrates mastery

Guessing is NOT a factor: guess 1 out of 4 25% guess 2 out of 4 6.25% guess 3 out of 4 1.6% guess 4 out of 4 0.4% … very few false positives !

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• Student scores and number correct per skill (left tally matrix)• Student mastery indicators (right OK matrix)

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1 2 3 4 5 O-E

OK OK 2OK OK 1

OK OK OK OK 2OK OK OK 3Ok OK OK 3OK OK OK 3OK OK 0

Name Score

1 2 3 4 5 %Jim E 11 2 4 0 3 2 55Mark G

9 0 2 0 4 3 45

Sue T 14 3 4 0 4 3 70Mike W

10 0 3 0 4 3 50

Jill D 12 1 4 1 3 3 60Paul R 12 0 4 1 3 3 60Dan H 11 2 4 0 3 2 45

Part 1 Equivalent common fractions [4]Part 2 Place value [4]

Part 3 Equivalent decimals [4]Part 4 Add & subtract [4]

Part 5 Multiply & divide [4]

Classroom Learning ProgressionsA learning progression is a carefully sequenced set of building blocks that students must master en route to mastering a more distant curricular aim.

-- James Popham, ASCD, 2007

Classroom learning progressions are: – often “customized”; – not etched in stone; – often referred to as learning trajectories;

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A scoop holds ¾ of a cup. How many scoops of bird seed are needed to fill a bird feeder that holds 3 cups?

Show how to use pictures to solve this problem. … or explain your solution in words.

Number line l__ı__ı__ı__l__ı__ı__ı__l__ı__ı__ı__l

1 1 1 2 2 2 3 3 3 4 4 4

Arrays 1 2 3 1 2 3 1 2 3 1 2 3

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Fractions and Word Problems

Length model … explained … Draw a model of 2/5 plus 1/3 using the number line.

0 ← 2/5 → ↑ ← 1/3 → ↑ 1 11/15

count unit fractions … 1/5 = 3/15 … 1/3 = 5/15

therefore ... 2/5 + 1/3 = 6/15 + 5/15 = 11/15

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Length model … explained …

What fraction is located at the point “X”?

0 X 2/3 1

2/3 equals 10 units1/3 equals 5 units

1 equals 15 units = 15/15

count 6 unit fractions of 1/15 to the “X” = 6/15 = 2/5

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ONE FINAL NOTE:BEWARE OF MISCONCEPTIONS

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Does the above sketch represent 4/6 + 4/6 = 8/12 ?

+ =

… what is wrong with this picture?

MISCONCEPTION FIX

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A major goal for K-8 math needs to be a proficiency with fractions …

students lost in math can recover if we can find a way to teach them

where they are at.

+ =

Bob Moses Re: The Algebra Project

Since 1985 Robert Moses has been building a consortium of Southern schools with the aim of graduating more students in the poor rural areas of Alabama and Mississippi.

http://flash.unctv.org/ncnow/ncn_rmoses_081408.html

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ReferencesAnderson, Charles W. and Mohan, Lindsey. “Learning Progressions to Inform the Development of

Standards,” Michigan State University, in Learning Progressions and Standards, Vol 2: AERA, 2009

Bahr, Damon L. and Kathleen, and DeGarcia, Lisa Ann. Elementary Mathematics is Anything but Elementary: Content and Methods from a Developmental Perspective, Paperback ISBN 0618928170: Cengage Brain, 2008. (Amazon $100.00)

Clements, Douglas H. and Sarama, Julie Learning and Teaching Early Math New York, NY: Routledge, 2008

English, Lyn D. and Graeme S. Halford. Mathematics Education: Models and Processes, Mahwah, NJ: Lawrence Erlbaum Associates, 1995.

Lamon, Susan. Teaching Fractions and Ratios for Understandin” 2nd Edition, New York, NY: Routledge, 2008.

Lamon, Susan. MORE In-depth Discussion of the Reasoning Activities in Teaching Fractions and Ratios for Understanding 2nd Edition, New York, NY: Routledge, 2009

Lamon, Susan. “Rational Numbers and Proportional Reasoning: Toward a Theoretical Framework for Research.” In Handbook of Research on Mathematics Teaching and Learning, edited by Douglas Grouws, pp. 629-67. Reston, VA: National Council of Teachers of Mathematics, 2007.

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ReferencesLoveless, Tom, “The Misplaced Math Student Lost in Eighth-Grade Algebra.” Brown Center on Educational

Policy, 2008

National Mathematics Advisory Panel. The Final Report of the National Mathematics Advisory Panel. Washington, DC: US Department of Education, 2008.

PARCC Math Frameworks for Grades 3 – 11 (76 pages) http://www.parcconline.org/parcc-content-frameworks

Popham, James W., (2007, April). “The Lowdown on Learning Progressions.” Educational Leadership, 64(7), pp. 83-84

Sanders, S., Riccomini, P. J., & Witzel, B. S. (2005). “The algebra readiness of high school students in South Carolina: Implications for middle school math teachers.” South Carolina Middle School Journal.

Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmentally. Boston, MA: Pearson Education, 2007.

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