Response to Intervention RTI Teams: Best Practices in Elementary Mathematics Interventions Jim...

Response to Intervention

www.interventioncentral.org

RTI Teams: Best Practicesin Elementary MathematicsInterventionsJim Wrightwww.interventioncentral.org


www.interventioncentral.org 2

PowerPoints from this workshop available at:

http://www.interventioncentral.org/math_workshop.php



Workshop Agenda…

Response to Intervention & Math Interventions: Brief Introduction

Foundations of Mathematical Skills

Assessing Math Interventions: Curriculum-Based Measurement

Math Computation: Strategies

Web Resources to Support Math Interventions

Big Ideas in Learning



RTI is a Model in Development

“Several proposals for operationalizing response to intervention have been made…The field can expect more efforts like these and, for a time at least, different models to be tested…Therefore, it is premature to advocate any single model.” (Barnett, Daly, Jones, & Lentz, 2004 )

Source: Barnett, D. W., Daly, E. J., Jones, K. M., & Lentz, F.E. (2004). Response to intervention: Empirically based special service decisions from single-case designs of increasing and decreasing intensity. Journal of Special Education, 38, 66-79.


www.interventioncentral.org 5Source: Georgia Dept of Education: http://www.doe.k12.ga.us/ Retrieved 13 July 2007

Georgia ‘Pyramid of Interventio

n’



Tier 1Tier 1

Tier 2Tier 2

Tier 3Tier 3

How can a school restructure to support RTI? The school can organize its intervention efforts into 4 levels, or Tiers, that represent a continuum of increasing intensity of support. (Kovaleski, 2003; Vaughn, 2003). In Georgia, Tier 1 is the lowest level of intervention, Tier 4 is the most intensive intervention level.

Standards-Based Classroom Learning: All students participate in general education learning that includes implementation of the Georgia Performance Standards through research-based practices, use of flexible groups for differentiation of instruction, & frequent progress-monitoring.

Tier 4Tier 4

Needs Based Learning: Targeted students participate in learning that is in addition to Tier 1 and different by including formalized processes of intervention & greater frequency of progress-monitoring.

SST Driven Learning: Targeted students participate in learning that is in addition to Tier I & II and different by including individualized assessments, interventions tailored to individual needs, referral for specially designed instruction if needed.

Specially Designed Learning: Targeted students participate in learning that includes specialized programs, adapted content, methodology, or instructional delivery; Georgia Performance standards access/extension.



The Purpose of RTI in Secondary Schools: What Students Should It Serve?

Early Identification. As students begin to show need for academic support, the RTI model proactively supports them with early interventions to close the skill or performance gap with peers.

Chronically At-Risk. Students whose school performance is marginal across school years but who do not qualify for special education services are identified by the RTI Team and provided with ongoing intervention support.

Special Education. Students who fail to respond to scientifically valid general-education interventions implemented with integrity are classified as ‘non-responders’ and found eligible for special education.



Math Intervention Planning: Some Challenges for Elementary RTI Teams

• There is no national consensus about what math instruction should look like in elementary schools

• Schools may not have consistent expectations for the ‘best practice’ math instruction strategies that teachers should routinely use in the classroom

• Schools may not have a full range of assessment methods to collect baseline and progress monitoring data on math difficulties



Focus of This Math Interventions Workshop…

• Intervention and assessment strategies that supplement the ‘core curriculum’

• NOTE: If greater than 20 percent of students in a classroom or grade level experience significant math difficulties, the focus should be on giving the teacher skills for effective whole-group instruction or on improving the ‘core curriculum’



‘Big Ideas’ About Student Learning



Big Ideas: Student Social & Academic Behaviors Are Strongly Influenced by the Instructional Setting

(Lentz & Shapiro, 1986)

• Students with learning problems do not exist in isolation. Rather, their instructional environment plays an enormously important role in these students’ eventual success or failure

Source: Lentz, F. E. & Shapiro, E. S. (1986). Functional assessment of the academic environment. School Psychology Review, 15, 346-57.



