Core Progress™ for Math - Renaissance Learning

Integralcomponents ofAccelerated MathLive andSTAR MathEnterprise Core Progress for MathEmpirically validated learning progressions

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ContentsExecutive Summary................................................................................................................................................iiIntroduction........................................................................................................................................................... 1What are learning progressions?.......................................................................................................................... 2Evolution of the Core Progress learning progression for mathematics................................................................. 3Phase I: Scope and sequence.............................................................................................................................. 3Phase II: Revised scope and sequence, addition of core objectives and prerequisites...................................... 5Phase III: Learning progression............................................................................................................................ 6Phase IV: Empirical analysis of Core Progress.................................................................................................... 10Mapping Core Progress to the Common Core State Standards......................................................................... 12Phase V: Building a new learning progression specifically for the Common Core............................................. 13Core Progress: an integral component of Accelerated Math Live and STAR Math Enterprise........................... 15Conclusion........................................................................................................................................................... 19References.......................................................................................................................................................... 32AppendicesCore Progress for Math Learning ProgressionAppendix A: Examples of skill progressions across grade levels...................................................................... 20Appendix B: Core Progress for Math includes four domains and 23 skill areas................................................. 21Appendix C: Core skills per grade, per domain................................................................................................. 22Appendix D: Example of how one core skill serves as a prerequisite for many other core skills....................... 23Appendix E: Common Core State Standards and Core Progress...................................................................... 24Core Progress Learning Progression for Math - Built for the Common Core State StandardsAppendix F: Organization of Skill Areas within the 11 Domains for K-8............................................................. 25Appendix G: Organization of Skill Areas within the 21 Domains for high school................................................ 26Appendix H: Examples of skill progressions across grade levels, Whole Numbers: Place Value...................... 28Appendix I: Core skills per grade, per domain................................................................................................... 29Appendix J: Example of mapping a core skill to many other core skills, Fractions............................................ 31FiguresFigure 1: Core Progress for Math.......................................................................................................................... 7Figure 2: Prerequisite map of place value............................................................................................................. 9Figure 3: Correlation of STAR Math Enterprise to Core Progress........................................................................ 11Figure 4: Correlation of STAR Math Enterprise to Core Progress Math built for CCSS....................................... 13Figure 5: Accelerated Math Live Student Record Report................................................................................... 16Figure 6: STAR Math Enterprise provides your entry point into Core Progress................................................... 17Figure 7: Example of Instructional Planning Report generated by STAR Math Enterprise................................. 18Figure 8: STAR Record Book............................................................................................................................... 18TablesTable 1: Cross-grade progression of Fraction Concepts and Operations............................................................ 5Table 2: Example of how one core skill serves as a prerequisite for many other core skills................................. 8Table 3: Example of Core Progress alignment to Common Core State Standard............................................... 12i

Executive SummaryLearning progressions are descriptions of how learning typically advances in a subject area. “Empiricallybased learning progressions can visually and verbally articulate a hypothesis, or an anticipated path, of howstudent 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 learningprogression 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 acontinuous process of research, expert review, and iterative revision. Continually refined since 2007, the CoreProgress for Math learning progression is an interconnected web of prerequisite skills.The skills and understandings in the Core Progress learning progressions provide the intermediate stepsalong with prerequisite skills necessary to reach the levels of expertise identified through the Common CoreState Standards. They begin with early numeracy and progress to the level of competence in mathematicsrequired to be college and career ready.Core Progress was originally developed to provide a research-based framework for Accelerated Mathpersonalized practice software. Once built, the Core Progress skills were field tested through the STARMath assessment.The results were remarkable. As illustrated in the graph below, the order of skills in Core Progress are highlycorrelated with the difficulty level of STAR Math assessment items. With a strong correlation, the natural nextstep was to statistically link Core Progress to the STAR Math Enterprise assessment.1000Core Progress Skill Difficulty800Scaled Difficulty 70600400Numbers andOperationsAlgebraGeometry andMeasurementData Analysis, Statistics,and Probabilityy = 253.5Ln(x) + 324.85r = 0.9059y = 251.45Ln(x) + 333.35r = 0.9440y = 271.68Ln(x) + 313.65r = 0.9104y = 240.13Ln(x) + 334.27r = 0.89602000 2 4 6 8 10 12Grade Level OrderAs a result of the statistical link between STAR Math Enterprise and Core Progress, a student’s STAR Mathscore provides insight into her achievement level, as well as skills and understandings she is ready to developnext. Core Progress is now an integral component of both Accelerated Math Live and STAR Math Enterprise—a true bridge between assessment, instruction, and practice.ii

IntroductionOver the last decade, much of the focus of educational reform in the United States has been on the creationand improvement of standards of learning. A watershed moment of this movement was the 2010 publicationof the Common Core State Standards (CCSS) for learning in Mathematics and English language arts. Asthe CCSS mission statement explains, “The Common Core State Standards provide a consistent, clearunderstanding of what students are expected to learn, so teachers and parents know what they need todo to help them.”At the same time, within the field of education, the idea of learning progressions has received increasingattention (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 thisinterest is the desire to provide descriptions ofincremental steps of learning. These steps, moreprecise than are currently represented in standards,can be used to guide design of curriculum, instruction,and assessment. Learning occurs when students seethese incremental steps as special cases of moregeneral and basic processes and principles.The next step, clarified bythe CCSS, is the developmentof learning progressions thatmirror the CCSS.While the Common Core State Standards represent a clear step toward providing a more coherent pathwayto meeting educational goals than many prior standards, the CCSS do not describe a fully formed pathwayalong which students are expected to progress. The next step, clarified by the CCSS, is the development oflearning progressions that mirror the CCSS.Originally built to provide a framework for Accelerated Math personalized practice software, Core Progressnow serves as an integral component for both Accelerated Math Live and the STAR Math Enterpriseassessment. Now, with all three pieces linked, there is a true bridge between assessment, instruction,and practice.1

