Workshop PPTs and handout available at: interventioncentral/rtimath

Response to Intervention

www.interventioncentral.org

Best Practices in Classroom Math Interventions (Elementary)

Jim Wrightwww.interventioncentral.org



Workshop PPTs and handout available at:

http://www.interventioncentral.org/rtimath



Workshop Agenda: RTI Challenges…

Defining Research-Based Principles of Effective Math Instruction & Intervention

Finding Effective, Research-Based Math Interventions

Understanding the Student With ‘Math Difficulties’

Screening and Progress-Monitoring for Students With Math Difficulties

Finding Web Resources to Support Math Assessment & Interventions



Core Instruction & Tier 1 Intervention

Focus of Inquiry: What are the indicators of high-quality core instruction and classroom (Tier 1) intervention for math?

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www.interventioncentral.org 5

“ ”“Tier I of an RTI model involves quality core instruction in general education and benchmark assessments to screen students and monitor progress in learning.” p. 9

Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York.

“ ”“It is no accident that high-quality intervention is listed first [in the RTI model], because success in tiers 2 and 3 is quite predicated on an effective tier 1. “ p. 65



Common Core State Standards Initiativehttp://www.corestandards.org/

View the set of Common Core Standards for English Language Arts (including writing) and mathematics being adopted by states across America.



Common Core Standards. Provide external instructional goals that guide the development and mapping of the school’s curriculum. However, the sequence in which the standards are taught is up to the district and school.

School Curriculum. Outlines a uniform sequence shared across instructors for attaining the Common Core Standards’ instructional goals. Scope-and-sequence charts bring greater detail to the general curriculum. Curriculum mapping ensures uniformity of practice across classrooms, eliminates instructional gaps and redundancy across grade levels.

Commercial Instructional and Intervention Programs. Provide materials for teaching the curriculum. Schools often piece together materials from multiple programs to help students to master the curriculum. It should be noted that specific programs can change, while the underlying curriculum remains unchanged.

Common Core Standards, Curriculum, and Programs: How Do They Interrelate?



An RTI Challenge: Limited Research to Support Evidence-Based Math Interventions

“… in contrast to reading, core math programs that are supported by research, or that have been constructed according to clear research-based principles, are not easy to identify. Not only have exemplary core programs not been identified, but also there are no tools available that we know of that will help schools analyze core math programs to determine their alignment with clear research-based principles.” p. 459

Source: Clarke, B., Baker, S., & Chard, D. (2008). Best practices in mathematics assessment and intervention with elementary students. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 453-463).



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



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

The Elements of Mathematical Proficiency: What the Experts Say…



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.




Five Strands of Mathematical

Proficiency (NRC, 2002)1. 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.

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.

Table Activity: Evaluate Your School’s Math Proficiency…

• As a group, review the National Research Council ‘Strands of Math Proficiency’.

• Which strand do you feel that your school / curriculum does the best job of helping students to attain proficiency?

• Which strand do you feel that your school / curriculum should put the greatest effort to figure out how to help students to attain proficiency?

• Be prepared to share your results.



What Works Clearinghouse Practice Guide: Assisting Students Struggling with Mathematics: Response to Intervention (RtI) for Elementary and Middle Schoolshttp://ies.ed.gov/ncee/wwc/

This publication provides 8 recommendations for effective core instruction in mathematics for K-8.



Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations

• Recommendation 1. Screen all students to identify those at risk for potential mathematics difficulties and provide interventions to students identified as at risk

• Recommendation 2. Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in kindergarten through grade 5 and on rational numbers in grades 4 through 8.

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Assisting Students Struggling with Mathematics: RtI for Elementary & Middle Schools: 8 Recommendations (Cont.)

• Recommendation 3. Instruction during the intervention should be explicit and systematic. This includes providing models of proficient problem solving, verbalization of thought processes, guided practice, corrective feedback, and frequent cumulative review

• Recommendation 4. Interventions should include instruction on solving word problems that is based on common underlying structures.

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• Recommendation 5. Intervention materials should include opportunities for students to work with visual representations of mathematical ideas and interventionists should be proficient in the use of visual representations of mathematical ideas

• Recommendation 6. Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts

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• Recommendation 7. Monitor the progress of students receiving supplemental instruction and other students who are at risk

• Recommendation 8. Include motivational strategies in tier 2 and tier 3 interventions.

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How Do We Reach Low-Performing Math Students?: Instructional Recommendations

Important elements of math instruction for low-performing students:

– “Providing teachers and students with data on student performance”

– “Using peers as tutors or instructional guides”– “Providing clear, specific feedback to parents on their children’s

mathematics success”– “Using principles of explicit instruction in teaching math concepts

and procedures.” p. 51

Source: Baker, S., Gersten, R., & Lee, D. (2002).A synthesis of empirical research on teaching mathematics to low-achieving students. The Elementary School Journal, 103(1), 51-73..



Activity: How Do We Reach Low-Performing Students? p.5

• Review the handout on p. 5 of your packet and consider each of the elements found to benefit low-performing math students.

• For each element, brainstorm ways that you could promote this idea in your math classroom.

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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..



Math Challenge: The student can not yet reliably access an internalnumber-line of numbers 1-10.

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What Does the Research Say?...



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 (or neurological ‘foundation structures’) in 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 Number-LineAs students internalize the number-Line, 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



Math Challenge: The student can not yet reliably access an internalnumber-line of numbers 1-10.

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Solution: Use this strategy:• Building Number Sense Through a Counting Board Game (Supplemental Packet)



Building Number Sense Through a Counting Board Game

DESCRIPTION: The student plays a number-based board game to build skills related to 'number sense', including number identification, counting, estimation skills, and ability to visualize and access specific number values using an internal number-line (Siegler, 2009).

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Source: Siegler, R. S. (2009). Improving the numerical understanding of children from low-income families. Child Development Perspectives, 3(2), 118-124.



MATERIALS: • Great Number Line Race! form• Spinner divided into two equal regions marked "1" and

"2" respectively. (NOTE: If a spinner is not available, the interventionist can purchase a small blank wooden block from a crafts store and mark three of the sides of the block with the number "1" and three sides with the number "2".)

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INTERVENTION STEPS: A counting-board game session lasts 12 to 15 minutes, with each game within the session lasting 2-4 minutes. Here are the steps:

1. Introduce the Rules of the Game. The student is told that he or she will attempt to beat another player (either another student or the interventionist). The student is then given a penny or other small object to serve as a game piece. The student is told that players takes turns spinning the spinner (or, alternatively, tossing the block) to learn how many spaces they can move on the Great Number Line Race! board. Each player then advances the game piece, moving it forward through the numbered boxes of the game-board to match the number "1" or "2" selected in the spin or block toss.