Big Ideas: Learn Unit (Heward, 1996)

The three essential elements of effective student learning include:1. Academic Opportunity to Respond. The student is presented with

a meaningful opportunity to respond to an academic task. A question posed by the teacher, a math word problem, and a spelling item on an educational computer ‘Word Gobbler’ game could all be considered academic opportunities to respond.

2. Active Student Response. The student answers the item, solves the problem presented, or completes the academic task. Answering the teacher’s question, computing the answer to a math word problem (and showing all work), and typing in the correct spelling of an item when playing an educational computer game are all examples of active student responding.

3. Performance Feedback. The student receives timely feedback about whether his or her response is correct—often with praise and encouragement. A teacher exclaiming ‘Right! Good job!’ when a student gives an response in class, a student using an answer key to check her answer to a math word problem, and a computer message that says ‘Congratulations! You get 2 points for correctly spelling this word!” are all examples of performance feedback.

Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.



Big Ideas: The Four Stages of Learning Can Be Summed Up in the ‘Instructional Hierarchy’

(Haring et al., 1978)

Student learning can be thought of as a multi-stage process. The universal stages of learning include:

• Acquisition: The student is just acquiring the skill.• Fluency: The student can perform the skill but

must make that skill ‘automatic’.• Generalization: The student must perform the skill

across situations or settings.• Adaptation: The student confronts novel task

demands that require that the student adapt a current skill to meet new requirements.

Source: Haring, N.G., Lovitt, T.C., Eaton, M.D., & Hansen, C.L. (1978). The fourth R: Research in the classroom. Columbus, OH: Charles E. Merrill Publishing Co.



National Mathematics Advisory Panel Report13 March 2008



Math Advisory Panel Report at:

http://www.ed.gov/mathpanel



2008 National Math Advisory Panel Report: Recommendations• “The areas to be studied in mathematics from pre-kindergarten through

eighth grade should be streamlined and a well-defined set of the most important topics should be emphasized in the early grades. Any approach that revisits topics year after year without bringing them to closure should be avoided.”

• “Proficiency with whole numbers, fractions, and certain aspects of geometry and measurement are the foundations for algebra. Of these, knowledge of fractions is the most important foundational skill not developed among American students.”

• “Conceptual understanding, computational and procedural fluency, and problem solving skills are equally important and mutually reinforce each other. Debates regarding the relative importance of each of these components of mathematics are misguided.”

• “Students should develop immediate recall of arithmetic facts to free the “working memory” for solving more complex problems.”

Source: National Math Panel Fact Sheet. (March 2008). Retrieved on March 14, 2008, from http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-factsheet.html



‘Elbow Group’ Activity: What are common student mathematics concerns in your school?

In your ‘elbow groups’:

• Discuss the most common student mathematics problems that you encounter in your school(s). At what grade level do you typically encounter these problems?

• Be prepared to share your discussion points with the larger group.



“Mathematics is made of 50 percent formulas, 50 percent proofs, and 50 percent imagination.”

–Anonymous



Who is At Risk for Poor Math Performance?: A Proactive Stance

“…we use the term mathematics difficulties rather than mathematics disabilities. Children who exhibit mathematics difficulties include those performing in the low average range (e.g., at or below the 35th percentile) as well as those performing well below average…Using higher percentile cutoffs increases the likelihood that young children who go on to have serious math problems will be picked up in the screening.” p. 295

Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics difficulties. Journal of Learning Disabilities, 38, 293-304.



Profile of Students with Math Difficulties (Kroesbergen & Van Luit, 2003)

[Although the group of students with difficulties in learning math is very heterogeneous], in general, these students have memory deficits leading to difficulties in the acquisition and remembering of math knowledge.

