Accelerated Math for Intervention™ - Renaissance Learning

THE RESEARCH FOUNDATION FORAccelerated Mathfor Intervention Evidence-Based Strategies to Help StudentsStruggling in Mathematics

Accelerated Math, MathFacts in a Flash, and STAR Math, are highly rated for progress-monitoring bythe National Center on Intensive Intervention.Accelerated Math and MathFacts in a Flash are highest rated for progress-monitoring masterymeasurement by the National Center on Response to Intervention, with perfect scores in all categories.STAR Math is highest rated for math screening and progress monitoring by the National Center onResponse to Intervention, with perfect scores in all categories.Accelerated Math and STAR Math meet all criteria for scientifically based progress-monitoring tools setby the National Center on Student Progress Monitoring.Accelerated Math has earned the top rating for Prevention and Intervention at all grade levels from theNational Dropout Prevention Center.Accelerated Math, Accelerated Math Best Practices, Accelerated Math for Intervention, Accelerating learning for all, AccelScan,MathFacts in a Flash, NEO 2, Renaissance, Renaissance Home Connect, Renaissance Learning, the Renaissance Learning logo,Renaissance Responder, and STAR Math are trademarks of Renaissance Learning, Inc., and its subsidiaries, registered, commonlaw, or pending registration in the United States and other countries.© 2010 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 copyrightedmaterial without authorization from the copyright holder. For more information, contact:RENAISSANCE LEARNINGP.O. Box 8036Wisconsin Rapids, WI 54495-8036(800) 338-4204www.renlearn.comanswers@renlearn.com02/13

ContentsIntroduction........................................................................................................................................................... 1Targeted Instruction With Differentiated Student Practice.................................................................................... 6Critical Math Skills Mastery................................................................................................................................. 14Motivation and Engagement............................................................................................................................... 18Informative Assessment...................................................................................................................................... 22AppendicesAppendix A: Overview of Accelerated Math, MathFacts in a Flash, and STAR Math......................................... 25Appendix B: Sample Key Reports...................................................................................................................... 29STAR Math Student Progress Monitoring Report.......................................................................................... 30STAR Math Screening Report....................................................................................................................... 31Accelerated Math Diagnostic Report........................................................................................................... 32Accelerated Math Status of the Class Report.............................................................................................. 33MathFacts in a Flash Student Progress Report............................................................................................ 34References.......................................................................................................................................................... 35Acknowledgements............................................................................................................................................. 40FiguresFigure 1: NAEP Math Results 2005, 2009............................................................................................................. 1Figure 2: Five Strands of Math Proficiency............................................................................................................ 2Figure 3: Accelerated Math Achievement Gains by Subgroup............................................................................ 9Figure 4: Prerequisites for a Sample Fifth-Grade Objective in Accelerated Math.............................................. 11Figure 5: Sample Worked Example..................................................................................................................... 13Figure 6: Example Accelerated Math for Intervention Classroom....................................................................... 19Figure 7: Example Renaissance Home Connect Screen.................................................................................... 20Figure 8: Sample STAR Math Student Progress Monitoring Report.................................................................... 21Figure 9: Sample STAR Math Screening Report................................................................................................. 23Figure A1: Accelerated Math Cycle Overview.................................................................................................... 26Figure A2: Sample MathFacts in a Flash Practice Screen.................................................................................. 27Figure A3: Sample STAR Math Item.................................................................................................................... 28TableTable 1: OECD Rankings on 2006 PISA Mathematics Test................................................................................... 2i

Introduction“I just can’t do math!” Teachers and parents hear it time and time again. Some of us have likely even said itourselves at some point. The problem with this statement is that it is simply not true. “Learning mathematicsdoes not come as naturally as learning to speak, but our brains do have the necessary equipment” and it canbe done—it just takes time, effort, and practice (Willingham, 2009–10, p. 14). Students struggling with mathmay feel they just “weren’t born to do math” becauseno matter how hard they try, they cannot do thework. And before long, these feelings result in lowmotivation and lack of engagement related tomath, making the challenge for educators thatmuch greater.The good news is that the difficulties withmathematics so many students suffer from are notcaused by lack of either smarts or the mythical “mathgene” (Dehaene, 1999). For most students thatstruggle, the problem lies in not having fully masteredthe previous knowledge. Students struggling with math likely received instruction on the skills they arelacking, but they may not have spent enough time practicing them to reach mastery and/or their struggleswent unnoticed as the class moved on to the next lesson. “Because math is highly proceduralized andcontinually builds on previous knowledge for successful learning, early deficits have enduring anddevastating effects on later learning” (VanDerHeyden, 2009, p. 5).To successfully tackle any new math skill, a student needs to have mastered the prerequisite skills—butthis takes time, effort, and practice. For example, in order to do double-digit division, a student needsknowledge of basic facts, concepts, and procedures related to division, multiplication, and subtraction. Andthese prerequisite skills, in turn, have their own prerequisites. Students that are either partially or completelylacking proficiency in some important prerequisite skills are going to struggle with grade-level material atsome point—and they are at risk of falling farther and farther behind without intervention. A robust interventionis needed that will identify what each student knows while providing teachers with the time and toolsnecessary to efficiently build upon students’ existing skills to bring them up to grade level.Low performance + disappointing growthFigure 1 shows the 2009 results from the National Assessment of Educational Progress (NAEP), called the“nation’s report card,” which found U.S. fourth graders had made no learning gains since the last time theNAEP math test was given, in 2007, while U.S. eighth graders showed slight progress (National Center forEducation Statistics, 2009). Even with the eighth graders’ modest improvement, overall results show thatFigure 1: NAEP Math Results 2005, 2009*A robust intervention is neededthat will identify what each studentknows while providing teacherswith the time and tools necessaryto efficiently build upon students’existing skills to bring them up tograde level.% Below Basic % at Basic % at Proficient % at AdvancedGrade 418% 43% 33% 6%42%Grade 827%39%26%8%Grade 1239% 38% 21% 2%0 20 40 60 80 100Percent of Students*The NAEP assessment was last administered to Grade 12 in 2005.1

INTRODUCTIONproficiency levels decline as students move through the grades. Less than 40% of U.S. math students infourth grade and less than 35% in eighth grade display “proficiency,” defined as “competency overchallenging subject matter, including subject-matter knowledge, application of such knowledge to real-worldsituations, and analytical skills appropriate to the subject matter” (Winick et al., 2008). By 12th grade, NAEPresults show that less than one quarter, only 23%, of U.S. math students are at or above proficiency (Grigg,Donahue, & Dion, 2007).U.S. students are also not performing competitively in mathematics compared to students internationally(see Table 1). The 2006 mathematics results from the Program for International Student Assessment (PISA),a system of international assessments focused on 15-year-olds’ capabilities in reading, mathematics, andscience, show that students in 23 out of 29 participating Organization for Economic Co-operation andDevelopment (OECD) countries outperformed their U.S. peers (Baldi, Jin, Skemer, Green, & Herget, 2007).These results place the U.S. in the bottom quartile, a position that has remained relatively unchanged since2003 (Provasnik, Gonzales, & Miller, 2009).Table 1: OECD Rankings on 2006 PISA Mathematics TestAbove OECD Average OECD Average Below OECD AverageFinland Germany United StatesKorea Sweden PortugalNetherlands Ireland ItalySwitzerland France GreeceCanada United Kingdom TurkeyJapan Poland MexicoFinally, U.S. students are graduating high school without sufficient mathematics proficiency to be college andcareer ready. Results from the 2009 ACT exam, designed to measure academic skills taught in schools anddeemed important for success in first-year collegecourses, show only 42% of students are preparedfor college-level coursework in mathematics, down1 percent from 2008 (The ACT, 2009).Clearly, such consistently lackluster performance onnational and international measures indicates theU.S. student math problem is systemic. It is time tointervene. But first, a closer look at what it means tobe struggling with mathematics.What is mathematics proficiency?Mathematics proficiency is of critical importance,both to succeed academically and ultimately forsuccess in life. Experts describe proficiency as “fiveinterrelated strands of knowledge, skills, abilities,and beliefs that allow for mathematics manipulationFigure 2: Five Strands of Math Proficiency1. Conceptual understanding: comprehension ofmathematical concepts, operations, and relations2. Procedural fluency: ability to carry out proceduresflexibly, accurately, efficiently, and appropriately3. Strategic competence: ability to formulate,represent, and solve mathematical problems4. Adaptive reasoning: capacity for logical thought,reflection, explanation, and justification5. Productive disposition: habitual inclination to seemathematics as sensible, useful, and worthwhile,coupled with a belief in diligence and one’sown efficacy(National Research Council, 2001, p. 5)and achievement across all mathematical domains”: conceptual understanding, procedural fluency, strategiccompetence, adaptive reasoning, and productive disposition (Steedly, Dragoo, Arefeh, & Luke, 2008, p. 10;see also National Research Council, 2001; Figure 2).2

INTRODUCTIONFor students to learn successfully, these interconnected strands must work together. “For example, as astudent gains conceptual understanding, computational procedures are remembered better and used moreflexibly to solve problems. In turn, as a procedure becomes more automatic, the student is enabled to thinkabout other aspects of a problem and to tackle new kinds of problems, which leads to understanding”(National Research Council, 2002, p. 17).The phrase students struggling with mathematics does not have a specific, formal definition, but broadlyspeaking it may be thought of as those students who are unable to meet state grade-level standards.And teachers are likely to define struggling students as those who struggle to keep up with grade-levelassignments, perform poorly on math assessments, and/or are perceived as needing extra help (Louie,Brodesky, Brett, Yang, & Tan, 2008).Accelerated Math for Intervention is evidence based to help students succeedBecause we learn new skills in the context of what we already know (Willingham, 2009), it is safe to assumethat students struggling with mathematics are missing critical foundational skills that, in math, form thebackground knowledge needed to successfullycomplete grade-level work. In addition, these studentshave likely not mastered math facts automaticity—theautomatic recall of math facts that frees students toconcentrate on more complex mathematical tasks.Without a robust intervention to identify what studentsknow and what they need to master in order to build afoundation for learning grade-level material, it is nearlyimpossible to help them catch up to their peers andexperience success in a core mathematics course.Accelerated Math for Intervention is a dynamic,evidence-based math intervention for grades 3–12that provides diagnostic tools to help teachersidentify both students’ strengths and critical skillsdeficiencies. At the heart of the program are threeproven Renaissance Learning tools for differentiatingstudent math practice: Accelerated Math, MathFactsin a Flash, and the STAR Math assessment. Workingin concert and supported by professionaldevelopment, these tools help teachers implementa formative assessment process within theirAccelerated Math for Intervention comprises threeproven programs working in concert to provide• Tools forIdentification of skills gaps anddiagnostic assessment (STAR Math,Accelerated Math, MathFacts in a Flash)Personalized, differentiated mathpractice with feedback (AcceleratedMath, MathFacts in a Flash)Progress monitoring (Accelerated Math,MathFacts in a Flash, STAR Math)• Intensive professional development tohelp teachersImplement a formative assessmentprocess within their intervention frameworkUtilize routines and strategies todifferentiate learningDevelop students’ self-efficacyFor a brief overview of the Accelerated Math for Interventioncomponents, see Appendix A, p. 25.intervention framework, allowing them to deliver targeted instruction based on data about the specific skillsstudents are lacking mastery in and provide invaluable time for differentiated student math practice, withfeedback, of these same skills.Increasing time for learningAcademic Learning Time (ALT)—the amount of time students spend on actual learning activities—has longbeen identified as a critical contributor to academic growth (Batsche, 2007; Berliner, 1991; Gettinger &Stoiber, 1999; Karweit, 1982). An important, but often underemphasized, aspect of ALT is time for studentpractice of learned skills—which is as important as explicit instruction (Szadokierski & Burns, 2008). SinceALT is the time when most learning actually takes place, increasing ALT is a powerful tool for improvingacademic results (Aronson, Zimmerman, & Carlos, 1998; Berliner, 1978; Smith, 1998). The challenge is thatin order to increase ALT it is necessary to directly measure and manage it, which is exceedingly difficult,especially in a diverse classroom where learners are working at many different levels at the same time.3

