# RTI Toolkit: A Practical Guide For Schools RTI: Resource .

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RTI Toolkit: A Practical Guide for SchoolsRTI: Resource Materials for Math Interventions& AssessmentJim Wright, Presenter9 September 2008SSTAGE Fall ConferenceDublin, GAJim Wright364 Long RoadTully, NY 13159Email: jim@jimwrightonline.comWebsite: www.interventioncentral.org

Curriculum-Based Measurement Administration & ScoringGuidelines for Math ComputationCBM MATHDescriptionThere are 2 types of CBM math probes, single-skill worksheets (those containing like problems)and multiple-skill worksheets (those containing a mix of problems requiring different math operations).Single-skill probes give instructors good information about students' mastery of particular problem-types,while multiple-skill probes allow the teacher to test children's math competencies on a range ofcomputational objectives during a single CBM session.Both types of math probes can be administered either individually or to groups of students. Theexaminer hands the worksheet(s) out to those students selected for assessment. Next, the examiner readsaloud the directions for the worksheet. Then the signal is given to start, and students proceed to completeFigure 5: A Sampling of Math Computational Goals for Addition, Subtraction, Multiplication, and Division (from Wright,2002).AdditionTwo 1-digit numbers: sums to 10Two 3-digit numbers: no regrouping1- to 2-digit number plus 1- to 2-digit number: regroupingSubtractionTwo 1-digit numbers: 0 to 92-digit number from a 2-digit number: no regrouping2-digit number from a 2-digit number: regroupingMultiplicationMultiplication facts: 0 to 92-digit number times 1-digit number: no regrouping3-digit number times 1-digit number: regroupingDivisionDivision facts: 0 to 92-digit number divided by 1-digit number: no remainder2-digit number divided by 1-digit number: remainderWright, J. (2002) Curriculum-Based Assessment Math Computation Probe Generator: Multiple-Skill Worksheets inMixed Skills. Retrieved August 13, 2006, from be/allmult.shtmlas many items as possible within 2 minutes. The examiner collects the worksheets at the end of theassessment for scoring.Creating a measurement pool for math computational probesThe first task of the instructor in preparing CBM math probes is to define the computational skills tobe assessed. Many districts have adopted their own math curriculum that outlines the various computationalJim Wright, Presenterwww.interventioncentral.org2

skills in the order in which they are to be taught. Teachers may also review scope-and-sequence charts thataccompany math textbooks when selecting CBM computational objectives.The order in which math computational skills are taught, however, probably does not vary a greatdeal from district to district. Figure 5 contains sample computation goals for addition, subtraction,multiplication, and division.Instructors typically are interested in employing CBM to monitor students' acquisition of skills inwhich they are presently being instructed. However, teachers may also want to use CBM as a skills checkup to assess those math objectives that students have been taught in the past or to "preview" a mathgroup's competencies in computational material that will soon be taught.Preparing CBM Math ProbesAfter computational objectives have been selected, the instructor is ready to prepare math probes.The teacher may want to create single-skills probes, multipleskill probes, or both types of CBM mathworksheets.Creating the Single-skill Math ProbeAs the first step in putting together a single-skill math probe, the teacher will select onecomputational objective as a guide. The measurement pool, then, will consist of problems -------Figure 6: Example of a single-skill math probe: Three to five 3- and 4-digit numbers: no regrouping 1056002932031 531 2322 111717260 634 8240 203 onstructed that conform to the computational objective chosen. For example, the instructor may select thefollowing computational objective (Figure 6) as the basis for a math probe.The teacher would then construct a series of problems that match the computational goal, as in Figure 6. Ingeneral, single-skill math probes should contain between 80 and 200 problems, and worksheets shouldhave items on both the front and back of the page. Adequate space should also be left for the student'scomputations, especially with more complex problems such as long division.Creating the Multiple-skill Math ProbeTo assemble a multiple-skill math probe, the instructor will first select the range of math operationsand of problem-types that will make up the probe. The teacher will probably want to consult the district ---Figure 7: Example of a multiple-skill math probe:Division: 3-digit number divided by 1-digit number: no remainderSubtraction: 2-digit number from a 2-digit number: regroupingMultiplication” 3-digit number times 1-digit number: no regroupingDivision: Two 3-digit numbers: no regrouping9/431 20-18 113x 2 106172200600 im Wright, Presenterwww.interventioncentral.org3