Moreover, they often show inadequate use of strategies for solving math tasks, caused by problems with the acquisition and the application of both cognitive and metacognitive strategies.

Because of these problems, they also show deficits in generalization and transfer of learned knowledge to new and unknown tasks.

Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..



The Elements of Mathematical Proficiency: What the Experts Say…



Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press.

5 Strands of Mathematical Proficiency

1. Understanding

2. Computing

3. Applying

4. Reasoning

5. Engagement

5 Big Ideas in Beginning Reading

1. Phonemic Awareness

2. Alphabetic Principle

3. Fluency with Text

4. Vocabulary

5. ComprehensionSource: Big ideas in beginning reading. University of Oregon. Retrieved September 23, 2007, from http://reading.uoregon.edu/index.php



Five Strands of Mathematical Proficiency1. Understanding: Comprehending mathematical concepts,

operations, and relations--knowing what mathematical symbols, diagrams, and procedures mean.

2. Computing: Carrying out mathematical procedures, such as adding, subtracting, multiplying, and dividing numbers flexibly, accurately, efficiently, and appropriately.

3. Applying: Being able to formulate problems mathematically and to devise strategies for solving them using concepts and procedures appropriately.




Five Strands of Mathematical Proficiency (Cont.)

4. Reasoning: Using logic to explain and justify a solution to a problem or to extend from something known to something less known.

5. Engaging: Seeing mathematics as sensible, useful, and doable—if you work at it—and being willing to do the work.




Three General Levels of Math Skill Development (Kroesbergen & Van Luit, 2003)

As students move from lower to higher grades, they move through levels of acquisition of math skills, to include:

• Number sense• Basic math operations (i.e., addition, subtraction,

multiplication, division)• Problem-solving skills: “The solution of both verbal

and nonverbal problems through the application of previously acquired information” (Kroesbergen & Van Luit, 2003, p. 98)

Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114..



What is ‘Number Sense’? (Clarke & Shinn, 2004)

“… the ability to understand the meaning of numbers and define different relationships among numbers.

Children with number sense can recognize the relative size of numbers, use referents for measuring objects and events, and think and work with numbers in a flexible manner that treats numbers as a sensible system.” p. 236

Source: Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics curriculum-based measurement. School Psychology Review, 33, 234–248.



What Are Stages of ‘Number Sense’?

(Berch, 2005, p. 336)

1. Innate Number Sense. Children appear to possess ‘hard-wired’ ability (neurological ‘foundation structures’) to acquire number sense. Children’s innate capabilities appear also to be to ‘represent general amounts’, not specific quantities. This innate number sense seems to be characterized by skills at estimation (‘approximate numerical judgments’) and a counting system that can be described loosely as ‘1, 2, 3, 4, … a lot’.

2. Acquired Number Sense. Young students learn through indirect and direct instruction to count specific objects beyond four and to internalize a number line as a mental representation of those precise number values.

Source: Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of Learning Disabilities, 38, 333-339...



The Basic Number Line is as Familiar as a Well-Known Place to People Who Have Mastered Arithmetic

Combinations

Moravia, NY Number Line: 0-144 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 2021 22 23 24 25 26 27 28 29 3031 32 33 34 35 36 37 38 39 4041 42 43 44 45 46 47 48 49 5051 52 53 54 55 56 57 58 59 6061 62 63 64 65 66 67 68 69 7071 72 73 74 75 76 77 78 79 8081 82 83 84 85 86 87 88 89 9091 92 93 94 95 96 97 98 99 100101 102 103 104 105 106 107 108 109 110111 112 113 114 115 116 117 118 119 120121 122 123 124 125 126 127 128 129 130131 132 133 134 135 136 137 138 139 140141 142 143 144



Internal NumberlineAs students internalize the numberline, they are better able to perform ‘mental arithmetic’ (the manipulation of numbers and math operations in their head).

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

2 + 4 = 628 ÷ 4 = 79 – 7 = 23 X 7 = 21



Mental Arithmetic: A Demonstration

332 x 420 = ?

