The DevelopmenT of The

Accelerated Math Second-Edition Libraries

New library content aligned

with National Math Panel

recommendations and NCTM

Curriculum Focal Points!

Accelerated Math is the only mastery measurement tool for progress monitoring to be reviewed by the National Center on Response to Intervention (NCRTI). Its high marks confirm what teachers nationwide have known all along—Accelerated Math is truly in a class by itself.

Accelerated Math meets all National Center on Student Progress Monitoring criteria for scientifically based progress-monitoring tools, and was favorably reviewed by the Comprehensive School Reform Quality Center and the Northwest Regional Educational Development Laboratory.

CSRQCenter

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.

© 2009 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) 338-4204www.renlearn.comanswers@renlearn.com

09/09

Contents

Introduction ............................................................................................................................................................ 1Recent History of Math Education ................................................................................................................ 1

Accelerated Math Content Revision Overview ....................................................................................................... 2

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

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

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

Phase IV: Expert Review ............................................................................................................................... 5 Phase V: Content Development .................................................................................................................... 6 Phase VI: State Standard Alignments .......................................................................................................... 7

Conclusion .............................................................................................................................................................. 7

References ............................................................................................................................................................. 8

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

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IntroduCtIon

Accelerated Math is a scientifically based, continuous progress-monitoring system for educators that produces daily, personalized, student math practice and assessments aligned to state standards and national guidelines. Teachers use the data generated by Accelerated Math 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-Calcu-lus, Calculus, and Basic Math.

To ensure Accelerated Math Second Edition content 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, 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, and Service Subscription users may purchase the new libraries individually.

To develop the new scope and sequence, the Renaissance Learning mathematics team conducted extensive research. We also convened a panel of mathematicians and researchers including Dr. Sybilla Beckmann, University of Georgia; Dr. Richard Bisk, Worcester State College; Dr. R. James Milgram, Stanford University; Dr. Tom Hogan, University of Scranton; Dr. Sharif M. Shakrani, Michigan State University; and Dr. Amanda VanDerHeyden, a private consultant and researcher living in Fairhope, Alabama. In addition, our team sought input from the U.S. Department of Education’s Northwest Regional Educational Laboratory, and from mathematics teachers in several states. Finally, Renaissance Learning staff relied heavily on National Council of Teachers of Mathematics (NCTM) Curriculum Focal Points, the National Mathematics Advisory Panel final report, and state and international standards.

This white paper documents the extensive revision process Renaissance Learning followed.

recent history of mathematics education

The 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 2003 Trends in International Mathematics and Science Study (TIMSS) report (Mullis, Martin, Gonzalez, & Chrostowski, 2004). In contrast, Singapore’s eighth-graders took the top position in math on this report 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 US 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” (Milgram & Wu, 2005, 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, 2007).

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 (National Council of Teachers of

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

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Mathematics [NCTM], 2006, p. 9). Both events are driving critical changes to state standards and publishers’ treatment of mathematics topics.

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

The NCTM 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, “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 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 repetitious and inefficient reteaching” (NCTM, 2006, p. 5).

In addition to this “deep understanding” of topics, the NCTM also emphasized the importance of mathematical fluency (NCTM, 2006, p. 5). The NMP 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” (National Mathematics Advisory Panel [NMP], 2008, p. 30).

The call is clear. New systems for teaching mathematics in the US must be structured with a focused, coherent progression, 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 be a daunting and Herculean task. Renaissance Learning’s Accelerated Math daily practice and diagnostic software was developed with 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.

ACCelerAted MAth Content revIsIon overvIew

To ensure Accelerated Math Second Edition content 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.

After the release of the other libraries, it became clear that the Accelerated Math Second-Edition Libraries must reach further to provide practice for math objectives associated with early numeracy skills. The NCTM Curriculum Focal Points had already identified foundational prekindergarten and kindergarten skills. And the prerelease of the Committee on Early

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Childhood Mathematics and National Research Council’s (2009) report Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity gave a resounding call for a greater emphasis on early numeracy. The Accelerated Math Early Numeracy Library was developed with objectives that met math practice needs for early or struggling math learners.