If the answers in Figure 8 were scored as either correct or wrong, the child would receive a score of 1correct answer out of 4 possible answers (25 percent). However, when each individual digit is scored, itbecomes clear that the student actually correctly computed 12 of 15 possible digits (80 percent). Thus, theCBM procedure of assigning credit to each correct digit demonstrates itself to be quite sensitive to astudent's emerging, partial competencies in math computation.The following scoring rules will aid the instructor in marking single- and multiple-skill math probes: Individual correct digits are counted as correct.Reversed or rotated digits are not counted as errors unless their change in position makes themappear to be another digit (e.g., 9 and 6). Incorrect digits are counted as errors.Digits that appear in the wrong place value, even if otherwise correct, are scored as errors.Example"873" is the correct answer to this problem, but nocredit can be given since the addition of the 097pushes the other digits out of their proper placex98730value positions. The student is given credit for "place-holder" numerals that are included simply to correctly alignthe problem. As long as the student includes the correct space, credit is given whether or not a "0"has actually been inserted.Example55x 8211044004510Since the student correctly placed 0 in the "placeholder" position, it is given credit as a correct digit.Credit would also have been given if the spacewere reserved but no 0 had been inserted. In more complex problems such as advanced multiplication, the student is given credit for allcorrect numbers that appear below the line.Example33x 28Credit is given for all work below the line. In this264example, the student earns credit for 9 correct660digits.924 Credit is not given for any numbers appearing above the line (e.g., numbers marked at the top ofnumber columns to signify regrouping).Example1Credit is given for the 2 digits below the line.46However,the carried "1" above the line does not 39receive credit.85Reference: Wright, J. (n.d.). Curriculum-based measurement: A manual for teachers. Retrieved September 23, 2006,from dfJim Wright, Presenterwww.interventioncentral.org5

Appendix D: Computational GoalsMID22. Add a three-digit number to a three-digit number with regroupingfrom the units to the tens column and from the tens to the hundredscolumn.23. Add a four-digit number to a four-digit number with regrouping inone to three columns.24. Subtract two four-digit numbers-no regrouping.25. Subtract a one-digit number from a two-digit number withregrouping.26. Subtract a two-digit number from a two-digit number withregrouping.27. Subtract a two-digit number from a three-digit number withregrouping from the units to the tens column only.28. Subtract a two-digit number from a three-digit number withregrouping from the tens to the hundreds column only.29. Subtract a two-digit number from a three-digit number withregrouping from the units to the tens column and from the tens tothe hundreds column.30. Subtract a three-digit from a three-digit number with regroupingfrom the units to the tens column only.31. Subtract a three-digit number from a three-digit number withregrouping from the tens to the hundreds column only.32. Subtract a three-digit number from a three-digit number withregrouping from the units to the tens column and from the tens tothe hundreds column.33. Multiplication facts--3 to 9.Grade 434. Add a five- or six-digit number to a five- or six-digit number withregrouping in any columns.35. Add three or more two-digit numbers with regrouping.36. Add three or more three-digit numbers with regroupingwith regrouping in any columns.37. Subtract a five- or six-digit number from a five- or six-digitnumber with regrouping in any columns.38. Multiply a two-digit number by a one-digit number with noregrouping.39. Multiply a three-digit number by a one-digit number with noregrouping.40. Multiply a two-digit number by a one-digit number with noregrouping.41. Multiply a three-digit number by a one-digit number with regrouping.42. Division facts--0 to 9.43. Divide a two-digit number by a one-digit number with no remainder.44. Divide a two-digit number by a one-digit number with remainder.45. Divide a three-digit number by a one digit number with remainder.46. Divide a four-digit number by a one-digit number with remainder.CBM Workshop ManualJim Wright, PresenterJim Wrightwww.interventioncentral.orgAppendix D-27

Appendix D: Computational GoalsMIDGrade 547. Multiply a two-digit number by a two-digit number with regrouping.48. Multiply a three-digit number by a two-digit number withregrouping.49. Multiply a three-digit number by a three-digit number withregrouping.List of computational goals taken from Shapiro, Edward S. (1989). Academicskills problems: Direct assessment and intervention. New York: GuilfordPress.CBM Workshop ManualJim Wright, PresenterJim Wrightwww.interventioncentral.orgAppendix D-38

Curriculum-Based Assessment MathematicsMultiple-Skills Computation Probe: Student CopyStudent:727,162 30,484146,569 532,260Date: 42,286-29,75633,516-21,366 156x623192x371 52/2207 43/4742 www.interventioncentral.orgJim Wright, Presenterwww.interventioncentral.org9

