the development of the recommendations accelerated math...

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THE DEVELOPMENT OF THE

Accelerated Math Second-Edition Libraries

With content aligned to National Math Panel

recommendations and NCTM Curriculum

Focal Points

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

© 2011 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 LEARNING, INC.P.O. Box 8036Wisconsin Rapids, WI 54495-8036(800) [email protected]

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Accelerated Math was the first progress-monitoring tool reviewed by the National Center on Response to Intervention to be highly rated as a mastery measurement tool. Its high marks confirm what teachers nationwide have said all along—Accelerated Math is truly in a class by itself.

Accelerated Math meets all criteria for scientifically based progress-monitoring tools set by the National Center on Student Progress Monitoring.

Accelerated Math has earned the top rating for Prevention and Intervention at all grade levels by the National Dropout Prevention Center.

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ContentsIntroduction ...........................................................................................................................................................1

Recent History of Mathematics Education ............................................................................................................1

Accelerated Math Content Revision Overview ......................................................................................................3

Phase I: Accelerated Math Content Audit ......................................................................................................3

Phase II: Core Objectives Identification ........................................................................................................ 4

Phase III: Scope and Sequence Development ..............................................................................................5

Phase IV: Expert Review .................................................................................................................................6

Phase V: Content Development ......................................................................................................................7

Phase VI: State Standards Alignment .............................................................................................................7

Conclusion .............................................................................................................................................................8

References ............................................................................................................................................................9

Acknowledgements .............................................................................................................................................10

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IntroductionAccelerated Math is a computerized tool for efficient progress monitoring and management of students’ personalized daily math practice for grades 1–12 within a formative assessment process. The program’s content is aligned to state standards and national guidelines, and teachers use data generated by this tool to differentiate instruction and address individual student needs. Accelerated Math includes separate libraries for Early Numeracy, Grades 1 through 8, Algebra 1 and 2, Geometry, Probability and Statistics, Pre-Calculus, Calculus, and Basic Math.

To ensure the content in the second edition of Accelerated Math meets today’s highest standards, Renaissance Learning recently developed a new scope and sequence for Early Numeracy, Grades 1 through 8, Algebra 1, and Geometry that includes core objectives, learning progressions, and prerequisite skills. This new material is collectively called the Accelerated Math Second-Edition Libraries. The new Second-Edition Libraries are available with Renaissance Place versions of Accelerated Math. For Accelerated Math Enterprise users, the new libraries are available at no additional cost; Service Subscription users may purchase the new libraries individually.

Several resources were utilized to develop the new scope and sequence. The Renaissance Learning mathematics team conducted extensive research, relying heavily on the National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points, the National Mathematics Advisory Panel final report, and state and international standards. The math team also convened a reviewer panel of mathematicians and researchers (Dr. Sybilla Beckmann, University of Georgia; Dr. Richard Bisk, Worcester State College; Dr. Tom Hogan, University of Scranton; Dr. R. James Milgram, Stanford University; Dr. Sharif M. Shakrani, a private consultant and researcher; and Dr. Amanda VanDerHeyden, a private consultant and researcher), sought input from the U.S. Department of Education’s Northwest Regional Educational Laboratory, and consulted mathematics teachers in several states.

This white paper documents the extensive revision process that was followed.

Recent history of mathematics educationThe transformation of U.S. math standards, curriculum, instruction, and practice is long overdue. American students have not merely fallen behind world-class math students like those from Singapore—they have been seriously outpaced.

In 2003, U.S. eighth graders placed 15th in math on the Trends in International Mathematics and Science Study (TIMSS) (Mullis, Martin, Gonzalez, & Chrostowski, 2004). In contrast, Singapore’s eighth graders took the top position three times in a row—in 1995, 1999, and most recently in 2003 (International Association for Evaluation of Education Achievement, 1995, 1999, 2003).1

In a seminal 2005 article, “The Key Topics in a Successful Math Curriculum,” mathematicians Milgram and Hung-Hsi Wu joined a chorus of researchers led by Schmidt, McKnight, and Raizen (1997) that have described the U.S. mathematics curriculum as “a mile wide and an inch deep.” They explained that, “It covers too many topics and each topic is treated superficially. By contrast, the structure of mathematics instruction in countries that outperformed the U.S. follows a strikingly different pattern. In all cases, only a few carefully selected focus topics are taught and learned to mastery by students in the early grades” (p.1). In 2006, 15-year-old American math students ranked 32nd on the 2006 Programme for International Student Assessment (PISA) test, while Chinese Taipei, Finland, the Republic of Korea, the Republic of Hong Kong-China, and the Netherlands were in the top five (Organisation for Economic Co-operation and Development, 2007a, 2007b).