The following section highlights the key steps in each phase of our extensive Accelerated Math content revision process. Phase I: Accelerated Math content audit

The 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. This initial Accelerated Math content predated not only the National Math Panel recommendations and the NCTM 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. A great deal has occurred in education in the US since 1998, and much professional thought has gone into what is important in mathematics. Renaissance Learning has taken 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 began in fall 2006 with a review of selected content of grades 3, 4, and 5 by the Northwest Regional Educational Laboratory (NWREL). NWREL, based in Portland, Oregon, is part of a network of ten regional educational laboratories and “provides research-based products, technical assistance, and training to improve educational systems and learning.” The NWREL audit staff included mathematics curriculum experts and assessment experts. The audit focused on 163 Accelerated Math objectives for grades 3, 4, and 5, and 99 specific problems for grade 5. All reviewed content was in whole-number skills. The K–12 curriculum experts 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 the NCTM standards and focal points. The curriculum and assessment experts analyzed 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-life 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: • 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.

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.

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Audit feedback from NWREL and the teachers was distilled in 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. Renaissance Learning’s math experts then gave serious consideration to how the objectives in the original Accelerated Math content should be revised. What follows is the process and resources these experts used.

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 NCTM focal points, National Math Panel recommendations, state standards, top international standards, Accelerated Math student achievement data, and information from recognized mathematicians. The 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 article “The Key Topics in a Successful Math Curriculum,” mentioned earlier.

NCTM Curriculum Focal Points: The NCTM’s three focal points for each grade served as the basis for decisions on which essential math topics to include at each grade level.

National Math Panel recommendations: The core objectives were cross-checked with the NMP’s essential mathematical concepts and skills, found in their “Report of the Task Group on Conceptual Knowledge and Skills,” to determine if any skills at any grade level were missing. Also closely followed were the NMP’s “Benchmarks for the Critical Foundations” and 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, Renaissance Learning’s math experts used three sources to select a group of states with model math standards. First, from the Thomas B. Fordham Foundation report, 2006: The State of State Standards, the math team 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. Next, 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. Then, the states being considered as the core model states were checked against the 2007 U.S. Chamber of Commerce Education Report Card. In that report card, California, Indiana, and Massachusetts received high marks for their math standards, and Georgia received above-average marks for its math standards.

Finally, model state standards were selected. 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, 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: Renaissance Learning math experts studied the standards of countries with top-performing math students to learn from their approach. The math experts 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. A greater emphasis is placed on using concrete representations to teach a concept.

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Accelerated Math data-mining project: This project looked at data from 66,000 Accelerated Math students from 88 schools/districts over a 3-year period. The study yielded data about which objectives these students struggle with the 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

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” (NMP, 2008, p. xvii).

After auditing existing content and identifying a new set of core objectives for grades 1 through 8, the Renaissance Learning mathematics team began work on a new scope and sequence of Accelerated Math objectives. To accomplish this task, math 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.

Building of the scope and sequence: The process of placing the core objectives into their respective strands required consulting all of the sources that Renaissance Learning’s math experts had studied during the development process—all of the knowledge that would put Accelerated Math at the forefront of improving student mathematics achievement in the US. As the math team members placed the core objectives into the appropriate strands, they continually consulted the NMP’s work, including its final report and “Benchmarks for the Critical Foundations,” the NCTM focal points, and the Singapore primary and secondary mathematics standards. The math experts then analyzed the math standards of each of the selected model states to align with their standards. After initial definition of the scope and sequence was complete, it was submitted to NWREL for external review.

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

Learning progressions: An additional feature of the scope and sequence is the identification of learning progressions and prerequisite objectives. A learning progression is a carefully sequenced set of building blocks that 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.

Phase Iv: expert review Renaissance Learning sought input during every phase of the content revision process. The previous sections outline the steps followed before and during initial development, Phases I through III. In Phase IV of the process, Renaissance Learning presented sample items to many groups of experts for input including mathematicians, researchers, mathematics teachers, and the Northwest Regional Educational Laboratory.

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Grades 1 through 8 expert reviewA panel of prominent university-based mathematicians was convened in Menlo Park, California to review 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. Renaissance Learning math editors then applied the feedback received to make adjustments as appropriate to the content libraries for grades 1 through 8, Algebra 1, and Geometry.