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 moresophisticated ways of thinking about key disciplinary concepts and practices across multiple grades” whichoutline “the intermediate steps toward expertise.” Leahy and Wiliam (2011, p. 1) view learning progressionsas descriptions of “what it is that gets better when someone gets better at something.” “Empirically basedlearning progressions can visually and verbally articulate a hypothesis, or an anticipated path, of how studentlearning 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 thatmove from simple to complex and, while not necessarily linear, the progression is not random, but rather issequenced and ordered as “expectedtendencies” or “likely probabilities” of howlearning develops (Confrey and Maloney, 2010).Masters and Forster (1996, p. 1) describeprogressions as “a picture of what it means to‘improve’ in an area of learning.”Finally, Heritage (2011, p. 3) suggests thatlearning progressions provide descriptions of“how students’ learning of important conceptsand skills in a domain develops from its mostrudimentary state through increasinglysophisticated states over a period of schooling.”“Empirically based learningprogressions can visually andverbally articulate a hypothesis, oran anticipated path, of how studentlearning will typically move towardincreased understanding over timewith good instruction.”Hess, Kurizaki, and Holt, 2009Inherent in these views of progressions is the idea of a coherent and continuous pathway along whichstudents move incrementally through states of increasing competence in a domain. Every incremental statebuilds on and integrates the previous one as students accrue new levels of expertise with each successivestep in the progression. It is important to note, however, that while progressions may provide cleardescriptions of how learning develops in a domain, they are not developmentally inevitable. Rather, theyare 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 suchprogressions absolutely necessary.As Herman (2006, p. 122) observes, “whether and how children are able to engage in particular learningperformances and the sequence in which they are able to do so are very much dependent on previousopportunities to learn.” The benefit of progressions is that they lay out a continuum to guide teaching andlearning 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 andSteedle, 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). Commonto these perspectives is the idea that the development of learning progressions is an iterative process. Itbegins with a hypothesis, informed by what we know about student learning, which undergoes empiricaltesting and subsequent refinement based on the data. Core Progress for Math was developed according tothis iterative model.2

Evolution of the Core Progress learning progression for mathematicsCore Progress began as a scope and sequence and evolved into an empirically validated learningprogression. 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 personalizedpractice software. Once built, the Core Progress skills were field tested through the STAR Math assessment 1 .The results were noteworthy and gratifying. The order of skills in Core Progress was highly correlated withthe difficulty level of STAR Math Enterprise assessment items. With a strong correlation, the natural next stepwas to statistically link Core Progress to the STAR Math assessment. As a result, a student’s STAR Math scoreprovides insight into his/her achievement level, as well as skills he/she is ready to learn next. Core Progress isnow an integral component of both Accelerated Math Live and STAR Math Enterprise—a true bridge betweenassessment, instruction, and practice.Phase I: Scope and sequenceResearchThe origin of the Core Progress learning progression dates back to 2007. It started as a scope and sequencefor Accelerated Math Enterprise 2 , spanning grade 1 to algebra.To develop the original scope and sequence, Renaissance Learning’s mathematics team relied heavily onresearch and standards including the National Council of Teachers of Mathematics (NCTM) CurriculumFocal Points (2006), the early work of the National Mathematics Advisory Panel (2008), state and internationalmathematics standards, and the American Diploma Project Benchmarks (Achieve, Inc., 2007) which provideone 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 researchlaboratory funded by the U.S. Department of Education; a panel of mathematics teachers; and a panel ofprominent 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)1Over 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 wasfirst 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 in2012-2013, now includes content libraries based on the Core Progress Math built for CCSS learning progression.3Formerly the Northwest Regional Educational Laboratory3

RevisionThe initial review focused on Numbers and Operations. Items were analyzed for difficulty, alignment toobjectives, accuracy, item quality, and relationship to current pedagogy. Based on the reviews, RenaissanceLearning’s mathematics team identified two principal goals: (1) reduce the overlap of objectives betweengrades, 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 mustmaster at each grade in order to advance to the next grade. The NCTM’s Curriculum Focal Points served asthe basis for decisions about which topics to include at each grade level. The team also referred to severalseminal 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 onmastery of key topics and provide a progression of increasing difficulty, rather than use the spiraling approachof revisiting topics from previous grades.In addition, researchers at Renaissance Learning examined empirical Accelerated Math data that included66,000 students in 88 schools over three years. The analysis provided real-world insight into the objectives inmathematics students struggle with the most. As a result of this analysis, additional objectives were identifiedfor possible inclusion as core objectives.When the draft core objectives were complete, Dr. Tom Hogan from the University of Scranton providedexpert review. Renaissance Learning incorporated Dr. Hogan’s objective-by-objective feedback andgeneral comments.4

Phase II: Revised scope and sequence, addition of core objectivesand prerequisitesResearchWith the core objectives for grades 1 through 8 identified, Renaissance Learning began work on a new andimproved scope and sequence. Development of the scope and sequence reflected the second goal identifiedin 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 continuallyconsulting the National Mathematics Advisory Panel (2008), NCTM focal points (2006), the Singapore primaryand secondary mathematics standards, and the American Diploma Project Benchmarks (Achieve, Inc., 2007).After the core objectives were identified and putinto skill areas, the team distilled each objective toits 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 progressionof associated skills.For example, as illustrated in Table 1, in theskill area Fraction Concepts and Operations,third-grade students are expected to developan understanding of the meaning of a fraction.Having established this understanding, studentsmove incrementally through successive steps ofincreasing competence. By fourth grade, studentsshould understand that fraction addition andsubtraction for fractions with like denominators isa c a ++c= -defined by the rule b - b bTable 1: Cross-grade progression of Fraction Conceptsand OperationsDomain: Numbers and OperationsSkill Area: Fraction Concepts and OperationsGrade. By fifth grade, students should add and subtract fractions and mixed numberswith unlike denominators. In sixth grade, students should progress incrementally through multiplication anddivision of fractions. By seventh grade, students should be able to solve multistep problems involving fractionsor 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 processbegan. Renaissance Learning uses an alignment process developed with input from Mid-continent Researchfor Education and Learning (McRel) and Education Northwest. 434567SkillStudents develop an understanding of themeaning of a fractionStudents are able to add and subtract fractionswith like denominatorsStudents are able to add and subtract fractionsand mixed numbers with unlike denominatorsStudents progress incrementally throughmultiplication and division of fractionsStudents are solving multistep problemsinvolving fractions or mixed numbersThis 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, suchas 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’sstandards team received extensive training, including training in how to calibrate alignment results. Afterthe 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 andprerequisite objectives, was finalized. This new and improved scope and sequence became the basis forthe development of the learning progression for mathematics.4McRel and Education Northwest are part of the Regional Educational Laboratory Program funded by the US Department of Education’s Institute ofEducation Sciences.5

