core progress for mathcore progress for math has four domains, which form the base of the learning...

Core Progress® for Math Empirically validated learning progressions

WHITE PAPER | JULY 2013

Integral components of Renaissance Accelerated Math® and Renaissance Star Math®

All logos, designs, and brand names for Renaissance Learning’s products and services, including but not limited to 2Know!, Accelerated Math, Accelerated Reader, AccelScan, AccelTest, AR, ATOS, Core Progress, DEEP, English in a Flash, KeyWords, Learnalytics, MathFacts in a Flash, NEO, Renaissance Home Connect, Renaissance Learning, Renaissance Place Real Time, Renaissance School Excellence, Renaissance Training Center, STAR, STAR Early Literacy, STAR Math, STAR Reading, STAR Reading Spanish, Successful Reader, and Subtext are trademarks of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other countries. All other product and company names should be considered the property of their respective companies and organizations.

© 2013 by Renaissance Learning, Inc. All rights reserved. Printed in the United States of America.

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RENAISSANCE LEARNINGP.O. Box 8036Wisconsin Rapids, WI 54495-8036(800) [email protected]

07/13

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ContentsExecutive Summary ...............................................................................................................................................ii

Introduction .......................................................................................................................................................... 1

What are learning progressions? ......................................................................................................................... 2

Evolution of the Core Progress learning progression for mathematics ................................................................ 3

Phase I: Scope and sequence ............................................................................................................................. 3

Phase II: Revised scope and sequence, addition of core objectives and prerequisites ..................................... 5

Phase III: Learning progression ........................................................................................................................... 6

Phase IV: Empirical analysis of Core Progress ................................................................................................... 10

Mapping Core Progress to the Common Core State Standards ........................................................................ 12

Phase V: Building a new learning progression specifically for the Common Core ............................................ 13

Core Progress: an integral component of Accelerated Math Live and STAR Math Enterprise .......................... 15

Conclusion .......................................................................................................................................................... 19

References ......................................................................................................................................................... 32

Appendices

Core Progress for Math Learning Progression

Appendix A: Examples of skill progressions across grade levels ..................................................................... 20

Appendix B: Core Progress for Math includes four domains and 23 skill areas ................................................ 21

Appendix C: Core skills per grade, per domain ................................................................................................ 22

Appendix D: Example of how one core skill serves as a prerequisite for many other core skills ...................... 23

Appendix E: Common Core State Standards and Core Progress...................................................................... 24

Core Progress Learning Progression for Math - Built for the Common Core State Standards

Appendix F: Organization of Skill Areas within the 11 Domains for K-8 ............................................................ 25

Appendix G: Organization of Skill Areas within the 21 Domains for high school ............................................... 26

Appendix H: Examples of skill progressions across grade levels, Whole Numbers: Place Value ..................... 28

Appendix I: Core skills per grade, per domain .................................................................................................. 29

Appendix J: Example of mapping a core skill to many other core skills, Fractions ........................................... 31

Figures

Figure 1: Core Progress for Math ......................................................................................................................... 7

Figure 2: Prerequisite map of place value ............................................................................................................ 9

Figure 3: Correlation of STAR Math Enterprise to Core Progress ....................................................................... 11

Figure 4: Correlation of STAR Math Enterprise to Core Progress Math built for CCSS ...................................... 13

Figure 5: Accelerated Math Live Student Record Report .................................................................................. 16

Figure 6: STAR Math Enterprise provides your entry point into Core Progress .................................................. 17

Figure 7: Example of Instructional Planning Report generated by STAR Math Enterprise ................................ 18

Figure 8: STAR Record Book .............................................................................................................................. 18

Tables

Table 1: Cross-grade progression of Fraction Concepts and Operations ........................................................... 5

Table 2: Example of how one core skill serves as a prerequisite for many other core skills ................................ 8

Table 3: Example of Core Progress alignment to Common Core State Standard .............................................. 12

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Executive SummaryLearning progressions are descriptions of how learning typically advances in a subject area. “Empirically based learning progressions can visually and verbally articulate a hypothesis, or an anticipated path, of how student learning will typically move toward increased understanding over time with good instruction” (Hess, Kurizaki, and Holt, 2009). This paper describes Core Progress for mathematics, the learning progression developed by Renaissance Learning. We begin by explaining what learning progressions are, how they operate in relation to the standards, and how they support assessment, instruction, and practice. This paper then describes the research-based approach used to develop Core Progress.

A learning progression as comprehensive and interrelated as Core Progress takes years to develop through a continuous process of research, expert review, and iterative revision. Continually refined since 2007, the Core Progress for Math learning progression is an interconnected web of prerequisite skills.

The skills and understandings in the Core Progress learning progressions provide the intermediate steps along with prerequisite skills necessary to reach the levels of expertise identified through the Common Core State Standards. They begin with early numeracy and progress to the level of competence in mathematics required to be college and career ready.

Core Progress was originally developed to provide a research-based framework for Accelerated Math personalized practice software. Once built, the Core Progress skills were field tested through the STAR Math assessment.

The results were remarkable. As illustrated in the graph below, the order of skills in Core Progress are highly correlated with the difficulty level of STAR Math assessment items. With a strong correlation, the natural next step was to statistically link Core Progress to the STAR Math Enterprise assessment.

As a result of the statistical link between STAR Math Enterprise and Core Progress, a student’s STAR Math score provides insight into her achievement level, as well as skills and understandings she is ready to develop next. Core Progress is now an integral component of both Accelerated Math Live and STAR Math Enterprise—a true bridge between assessment, instruction, and practice.

0 2 4 6 8 10 12200

400

600

800

1000

Data Analysis, Statistics, and Probability

Geometry and Measurement

Algebra

Numbers and Operations

Core Progress Skill Difficulty

Sca

led

Dif

ficu

lty

70

Grade Level Order

y = 240.13Ln(x) + 334.27r = 0.8960

y = 271.68Ln(x) + 313.65r = 0.9104

y = 251.45Ln(x) + 333.35r = 0.9440

y = 253.5Ln(x) + 324.85r = 0.9059

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IntroductionOver the last decade, much of the focus of educational reform in the United States has been on the creation and improvement of standards of learning. A watershed moment of this movement was the 2010 publication of the Common Core State Standards (CCSS) for learning in Mathematics and English language arts. As the CCSS mission statement explains, “The Common Core State Standards provide a consistent, clear understanding of what students are expected to learn, so teachers and parents know what they need to do to help them.”

At the same time, within the field of education, the idea of learning progressions has received increasing attention (for example, Alonzo and Gearhart, 2006; Corcoran, Mosher, and Rogat, 2009; Heritage, 2008, 2009, 2011; Hess, 2010; Hess, Kurizaki, and Holt, 2009; Leahy and Wiliam, 2011). One of the reasons for this interest is the desire to provide descriptions of incremental steps of learning. These steps, more precise than are currently represented in standards, can be used to guide design of curriculum, instruction, and assessment. Learning occurs when students see these incremental steps as special cases of more general and basic processes and principles.

While the Common Core State Standards represent a clear step toward providing a more coherent pathway to meeting educational goals than many prior standards, the CCSS do not describe a fully formed pathway along which students are expected to progress. The next step, clarified by the CCSS, is the development of learning progressions that mirror the CCSS.

Originally built to provide a framework for Accelerated Math personalized practice software, Core Progress now serves as an integral component for both Accelerated Math Live and the STAR Math Enterprise assessment. Now, with all three pieces linked, there is a true bridge between assessment, instruction, and practice.

The next step, clarified by the CCSS, is the development of learning progressions that mirror the CCSS.

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What are learning progressions?

Simply put, learning progressions are descriptions of how learning typically advances in a subject area. Specifically, Pellegrino (2011, p. 9) defines learning progressions as “descriptions of successively more sophisticated ways of thinking about key disciplinary concepts and practices across multiple grades” which outline “the intermediate steps toward expertise.” Leahy and Wiliam (2011, p. 1) view learning progressions as descriptions of “what it is that gets better when someone gets better at something.” “Empirically based learning progressions can visually and verbally articulate a hypothesis, or an anticipated path, of how student learning will typically move toward increased understanding over time with good instruction” (Hess, Kurizaki, and Holt, 2009).