Curriculum-Based Assessment MathematicsMultiple-Skills Computation Probe: Examiner CopyItem 1:6 CD/6 CD TotalADDITION: 5- to 6digit number plus 5to 6-digit number:Regrouping in anycolumn727,162 30,484757,646Item 5:6 CD/49 CD TotalADDITION: 5- to 6digit number plus 5to 6-digit number:Regrouping in anycolumn146,569 532,260678,829 Item 2:5 CD/11 CD TotalSUBTRACTION: 5digit number from5-digit number:regrouping in anycolumn42,286- 29,75612,530Item 6:5 CD/54 CD TotalSUBTRACTION: 5digit number from5-digit number:regrouping in anycolumn33,516- 21,36612,150Item 3:17 CD/28 CD TotalMULTIPLICATION: 3digit number times3-digit number:regrouping 156x 623468312936-97,188Item 4:15 CD/43 CD TotalDIVISION: 4-digit numberdivided by 2-digitnumber: remainder Item 7:18 CD/72 CD TotalMULTIPLICATION: 3digit number times3-digit number:regrouping 192x 3711921344576-71,232 42 r2352/2207-208127-10423Item 8:13 CD/85 CD TotalDIVISION: 4-digit numberdivided by 2-digitnumber: remainder 110 r1243/4742-4344-4312www.interventioncentral.orgJim Wright, Presenterwww.interventioncentral.org10

www.interventioncentral.orgCurriculum-Based Measurement(CBM) GraphMath Computation:0-80:12 WeeksSetting up the graph At the top of the graph, fill out the student’s name, his or her classroom and/or grade, andinformation about the level at which the student is being monitored withCBM.Figure 1 After you have collected baseline CBM information, fill out the start date and-6 5 -6 16end date in the Baseline date section for the time span during which you-6 13 -6 20collected baseline data (Figure 1). Then decide how many instructionalweeks that you plan to monitor the student’s progress. Fill out the start date (Monday) and enddate (Friday) in the Monitoring date section for each instructional week during which monitoringwill take place (Figure 1). If possible, you should try to collect at least one CBM observationper week for your target student. It is a good idea to fill in the weekly start- and end-dates inadvance to give yourself an incentive to stay up-to-date on your CBM monitoring.Figure 2BASELINE WEEK 1Entering information onto the graph Baseline datapoints. Collect at least 3-5 baseline datapoints. (Baseline data arecollected to get a sense of the student’s current performance level and rate ofprogress. It is a good idea to collect them within a 1- to 2-week span.) Plot thesedatapoints in the ‘baseline’ column on the graph, as shown in Figure 2. Next toeach plotted datapoint, write the date on which it was collected. Connect allbaseline datapoints with lines to identify them as a single data-series.6/96/56/12Progress-monitoring datapoints. When graphing a CBM datapoint collected during progressmonitoring, find the week whose date span includes the date on which the CBM assessmentwas completed. At the bottom of the graph, circle the weekday (‘MTWTF’) on which theassessment was conducted. Then plot the datapoint above that circled day. (See Figure 3 foran example.) Connect all monitoring datapoints with lines to identify them as a single dataseries. Do not connect the baseline and monitoring data-series, however, as each should beconsidered separate data ‘phases’.Figure 3Want additional guidelines for setting up your data chart?For more information about how to set up and use a CBM progress-monitoring chart,consult the free book Curriculum-Based Measurement: A Manual for Teachers. Thismanual provides a complete introduction to CBM and its use in schools. Find it on theweb at: dfJim Wright, Presenterwww.interventioncentral.org11

WEEK 1 WEEK 2BASELINE- - -- - -WEEK 3 WEEK 4- -WEEK 5-- --WEEK 6-WEEK 7-- -WEEK 8-WEEK 9-WEEK 10 WEEK 11 WEEK 12- - ----- -807050Instructional 4 4030(Norms from Shapiro, 1996)Mastery 1-3Mastery 4 6020Frustr’l 1-3 Instr’l 1-3Frustraitional 4 Correct Digits Per 2 Minutes: Problem Type(s):Student: Classrm/Grade: Monitoring Level:100BASELINEM T W T F M T W TFM T W T F M T W TFM T W T F M T W TFM T W T F M T W TFInstructional DaysJim Wright, Presenterwww.interventioncentral.orgM T W T F M T W TFM T W T F M T W TFMath 80-12 2003 Jim Wright www.interventioncentral.org12