INTRODUCTIONAccelerated Math for Intervention overcomes this difficulty, and increases learning time, by focusing on thefour components of ALT that distinguish it from simple measures of classroom time or “time on task”:1. Students are actually engaged with material: The personalized problem sets of Accelerated Mathand the highly interactive nature of MathFacts in a Flash engage students, while providing animmediate and direct measurement of how much instructional time each student actually devotedto the work.2. Material is at the proper level of challenge, or zone of proximal development (ZPD): 1 Assessmentvia STAR Math and ongoing measurement of results from Accelerated Math and MathFacts in a Flashensure that each assignment is tailored to each student’s ZPD, which is continuously recalculated asthe student progresses.3. Students experience a high rate of success: Accelerated Math and MathFacts in a Flashgenerate personalized assignments at a level calculated for high success rates, which areautomatically monitored.4. Both student and teacher receive regular feedback about performance: Both Accelerated Mathand MathFacts in a Flash instantly provide students with results on their assignments, and teachersreceive concise, real-time reports so they know each student is progressing satisfactorily and canintervene promptly when necessary.The management and measurement functions of Accelerated Math for Intervention amount to direct controlover increasing ALT—for virtually the first time in educational history.Getting students back on the track to successThe goal of any intervention is to get struggling students back on grade level and performing successfully inthe general classroom as quickly as possible. Because no two students have the same learning profile,Accelerated Math for Intervention helps teachers make the best use of limited time and resources. Studentwork in the program is closely monitored by the teacher to ensure they are always working at an appropriatelevel of challenge, receiving immediate feedback, and experiencing success—with prompt intervention whendifficulties arise.Accelerated Math for Intervention helps studentsdevelop self-efficacy and self-confidence inmath as well as personal responsibility forlearning and success. And for teachers, theprogram provides technology and professionaldevelopment to create and support anenvironment where instruction is truly targetedto students’ needs.Support for Accelerated Math for InterventionThe effectiveness of this program is supported by morethan 114 independent studies, including 26 articlespublished in peer-reviewed journals, and by theU.S. Department of Education’s National Center onIntensive Intervention (2012a, 2012b, 2012c), NationalCenter on Response to Intervention (2009, 2010,2011a, 2011b), and National Center on StudentProgress Monitoring (2006, 2007).Accelerated Math for Intervention is built around the following research-based principles:• Targeted instruction with differentiated student practice• Critical math skills mastery• Motivation and engagement• Informative assessment1A student’s ZPD, a theoretical concept inspired by Russian psychologist Lev Vygotsky (1978), is based on an appropriate level of difficulty—neithertoo easy nor too hard—where the student is challenged without being frustrated.4

INTRODUCTIONA section of this paper is devoted to each of these principles, including a summary of comprehensiveevidence from experts and results of research studies called, “What the research says,” as well as anexplanation of how the intervention carries out these research findings titled, “Putting the research intopractice with Accelerated Math for Intervention.”For those familiar with Response to Intervention (RTI), it will be clear that the data-rich nature of AcceleratedMath for Intervention (and likewise Accelerated Math) makes this program a perfect fit for an RTI frameworkwhere teachers need to be able to monitor a student’s response to assistance and make informed decisionsbased on that response. As a small-group intervention, it will commonly be most appropriate for use at RTITiers 2 or above.5

Targeted Instruction With Differentiated Student PracticeWith Accelerated Math for Intervention, teachers use formative feedback from individualized student practiceto target instruction matched to students’ specific prerequisite skills deficiencies. Because the key to mathskill mastery is having time to practice skills that have been taught, Accelerated Math for Intervention providesstudents with deliberate, differentiated practice at an appropriate challenge level and aligned to students’unique needs.What the research saysBecause no two students are identical, the first important step in serving struggling learners is to determinewhy each student has been struggling—in this case, with grade-level mathematics. Given the sequentialnature of mathematics, once a student misses mastering an underlying prerequisite skill, it becomes moreand more difficult to master subsequent skills.The more gaps in knowledge, the more likely it is thatthese missing skills will impede a student’s success.As noted previously, these gaps are likely due to alack of competence in the five interrelated strands ofmathematical proficiency (conceptual understanding,procedural fluency, strategic competence, adaptivereasoning, and productive disposition), though somestrands may be weaker than others.Once tasks are matched to thecapability of each student, theyneed extensive opportunitiesfor practice, along with constantfeedback on their performance.(VanDerHeyden, n.d.)Students struggling with mathematics may need intervention in one or a combination of these areas, but asthe National Mathematics Advisory Panel (2008b) notes,Debates regarding the relative importance of conceptual knowledge, procedural skills (e.g.,the standard algorithms), and the commitment of addition, subtraction, multiplication, anddivision facts to long-term memory are misguided. These capabilities are mutually supportive,each facilitating learning of the others. Conceptual understanding of mathematical operations,fluent execution of procedures, and fast access to number combinations together supporteffective and efficient problem solving. (p. 26)Once skills gaps are identified, research shows that explicit, systematic instructional strategies are the mosteffective for students struggling in mathematics (National Council of Teachers of Mathematics [NCTM], 2007).The Institute of Education Sciences (IES) Practice Guide (Gersten, Beckmann, et al., 2009) lists providing thefollowing as vital to instruction during intervention:• Models of proficient problem solving• Verbalization of thought processes• Guided practice• Corrective feedback• Cumulative reviewTargeted instructionIt is important to understand the diverse needs of each student in an intervention setting. An effectiveintervention is not “one size fits all.” Teachers need an efficient process to isolate the learning gaps preventingstudents from experiencing success in math, and once tasks are matched to the capability of each student,they need extensive opportunities for practice, along with constant feedback on their performance(VanDerHeyden, n.d.).6

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEDuring intervention, as students are given time for math practice on individualized tasks at appropriatelevels of difficulty (increased Academic Learning Time), they may be able to work through several objectiveswith success. But, undoubtedly, students will encounter objectives that present difficulties. This createsopportunities for the teacher to intervene with appropriate and targeted instruction, including instructionalstrategies specific to struggling students.Teacher modeling of problem-solving strategies is an effective technique, particularly for populationsstruggling with mathematics. The process should include verbalization of the teacher’s thinking whilesolving a problem (Gersten, Beckmann, et al., 2009). Montague (2004) suggests that both correct andincorrect problem-solving behaviors be modeled:Modeling of correct behaviors helps students understand how good problem solvers use theprocesses and strategies appropriately. Modeling of incorrect behaviors allows students tolearn how to use self-regulation strategies to monitor their performance and locate and correcterrors….When students learn the modeling routine, they then can exchange places with theteacher and become models for their peers. (p. 8)In addition, “when teaching a new procedure or concept, teachers should begin by modeling and/or thinkingaloud and working through several examples….While modeling the steps in the problem…the teacher shouldverbalize the procedures, note the symbols used andwhat they mean, and explain any decision-making andthinking processes (for example, ‘That is a plus sign.That means I should…’)” (Jayanthi, Gersten, & Baker,2008, p. 5).Similarly, student verbalization, or “think-aloud,” is anevidence-based approach that helps build studentunderstanding while providing the teacher with anopportunity to diagnose misconceptions (Gersten,Beckmann, et al., 2009). The National MathematicsAdvisory Panel (2008b) recommends that teachers, aspart of explicit instruction, allow students to think aloudabout decisions made while solving problems.“Verbalization may help to anchor skills and strategiesboth behaviorally and mathematically. Verbalizing stepsin problem solving may address students’ impulsivityEffective Strategies for StudentsStruggling With Mathematics• Teacher modeling of problem-solvingstrategies, using verbalization (Gersten,Beckmann, et al., 2009), or thinkingaloud, about procedures, symbols, anddecision making (Jayanthi, Gersten, &Baker, 2008), including both correct andincorrect behaviors (Montague, 2004)• Student verbalization, or thinkalouds(NCTM, 2007), to help buildunderstanding while providing theteacher with an opportunity to diagnosemisconceptions (Gersten, Beckmann,et al., 2009)directly, thus suggesting that verbalization may facilitate students’ self-regulation during problem solving”(Jayanthi et al., 2008, p. 7). And student verbalization is beneficial for the teacher to pinpoint any areas ofconfusion or misunderstanding as part of a formative process to target additional instruction.Differentiated student practice with feedbackPractice is critical to a student solidifying knowledge of a skill, but experts say that it is important a student istaught and understands a skill before practicing it (Shellard, 2004). Depending on the skill and the student’slevel of competence, it may be more appropriate to develop a conceptual understanding before extensivepractice. In some circumstances, experts say that concepts and procedures can be taught simultaneously,and practice with feedback will strengthen that knowledge. As the National Research Council (2001) notes,Practice is important in the development of mathematical proficiency. When students havemultiple opportunities to use computational procedures, reasoning processes, and problemsolvingstrategies they are learning, the methods they are using become smoother, morereliable, and better understood. Practice alone does not always suffice; it needs to be built onconceptual understanding and instruction on procedures and strategies. (p. 422)7