Confrey and colleagues suggest that learning progressions assume a progression of cognitive states that move from simple to complex and, while not necessarily linear, the progression is not random, but rather is sequenced and ordered as “expected tendencies” or “likely probabilities” of how learning develops (Confrey and Maloney, 2010). Masters and Forster (1996, p. 1) describe progressions as “a picture of what it means to ‘improve’ in an area of learning.”

Finally, Heritage (2011, p. 3) suggests that learning progressions provide descriptions of “how students’ learning of important concepts and skills in a domain develops from its most rudimentary state through increasingly sophisticated states over a period of schooling.”

Inherent in these views of progressions is the idea of a coherent and continuous pathway along which students move incrementally through states of increasing competence in a domain. Every incremental state builds on and integrates the previous one as students accrue new levels of expertise with each successive step in the progression. It is important to note, however, that while progressions may provide clear descriptions of how learning develops in a domain, they are not developmentally inevitable. Rather, they are dependent on well-mapped curriculum and sound instruction (Duschl, Schweingruber, and Shouse, 2007; Pellegrino, 2011). It should also be noted that the strong hierarchical nature of mathematics makes such progressions absolutely necessary.

As Herman (2006, p. 122) observes, “whether and how children are able to engage in particular learning performances and the sequence in which they are able to do so are very much dependent on previous opportunities to learn.” The benefit of progressions is that they lay out a continuum to guide teaching and learning over time so that student competence in the domain can be advanced coherently and continuously.

Several views of how learning progressions can be developed have been set forth (for example, Alonzo and Steedle, 2008; Anderson, 2008a; Corcoran, Mosher, and Rogat, 2009; Confrey and Maloney, 2010; Hess, 2010; Hess, Kurizaki, and Holt, 2009; Pellegrino, 2011; Smith, Wiser, Anderson, and Krajcik, 2006). Common to these perspectives is the idea that the development of learning progressions is an iterative process. It begins with a hypothesis, informed by what we know about student learning, which undergoes empirical testing and subsequent refinement based on the data. Core Progress for Math was developed according to this iterative model.

“Empirically based learning progressions can visually and verbally articulate a hypothesis, or an anticipated path, of how student learning will typically move toward increased understanding over time with good instruction.”

Hess, Kurizaki, and Holt, 2009

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Evolution of the Core Progress™ learning progression for mathematicsCore Progress began as a scope and sequence and evolved into an empirically validated learning progression. Since its inception in 2007, Core Progress has gone through a continuous cycle of research, review, and revision.

Core Progress was developed to provide a research-based framework for Accelerated Math personalized practice software. Once built, the Core Progress skills were field tested through the STAR Math assessment1. The results were noteworthy and gratifying. The order of skills in Core Progress was highly correlated with the difficulty level of STAR Math Enterprise assessment items. With a strong correlation, the natural next step was to statistically link Core Progress to the STAR Math assessment. As a result, a student’s STAR Math score provides insight into his/her achievement level, as well as skills he/she is ready to learn next. Core Progress is now an integral component of both Accelerated Math Live and STAR Math Enterprise—a true bridge between assessment, instruction, and practice.

Phase I: Scope and sequence ResearchThe origin of the Core Progress learning progression dates back to 2007. It started as a scope and sequence for Accelerated Math Enterprise2, spanning grade 1 to algebra.

To develop the original scope and sequence, Renaissance Learning’s mathematics team relied heavily on research and standards including the National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points (2006), the early work of the National Mathematics Advisory Panel (2008), state and international mathematics standards, and the American Diploma Project Benchmarks (Achieve, Inc., 2007) which provide one of the key foundations for the Common Core State Standards.

ReviewThe scope and sequence was reviewed by several experts including the Education Northwest,3 a research laboratory funded by the U.S. Department of Education; a panel of mathematics teachers; and a panel of prominent mathematicians:

• Dr. Sybilla Beckmann, University of Georgia, (grade 5 review)

• Dr. Richard Bisk, Worcester State College, (grade 6 review)

• Dr. Tom Hogan, University of Scranton (all core objectives)

• Dr. James Milgram, Stanford University (grade 3 core review)

• Dr. Sharif Shakrani, Michigan State University (grade 8, Algebra 1, and Geometry review)

1 Over 9,500 items were field tested between June 2008 and February 2012. See pages 10-11 for more information.

2 Accelerated Math enables differentiated practice in mathematics and provides daily information on every skill students master. Accelerated Math was first released in 1998 with a scope and sequence that reflected the standards and curricula of the time. The second edition of Accelerated Math, developed in 2007 and released in 2008, was built on the Core Progress for Math learning progression, and Accelerated Math Live, developed in 2012-2013, now includes content libraries based on the Core Progress Math built for CCSS learning progression.

3 Formerly the Northwest Regional Educational Laboratory

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RevisionThe initial review focused on Numbers and Operations. Items were analyzed for difficulty, alignment to objectives, accuracy, item quality, and relationship to current pedagogy. Based on the reviews, Renaissance Learning’s mathematics team identified two principal goals: (1) reduce the overlap of objectives between grades, and (2) establish a clear progression of difficulty levels through the grades.

To reduce the grade-level overlap, the team decided to develop a set of core objectives that students must master at each grade in order to advance to the next grade. The NCTM’s Curriculum Focal Points served as the basis for decisions about which topics to include at each grade level. The team also referred to several seminal works to inform their decisions (e.g. Ma, 1999; Milgram and Wu, 2005).

The core objectives closely follow the Mathematics Advisory Panel’s recommendations that curricula focus on mastery of key topics and provide a progression of increasing difficulty, rather than use the spiraling approach of revisiting topics from previous grades. In addition, researchers at Renaissance Learning examined empirical Accelerated Math data that included 66,000 students in 88 schools over three years. The analysis provided real-world insight into the objectives in mathematics students struggle with the most. As a result of this analysis, additional objectives were identified for possible inclusion as core objectives.

When the draft core objectives were complete, Dr. Tom Hogan from the University of Scranton provided expert review. Renaissance Learning incorporated Dr. Hogan’s objective-by-objective feedback and general comments.

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Phase II: Revised scope and sequence, addition of core objectives and prerequisites ResearchWith the core objectives for grades 1 through 8 identified, Renaissance Learning began work on a new and improved scope and sequence. Development of the scope and sequence reflected the second goal identified in the review process: to establish a clear progression of achievable difficulty levels through the grades.

To begin this process, Renaissance Learning’s mathematics team identified core objectives by continually consulting the National Mathematics Advisory Panel (2008), NCTM focal points (2006), the Singapore primary and secondary mathematics standards, and the American Diploma Project Benchmarks (Achieve, Inc., 2007).

After the core objectives were identified and put into skill areas, the team distilled each objective to its most basic elements including concepts, skills, and terminology needed to learn that objective. The team also identified prerequisite objectives, which were then linked together in a progression of associated skills. For example, as illustrated in Table 1, in the skill area Fraction Concepts and Operations, third-grade students are expected to develop an understanding of the meaning of a fraction. Having established this understanding, students move incrementally through successive steps of increasing competence. By fourth grade, students should understand that fraction addition and subtraction for fractions with like denominators is defined by the rule . By fifth grade, students should add and subtract fractions and mixed numbers with unlike denominators. In sixth grade, students should progress incrementally through multiplication and division of fractions. By seventh grade, students should be able to solve multistep problems involving fractions or mixed numbers. Additional examples of cross-grade progressions are in Appendix A.

Review and standards alignmentOnce the core and prerequisite objectives in mathematics were identified, the standards alignment process began. Renaissance Learning uses an alignment process developed with input from Mid-continent Research for Education and Learning (McRel) and Education Northwest.4

This alignment process balances the objective and subjective aspects of alignments to standards. The strategy is documented with definitions and examples for each specific purpose of the alignment, such as practice or assessment, and incorporates an “unpacking process” of separating the standard into skill, action, vocabulary, and context. To standardize the quality of the alignments, Renaissance Learning’s standards team received extensive training, including training in how to calibrate alignment results. After the scope and sequence was complete, it was submitted to Education Northwest for external review.