Directions: As you watch this video of a person using mental arithmetic to solve a computation problem, note the strategies and ‘shortcuts’ that he employs to make the task more manageable.



\Mental Arithmetic Demonstration: What Tools Were Used?

132,800

x 6,640

132,800 + 6000 = 138,800

132,800 + 640 = 139,440

7. Add Intermediate Products: Chunk into Smaller Computation Tasks

6,640

’66 is a famous

national road’ & ’40 is speed limit in front

of house’

6. Use Mnemonic Strategy to Remember Intermediate Product

332

x 20

332 x 10 = 3320

3320 x 2 = 6640

5. Continue with Next ‘Chunk’ of Problem: Math Shortcut

132,800

‘1=3-2’ & ‘800 is a toll-free

number’

4. Use Mnemonic Strategy to Remember Intermediate Product

1,328

x 100

132,800

3. Apply Math Shortcut: Add Zeros in One’s Place for Each Multiple of Ten332

x 4

1,328

2. Break Problem into Manageable Chunks

300

x 400

120,000

1. Estimate Answer332

X 420

Solving for…



Math Computation: Building FluencyJim Wrightwww.interventioncentral.org



"Arithmetic is being able to count up to twenty without taking off your shoes."

–Anonymous



Benefits of Automaticity of ‘Arithmetic Combinations’ (Gersten, Jordan, & Flojo, 2005)

• There is a strong correlation between poor retrieval of arithmetic combinations (‘math facts’) and global math delays

• Automatic recall of arithmetic combinations frees up student ‘cognitive capacity’ to allow for understanding of higher-level problem-solving

• By internalizing numbers as mental constructs, students can manipulate those numbers in their head, allowing for the intuitive understanding of arithmetic properties, such as associative property and commutative property




How much is 3 + 8?: Strategies to Solve…Least efficient strategy: Count out and group 3 objects; count out and group 8 objects; count all objects:

+ =11

More efficient strategy: Begin at the number 3 and ‘count up’ 8 more digits (often using fingers for counting): 3 + 8More efficient strategy: Begin at the number 8 (larger number) and ‘count up’ 3 more digits: 8 + 3Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in memory and automatically retrieved: Answer = 11




Math Skills: Importance of Fluency in Basic Math Operations

“[A key step in math education is] to learn the four basic mathematical operations (i.e., addition, subtraction, multiplication, and division). Knowledge of these operations and a capacity to perform mental arithmetic play an important role in the development of children’s later math skills. Most children with math learning difficulties are unable to master the four basic operations before leaving elementary school and, thus, need special attention to acquire the skills. A … category of interventions is therefore aimed at the acquisition and automatization of basic math skills.”

Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs. Remedial and Special Education, 24, 97-114.



Big Ideas: Learn Unit (Heward, 1996)

The three essential elements of effective student learning include:1. Academic Opportunity to Respond. The student is presented with

a meaningful opportunity to respond to an academic task. A question posed by the teacher, a math word problem, and a spelling item on an educational computer ‘Word Gobbler’ game could all be considered academic opportunities to respond.

2. Active Student Response. The student answers the item, solves the problem presented, or completes the academic task. Answering the teacher’s question, computing the answer to a math word problem (and showing all work), and typing in the correct spelling of an item when playing an educational computer game are all examples of active student responding.

3. Performance Feedback. The student receives timely feedback about whether his or her response is correct—often with praise and encouragement. A teacher exclaiming ‘Right! Good job!’ when a student gives an response in class, a student using an answer key to check her answer to a math word problem, and a computer message that says ‘Congratulations! You get 2 points for correctly spelling this word!” are all examples of performance feedback.

Source: Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.



Math Intervention: Tier I or II: Elementary & Secondary: Self-Administered Arithmetic Combination Drills With Performance

Self-Monitoring & Incentives

1. The student is given a math computation worksheet of a specific problem type, along with an answer key [Academic Opportunity to Respond].

