An integral Core Progress for Math - esboces.org€¦ · to these perspectives is the idea that the development of learning progressions is an iterative process. It begins with a

Core Progress for MathAn empirically validated learning progression

An integral component of

Accelerated Math Enterprise and

STAR MathEnterprise.

Seamless transition to Common Core

Accelerated Math, Accelerated Math Enterprise, Advanced Technology for Data-Driven Schools, Core Progress, Renaissance Learning, the Renaissance Learning logo, STAR Enterprise, STAR Math, and STAR Math Enterprise are trademarks of Renaissance Learning, Inc., and its subsidiaries, registered, common law, or pending registration in the United States and other countries.

© 2012 by Renaissance Learning, Inc. All rights reserved. Printed in the United States of America.

This publication is protected by U.S. and international copyright laws. It is unlawful to duplicate or reproduce any copyrighted material without authorization from the copyright holder. For more information, contact:

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

04/12

i

ContentsExecutive summary ...............................................................................................................................................ii

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

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

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

Phase I .................................................................................................................................................................. 3

Phase II ................................................................................................................................................................. 5

Phase III ................................................................................................................................................................ 6

Phase IV ............................................................................................................................................................. 10

Mapping the Common Core State Standards .................................................................................................... 11

Core Progress: an integral component of Accelerated Math and STAR Math Enterprise.................................. 12

Conclusion .......................................................................................................................................................... 16

References ......................................................................................................................................................... 22

Appendices

Appendix A. Examples of skill progressions across grade levels ..................................................................... 17

Appendix B. Core Progress for math includes four domains and 23 skill areas ................................................ 18

Appendix C. Core skills per grade, per domain ................................................................................................ 19

Appendix D. Example of how one core skill serves as a prerequisite for many other skills .............................. 20

Appendix E. Common Core State Standards and Core Progress...................................................................... 21

Figures

Figure 1: Core Progress for math ......................................................................................................................... 6

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

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

Figure 4: Accelerated Math Student Record Report .......................................................................................... 13

Figure 5: STAR provides your entry point into Core Progress ............................................................................ 14

Figure 6: Example of Instructional Planning Report generated by STAR Math Enterprise ................................ 15

Figure 7: Example of Core Progress search tool ................................................................................................ 15

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 skills ........................................ 8

Table 3: Example of Core Progress alignment to Common Core State Standard .............................................. 10

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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).

Continually refined since 2007, the Core Progress for math learning progression is an interconnected web of prerequisite skills. 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.

The 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. It begins with early numeracy and progresses to the level of math ability 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 astounding. 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 his achievement level, as well as skills he is ready to learn next. Core Progress is now an integral component of both Accelerated Math Enterprise 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

1

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 math 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.

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 and largely made possible by the CCSS, is the development of fully formed and validated learning progressions.

This paper describes Core Progress for math, the learning progression developed by Renaissance Learning. It starts by explaining what learning progressions are, how they operate in relation to standards, and how they support assessment, instruction, and practice. The paper then describes the research-based approach used to develop Core Progress.

Originally built to provide a framework for Accelerated Math personalized practice software, Core Progress now serves as an integral component for both Accelerated Math Enterprise 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 and largely made possible by the Common Core State Standards, is the development of fully formed learning progressions.

2

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).

Masters and Forster (1996, p. 1) describe progressions as “a picture of what it means to ‘improve’ in an area of learning.” 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).

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).

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

3

Evolution of the Core Progress learning progression for mathCore 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 astounding. 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 Enterprise 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 math 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 math 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 page 11 for more information.

2 Accelerated Math enables differentiated math practice 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, is built on the Core Progress learning progression.

3 Formerly the Northwest Regional Educational Laboratory

R eviewed by:

• Dr. Sybilla Beckmann University of Georgia

• Dr. Richard Bisk Worcester State College

• Dr. Tom Hogan University of Scranton

• Dr. James Milgram Stanford University

• Dr. Sharif Shakrani Michigan State University

4

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 math 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 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).