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1. Introduce the Rules of the Game (cont.).

When advancing the game piece, the player must call out the number of each numbered box as he or she passes over it. For example, if the player has a game piece on box 7 and spins a "2", that player advances the game piece two spaces, while calling out "8" and "9" (the names of the numbered boxes that the game piece moves across during that turn).

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2. Record Game Outcomes. At the conclusion of each game, the interventionist records the winner using the form found on the Great Number Line Race! form. The session continues with additional games being played for a total of 12-15 minutes.

3. Continue the Intervention Up to an Hour of Cumulative Play. The counting-board game continues until the student has accrued a total of at least one hour of play across multiple days. (The amount of cumulative play can be calculated by adding up the daily time spent in the game as recorded on the Great Number Line Race! form.)

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Math Challenge: The student has not yet acquired math facts.

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

(Supplemental Packet)(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.



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

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.



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




Math Short-Cuts: Addition (Supplemental Packet)

• The order of the numbers in an addition problem does not affect the answer.

• When zero is added to the original number, the answer is the original number.

• When 1 is added to the original number, the answer is the next larger number.

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Source: Miller, S.P., Strawser, S., & Mercer, C.D. (1996). Promoting strategic math performance among students with learning disabilities. LD Forum, 21(2), 34-40.



Math Short-Cuts: Subtraction (Supplemental Packet)

• When zero is subtracted from the original number, the answer is the original number.

• When 1 is subtracted from the original number, the answer is the next smaller number.

• When the original number has the same number subtracted from it, the answer is zero.

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Math Short-Cuts: Multiplication (Supplemental Packet)

• When a number is multiplied by zero, the answer is zero.• When a number is multiplied by 1, the answer is the

original number.• When a number is multiplied by 2, the answer is equal to

the number being added to itself.• The order of the numbers in a multiplication problem

does not affect the answer.

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Math Short-Cuts: Division (Supplemental Packet)

• When zero is divided by any number, the answer is zero.• When a number is divided by 1, the answer is the

original number.• When a number is divided by itself, the answer is 1.

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Math Challenge: The student has not yet acquired math facts.

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Solution: Use these strategies:•Strategic Number Counting Instruction (Supplemental Packet)•Incremental Rehearsal• Cover-Copy-Compare• Peer Tutoring in Math Computation with Constant Time Delay



DESCRIPTION: The student is taught explicit number counting strategies for basic addition and subtraction. Those skills are then practiced with a tutor (adapted from Fuchs et al., 2009).

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Strategic Number Counting Instruction

Source: Fuchs, L. S., Powell, S. R., Seethaler, P. M., Cirino, P. T., Fletcher, J. M., Fuchs, D., & Hamlett, C. L. (2009). The effects of strategic counting instruction, with and without deliberate practice, on number combination skill among students with mathematics difficulties. Learning and Individual Differences 20(2), 89-100.



MATERIALS: • Number-line (attached)• Number combination (math fact) flash cards for basic

addition and subtraction• Strategic Number Counting Instruction Score Sheet

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PREPARATION: The tutor trains the student to use these two counting strategies for addition and subtraction:

• ADDITION: The student is given a copy of the number-line. When presented with a two-addend addition problem, the student is taught to start with the larger of the two addends and to 'count up' by the amount of the smaller addend to arrive at the answer to the problem.

E..g., 3 + 5= ___

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0 1 2 3 4 5 6 7 8 9 10



PREPARATION: The tutor trains the student to use these two counting strategies for addition and subtraction:

• SUBTRACTION: With access to a number-line, the student is taught to refer to the first number appearing in the subtraction problem (the minuend) as 'the number you start with' and to refer to the number appearing after the minus (subtrahend) as 'the minus number'. The student starts at the minus number on the number-line and counts up to the starting number while keeping a running tally of numbers counted up on his or her fingers. The final tally of digits separating the minus number and starting number is the answer to the subtraction problem.

E..g., 6 – 2 = ___

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0 1 2 3 4 5 6 7 8 9 10



INTERVENTION STEPS: For each tutoring session, the tutor follows these steps:

1. Create Flashcards. The tutor creates addition and/or subtraction flashcards of problems that the student is to practice. Each flashcard displays the numerals and operation sign that make up the problem but leaves the answer blank.

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2. Review Count-Up Strategies. At the opening of the session, the tutor asks the student to name the two methods for answering a math fact. The correct student response is 'Know it or count up.' The tutor next has the student describe how to count up an addition problem and how to count up a subtraction problem. Then the tutor gives the student two sample addition problems and two subtraction problems and directs the student to solve each, using the appropriate count-up strategy.

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3. Complete Flashcard Warm-Up. The tutor reviews addition/subtraction flashcards with the student for three minutes. Before beginning, the tutor reminds the student that, when shown a flashcard, the student should try to 'pull the answer from your head'—but that if the student does not know the answer, he or she should use the appropriate count-up strategy. The tutor then reviews the flashcards with the student. Whenever the student makes an error, the tutor directs the student to use the correct count-up strategy to solve. NOTE: If the student cycles through all cards in the stack before the three-minute period has elapsed, the tutor shuffles the cards and begins again. At the end of the three minutes, the tutor counts up the number of cards reviewed and records the total correct responses and errors.

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4. Repeat Flashcard Review. The tutor shuffles the math-fact flashcards, encourages the student to try to beat his or her previous score, and again reviews the flashcards with the student for three minutes. As before, whenever the student makes an error, the tutor directs the student to use the appropriate count-up strategy. Also, if the student completes all cards in the stack with time remaining, the tutor shuffles the stack and continues presenting cards until the time is elapsed.

At the end of the three minutes, the tutor once again counts up the number of cards reviewed and records the total correct responses and errors.

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5. Provide Performance Feedback. The tutor gives the student feedback about whether (and by how much) the student's performance on the second flashcard trial exceeded the first. The tutor also provides praise if the student beat the previous score or encouragement if the student failed to beat the previous score.

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Score Sheet

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Acquisition Stage: 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.



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

answerSource: 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.




Here is one way to create CCC math worksheets, using the math worksheet generator on www.interventioncentral.org:

1. From any math operations page, select the computation target.

2. Then click the ‘Cover-Copy-Compare’ button. A ‘scaffolded’ version of the CCC worksheet will be created that provides the student with both a completed model and a partially completed model.

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.




Here is another way to create CCC math worksheets, using the math worksheet generator on www.interventioncentral.org:

1. From any math operations page, select a computation skill for the CCC worksheet.

2. Next, set the ‘Number of Columns’ setting to ‘1’.3. Then set the ‘Number of Rows’ setting to the number of CCC

problems that you would like the student to complete.4. Click the ‘Single-Skill Computation Probe’ button.5. Print off only the answer key—and use it as your student’s CCC

worksheet.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.