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEResearch shows not all types of student practice are equally effective. Practice is absolutely essential forbuilding any skill, but research shows that to be effective, practice should embody certain characteristics,which parallel the characteristics of Academic Learning Time in general (see p. 3). It must be personalizedand coupled with instruction, as mentioned previously.Personalized practice means practice matched to student abilityso students are challenged but not frustrated. There is a strongrelationship between deliberate practice and the development ofexpertise in many domains including mathematical calculation(Butterworth, 2006).It also means practice must be accountable. Teachers andstudents must receive frequent feedback, and teachers mustintervene as necessary to assure students are successful at aKey Characteristics ofEffective Practice• Matched to student ability• Coupled with instruction• Accompanied by frequentfeedback for teachersand students• Intervened by teacher as neededhigh level (Coyle, 2009; Ericsson, Krampe, & Tesch-Römer, 1993). The National Research Council (2001)recommends that “practice should be used with feedback to support all strands of mathematical proficiencyand not just procedural fluency. In particular, practice on computational procedures should be designed tobuild on and extend understanding” (p. 423).According to Sadler (1989, p. 121), “Feedback is a key element in formative assessment,” but information“is considered as feedback only when it is used to alter the gap” between a student’s goal and actualperformance. Differentiated student practice is facilitated by the teacher utilizing ongoing formativeassessment information to help students answer three questions related to their learning (Atkin, Black, &Coffey, 2001): Where am I going? Where am I now? How can I close the gap between where I am now andwhere I am going? Formative assessment is an ongoing, dynamic process used to guide instruction and helpdetermine the next steps in learning for each individual student (Chappuis & Chappuis, 2007/2008).The feedback provided in a formative assessment process occurs while there is still time to take action.“Only by keeping a very close eye on emerging learning through formative assessment can teachers beprospective, determining what is within the students’ reach, and providing them experiences to support andextend learning” (Heritage, 2010, p. 8). “Consistent and ongoing feedback has been shown to be quiteeffective in improving student performance.…In particular, the value of immediate feedback stands out.Regular feedback helps students guide and improve their own practice, even as giving feedback helpsteachers guide and tailor their own instruction” (Steedly et al., 2008, p. 9; see also NCTM, 2007).A meta-analysis shows that providing ongoing feedback using formative data about how each student wasperforming enhanced student math achievement (Gersten, Chard, et al., 2009). “The regular use of formativeassessment data by teachers, especially if teachers have additional guidance on using the data to designand individualize instruction, has been shown to be an effective instructional practice” (National MathematicsAdvisory Panel [NMAP], 2008b, p. xxiii).Putting the research into practice with Accelerated Math for InterventionRenaissance Learning’s trusted Accelerated Math software is the driving force behind Accelerated Math forIntervention. As described by the National Math Panel, Accelerated Math is a “mathematics program withassessment of skill level, tailoring of the instruction to match skill level, individual pacing and goal setting,ample practice, and immediate feedback to student and teacher on performance” (2008a, p. 160).Accelerated Math meets five critical criteria identified by the Formative Assessment for Teachers andStudents (FAST) State Collaborative on Assessment and Student Standards (SCASS) for effective formativeassessment: “(1) learning progressions, (2) learning goals and criteria for success, (3) descriptive feedback,(4) self- and peer-assessment, and (5) collaboration” (McManus, 2008, pp. 4–5).8

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEWithin the Accelerated Math for Intervention framework, Accelerated Math’s diagnostic tools enable optimaldifferentiation, providing students with efficient practice of critical skills coupled with immediate feedback.For more than 10 years, Accelerated Math has helped educators differentiate instruction and make time fordedicated student practice as well as helped students experience success and consequently get moreenjoyment out of math. For example, in an Ysseldyke & Tardrew (2007) study, 2,202 students in grades 3–10at 47 schools across 24 U.S. states gained 7 to 18 percentiles more than comparison students. In everygrade and subgroup identified, such as eligibility for Title I and free or reduced-lunch programs, students inAccelerated Math classes performed better than students in classes not using the software (see Figure 3).Figure 3: Accelerated Math Achievement Gains Math by Subgroup Achievement GainsPercentile Rank Gain2220181614121086420-2-4Non-AMAM810511214LD GT FRL Title 1 ELL LA0Subgroup15-318820LD=Learning Disabled or Special Needs GT=Gifted and TalentedFRL=Free/Reduced Lunch Title 1=Students in Title 1 ProgramsELL=English Language Learners LA=Low AchieversAdditionally, students who more closely followed Accelerated Math Best Practices by scoring greater than85% correct and completing more objectives, gained even more than students who did not. Accelerated Matheducators reported qualitative improvements in their classrooms as well—teachers spent more time providingindividual instruction, students spent more time academically engaged, and students enjoyed math more andtook responsibility for their work. In total, 80% of Accelerated Math educators stated that students werelearning basic math skills better.Accelerated Math facilitates formative assessment and differentiation within Accelerated Math for Intervention,in line with Heritage’s (2010) definition of a formative assessment process that provides “indications ofstudents’ learning status relative to the ‘gap’ that teachers and students can use to make adjustments tolearning while that learning is developing,” and ”consistently working from students’ emerging understandingswithin the ZPD, supporting learning through the instructional scaffolding, including feedback, and the activeinvolvement of students in the assessment/learning process” (p. 15).9

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEIn addition to being confirmed by several leadingmath experts (see box), the scope and sequence ofthe Accelerated Math learning objectives parallelsrecommendations found in the National Council ofTeachers of Mathematics (NCTM) 2006 CurriculumFocal Points, the National Mathematics Advisory Panel2008 final report, and the 2010 Common Core StateStandards for Mathematics.Accelerated Math for Intervention Scopeand Sequence Is Validated by Math ExpertsSybilla Beckmann, Ph.D., University of GeorgiaRichard Bisk, Ph.D., Worcester State CollegeThomas P. Hogan, Ph.D., University of ScrantonR. James Milgram, Ph.D., Stanford UniversitySharif M. Shakrani, Ph.D., private consultantAmanda VanDerHeyden, Ph.D., private consultantFor Accelerated Math for Intervention, the power ofAccelerated Math has been combined with MathFactsin a Flash and STAR Math, and supported by focused professional development on proven instructionalstrategies, to provide educators with a comprehensive intervention that includes everything they need toaffect change in students struggling with mathematics.To learn math, students must continually climb a staircase of learning progressions that leads tounderstanding. Critical skills and concepts that students miss become gaps in this foundational staircasethat make progressing to the next step more difficult. The more steps that are missed, the more difficult it is tosuccessfully move forward.Accelerated Math for Intervention gives teachers several ways to identify the critical steps students havemissed. This is important to note, as students needing intervention will likely have diverse needs. Likewise,instruction and student practice within Accelerated Math for Intervention is fully differentiated; each next stepis triggered by student performance.Initial assessmentTeachers using Accelerated Math for Intervention administer STAR Math, a reliable, valid, and efficient,computer-adaptive assessment of general math achievement, to gather baseline data for students and initiallydetermine their placement for practice in the Accelerated Math libraries of objectives. Results from STAR Mathgive teachers an idea about which grade-level math objectives a student probably already knows and whichobjectives a student will likely need more instruction and practice with. Since 2009, STAR Math has beenhighly rated as a screening tool by the National Center on Response to Intervention, earning perfect scores inall categories in the most recent review (U.S. Department of Education [U.S. DOE], 2011b), meaning thisassessment can support decisions regarding whether a particular student needs intervention. (Please seeAppendix B, p. 29, for the STAR Math Screening Report or a sampling of key Accelerated Math forIntervention reports.)Determining the skills a student has and has not masteredTeachers may begin students’ work in Accelerated Math by having them take diagnostic tests on core mathobjectives—those that are gateway skills to advanced objectives—to confirm which objectives have beenmastered and which need attention. Because each objective has been meticulously designed, validated,and sequenced into learning progressions, after determining the skills a student in an Accelerated Math forIntervention classroom is struggling with, educators are able to easily determine the prerequisite skills thatstudent needs to master by reviewing the Accelerated Math Learning Progressions for Instructional Planningdocument (Renaissance Learning, 2009a). Figure 4 illustrates the prerequisite objectives for a samplefifth-grade objective.10

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEFigure 4: Prerequisites for a Sample Fifth-Grade Objective in Accelerated MathCurrent learning objectiveGrade 5, objective 21:Divide a multi-digit wholenumber by a 2-digit number, with aremainder and at least one zeroin the quotientPrerequisite objectivesGrade 5, objective 14:Divide a multi-digit wholenumber by a 1-digit number withno remainder and at least onezero in the quotientGrade 4, objective 30:Estimate a product of wholenumbers by roundingGrade 3, objective 37:Know basic division facts to100 divided by 10Grade 3, objective 35:Know basic multiplicationfacts to 10 X 10Grade 3, objective 8:Determine the 4- or 5-digitwhole number representedin expanded formGrade 2, objective 38:Subtract a 1- or 2-digit numberfrom a 2-digit numberwith one regroupingGrade 1, objective 43:Know basic subtractionfacts to 20 minus 10Prerequisite objectives include concepts, terminology, and skills that are essential for mastering later objectives. Accelerated Math for Interventionhelps teachers quickly identify the gaps in each student’s knowledge that are preventing mastery of grade-level material.Based on the results of the diagnostic test, students begin individualized student practice and practice/testcycles through mastery and review of each needed objective. When students need help, the teacher andstudent confer to diagnose errors and use modeling, student verbalization, or think-alouds, and otherstrategies to address individual needs.Identifying need for math fact fluencyBecause students struggling with mathematics are frequently lacking math facts fluency, teachers usingAccelerated Math for Intervention also have students begin working with Renaissance Learning’s provenMathFacts in a Flash software to identify the key math facts they know with automaticity and those they needto work on. Placement is based on grade-level benchmarks for fact fluency, but teachers may use MathFactsin a Flash review tests to help with initial placement if necessary. Teachers never assign fact fluency practiceon operations and facts that have not yet been taught.Practice, practice, instruction, and more practice!One characteristic shared by nearly all studentsstruggling with math—and this is likely true of moststudents in the country—is that they have not devotedenough time to practicing core math skills along withfeedback (i.e. Academic Learning Time, or ALT) to trulybecome proficient. Effective, personalized practice requirescontinuous daily—even hourly—instruction,Accelerated Math for Interventionand School Improvement ModelsAccelerated Math for Intervention increasesAcademic Learning Time (ALT), an importantcomponent of most school improvementmodels, including Response to Intervention,the turnaround and transformational models ofthe federal Race to the Top and SchoolImprovement Grant programs, among others.11