Phase III: Learning progressionThe shift from scope and sequence to learning progression began in Phase II with the identification of coreobjectives, prerequisite skills, and the progression of associated skills. Now firmly down the learningprogression path, Renaissance Learning wasready to go farther.Two critical events led to the next breakthroughin Renaissance Learning’s learning progressionwork. First, the Common Core State StandardsInitiative (CCSSI) began. Second, theMathematics Framework for the 2011National Assessment of Educational Progresswas published.Since 2007, Renaissance Learninghas been studying the Achievestandards, which are a keyfoundation of the Common CoreState Standards. As a result,aligning our learning progressionwith the CCSS was a natural andfamiliar process.Since 2007, Renaissance Learning had beenstudying and aligning to the Achieve standards,which are a key foundation of the Common Core State Standards. Then, when the CCSSI began, thestandards team closely followed every stage of CCSS development. As a result, aligning the Core Progresslearning 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 andoperations; 2) algebra; 3) geometry and measurement; and 4) data analysis, statistics, and probability. Thefour domains are represented by the four different colors in Figure 1 (next page).The skills areas (e.g. whole numbers, placevalue, symbols and expressions, time, etc.)represent the various skills and conceptsstudents acquire as they progress thedevelopment of mathematics. There are23 skill areas, which can be found inAppendix B.The network of interrelated skillsand prerequisites in Core Progressis extensive. Many core skills for onegrade serve as prerequisite skills forsubsequent grades, reflecting thehierarchial nature of mathematics.The core skills and prerequisites act asbuilding blocks, each representing a specificlevel of competency of a skill or understanding that rests on prior development and that also provides afoundation for the next level of learning. There are 1,326 skills in the Core Progress learning progression. Ofthese, 398 are core skills, and many of these serve as prerequisites within and across domains, not to belearned in isolation, but as important parts of a single whole. See Appendix C for a complete count of skillsper grade, for each domain.The skill areas and skills were reviewed for coherence and continuity across grade levels to ensure thateach contributed to the larger goal of improving student mathematical understanding. In addition to internalanalysis, a focus group of teachers across various grade levels was convened. This group provided feedbackon 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.6

Prerequisite mapping in Core ProgressThe Core Progress learning progression is an interconnectedweb of prerequisite skills. Moving toward increasedunderstanding over time requires continually building upand building on a solid foundation of knowledge, concepts,and skills.The Core Progresslearning progression isan interconnected webof prerequisite skills.One indication of the interrelated network of concepts in Core Progress is the number of skills that build upand build on each other. Specifically, 121 of the 398 core skills in Core Progress serve as prerequisites toothers in subsequent grades.Figure 1: Core Progress for Math= Approx. 5 skills= Approx. 10 skillsData Analysis, Statistics,and ProbabilityGeometry andMeasurementAlgebraNumbers and OperationsEarly NumeracyGrade 1Grade 2Grade 3Grade 4Grade 5Grade 6Grade 7Grade 8Algebra IGeometryCore Progress for Math is an empirically validated continuum to guide teaching and learning over time so that student competence inmathematics can be advanced coherently and continuously.7

To illustrate the interrelated nature of the core skills and how they serve as prerequisites to each other, seeTable 2. In this example, the seventh grade core skill, subtract integers, serves as a prerequisite for sevencore 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 skillsSubtract integers is a prerequisite for the following:Grade Core Skills DomainGrade 7 WP: Add and subtract using integers Numbers and operationsGrade 7Evaluate a 2-variable expression, with two or threeoperations, using integer substitutionAlgebraGrade 7 Solve a 1-step linear equation involving integers AlgebraGrade 8 Simplify an algebraic expression by combining like terms AlgebraAlgebra 1 Determine the slope of a line given two points on the line AlgebraAlgebra 1Apply the quotient of powers property to monomialalgebraic expressionsAlgebraGeometry Solve a problem involving the distance formula GeometryExample of how one core skill serves as a prerequisite for seven skills across four grade levels in three domains.8

Figure 2 offers a different way to think about thedeeply interrelated nature of Core Progress forMath. This figure shows a true mapping of skills,illustrating how skills build on each other, serving asprerequisites to one another.As Figure 2 illustrates, Core Progress is aninterconnected web of prerequisite skills.It’s important to recognize that a learningprogression as comprehensive and interrelated asCore Progress for Math takes years to develop and could only come to fruition through a continuous processof research, expert review, and iterative revision.Figure 2: Prerequisite map of place valueA comprehensive and interrelatedlearning progression like CoreProgress takes years to developthrough a continuous process ofresearch, expert review, anditerative revision.Grade 1 Grade 2Skill 1Read a whole numberto 30Skill 1Read a whole numberto 1,000Grade 3Skill 2Read a whole numberfrom 31 to 100Skill 2Determine the wordform of a whole numberto 1,000Skill 1Read a 4- or 5-digitwhole numberGrade 3Skill 3Determine the word formof a whole number to 30Skill 11Determine the result ofchanging a digit in a3-digit whole numberSkill 2Determine the wordform of a 4- or 5-digitwhole numberSkill 9Determine anequivalent form of a4-digit whole numberusing thousands,hundreds, tens, and onesSkill 4Determine the word formof a whole numberfrom 31 to 100Skill 12Model a number usinghundreds, tens, andones to 1,000Skill 5Represent a 4-digit wholenumber as thousands,hundreds, tens, and onesSkill 13Recognize a numberfrom a model of hundreds,tens, and ones to 1,000Skill 6Determine the 4-digitwhole numberrepresented in thousands,hundreds, tens, and onesGrade 2Skill 14Represent a 3-digitnumber as hundreds,tens, and onesSkill 15Determine the 3-digitnumber represented ashundreds, tens, and onesCore Progress is a true map of skills: new learning is built on previous, foundational understandings. The arrows identify a typicaldevelopmental path within the learning progression. In the Common Core Standards the student is expected to see the expanded form asthe “name” of a number.9

Phase IV: Empirical analysis of Core Progress MethodIn 2008, Renaissance Learning began Phase IV of Core Progress development: empirical analysis. The orderof skills in the learning progression was re-examined empirically through a calibration process used to analyzeassessment items. The purpose was to compare the empirically observed order of skills (i.e. where skills fallon an assessment scale) to a pedagogically determined ordering of skills (i.e. the most productive order ofskills for teaching, mastering, and learning a concept).Between June 2008 and February 2012 over 9,500 items were field tested, calibrated, and analyzed using aprocess called dynamic calibration. 5 In this process, a small number of experimental items (one to three) wereadded onto each student’s STAR Math Enterprise assessment nationwide. Response data from thousands ofstudents were collected for each of these experimental items, and the items were then calibrated by fitting alogistic regression model (the Rasch model) to the relationship between scores on each item and a student’sRasch ability scores on STAR Math. The result was to calibrate the difficulty of each new item on the sameRasch 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 theitems assessing a skill was determined; the average is a “skill characteristic curve.” For each skill, a skilldifficulty parameter was then calculated: the point on the Rasch scale at which a student of the same Raschability would have an expected percent correct of 70 if tested on all of the items that measure the skill. Thisparameter is designated SD70, or Scaled Difficulty 70. Finally, the relationship between the empiricallycalibrated SD70 skill difficulties and the sequential order of STAR Math skills in the learning progressionwas 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 theirdifficulty level on the STAR Math Enterprise assessment scale and their instructional order according to thelearning progression.Each datapoint in Figure 3 represents a skill on the learning progression. The difficulty value (vertical scale) ofeach skill is derived from the calibrated difficulty of the test items from STAR Math that assess that skill. Thereare several assessment items per skill, called an item-set.Best-fitting logarithmic functions relating the SD70 value of each item-set to instructional order werecalculated for each of the four domains of STAR Math. These are plotted in Figure 3 as color-coded curvedlines superimposed on the scatter plot. For each domain, the parameters of the fitted logarithmic functionare displayed, along with the correlation between the skill difficulty parameters and instructional order. Thesecorrelations range from approximately 0.90 (for the Data Analysis, Statistics and Probability domain) to 0.94for the Algebra domain. These correlations may be thought of as measures of the validity of the Core Progresslearning progression for describing the developmental sequence of the hundreds of skills that make up theSTAR Math domains.5New assessment items aligned to the Common Core will continue to be tested on an ongoing basis.10