RevisionAfter the review by Education Northwest was complete, the scope and sequence, including core and prerequisite objectives, was finalized. This new and improved scope and sequence became the basis for the development of the learning progression for mathematics.

Table 1: Cross-grade progression of Fraction Concepts and Operations

Domain: Numbers and Operations

Skill Area: Fraction Concepts and Operations

Grade Skill

3 Students develop an understanding of the meaning of a fraction

4 Students are able to add and subtract fractions with like denominators

5 Students are able to add and subtract fractions and mixed numbers with unlike denominators

6 Students progress incrementally through multiplication and division of fractions

7 Students are solving multistep problems involving fractions or mixed numbers

4 McRel and Education Northwest are part of the Regional Educational Laboratory Program funded by the US Department of Education’s Institute of Education Sciences.

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a c b

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Phase III: Learning progressionThe shift from scope and sequence to learning progression began in Phase II with the identification of core objectives, prerequisite skills, and the progression of associated skills. Now firmly down the learning progression path, Renaissance Learning was ready to go farther.

Two critical events led to the next breakthrough in Renaissance Learning’s learning progression work. First, the Common Core State Standards Initiative (CCSSI) began. Second, the Mathematics Framework for the 2011 National Assessment of Educational Progress was published.

Since 2007, Renaissance Learning had been studying and aligning to the Achieve standards, which are a key foundation of the Common Core State Standards. Then, when the CCSSI began, the standards team closely followed every stage of CCSS development. As a result, aligning the Core Progress learning progression with the CCSS was a natural and familiar process.

The refinements to the learning progression, made as a result of studying the Common Core State Standards, led to a new organizational structure: domains (4), skill areas (23), and core skills (398).

Core Progress for Math has four domains, which form the base of the learning progression: 1) numbers and operations; 2) algebra; 3) geometry and measurement; and 4) data analysis, statistics, and probability. The four domains are represented by the four different colors in Figure 1 (next page).

The skills areas (e.g. whole numbers, place value, symbols and expressions, time, etc.) represent the various skills and concepts students acquire as they progress the development of mathematics. There are 23 skill areas, which can be found in Appendix B.

The core skills and prerequisites act as building blocks, each representing a specific level of competency of a skill or understanding that rests on prior development and that also provides a foundation for the next level of learning. There are 1,326 skills in the Core Progress learning progression. Of these, 398 are core skills, and many of these serve as prerequisites within and across domains, not to be learned in isolation, but as important parts of a single whole. See Appendix C for a complete count of skills per grade, for each domain.

The skill areas and skills were reviewed for coherence and continuity across grade levels to ensure that each contributed to the larger goal of improving student mathematical understanding. In addition to internal analysis, a focus group of teachers across various grade levels was convened. This group provided feedback on how well the progressions align with their own knowledge of students’ development of mathematics. Feedback on Core Progress will continue to be solicited in this way from teachers and administrators.

Since 2007, Renaissance Learning has been studying the Achieve standards, which are a key foundation of the Common Core State Standards. As a result, aligning our learning progression with the CCSS was a natural and familiar process.

The network of interrelated skills and prerequisites in Core Progress is extensive. Many core skills for one grade serve as prerequisite skills for subsequent grades, reflecting the hierarchial nature of mathematics.

7

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Gra

de 4

Gra

de 5

Gra

de 6

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Gra

de 8

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Geometry andMeasurement

Algebra

Data Analysis, Statistics,and Probability


= Approx. 10 skills

= Approx. 5 skills

Core Progress for Math is an empirically validated continuum to guide teaching and learning over time so that student competence in mathematics can be advanced coherently and continuously.

Prerequisite mapping in Core Progress The Core Progress learning progression is an interconnected web of prerequisite skills. Moving toward increased understanding over time requires continually building up and building on a solid foundation of knowledge, concepts, and skills.

One indication of the interrelated network of concepts in Core Progress is the number of skills that build up and build on each other. Specifically, 121 of the 398 core skills in Core Progress serve as prerequisites to others in subsequent grades.

Figure 1: Core Progress for Math

The Core Progress learning progression is an interconnected web of prerequisite skills.

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To illustrate the interrelated nature of the core skills and how they serve as prerequisites to each other, see Table 2. In this example, the seventh grade core skill, subtract integers, serves as a prerequisite for seven core skills spanning four grades and three domains. For an additional example, see Appendix D.

Table 2: Example of how one core skill serves as a prerequisite for many other core skills

Grade Core Skills Domain

Grade 7 WP: Add and subtract using integers Numbers and operations

Grade 7 Evaluate a 2-variable expression, with two or three operations, using integer substitution Algebra

Grade 7 Solve a 1-step linear equation involving integers Algebra

Grade 8 Simplify an algebraic expression by combining like terms Algebra

Algebra 1 Determine the slope of a line given two points on the line Algebra

Algebra 1 Apply the quotient of powers property to monomial algebraic expressions Algebra

Geometry Solve a problem involving the distance formula Geometry

Example of how one core skill serves as a prerequisite for seven skills across four grade levels in three domains.

Subtract integers is a prerequisite for the following:

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Skill 9Determine an equivalent form of a 4-digit whole number using thousands,hundreds, tens, and ones

Skill 2Determine the word form of a 4- or 5-digit whole number

Skill 5Represent a 4-digit wholenumber as thousands, hundreds, tens, and ones

Skill 14Represent a 3-digit number as hundreds, tens, and ones

Skill 1Read a 4- or 5-digit whole number

Skill 2Determine the word form of a whole number to 1,000

Skill 1Read a whole numberto 30

Skill 2Read a whole numberfrom 31 to 100

Skill 3Determine the word formof a whole number to 30

Skill 4Determine the word formof a whole numberfrom 31 to 100

Skill 12Model a number usinghundreds, tens, and ones to 1,000

Skill 13Recognize a number from a model of hundreds,tens, and ones to 1,000

Skill 15Determine the 3-digit number represented as hundreds, tens, and ones

Skill 6Determine the 4-digitwhole number represented in thousands, hundreds, tens, and ones

Grade 1 Grade 2

Grade 2

Grade 3

Grade 3

Skill 1Read a whole numberto 1,000

Skill 11Determine the result ofchanging a digit in a3-digit whole number

Figure 2 offers a different way to think about the deeply interrelated nature of Core Progress for Math. This figure shows a true mapping of skills, illustrating how skills build on each other, serving as prerequisites to one another.

As Figure 2 illustrates, Core Progress is an interconnected web of prerequisite skills. It’s important to recognize that a learning progression as comprehensive and interrelated as Core Progress for Math takes years to develop and could only come to fruition through a continuous process of research, expert review, and iterative revision. Figure 2: Prerequisite map of place value

Core Progress is a true map of skills: new learning is built on previous, foundational understandings. The arrows identify a typical developmental path within the learning progression. In the Common Core Standards the student is expected to see the expanded form as the “name” of a number.

A comprehensive and interrelated learning progression like Core Progress takes years to develop through a continuous process of research, expert review, and iterative revision.

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5 New assessment items aligned to the Common Core will continue to be tested on an ongoing basis.

Phase IV: Empirical analysis of Core Progress™ MethodIn 2008, Renaissance Learning began Phase IV of Core Progress development: empirical analysis. The order of skills in the learning progression was re-examined empirically through a calibration process used to analyze assessment items. The purpose was to compare the empirically observed order of skills (i.e. where skills fall on an assessment scale) to a pedagogically determined ordering of skills (i.e. the most productive order of skills for teaching, mastering, and learning a concept).