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEpractice, and assessment within a formative assessment process. The most salient characteristic ofAccelerated Math for Intervention is that students needing intervention must be provided with sufficient timeto practice mathematics problems at their individual levels. Once critical skills gaps are identified using theprogram assessment components, students will spend an extensive amount of time practicing—withimmediate feedback from the software, and instructional guidance from the teacher as necessary. The fact isthat this kind of hard, dedicated work is the only way for students who are significantly working below gradelevel to become proficient. The good news is that this kind of differentiated instruction and practice has beenshown to produce remarkable gains in achievement.Research on Accelerated Math shows that overall student mathematics achievement improves significantly ifsufficient time is spent practicing mathematics at the appropriate level of challenge (Ysseldyke & Bolt, 2007;Ysseleyke & Tardrew, 2007). Ensuring that students are practicing solving mathematics problems at theirindividual levels—so students can and do understand the reasoning and processes used—is a fundamentalprinciple underlying Accelerated Math for Intervention. Like any other skill, Academic Learning Time is vital:Time must be spent actually practicing mathematics in order to improve.Why is differentiated instruction & practice necessary?Intervention classrooms tend to be quite diversein terms of mastery and achievement levels ofthe students. Some may have mixed grades inthe same class and most certainly havesignificant variance in terms of achievement.The students in a given class usually do notneed the same thing, or at the same time, soone-size-fits-all, whole-group instruction willgenerally not be an effective technique. Recallthe definition of ALT on p. 3: Proper level forindividual success is just as important as totalminutes on task, as is personalized feedback.Accelerated Math for Intervention gives teachersa toolbox to re-teach concepts that students havestruggled with in the past as well as the tools neededto identify weak areas and target appropriate instruction to specific students.Accelerated Math for Interventionprovides strategic use of technologyfor differentiated student practiceand to support—not replace—teacher-led instruction. This programrevolves around student-to-teacherinteraction, which research showsis most effective.(Kroesbergen & Van Luit, 2003)How can you provide all this differentiated attention without burdening the teacher?The Accelerated Math software puts technology to use to do what it does best, enable efficient classroommanagement with individualized student practice, automatic scoring, immediate feedback for both studentand teacher, and comprehensive tracking of student data. With these management tasks out of the way,intervention instruction revolves around student-to-teacher interaction. The teacher’s time is freed to dowhat a teacher does best—help students fill their critical learning gaps with targeted instruction specific totheir needs.Rich data from Accelerated Math enhances student-to-teacher interaction by helping the teacher use bothstrategic groups and independent, targeted one-to-one attention in the most effective manner possible. Thestrategic use of computers to enhance teacher-centric strategies in an intervention setting is supported byresearch (Kroesbergen & Van Luit, 2003):12

TARGETED INSTRUCTION WITH DIFFERENTIATED STUDENT PRACTICEComputer-aided instruction (CAI) can be very helpful when students have to be motivatedto practice with certain kinds of problems. With the use of a computer, it is possible to letchildren practice and automatize math facts and also to provide direct feedback. However, thecomputer cannot remediate the basic difficulties that the children encounter. The results…show that in general, traditional interventions with humans as teachers, and not computers,are most effective. (p. 112)Effective instructional strategiesAccelerated Math for Intervention provides teachers with thetraining and tools to implement a formative assessment processwithin their intervention framework. Focused professionaldevelopment includes classroom management and monitoringtechniques as well as instructional strategies proven to work forstudents struggling with mathematics. Data from Accelerated Mathfacilitates using formative feedback for activities including progressmonitoring;goal setting; strategic grouping; problem-solvingstrategies, such as verbalization and modeling; schema-basedinstruction; and corrective feedback. Also included in the interventionpackage are number sense activities—a common skill gapfor students struggling with math—that may be conducted withsmall-groups of students or as peer-to-peer interactions (NCTM,2007), which are found in the Numeracy Development andIntervention Guide and the Fractions, Decimals, and PercentsDevelopment and Intervention Guide, both by Dr. Kenneth E. Vos.Figure 5: Sample Worked ExampleWith Accelerated Math for Intervention, teachers and students have regular, purposeful interactions—shortconferences focused on the student’s current work and feedback. These two-way interactions incorporateboth teacher modeling and student verbalization, or think-aloud, techniques. The teacher models strategiesand mathematical thinking for the student, including using visual aids such as Worked Examples (seeFigure 5), and students verbalize their decision-making processes and explain how they arrived at thesolution to a math problem. Each strategy and technique built into Accelerated Math for Intervention allowsfor the differentiated practice and targeted instruction required for students struggling with mathematics.13

Critical Math Skills MasteryAccelerated Math for Intervention is focused on student mastery of critical mathematics skills—the core mathobjectives, concepts, and procedures as well as math facts automaticity—that students who struggle withmathematics have never mastered sufficiently to be able to tackle more complex concepts and procedures.What the research saysComputation and problem-solving masteryAs mentioned previously, the knowledge gaps that students struggling with mathematics have may becharacterized as deficiencies in conceptual or procedural understanding, strategic competence, adaptivereasoning, or a student’s disposition toward math—and most likely are a combination of these five strands.“Providing a mix of direct instruction of new skills and concepts, guided practice, opportunities for complexthinking and problem solving, and time for discussion is even more important for the struggling student thanfor students in general” (Shellard, 2004, p. 41).A meta-analysis of research summarized in the Institute of Education Sciences (IES) Practice Guide forAssisting Students Struggling with Mathematics recommends that a math intervention should include bothfoundational concepts and skills introduced earlier in the student’s career but not fully understood andmastered as well as systematic, explicit instruction (Gersten, Beckmann, et al., 2009).The National Mathematics Advisory Panel (2008b) found that providing “explicit instruction with students whohave mathematical difficulties has shown consistently positive effects on performance with word problemsand computation….This finding does not mean that all of a student’s mathematics instruction should bedelivered in an explicit fashion. However, the Panel recommends that struggling students receive someexplicit mathematics instruction regularly. Some of this time should be dedicated to ensuring that thesestudents possess the foundational skills and conceptual knowledge necessary for understanding themathematics they are learning at their grade level” (p. xxiii).Providing teachers with precise information on student progress and specific areas of strengths andweaknesses in mathematics is an effective practice. “Before mastery instruction or techniques are used, itis essential that the student possess the preskills andunderstand the concept related to the targeted skill” (Mercer& Miller, 1992, p. 22). Using effective strategies, teacherscan help to close skills gaps by having each student workon what they know and build upon that foundation.Math facts masteryA common critical skill students struggling withmathematics lack is math facts knowledge andautomaticity. “Automaticity refers to the phenomenon thatUpon entering sixth grade,fewer than a third of studentshave demonstrated mastery ofmultiplication facts, and fewerthan a fifth have demonstratedmastery of division.a skill can be performed with minimal awareness of its use” (Hartnedy, Mozzoni, & Fahoum, 2005; Howell &Lorson-Howell, 1990). While practicing single-digit calculations is essential for developing fluency (NationalResearch Council, 2001), it is widely recognized that, “Few curricula in the United States provide sufficientpractice to ensure fast and efficient solving of basic fact combinations and execution of the standardalgorithms” (NMAP, 2008b, p. 27).Upon entering sixth grade, fewer than a third of students have demonstrated mastery of multiplication facts,and fewer than a fifth have demonstrated mastery of division. And since there is little if any focus on thesefacts after fifth grade, it is a safe assumption that many never master them at all (Baroody, 1985; Isaacs &Carroll, 1999).14

CRITICAL MATH SKILLS MASTERYThis is not to say that most students do not know how to add, subtract, multiply, and divide. Clearly they do,or they would not score even as well as they do on benchmark assessments. But they have not achievedmastery—or more strictly speaking, they have not achieved automaticity, the essential foundation ofcomputational fluency. Their knowledge of these core operations, which undergird all of mathematics, is“procedural” rather than “declarative.” That is, students know how to multiply 8 x 7, but they do not knowthat 8 x 7 is 56, so they must calculate or use strategies that take time and mental resources away fromhigher-level operations (Ashlock, 2009).There is a growing consensus that automatic recall of math facts isan indispensable element in building computational fluency,preparing students for math success, both present and future(National Research Council, 2001). Just as phonemic awarenessand decoding are the crucial elements in learning to read,automaticity and conceptual understanding go hand in hand inThe key to automaticity ispractice, and lots of it.(Willingham, 2009)mathematics development (Gersten & Chard, 1999). Failure to develop automatic retrieval, on the other hand,leads to mathematical difficulties (Bryant, Bryant, Gersten, Scammacca, & Chavez, 2008).It is not enough that students simply “learn” their number facts—they must be committed to memory, justas letter sounds must be memorized in development of phonics automaticity (Willingham, 2009).Automaticity is a different type of knowledge; it is “based on memory retrieval, whereas nonautomaticperformance is based on an algorithm” (Logan, 1988, p. 494). Learning starts with understanding ofconcepts, to be sure, but memory skills must develop simultaneously. “Children need both proceduralknowledge about how to do things and declarative knowledge of facts” (Pellegrino & Goldman, 1987, p. 31).Declarative memory (which recalls that things are so) not only speeds up the basic arithmetic operationsthemselves (Garnett & Fleischner, 1983); it also acts to “free up working memory capacity that then becomesavailable to address more difficult mathematical tasks” (Pegg, Graham, & Bellert, 2005, p. 50; see alsoGersten, Jordan, & Flojo, 2005; Sousa, 2006). This automatic retrieval of basic math facts is critical to solvingcomplex problems that have simpler problems embedded in them (Willingham, 2009–10). The NationalMathematics Advisory Panel final report in 2008 put it this way:To prepare students for Algebra, the curriculum must simultaneously develop conceptualunderstanding, computational fluency, and problem-solving skills …. Computationalproficiency with whole number operations is dependent on sufficient and appropriate practiceto develop automatic recall of addition and related subtraction facts, and of multiplication andrelated division facts. (p. xix)The key to automaticity is practice, and lots of it (Willingham, 2009). To move a fact (or skill) from short-termto long-term memory requires “overlearning”—not just getting an item right, but getting it right repeatedly(Willingham, 2004). And retaining the memory for a long interval requires spacing out additional practice afterinitial mastery—emphasizing the importance of regular review of learned material (Rohrer, Taylor, Pashler,Wixted, & Cepeda, 2005). Brain research indicates that repetitions actually produce changes in the brain,thickening the neurons’ myelin sheath and creating more “bandwidth” for faster retrieval (Hill & Schneider,2006; see also Coyle, 2009).For all of these reasons, the federal Institute of Education Sciences (IES) recommends math facts fluencyintervention for students at all grade levels in Response to Intervention (RTI) schools. The IES Practice Guide(Gersten, Beckmann, et al., 2009) states:Quick retrieval of basic arithmetic facts is critical for success in mathematics. Yet researchhas found that many students with difficulties in mathematics are not fluent in such facts….We recommend that about 10 minutes be devoted to building this proficiency during eachintervention session. (p. 37)15