Peer Tutoring in Math Computation with

Constant Time Delay pp. 20-26



Peer Tutoring in Math Computation with Constant Time Delay

• DESCRIPTION: This intervention employs students as reciprocal peer tutors to target acquisition of basic math facts (math computation) using constant time delay (Menesses & Gresham, 2009; Telecsan, Slaton, & Stevens, 1999). Each tutoring ‘session’ is brief and includes its own progress-monitoring component--making this a convenient and time-efficient math intervention for busy classrooms.




MATERIALS: Student Packet: A work folder is created for each tutor pair. The

folder contains:

10 math fact cards with equations written on the front and correct answer appearing on the back. NOTE: The set of cards is replenished and updated regularly as tutoring pairs master their math facts.

Progress-monitoring form for each student. Pencils.



PREPARATION: To prepare for the tutoring program, the teacher selects students to participate and trains them to serve as tutors.

Select Student Participants. Students being considered for the reciprocal peer tutor program should at minimum meet these criteria (Telecsan, Slaton, & Stevens, 1999, Menesses & Gresham, 2009):

Is able and willing to follow directions; Shows generally appropriate classroom behavior;Can attend to a lesson or learning activity for at least 20

minutes.




Peer Tutoring in Math Computation with Constant Time DelaySelect Student Participants (Cont.). Students being considered for the reciprocal

peer tutor program should at minimum meet these criteria (Telecsan, Slaton, & Stevens, 1999, Menesses & Gresham, 2009):

Is able to name all numbers from 0 to 18 (if tutoring in addition or subtraction math facts) and name all numbers from 0 to 81 (if tutoring in multiplication or division math facts).

• Can correctly read aloud a sampling of 10 math-facts (equation plus answer) that will be used in the tutoring sessions. (NOTE: The student does not need to have memorized or otherwise mastered these math facts to participate—just be able to read them aloud from cards without errors).

• [To document a deficit in math computation] When given a two-minute math computation probe to complete independently, computes fewer than 20 correct digits (Grades 1-3) or fewer than 40 correct digits (Grades 4 and up) (Deno & Mirkin, 1977).



Peer Tutoring in Math Computation: Teacher

Nomination Form



Tutoring Activity. Each tutoring ‘session’ last for 3 minutes. The tutor: – Presents Cards. The tutor presents each card to the tutee for 3

seconds. – Provides Tutor Feedback. [When the tutee responds correctly] The

tutor acknowledges the correct answer and presents the next card.

[When the tutee does not respond within 3 seconds or responds incorrectly] The tutor states the correct answer and has the tutee repeat the correct answer. The tutor then presents the next card.

– Provides Praise. The tutor praises the tutee immediately following correct answers.

– Shuffles Cards. When the tutor and tutee have reviewed all of the math-fact carts, the tutor shuffles them before again presenting cards.




Progress-Monitoring Activity. The tutor concludes each 3-minute tutoring session by assessing the number of math facts mastered by the tutee. The tutor follows this sequence:– Presents Cards. The tutor presents each card to the tutee for 3

seconds.– Remains Silent. The tutor does not provide performance feedback or

praise to the tutee, or otherwise talk during the assessment phase.– Sorts Cards. Based on the tutee’s responses, the tutor sorts the

math-fact cards into ‘correct’ and ‘incorrect’ piles.– Counts Cards and Records Totals. The tutor counts the number of

cards in the ‘correct’ and ‘incorrect’ piles and records the totals on the tutee’s progress-monitoring chart.




Tutoring Integrity Checks. As the student pairs complete the tutoring activities, the supervising adult monitors the integrity with which the intervention is carried out. At the conclusion of the tutoring session, the adult gives feedback to the student pairs, praising successful implementation and providing corrective feedback to students as needed. NOTE: Teachers can use the attached form Peer Tutoring in Math Computation with Constant Time Delay: Integrity Checklist to conduct integrity checks of the intervention and student progress-monitoring components of the math peer tutoring.




Peer Tutoring in Math

Computation: Intervention

Integrity Sheet:(Part 1:

Tutoring Activity)




Computation: Intervention

Integrity Sheet(Part 2:

Progress-Monitoring)




Computation: Score Sheet



Math Challenge: The student has acquired math computation skills but is not yet fluent.

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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




Associative Property

• “within an expression containing two or more of the same associative operators in a row, the order of operations does not matter as long as the sequence of the operands is not changed”

• Example:

–(2+3)+5=10– 2+(3+5)=10

Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Associative



Commutative Property

• “the ability to change the order of something without changing the end result.”

• Example:

– 2+3+5=10– 2+5+3=10

Source: Associativity. Wikipedia. Retrieved September 5, 2007, from http://en.wikipedia.org/wiki/Commutative



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 Challenge: The student has acquired math computation skills but is not yet fluent.

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Solution: Use these strategies:• Explicit Time Drills• Self-Administered Arithmetic Combination Drills With Performance Self-Monitoring & Incentives



Explicit Time Drills: p. 25Math 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.



Fluency Stage: Math Computation p. 30Math Computation: Increase Accuracy and ProductivityRates Via Self-Monitoring and Performance Feedback

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-Monitoring & Performance FeedbackExamples 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-Monitoring & Performance Feedback

No Reward

Reward GivenReward GivenReward Given

No RewardNo Reward

Reward Given



Student Self-Monitoring of Productivity to Increase Fluency on Math Computation Worksheets (Supplemental Packet)

DESCRIPTION: The student monitors and records her or his work production on math computation worksheets on a daily basis—with a goal of improving overall fluency (Maag, Reid, R., & DiGangi, 1993). This intervention can be used with a single student, a small group, or an entire class.

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Source: Maag, J. W., Reid, R., & DiGangi, S. A. (1993). Differential effects of self-monitoring attention, accuracy, and productivity. Journal of Applied Behavior Analysis, 26, 329-344.



Student Self-Monitoring of Productivity to Increase Fluency on Math Computation Worksheets

MATERIALS: • Student self-monitoring audio prompt: Tape / audio file

with random tones or dial-style kitchen timer• Math computation worksheets containing problems

targeted for increased fluency• Student Speed Check! recording form

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Student Speed Check! Form

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Preparation: To prepare for the intervention the teacher:

1. Decides on the Length and Frequency of Each Self-Monitoring Period. The instructor decides on the length of session and frequency of the student's self-monitoring intervention. NOTE: One good rule of thumb is to set aside at least 10 minutes per day for this or other interventions to promote fluent student retrieval of math facts (Gersten et al., 2009).