Between June 2008 and February 2012 over 9,500 items were field tested, calibrated, and analyzed using a process called dynamic calibration.5 In this process, a small number of experimental items (one to three) were added onto each student’s STAR Math Enterprise assessment nationwide. Response data from thousands of students were collected for each of these experimental items, and the items were then calibrated by fitting a logistic regression model (the Rasch model) to the relationship between scores on each item and a student’s Rasch ability scores on STAR Math. The result was to calibrate the difficulty of each new item on the same Rasch scale that is used for adaptive item selection in STAR Math.

Following the calibration process, the average of the calibrated Rasch response functions for each of the items assessing a skill was determined; the average is a “skill characteristic curve.” For each skill, a skill difficulty parameter was then calculated: the point on the Rasch scale at which a student of the same Rasch ability would have an expected percent correct of 70 if tested on all of the items that measure the skill. This parameter is designated SD70, or Scaled Difficulty 70. Finally, the relationship between the empirically calibrated SD70 skill difficulties and the sequential order of STAR Math skills in the learning progression was evaluated, as a means of validating the Core Progress for Math learning progression. ResultsCore Progress for Math includes 1,326 skills. Figure 3 (next page) shows 626 of the skills plotted by their difficulty level on the STAR Math Enterprise assessment scale and their instructional order according to the learning progression.

Each datapoint in Figure 3 represents a skill on the learning progression. The difficulty value (vertical scale) of each skill is derived from the calibrated difficulty of the test items from STAR Math that assess that skill. There are several assessment items per skill, called an item-set.

Best-fitting logarithmic functions relating the SD70 value of each item-set to instructional order were calculated for each of the four domains of STAR Math. These are plotted in Figure 3 as color-coded curved lines superimposed on the scatter plot. For each domain, the parameters of the fitted logarithmic function are displayed, along with the correlation between the skill difficulty parameters and instructional order. These correlations range from approximately 0.90 (for the Data Analysis, Statistics and Probability domain) to 0.94 for the Algebra domain. These correlations may be thought of as measures of the validity of the Core Progress learning progression for describing the developmental sequence of the hundreds of skills that make up the STAR Math domains.

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0 2 4 6 8 10 12200

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Data Analysis, Statistics, and Probability


Algebra


Core Progress Skill DifficultyS

cale

d D

iffi

cult

y 70

Grade Level Order

y = 240.13Ln(x) + 334.27r = 0.8960

y = 271.68Ln(x) + 313.65r = 0.9104

y = 251.45Ln(x) + 333.35r = 0.9440

y = 253.5Ln(x) + 324.85r = 0.9059

The high correlation between STAR Math Enterprise and Core Progress provides empirical evidence of the bridge between assessment and instruction.

Figure 3: Correlation of STAR Math Enterprise to Core Progress

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Mapping Core Progress™ to the Common Core State Standards The Common Core State Standards represent a clear step toward providing a more coherent pathway to meeting educational goals than many prior state standards. At the same time, they do not describe a fully formed pathway along which students are expected to progress. The next step, clarified and made possible by the CCSS, is the development of such fully formed learning progressions.

The concepts, skills, and understandings in Core Progress align with the Common Core State Standards, and also provide the intermediate steps and prerequisite skills necessary to reach the levels of expertise identified through the standards. Core Progress begins with early numeracy and progresses to the minimal level of ability in mathematics required to be college and career ready.

Our process of analyzing and mapping the Common Core State Standards began before the final draft of the standards was released. As the movement to create the Common Core State Standards was getting underway, Renaissance Learning was already reviewing and learning from the work of independent educational organizations such as Achieve. Then, as the Common Core State Standards entered into various stages of completion, Renaissance Learning carefully monitored them in draft form and provided public commentary. Core Progress was developed with a deep understanding of the CCSS.

Table 3 illustrates the Core Progress skills needed to master the Common Core State Standard CC A-REI.3: “Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.” For another example, see Appendix E.

Table 3: Example of Core Progress alignment to Common Core State Standard

Grade Skill Domain



Grade 8 Solve a 1-step equation involving rational numbers Algebra

Grade 8 Solve a 2-step equation involving rational numbers Algebra

Grade 8 Solve a 2-step linear inequality in one variable Algebra

Algebra 1 Solve a 1-variable linear equation with the variable on both sides Algebra

Algebra 1 Solve a 1-variable linear inequality with the variable on one side Algebra

Algebra 1 Solve a 1-variable linear inequality with the variable on both sides Algebra

Algebra 1 Solve a 1-variable compound inequality Algebra

The Common Core State Standards set the bar. Core Progress provides the prerequisite and intermediary steps for achieving the standards.

Students work through each incremental skill involving linear equations and inequalities, developing and expanding these skills at each grade level.

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Phase V: Building a new learning progression specifically for the Common CoreThe Core Progress for Math learning progression was originally developed to provide a research-based framework for Accelerated Math personalized practice. When development of the learning progression began in 2007, the content team at Renaissance Learning drew heavily on the American Diploma Project Benchmarks, which provided the foundation for the Common Core State Standards (CCSS). Since then, the new standards were published and have been adopted by the majority of states. The need for a learning progression built specifically for the CCSS was recognized, and the content team embarked on this project. In July 2013, the Core Progress Learning Progression for Math - Built for the Common Core State Standards was released. It included incremental steps of learning that fulfill the intent and specifics of the standards, culminating in college and career readiness.

To create a learning progression built on the CCSS, the content team started with an analysis of each set of standards by grade. They identified the intent of each standard and the inherent skills, as well as key terminology used to describe the standard. Developers also immersed themselves in the literature and resources regarding the Common Core to inform them as they worked to interpret the standards. The team then evaluated how states and relevant consortia implemented the standards.

The organization of the learning progression is identical to the framework of the standards. Grades K-8 have 11 domains and 49 skill areas. Grades 9-11 have 21 domains and 44 skill areas. For a list of the skill areas within each domain, see Appendices F and G.

With an overall structure in place, the content team began the process of identifying skills for each standard—within each grade and from grade to grade. Many of the skill statements from the original Core Progress for Math were perfect matches to the standards in the Common Core. These skill statements have been quantitatively analyzed in the calibration process (see Phase IV—Empirical analysis of Core Progress) so they were known to be accurate grade-level indicators of student learning. Figure 4 shows a sampling of the skills plotted by their difficulty level on the STAR Math Enterprise assessment scale and their instructional order according to the Core Progress Math built for CCSS learning progression. As new skills were identified, they were written to meet the specific needs of the CCSS. Throughout the process of developing the CCSS learning progression, the content team verified that skills within a grade were presented in a teachable order. To see how skill areas progress across grades, see Appendix H.

Figure 4: Correlation of STAR Math Enterprise to Core Progress Math built for CCSS

y = -4.780x2 + 111.684x + 299.407 R² = 0.850

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Core Progress Math built for CCSS: Skill Difficulty (June'13)

Numbers/Operations

Algebra

Functions

Geometry

Data/Statistics/Probability

The correlation between STAR Math Enterprise and Core Progress Math built for CCSS provides empirical evidence of the bridge between assessment and instruction.

14

For each grade, the team cross-referenced several CCSS guides. The Common Core State Standards for Mathematics (CCSSM) was the deciding factor for the placement and pace of skill development. Skills from adopter states and other states such as Texas and Minnesota were added when they enhanced and clarified the meaning of the CCSSM. Before an augmented skill was included in the learning progression, the team verified the skill did not contradict the Common Core. The K-8 and High School Publishers’ Criteria were used to identify clusters, which were tagged as major, supportive, or additional. The K-8 Publishers’ Criteria for the Common Core State Standards for Mathematics specifies clusters as indicators of algebra readiness—a crucial level needed for high school success. This information was compared against core skills and their prerequisites. If a prerequisite was missing, it was added into the learning progression. See Appendix I for a list of core skills per grade and Appendix J for an example of how one core skill serves as a prerequisite for many other skills. The Common Core typically describes the ultimate way students are expected to use and understand key concepts, but the detailed steps needed for them to attain this level are usually not discussed there. This is where the true value of a learning progression becomes apparent.

Once a grade band (K-2, 3-5, 6-8) was completed, it was reviewed holistically to ensure the difficulty of skills made sense across the grades. Skills were reviewed by domain to verify the progression was accurate. The team also confirmed that the skills met the conditions of the K-8 Publishers’ Criteria for the Common Core State Standards for Mathematics.