After the core objectives were identified, they were cross-checked with the findings of the National Mathematics Advisory Panel (2008). The team closely followed the National Math Panel’s Benchmarks for the Critical Foundations, which strongly recommend that curricula focus on mastery of key topics and provide a progression of difficulty, rather than use the spiraling approach of revisiting topics from grade to grade. 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 math objectives 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.

5

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 difficulty levels through the grades.

To begin this process, Renaissance Learning’s math 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 down to its most basic elements including concepts, skills, and terminology needed to learn that objective. This process is known as “unpacking.” Once unpacked, the team began identifying prerequisite objectives. These prerequisites were then linked together to identify a progression of associated skills. For example, as illustrated in Table 1, in the skill area Fraction Concepts and Operations, third-grade students 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 be able to add and subtract fractions with like denominators. 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 math objectives 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.

ReviseAfter 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 math learning progression.

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.

6

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.

Figure 1. Core Progress for math

Ear

ly N

umer

acy

Gra

de 1

Gra

de 2

Gra

de 3

Gra

de 4

Gra

de 5

Gra

de 6

Gra

de 7

Gra

de 8

Alg

ebra

I

Geo

met

ry

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 math can be advanced coherently and continuously.

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.

7

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 a different color in Figure 1.

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 in math development. 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. 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 student’s math development. Feedback on Core Progress will continue to be solicited in this way from teachers and administrators.

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.

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.

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

8

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:

9

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 and related skills 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.

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

10

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 and mastering and learning a concept).

Over 9,500 items were field tested, calibrated, and analyzed between June 2008 and February 2012.5 Through the calibration process, information on the psychometric properties of an item is collected, including reliability, correlation with test scores, and calibration of an item’s Rasch threshold parameters.

Using a process called dynamic calibration, researchers at Renaissance Learning added one to three experimental items to students’ STAR Math Enterprise assessments nationwide. Response data from a minimum of 1,000 students was collected on each item. The items were spread throughout the test, so even if students were aware they were getting additional items, they did not know which items were experimental.

ResultsEach 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. Broadly speaking, the difficulty level of each skill is obtained by averaging the Rasch difficulty level of all the test items in the item-set. “Scaled difficulty 70” represents a 70% chance of getting the item correct. The actual derivation is somewhat more complex, and is grounded in the fundamentals of Item Response Theory.

Core Progress for math includes 1,326 skills. Figure 3 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. The correlations are remarkably high ranging from 0.90 to 0.94.

Figure 3. Correlation of STAR Math Enterprise to Core Progress

0 2 4 6 8 10 12200

400

600

800

1000

Data Analysis, Statistics, and Probability


Algebra


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

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

11

Mapping 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 largely made possible by the CCSS, is the development of fully formed learning progressions.

The concepts and 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 level of math ability 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 even 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

The fastest, easiest way to know if students are on track with CCSS.

Grade Skill Domain

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

Grade 7 Solve a 2-step linear equation involving integers Algebra

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.

12

Core Progress: an integral component of Accelerated Math Enterprise and STAR Math EnterpriseThe 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 Enterprise. As a result, there is now a true bridge between assessment (STAR Math Enterprise), instruction (Core Progress learning progression), and practice (Accelerated Math Enterprise).

Core Progress was developed to provide a research-based framework for Accelerated Math Enterprise 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 astounding. 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 Enterprise assessment. As a result, a student’s STAR Math Enterprise score now provides insight into his/her achievement level, as well as skills he/she is ready to learn next.

Likewise, Accelerated Math Enterprise generates personalized practice assignments for each student based on the skills that they are ready to learn next or need to review. The order in which the Accelerated Math objectives are assigned to students is based on the Core Progress learning progression.

Accelerated Math Enterprise Accelerated Math Enterprise software enables monitored, differentiated math practice. It provides daily information to teachers about student progress toward math 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 Enterprise generates personalized practice assignments for each student based on the skills that they are ready to learn next or need to review. The order in which the objectives are assigned to students is based on the Core Progress learning progression.