CRITICAL MATH SKILLS MASTERY“Because progress in math builds heavily upon previously learned skills, it is important for instruction to beclear, unambiguous, and systematic, with key prerequisite skills taught in advance. For instance, childrenshould not be expected to develop automatic recall of addition facts if they do not understand the basicconcept of addition or the meaning of the addition sign” (Spear-Swerling, 2005, p. 1).Traditional methods of math facts practice—such as flash cards—are not bad in themselves, but they areinsufficient. They require far too much paperwork and teacher time to administer the necessary number ofitems at the desired frequency to produce mastery. Such methods also do not keep track of which facts havebeen mastered and should be reinforced over time, and which new facts to introduce next, so that the studentcan move through the full sequence on a timely basis. For this reason, properly designed software tools arerecommended by the National Mathematics Advisory Panel (2008b).Putting the research into practice with Accelerated Math for InterventionSupporting mastery with appropriate toolsAccelerated Math for Intervention supports explicit, targeted instruction on and dedicated student practiceof critical math skills, so that students struggling with mathematics are able to fill any gaps in prerequisiteknowledge that are impeding their grade-level success. The intervention is focused on math fact automaticityand core math skills, based on a logical progression through mastery of math skills.Because automaticity of basic addition, subtraction, multiplication, anddivision facts enables fast and accurate computation as well as efficientproblem solving, it is fundamental to improving math achievement.Accelerated Math for Intervention was designed based on automaticityresearch. The professional development materials include guidance forteachers on building students’ math fact fluency, including devoting 10minutes of each class period to deliberate, measurable practice withMathFacts in a Flash. Research shows MathFacts in a Flash can help todouble math gains for students who use the program regularly (Burns,Kanive, & DeGrande, 2012).NumeracyDevelopment andIntervention GuideIf a student is lacking the prerequisite early-number skills needed toeven begin working on math fact automaticity, teachers can refer tothe Numeracy Development and Intervention Guide, by Dr. Kenneth E.Vos, also included with the program. Likewise, to help teachers developstudents’ conceptual understanding of fractions—a number sense skill ofparticular importance for algebra readiness—Accelerated Math forIntervention also includes the Fractions, Decimals, and PercentsDevelopment and Intervention Guide (also by Dr. Vos).To help students develop core math skills, the Accelerated Math softwareprovides differentiated practice on math objectives until mastery is achieved, including review of masteredobjectives to ensure retention. Accelerated Math objectives focus mostly on concepts (e.g., questions aimedat understanding what a fraction is) and procedures (e.g., how to add multi-digit numbers) but also includemultistep word problems that require use of both basic problem-solving strategies and conceptual/procedural knowledge.Data from MathFacts in a Flash and Accelerated Math helps the teacher identify core objectives, learningprogressions, and prerequisite skills for each student based on the skills a student has and has not mastered,ensuring that teachers are able to target instruction specific to individual student needs. This information isalso beneficial to determine if students can be grouped for instructional support on a specific skill, or if a16

CRITICAL MATH SKILLS MASTERYparticular student requires additional individualattention. Instructional tools help teachers prioritizeand prepare instruction on skills as well as provideadditional information such as key concepts(terminology), sample practice items, workedexamples, and a math glossary. And with theAccelerated Math Learning Progressions forInstructional Planning document (RenaissanceLearning, 2009a), teachers are able to visually assesswhere objectives sit within learning progressions thatspan grade levels, enhancing their ability to ensuremastery of critical skills.To help students develop coremath skills, the Accelerated Mathsoftware provides differentiatedpractice on math objectives untilmastery is achieved, includingreview of mastered objectives toensure retention.Support mastery through consistent routinesA core routine for the Accelerated Math for Intervention program involves teacher-student conferences. Theseshort, purposeful meetings are guided by professional development and help teachers provide feedback aswell as probe into students’ thought processes and diagnose misconceptions as they arise. Studentscomplete Accelerated Math assignments and tests on paper before submitting their answers for scoringvia handheld responder or scanner, which provides the teacher with a window into a student’s thinking andproblem-solving patterns. The teacher can use students’ worked problems as a jumping-off point forutilizing verbalization, or think-aloud, strategies, by asking students to talk about the steps they took tosolve a problem.17

Motivation and EngagementStudents struggling with mathematics are often lacking in motivation and engagement, stemming from theirbelief that they “just can’t do math” and a continued struggle with coursework that seems to get increasinglymore difficult. Because these students are missing critical foundational skills, they fall further and furtherbehind as the class continues to move on to each subsequent topic. Teachers can motivate students to strivefor mathematical proficiency by supporting their expectations for achieving success through effort and byhelping students appreciate the value of what they are learning.What the research saysMany students who experience difficulty in math develop negative attitudes toward the subject matter, whichmakes it increasingly important to utilize strategies that encourage positive attitudes, such as involvingstudents in setting challenging but attainable goals, ensuring instruction builds on previously learned skills,and modeling an enthusiastic and positive attitude about math (Mercer & Miller, 1992). According to theNational Research Council (2001), “students need continued confidence that they can meet the challengesof school mathematics” (p. 339). Helping students achieve self-efficacy can be done through two basicstrategies: assign tasks students can successfully completewith reasonable effort, and provide necessary scaffolding tosupport students as they work to attain and put to useconcepts, skills, and abilities.Student academic engagement is correlated with improvedacademic achievement (Batsche, 2007; Berliner, 1991;Gettinger & Stoiber, 1999; Karweit, 1982). Frederick (1977)found that high-achieving students in secondary classroomswere academically engaged 75% of the time, compared toHelp Students Achieve Self-Efficacy• Assign tasks where they cansucceed with reasonable effort• Provide scaffolding so they canattain and use concepts, skills,and abilities(National Reseach Council, 2001)51% for low-achieving students. Thus, it is vital to engage students that are struggling with mathematics inorder to improve their mathematics proficiency. Students will become engaged as they begin to experiencesuccess, see the value of mathematics, and recognize that effort matters; any math intervention needs toinclude strategies to address these needs. The National Mathematics Advisory Panel (2008b) found,Experimental studies have demonstrated that children’s beliefs about the relative importanceof effort and ability or inherent talent can be changed, and that increased emphasis on theimportance of effort is related to greater engagement in mathematics learning and, throughthis engagement, improved mathematics grades and achievement. Research demonstratingthat beliefs about effort matter and that these beliefs can be changed is critical. Much of thepublic’s resignation about mathematics education (together with the common tendencies todismiss weak achievement and to give up early) seems rooted in the idea that success inmathematics is largely a matter of inherent talent, not effort. (p. 31)In order to improve students’ motivation, the IES Practice Guide (Gersten, Beckmann, et al., 2009)recommends teachers “reinforce or praise students for their effort and for attending to and being engaged inthe lesson” (p. 12). A student’s ability to self monitor by graphing or charting their own progress and settingtheir own goals can also have potentially positive effects. “The panel believes that praise should beimmediate and specific to highlight student effort and engagement....Praise is most effective when it pointsto specific progress that students are making” (p. 45).Putting the research into practice with Accelerated Math for InterventionProviding opportunities for successHigh levels of success are not only vital to improving math achievement but also critical for motivation.Success motivates. Continual failure de-motivates and frustrates. This is why student practice in AcceleratedMath for Intervention is aligned to each student’s zone of proximal development (ZPD), the difficulty level that18

MOTIVATION AND ENGAGEMENTis neither too difficult and thus frustrating nor so easy that no new knowledge and skills are learned. Data fromthe program ensures teachers understand both what students know and what they need to learn, so they areable to build on students’ current knowledge. Practicing math in individualized ZPDs set students up forsuccess and, in turn, motivates more practice.The key to successful student practice in Accelerated Math for Intervention is for the students to see thatthey are, in fact, successful. The intervention components lend themselves well to ensuring students receivefrequent, immediate feedback. Students receive immediate feedback in the form of informative reports uponsubmitting Accelerated Math assignments (which students first complete on paper and then submit forscoring either via handheld responder or scanner). In MathFacts in a Flash, student progress is displayedon screen and in reports for work completed. It is often motivating for students to attempt to beat previousresponse rates as facts become automatically retrieved from memory, and this, in turn, encourages automaticretrieval of facts as well as improved attitudes about math.Accelerated Math for Intervention incorporates social interaction for students struggling with mathematics.Teachers may group students struggling with similar skills for targeted instruction or other activities (seeFigure 6). In these small groups, students can discuss problem-solving strategies and share ideas.Figure 6: Example Accelerated Math for Intervention ClassroomDifferentiatedinstruction ensuressuccess for everystudent. The teachermonitors each student’sprogress andprovides instructionas needed.Number senseactivities helpstudents attainmissing prerequisitemath skills.Learning Station 1StudentTeacher’s AideTeacherPersonalizedpracticeassignmentskeep studentsmotivated andchallenged.DifferentiatedInstructionPersonalized Math PracticeMath Fact Fluency and AssessmentMath factpractice leads toautomaticity for fastand accuratecomputation andproblem solving.Learning Station 219

MOTIVATION AND ENGAGEMENTParental involvement—connecting school to home—is a very motivating social factor for students.Accelerated Math for Intervention establishes open communication with the Renaissance Home Connectwebsite, which parents can access to view details of their child’s completed math work, and students can usefor additional math practice at home (see Figure 7). Renaissance Home Connect also allows students to haveresults sent via email to parents and other caregivers upon completion of an Accelerated Math test orMathFacts in a Flash level.Figure 7: Example Renaissance Home Connect ScreenIn the Accelerated Math for Intervention classroom,teachers build in a range of motivators. What worksfor different classrooms and ages varies, but theprofessional development that supports theintervention includes examples of what actualteachers have found to be successful, tied tomeasurable effort on student practice, tests, orindividual goals. For example, professionaldevelopment recommendations for AcceleratedMath for Intervention suggest teachers capitalizeon immediate feedback—which is essential in aformative assessment process—by having studentscheck in with the teacher upon its receipt foradditional quick, efficient, and genuine personalfeedback that includes congratulations,encouragement, and additional instruction asneeded. Training materials also emphasize that effort,not only success, should be highly and frequentlypraised by teachers.Accelerated Math for Intervention routines, which teachers learn through focused professional developmenttraining, emphasize that students taking part in the intervention always remain engaged in an activity (be itcompleting Accelerated Math assignments, practicing math facts fluency with MathFacts in a Flash,referencing classroom resources to aid problem solving—such as Worked Examples provided as part ofthe intervention—and so forth). When students complete an activity, they should know what the teacherexpects them to work on next. These routines develop and promote students self-efficacy, which is tied toincreased motivation and improved attitudes towards math.In 2001, students were asked to respond to attitudinal surveys at the beginning and end of an AcceleratedMath study (Gaeddert). At the end of the study, students using Accelerated Math showed greaterimprovement in their attitudes toward math than students in the control classes. And surveys of parents withchildren in the intervention classes also indicated more positive attitudes toward math than parents of thechildren in the control classes. Likewise, in a 2005 study, almost 60% of the students responding to a surveyindicated that they liked math better after using MathFacts in a Flash (Ysseldyke, Thill, Pohl, & Bolt).20