1 The 2007 TIMSS report was not available at the time the Accelerated Math Second-Edition Libraries were developed.

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A turning point came in the United States in 2006, when two critical events paved the way to success for American math students. In April 2006, President Bush created the National Mathematics Advisory Panel, commonly referred to as the National Math Panel (NMP), and charged them to provide recommendations on the “best use of scientifically based research to advance the teaching and learning of mathematics” (Executive Order No. 13,398, 2006). In September 2006, the NCTM released Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics, which has become a cornerstone document in identifying “what mathematics students should know and be able to do” (p. 9). Both events are driving critical changes to state standards and publishers’ treatment of mathematics topics.

A new direction. Both the NMP and NCTM recommendations were informed by nations on the cutting edge of mathematics instruction, and they now serve as blueprints for the new way mathematics will be taught in the U.S. The authors of these reports have, in effect, put forth a national challenge. These blueprints must be read, interpreted, and implemented—soon—throughout the entire country.

The NCTM Curriculum Focal Points present the three most important mathematical topics for each grade level, prekindergarten through grade 8 (Fennell, 2006). The National Math Panel’s final report, released in March 2008, includes the “Benchmarks for the Critical Foundations.” The NMP recommended that these benchmarks be used “to guide classroom curricula, mathematics instruction, and state assessments. They should be interpreted flexibly, to allow for the needs of students and teachers” (p. 20).

According to the NMP (2008), “A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided” (p. 22). Likewise, the NCTM Curriculum Focal Points (2006) asserted, “When instruction focuses on a small number of key areas of emphasis, students gain extended experience with core concepts and skills…The decision to organize instruction around focal points assumes that the learning of mathematics is cumulative, with work in the later grades building on and deepening what students have learned in the earlier grades, without repeti-tious and inefficient reteaching” (p. 5).

In addition to a “deep understanding” of topics, the NCTM (2006) emphasized the importance of mathematical fluency (p. 5). The NMP (2008) also stressed that “practice allows students to achieve automaticity of basic skills—the fast, accurate, and effortless processing of content information—which frees up working memory for more complex aspects of problem solving” (p. 30).

The call is clear. New systems for teaching mathematics in the U.S. must be structured with a focused, coherent progression, with an emphasis on mastery of fewer key topics at each grade, and without inefficient repetition. This new system must give students extended experience with core concepts and skills, and work in the later grades must build upon and deepen what students have learned in the earlier grades. The system must emphasize mathematical fluency and early automaticity in order to free mathematical learning capacity. This model of mathematics teaching is one that educators admire and welcome, but the process of transformation to the new model can seem a daunting, Herculean task. Renaissance Learning’s Accelerated Math daily practice and diagnostic software was first developed based on all of these broad philosophical underpinnings. And the Accelerated Math Second-Edition Libraries were created with the utmost attention to these recommendations, the latest research, and the needs of American math educators and students, in order to aid in this transformation.

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Accelerated Math Content Revision OverviewThis section highlights key steps taken during each phase of the extensive Accelerated Math content revision for the second edition, which resulted in the creation of the Accelerated Math Second-Edition Libraries.