Figure 3: Correlation of STAR Math Enterprise to Core Progress1000Core Progress Skill DifficultyScaled Difficulty 70800600400Numbers andOperationsAlgebraGeometry andMeasurementData Analysis, Statistics,and Probabilityy = 253.5Ln(x) + 324.85r = 0.9059y = 251.45Ln(x) + 333.35r = 0.9440y = 271.68Ln(x) + 313.65r = 0.9104y = 240.13Ln(x) + 334.27r = 0.89602000 2 4 6 8 10 12Grade Level OrderThe high correlation between STAR MathEnterprise and Core Progress providesempirical evidence of the bridge betweenassessment and instruction.11

Mapping Core Progress to the Common Core State StandardsThe Common Core State Standards represent a clear step toward providing a more coherent pathway tomeeting educational goals than many prior state standards. At the same time, they do not describe a fullyformed pathway along which students are expected to progress. The next step, clarified and made possibleby 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, andalso provide the intermediate steps and prerequisite skills necessary to reach the levels of expertiseidentified through the standards. Core Progress begins with early numeracy and progresses to the minimallevel 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 ofthe standards was released. As the movement to create the Common Core State Standards was gettingunderway, Renaissance Learning was already reviewing and learning from the work of independenteducational organizations such as Achieve. Then, as the Common Core State Standards entered into variousstages of completion, Renaissance Learning carefully monitored them in draft form and provided publiccommentary. 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 byletters.” For another example, see Appendix E.Table 3: Example of Core Progress alignment to Common Core State StandardThe Common Core State Standards set the bar. Core Progress provides the prerequisite and intermediarysteps for achieving the standards.Grade Skill DomainGrade 7 Solve a 1-step linear equation involving integers AlgebraGrade 7 Solve a 2-step linear equation involving integers AlgebraGrade 8 Solve a 1-step equation involving rational numbers AlgebraGrade 8 Solve a 2-step equation involving rational numbers AlgebraGrade 8 Solve a 2-step linear inequality in one variable AlgebraAlgebra 1Algebra 1Algebra 1Solve a 1-variable linear equation with the variable onboth sidesSolve a 1-variable linear inequality with the variable onone sideSolve a 1-variable linear inequality with the variable onboth sidesAlgebraAlgebraAlgebraAlgebra 1 Solve a 1-variable compound inequality AlgebraStudents work through each incremental skill involving linear equations and inequalities, developing and expanding these skills at eachgrade level.12

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-basedframework for Accelerated Math personalized practice. When development of the learning progressionbegan in 2007, the content team at Renaissance Learning drew heavily on the American Diploma ProjectBenchmarks, which provided the foundation for the Common Core State Standards (CCSS). Since then, thenew standards were published and have been adopted by the majority of states. The need for a learningprogression 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 Standardswas 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 setof standards by grade. They identified the intent of each standard and the inherent skills, as well as keyterminology used to describe the standard. Developers also immersed themselves in the literature andresources regarding the Common Core to inform them as they worked to interpret the standards. The teamthen 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 have11 domains and 49 skill areas. Grades 9-11 have 21 domains and 44 skill areas. For a list of the skill areaswithin 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 Progressfor Math were perfect matches to the standards in the Common Core. These skill statements have beenquantitatively analyzed in the calibration process (see Phase IV—Empirical analysis of Core Progress) so theywere known to be accurate grade-level indicators of student learning. Figure 4 shows a sampling of the skillsplotted by their difficulty level on the STAR Math Enterprise assessment scale and their instructional orderaccording to the Core Progress Math built for CCSS learning progression. As new skills were identified, theywere written to meet the specific needs of the CCSS. Throughout the process of developing the CCSSlearning 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 CCSS14001200Core Progress Math built for CCSS: Skill Difficulty (June'13)y = -4.780x 2 + 111.684x + 299.407R² = 0.850Scaled Difficulty 701000800600400Numbers/OperationsAlgebraFunctionsThe correlation between STARMath Enterprise and Core ProgressMath built for CCSS providesempirical evidence of the bridgebetween assessment and instruction.Geometry200Data/Statistics/Probability00 1 2 3 4 5 6 7 8 9 10 11 12Grade Equivalent Order13

For each grade, the team cross-referenced several CCSS guides. The Common Core State Standards forMathematics (CCSSM) was the deciding factor for the placement and pace of skill development. Skills fromadopter states and other states such as Texas and Minnesota were added when they enhanced and clarifiedthe meaning of the CCSSM. Before an augmented skill was included in the learning progression, the teamverified the skill did not contradict the Common Core.The K-8 and High School Publishers’ Criteria were used to identify clusters, which were tagged asmajor, supportive, or additional. The K-8 Publishers’ Criteria for the Common Core State Standards forMathematics specifies clusters as indicators of algebra readiness—a crucial level needed for high schoolsuccess. This information was compared against core skills and their prerequisites. If a prerequisite wasmissing, it was added into the learning progression. See Appendix I for a list of core skills per grade andAppendix J for an example of how one core skill serves as a prerequisite for many other skills. The CommonCore typically describes the ultimate way students are expected to use and understand key concepts, but thedetailed steps needed for them to attain this level are usually not discussed there. This is where the true valueof a learning progression becomes apparent.Once a grade band (K-2, 3-5, 6-8) was completed,it was reviewed holistically to ensure the difficultyof skills made sense across the grades. Skills werereviewed by domain to verify the progression wasaccurate. The team also confirmed that theskills met the conditions of the K-8 Publishers’Criteria for the Common Core State Standardsfor Mathematics.As with the original Core Progress for Math,Renaissance Learning worked with external expertsthroughout the development of the learning progression. These experts provided guidance and suggestionsfor each grade band in the Core Progress Math built for CCSS. They considered the adequacy of each skill inaddressing the standards, the progression of skills from grade to grade, and the language used to describeskills. 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)Each skill was reviewed from theperspective and stated philosophyof the Common Core. Theprogression includes skills thatmay not be explicitly stated in theCCSS but are considered keylogical steps to student learning.Core Progress Math built for CCSS embodies the Common Core and provides a teachable order of skillsgrade 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 logicalsteps 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.14