2. The student consults his or her performance chart and notes previous performance. The student is encouraged to try to ‘beat’ his or her most recent score.

3. The student is given a pre-selected amount of time (e.g., 5 minutes) to complete as many problems as possible. The student sets a timer and works on the computation sheet until the timer rings. [Active Student Responding]

4. The student checks his or her work, giving credit for each correct digit (digit of correct value appearing in the correct place-position in the answer). [Performance Feedback]

5. The student records the day’s score of TOTAL number of correct digits on his or her personal performance chart.

6. The student receives praise or a reward if he or she exceeds the most recently posted number of correct digits.

Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi (Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.



Self-Administered Arithmetic Combination Drills:Examples of Student Worksheet and Answer Key

Worksheets created using Math Worksheet Generator. Available online at:http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php



Self-Administered Arithmetic Combination Drills…

No Reward

Reward GivenReward GivenReward Given

No RewardNo Reward

Reward Given



How to… Use PPT Group Timers in the Classroom



Cover-Copy-Compare: Math Computational Fluency-Building Intervention

The student is given sheet with correctly completed math problems in left column and index card.

For each problem, the student:– studies the model– covers the model with index card– copies the problem from memory– solves the problem– uncovers the correctly completed model to check answer

Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing multiplication performance. School Psychology Review, 18, 412-420.



Math Shortcuts: Cognitive Energy- and Time-Savers

“Recently, some researchers…have argued that children can derive answers quickly and with minimal cognitive effort by employing calculation principles or “shortcuts,” such as using a known number combination to derive an answer (2 + 2 = 4, so 2 + 3 =5), relations among operations (6 + 4 =10, so 10 −4 = 6) … and so forth. This approach to instruction is consonant with recommendations by the National Research Council (2001). Instruction along these lines may be much more productive than rote drill without linkage to counting strategy use.” p. 301




9 x 19 x 29 x 39 x 49 x 59 x 69 x 79 x 89 x 99 x 10

Math Multiplication Shortcut: ‘The 9 Times Quickie’• The student uses fingers as markers to find the product of single-

digit multiplication arithmetic combinations with 9. • Fingers to the left of the lowered finger stands for the ’10’s place

value. • Fingers to the right stand for the ‘1’s place value.

Source: Russell, D. (n.d.). Math facts to learn the facts. Retrieved November 9, 2007, from http://math.about.com/bltricks.htm



Students Who ‘Understand’ Mathematical Concepts Can Discover Their Own ‘Shortcuts’

“Students who learn with understanding have less to learn because they see common patterns in superficially different situations. If they understand the general principle that the order in which two numbers are multiplied doesn’t matter—3 x 5 is the same as 5 x 3, for example—they have about half as many ‘number facts’ to learn.” p. 10




Application of Math Shortcuts to Intervention Plans

• Students who struggle with math may find computational ‘shortcuts’ to be motivating.

• Teaching and modeling of shortcuts provides students with strategies to make computation less ‘cognitively demanding’.



Math Computation: Motivate With ‘Errorless Learning’ Worksheets

In this version of an ‘errorless learning’ approach, the student is directed to complete math facts as quickly as possible. If the student comes to a number problem that he or she cannot solve, the student is encouraged to locate the problem and its correct answer in the key at the top of the page and write it in.

Such speed drills build computational fluency while promoting students’ ability to visualize and to use a mental number line.

TIP: Consider turning this activity into a ‘speed drill’. The student is given a kitchen timer and instructed to set the timer for a predetermined span of time (e.g., 2 minutes) for each drill. The student completes as many problems as possible before the timer rings. The student then graphs the number of problems correctly computed each day on a time-series graph, attempting to better his or her previous score.

Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282



‘Errorless Learning’ Worksheet Sample

Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282



Math Computation: Two Ideas to Jump-Start Active Academic Responding

Here are two ideas to accomplish increased academic responding on math tasks.