Initially, a new scope and sequence, including core objectives, learning progressions, and prerequisite skills, was created for Grades 1 through 8, Algebra 1, and Geometry. After the release of these libraries, however, it became clear that Accelerated Math Second Edition must reach further to provide practice for math objectives associated with early numeracy skills. The NCTM Curriculum Focal Points had identified foundational prekindergarten and kindergarten skills, and the prerelease of the 2009 Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity report (by the National Research Council and the Committee on Early Childhood Mathematics) gave a resounding call for greater emphasis on early numeracy. Thus, the Accelerated Math Early Numeracy Library was developed with objectives to meet math practice needs for early or struggling math learners. Phase I: Accelerated Math content auditThe Accelerated Math libraries were first published in 1998, with a scope and sequence based on commonalities between the 1989 NCTM standards and leading publisher textbooks, National Assessment of Educational Progress (NAEP) editions, and math editor teaching experience from the 1990s. A great deal has occurred in education in the U.S. since 1998, and much professional thought has gone into what is important in mathematics. The initial Accelerated Math content predated not only the National Math Panel recommendations and the NCTM Curriculum Focal Points, but also the NCTM’s 2000 release of Principles and Standards for School Mathematics, the No Child Left Behind Act of 2001, and revised grade-level expectations in nearly every state’s mathematics content standards. Renaissance Learning took all of these landmark changes into account in the new Accelerated Math Second-Edition Libraries.

The first phase of the extensive content upgrade for Accelerated Math Second Edition began in fall 2006 with a review of selected content for grades 3, 4, and 5 by the Northwest Regional Educational Laboratory (NWREL). NWREL, based in Portland, Oregon, is part of a network of 10 regional educational laboratories and “provides research-based products, technical assistance, and training to improve educational systems and learning.” The audit focused on 163 Accelerated Math objectives on whole-number skills for grades 3, 4, and 5, and 99 specific problems for grade 5. K–12 curriculum experts from NWREL analyzed the objectives for consistency with accepted, common expectations for U.S. students based on standards for 10 states, representing nearly 50 percent of the U.S. student population: Arizona, California, Florida, Maryland, Michigan, New York, North Carolina, Ohio, Texas, and Washington. They also considered NCTM standards and focal points. NWREL curriculum and assessment experts analyzed the items for difficulty, alignment to Accelerated Math objectives, accuracy, item quality, and relationship to current pedagogy.

The second phase of the content audit was a real-world classroom-experience review by active classroom teachers. Renaissance Learning selected six teachers representing diverse geographic and marketing regions who had used Accelerated Math in at least two of the three grades reviewed—3, 4, and 5. Four of the states the teachers represented, California, North Carolina, Texas, and Washington, were also included in the standards-alignment survey of the NWREL report.

The auditors found three major areas of strength in the Accelerated Math content:

• Objectives matched approximately 85 percent of skills that are actually taught by teachers in the six target states.

• Number and numeration skills were well covered.

• Objectives presented a good balance of difficulty, with more than 75 percent falling between being too easy and too difficult.

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The auditors also suggested seven areas for improvement that were incorporated into the revision of the Accelerated Math Second-Edition Libraries:

• Improve the degree of alignment between objectives and state standards.

• Vary and increase the level of cognitive demand.

• Create items that require an understanding of math properties and the relationships among them.

• Increase the number of word problems and problems with context and real-life situations, and revise the approach to word problems to require problem-solving strategies.

• Remove the current objectives for estimation and exponents, and revise the approach to presenting estimation problems.

• Increase the 50-character limit for objective names, and include a verb indicating what the student should know or do.

• Revise objectives/items so they do not dictate a single method to arrive at the correct answer.

Audit feedback from experts at NWREL and the teacher panel was distilled into an executive summary to be applied to the revision of the content libraries for Grades 1 through 8, Algebra 1, and Geometry. Two principal goals were identified: (1) Reduce the overlap of objectives between grades, and (2) establish a clear progression of difficulty level through the grades. The Renaissance Learning math team then gave serious consideration as to how the objectives in the original Accelerated Math content should be revised.

Phase II: Core objectives identification The goal was to develop a set of core objectives students need to achieve to be successful in math. This included researching several sources, including the NCTM Curriculum Focal Points, National Math Panel recommendations, state standards, top international standards, Accelerated Math student achievement data, and information from recognized mathematicians. What follows is an explanation of how these resources were utlized. The Renaissance Learning math team also studied Liping Ma’s 1999 book, Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China and the United States and Milgram and Wu’s (2005) article, “The Key Topics in a Successful Math Curriculum.”

• NCTM Curriculum Focal Points: For each grade level, the NCTM’s three focal points served as the basis for decisions about the essential math topics to include.