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2. Selects a Math Computation Skill Target. The instructor chooses one or more problem types that are to appear in intervention worksheets.

For example, a teacher may select two math computation problem-types for a student: Addition—double-digit plus double-digit with regrouping and Subtraction—double-digit plus double-digit with no regrouping.

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3. Creates Math Computation Worksheets. When the teacher has chosen the problem types, he or she makes up sufficient equivalent worksheets (with the same number of problems and the same mix of problem-types) to be used across the intervention days. Each worksheet should have enough problems to keep the student busy for the length of time set aside for a self-monitoring intervention session.

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4. Determines How Many Audio Prompts the Student Will Receive. This intervention relies on student self-monitoring triggered by audio prompts. Therefore, the teacher must decide on a fixed number of audio prompts the student is to receive per session. NOTE: On the attached Student Speed Check! form, space is provided for the student to record productivity for up to five audio prompts per session.

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5. Selects a Method to Generate Random Audio Prompts. Next, the teacher must decide on how to generate the audio prompts (tones) that drive this intervention. There are two possible choices: (A) The teacher can develop a tape or audio file that has several random tones spread across the time-span of the intervention session, with the number of tones equaling the fixed number of audio prompts selected for the intervention. For example, the instructor may develop a 10-minute tape with five tones randomly sounding at 2 minutes, 3 minutes, 5 minutes, 7 minutes, and 10 minutes.

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5. (B) The instructor may purchase a dial-type kitchen timer. During the intervention period, the instructor turns the dial to a randomly selected number of minutes. When the timer expires and chimes as a student audio prompt, the teacher resets the timer to another random number of minutes and repeats this process until the intervention period is over.

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INTERVENTION STEPS: Sessions of the productivity self-monitoring intervention for math computation include these steps:

1. [Student] Set a Session Computation Goal. The student looks up the total number of problems completed on his or her most recent timed worksheet and writes that figure into the 'Score to Beat' section of the current day's Student Speed Check! form.

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2. [Teacher] Set the Timer or Start the Tape. The teacher directs the student to begin working on the worksheet and either starts the tape with tones spaced at random intervals or sets a kitchen timer. If using a timer, the teacher randomly sets the timer randomly to a specific number of minutes. When the timer expires and chimes as a student audio prompt, the teacher resets the timer to another random number of minutes and repeats this process until the intervention period is over.

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3. [Student] At Each Tone, Record Problems Completed. Whenever the student hears an audio prompt or at the conclusion of the timed intervention period, the student pauses to:A. circle the problem that he or she is currently working onB. count up the number of problems completed since the previous

tone (or in the case of the first tone, the number of problems completed since starting the worksheet)

C. record the number of completed problems next to the appropriate tone interval on the attached Student Speed Check! form.

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4. [Teacher] Announce the End of the Intervention Period. The teacher announces that the intervention period is over and that the student should stop working on the worksheet.

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5. [Student] Tally Day's Performance. The student adds up the problems completed at the tone-intervals to give a productivity total for the day. The student then compares the current day's figure to that of the previous day to see if he or she was able to beat the previous score.

If YES, the student receives praise from the teacher; if NO, the student receives encouragement from the teacher.

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Math Challenge: The student is often inconsistent in performance on computation or word problems—and may make a variety of hard-to-predict errors.

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Profile of Students With Significant Math Difficulties p. 4 Spatial organization. The student commits errors such as misaligning numbers in columns in a

multiplication problem or confusing directionality in a subtraction problem (and subtracting the original number—minuend—from the figure to be subtracted (subtrahend).

Visual detail. The student misreads a mathematical sign or leaves out a decimal or dollar sign in the answer.

Procedural errors. The student skips or adds a step in a computation sequence. Or the student misapplies a learned rule from one arithmetic procedure when completing another, different arithmetic procedure.

Inability to ‘shift psychological set’. The student does not shift from one operation type (e.g., addition) to another (e.g., multiplication) when warranted.

Graphomotor. The student’s poor handwriting can cause him or her to misread handwritten numbers, leading to errors in computation.

Memory. The student fails to remember a specific math fact needed to solve a problem. (The student may KNOW the math fact but not be able to recall it at ‘point of performance’.)

Judgment and reasoning. The student comes up with solutions to problems that are clearly unreasonable. However, the student is not able adequately to evaluate those responses to gauge whether they actually make sense in context.

Source: Rourke, B. P. (1993). Arithmetic disabilities, specific & otherwise: A neuropsychological perspective. Journal of Learning Disabilities, 26, 214-226.



Activity: Profile of Math Difficulties p. 4

• Review the profile of students with significant math difficulties that appears on p. 4 of your handout.

• For each item in the profile, discuss what methods you might use to discover whether a particular student experiences this difficulty. Jot your ideas in the ‘NOTES’ column.

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Math Challenge: The student is often inconsistent in performance on computation or word problems—and may make a variety of hard-to-predict errors.

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Solution: Use this strategy:• Increase Student Math Success with Customized Math Self- Correction Checklists (Supplemental Packet)



Increase Student Math Success with Customized Math Self-Correction Checklists

DESCRIPTION: The teacher analyzes a particular student's pattern of errors commonly made when solving a math algorithm (on either computation or word problems) and develops a brief error self-correction checklist unique to that student. The student then uses this checklist to self-monitor—and when necessary correct—his or her performance on math worksheets before turning them in.

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Sources: Dunlap, L. K., & Dunlap, G. (1989). A self-monitoring package for teaching subtraction with regrouping to students with learning disabilities. Journal of Applied Behavior Analysis, 229, 309-314.

Uberti, H. Z., Mastropieri, M. A., & Scruggs, T. E. (2004). Check it off: Individualizing a math algorithm for students with disabilities via self-monitoring checklists. Intervention in School and Clinic, 39(5), 269-275.




MATERIALS: • Customized student math error self-correction checklist• Worksheets or assignments containing math problems

matched to the error self-correction checklist

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Sample Self-Correction Checklist

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INTERVENTION STEPS: The intervention includes these steps (adapted from Dunlap & Dunlap, 1989; Uberti et al., 2004):

1. Develop the Checklist. The teacher draws on multiple sources of data available in the classroom to create a list of errors that the student commonly makes on a specific type of math computation or word problem. Good sources of information for analyzing a student's unique pattern of math-related errors include review of completed worksheets and other work products, interviewing the student, asking the student to solve a math problem using a 'think aloud' approach to walk through the steps of an algorithm, and observing the student completing math problems in a cooperative learning activity with other children.