• Break longer assignments into shorter assignments with performance feedback given after each shorter ‘chunk’ (e.g., break a 20-minute math computation worksheet task into 3 seven-minute assignments). Breaking longer assignments into briefer segments also allows the teacher to praise struggling students more frequently for work completion and effort, providing an additional ‘natural’ reinforcer.

• Allow students to respond to easier practice items orally rather than in written form to speed up the rate of correct responses.

Source: Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing opportunities for responding and influencing students to choose to respond. Psychology in the Schools, 42, 389-403.



Math Computation: Problem Interspersal Technique• The teacher first identifies the range of ‘challenging’ problem-types

(number problems appropriately matched to the student’s current instructional level) that are to appear on the worksheet.

• Then the teacher creates a series of ‘easy’ problems that the students can complete very quickly (e.g., adding or subtracting two 1-digit numbers). The teacher next prepares a series of student math computation worksheets with ‘easy’ computation problems interspersed at a fixed rate among the ‘challenging’ problems.

• If the student is expected to complete the worksheet independently, ‘challenging’ and ‘easy’ problems should be interspersed at a 1:1 ratio (that is, every ‘challenging’ problem in the worksheet is preceded and/or followed by an ‘easy’ problem).

• If the student is to have the problems read aloud and then asked to solve the problems mentally and write down only the answer, the items should appear on the worksheet at a ratio of 3 ‘challenging’ problems for every ‘easy’ one (that is, every 3 ‘challenging’ problems are preceded and/or followed by an ‘easy’ one).

Source: Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal ratios on fifth-grade students’ mathematics accuracy. School Psychology Review, 34, 543-555..



How to… Create an Interspersal-Problems Worksheet



Additional Math InterventionsJim Wrightwww.interventioncentral.org



Math Review: Incremental Rehearsal of ‘Math Facts’

2 x 6 =__Step 1: The tutor writes down on a series of index cards the math facts that the student needs to learn. The problems are written without the answers.

3 x 8 =__

9 x 2 =__

4 x 7 =__

7 x 6 =__

5 x 5 =__

5 x 3 =__

3 x 6 =__

8 x 4 =__

3 x 5 =__

4 x 5 =__

3 x 2 =__

6 x 5 =__

8 x 2 =__

9 x 7 =__



Math Review: Incremental Rehearsal of ‘Math Facts’

2 x 6 =__Step 2: The tutor reviews the ‘math fact’ cards with the student. Any card that the student can answer within 2 seconds is sorted into the ‘KNOWN’ pile. Any card that the student cannot answer within two seconds—or answers incorrectly—is sorted into the ‘UNKNOWN’ pile.

3 x 8 =__

4 x 7 =__

7 x 6 =__

5 x 3 =__

3 x 6 =__ 8 x 4 =__

4 x 5 =__

3 x 2 =__

6 x 5 =__

9 x 7 =__

9 x 2 =__

3 x 5 =__

8 x 2 =__

5 x 5 =__

‘KNOWN’ Facts ‘UNKNOWN’ Facts



Math Review: Incremental Rehearsal of ‘Math Facts’Step 3: The tutor is now ready to follow a nine-step incremental-rehearsal sequence: First, the tutor presents the student with a single index card containing an ‘unknown’ math fact. The tutor reads the problem aloud, gives the answer, then prompts the student to read off the same unknown problem and provide the correct answer.

3 x 8 =__ 2 x 6 =__

4 x 7 =__

5 x 3 =__3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__

Step 3: Next the tutor takes a math fact from the ‘known’ pile and pairs it with the unknown problem. When shown each of the two problems, the student is asked to read off the problem and answer it.