Core Progress : an integral component of Accelerated Math Live and STARMath Enterprise The more comprehensive a learning progression is, the more ways it can be used. Because of the depth andbreadth of Core Progress, it now serves as an integral component for STAR Math Enterprise and AcceleratedMath 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), andpractice (Accelerated Math Live).Core Progress was developed to provide a research-based step-by-step framework for Accelerated Mathpersonalized practice software. Once built, the Core Progress skills were translated into assessment itemsand field tested via STAR Math Enterprise. As illustrated in Figure 3 (p.11), the results were gratifying. Theorder of skills in the learning progression was highly correlated with the difficulty level of the skills-turned-STARMath Enterprise items.With a strong correlation, the natural next step wasto statistically link Core Progress to the STAR Mathassessment. As a result, students’ STAR MathEnterprise score now provides insight into theirachievement level, as well as skills they are readyto develop next.Because of the depth and breadthof Core Progress, it now servesas an integral component for STARMath Enterprise and AcceleratedMath Live.Accelerated Math LiveAccelerated Math Live software enables monitored, differentiated practice in mathematics. It provides dailyinformation to teachers about student progress toward mastery, skill by skill. Accelerated Math is recognizedas a “mastery measure” by the U.S. Department of Education 6 (U.S. DOE). A mastery measure tracks “astudent’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 skillsthat they are ready to learn next and/or need to review. The order in which Accelerated Math the objectivesare arranged and typically assigned to students is based on the Core Progress learning progression. If a skillneeds to be reviewed, the instructor can use the learning progression to quickly locate prerequisite skills. Theprerequisite skills the student needs to review, in order to successfully complete the assignment, can then beassociated 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 inmathematics to meet the student’s immediate learning needs.Figure 5 (next page) shows an Accelerated Math Live report for hypothetical student, Derek Adams. He hasmastered the first 13 skills in the Core Progress learning progression for his grade (numbers 1-13) and isworking on the next two (numbers 14-15). Typically, a teacher will run this report weekly to monitor eachstudent’s progress and pace.6 The U.S. Department of Education’s National Center on Response to Intervention (NCRTI) conducts rigorous, research-based reviews of assessmentsand interventions. Accelerated Math met NCRTI’s highest standards.15

Figure 5: Accelerated Math Live Student Record ReportStudent Record ReportPrinted Thursday, October 6, 2011 12:22:20 PMSchool: East Elementary School Reporting Period: 09/01/2011 - 10/06/2011(2011-2012)Adams, DerekID: DADAM Class: Grade 2Grade: 2 Teacher: DeMarco, C.Active ObjectivesLibraryObjective Ready TestAverage Percent CorrectRegular DiagnosticObjectivesCode To Test Completed Practice Exercise TestTest14. ˜ 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 ObjectivesLibraryAverage Percent CorrectObjectivesObjectiveCodeDateMastered Practice ExerciseRegularTestDiagnosticTest Review1. ˜ 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˜Designates a core objective. Core objectives identify the most critical objectives to learn at each grade level.The Student Record Report in Accelerated Math Live enables teachers to monitor students’ mastery of successive skills from the CoreProgress learning progression.16

School: Pine Hill Middle SchoolClass: 5th Hour MathCurrent SS (Scaled Score): 741 Test Date: 9/5/2013Jasmine's Current PerformanceSchool BenchmarksaThis student was given extra time to complete the test.Printed Thursday, September 5, 2013 4:15:12 PMCurrentTeacher: Mrs. T. Wi liamsSTAR Math EnterpriseIn the landmark report, Knowing What Students Know, the authors establish learning progressions as thefoundation for assessment. Specifically, the authors state, “models of student progression in learning shouldunderlie the assessment system, and tests should be designed to provide information that maps back to theprogression” (Pellegrino, Chudowsky, and Glaser, 2001, p. 256).In a 2011 paper, Pellegrino, one of the report’s authors, suggested that learning progressions can guidethe specification of learning performances, which in turn can guide the development of tasks that enableeducators to infer students’ level of competence for the major constructs that are the target of instruction andassessment. If assessments are developed from a progression, they can provide a continuous source ofevidence 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, enablingresearch-based inferences about which skills they have likely already developed, which skills are ready to bedeveloped, which skills and understandings need remediation, and which skills will likely develop soon. Thinkof 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 Progress1400College &CareerReadySkillsRemainingto LearnInstructional Planning Reportfor Jasmine Major1 of 2STAR Math Test ResultsGrade: 7Algebra Readiness: Jasminehas not yet me the algebra readiness grade level expectations.Projected SS for 06/16/14: 785 Based on research, 50% of students a this student's level wi l achieve this much growthCurrentProjectedMr. Steward is buying a house. He can spend no more than31% of his income on monthly house payments. If he earns$4,600 per month, what is the largest monthly housepayment he can make?A $148B $1,480C$1,426741SkillsReadyto LearnScaled Score 550 600 650 700 750 800 850Skills to LearnProjectedûUrgent Intervention ûIntervention ûOn Watch ûAt/Above BenchmarkJasmine's recent STAR Math scaled score(s) suggests these ski ls from Core Progress learning progressions would bechallenging, but no too di ficult for her. Combine this information with your own knowledge of the student and use yourprofessional judgment when designing an instructional program. Use the Core Progress learning progressions to see howthese ski ls fit within the larger context of the progression.K-8GR7Students draw and construct geometrical figures and describe the relationship between figures with di ferent a tributes.GeometryThey solve real-world and mathematical problems involving angle measure and the area of 2-dimensional figures. Theysolve real-world and mathematical problems involving the surface area and volume of 3-dimensional figures.7 Relate volume found using unit cubes to multiplication of edge lengths in a right rectangular prism77 WP: Find the volume of a right rectangular prism with fractional edge lengths using formulas7 Plot coordinates to form a polygon on the coordinate plane7 Find a side length of a polygon on the coordinate plane» Find the volume of a right rectangular prism with fractional edge lengths using formulasD $1,3267 Ratios and Proportional RelationshipsStudents analyze proportional relationships in real-world and mathematical problems. They write equations to representa proportional relationship. They solve multi-step real-world and mathematical ratio and percent problems.7 Understand the concept of a unit rate77 Identify an input-outpu table that contains values for a given ratio7 Find missing values in a ratio table7 Graph in the coordinate plane the values of a ratio table7 WP: Use tables to compare ratios7» Determine a unit rate» WP: Solve a unit rate problem7 The Number System» Designates a core ski l. Core ski ls identify the most critical ski ls to learn at each grade level.SkillsMastered0EarlyNumeracy17