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1. Develop the Checklist (cont.). Based on this error analysis, the teacher creates a short (4-to-5 item) student self-correction checklist that includes the most common errors made by that student. Items on the checklist are written in the first person and when possible are stated as 'replacement' or goal behaviors.

NOTE: To reduce copying costs, the teacher can laminate the self-correction checklist and provide the student with an erasable marker to allow for multiple re-use of the form.

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2. Introduce the Checklist. The teacher shows the student the self-correction checklist customized for that student. The teacher states that the student is to use the checklist to check his or her work before turning it in so that the student can identify and correct the most common errors.

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3. Prompt the Student to Use the Checklist to Evaluate Each Problem. The student is directed to briefly review all items on the checklist before starting any worksheet or assignment containing the math problems that it targets. The student uses the checklist after every problem to check the work—marking each checklist item with a plus sign ( '+') if correctly followed or a minus sign ( '-') if not correctly followed. If any checklist item receives a minus rating, the student leaves the original solution to the problem untouched, solves the problem again, and again uses the checklist to check the work.

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4. Provide Performance Feedback, Praise, and Encouragement. Soon after the student submits any math worksheets associated with the intervention, the teacher should provide him or her with timely feedback about errors, praise for correct responses, and encouragement to continue to apply best effort.

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5. [OPTIONAL] Provide Reinforcement for Checklist Use. If the student appears to need additional incentives to increase motivation for the intervention, the teacher can assign the student points for intervention compliance: (1) the student earns one point on any assignment for each correct answer, and (2) the student earns an additional point for each problem on which the student committed none of the errors listed on the self-correction checklist. The student is allowed to collect points and to redeem them for privileges or other rewards in a manner to be determined by the teacher.

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6. Fade the Intervention. The error self-correction checklist can be discontinued when the student is found reliably to perform on the targeted math skill(s) at a level that the teacher defines as successful (e.g., 90 percent success or greater).

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Math School-Wide Screenings

Focus of Inquiry: What math school-wide screening and progress-monitoring tools are available and how is that information used in RTI?

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Building-Wide Screening: Assessing All Students (Stewart & Silberglit, 2008)

Screening data in basic academic skills are collected at least 3 times per year (fall, winter, spring) from all students.

• Schools should consider using ‘curriculum-linked’ measures such as Curriculum-Based Measurement that will show generalized student growth in response to learning.

• If possible, schools should consider avoiding ‘curriculum-locked’ measures that are tied to a single commercial instructional program.

Source: Stewart, L. H. & Silberglit, B. (2008). Best practices in developing academic local norms. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 225-242). Bethesda, MD: National Association of School Psychologists.



Applications of Screening Data (Stewart & Silberglit, 2008)

Math screening data can be used to:• Evaluate and improve the current core math instructional

program: How well are our children learning?• Allocate resources to classrooms, grades, and buildings where

student academic needs are greatest: Where can we best put our scarce resources to help struggling students?

• Guide the creation of targeted Tier 2/3 (supplemental intervention) groups: What students need supplemental math interventions—and what kinds of interventions do they need?

• Set academic goals for improvement for students on Tier 2 and Tier 3 interventions: Using local or research norms for math, what progress do we expect for students on intervention?

Source: Stewart, L. H. & Silberglit, B. (2008). Best practices in developing academic local norms. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 225-242). Bethesda, MD: National Association of School Psychologists.



Clearinghouse for RTI Screening and Progress-Monitoring Tools

• The National Center on RTI (www.rti4success.org) maintains pages rating the technical adequacy of RTI screening and progress-monitoring tools.

• Schools should strongly consider selecting screening tools that have national norms or benchmarks to help them to assess the academic-risk level of their students.

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Methods of RTI Math Screening/Progress-Monitoring

Description

Early Math Fluency Kdg and Grade 1: One-minute measures of numberline: Quantity Discrimination, Missing Number, Number Identification, Oral Counting Fluency

Math Computation Fluency Grades 1-8: Two-minute assessments of math computation skills.

Math Concepts & Applications Grades K-8: Mixed problems that map to the Math Focal Points from the NCTM.



Early Math Fluency: Measuring ‘Number Sense’• Early Math Fluency measures track primary-

grade students’ acquisition of number sense (defined as mastery of internal number line)

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0 1 2 3 4 5 6 7 8 9 10



Early Math Fluency: Measuring ‘Number Sense’• Quantity Discrimination [1 minute]: The student is given a

worksheet with number pairs and, for each pair, identifies the larger of the two numbers.

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• Missing Number [1 minute]: The student is given a worksheet with 4-digit number series with one digit randomly left blank and, for each series, names the missing number.

• Number Identification [1 minute]: The student is given a worksheet randomly generated numbers and reads off as many as possible within the time limit.



Numberfly Early Math Fluency Generatorhttp://www.interventioncentral.org

Use this free online application to design and create Early Math Fluency Probes, including:

•Quantity Discrimination•Missing Number•Number Identification



Math Computation Fluency: Computation Speed and Accuracy

Math Computation Fluency [2 minutes]: • The student is given a worksheet of computation

problems that either is a mix of different problem-types (mixed-skill worksheet) or has problems all of the same type (single-skill worksheet).

• The student has two minutes to answer as many problems as possible.

• The computation probe is then scored, with the student getting ‘credit’ for every correct digit.

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Computation Fluency: 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




Math Computation Fluency Generatorhttp://www.interventioncentral.org

Use this free online application to design and create curriculum-based measurement Math Computation Probes, for the basic math operations, including:

•Addition•Subtraction•Multiplication•Division

NOTE: See pp. 20-22 for a listing of math computation goals by grade level.



Math Computation Fluency: Computation Speed and Accuracy

• Strength: Computation Fluency provides good information about a student’s proficiency with math facts, a strong indicator of his or her ability to do mental arithmetic.

• Drawback: Computation Fluency taps only a narrow set of math competencies and is not a good ‘general outcome measure’ or predictor of more global math performance.

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Math Concepts & ApplicationsMath Concepts & Applications [www.easycbm.com online administration]: The student goes online to complete a mixed-skills series of ‘concepts & applications’ in mathematics.

.

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EasyCBM Math Concepts & Applicationshttp://www.easycbm.com

This website provides two levels of support:

• Teacher Version [free]: Any teacher can create a free account and use easycbm tools to monitor student progress on interventions. NOTE: There are 16 items on the C&A Teacher Version probes.

• District Version [pay]: Allows schools to screen student populations 3 times per year. NOTE: There are 45 items on the C&A District Version probes.



Math Concepts & Applications

• Strengths: Concepts & Applications measures sample a broad array of math skills and concepts.

They also tap into a student’s conceptual knowledge of mathematics, not just procedural knowledge.