3 x 8 =__ 4 x 5 =__

Step 3: The tutor then repeats the sequence--adding yet another known problem to the growing deck of index cards being reviewed and each time prompting the student to answer the whole series of math facts—until the review deck contains a total of one ‘unknown’ math fact and nine ‘known’ math facts (a ratio of 90 percent ‘known’ to 10 percent ‘unknown’ material )

3 x 8 =__



Math Review: Incremental Rehearsal of ‘Math Facts’Step 4: The student is then presented with a new ‘unknown’ math fact to answer--and the review sequence is once again repeated each time until the ‘unknown’ math fact is grouped with nine ‘known’ math facts—and on and on. Daily review sessions are discontinued either when time runs out or when the student answers an ‘unknown’ math fact incorrectly three times.

2 x 6 =__

5 x 3 =__

3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__3 x 8 =__9 x 2 =__ 2 x 6 =__

4 x 7 =__

5 x 3 =__3 x 6 =__

8 x 4 =__

3 x 2 =__

6 x 5 =__

4 x 5 =__3 x 8 =__

Step 4: At this point, the last ‘known’ math fact that had been added to the student’s review deck is discarded (placed back into the original pile of ‘known’ problems) and the previously ‘unknown’ math fact is now treated as the first ‘known’ math fact in new student review deck for future drills.



Measuring the ‘Intervention Footprint’: Issues of Planning, Documentation, & Follow-ThroughJim Wrightwww.interventioncentral.org



Essential Elements of Any Academic or Behavioral Intervention (‘Treatment’) Strategy:

• Method of delivery (‘Who or what delivers the treatment?’)Examples include teachers, paraprofessionals, parents, volunteers, computers.

• Treatment component (‘What makes the intervention effective?’)Examples include activation of prior knowledge to help the student to make meaningful connections between ‘known’ and new material; guide practice (e.g., Paired Reading) to increase reading fluency; periodic review of material to aid student retention. As an example of a research-based commercial program, Read Naturally ‘combines teacher modeling, repeated reading and progress monitoring to remediate fluency problems’.



Interventions, Accommodations & Modifications: Sorting Them Out

• Interventions. An academic intervention is a strategy used to teach a new skill, build fluency in a skill, or encourage a child to apply an existing skill to new situations or settings.

An intervention is said to be research-based when it has been demonstrated to be effective in one or more articles published in peer–reviewed scientific journals. Interventions might be based on commercial programs such as Read Naturally. The school may also develop and implement an intervention that is based on guidelines provided in research articles—such as Paired Reading (Topping, 1987).




• Accommodations. An accommodation is intended to help the student to fully access the general-education curriculum without changing the instructional content. An accommodation for students who are slow readers, for example, may include having them supplement their silent reading of a novel by listening to the book on tape.

An accommodation is intended to remove barriers to learning while still expecting that students will master the same instructional content as their typical peers. Informal accommodations may be used at the classroom level or be incorporated into a more intensive, individualized intervention plan.




• Modifications. A modification changes the expectations of what a student is expected to know or do—typically by lowering the academic expectations against which the student is to be evaluated.

Examples of modifications are reducing the number of multiple-choice items in a test from five to four or shortening a spelling list. Under RTI, modifications are generally not included in a student’s intervention plan, because the working assumption is that the student can be successful in the curriculum with appropriate interventions and accommodations alone.



Writing Quality ‘Problem Identification’ Statements



Writing Quality ‘Problem Identification’ Statements

• A frequent problem at RTI Team meetings is that teacher referral concerns are written in vague terms. If the referral concern is not written in explicit, observable, measurable terms, it will be very difficult to write clear goals for improvement or select appropriate interventions.

• Use this ‘test’ for evaluating the quality of a problem-identification (‘teacher-concern’) statement: Can a third party enter a classroom with the problem definition in hand and know when they see the behavior and when they don’t?



Writing Quality ‘Teacher Referral Concern’ Statements: Examples

• Needs Work: The student is disruptive.• Better: During independent seatwork , the

student is out of her seat frequently and talking with other students.