MOTIVATION AND ENGAGEMENTStudent-centered goal settingThe STAR Math component of Accelerated Math for Interventionincludes the ability for teachers to set individual goals for eachstudent and then monitor students’ progress towards these goals.Teachers can include students in the goal-setting process and thenuse a report generated by the assessment, presented in a userfriendly,graphical display, to share student progress both withstudents and parents. Seeing a visual representation of growth, likethe Student Progress Monitoring Report, can be very motivational forstudents that may be experiencing success in mathematics for thevery first time (see Figure 8 2 ).Additionally, Accelerated Math for Intervention routines includehaving students chart their own progress. Teacher materials includereproducible charts specifically for this purpose. Research onAccelerated Math has found students’ attitudes about math becomemore positive as they see personal success (Cosenza, 2004;Ysseldyke et al., 2007).Figure 8: Sample STAR Math StudentProgress Monitoring Report1 of 2Student Progress Monitoring ReportTuesday, February 21, 2012 1:26:12 PMSchool: Pine Hill Middle School Reporting Period: 9/1/2011 - 6/12/2012(School Year)Major, JasmineGrade: 7Class: 5th Hour MathID: JMAJORTeacher: Williams, T.STAR Math Scaled Score900850800750700650Aug-11 Sep-11 Oct-11 Nov-11 Dec-11 Jan-12 Feb-12 Mar-12 Apr-12 May-12Test scoreJasmine's Current GoalGoal: 772 SS 45 PR (Moderate) Goal End Date: 6/11/2012Trend line Goal line GoalIntervention changeExpected Growth Rate: 1.1 SS/WeekFluctuation of scores is typical with any test administered multiple times within a short period. Focus on the general direction emerging aftermultiple administrations of the test rather than on the ups and downs between individual scores.2For a full-sized example of this report, or a look at a sampling of key Accelerated Math for Intervention reports, see Appendix B, p. 29. The completemenu of reports available for the Accelerated Math for Intervention software components is found in a separate publication from RenaissanceLearning, Key Report Samples, available online from http://doc.renlearn.com/KMNET/R003563228GE7E80.pdf21

Informative AssessmentThe Accelerated Math for Intervention components—Accelerated Math, MathFacts in a Flash, and STARMath—are all reliable, valid, and efficient measures that provide actionable data about student achievementin mathematics for screening, progress monitoring, skills diagnosis, and formative assessment.What the research saysAssessment is a critical component of intervention. Data from assessments is used both to identify students’knowledge gaps in critical skills mastery and to regularly monitor the progress of students receiving anintervention. The National Council of Teachers of Mathematics (2000) recommends that assessment “be anintegral part of instruction that informs and guides teachers as they make instructional decisions” (p. 22) andthat teachers use a variety of assessment methods, including formative techniques that guide instruction andsummative methods that measure progress. “Assessment…involves both teacher and students in reciprocalactivity to move learning forward within a community of practice” (Heritage, 2010, p. 8).The IES Practice Guide (Gersten, Beckmann, et al., 2009) recommendsscreening students to identify those at risk for potential mathematicsdifficulties, monitoring the progress of students receiving supplementalinstruction at least once a month, and using curriculum-embeddedassessments to determine whether students are learning from theintervention as often as daily or as infrequently as every other week.The Guide advises that a complete intervention needs to incorporate allthree types of assessment to ensure these purposes are met.Accelerated Math,MathFacts in a Flash,and STAR Math arehighly rated by NCIIand NCRTI experts.After students are identified for intervention, formative assessment is used frequently in an intervention settingto monitor student progress toward proficiency and inform targeted instruction and differentiated practice.Research indicates the importance of two general approaches to formative evaluation (Fuchs & Deno, 1991;Gerston, Chard, et al., 2009), which provide different, but equally vital, types of information:• Mastery measures, sometimes thought of as curriculum-embedded assessments, meant to gauge howwell students have learned material in a daily lesson or have mastered a specific skill or subskill• General outcome measures, or broader measures of mathematics proficiency, meant to provide theschool with a sense of how the overall mathematics program is affecting a studentMastery measurement, in particular, is an instructional process with extensive research support as beingeffective for all students. Because mastery measures are focused on specific skills, they are well suited fordiagnostic evaluation, meaning teachers can pinpoint areas needing attention (Burns, 2010). BenjaminBloom’s (1980) “mastery learning” experiments demonstrated that using formative assessments as a basisto modify curriculum and instruction improved average student performance dramatically—in effect, shiftingthe entire achievement distribution curve in a positive direction. Black and Wiliam’s 1998 meta-analysis furtherdocumented how using assessment results to set goals and determine whether interventions improveperformance is particularly effective in reducing achievement gaps between subgroups. Other researchershave demonstrated that lower achieving students were less likely to require special-education referrals, orremained in special education less time, when these techniques were applied system wide (Bollman,Silberglitt, & Gibbons, 2007; Marston, Muyskens, Lau, & Canter, 2003).Putting the research into practice with Accelerated Math for InterventionEfficiently generating necessary achievement data is of primary importance within an intervention setting.Teachers rely on up-to-date information for each student to ensure that all students are receiving thenecessary help to make them successful. Accelerated Math for Intervention provides a complete, cohesive22

INFORMATIVE ASSESSMENTassessment package that includes screening, progress monitoring, and formative evaluation of mastery ofskills as well as general outcome, diagnostic, and automaticity measurement.The U.S. Department of Education’s National Center on Intensive Intervention (NCII) and National Center onResponse to Intervention (NCRTI) conduct frequent reviews of educational tools. The program components—Accelerated Math, MathFacts in a Flash, and STAR Math—are highly rated by NCII (2012a, 2012b, 2012c) forprogress monitoring and by NCRTI (2009, 2010, 2011a, 2011b) for screening and progress monitoring.Screening and progress monitoringSTAR Math is the first step in the Accelerated Math for Intervention program. It provides teachers withactionable data for screening as well as initial placement in and progress monitoring the efficacy of theintervention (see Figure 9 3 ). STAR Math is accurate, valid, reliable, efficient, and repeatable, as researchindicates is essential (Johnson, Mellard, Fuchs, & McKnight, 2006; Mellard & Johnson, 2008; Tilly, 2007).The computer-adaptive STAR Math assessment is easy to use and saves teachers significant time overadministering paper and pencil tests, and it provides much more information. Test administration time forSTAR Math is only about 15 minutes, and students quickly learn to take the test with minimal monitoring by theteacher. Computer adaptive means that the software adapts to the student’s responses while the student istaking the test, so each administration is neither too difficult nor too easy.Within the intervention setting, STAR Math is even more powerfulby providing a way to monitor growth for each student receivingintervention. Teachers can easily set performance goals and assessstudents periodically to determine if their growth is progressing at asufficient rate. Research consistently demonstrates using progressmonitoring techniques to collect, analyze, and make decisions onstudent performance data is associated with greater gains in studentachievement (Jimerson, Burns, & VanDerHeyden, 2007).Figure 9: Sample STAR MathScreening ReportSTAR Math includes a Goal-Setting Wizard that provides teacherswith a scientific method of goal setting using growth normsmodeling. STAR Math data also supports teacher decisions aboutwhether students are ready to exit the intervention or whetherinstructional changes are necessary.Additionally, the STAR Math assessment guides the teacher bysuggesting the appropriate Accelerated Math content library to havestudents begin working in based on their current performance. Akey analytic benefit of STAR Math is the ability to compare growthbetween students to guide instructional decisions. For example, if themajority of students are not showing growth in STAR Math, the teacher should make sure the intervention isbeing implemented with fidelity. On the other hand, if most students are showing growth it can be assumedthat the intervention is working overall. Data from STAR Math will then help the teacher identify any studentsnot making progress and provide guidance about what is stalling success. This is much more useful to informand adapt instruction and student practice than waiting for mid-year or end-of-year scores to see howprogress is developing when it is too late to make changes that could impact a student’s growth.3For a full-sized example of this report, or a look at a sampling of key Accelerated Math for Intervention reports, see Appendix B, p. 29. The completemenu of reports available for the Accelerated Math for Intervention software components is found in a separate publication from RenaissanceLearning, Key Report Samples, available online from http://doc.renlearn.com/KMNET/R003563228GE7E80.pdf23

INFORMATIVE ASSESSMENTFormative assessment and monitoring of students’ skills masteryDaily progress-monitoring assessments are ideal within a formative assessment process because theyinform instruction, provide immediate performance feedback, help monitor progress, and increase studentmotivation. Accelerated Math for Intervention facilitates formative assessment defined as “a process used byteachers and students during instruction that provides feedback to adjust ongoing teaching and learning toimprove students’ achievement of intended instructional outcomes” (McManus, 2008, p. 3). The interventionincorporates the formative evaluation measurement tools necessary to monitor students’ mastery of skills andprogress towards overall mathematics proficiency. As a result, teachers have the ability to measure students’retention of skills, individual growth, and growth in comparison to peers.Accelerated Math and MathFacts in a Flash provide immediate formative feedback for teachers usingAccelerated Math for Intervention. The programs suggest next steps for the teacher and students on a dailybasis; having a fully differentiated classroom—both in instruction and student practice—would be impossiblewithout this kind of daily, formative assessment.With the Accelerated Math and MathFacts in a Flash software, this intervention incorporates the first progressmonitoringmastery measurement systems to meet federal standards of quality set by the National Centeron Response to Intervention (U.S. DOE, 2009, 2011a). Accelerated Math and MathFacts in a Flash are alsohighly rated for progress-monitoring by the National Center on Intensive Intervention (U.S. DOE, 2012a,2012b). As mentioned, mastery measurement, a type of formative assessment, is a long-standing educationalpractice with a strong basis of research supporting its effectiveness. Accelerated Math and MathFacts in aFlash, as mastery measurement systems, do two things effectively: (1) provide students with an opportunity topractice math at individualized levels and receive immediate feedback, and (2) provide teachers with timelyinformation on student mastery that can help them make the targeted instructional decisions.In Accelerated Math for Intervention classrooms, students work on a hierarchical series of objectivesand must demonstrate mastery of each objective before proceeding to the next one. Teachers can use datafrom Accelerated Math for diagnostic evaluation of the math objectives students may be struggling with andthen adapt instruction as necessary. AcceleratedMath for Intervention allows for a fully differentiatedclassroom because it identifies the specific skillseach student is lacking mastery of that are ultimatelypreventing them from grade-level success. Essentially,the program helps teachers find the boundaries ofeach student’s knowledge and locate whereopportunities lie to build upon that knowledge.Teachers using Accelerated Math for Interventionknow, specifically, which prerequisite skills to assigneach student for more practice time, and whichstudents need targeted instruction on those sameskills. And built-in, frequent formative assessmentAccelerated Math for Interventionsuggests next steps for theteacher and students on a dailybasis; having a fully differentiatedclassroom—both in instructionand student practice—would beimpossible without this kind ofdaily, formative assessment.continuously monitors students’ work to ensure that mastery of these skills is attained. Retention of skills isassured through built-in review of previously mastered objectives on practice assignments.Achievement of foundational math facts automaticity is also monitored in Accelerated Math for Intervention.MathFacts in a Flash tracks the speed and accuracy of each student working towards math facts fluency,allowing for close monitoring of learned facts and the ability to set increasingly challenging time goals asautomaticity progresses. Educators also have the ability to measure progress against local benchmarks.24