Core Progress learning progressions help to informinstruction within STAR Math Enterprise. Educatorsaccess a learning progression through InstructionalPlanning Reports (which are generated by STARMath Enterprise in real time), or the STARRecord Book.Instructional Planning ReportsSkills-based information for students is providedby the Instructional Planning Reports producedinstantly by STAR Math Enterprise after a studentcompletes a test. These reports use the CoreProgress learning progression to identify the rangeof skills students are ready to develop next. TheStudent Instructional Planning Report also showsan individual student’s current performance inrelation to pre-selected benchmarks, so teacherscan keep an eye on whether a student is on trackto meet state or locally established proficiencygoals. (See Figure 7)STAR Record BookThe STAR Record Book highlights suggestedskills from Core Progress that a student is readyto learn. It is a tool that bridges assessment andinstruction. The student’s STAR scaled scoreis placed on the learning progression andsuggests skills that are appropriate to focus on.To further expand understanding of skills andsupport instruction, educators willfind Sample Items and WorkedExamples associated with skills.Each skill includes prerequisiteskill mapping, ELL support, andcontent-area vocabulary. Inaddition, the Record Book includesDepth of Knowledge 3 (DOK3)items and performance tasks.These resources are designed tohelp teachers probe more deeplyinto the skills and knowledge ofeach student. (See Figure 8)Figure 7: Example of Instructional Planning Reportgenerated by STAR Math EnterpriseSchool: Pine Hill Middle SchoolClass: 5th Hour MathSTAR Math Test ResultsCurrent SS (Scaled Score): 741 Test Date: 9/5/2013Instructional Planning Reportfor Jasmine MajorPrinted Thursday, September 5, 2013 4:15:12 PMAlgebra Readiness: Jasminehas not yet met the algebra readiness grade level expectations.1 of 2Teacher: Mrs. T. WilliamsGrade: 7Projected SS for 06/16/14: 785 Based on research, 50% of students at this student's level will achieve this much growthJasmine's Current PerformanceSchool BenchmarksCurrentProjectedScaled Score 550 600 650 700 750 800 850Skills to LearnCurrentProjectedûUrgent Intervention ûIntervention ûOn Watch ûAt/Above BenchmarkJasmine's recent STAR Math scaled score(s) suggests these skills from Core Progress learning progressions would bechallenging, but not too difficult for her. Combine this information with your own knowledge of the student and use yourprofessional judgment when designing an instructional program. Use the Core Progress learning progressions to see howthese skills fit within the larger context of the progression.K-8GR7Students draw and construct geometrical figures and describe the relationship between figures with different attributes.77777777777777GeometryThey solve real-world and mathematical problems involving angle measure and the area of 2-dimensional figures. Theysolve 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» Find the volume of a right rectangular prism with fractional edge lengths using formulasWP: Find the volume of a right rectangular prism with fractional edge lengths using formulasPlot coordinates to form a polygon on the coordinate planeFind a side length of a polygon on the coordinate planeRatios and Proportional RelationshipsStudents analyze proportional relationships in real-world and mathematical problems. They write equations to representa proportional relationship. They solve multi-step real-world and mathematical ratio and percent problems.Understand the concept of a unit rate» Determine a unit rateIdentify an input-output table that contains values for a given ratioFind missing values in a ratio tableGraph in the coordinate plane the values of a ratio tableWP: Use tables to compare ratios» WP: Solve a unit rate problemThe Number System» 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 8: STAR Record Book18

ConclusionThe goal of school districts is to ensure that allstudents in all schools be fully prepared forcollege or career by the time they graduatefrom high school. The benefit of learningprogressions is that they lay out a pathway toguide teaching and learning over time so thatstudent competence in the domain can beadvanced coherently and continuously. CoreProgress for Math and Core ProgressLearning Progression for Math - Built forThe benefit of learning progressionsis that they lay out a pathway to guideteaching and learning over time sothat student competence in thedomain can be advanced coherentlyand continuously.the Common Core State Standards describe fully formed progressions of learning within the domain ofmathematics, including the intermediate steps not evident in state standards and the Common Core. Theyhelp educators locate where students are on their pathway, not only pointing in the right direction, but alsoproviding 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 educatorsachieve the intent and spirit of the standards their state has adopted, while propelling their students towardcollege and career readiness.19

CORE PROGRESS FOR MATH LEARNING PROGRESSIONAppendix A: Examples of skill progressions across grade levelsGrade123Domain: Numbers and OperationsSkill Area: Whole Numbers: Place ValueSkill• 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• 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• 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 form4 • Round a 4- to 6-digit number to a specified placeGrade4Domain: Numbers and OperationsSkill Area: Decimal Concepts and OperationsSkill• 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 hundredths5• 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 forms7 • Solve a multi-step word problem involving decimal numbers8 • Convert between standard form and scientific notation of decimal numbers20

CORE PROGRESS FOR MATH LEARNING PROGRESSIONAppendix B: Core Progress for Math includes four domains and 23 skill areasDomainNumbers and OperationsSkill 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 RootsAlgebra• Algebra: Variable Equations and Expressions• Symbols and Expressions• Functions• Linear Equations• Nonlinear Equations• Algebra of Polynomials• Quadratic EquationsGeometry and Measurement• Measurement• Time• Geometry: 2-Dimensional• Geometry: 3-DimensionalData Analysis, Statistics,and Probability• Data Representation and Analysis21

CORE PROGRESS FOR MATH LEARNING PROGRESSIONAppendix C: Core skills per grade, per domainGrade Domain Core skills (398) Total skills (1,326)Numbers and Operations 0 58Early NumeracyAlgebra 0 5Geometry and Measurement 0 9Totals 0 72Numbers and Operations 27 59Algebra 4 10Grade 1 Geometry and Measurement 7 16Data Analysis, Statistics, and Probability 0 14Totals 38 99Numbers and Operations 23 64Algebra 3 11Grade 2 Geometry and Measurement 4 14Data Analysis, Statistics, and Probability 4 9Totals 34 98Numbers and Operations 27 55Algebra 2 11Grade 3 Geometry and Measurement 1 38Data Analysis, Statistics, and Probability 0 10Totals 30 114Numbers and Operations 24 83Algebra 5 9Grade 4 Geometry and Measurement 13 42Data Analysis, Statistics, and Probability 6 9Totals 48 143Numbers and Operations 30 97Algebra 4 14Grade 5 Geometry and Measurement 8 40Data Analysis, Statistics, and Probability 1 16Totals 43 167Numbers and Operations 42 91Algebra 7 16Grade 6 Geometry and Measurement 2 32Data Analysis, Statistics, and Probability 0 18Totals 51 157Numbers and Operations 24 60Algebra 7 23Grade 7 Geometry and Measurement 9 41Data Analysis, Statistics, and Probability 5 14Totals 45 138Numbers and Operations 14 30Algebra 17 25Grade 8 Geometry and Measurement 4 20Data Analysis, Statistics, and Probability 0 24Totals 35 99Algebra 1Algebra 43 127Total 43 127GeometryGeometry and Measurement 31 112Total 31 11222