• National Math Panel recommendations: The NMP’s “essential mathematical concepts and skills,” found in the “Report of the Task Group on Conceptual Knowledge and Skills” (Fennell et al., 2008), were cross-referenced to determine if any grade-level skills were missing from the selected core objectives. Also closely followed were the NMP’s “Benchmarks for the Critical Foundations” and the recommendation that curricula bring students to the mastery of key topics and a progression of difficulty level, rather than the circular approach of revisiting topics from grade to grade.

• Model state standards: It was crucial that the core objectives chosen for the Accelerated Math Second-Edition Libraries be aligned with state standards. To research commonalities among standards, the math team used three sources to select a group of states with model math standards. From the Thomas B. Fordham Foundation report, 2006: The State of State Standards (Finn, Julian, & Petrilli, 2006), they culled the states of California, Indiana, and Massachusetts, which the report gave an A in math standards, and Alabama, Georgia, and New Mexico, which received a B. The math team cross- referenced this list with the states meeting the American Federation of Teachers content standards in math in that organization’s 2006 policy brief, “Smart Testing: Let’s Get it Right.” In that listing, all but nine states were deemed to have strong math standards. The states being considered as the core model states were then checked against the 2007 U.S. Chamber of Commerce Education Report Card. In that

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report, California, Indiana, and Massachusetts received high marks for their math standards, and Georgia received above-average marks.

The team then made their final selection of model states. California and Indiana were included in the final pool because they were mentioned by all three sources. Alabama, Georgia, and Massachusetts, which were on the Fordham Foundation and the Chamber of Commerce lists, were also selected. Texas was high on the American Federation of Teachers list and received an A for rigors of standards on the Chamber of Commerce list, but did not rank high on the Fordham Foundation report; however, the population of Texas as the second-largest state tipped the balance in favor of including it. Lastly, Florida was included because the state, also populous, was in the process of incorporating the NCTM’s Curriculum Focal Points into its standards. The standards of Florida and Texas, like those of California, also had the advantage of having been used for comparison with Accelerated Math objectives in the NWREL audit.

• Singapore Math standards and other top international standards: The math team studied the standards of countries with top-performing math students to learn from their approach. They learned that the primary goal of these countries’ math systems is to develop students’ mathematical problem-solving abilities. There are fewer standards, and standards are not repeated across grade levels. Each year, students are expected to build upon and deepen their understanding of the skills and concepts learned in previous grades. And greater emphasis is placed on using concrete representations to teach concepts.

• Accelerated Math data-mining project: This project looked at data for 66,000 Accelerated Math students from 88 schools/districts over a 3-year period. The study yielded data about which objectives these students struggled with most and resulted in a set of objectives to cross-check for possible inclusion in the core objectives.

• Initial review of core objectives: When a draft of the core objectives was complete, University of Scranton professor Dr. Tom Hogan provided expert review. The math team incorporated Professor Hogan’s objective-by-objective feedback and general comments.

Phase III: Scope and sequence development In their 2008 report, the National Math Panel made strong recommendations about scope and sequence, specifically that “a focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm….By the term focused, the Panel means that curriculum must include…the most important topics. By the term coherent, the Panel means that the curriculum is marked by effective, logical progressions from earlier, less sophisticated topics into later, more sophisticated ones” (p. xvii).

After auditing existing content and identifying a new set of core objectives for Grades 1 through 8, the Renaissance Learning math team began work on a new scope and sequence of Accelerated Math objectives. To accomplish this, team members studied the math standards of the seven model states, Alabama, California, Florida, Georgia, Indiana, Massachusetts, and Texas, and identified six key strands. These became the topics for the new libraries for Grades 1 through 8, Algebra 1, and Geometry.

The process of placing the core objectives into their respective strands required consulting all of the sources the math team had studied during the development process—the NMP’s work, including its final report and “Benchmarks for the Critical Foundations,” the NCTM Curriculum Focal Points, the Singapore primary and secondary mathematics standards, and the math standards of each of the selected model states. After initial definition of the scope and sequence was complete, it was submitted to NWREL for external review.

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Later, when the team developed the Early Numeracy scope and sequence, they incorporated key recommendations from the National Research Council’s (2009) report, Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, which advises that early numeracy math experiences should focus on numbers, including whole numbers, operations, and relations, as well as geometry, spatial relations, and measurement. The greater emphasis, however, is to be on numbers.

While work was being done to develop the new scope and sequence, members of the math team also researched effective development of algorithm-generated dynamic items and applied their findings to the creation of new items.