• Needs Work: The student doesn’t do his math.• Better: When math homework is assigned, the

student turns in math homework only about 20 percent of the time. Assignments turned in are often not fully completed.



Math Computation Fluency: RTI Case Study



RTI: Individual Case Study: Math Computation

• Jared is a fourth-grade student. His teacher, Mrs. Rogers, became concerned because Jared is much slower in completing math computation problems than are his classmates.



Tier 1: Math Interventions for Jared

• Jared’s school uses the Everyday Math curriculum (McGraw Hill/University of Chicago). In addition to the basic curriculum the series contains intervention exercises for students who need additional practice or remediation.

The instructor, Mrs. Rogers, works with a small group of children in her room—including Jared—having them complete these practice exercises to boost their math computation fluency.



Tier 2: Standard Protocol (Group): Math Interventions for Jared

• Jared did not make sufficient progress in his Tier 1 intervention. So his teacher referred the student to the RTI Intervention Team. The team and teacher decided that Jared would be placed on the school’s educational math software, AMATH Building Blocks, a ‘self-paced, individualized mathematics tutorial covering the math traditionally taught in grades K-4’.

Jared worked on the software in 20-minute daily sessions to increase computation fluency in basic multiplication problems.



Tier 2: Math Interventions for Jared (Cont.)

• During this group-based Tier 2 intervention, Jared was assessed using Curriculum-Based Measurement (CBM) Math probes. The goal was to bring Jared up to at least 40 correct digits per 2 minutes.



Tier 2: Math Interventions for Jared (Cont.)• Progress-monitoring worksheets were created using

the Math Computation Probe Generator on Intervention Central (www.interventioncentral.org).

Example of Math Computation

Probe: Answer Key



Tier 2: Math Interventions for Jared: Progress-Monitoring



Tier 3: Individualized Plan: Math Interventions for Jared

• Progress-monitoring data showed that Jared did not make expected progress in the first phase of his Tier 2 intervention. So the RTI Intervention Team met again on the student. The team and teacher noted that Jared counted on his fingers when completing multiplication problems. This greatly slowed down his computation fluency. The team decided to use a research-based strategy, Explicit Time Drills, to increase Jared’s computation speed and eliminate his dependence on finger-counting.During this individualized intervention, Jared continued to be assessed using Curriculum-Based Measurement (CBM) Math probes. The goal was to bring Jared up to at least 40 correct digits per 2 minutes.



Explicit Time Drills: Math Computational Fluency-Building Intervention

Explicit time-drills are a method to boost students’ rate of responding on math-fact worksheets.

The teacher hands out the worksheet. Students are told that they will have 3 minutes to work on problems on the sheet. The teacher starts the stop watch and tells the students to start work. At the end of the first minute in the 3-minute span, the teacher ‘calls time’, stops the stopwatch, and tells the students to underline the last number written and to put their pencils in the air. Then students are told to resume work and the teacher restarts the stopwatch. This process is repeated at the end of minutes 2 and 3. At the conclusion of the 3 minutes, the teacher collects the student worksheets.

Source: Rhymer, K. N., Skinner, C. H., Jackson, S., McNeill, S., Smith, T., & Jackson, B. (2002). The 1-minute explicit timing intervention: The influence of mathematics problem difficulty. Journal of Instructional Psychology, 29(4), 305-311.



Tier 3: Math Interventions for Jared: Progress-Monitoring



Tier 3: Math Interventions for JaredExplicit Timed Drill Intervention: Outcome• The progress-monitoring data showed that Jared was well

on track to meet his computation goal. At the RTI Team follow-up meeting, the team and teacher agreed to continue the fluency-building intervention for at least 3 more weeks. It was also noted that Jared no longer relied on finger-counting when completing number problems, a good sign that he had overcome an obstacle to math computation.

Response to Intervention RTI Teams: Best Practices in Elementary Mathematics Interventions Jim...

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