Appendix A: Overview of Accelerated Math, MathFacts in a Flash, and STAR MathAccelerated MathAccelerated Math software personalizes student math practice and helps teachers generate assignments/tests, monitor progress, and motivate students to succeed. Teachers use progress-monitoring informationprovided by Accelerated Math to do what they do best—provide individualized, differentiated instruction toeach student.Content developmentThe Accelerated Math libraries were first published in 1998, with a scope and sequence based oncommonalities between the 1989 National Council of Teachers of Mathematics (NCTM) standards, leadingpublisher textbooks, National Assessment of Educational Progress editions, and math editor teachingexperience from the 1990s.A great deal has occurred in U.S. education since 1998, and much professional thought has gone into whatis important in mathematics. For the 2008 Accelerated Math content revision, Renaissance Learning (2009b)took into account all of the landmark changes and created a new scope and sequence for the AcceleratedMath Second-Edition Libraries for Grades 1 through 8, Algebra 1, and Geometry that incorporated• National Mathematics Advisory Panel (2008b) essential concepts and skills, andother recommendations• National Council of Teachers of Mathematics (2006) Curriculum Focal Points• Math curricular profiles of Singapore and other top-performing countries• Alignments with model state standards• Content reviews by the Northwest Regional Educational Laboratory, mathematics educators,mathematicians, and university researchers• Research on effective development of algorithm-generated dynamic itemsThen, in 2009, after the prerelease of the Committee on Early Childhood Mathematics and National ResearchCouncil’s report, Mathematics Learning in Early Childhood: Paths Toward Excellence and Equity, it becameclear there was a resounding call for greater emphasis on early numeracy in mathematics. The NCTMCurriculum Focal Points had already identified foundational prekindergarten and kindergarten skills, andRenaissance Learning developed the Accelerated Math Early Numeracy Library with objectives to meet theneeds of early or struggling math learners.How Accelerated Math worksAccelerated Math assignments are individually generated and printed for each student. After students workthem on paper, they record and submit their responses using a scan card and AccelScan scanner, a handheldRenaissance Responder, or a NEO 2 mobile e-learning device. The software automatically scores eachassignment and shares immediate feedback via informative reports for the teacher and student.Students follow various pathways for mastering objectives while using Accelerated Math (see Figure A1, nextpage), with the teacher monitoring progress regularly and providing instruction as needed. Upon completionof an assignment, the student continues on a path based on whether he or she was successful or not. If astudent continues to be unsuccessful, the teacher receives a notification to intervene (both on reports 4 printedfrom the program and in the software’s assignment book), and following teacher intervention, the student mustattempt to master the objective again by practicing and testing. Once an objective is practiced, tested, andmastered, it is reviewed on practice assignments after two weeks. If the student is unsuccessful upon review,the objective status changes to intervene and the student repeats the practice, test, and review cycle.4For a sampling of key Accelerated Math for Intervention reports, see Appendix B, p. 29.25

APPENDIX A: OVERVIEW OF ACCELERATED MATH, MATHFACTS IN A FLASH, AND STAR MATHFigure A1: Accelerated Math Cycle OverviewTeacher prints diagnostictest that includes one ormore objectives. Student isDiagnostic Testing4 out of 5 requiredYesFor each objective,did the student pass?Practice PrintedStudent is Working5 out of 6 requiredExerciseTeacher prints if objective is inintervene state due to student beingunsuccessful while practicing.Student is Working5 out of 6 requiredDid the student pass?YesStudent is Ready to TestTeacher prints testStudent is Testing4 out of 5 requiredDiagnostic TestTeacher prints if objective is inintervene state due to student beingunsuccessful while testing or reviewing.Student is Diagnostic Testing4 out of 5 requiredDid the student pass?YesNoNoObjective AssignedTeacher assigns additionalobjectives as necessary.Objective goes to theReady to Work stateNoNoafter 3 attemptsInterveneDid the student pass?YesYes / ExerciseNoafter 2 attemptsObjective MasteredAfter two weeks, theobjective begins to appearon practices as Review.3 out of 4 requiredYes / Diagnostic TestDid the student pass?YesObjective ReviewedObjective has beenreviewed and will no longerappear on assignments.Noafter 2 attemptsTo help teachers identifystudents’ skill gaps,To helpwork inteachersAcceleratedMath for Intervention begins with diagnostictesting. This allows students to demonstratemastery of known objectives, and thesoftware to place those objectives not masteredor known in the Ready to Work state foradditional practice after instruction.Teachers assign additional objectivesfor practice as necessary.identify students’ skill gaps,work in Accelerated Math forIntervention begins with diagnostictesting. This allows students todemonstrate mastery of knownobjectives, and allows the software toplace objectives that are not mastered orknown in the Ready to Work statefor additional practice afterinstruction. Teachers assignadditional objectives forpractice as necessary.26

APPENDIX A: OVERVIEW OF ACCELERATED MATH, MATHFACTS IN A FLASH, AND STAR MATHThere are four types of assignments in Accelerated Math:• Practice assignments consist of multiple-choice questions, include teacher-assigned and Ready toWork objectives, 5 and are designed to give students an opportunity to practice math concepts that havebeen previously taught.• Exercise assignments consist of multiple-choice or free-response questions, include any objectivespecified by the teacher, and are designed to be used to supplement daily lessons or to provide morepractice on specific objectives after a lesson or intervention.• Tests consist of multiple-choice or free-response questions, include Ready to Test objectives, 6 and aredesigned to allow students to demonstrate mastery of an objective.• Diagnostic tests consist of multiple-choice or free-response questions, include any objectivesspecified by the teacher, and are designed to test students on any objective, even those not recentlypracticed. Teachers can use a diagnostic test to place incoming students or to allow students to masterobjectives directly when they have previous knowledge of certain objectives.MathFacts in a FlashMathFacts in a Flash provides students at all levels with essential personalized practice of addition,subtraction, multiplication, and division facts, as well as other math skills, including squares and conversionsbetween fractions, decimals, and percentages. Timed tests at appropriate skill levels accurately measurestudents’ practice and mastery, with feedback provided both onscreen and via a variety of detailed reports.Feedback motivates students and helps teachers inform instruction and monitor student progress towardsbenchmarks throughout the year.There are five basic steps to implementing MathFacts ina Flash:1. Baseline test. Students complete a 40-item timed testat the computer for each new math level. Immediateonscreen feedback provides time and accuracy dataand shows any missed facts. If a student answers all 40items correctly within the time limit, he or she moves onto the next math level. This allows all students to work attheir own level of challenge.Figure A2: Sample MathFacts in a FlashPractice Screen2. Personalized practice. Math facts not mastered on thebaseline test are presented to the student in practicesessions (see example, Figure A2). These are intermixedwith mastered facts that are typically difficult for studentsat each level and other known facts, for a minimum of 20 items per practice session. Time allottedper day ranges between 5 and 15 minutes, with a recommended frequency of at least three times perweek. Research shows frequent use ofMathFacts in a Flash can help students to double math gains (Burns, Kanive, & DeGrande, 2012).3. Timed test for mastery. Once students have successfully completed the practice sessions at a level,they complete a 40-item timed test for that level. Students master a level in MathFacts in a Flash whenthey are able to complete a test within the time goal with 100% accuracy.5Ready to Work objectives are those that students have encountered on at least one practice or exercise assignment or diagnostic testwithout mastering.6Ready to Test objectives are those for which the student has successfully practiced, so he or she is ready to test for mastery on these objectives.27

APPENDIX A: OVERVIEW OF ACCELERATED MATH, MATHFACTS IN A FLASH, AND STAR MATH4. Instant feedback. Immediate onscreen feedback after each completed MathFacts in a Flash practiceor test provides students with results on accuracy and time goals, allowing them to monitor their ownprogress. Additional corrective feedback is provided after each question answered incorrectly.5. Automatic advancement. After students master their current level, the software automatically assignsthe next math level. The sequence of math levels can be reordered to fit any curriculum.STAR MathThe STAR Math assessment—used for screening, progressmonitoring,and diagnostic assessment—is a reliable, valid,and efficient, computer-adaptive assessment of general mathachievement for grades 1–12. 7 STAR Math reports nationallynorm-referenced math scores and criterion-referencedevaluations of skill levels and provides these results via avariety of informative, easy-to-understand reports. A STAR Mathassessment can be completed without teacher assistance inless than 15 minutes and repeated as often as weekly forprogress monitoring. As part of the Accelerated Math forIntervention program, Renaissance Learning recommendsmonitoring students’ progress with STAR Math at least oncea month.Figure A3: Sample STAR Math ItemThe content for STAR Math is based on an analysis of professional standards, curriculum materials, testframeworks, and content-area research, including best practices for mathematics instruction. The STAR Mathitem bank includes 214 core math objectives, with multiple items available to measure each objective. FigureA3 shows a sample assessment item.7STAR Math may also be used with kindergarten students, though the assessment has not been normed for this age group. Students are ready totest in STAR Math when they have a 100-word sight vocabulary, which for some students may not occur until later in first grade.28