Learning progressions.A key feature of the Accelerated Math scope and sequence is the identification of learning progressions and prerequisite objectives. A learning progression is a carefully sequenced set of building blocks students must master en route to mastering a more distant curricular aim. These building blocks consist of subskills and bodies of enabling knowledge (Popham, 2007). The objectives are linked together in a manner that indicates the progression of the associated skills. A prerequisite objective is an objective that provides practice with the prerequisite skills needed to become proficient with the requisite objective. The goal of the learning progressions and prerequisite objectives is to provide a structure that facilitates identifying appropriate practice for students, enabling them to increase their math achievement.

A copy of the Accelerated Math Second-Edition Libraries scope and sequence is available by request to (800) 338-4204.

Phase IV: Expert review As mentioned previously, Renaissance Learning sought input during each phase of the content revision— the Accelerated Math content audit, the core objectives identification, and the scope and sequence development. During Phase IV, the Renaissance Learning math team presented sample items to several experts for input, including mathematicians, researchers, mathematics teachers, and the Northwest Regional Educational Laboratory.

• Grades 1 through 8 expert review: A panel of prominent university-based mathematicians was convened in Menlo Park, California, to review the core objectives and scope and sequence for a sample of grade levels. Dr. R. James Milgram, professor of mathematics at Stanford University, analyzed the Grade 3 core objectives and scope and sequence; Dr. Sybilla Beckmann, professor of mathematics at the University of Georgia and a member of the NCTM Curriculum Focal Points writing team, reviewed Grade 5; and Dr. Richard Bisk, professor of mathematics at Worcester State College in Massachusetts, reviewed Grade 6. Based on this feedback, adjusments were made as appropriate to the content libraries for Grades 1 through 8, Algebra 1, and Geometry.

• Algebra 1 expert review: The Algebra 1 scope and sequence was submitted for review to Dr. Sharif M. Shakrani, a private consultant and researcher as well as former professor of measurement and quantitative methods at Michigan State University. Dr. Shakrani called the scope and sequence “very comprehensive and rigorous in depth and content.” Dr. Shakrani also compared the Algebra 1 topics and objectives to the Achieve/American Diploma Project Algebra 1 content standards and found “a great deal of commonality in the algebra topics. Of the 27 objectives listed by Achieve, 22 are common to AM Algebra 1.”

• Early Numeracy expert review: The Early Numeracy scope and sequence and sample items were submitted for review to Dr. Amanda VanDerHeyden, a private consultant and researcher. The Renaissance Learning math team then met with Dr. VanDerHeyden to discuss the feedback she submitted. Appropriate adjustments were made to the Early Numeracy content.

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• Mathematics teachers: A group of teachers from Delaware, Texas, Washington, and Wisconsin reviewed items for grades 1 through 4 for grade-level appropriateness.

• Northwest Regional Educational Laboratory: NWREL, which had performed the initial audit of Accelerated Math, also reviewed the learning progressions, core objectives, scope and sequence, and prerequisite mappings for the Accelerated Math Second-Edition Libraries. In their report on the Algebra 1 scope and sequence, NWREL reviewers praised the appropriateness of the course’s strands and topics. In addition, they noted the Algebra 1 objectives were clearly stated and included a proper level of specificity.

Phase V: Content development As new items were created, a strict process was maintained to ensure quality item development. Math editors made certain that each new Second-Edition library item was aligned to but did not exceed the skills indicated by the item’s objective statement. Items were then analyzed to ensure accuracy. Item stems and answer choices were also checked to ensure the cognitive load, math vocabulary, and readability were grade-level appropriate. Common mistakes students make when performing an operation were chosen as incorrect answer choices, or distractors, to aid teachers in determining a student’s missteps while solving a problem.

The Renaissance Learning content-development process also required all newly written items to be examined closely by two math editors to (1) eliminate distractors that could allow students to guess to arrive at an answer, and (2) assess for bias and content appropriateness. Then the items were edited by a language arts editor and a math proofreader to ensure that all grammar, usage, mechanics, and spelling adhered to rules in designated content-reference books.