Early Math Fluency CBM Probe: Quantity DiscriminationThis introduction to the Quantity Discrimination probe provides information about the preparation,administration, and scoring of this Early Math CBM measure. Additionally, it offers brief guidelinesfor integrating this assessment into a school-wide ‘Response-to-Intervention’ model.Quantity Discrimination: Description (Clarke & Shinn, 2005; Gersten, Jordan & Flojo, 2005)The student is given a sheet containing pairs of numbers. In each number pair, one number islarger than the other. The numbers in each pair are selected from within a predefined range (e.g.,no lower than 0 and no higher than 20). During a one-minute timed assessment, the studentidentifies the larger number in each pair, completing as many items as possible while the examinerrecords any Quantity Discrimination errors.Quantity Discrimination: PreparationThe following materials are needed to administer Quantity Discrimination (QD) Early Math CBMprobes: Student and examiner copies of a QD assessment probe. (Note: Customized QD probes canbe created conveniently and at no cost using Numberfly, a web-based application. VisitNumberfly at umberfly.php). A pencil, pen, or marker A stopwatchQuantity Discrimination: Directions for Administration1. The examiner sits with the student in a quiet area without distractions. The examiner sits at atable across from the student.2. The examiner says to the student:“The sheet on your desk has pairs of numbers. In each set, one number is bigger than theother.”“When I say, 'start,' tell me the name of the number that is larger in each pair. Start at the topof this page and work across the page [demonstrate by pointing]. Try to figure out the largernumber for each example. When you come to the end of a row, go to the next row. Are thereany questions? [Pause] Start. “3. The examiner begins the stopwatch when the student responds aloud to the first item. If thestudent hesitates on a number for 3 seconds or longer on a Quantity Discrimination item, theexaminer says, “Go to the next one.” (If necessary, the examiner points to the next number asa student prompt.)Jim Wright, Presenterwww.interventioncentral.org13

Jim Wrightwww.interventioncentral.org3 of 4Quantity Discrimination: Measurement StatisticsTest-Retest Reliability Correlations for Quantity Discrimination ProbesTime SpanCorrelationReference13-week interval0.85Clarke & Shinn (2005)26-week interval0.86Clarke & Shinn (2005)Predictive Validity Correlations for Quantity Discrimination ProbesPredictive Validity MeasureCorrelationReferenceCurriculum-Based Measurement Math0.67Clarke & Shinn (2005)Computation Fluency Probes: Grade 1Addition & Subtraction (Fall Administration ofQD Probe and Spring Administration of MathComputation Probe)0.79Clarke & Shinn (2005)Woodcock-Johnson Tests of Achievement:Applied Problems subtest (Fall Administrationof QD Probe and Spring Administration of WJACH subtest)Number Knowledge Test0.53Chard, Clarke, Baker, Otterstedt,Braun & Katz.(2005) cited inGersten, Jordan & Flojo (2005)ReferencesChard, D. J., Clarke, B., Baker, S., Otterstedt, J., Braun, D., & Katz, R. (2005). Using measures ofnumber sense to screen for difficulties in mathematics: Preliminary findings. Assessment ForEffective Intervention, 30(2), 3-14.Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and developmentof early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248.Gersten, R., Jordan, N.C., & Flojo, J.R. (2005). Early identification and interventions for studentswith mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.Jordan, N. C. & Hanich, L. B. (2003). Characteristics of children with moderate mathematicsdeficiencies: A longitudinal perspective. Learning Disabilities Research and Practice, 18(4), 213221.Berch, D. B. (2003). Making sense of number sense: Implications for children with mathematicaldisabilities. Journal of Learning Disabilities, 38, 333-339.Wright, J. (2007). The RTI toolkit: A practical guide for schools. Port Chester, NY: NationalProfessional Resources, Inc.Jim Wright, Presenterwww.interventioncentral.org15

Correct Quantity Discrimination Items Identified Per Min from Range of toStudent: Classrm/Grade: Monitoring Level:WEEK 1 WEEK 2BASELINE- - -- - -WEEK 3 WEEK 4- -- -WEEK 5--WEEK 6 WEEK 7- -- -WEEK 8 WEEK 9- -- -WEEK 10 WEEK 11 WEEK 12- - --- -80706050403020100BASELINEJim Wright, PresenterM T W T F M T W TFM T W T F M T W TFM T W T F M T W TFM T W T F M T W TFInstructional Dayswww.interventioncentral.orgM T W T F M T W TFM T W T F M T W TF 2007 Jim Wright www.interventioncentral.org16

Early Math Fluency CBM Probe: Missing NumberThis introduction to the Missing Number probe provides information about the preparation,administration, and scoring of this Early Math CBM measure. Additionally, it offers brief guidelinesfor integrating this assessment into a school-wide ‘Response-to-Intervention’ model.Missing Number: Description (Clarke & Shinn, 2005; Gersten, Jordan & Flojo, 2005)The student is given a sheet containing multiple number series. Each series consists of 3-4numbers that appear in sequential order. The numbers in each short series are selected to fallwithin a predefined range (e.g., no lower than 0 and no high

Two 1-digit numbers: 0 to 9 2-digit number from a 2-digit number: no regrouping 2-digit number from a 2-digit number: regrouping Multiplication Multiplication facts: 0 to 9 2-digit number times 1-digit number: no regrouping 3-digit number times 1-digit number: regrouping Division Division facts: 0 to 9 2-digit

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