As with the original Core Progress for Math, Renaissance Learning worked with external experts throughout the development of the learning progression. These experts provided guidance and suggestions for each grade band in the Core Progress Math built for CCSS. They considered the adequacy of each skill in addressing the standards, the progression of skills from grade to grade, and the language used to describe skills. The expert reviewers are:

• Dr. Amanda VanDerHeyden (Kindergarten-grade 2)

• Dr. Karin Hess (grades 3-5)

• Dr. James Milgram (grades 6-8 plus algebra I, geometry, and algebra II)

Core Progress Math built for CCSS embodies the Common Core and provides a teachable order of skills grade by grade. Each skill was reviewed from the perspective and stated philosophy of the Common Core. The progression includes skills that may not be explicitly stated in the CCSS but are considered key logical steps to student learning. In following the skills progression in Core Progress Math built for CCSS, students will be on the path to achieving the common goal of attaining college and career readiness.

Each skill was reviewed from the perspective and stated philosophy of the Common Core. The progression includes skills that may not be explicitly stated in the CCSS but are considered key logical steps to student learning.

15

Core Progress™: an integral component of Accelerated Math Live™ and STAR Math Enterprise™

The more comprehensive a learning progression is, the more ways it can be used. Because of the depth and breadth of Core Progress, it now serves as an integral component for STAR Math Enterprise and Accelerated Math Live. As a result, there is now a true bridge between assessment (STAR Math Enterprise), instruction (Core Progress for Math or Core Progress Math built for CCSS learning progressions), and practice (Accelerated Math Live).

Core Progress was developed to provide a research-based step-by-step framework for Accelerated Math personalized practice software. Once built, the Core Progress skills were translated into assessment items and field tested via STAR Math Enterprise. As illustrated in Figure 3 (p.11), the results were gratifying. The order of skills in the learning progression was highly correlated with the difficulty level of the skills-turned-STAR Math Enterprise items.

With a strong correlation, the natural next step was to statistically link Core Progress to the STAR Math assessment. As a result, students’ STAR Math Enterprise score now provides insight into their achievement level, as well as skills they are ready to develop next.

Accelerated Math Live Accelerated Math Live software enables monitored, differentiated practice in mathematics. It provides daily information to teachers about student progress toward mastery, skill by skill. Accelerated Math is recognized as a “mastery measure” by the U.S. Department of Education6 (U.S. DOE). A mastery measure tracks “a student’s successive mastery of a hierarchy of objectives” (NCRTI, 2010). Accelerated Math met the U.S. DOE’s strict definition of “mastery measure” because of the instructional hierarchy provided by Core Progress.

Accelerated Math Live generates personalized practice assignments for each student based on the skills that they are ready to learn next and/or need to review. The order in which Accelerated Math the objectives are arranged and typically assigned to students is based on the Core Progress learning progression. If a skill needs to be reviewed, the instructor can use the learning progression to quickly locate prerequisite skills. The prerequisite skills the student needs to review, in order to successfully complete the assignment, can then be associated with objectives in Accelerated Math Live to fill any gaps in understanding the student might have. Completing the review of the prerequisite skills can then lead to success with completing the assignment. In this way, by utilizing the learning progression, Accelerated Math Live personally tailors daily practice in mathematics to meet the student’s immediate learning needs.

Figure 5 (next page) shows an Accelerated Math Live report for hypothetical student, Derek Adams. He has mastered the first 13 skills in the Core Progress learning progression for his grade (numbers 1-13) and is working on the next two (numbers 14-15). Typically, a teacher will run this report weekly to monitor each student’s progress and pace.

Because of the depth and breadth of Core Progress, it now serves as an integral component for STAR Math Enterprise and Accelerated Math Live.

6 The U.S. Department of Education’s National Center on Response to Intervention (NCRTI) conducts rigorous, research-based reviews of assessments and interventions. Accelerated Math met NCRTI’s highest standards.

16

Figure 5: Accelerated Math Live Student Record Report

Student Record Report Printed Thursday, October 6, 2011 12:22:20 PM

School: East Elementary School Reporting Period: 09/01/2011 - 10/06/2011 (2011-2012)

˜Designates a core objective. Core objectives identify the most critical objectives to learn at each grade level.

Adams, Derek ID: DADAM Class: Grade 2 Grade: 2 Teacher: DeMarco, C.

Active Objectives

Objectives

Library Objective

Code Ready To Test

Test Completed

Average Percent Correct

Practice Exercise Regular

Test Diagnostic

Test 14. ˜ Represent a 3-digit number as hundreds, tens, and ones DMG2-014 67 4 / 6 - - - 15. ˜ Determine the 3-digit number represented as hundreds, tens, and ones DMG2-015 67 4 / 6 - - -

Summary: 2 Objectives 67% - - -

Mastered Objectives

Objectives

Library Objective

Code Date

Mastered

Average Percent Correct

Practice Exercise Regular

Test Diagnostic

Test Review

1. ˜ Read a whole number to 1,000 DMG2-001 09/06/11 100 6 / 6 - 80 4 / 5 - - 2. ˜ Determine the word form of a whole number to 1,000 DMG2-002 09/08/11 75 9 / 12 - 70 7 / 10 80 4 / 5 - 3. ˜ Complete a skip pattern starting from a multiple of 2, 5, or 10 DMG2-003 09/09/11 83 5 / 6 - 80 4 / 5 - - 4. ˜ Complete a skip pattern of 2, 5, or 10 starting from any number DMG2-004 09/13/11 100 6 / 6 - 100 5 / 5 - -5. ˜ Count on by 100s from any number DMG2-005 09/15/11 83 5 / 6 - 70 7 / 10 80 4 / 5 -6. Identify odd and even numbers between 100 and 1,000 DMG2-006 09/19/11 83 5 / 6 - 80 4 / 5 - -7. Solve problems involving the concept of odd and even numbers DMG2-007 09/21/11 75 9 / 12 - 60 6 / 10 80 4 / 5 - 8. Answer a question using an ordinal number up to "twentieth" DMG2-008 09/22/11 83 5 / 6 - 80 4 / 5 - - 9. Determine the value of a digit in a 3-digit number DMG2-009 09/26/11 100 6 / 6 - 80 4 / 5 - -

10. Determine which digit is in a specified place in a 3-digit whole number DMG2-010 09/27/11 83 10 / 12 - 100 5 / 5 - -11. Determine the result of changing a digit in a 3-digit whole number DMG2-011 09/28/11 83 10 / 12 - 80 4 / 5 - -12. Model a number using hundreds, tens, and ones to 1,000 DMG2-012 10/03/11 91 10 / 11 - 100 5 / 5 - -13. Recognize a number from a model of hundreds, tens, and ones to 1,000 DMG2-013 10/05/11 83 5 / 6 - 80 8 / 10 80 4 / 5 -

Summary: 13 Objectives 85% - 79% 80% -

1 of 1

The Student Record Report in Accelerated Math Live enables teachers to monitor students’ mastery of successive skills from the Core Progress learning progression.

17

STAR Math EnterpriseIn the landmark report, Knowing What Students Know, the authors establish learning progressions as the foundation for assessment. Specifically, the authors state, “models of student progression in learning should underlie the assessment system, and tests should be designed to provide information that maps back to the progression” (Pellegrino, Chudowsky, and Glaser, 2001, p. 256).

In a 2011 paper, Pellegrino, one of the report’s authors, suggested that learning progressions can guide the specification of learning performances, which in turn can guide the development of tasks that enable educators to infer students’ level of competence for the major constructs that are the target of instruction and assessment. If assessments are developed from a progression, they can provide a continuous source of evidence as student learning evolves toward increasingly sophisticated levels of understanding and skills.