Phase VI: State standards alignment Renaissance Learning recognizes the impact of standards-based reform and high-stakes standardized testing on schools, and we share the concern of both educators and administrators that students perform well on high-stakes assessments. Thus, after developing the content for the Accelerated Math Second-Edition Libraries, Renaissance Learning standards staff aligned the new content to the mathematics standards of all 50 states and the District of Columbia.

Working with Mid-continent Research for Education and Learning and the Northwest Regional Educational Laboratory, Renaissance Learning developed an alignment strategy to balance both the objective and subjective aspects of alignments to standards. This strategy is documented with definitions and examples for each specific type of 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, all standards staff received extensive training, including how to calibrate alignment results.

State alignment reports are available online from http://www.renlearn.com/fundingcenter/statestandardalignments or by request to (800) 338-4204. The Renaissance Learning alignment document is available by request to (800) 338-4204.

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Conclusion Teachers nationwide have said all along that Accelerated Math is in a class by itself. The new Accelerated Math Second-Edition Libraries build upon the many strengths of this program to marry the best research available to the mathematics standards school districts across the U.S. are demanding now—and will be demanding for years to come. Accelerated Math Second Edition facilitates students’ progress in mathematics so they are ready each school year for more advanced levels of math challenge. Our hope is that Accelerated Math will help raise American students’ level of mathematics achievement and pave the way for the U.S. to be ranked among the “A+ countries” in math for the twenty-first century.

Interested school districts are encouraged to contact Renaissance Learning for more information or a product demonstration: (800) 338-4204.

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ReferencesAmerican Federation of Teachers, AFL-CIO. (2006, July). Smart testing: Let’s get it right: How assessment-savvy have states be come since NCLB (Policy Brief Number 19)? Washington, DC: AFT Teachers. Retrieved March 4, 2008, from http://www.aft.org/presscenter/releases/2006/smarttesting/Testingbrief.pdf Exec. Order No. 13,398, 3 C.F.R. 216 (2006).

Fennell, F. (2006, September). Skip Fennell announces curriculum focal points to NCTM. Message posted to http://hub.mspnet.org/index.cfm/13434

Fennell, F., Faulkner, L. R., Ma, L., Schmid, W., Stotsky, S., Wu, H., & Flawn, T. (2008). Foundations for success: Report of the task group on conceptual knowledge and skills. Washington, DC: U.S. Department of Education. Retrieved March 22, 2008,

from http://www.ed.gov/about/bdscomm/list/mathpanel/report/conceptual-knowledge.pdf

Finn, C. E., Jr., Julian, L., & Petrilli, M. J. (2006). 2006: The state of state standards. Retrieved March 4, 2008, from the Thomas B. Fordham Institute website: http://www.edexcellence.net/institute/publication/publication.cfm?id=358

International Association for Evaluation of Education Achievement. (1995). Trends in international mathematics and science study (TIMSS). Retrieved March 22, 2008, from http://www.questationmath.com/TIMSSReport.php



Ma, L. (1999). Knowing and teaching elementary mathematics: Teachers’ understanding of fundamental mathematics in China and the United States. New York: Lawrence Erlbaum Associates, Inc.

Milgram, R. J., & Wu, H. (2005). The key topics in a successful math curriculum. Retrieved March 22, 2008, from http://math.berkeley.edu/~wu/six-topics1.pdf

Mullis, I. V. S., Martin, M. O., Gonzalez, E. J., & Chrostowski, S. J. (2004). TIMSS 2003 international mathematics report: Findings from IEA’s Trends in International Mathematics and Science Study at the Fourth and Eighth Grades. Chestnut Hill, MA:

Boston College, Lynch School of Education, TIMSS & PIRLS International Study Center. Retrieved April 4, 2008, from http://timss.bc.edu/PDF/t03_download/T03INTLMATRPT.pdf

National Council of Teachers of Mathematics. (2006). Curriculum focal points for prekindergarten through grade 8 mathematics. Reston, VA: Author. Retrieved March 22, 2008, from http://www.nctm.org/standards/focalpoints.aspx?id=282

National Mathematics Advisory Panel. (2008). Foundations for success: The final report of the National Mathematics Advisory Panel. Washington, DC: U.S. Department of Education. Retrieved March 22, 2008, from

http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-report.pdf

National Research Council. (2009). Mathematics learning in early childhood: Paths toward excellence and equity. (Committee on Early Childhood Mathematics, C. T., Cross, T. A. Woods, & H. Schweingruber, Eds.). Washington, DC: The National

Academies Press.