Appendix B: Sample Key ReportsOn the following pages are sample key reports 8 for use with Accelerated Math for Intervention:• STAR Math Student Progress Monitoring Report• STAR Math Screening Report• Accelerated Math Diagnostic Report• Accelerated Math Status of the Class Report• MathFacts in a Flash Student Progress ReportThe complete menu of reports available for the Accelerated Math for Intervention software components isfound in a separate publication from Renaissance Learning, Key Report Samples, available online fromhttp://doc.renlearn.com/KMNET/R003563228GE7E80.pdf or by request to (800) 338-4204.8Reports are regularly reviewed and may vary from those shown as enhancements are made.29

APPENDIX B: SAMPLE KEY REPORTSRTIPage 1 of thisreport graphs astudent’s scores in relationto the goal, giving theteacher a picture of thestudent’s progress.(Moderate)After the teacherselects a goal for thestudent, the goal lineprojects an interventionoutcome, and the trendline shows the student’sactual progresstoward that goal.30

APPENDIX B: SAMPLE KEY REPORTSScreening isthe first step in RTI.Use this report forgrade-level planningand identifying studentswho need themost help.These studentsare all belowbenchmark.RTIUse these keyquestions to helpdeterminenext steps.In a typicalRTI implementation,about 80% of studentswill be served by Tier 1.In this school, about 20%need additional attentionin Tier 2 or Tier 3interventions.31

APPENDIX B: SAMPLE KEY REPORTSThis studentmay be struggling toomuch; Kristina would bemore successfulworking in a lowerlevel AM library.Students workingin lower-level librariesshould be masteringmore objectivesthan they would ifworking atgrade-level.Students workingin Accelerated Mathas part of their Tier 2 or3 program will mastermost of their initialobjectives onDiagnosticTests.This exampleshows 14 high schoolstudents in a Tier 2class period, usingappropriate grade levelAccelerated Mathlibraries as part oftheir interventionprogram.32

APPENDIX B: SAMPLE KEY REPORTSAcceleratedMath providesthe data todifferentiate studentpractice.Provide targetedinstruction to helpstudents havingproblems with specificobjectives.To aid withinstructional planning,the report helps identifywhich math objectivesare causing themost difficulty forstudents.33

APPENDIX B: SAMPLE KEY REPORTSThe targetdate for thebenchmark levelis April 1.Jeremy tookeach test multipletimes; Best Time showsthe lowest time it tookJeremy to meet themastery criteriafor the level.Date Masteredlists the date thelevel was originallymastered. This datematches the levelmastered symbol inthe graph.34

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AcknowledgmentsRenaissance Learning sincerely thanks the following individuals for sharing their expertise in consultation onour mathematics tools.Sybilla Beckmann, Ph.D., is a professor of mathematics at the University of Georgia. She is especiallyinterested in helping college faculty learn to teach mathematics content courses for elementary and middlegrades teachers and has developed three courses for prospective elementary school teachers at theUniversity 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 theNCTM’s Curriculum Focal Points for Prekindergarten Through Grade 8 Mathematics and has worked on thedevelopment 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 elementaryteachers. He has worked with K–12 teachers and students for 15 years and has taught and developednumerous professional development courses that focus on improving teacher understanding of mathematics,including the Singapore Math Project, which was developed in conjunction with the MassachusettsDepartment of Education. Bisk also presented testimony before the National Math Panel in September 2006regarding 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 ofelementary teachers.Matthew K. Burns, Ph.D., is a professor of educational psychology, coordinator of the School PsychologyProgram, and co-director of the Minnesota Center for Reading Research at the University of Minnesota. Burnshas published over 150 articles and book chapters in national publications, and has co-authored or co-editedseveral books. He is also the editor of School Psychology Review and past editor of Assessment for EffectiveIntervention. Specific areas in which Burns has conducted research include response to intervention,assessing the instructional level, academic interventions, and facilitating problem-solving teams.Thomas P. Hogan, Ph.D., is a professor of psychology and a Distinguished University Fellow at the Universityof 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 StanfordDiagnostic Mathematics Test, Stanford Modern Mathematics Test, and the Metropolitan Achievement Test.Hogan has published articles in the Journal for Research in Mathematics Education and MathematicalThinking and Learning, and has authored two textbooks and more than 100 scholarly publications in the areasof 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 mathematicseducation includes consulting with several states on math standards, including California. Milgram has givenlectures around the world and is a member of numerous boards and committees, including the National Boardof Education Sciences, created by the Education Sciences Reform Act of 2002 “to advise and consult with theDirector of the Institute of Education Sciences (IES) on agency policies,” and the Human Capital Committeeof the NASA Advisory Council, which “provides the NASA Administrator with counsel and advice on programsand issues of importance to the Agency.” Milgram is author of “An Evaluation of CMP,” “A Preliminary Analysisof SAT-I Mathematics Data for IMP Schools in California,” and “Outcomes Analysis for Core Plus Students atAndover High School: One Year Later.” Each of these papers identifies serious shortcomings in popularmathematics programs.40

ACKNOWLEDGEMENTSSharif M. Shakrani, Ph.D., is a private consultant and researcher specializing in measurement and quantitativemethods. Shakrani is a former co-director of the Education Policy Center at Michigan State University andprofessor of measurement and quantitative methods in the Department of Counseling, EducationalPsychology and Special Education. Before coming to Michigan State University, Shakrani served 8 yearsas the deputy executive director of the National Assessment Governing Board in the U.S. Department ofEducation. He was responsible for technical and policy direction for the National Assessment of EducationalPrograms (NAEP). He has also worked for the National Center for Education Statistics in the U.S. Departmentof Education where he guided the design and analysis of federal educational assessments. In his work inthe Michigan Department of Education, Shakrani was responsible for K–12 general curriculum andassessment and was instrumental in revising the Michigan Educational Assessment Program (MEAP).Edward S. Shapiro, Ph.D., is a professor of school psychology and director of the Center for PromotingResearch to Practice in the College of Education at Lehigh University. He is the 2006 winner of the AmericanPsychological Association’s Senior Scientist Award, was recognized in 2007 by the PennsylvaniaPsychological Association, and recently received Lehigh University’s Eleanor and Joseph Lipsch ResearchAward. Shapiro has authored 10 books and is best known for his work in curriculum-based assessmentand non-standardized methods of assessing academic skills problems. Among his many projects, Shapiroco-directs a federal project focused on the development of a multi-tiered, RTI model in two districts inPennsylvania, and he has recently been awarded a U.S. Department of Education grant to train schoolpsychologists as facilitators of RTI processes. He is also currently collaborating with the PennsylvaniaDepartment of Education in developing and facilitating the implementation of the state’s RTI methodology.Amanda M. VanDerHeyden, Ph.D., is a private consultant and researcher who has worked as a researcher,consultant, and national trainer in a number of school districts and published more than 60 scholarly articlesand book chapters related to RTI. In 2006, VanDerHeyden was named to a National Center for LearningDisabilities advisory panel to provide guidance related to RTI. She is associate editor of School PsychologyReview, serves on the editorial boards of several journals including School Psychology Quarterly and Journalof Early Intervention, author of Essentials of Response to Intervention (with Dr. Matthew Burns) and KeepingRtI on Track: How to Identify, Repair, and Prevent Mistakes that Derail Implementation RtI (with Dr. David Tilly),and co-editor of the Handbook of Response to Intervention. VanDerHeyden received the 2006 LightnerWitmer Early Career Contributions Award from the APA for her scholarship on early intervention, RTI, andmodels of data-based decision making. She serves as research advisor to iSTEEP, a web-based datamanagement system.Kenneth E. Vos, Ph.D., is a professor of education at St. Catherine University in St. Paul, Minnesota. For thelast 35 years, he has been preparing undergraduate and graduate students to teach mathematics, and intandem working with classroom teachers. Vos has directed numerous programs and institutes for teacherson mathematics, best practices in the mathematics classroom, and assessment. He holds a Ph.D. inmathematics education from the University of Minnesota.41

ACKNOWLEDGEMENTSJames Ysseldyke, Ph.D., is Emma Birkmaier Professor of Educational Leadership in the Department ofEducational Psychology at the University of Minnesota. He has been educating school psychologists andresearchers for more than 35 years. Ysseldyke has served the University of Minnesota as director of theMinnesota Institute for Research on Learning Disabilities, director of the National School Psychology Network,director of the National Center on Educational Outcomes, director of the School Psychology Program, andassociate dean for research. Professor Ysseldyke’s research and writing have focused on enhancing thecompetence of individual students and enhancing the capacity of systems to meet students’ needs. He is anauthor of major textbooks and more than 300 journal articles. Ysseldyke is conducting a set of investigationson the use of technology-enhanced progress-monitoring systems to track the performance and progress ofstudents in urban environments. Ysseldyke chaired the task forces that produced the three Blueprints on theFuture of Training and Practice in School Psychology, and he is former editor of Exceptional Children, theflagship journal of the Council for Exceptional Children. Ysseldyke has received awards for his research fromthe School Psychology Division of the American Psychological Association, the American EducationalResearch Association, and the Council for Exceptional Children. The University of Minnesota presented hima distinguished teaching award, and he received a distinguished alumni award from the University of Illinois.

AcknowledgementsRenaissance Learning sincerely thanks the following individuals for sharing their expertise in consultation onour mathematics tools. (To learn more, please see the detailed Acknowledgements section inside.)Sybilla Beckmann, Ph.D., is aprofessor of mathematics at theUniversity of Georgia.Sharif M. Shakrani, Ph.D., is aprivate consultant and researcherspecializing in measurement andquantitative methods.Richard Bisk, Ph.D., is chairand professor of mathematicsat Worcester State Collegein Massachusetts.Edward S. Shapiro, Ph.D., is aprofessor of school psychologyand director of the Center forPromoting Research to Practicein the College of Education atLehigh University.Matthew K. Burns, Ph.D., is aprofessor of educationalpsychology, coordinator of theSchool Psychology Program,and co-director of the MinnesotaCenter for Reading Research atthe University of Minnesota.Thomas P. Hogan, Ph.D., is aprofessor of psychology and aDistinguished University Fellowat the University of Scranton.Amanda M. VanDerHeyden,Ph.D., is a private consultantand researcher who has workedas a researcher, consultant, andnational trainer in a number ofschool districts.Kenneth E. Vos, Ph.D., is aprofessor of education at St.Catherine University in St.Paul, Minnesota.R. James Milgram, Ph.D., is aprofessor of mathematics atStanford University.James Ysseldyke, Ph.D., isEmma Birkmaier Professor ofEducational Leadership in theDepartment of EducationalPsychology at the Universityof Minnesota.L2681.0313.FP.2.5MR44687