Figure 4 shows an Accelerated Math Enterprise 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 Enterprise.

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.

13

Figure 4. Accelerated Math 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 Enterprise enables teachers to monitor student’s mastery of successive skills from the Core Progress learning progression.

14

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, a student’s scaled score (from STAR Math Enterprise) is his/her entry point into the learning progression, enabling research-based inferences about which skills the student has likely already developed, which skills are ready to be developed, 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 5.)

Figure 5. STAR provides your entry point into Core Progress

College &Career Ready

Early Numeracy

1400

0

Core Progress

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 Report for Jasmine MajorPrinted Wednesday, September 7, 2011 9:42:34 AM School: Pine Hill Middle School

Teacher: Mrs. T. Williams

Class: 5th Hour Math

Grade: 7

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

STAR Math Enterprise Test ResultsCurrent SS (Scaled Score): 741 Test Date: 09/02/2011Algebra Readiness: Jasmine has not yet met the end of year algebra readiness grade level expectations for grade 7.

Projected SS for 06/10/11: 821 Based on research, 50% of students at this student's level will achieve this much growth. Jasmine’s Current PerformanceSchool Benchmarks

Current

Projected

Scaled Score 600 650 700 750 800 850 900

Urgent Intervention Intervention On Watch At/Above Benchmark Skills to Learn Skills listed below are suggested skills Jasmine should work on based on her last STAR Math Enterprise Test. These skills

should be challenging, but not too difficult for Jasmine. Combine this information with your own knowledge of the student

and use your professional judgment when designing an instructional program. Use the Math Learning Progressions to find

additional information for each skill, worked examples, and example problems.Numbers and Operations This score suggests Jasmine has an understanding of how to divide with decimals; convert between decimals, fractions, and

percents; and solve problems involving percents and ratios. Based on this score, Jasmine should practice operations with

integers and solving problems involving percents, ratios, and proportions. Skills to Learn 1. Determine a percent of a whole number using less than 100% 2. Determine a percent of a whole number using more than 100% 3. Determine the percent of a whole number is of another whole number, with a result less than 100%

4. Determine a whole number given a part and a percentage less than 100% 5. ˜ WP: Determine a percent of a whole number using less than 100% Algebra

This score suggests Jasmine has an understanding of how to relate tables of paired numbers to variable expressions and

graphs. Based on this score, Jasmine should practice relating verbal expressions and situations to variable expressions and

equations. Skills to Learn 1. Use a variable expression with two operations to represent a verbal expression 2. Use a verbal expression to represent a variable expression with two operations 3. WP: Use a variable expression with two operations to represent a situation 4. ˜ WP: use a 2-variable equation to represent a situation involving a direct proportion 5. ˜ WP: Use a 2-variable linear equation to represent a situation Geometry and Measurement This score suggests Jasmine has an understanding of how to solve problems involving area, volume, and surface area of

shapes. Based on this score, Jasmine should practice relating solid shapes to nets, identifying properties of solid shapes,

and working with transformations on a grid.

1 of 2 1 of 2

Current

Projected

15

There are two ways to navigate the Core Progress learning progression. First, STAR Math Enterprise generates Instructional Planning reports in real time. Second, there is a web-based portal within STAR Math Enterprise to enter Core Progress. The portal provides access to a comprehensive search tool including prerequisite skill mapping, ELL support, worked examples, and sample items.

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 learn next. The Student Instructional Planning Report also shows an individual student’s current performance in relation to pre-selected benchmarks, so teachers can see whether a student is on track to meet state or locally established proficiency goals. (See Figure 6.)

Core Progress portal The Core Progress web-based portal is an interactive tool that enables teachers to navigate the learning progression. First, a teacher searches for a skill. The search engine then returns several pieces of information on that skill including: terminology, related concepts and skills, and related prerequisites skills. The search results also include instructional resources aligned to the skill and the prerequisite skill. These resources include worked examples and sample items. All materials can be printed or used with an interactive white board during instruction. Because the portal provides resources at multiple grade levels, teachers can more easily access materials to help fill gaps in knowledge. (See Figure 7.)