Because of the strong statistical correlation between STAR Math Enterprise and Core Progress, students’ scaled scores (from STAR Math Enterprise) are their entry point into the learning progression, enabling research-based inferences about which skills they have likely already developed, which skills are ready to be developed, which skills and understandings need remediation, and which skills will likely develop soon. Think of a student’s STAR Math Enterprise score as the entry point into the learning progression. (See Figure 6)

Figure 6: STAR Math Enterprise provides your entry point into Core Progress

College &Career Ready

Early Numeracy

1400

0

SkillsMastered

SkillsReady to Learn

SkillsRemainingto Learn

741

C

Mr. Steward is buying a house. He can spend no more than 31% of his income on monthly house payments. If he earns$4,600 per month, what is the largest monthly housepayment he can make?

$148

$1,480

$1,426

$1,326

A

B

C

D

Instructional Planning Reportfor Jasmine MajorPrinted Thursday, September 5, 2013 4:15:12 PM

1 of 2

Class: 5th Hour MathTeacher: Mrs. T. Williams

Grade: 7

School: Pine Hill Middle School

STAR Math Test ResultsCurrent SS (Scaled Score): 741 Test Date: 9/5/2013Projected SS for 06/16/14: 785 Based on research, 50% of students at this student's level will achieve this much growth

Algebra Readiness: Jasminehas not yet met the algebra readiness grade level expectations.

Jasmine's Current Performance

CurrentCurrent

School Benchmarks

Projected

Scaled Score550 600 650 700 750 800 850

Projected

ûUrgent Intervention ûIntervention ûOn Watch ûAt/Above BenchmarkJasmine's recent STAR Math scaled score(s) suggests these skills from Core Progress™ learning progressions would be

challenging, but not too difficult for her. Combine this information with your own knowledge of the student and use your

professional judgment when designing an instructional program. Use the Core Progress learning progressions to see how

these skills fit within the larger context of the progression.

Skills to Learn

K-8

GeometryGR7

Students draw and construct geometrical figures and describe the relationship between figures with different attributes.

They solve real-world and mathematical problems involving angle measure and the area of 2-dimensional figures. They

solve real-world and mathematical problems involving the surface area and volume of 3-dimensional figures.

Relate volume found using unit cubes to multiplication of edge lengths in a right rectangular prism

7Find the volume of a right rectangular prism with fractional edge lengths using formulas

»7WP: Find the volume of a right rectangular prism with fractional edge lengths using formulas

7Plot coordinates to form a polygon on the coordinate plane

7Find a side length of a polygon on the coordinate plane

7

Ratios and Proportional Relationships7

Students analyze proportional relationships in real-world and mathematical problems. They write equations to represent

a proportional relationship. They solve multi-step real-world and mathematical ratio and percent problems.

Understand the concept of a unit rate7

Determine a unit rate»7

Identify an input-output table that contains values for a given ratio

7Find missing values in a ratio table

7Graph in the coordinate plane the values of a ratio table

7WP: Use tables to compare ratios

7WP: Solve a unit rate problem

»7

The Number System7

» Designates a core skill. Core skills identify the most critical skills to learn at each grade level.aThis student was given extra time to complete the test.

18

Core Progress learning progressions help to inform instruction within STAR Math Enterprise. Educators access a learning progression through Instructional Planning Reports (which are generated by STAR Math Enterprise in real time), or the STAR Record Book.

Instructional Planning Reports Skills-based information for students is provided by the Instructional Planning Reports produced instantly by STAR Math Enterprise after a student completes a test. These reports use the Core Progress learning progression to identify the range of skills students are ready to develop next. The Student Instructional Planning Report also shows an individual student’s current performance in relation to pre-selected benchmarks, so teachers can keep an eye on whether a student is on track to meet state or locally established proficiency goals. (See Figure 7)

STAR Record Book The STAR Record Book highlights suggested skills from Core Progress that a student is ready to learn. It is a tool that bridges assessment and instruction. The student’s STAR scaled score is placed on the learning progression and suggests skills that are appropriate to focus on. To further expand understanding of skills and support instruction, educators will find Sample Items and Worked Examples associated with skills. Each skill includes prerequisite skill mapping, ELL support, and content-area vocabulary. In addition, the Record Book includes Depth of Knowledge 3 (DOK3) items and performance tasks. These resources are designed to help teachers probe more deeply into the skills and knowledge of each student. (See Figure 8)

Instructional Planning Reportfor Jasmine MajorPrinted Thursday, September 5, 2013 4:15:12 PM

1 of 2

Class: 5th Hour MathTeacher: Mrs. T. Williams

Grade: 7

School: Pine Hill Middle School

STAR Math Test ResultsCurrent SS (Scaled Score): 741 Test Date: 9/5/2013Projected SS for 06/16/14: 785 Based on research, 50% of students at this student's level will achieve this much growth

Algebra Readiness: Jasminehas not yet met the algebra readiness grade level expectations.

Jasmine's Current Performance

CurrentCurrent

School Benchmarks

Projected

Scaled Score550 600 650 700 750 800 850

Projected

ûUrgent Intervention ûIntervention ûOn Watch ûAt/Above BenchmarkJasmine's recent STAR Math scaled score(s) suggests these skills from Core Progress™ learning progressions would be

challenging, but not too difficult for her. Combine this information with your own knowledge of the student and use your

professional judgment when designing an instructional program. Use the Core Progress learning progressions to see how

these skills fit within the larger context of the progression.

Skills to Learn

K-8

GeometryGR7

Students draw and construct geometrical figures and describe the relationship between figures with different attributes.

They solve real-world and mathematical problems involving angle measure and the area of 2-dimensional figures. They

solve real-world and mathematical problems involving the surface area and volume of 3-dimensional figures.

Relate volume found using unit cubes to multiplication of edge lengths in a right rectangular prism

7Find the volume of a right rectangular prism with fractional edge lengths using formulas

»7WP: Find the volume of a right rectangular prism with fractional edge lengths using formulas

7Plot coordinates to form a polygon on the coordinate plane

7Find a side length of a polygon on the coordinate plane

7

Ratios and Proportional Relationships7

Students analyze proportional relationships in real-world and mathematical problems. They write equations to represent

a proportional relationship. They solve multi-step real-world and mathematical ratio and percent problems.

Understand the concept of a unit rate7

Determine a unit rate»7

Identify an input-output table that contains values for a given ratio

7Find missing values in a ratio table

7Graph in the coordinate plane the values of a ratio table

7WP: Use tables to compare ratios

7WP: Solve a unit rate problem

»7

The Number System7

» Designates a core skill. Core skills identify the most critical skills to learn at each grade level.aThis student was given extra time to complete the test.

Figure 7: Example of Instructional Planning Report generated by STAR Math Enterprise

Figure 8: STAR Record Book

19

ConclusionThe goal of school districts is to ensure that all students in all schools be fully prepared for college or career by the time they graduate from high school. The benefit of learning progressions is that they lay out a pathway to guide teaching and learning over time so that student competence in the domain can be advanced coherently and continuously. Core Progress for Math and Core Progress Learning Progression for Math - Built for the Common Core State Standards describe fully formed progressions of learning within the domain of mathematics, including the intermediate steps not evident in state standards and the Common Core. They help educators locate where students are on their pathway, not only pointing in the right direction, but also providing tangible and achievable next steps for getting there. Together, STAR Math Enterprise assessments, Core Progress learning progressions for mathematics, and Accelerated Math Live content help educators achieve the intent and spirit of the standards their state has adopted, while propelling their students toward college and career readiness.

The benefit of learning progressions is that they lay out a pathway to guide teaching and learning over time so that student competence in the domain can be advanced coherently and continuously.