Northwest Regional Educational Laboratory. (2008). About the Northwest Regional Educational Laboratory. Retrieved March 22, 2008, from http://www.nwrel.org/comm/index.php

Organisation for Economic Co-operation and Development. (2007a, March). PISA 2006: Science competencies for tomorrow’s world: Volume 1: Analysis. Paris: OECD Publishing. Available online from

http://www.pisa.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html

Organisation for Economic Co-operation and Development. (2007b, March). PISA 2006: Science competencies for tomorrow’s world: Volume 2: Data. Paris: OECD Publishing. Available online from

http://www.pisa.oecd.org/document/2/0,3343,en_32252351_32236191_39718850_1_1_1_1,00.html

Popham, W. J. (2007). The lowdown on learning progressions. Educational Leadership, 64(7), 83–84.

Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1997). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, The Netherlands: Kluwer.

U.S. Chamber of Commerce. (2007). Leaders and laggards: A state-by-state report card on educational effectiveness. Retrieved March 5, 2008, from http://www.uschamber.com/icw/reportcard/default

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AcknowledgmentsThe Renaissance Learning mathematics team extends deep gratitude to the individuals who have helped further our work on the Accelerated Math Second-Edition Libraries. We have great esteem for these individuals’ knowledge and accomplishments.

Mathematicians and researchersRenaissance Learning wishes to express appreciation to the panel of mathematicians who analyzed and provided expert feedback on the content of the Accelerated Math Second-Edition Libraries.

Sybilla Beckmann, Ph.D., is a professor of mathematics at the University of Georgia. She is especially interested in helping college faculty learn to teach mathematics content courses for elementary and middle grades teachers and has developed three courses for prospective elementary school teachers at the University of Georgia. She has written a book for such courses, Mathematics for Elementary Teachers, published by Addison-Wesley, now in a second edition. Beckmann was a member of the writing team for the NCTM’s Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics and has worked on the development of several state mathematics standards.

Richard Bisk, Ph.D., is chair and professor of mathematics at Worcester State College in Massachusetts, where he teaches mathematical modeling, linear algebra, number theory, and mathematics for elementary teachers. He has worked with K–12 teachers and students for 15 years and has taught and developed numerous professional development courses that focus on improving teacher understanding of mathematics, including the Singapore Math Project, which was developed in conjunction with the Massachusetts Department of Education. Bisk also presented testimony before the National Math Panel in September 2006 regarding the need to improve preservice elementary teacher knowledge of the mathematics they teach. And he assisted in the development of Massachusetts’s new guidelines for the mathematical preparation of elementary teachers.

Thomas P. Hogan, Ph.D., is a professor of psychology and a Distinguished University Fellow at the University of Scranton. He has more than 40 years of experience conducting reviews of mathematics curricular content, principally in connection with the preparation of a wide variety of educational tests, including the Stanford Diagnostic Mathematics Test, Stanford Modern Mathematics Test, and the Metropolitan Achievement Test. Hogan has published articles in the Journal for Research in Mathematics Education and Mathematical Thinking and Learning, and has authored two textbooks and more than 100 scholarly publications in the areas of measurement and evaluation. He has also served as consultant to a wide variety of school systems, states, and other organizations on matters of educational assessment, program evaluation, and research design.

R. James Milgram, Ph.D., is a professor of mathematics at Stanford University. His work in mathematics education includes consulting with several states on math standards, including California. Milgram has given lectures around the world and is a member of numerous boards and committees, including the National Board of Education Sciences, created by the Education Sciences Reform Act of 2002 “to advise and consult with the Director of the Institute of Education Sciences (IES) on agency policies,” and the Human Capital Committee of the NASA Advisory Council, which “provides the NASA Administrator with counsel and advice on programs and issues of importance to the Agency.” Milgram is author of “An Evaluation of CMP,” “A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California,” and “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later.” Each of these papers identifies serious shortcomings in popular mathematics programs.