Instructional Planning Report for Jasmine MajorPrinted Wednesday, September 7, 2011 9:42:34 AM School: Pine Hill Middle School

Teacher: Mrs. T. Williams

Class: 5th Hour Math

Grade: 7

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

STAR Math Enterprise Test ResultsCurrent SS (Scaled Score): 741 Test Date: 09/02/2011Algebra Readiness: Jasmine has not yet met the end of year algebra readiness grade level expectations for grade 7.

Projected SS for 06/10/11: 821 Based on research, 50% of students at this student's level will achieve this much growth. Jasmine’s Current PerformanceSchool Benchmarks

Current

Projected

Scaled Score 600 650 700 750 800 850 900

Urgent Intervention Intervention On Watch At/Above Benchmark Skills to Learn Skills listed below are suggested skills Jasmine should work on based on her last STAR Math Enterprise Test. These skills

should be challenging, but not too difficult for Jasmine. Combine this information with your own knowledge of the student

and use your professional judgment when designing an instructional program. Use the Math Learning Progressions to find

additional information for each skill, worked examples, and example problems.Numbers and Operations This score suggests Jasmine has an understanding of how to divide with decimals; convert between decimals, fractions, and

percents; and solve problems involving percents and ratios. Based on this score, Jasmine should practice operations with

integers and solving problems involving percents, ratios, and proportions. Skills to Learn 1. Determine a percent of a whole number using less than 100% 2. Determine a percent of a whole number using more than 100% 3. Determine the percent of a whole number is of another whole number, with a result less than 100%

4. Determine a whole number given a part and a percentage less than 100% 5. ˜ WP: Determine a percent of a whole number using less than 100% Algebra

This score suggests Jasmine has an understanding of how to relate tables of paired numbers to variable expressions and

graphs. Based on this score, Jasmine should practice relating verbal expressions and situations to variable expressions and

equations. Skills to Learn 1. Use a variable expression with two operations to represent a verbal expression 2. Use a verbal expression to represent a variable expression with two operations 3. WP: Use a variable expression with two operations to represent a situation 4. ˜ WP: use a 2-variable equation to represent a situation involving a direct proportion 5. ˜ WP: Use a 2-variable linear equation to represent a situation Geometry and Measurement This score suggests Jasmine has an understanding of how to solve problems involving area, volume, and surface area of

shapes. Based on this score, Jasmine should practice relating solid shapes to nets, identifying properties of solid shapes,

and working with transformations on a grid.

1 of 2 1 of 2

Current

Projected

Figure 6. Example of Instructional Planning Report generated by STAR Math Enterprise

Figure 7. Example of Core Progress search tool

16

ConclusionAmong the first of its kind, the new mixed-method approach used to develop Core Progress for math has made a contribution to the field of learning progressions.

Developed by math content-area and learning-progression experts, supported by a framework of assessment and instructional tools, and designed to reflect the reality of student learning based on data, Core Progress for math fills a need that is only now becoming fully realized within the field of education. It describes a fully formed progression of learning within the domain of math, including the intermediate steps not evident in state standards and Common Core State Standards. It provides detailed assessment data that locates where students are on that pathway, not only pointing in the right direction, but also providing tangible and achievable next steps for getting there.

Among the first of its kind, the new mixed-method approach used to develop Core Progress for math has made a contribution to the field of learning progressions.

17

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

18

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

19

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

20

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:

21

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 CC 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:

22

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Dr. Karin Hess, Senior Associate with the National Center for the Improvement of Educational Assessment (NCIEA) since 2002, brings to the Center’s work over 30 years of deep experience in curriculum, instruction, and assessment. She has assisted more than a dozen states in major development of grade level expectations, revisions to state content standards, and in creating detailed assessment specifications aligned to content standards that are both educationally and technically sound. Dr. Hess is recognized nationally for her research and work with learning progressions, performance assessment, and cognitive rigor.

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Acknowledgements

L2791.0512.RN.2.5M R55248

Reviewer

Reviewer