• Drawback: While Concepts & Applications measures sample a broader array of math skills and concepts, they do not provide deeper information about the student’s performance on any one skill. (Nor were they designed to!)

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Using Research Math Norms to Estimate Risk: Example Using EasyCBM

• Low Risk: At or above the 20th percentile: Core instruction alone is sufficient for the student.

• Some Risk: 10th to 20th percentile: The student will benefit from additional intervention, which may be provided by the classroom teacher or other provider.

• At Risk: Below 10th percentile : The student requires intensive intervention, which may be provided by the classroom teacher or other provider.

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Creating a School-Wide RTI Math Screening Plan: Recommendations

1. Analyze your student demographics and academic performance and select math (or other) academic screeners matched to those demographics.

2. Consider piloting new screening tools (e.g., at single grade levels or in selected classrooms) before rolling out through all grade levels.

3. Ensure that any discussion about grade- or school- or district-wide adoption of RTI screening tools includes general education and special education input.

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Creating a School-Wide RTI Math Screening Plan: Recommendations (Cont.)

4. When adopting a screening tool, inventory all formal assessments administered in your school. Discuss whether any EXISTING assessments can be made optional or dropped whenever new screening tools are being added.

5. If possible, use screening tools found by the National Center on RTI to have ‘technical adequacy’.

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Team Activity: Creating an RTI Math Screening Plan for Your School

• Review the recommendations just presented on school-wide screening tools in math, including Early Math Fluency, Math Computation Fluency, and Concepts & Applications.

• If your school is currently using a set of math school-wide screeners, discuss how you might evaluate them to ensure that they are adequate and meet your needs.

• If your school does NOT yet have a set of school-wide screeners, discuss how you might begin to select and pilot these screeners.

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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 brought the student up at a grade-level ‘Data Analysis Team meeting’ held once per month to consider students for entry into Tier 2 interventions. 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 ‘standard-treatment protocol’ 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 his Tier 2 intervention. So the school’s RTI Problem-Solving Team met 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.



Core Instruction & Tier 1 Intervention

Focus of Inquiry: What are the indicators of high-quality core instruction for writing?

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Graham, S., & Perin, D. (2007). Writing next: Effective strategies to improve writing of adolescents in middle and high schools – A report to Carnegie Corporation of New York. Washington, DC Alliance for Excellent Education. Retrieved from http://www.all4ed.org/files/WritingNext.pdf



The Effect of Grammar Instruction as an Independent Activity“Grammar instruction in the studies reviewed [for the Writing Next report] involved the explicit and systematic teaching of the parts of speech and structure of sentences. The meta-analysis found an effect for this type of instruction for students across the full range of ability, but …surprisingly, this effect was negative…Such findings raise serious questions about some educators’ enthusiasm for traditional grammar instruction as a focus of writing instruction for adolescents….Overall, the findings on grammar instruction suggest that, although teaching grammar is important, alternative procedures, such as sentence combining, are more effective than traditional approaches for improving the quality of students’ writing.” p. 21

Source: Graham, S., & Perin, D. (2007). Writing next: Effective strategies to improve writing of adolescents in middle and high schools – A report to Carnegie Corporation of New York. Washington, DC Alliance for Excellent Education.



Evaluating the Impact of Effect Size Coefficients

• 0.20 Effect Size = Small• 0.50 Effect Size = Medium• 0.80 Effect Size = Large

Source: Cohen,J. (1988). Statistical power analysis for the behavioral sciences (2nded.). Hillsdale,NJ:Erlbaum.



Elements of effective writing instruction for adolescents:

1. Writing Process (Effect Size = 0.82): Students are taught a process for planning, revising, and editing.

2. Summarizing (Effect Size = 0.82): Students are taught methods to identify key points, main ideas from readings to write summaries of source texts.

3. Cooperative Learning Activities (‘Collaborative Writing’) (Effect Size = 0.75): Students are placed in pairs or groups with learning activities that focus on collaborative use of the writing process.

4. Goal-Setting (Effect Size = 0.70): Students set specific ‘product goals’ for their writing and then check their attainment of those self-generated goals.

Source: Graham, S., & Perin, D. (2007). Writing next: Effective strategies to improve writing of adolescents in middle and high schools – A report to Carnegie Corporation of New York. Washington, DC Alliance for Excellent Education. Retrieved from http://www.all4ed.org/files/WritingNext.pdf




5. Writing Processors (Effect Size = 0.55): Students have access to computers/word processors in the writing process.

6. Sentence Combining (Effect Size = 0.50): Students take part in instructional activities that require the combination or embedding of simpler sentences (e.g., Noun-Verb-Object) to generate more advanced, complex sentences.

7. Prewriting (Effect Size = 0.32): Students learn to select, develop, or organize ideas to incorporate into their writing by participating in structured ‘pre-writing’ activities.

8. Inquiry Activities (Effect Size = 0.32): Students become actively engaged researchers, collecting and analyzing information to guide the ideas and content for writing assignments.





9. Process Writing (Effect Size = 0.32): Writing instruction is taught in a ‘workshop’ format that “ stresses extended writing opportunities, writing for authentic audiences, personalized instruction, and cycles of writing” (Graham & Perin, 2007; p. 4).

10. Use of Writing Models (Effect Size = 0.25): Students read and discuss models of good writing and use them as exemplars for their own writing.

11. Writing to Learn Content (Effect Size = 0.23): The instructor incorporates writing activities as a means to have students learn content material.




Writing ‘Blockers’ pp. 51-52



Physical Production of Writing

___Y ___N Writing Speed. Writes words on the page at a rate equal or nearly equal to that of classmates

•Teach keyboarding skills•Allow student to dictate ideas into a tape-recorder and have a volunteer (e.g., classmate, parent, school personnel) transcribe them.

___Y ___N Handwriting. Handwriting is legible to most readers

•Provide training in handwriting•Teach keyboarding skills.



Mechanics & Conventions of Writing

___Y ___N Grammar & Syntax. Knowledge of grammar (rules governing use of language) and syntax (grammatical arrangement of words in sentences) is appropriate for age and/or grade placement

•Teach rules of grammar, syntax•Have students compile individualized checklists of their own common grammar/syntax mistakes; direct students to use the checklist to review work for errors before turning in.

___Y ___N Spelling. Spelling skills are appropriate for age and/or grade placement

•Have student collect list of own common misspellings; assign words from list to study; quiz student on list items.•Have student type assignments and use spell-check.