20

Appendix A: Examples of skill progressions across grade levels


Skill Area: Whole Numbers: Place Value

Grade Skill

1 • Write and identify a 2-digit number from a model of tens and ones• Determine a value of a digit in a 2-digit number

2 • Write and identify a 3-digit number as hundreds, tens, and ones• Recognize equivalent forms of a 3-digit number using hundreds, tens, and ones

3• Write and identify a 4- or 5-digit number as thousands, hundreds, tens, and ones• Recognize equivalent forms of a 4-digit number using thousands, hundreds, tens, and ones• Write and identify a 4- or 5-digit number in expanded form

4 • Round a 4- to 6-digit number to a specified place


Skill Area: Decimal Concepts and Operations

Grade Skill

4

• Write and identify a decimal number from a model of tenths and hundredths• Represent a decimal number to tenths by a point on a number line• Recognize an equivalent form of a decimal number and a fraction• Compare and order decimal numbers through hundredths • Round a decimal to a specified place through hundredths

5

• Compare and order decimal numbers of differing places to thousandths• Add and subtract decimal numbers to differing places to thousandths• Solve word problems involving addition and subtraction of decimal numbers through thousandths• Estimate decimal sums and differences through thousandths.

6

• Divide whole numbers resulting in a decimal quotient through thousandths• Recognize and represent decimal numbers in expanded form using powers of ten• Multiply a decimal number through thousandths by a whole number• Divide a decimal number by 10, 100, or 1,000• Divide a decimal number through thousandths by a whole number• Divide a whole number by a decimal number to tenths• Multiply and divide decimal numbers through thousandths • Solve word problems involving multiplication and division of decimal numbers through thousandths• Estimate decimal products and quotients• Compare and order numbers in decimal and fraction forms

7 • Solve a multi-step word problem involving decimal numbers

8 • Convert between standard form and scientific notation of decimal numbers

CORE PROGRESS FOR MATH LEARNING PROGRESSION

21

Appendix B: Core Progress™ for Math includes four domains and 23 skill areas

Domain Skill Area


• Whole Numbers: Counting, Comparing, and Ordering• Whole Numbers: Place Value • Patterns, Relations, and Functions• Whole Numbers: Addition and Subtraction• Money• Whole Numbers: Multiplication and Division• Fraction Concepts and Operations• Decimal Concepts and Operations• Percents, Ratios, and Proportions• Integers• Powers and Roots

Algebra

• Algebra: Variable Equations and Expressions• Symbols and Expressions • Functions• Linear Equations• Nonlinear Equations• Algebra of Polynomials• Quadratic Equations


• Measurement• Time • Geometry: 2-Dimensional• Geometry: 3-Dimensional

Data Analysis, Statistics, and Probability • Data Representation and Analysis


22

Appendix C: Core skills per grade, per domain

Grade Domain Core skills (398) Total skills (1,326)

Early Numeracy

Numbers and Operations 0 58

Algebra 0 5

Geometry and Measurement 0 9

Totals 0 72

Grade 1


Algebra 4 10


Data Analysis, Statistics, and Probability 0 14

Totals 38 99

Grade 2


Algebra 3 11



Totals 34 98

Grade 3


Algebra 2 11



Totals 30 114

Grade 4


Algebra 5 9



Totals 48 143

Grade 5


Algebra 4 14



Totals 43 167

Grade 6


Algebra 7 16



Totals 51 157

Grade 7


Algebra 7 23



Totals 45 138

Grade 8


Algebra 17 25



Totals 35 99

Algebra 1Algebra 43 127

Total 43 127

GeometryGeometry and Measurement 31 112

Total 31 112


23

Appendix D: Example of how one core skill serves as a prerequisite for many other core skills

Grade Core Skill Domain

Grade 1 Count objects grouped in tens and ones (grade 1) Numbers and operations

Grade 1 Tell time to the half hour (grade 1) Geometry and measurement

Grade 2 Complete a skip pattern starting from a multiple of 2, 5, or 10 Numbers and operations

Grade 2 Complete a skip pattern of 2, 5, or 10 starting from any number Numbers and operations

Grade 2 Count on by 100s from any number Numbers and operations

Grade 2 Use a pictograph to represent data (1 symbol = more than 1 object)

Data analysis, statistics and probability

Grade 3 Tell time to the minute Geometry and measurement

Grade 4 Answer a question using information from a line graph Data analysis, statistics and probability

Grade 4 Use a double-bar graph to represent data Data analysis, statistics and probability

Grade 4 Answer a question using information from a double-bar graph

Data analysis, statistics and probability

Example of how one core skill serves as a prerequisite for 10 skills across four grade levels in three domains.

Count by 5s or 10s to 100 starting from a multiple of 5 or 10, respectively is a prerequisite for the following:


24

Appendix E: Common Core State Standards and Core Progress™

Grade Skill Domain

Grade 4 WP: Solve a 2-step problem involving addition and/or subtraction of multi-digit whole numbers Numbers and operations

Grade 4 WP: Solve a 2-step whole number problem using more than 1 operation Numbers and operations

Grade 5 WP: Solve a 2-step problem involving whole numbers Numbers and operations

Grade 6 WP: Solve a multi-step problem involving whole numbers Numbers and operations

Grade 6 WP: Solve a 2-step problem involving fractions Numbers and operations

Grade 6 WP: Solve a 2-step problem involving decimals Numbers and operations

Grade 7 WP: Solve a multi-step problem involving decimals Numbers and operations

Grade 7 WP: Solve a multi-step problem involving fractions or mixed numbers Numbers and operations

Grade 7 WP: Estimate the result of dividing or multiplying a whole number by a fraction Numbers and operations

The Common Core State Standards set the bar. Core Progress provides the prerequisites and intermediary steps for achieving the standard.

CCSS performance standard Grade 7, Expressions and Equations Domain, Standard 3 (7.EE.3) “Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form” is mapped to the following Core Progress skills:


25

Appendix F: Organization of Skill Areas within the 11 Domains for K-8

Domain Skill Area

Counting and Cardinality Whole Numbers: Counting, Comparing, and Ordering

Operations and Algebraic Thinking

Algebraic ThinkingEvaluate Numerical ExpressionsWhole Numbers: Addition and SubtractionWhole Numbers: Counting, Comparing, and OrderingWhole Numbers: Multiplication and Division

Number and Operations in Base Ten

Decimal Concepts and OperationsPowers, Roots, and RadicalsWhole Numbers: Addition and SubtractionWhole Numbers: Counting, Comparing, and OrderingWhole Numbers: Multiplication and DivisionWhole Numbers: Place Value

Number and Operations — Fractions

Decimal Concepts and OperationsFraction Concepts and Operations

Ratios and Proportional Relationships Percents, Ratios, and Proportions

The Number System

Coordinate GeometryDecimal Concepts and OperationsFraction Concepts and OperationsIntegersWhole Numbers: Multiplication and Division

Expressions and Equations

Evaluate and Use Variable ExpressionsEvaluate Numerical ExpressionsLinear Equations and InequalitiesPowers, Roots, and RadicalsQuadratic and Nonlinear Equations and InequalitiesSystems of Equations and Inequalities

Functions Relations and Functions

Geometry

Angles, Segments, and LinesCongruence and SimilarityCoordinate GeometryFraction Concepts and OperationsGeometry: Three-Dimensional Shapes and AttributesGeometry: Two-Dimensional Shapes and AttributesPerimeter, Circumference, and AreaRight Triangles and TrigonometrySurface Area and VolumeTransformations

Measurement and Data

Angles, Segments, and LinesData Representation and AnalysisGeometry: Two-Dimensional Shapes and AttributesMeasurementMoneyPerimeter, Circumference, and AreaSurface Area and VolumeTimeWhole Numbers: Addition and SubtractionWhole Numbers: Counting, Comparing, and Ordering

Statistics and ProbabilityCombinatorics and ProbabilityData Representation and Analysis

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDS

26

Appendix G: Organization of Skill Areas within the 21 Domains for high school

Domain Skill Area

The Real Number SystemFraction Concepts and OperationsPowers, Roots, and Radicals

Quantities* Data Representation and Analysis

Seeing Structure in Expressions

Algebra of PolynomialsLinear Equations and InequalitiesQuadratic and Nonlinear Equations and InequalitiesRelations and Functions

Arithmetic with Polynomials and Rational Expressions Algebra of Polynomials

Creating Equations* Linear Equations and Inequalities

Reasoning with Equations and Inequalities

Linear Equations and InequalitiesQuadratic and Nonlinear Equations and InequalitiesRelations and FunctionsSystems of Equations and Inequalities