Algebra 1 expert reviewThe Accelerated Math Second-Edition Algebra 1 scope and sequence was later submitted for review to Dr. Sharif M. Shakrani, 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 reviewThe Accelerated Math Second-Edition Early Numeracy scope and sequence and sample items were submitted for review to Dr. Amanda VanDerHeyden, private consultant and researcher. Renaissance Learning math experts received feedback in writing from Dr. VanDerHeyden. After each of the math experts had the opportunity to consider Dr. VanDerHeyden’s comments and to formulate questions, the Renaissance Learning group met with her. Appropriate adjustments were then made to the Early Numeracy content.

Mathematics teachersAnalysis of items from Accelerated Math included a teacher review of grades 1 through 4 items for grade-level appropriateness. The teachers who reviewed items were from Delaware, Texas, Washington, and Wisconsin. Simultaneous to work on the scope and sequence, members of the Renaissance Learning math team researched effective development of algorithm-generated dynamic items. They applied their findings to the creation of new items.

Northwest Regional Educational LaboratoryThe Northwest Regional Educational Laboratory, 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, the reviewers judged that the Algebra 1 objectives were clearly stated and included a proper level of specificity.

Phase v: Content development A strict development process was maintained to ensure quality item development. Accelerated Math Second Edition editors ensured that each item was aligned to the skills indicated by the objective statement and did not exceed those skills. Items were then analyzed to ensure the accuracy of the math. Item stems and answer choices were checked to ensure the cognitive load, math vocabulary, and readability were grade-level appropriate. Common mistakes that students might make when performing an operation were chosen as the incorrect answer choices, known as distractors, to help teachers determine the type of mistake a student is making.

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

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Phase vI: state standard alignment

Renaissance Learning recognizes the impact of standards-based reform and high-stakes standardized testing on schools, and we share the concerns of educators and administrators that students perform well on high-stakes assessments. To this end, after completing the content for the Accelerated Math Second-Edition Libraries, Renaissance Learning standards staff aligned Accelerated Math Second Edition to the mathematics standards of all 50 states and the District of Columbia.

Based on working with Mid-continent Research for Education and Learning and the Northwest Regional Educational Laboratory, Renaissance Learning developed an alignment strategy that balances the objective and subjective aspects of alignments to standards. The strategy is documented with definitions and examples for each specific purpose of the alignment, such as practice or assessment, and incorporates an “unpacking process” of separating the standard into skill, action, vocabulary, and context. To standardize the quality of the alignments, standards staff received extensive training, including how to calibrate alignment results.

State alignment reports may be accessed at the Renaissance Learning website at http://www.renlearn.com/fundingcenter/statestandardalignments/. To request alignment reports or the Renaissance Learning alignment document, please contact a local sales representative at (800) 338-4204.

ConClusIon

The Second-Edition Libraries were developed to ensure Accelerated Math is the personalized practice and progress- monitoring system to help raise American students’ level of mathematics achievement. With the Accelerated Math Second-Edition Libraries, students learn to solve problems and progress so they are ready each school year for more advanced levels of mathematics. The new Second-Edition Libraries marry the best research available to the mathematics standards school districts across the US are demanding now—and will be demanding for years to come. Our hope is that Accelerated Math can help pave the way for the US to be ranked among the “A+ countries” in math for the twenty-first century.

Interested school districts are encouraged to contact Renaissance Learning, Inc., to request a copy of the Accelerated Math Second-Edition Libraries scope and sequence. Product demonstrations are also available upon request.

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referenCes

American Federation of Teachers, AFL-CIO. (2006, July). Smart testing: Let’s get it right: How assessment-savvy have states become 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., et al. (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

International Association for Evaluation of Education Achievement. (1999). Trends in international mathematics and science study (TIMSS). Retrieved March 22, 2008, from http://www.questationmath.com/TIMSSReport.php

International Association for Evaluation of Education Achievement. (2003). 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. (2007, March). PISA 2006: Science competencies for tomorrow’s world: Volume 1: Analysis. Paris: OECD Publishing. Available 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. (2007, March). PISA 2006: Science competencies for tomorrow’s world: Volume 2: Data. Paris: OECD Publishing. Available from

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

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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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ACknowledgMents

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

Mathematicians and researchers

Renaissance 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 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 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.

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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 laboratory

The 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 teachers

Finally, 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

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ACknowledgeMents

Renaissance Learning extends a sincere “thank you” to the following individuals for lending their expertise in mathematics and developing assessments to reviewing the revised content for Accelerated Math Second Edition. (To learn more about all of the expert reviewers involved in the Accelerated Math 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 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.

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