RTI for Math & Writing: Grades K-8 Jim Wright interventioncentral

Response to Intervention

www.interventioncentral.org

RTI for Math & Writing: Grades K-8

Jim Wrightwww.interventioncentral.org



Workshop PPTs and handout available at:

http://www.interventioncentral.org/gisd



RTI: Key Concepts

Focus of Inquiry: What are central concepts of RTI that are helpful in understanding how to expand the model to include math and writing?

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www.interventioncentral.org 5

RTI Assumption: Struggling Students Are ‘Typical’ Until Proven Otherwise…

RTI logic assumes that:– A student who begins to struggle in general education is typical,

and that– It is general education’s responsibility to find the instructional

strategies that will unlock the student’s learning potential

Only when the student shows through well-documented interventions that he or she has ‘failed to respond to intervention’ does RTI begin to investigate the possibility that the student may have a learning disability or other special education condition.



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.



Target Student

Discrepancy 1: Skill Gap (Current Performance Level)

Avg Classroom Academic Performance Level

‘Dual-Discrepancy’: RTI Model of Learning Disability (Fuchs 2003)

Discrepancy 2:Gap in Rate of Learning (‘Slope of Improvement’)



Evaluating a Student’s ‘Non-Responder’ Status: An RTI

Checklist pp. 2-7



RTI ‘Non-Responder’ Checklist: Purpose

The document Evaluating a Student’s ‘Non Responder’ Status: An RTI Checklist was created to help schools to:

• audit the quality of their current RTI efforts in any academic area.

• create concrete guidelines for judging whether RTI intervention efforts for a particular student are of adequate quality.

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Evaluating a Student’s ‘Non-Responder’ Status: An RTI Checklist

Interventions: Evidence-Based & Implemented With Integrity

• Tier 1: High-Quality Core Instruction• Tier 1: Classroom Intervention• Tier 2 & 3 Interventions: Minimum Number & Length• Tier 2 & 3 Interventions: Essential Elements• Tier 1, 2, & 3 Interventions: Intervention Integrity

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Evaluating a Student’s ‘Non-Responder’ Status: An RTI Checklist

Academic Screenings: General Outcome Measures and Skill-Based Measures

• Selection of Academic Screening Measures• Local Norms Collected via Gradewide Academic

Screenings at Least 3 Times Per Year

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Evaluating a Student’s ‘Non-Responder’ Status: Activity

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At your table: Discuss how your school or district may use the document Evaluating a Student’s ‘Non-Responder’ Status: An RTI Checklist to:

• help you to identify the ‘non-negotiable’ elements of RTI as you extend it to cover math and writing.

Be prepared to share the main points of your discussion with the large group.



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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“ ”“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 reading 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).



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.23

• Review the handout on p. 23 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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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.12-13



• 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 13 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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Core Instruction & Tier 1 Intervention

Focus of Inquiry: How does a school select ‘research-based’ math instruction and intervention ideas for use in the classroom?

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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) .



Intervention Centralwww.interventioncentral.org



Math School-Wide Screenings

Focus of Inquiry: What math school-wide screenings 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

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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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

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.



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 Interventions

Focus of Inquiry: What are sample intervention programs or ideas to address math delays?

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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.



Profile of Students With Significant Math Difficulties p. 15 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. 15

• Review the profile of students with significant math difficulties that appears on p. 15 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 has not yet acquired math facts.

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Solution: Use these strategies:• Incremental Rehearsal• Cover-Copy-Compare• Peer Tutoring in Math Computation with Constant Time Delay



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 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.



Peer Tutoring in Math Computation with

Constant Time Delay



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



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.



Fluency Stage: Math ComputationSelf-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







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 meeting’ held once per month to consider students for 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.

Documents

RTI for Math & Writing: Grades K-8 Jim Wright interventioncentral