Interpreting Functions Relations and Functions

Building Functions Relations and Functions

Linear, Quadratic, and Exponential Models*

Linear Equations and InequalitiesQuadratic and Nonlinear Equations and Inequalities

The Complex Number SystemAlgebra of PolynomialsComplex Numbers

Trigonometric Functions Right Triangles and Trigonometry

Congruence

Angles, Segments, and LinesCongruence and SimilarityGeometry: Two-Dimensional Shapes and AttributesPolygons and CirclesTransformations

Similarity, Right Triangles, and Trigonometry

Congruence and SimilarityRight Triangles and TrigonometryTransformations

Circles Polygons and Circles

Expressing Geometric Properties with Equations

Coordinate GeometryPolygons and Circles

Geometric Measure and Dimension

Geometry: Three-Dimensional Shapes and AttributesPerimeter, Circumference, and AreaSurface Area and Volume

Modeling with Geometry

Coordinate GeometryGeometry: Three-Dimensional Shapes and AttributesPerimeter, Circumference, and AreaPolygons and CirclesRight Triangles and TrigonometrySurface Area and Volume

Conditional Probability and the Rules of Probability Combinatorics and Probability

Using Probability to Make Decisions Combinatorics and Probability

* Modeling Standards: Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice.


27

Domain Skill Area

Interpreting Categorical and Quantitative Data Data Representation and Analysis

Making Inferences and Justifying Conclusions Data Representation and Analysis


28

Appendix H: Examples of skill progressions across grade levels, Whole Numbers: Place Value

Standard Skill

K Number and Operations in Base Ten – Work with numbers 11-19 to gain foundations for place value

CC K.NBT.1 Decompose a number from 11 to 19 into a group of ten ones and some further ones using objects

CC K.NBT.1 Decompose a number from 11 to 19 into a group of ten ones and some further ones using pictures

CC K.NBT.1 Compose a number from 11 to 19 from ten ones and some further ones using objects

CC K.NBT.1 Compose a number from 11 to 19 from ten ones and some further ones using pictures

CC K.NBT.1 Understand that the numbers 11 to 19 are composed of 10 ones and some ones

Grade 1 Number and Operations in Base Ten – Understand place value

CC 1.NBT.2.a Understand that 10 represents a collection of ten ones

C 1.NBT.2.b Understand that the numbers 11 to 19 are composed of a ten and some ones

C 1.NBT.2.c Understand that a number ending in zero from 10 to 90 is a group of tens and no ones

C 1.NBT.2 Understand that the digits of a 2-digit number represent amounts of tens and ones

Grade 2 Number and Operations in Base Ten – Understand place value

CC 2.NBT.1.a Understand that 100 represents a collection of 10 tens

CC 2.NBT.1.b Understand that a number ending in two zeros from 100 to 900 is a group of hundreds and zero tens and zero ones

CC 2.NBT.1 Understand that the digits of a 3-digit number represent amounts of hundreds, tens, and ones


29

Appendix I: Core skills per grade, per domain

Grade Domain Core skills (255) Total Skills (1028)

Kindergarten

Counting and Cardinality 6 18

Operations and Algebraic Thinking 6 28

Number and Operations in Base Ten 1 5

Measurement and Data 1 8

Geometry 1 12

Totals 15 71

Grade 1




Geometry 1 11

Totals 19 71

Grade 2




Geometry 1 9

Totals 15 67

Grade 3



Number and Operations — Fractions 4 15


Geometry 1 6

Totals 19 76

Grade 4





Geometry 2 7

Totals 20 75

Grade 5





Geometry 1 7

Totals 20 75

Grade 6

Ratios and Proportional Relationships 4 14

The Number System 4 30

Expressions and Equations 5 23

Geometry 3 12

Statistics and Probability 2 12

Totals 18 91

Grade 7

Ratios and Proportional Relationships 5 9



Geometry 4 13


Totals 25 80


30

Grade Domain Core skills (255) Total Skills (1028)

Grade 8



Functions 5 11

Geometry 7 23


Totals 21 77

Grade 9

The Real Number System 2 5

Quantities* 3 5

Seeing Structure in Expressions 3 9

Arithmetic with Polynomials and Rational Expressions 0 3

Creating Equations* 3 12

Reasoning with Equations and Inequalities 6 19

Interpreting Functions 6 26

Building Functions 4 17

Linear, Quadratic, and Exponential Models* 2 9

Interpreting Categorical and Quantitative Data 2 17

Totals 31 122

Grade 10

Congruence 3 41

Similarity, Right Triangles, and Trigonometry 9 25

Circles 3 12

Expressing Geometric Properties with Equations 5 11

Geometric Measure and Dimension 1 8

Modeling with Geometry 0 6

Conditional Probability and the Rules of Probability 5 23

Using Probability to Make Decisions 1 2

Totals 27 128

Grade 11

The Complex Number System 1 9

Seeing Structure in Expressions 2 6

Arithmetic with Polynomials and Rational Expressions 3 19

Creating Equations* 1 7

Reasoning with Equations and Inequalities 2 5

Interpreting Functions 7 14

Building Functions 1 8

Linear, Quadratic, and Exponential Models* 1 2

Trigonometric Functions 3 6

Interpreting Categorical and Quantitative Data 1 4

Making Inferences and Justifying Conclusions 3 13

Using Probability to Make Decisions 0 2

Totals 25 95


* Modeling Standards: Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice.

31

Appendix J: Example of mapping a core skill to many other core skills, Fractions

Grade Core Skill Domain

3 Explain why two fractions are equivalent using a visual fraction model

Number and Operations -Fractions

4 Decompose a fraction into a sum of fractions with the same denominator in more than one way


4 Multiply a fraction by a whole number Number and Operations -Fractions

4 Understand a fraction as a multiple of a unit fraction Number and Operations -Fractions

5 Interpret a fraction as division of the numerator by the denominator


Grade 3 Core Skill: Understand the structure of a fraction


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AckowledgementsThe following experts have advised Renaissance in the development of Star Assessments®.

Contributing AdvisorsDr. Karin Hess, Senior Associate with the Center for Assessment (NCIEA) since 2002, brings to the Center’s work over 30 years of deep experience in curriculum, instruction, and assessment. Dr. Hess is recognized nationally for her research and work with learning progressions, text complexity, performance assessment, and cognitive rigor. In addition to ex-periences as a classroom teacher and school administrator, she has been a program evaluator for the Vermont Mathematics Project; a content specialist for development of the Vermont Science assessment; and as developer and editor of Science Exemplars K-8 performance assessments. Dr. Hess is the principal author of the content specifications for assessment of the CCSS ELA and Literacy standards and was a contributor to the math-ematics content specifications for the Smarter Balanced Assessment Consortium.

Dr. R. James Milgram is an emeritus professor of mathematics at Stanford University where he has taught since 1970. He is a former member of the National Board for Education Sciences, the NASA Advisory Council, Common Grounds Project, and Achieve Mathematics Advisory Panel. Dr. Milgram has helped author several states’ standards and recently served on the Validation Committee for the Common Core State Standards. He has published over 100 research papers in mathematics and four books. Dr. Milgram received his undergraduate and master’s degrees in mathematics from the University of Chicago, and his Ph.D. in mathematics from the University of Minnesota.

Dr. Amanda M. VanDerHeyden, is a private consultant and researcher who has directed and evaluated numerous school-wide intervention and reform efforts, most often in the area of mathematics. Dr. VanDerHeyden serves as advisor to the National Center for Learning Disabilities, iSTEEP (a web-based data management system), and is a standing panel member for the Institute for Education Sciences at the U.S. Department of Education. Dr. VanDerHeyden has published more than 60 scholarly articles and chapters, 5 books, and has given keynote addresses to state school psychology associations and state departments of education in 21 states. She is co-author of the Evidence-Based Mathematics Innovation Con-figuration for the National Comprehensive Center for Teacher Quality at Vanderbilt University and now the Collaboration for Effective Education Development, Accountability, and Reform at University of Florida.

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