"The difference between the right word and the almost right word is the difference between lightning and the lightning bug."– Mark Twain



Writing Content

___Y ___N Vocabulary. Vocabulary in written work is age/grade appropriate

•Compile list of key vocabulary and related definitions for subject area; assign words from list to study; quiz student on definitions of list items•Introduce new vocabulary items regularly to class; set up cooperative learning activities for students to review vocabulary.

___Y ___N Word Choice. Distinguishes word-choices that are appropriate for informal (colloquial, slang) discourse vs. formal written discourse

•Present examples to the class of formal vs. informal word choices•Have students check work for appropriate word choice as part of writing revision process.



"Your manuscript is both good and original. But the part that is good is not original, and the part that is original is not good."– Samuel Johnson



Writing Content (Cont.)

___Y ___N Audience. Identifies targeted audience for writing assignments and alters written content to match needs of projected audience

•Direct students to write a ‘targeted audience profile’ as a formal (early) step in the writing process; have students evaluate the final writing product to needs of targeted audience during the revision process.

___Y ___N Plagiarism. Identifies when to credit authors for use of excerpts quoted verbatim or unique ideas taken from other written works

•Define plagiarism for students. Use plentiful examples to show students acceptable vs. unacceptable incorporation of others’ words or ideas into written compositions.



"Nothing is particularly hard if you divide it into small jobs."– Henry Ford



Writing Preparation

__Y __N Topic Selection. Independently selects appropriate topics for writing assignments

•Have student generate list of general topics that that interest him or her; sit with the student to brainstorm ideas for writing topics that relate to the student’s own areas of interest.

__Y __N Writing Plan. Creates writing plan by breaking larger writing assignments into sub-tasks (e.g., select topic, collect source documents, take notes from source documents, write outline, etc.)

•Create generic pre-formatted work plans for writing assignments that break specific types of larger assignments (e.g., research paper) into constituent parts. Have students use these plan outlines as a starting point to making up their own detailed writing plans.



Writing Preparation (Cont.)

__Y __N Note-Taking. Researches topics by writing notes that capture key ideas from source material

•Teach note-taking skills; have students review note-cards with the teacher as quality check.



"When I sit at my table to write, I never know what it’s going to be until I'm under way. I trust in inspiration, which sometimes comes and sometimes doesn't. But I don't sit back waiting for it. I work every day."– Alberto Moravia



Writing Production & Revision

__Y __N Adequate ‘Seat Time’. Allocates realistic amount of time to the act of writing to ensure a quality final product

•Use teacher’s experience and information from proficient student writers to develop and share estimates of minimum writing ‘seat time’ needed to produce quality products for ‘typical’ writing assignments•Have students keep a writing diary to record amount of time spent in act of writing for each assignment. (Additional idea: Consider asking parents to monitor and record their child’s writing time.)

__Y __N Oral vs. Written Work. Student’s dictated and written passages are equivalent in complexity and quality

•Allow student to dictate ideas into a tape-recorder and have a volunteer (e.g., classmate, parent, school personnel) transcribe them•Permit the student to use speech-to-text software (e.g., Dragon Naturally Speaking) to dictate first drafts of writing assignments.



Writing Production & Revision

__Y __N Revision Process. Revises initial written draft before turning in for a grade or evaluation

•Create a rubric containing the elements of writing that students should review during the revision process; teach this rubric to the class; link a portion of the grade on writing assignments to students’ use of the revision rubric.

__Y __N Timely Submission. Turns in written assignments (class work, homework) on time

•Provide student incentives for turning work in on time.•Work with parents to develop home-based plans for work completion and submission.•Institute school-home communication to let parents know immediately when important assignments are late or missing.



Writing ‘Blockers’



Elbow Group Activity: What are the major writing concerns in your school?•Look over the Writing Skills Checklist

•As a group, select the TOP TWO areas that teachers in your school are most concerned about.

•Brainstorm possible intervention ideas to address these concerns.

•Appoint a spokesperson to share your group’s selections.



Writing Interventions

Focus of Inquiry: Where can schools find intervention programs or ideas to address writing delays?

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What Works Clearinghousehttp://ies.ed.gov/ncee/wwc/

The WWC website does not have a separate category for writing programs but does cite research-based reading programs that also target writing skills.

To find programs containing a writing component, the user goes to the home page of WWC and selects the intervention topic ‘literacy’. On the literacy-programs page, user then types in the keyword ‘writing’ to narrow the search to reading/writing programs.



Cognitive Strategy Instructionhttp://www.unl.edu/csi/

This website contains a series of cognitive strategies for writing (and other academic areas) that students can be taught to use on their own.



FreeReadeinghttp://www.freereading.net

This ‘open source’ website includes free lesson plans that target writing instruction and intervention.



Writing School-Wide Screenings

Focus of Inquiry: What school-wide screenings are available for writing and how is that information used in RTI?

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RTI: Status of School-Wide Skills for WritingThere are few RTI writing tools available for screening or progress-monitoring.

For instance, the National Center on RTI lists only reading and math screening and progress-monitoring tools on its ‘tools-ratings’ pages.

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CBM Writing: Group-Administered ProbeOne RTI-compliant progress-monitoring tool that can be used to track the mechanics and conventions of writing is Curriculum-Based Measurement (CBM) Writing. CBM Writing can be administered to groups of students.

• The student is given a story starter (story stem) and asked to think for one minute about a story he or she would like to write.

• The student is then given 3 minutes to produce a writing sample.• The CBM Writing probe is then scored for total words written,

correctly spelled words, or ‘correct writing sequences’.

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Sample CBM Writing Probe



CBM Writing Probe Generator




RTI for Math and Writing: Next Septs

In your teams:

• Review the resources and ideas shared at this workshop.

• Decide on 2-3 key ‘next steps’ that you would like to capitalize on the workshop content and move RTI forward in your school.







RTI at Tier 1: The Teacher as ‘First Responder’

Focus of Inquiry: What does Tier 1 intervention look like for the general-education classroom teacher who is supporting struggling students?

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RTI ‘Pyramid of Interventions’

Tier 1

Tier 2

Tier 3

Tier 1: Universal interventions. Available to all students in a classroom or school. Can consist of whole-group or individual strategies or supports.

Tier 2 Individualized interventions. Subset of students receive interventions targeting specific needs.

Tier 3: Intensive interventions. Students who are ‘non-responders’ to Tiers 1 & 2 are referred to the RTI Team for more intensive interventions.



Tier 1 Core InstructionTier I core instruction:• Is universal—available to all students.• Can be delivered within classrooms or throughout the school. • Is an ongoing process of developing strong classroom instructional

practices to reach the largest number of struggling learners.

All children have access to Tier 1 instruction/interventions. Teachers have the capability to use those strategies without requiring outside assistance.