Sharif M. Shakrani, Ph.D., is a private consultant and researcher specializing in measurement and quantitative methods. Shakrani is a former co-director of the Education Policy Center at Michigan State University and professor of measurement and quantitative methods in the Department of Counseling, Educational Psychology and Special Education. Before coming to Michigan State University, Dr. Shakrani served 8 years

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as the deputy executive director of the National Assessment Governing Board in the U.S. Department of Education. He was responsible for technical and policy direction for the National Assessment of Educational Programs (NAEP). He has also worked for the National Center for Education Statistics in the U.S. Department of Education where he guided the design and analysis of federal educational assessments. In his work in the Michigan Department of Education, Dr. Shakrani was responsible for K–12 general curriculum and assessment and was instrumental in revising the Michigan Educational Assessment Program (MEAP).

Amanda M. VanDerHeyden, Ph.D., is a private consultant and researcher, living in Fairhope, Alabama, who previously has held faculty positions, and has worked as a researcher, consultant, and national trainer in a number of school districts. In 2006, Dr. VanDerHeyden was named to a National Center for Learning Disabilities advisory panel to provide guidance related to RTI and the diagnosis of specific learning disability. She is associate editor of Assessment for Effective Intervention, serves on the editorial boards of several journals including School Psychology Quarterly and Journal of School Psychology, and has recently co- authored Essentials of Response to Intervention (with Dr. Matthew Burns). Dr. VanDerHeyden received the 2006 Lightner Witmer Early Career Contributions Award from the APA for her scholarship on early intervention, RTI, and models of data-based decision making. She serves as research advisor to iSTEEP, has published measures of early numeracy for preschool and kindergarten children, and currently directs research projects in early numeracy with preschool/kindergarten children as well as evaluation of Tier 2 mathematics intervention.

Northwest Regional Educational LaboratoryThe Northwest Regional Educational Laboratory (NWREL), based in Portland, Oregon, is a private, nonprofit corporation funded by the U.S. Department of Education that provides research-based products, technical assistance, and training to improve educational systems and learning. NWREL’s mission is to improve learning by building capacity in schools, families, and communities through applied research and development. The Renaissance Learning mathematics team benefited from the skills of mathematics curriculum and assessment experts at NWREL who were integral to the development process.

Classroom teachersFinally, in developing the Accelerated Math Second-Edition Libraries, Renaissance Learning was fortunate to be able to call on the experience and knowledge of a group of practicing mathematics teachers, diverse in both geography and background.

Lostra Burrow Temple Elementary School, Diboll, Texas

Denise Chicks Hockinson Primary School, Hockinson, Washington

Tanna Colwell Hockinson Intermediate School, Hockinson, Washington

Shellie Davis Temple Elementary School, Diboll, Texas

Wayne Dickey Sierra Elementary School, Tollhouse, California

Susie Martel Temple Elementary School, Diboll, Texas

Kay Mills Tannahill Intermediate School, Fort Worth, Texas

Terri Pink Highland Elementary School, Highland, Wisconsin

Rodney Rowe Prairie Elementary School, Worthington, Minnesota

Lauren Shouse Highland Renaissance Academy, Charlotte, North Carolina

Thomasina Simmons Iroquois Point Elementary School, Ewa Beach, Hawaii

Diane Smith Hockinson Primary School, Hockinson, Washington

Julie Smith Temple Elementary School, Diboll, Texas

Melissa Tuttle Benjamin Banneker Elementary School, Milford, Delaware

AcknowledgementsRenaissance Learning sincerely thanks the following individuals for lending their expertise in mathematics and developing assessments to reviewing the revised content for the Accelerated Math Second-Edition Libraries. (To learn more about all of the expert reviewers involved in the content revision, please see the detailed Acknowledgements section inside.)

Sybilla Beckmann, Ph.D., is a professor of mathematics at the University of Georgia.

Richard Bisk, Ph.D., is chair and professor of mathematics at Worcester State College in Massachusetts.

Thomas P. Hogan, Ph.D., is a professor of psychology and a Distinguished University Fellow at the University of Scranton.

R. James Milgram, Ph.D., is a professor of mathematics at Stanford University.

Sharif M. Shakrani, Ph.D., is a private consultant and researcher specializing in measurement and quantitative methods.

Amanda M. VanDerHeyden, Ph.D., is a private consultant and researcher living in Fairhope, Alabama.

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the development of the recommendations accelerated math...

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