CORE PROGRESS FOR MATH LEARNING PROGRESSIONAppendix D: Example of how one core skill serves as a prerequisite for manyother core skillsCount by 5s or 10s to 100 starting from a multiple of 5 or 10, respectively is a prerequisite for the following:Grade Core Skill DomainGrade 1 Count objects grouped in tens and ones (grade 1) Numbers and operationsGrade 1 Tell time to the half hour (grade 1) Geometry and measurementGrade 2Grade 2Complete a skip pattern starting from a multiple of 2, 5,or 10Complete a skip pattern of 2, 5, or 10 starting fromany numberNumbers and operationsNumbers and operationsGrade 2 Count on by 100s from any number Numbers and operationsGrade 2Use a pictograph to represent data (1 symbol = morethan 1 object)Data analysis, statisticsand probabilityGrade 3 Tell time to the minute Geometry and measurementGrade 4Grade 4Grade 4Answer a question using information from a line graphUse a double-bar graph to represent dataAnswer a question using information from adouble-bar graphData analysis, statisticsand probabilityData analysis, statisticsand probabilityData analysis, statisticsand probabilityExample of how one core skill serves as a prerequisite for 10 skills across four grade levels in three domains.23

CORE PROGRESS FOR MATH LEARNING PROGRESSIONAppendix E: Common Core State Standards and Core Progress CCSS performance standard Grade 7, Expressions and Equations Domain, Standard 3 (7.EE.3) “Solvemulti-step real-life and mathematical problems posed with positive and negative rational numbers in anyform” is mapped to the following Core Progress skills:Grade Skill DomainGrade 4Grade 4WP: Solve a 2-step problem involving addition and/orsubtraction of multi-digit whole numbersWP: Solve a 2-step whole number problem using morethan 1 operationNumbers and operationsNumbers and operationsGrade 5 WP: Solve a 2-step problem involving whole numbers Numbers and operationsGrade 6WP: Solve a multi-step problem involvingwhole numbersNumbers and operationsGrade 6 WP: Solve a 2-step problem involving fractions Numbers and operationsGrade 6 WP: Solve a 2-step problem involving decimals Numbers and operationsGrade 7 WP: Solve a multi-step problem involving decimals Numbers and operationsGrade 7Grade 7WP: Solve a multi-step problem involving fractions ormixed numbersWP: Estimate the result of dividing or multiplying awhole number by a fractionNumbers and operationsNumbers and operationsThe Common Core State Standards set the bar. Core Progress provides the prerequisites and intermediary steps forachieving the standard.24

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSAppendix F: Organization of Skill Areas within the 11 Domains for K-8DomainCounting and CardinalityOperations and AlgebraicThinkingNumber and Operations inBase TenNumber and Operations —FractionsRatios and ProportionalRelationshipsThe Number SystemExpressions and EquationsFunctionsGeometryMeasurement and DataStatistics and ProbabilitySkill AreaWhole Numbers: Counting, Comparing, and OrderingAlgebraic ThinkingEvaluate Numerical ExpressionsWhole Numbers: Addition and SubtractionWhole Numbers: Counting, Comparing, and OrderingWhole Numbers: Multiplication and DivisionDecimal Concepts and OperationsPowers, Roots, and RadicalsWhole Numbers: Addition and SubtractionWhole Numbers: Counting, Comparing, and OrderingWhole Numbers: Multiplication and DivisionWhole Numbers: Place ValueDecimal Concepts and OperationsFraction Concepts and OperationsPercents, Ratios, and ProportionsCoordinate GeometryDecimal Concepts and OperationsFraction Concepts and OperationsIntegersWhole Numbers: Multiplication and DivisionEvaluate and Use Variable ExpressionsEvaluate Numerical ExpressionsLinear Equations and InequalitiesPowers, Roots, and RadicalsQuadratic and Nonlinear Equations and InequalitiesSystems of Equations and InequalitiesRelations and FunctionsAngles, 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 VolumeTransformationsAngles, 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 OrderingCombinatorics and ProbabilityData Representation and Analysis25

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSAppendix G: Organization of Skill Areas within the 21 Domains for high schoolDomainThe Real Number SystemQuantities*Seeing Structure inExpressionsArithmetic with Polynomialsand Rational ExpressionsCreating Equations*Reasoning with Equations andInequalitiesInterpreting FunctionsBuilding FunctionsLinear, Quadratic, andExponential Models*The Complex Number SystemTrigonometric FunctionsCongruenceSimilarity, Right Triangles, andTrigonometryCirclesExpressing GeometricProperties with EquationsGeometric Measure andDimensionModeling with GeometryConditional Probability and theRules of ProbabilityUsing Probability to MakeDecisionsFraction Concepts and OperationsPowers, Roots, and RadicalsData Representation and AnalysisSkill AreaAlgebra of PolynomialsLinear Equations and InequalitiesQuadratic and Nonlinear Equations and InequalitiesRelations and FunctionsAlgebra of PolynomialsLinear Equations and InequalitiesLinear Equations and InequalitiesQuadratic and Nonlinear Equations and InequalitiesRelations and FunctionsSystems of Equations and InequalitiesRelations and FunctionsRelations and FunctionsLinear Equations and InequalitiesQuadratic and Nonlinear Equations and InequalitiesAlgebra of PolynomialsComplex NumbersRight Triangles and TrigonometryAngles, Segments, and LinesCongruence and SimilarityGeometry: Two-Dimensional Shapes and AttributesPolygons and CirclesTransformationsCongruence and SimilarityRight Triangles and TrigonometryTransformationsPolygons and CirclesCoordinate GeometryPolygons and CirclesGeometry: Three-Dimensional Shapes and AttributesPerimeter, Circumference, and AreaSurface Area and VolumeCoordinate GeometryGeometry: Three-Dimensional Shapes and AttributesPerimeter, Circumference, and AreaPolygons and CirclesRight Triangles and TrigonometrySurface Area and VolumeCombinatorics and ProbabilityCombinatorics and Probability* Modeling Standards: Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Makingmathematical models is a Standard for Mathematical Practice.26

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSDomainInterpreting Categorical andQuantitative DataMaking Inferences andJustifying ConclusionsData Representation and AnalysisData Representation and AnalysisSkill Area27