Tier 1 instruction encompasses:

• The school’s core curriculum.• All published or teacher-made materials used to deliver that curriculum.• Teacher use of ‘whole-group’ teaching & management strategies.

Tier I instruction addresses this question: Are strong classroom instructional strategies sufficient to help the student to achieve academic success?



Tier I (Classroom) InterventionTier 1 intervention:

• Targets ‘red flag’ students who are not successful with core instruction alone.

• Uses ‘evidence-based’ strategies to address student academic or behavioral concerns.

• Must be feasible to implement given the resources available in the classroom.

Tier I intervention addresses the question: Does the student make adequate progress when the instructor uses specific academic or behavioral strategies matched to the presenting concern?



The Key Role of Classroom Teachers as ‘Interventionists’ in RTI: 6 Steps

1. The teacher defines the student academic or behavioral problem clearly.

2. The teacher decides on the best explanation for why the problem is occurring.

3. The teacher selects ‘research-based’ interventions.4. The teacher documents the student’s Tier 1 intervention plan.5. The teacher monitors the student’s response (progress) to the

intervention plan.6. The teacher knows what the next steps are when a student fails

to make adequate progress with Tier 1 interventions alone.

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Available on conference webpage



RTI Interventions: What If There is No Commercial Intervention Package or Program Available?

“Although commercially prepared programs and … manuals and materials are inviting, they are not necessary. … A recent review of research suggests that interventions are research based and likely to be successful, if they are correctly targeted and provide explicit instruction in the skill, an appropriate level of challenge, sufficient opportunities to respond to and practice the skill, and immediate feedback on performance…Thus, these [elements] could be used as criteria with which to judge potential …interventions.” p. 88

Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools. Routledge: New York.



Motivation Deficit 1: The student is unmotivated because he or she cannot do the assigned work.

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• Profile of a Student with This Motivation Problem: The student lacks essential skills required to do the task.

Handout pp.2-3



• Profile of a Student with This Motivation Problem (Cont.):Areas of deficit might include:

• Basic academic skills. Basic skills have straightforward criteria for correct performance (e.g., the student defines vocabulary words or decodes text or computes ‘math facts’) and comprise the building-blocks of more complex academic tasks (Rupley, Blair, & Nichols, 2009).

• Cognitive strategies. Students employ specific cognitive strategies as “guiding procedures” to complete more complex academic tasks such as reading comprehension or writing (Rosenshine, 1995).

• Academic-enabling skills. Skills that are ‘academic enablers’ (DiPerna, 2006) are not tied to specific academic knowledge but rather aid student learning across a wide range of settings and tasks (e.g., organizing work materials, time management).

Motivation Deficit 1: Cannot Do the Work



Motivation Deficit 1: Cannot Do the Work (Cont.)

• What the Research Says: When a student lacks the capability to complete an academic task because of limited or missing basic skills, cognitive strategies, or academic-enabling skills, that student is still in the acquisition stage of learning (Haring et al., 1978). That student cannot be expected to be motivated or to be successful as a learner unless he or she is first explicitly taught these weak or absent essential skills (Daly, Witt, Martens & Dool, 1997).

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• How to Verify the Presence of This Motivation Problem: The teacher collects information (e.g., through observations of the student engaging in academic tasks; interviews with the student; examination of work products, quizzes, or tests) demonstrating that the student lacks basic skills, cognitive strategies, or academic-enabling skills essential to the academic task.

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• How to Fix This Motivation Problem: Students who are not motivated because they lack essential skills need to be taught those skills.

Direct-Instruction Format. Students learning new material, concepts, or skills benefit from a ‘direct instruction’ approach. (Burns, VanDerHeyden & Boice, 2008; Rosenshine, 1995; Rupley, Blair, & Nichols, 2009).

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• How to Fix This Motivation Problem: When following a direct-instruction format, the teacher:ensures that the lesson content is appropriately

matched to students’ abilities.opens the lesson with a brief review of concepts or

material that were previously presented.states the goals of the current day’s lesson.breaks new material into small, manageable increments,

or steps.

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• How to Fix This Motivation Problem: When following a direct-instruction format, the teacher:throughout the lesson, provides adequate explanations

and detailed instructions for all concepts and materials being taught. NOTE: Verbal explanations can include ‘talk-alouds’ (e.g., the teacher describes and explains each step of a cognitive strategy) and ‘think-alouds’ (e.g., the teacher applies a cognitive strategy to a particular problem or task and verbalizes the steps in applying the strategy).

regularly checks for student understanding by posing frequent questions and eliciting group responses.

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• How to Fix This Motivation Problem: When following a direct-instruction format, the teacher:verifies that students are experiencing sufficient success

in the lesson content to shape their learning in the desired direction and to maintain student motivation and engagement.

provides timely and regular performance feedback and corrections throughout the lesson as needed to guide student learning.

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• How to Fix This Motivation Problem: When following a direct-instruction format, the teacher:allows students the chance to engage in practice

activities distributed throughout the lesson (e.g., through teacher demonstration; then group practice with teacher supervision and feedback; then independent, individual student practice).

ensures that students have adequate support (e.g., clear and explicit instructions; teacher monitoring) to be successful during independent seatwork practice activities.

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Activity: ‘Good Instruction is Research-Based’• Review the elements of effective ‘direct

instruction’ that appear on page 3 of your handout.• Discuss how you can share this checklist with

others in your school to help them to realize that teacher-delivered instruction that follows these guidelines is ‘research-based’ and supports RTI, e.g.:– Whole-group: Tier 1 Core Instruction– Small-group: Tier 1 Intervention; Tier 2/3 Intervention– Individual student: Tier 3 Intervention

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Math Interventions

Focus of Inquiry: How can our school find intervention programs or ideas at Tiers 1-3 to address math delays?

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What Works Clearinghousehttp://ies.ed.gov/ncee/wwc/

This website reviews core instruction and intervention programs in math and reading/writing, as well as other academic areas.

The site reviews existing studies and draws conclusions about whether specific intervention programs show evidence of effectiveness.



Best Evidence Encyclopediahttp://www.bestevidence.org/

This site provides reviews of evidence-based reading and math programs.

The website is sponsored by the Johns Hopkins University School of Education's Center for Data-Driven Reform in Education (CDDRE) .



Recommended Instruction and Intervention Programs for ‘Number Sense’ (Clements & Sarama, 2011)

• RightStart Math• Building Blocks Math• Big Math for Little Kids• Number Worlds

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Source: Clements, D. H. & Sarama, J. (2011). Early childhood math intervention. Science, 333, 968-970.



Intervention Centralwww.interventioncentral.org

Documents

Workshop PPTs and handout available at: interventioncentral/rtimath