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSAppendix H: Examples of skill progressions across grade levels,Whole Numbers: Place ValueStandardK Number and Operations in Base Ten – Work with numbers 11-19 to gain foundations for place valueCC K.NBT.1CC K.NBT.1CC K.NBT.1CC K.NBT.1CC K.NBT.1SkillDecompose a number from 11 to 19 into a group of ten ones and some further onesusing objectsDecompose a number from 11 to 19 into a group of ten ones and some further onesusing picturesCompose a number from 11 to 19 from ten ones and some further ones using objectsCompose a number from 11 to 19 from ten ones and some further ones using picturesUnderstand that the numbers 11 to 19 are composed of 10 ones and some onesGrade 1 Number and Operations in Base Ten – Understand place valueCC 1.NBT.2.aC 1.NBT.2.bC 1.NBT.2.cC 1.NBT.2Understand that 10 represents a collection of ten onesUnderstand that the numbers 11 to 19 are composed of a ten and some onesUnderstand that a number ending in zero from 10 to 90 is a group of tens and no onesUnderstand that the digits of a 2-digit number represent amounts of tens and onesGrade 2 Number and Operations in Base Ten – Understand place valueCC 2.NBT.1.aCC 2.NBT.1.bCC 2.NBT.1Understand that 100 represents a collection of 10 tensUnderstand that a number ending in two zeros from 100 to 900 is a group of hundreds and zerotens and zero onesUnderstand that the digits of a 3-digit number represent amounts of hundreds, tens, and ones28

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSAppendix I: Core skills per grade, per domainGrade Domain Core skills (255) Total Skills (1028)Counting and Cardinality 6 18Operations and Algebraic Thinking 6 28KindergartenNumber and Operations in Base Ten 1 5Measurement and Data 1 8Geometry 1 12Totals 15 71Operations and Algebraic Thinking 6 24Number and Operations in Base Ten 10 27Grade 1 Measurement and Data 2 9Geometry 1 11Totals 19 71Operations and Algebraic Thinking 2 10Number and Operations in Base Ten 8 29Grade 2 Measurement and Data 4 19Geometry 1 9Totals 15 67Operations and Algebraic Thinking 7 19Number and Operations in Base Ten 3 5Grade 3Number and Operations — Fractions 4 15Measurement and Data 4 31Geometry 1 6Totals 19 76Operations and Algebraic Thinking 4 15Number and Operations in Base Ten 6 13Grade 4Number and Operations — Fractions 8 26Measurement and Data 0 14Geometry 2 7Totals 20 75Operations and Algebraic Thinking 1 6Number and Operations in Base Ten 9 31Grade 5Number and Operations — Fractions 6 19Measurement and Data 3 12Geometry 1 7Totals 20 75Ratios and Proportional Relationships 4 14The Number System 4 30Grade 6Expressions and Equations 5 23Geometry 3 12Statistics and Probability 2 12Totals 18 91Ratios and Proportional Relationships 5 9The Number System 7 21Grade 7 Expressions and Equations 5 13Geometry 4 13Statistics and Probability 4 24Totals 25 8029

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSGrade Domain Core skills (255) Total Skills (1028)The Number System 1 8Expressions and Equations 6 27Grade 8Functions 5 11Geometry 7 23Statistics and Probability 2 8Totals 21 77The Real Number System 2 5Quantities* 3 5Seeing Structure in Expressions 3 9Arithmetic with Polynomials and Rational Expressions 0 3Creating Equations* 3 12Grade 9 Reasoning with Equations and Inequalities 6 19Interpreting Functions 6 26Building Functions 4 17Linear, Quadratic, and Exponential Models* 2 9Interpreting Categorical and Quantitative Data 2 17Totals 31 122Congruence 3 41Similarity, Right Triangles, and Trigonometry 9 25Circles 3 12Expressing Geometric Properties with Equations 5 11Grade 10 Geometric Measure and Dimension 1 8Modeling with Geometry 0 6Conditional Probability and the Rules of Probability 5 23Using Probability to Make Decisions 1 2Totals 27 128The Complex Number System 1 9Seeing Structure in Expressions 2 6Arithmetic with Polynomials and Rational Expressions 3 19Creating Equations* 1 7Reasoning with Equations and Inequalities 2 5Interpreting Functions 7 14Grade 11 Building Functions 1 8Linear, Quadratic, and Exponential Models* 1 2Trigonometric Functions 3 6Interpreting Categorical and Quantitative Data 1 4Making Inferences and Justifying Conclusions 3 13Using Probability to Make Decisions 0 2Totals 25 95* Modeling Standards: Modeling is best interpreted not as a collection of isolated topics, but rather in relation to other standards. Makingmathematical models is a Standard for Mathematical Practice.30

CORE PROGRESS LEARNING PROGRESSION FOR MATH - BUILT FOR THE COMMON CORE STATE STANDARDSAppendix J: Example of mapping a core skill to many other core skills, FractionsGrade 3 Core Skill: Understand the structure of a fractionGrade Core Skill Domain34Explain why two fractions are equivalent using a visualfraction modelDecompose a fraction into a sum of fractions with thesame denominator in more than one way4 Multiply a fraction by a whole number4 Understand a fraction as a multiple of a unit fraction5Interpret a fraction as division of the numerator by thedenominatorNumber and Operations-FractionsNumber and Operations-FractionsNumber and Operations-FractionsNumber and Operations-FractionsNumber and Operations-Fractions31

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AcknowledgementsDr. 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 learningprogressions, text complexity, performance assessment, and cognitive rigor. In additionto experiences as a classroom teacher and school administrator, she has been a programevaluator for the Vermont Mathematics Project; a content specialist for development of theReviewer Vermont Science assessment; and as developer and editor of Science Exemplars K-8performance assessments. Dr. Hess is the principal author of the content specificationsfor assessment of the CCSS ELA and Literacy standards and was a contributor to the mathematics contentspecifications for the Smarter Balanced Assessment Consortium.ReviewerDr. R. James Milgram is an emeritus professor of mathematics at Stanford University wherehe has taught since 1970. He is a former member of the National Board for EducationSciences, the NASA Advisory Council, Common Grounds Project, and Achieve MathematicsAdvisory Panel. Dr. Milgram has helped author several states’ standards and recently servedon the Validation Committee for the Common Core State Standards. He has publishedover 100 research papers in mathematics and four books. Dr. Milgram received hisundergraduate 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 directedand evaluated numerous school-wide intervention and reform efforts, most often in thearea of mathematics. Dr. VanDerHeyden serves as advisor to the National Center forLearning Disabilities, iSTEEP (a web-based data management system), and is a standingpanel 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, andReviewer has given keynote addresses to state school psychology associations and state departmentsof education in 21 states. She is co-author of the Evidence-Based Mathematics Innovation Configuration for theNational Comprehensive Center for Teacher Quality at Vanderbilt University and now the Collaboration forEffective Education Development, Accountability, and Reform at University of Florida.L2791.0713.XX.XMR55248