Response to Intervention RTI for Math: Process Check Jim Wright

Response to Intervention

www.interventioncentral.org

RTI for Math: Process Check

Jim Wrightwww.interventioncentral.org



RTI: Math Interventions & Assessment: Workshop Agenda

‘Intervention Integrity’ for Math Interventions: Proactive Steps & Ways to Measure

Technical Assistance: RTI & Math

Math Interventions

Building Capacity for RTI in Math: Planning Time

Web Resources



Workshop Handouts and PPT Available at:

• http://www.jimwrightonline.com/monroe1.php

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RTI: Key Concepts


www.interventioncentral.org 5

Five Core Components of RTI Service Delivery

1. Student services are arranged in a multi-tier model 2. Data are collected to assess student baseline levels

and to make decisions about student progress 3. Interventions are ‘evidence-based’4. The ‘procedural integrity’ of interventions is measured5. RTI is implemented and developed at the school- and

district-level to be scalable and sustainable over time

Source: Glover, T. A., & DiPerna, J. C. (2007). Service delivery for response to intervention: Core components and directions for future research. School Psychology Review, 36, 526-540.



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.



Complementary RTI Models: Standard Treatment & Problem-Solving Protocols

“The two most commonly used RTI approaches are (1) standard treatment and (2) problem-solving protocol. While these two approaches to RTI are sometimes described as being very different from each other, they actually have several common elements, and both fit within a problem-solving framework. In practice, many schools and districts combine or blend aspects of the two approaches to fit their needs.”

Source: Duffy, H. (August 2007). Meeting the needs of significantly struggling learners in high school. Washington, DC: National High School Center. Retrieved from http://www.betterhighschools.org/pubs/ p. 5



RTI Interventions: Standard-Treatment vs. Problem-SolvingThere are two different vehicles that schools can use to deliver RTI interventions:

Standard-Protocol (Standalone Intervention). Programs based on scientifically valid instructional practices (‘standard protocol’) are created to address frequent student referral concerns. These services are provided outside of the classroom. A middle school, for example, may set up a structured math-tutoring program staffed by adult volunteer tutors to provide assistance to students with limited math skills. Students referred for a Tier II math intervention would be placed in this tutoring program. An advantage of the standard-protocol approach is that it is efficient and consistent: large numbers of students can be put into these group interventions to receive a highly standardized intervention. However, standard group intervention protocols often cannot be individualized easily to accommodate a specific student’s unique needs.

Problem-solving (Classroom-Based Intervention). Individualized research-based interventions match the profile of a particular student’s strengths and limitations. The classroom teacher often has a large role in carrying out these interventions. A plus of the problem-solving approach is that the intervention can be customized to the student’s needs. However, developing intervention plans for individual students can be time-consuming.



Tier I Instruction/InterventionsTier I instruction/interventions:

• Are universal—available to all students.• Can be delivered within classrooms or throughout the school. • Are likely to be put into place by the teacher at the first sign that a student is struggling.

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

Tier 1 instruction/interventions encompass:

• The school’s core curriculum and all published or teacher-made materials used to deliver that curriculum.

• Teacher use of ‘whole-group’ teaching & management strategies.• Teacher use of individualized strategies with specific students.

Tier I instruction/interventions attempt to answer the question: Are classroom instructional strategies & supports sufficient to help the student to achieve academic success?



Tier 1: Classroom-Level Interventions• Decision Point: Student is struggling and may face significant

high-stakes negative outcome if situation does not improve.• Collaboration Opportunity: Teacher can refer the student to a

grade-level, instruction team, or department meeting to brainstorm ideas – OR – teacher seeks out consultant in school to brainstorm intervention ideas.

• Documentation: Teacher completes ‘Classroom Intervention Form’ prior to carrying out intervention. Teacher collects classroom data.

• Decision Rule [Example]: Teacher should refer student to the next level of RTI support if the intervention is not successful within 8 instructional weeks.





Tier 2: Supplemental (Standard-Protocol Model) Interventions

Tier 2 interventions are typically delivered in small-group format. About 15% of students in the typical school will require Tier 2/supplemental intervention support.

Group size for Tier 2 interventions is limited to 4-6 students.

Students placed in Tier 2 interventions should have a shared profile of intervention need.

The reading progress of students in Tier 2 interventions are monitored at least 1-2 times per month.

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



Tier 2: Supplemental Interventions• Decision Point: Building-wide academic screenings• Collaboration Opportunity: After each building-wide academic screening, ‘data

teams’ meet (teachers at a grade level; building principal; reading teacher, etc.) At the meeting, the group considers how the assessment data should shape/inform core instruction. Additionally, the data team sets a cutpoint to determine which students should be recruited for Tier 2 group interventions. NOTE: Team may continue to meet every 5 weeks to consider student progress in Tier 2; move students into and out of groups.

• Documentation: Tier 2 instructor completes a Tier 2 Group Assignment Sheet listing students and their corresponding interventions. Progress-monitoring occurs 1-2 times per month.

• Decision Rules [Example]: Student is returned to Tier 1 support if they perform above the 25th percentile in the next school-wide screening. Student is referred to Tier 3 (RTI Team) if they fail to make expected progress despite two Tier 2 (group-based) interventions.





Scheduling Elementary Tier 2 Interventions

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

Classroom 1 Classroom 2 Classroom 3Grade K

Classroom 1 Classroom 2 Classroom 3Grade 1





Anyplace Elementary School: RTI Daily Schedule

Option 3: ‘Floating RTI’:Gradewide Shared Schedule. Each grade has a scheduled RTI time across classrooms. No two grades share the same RTI time. Advantages are that outside providers can move from grade to grade providing push-in or pull-out services and that students can be grouped by need across different teachers within the grade.

9:00-9:30

9:45-10:15

10:30-11:00

12:30-1:00

1:15-1:45

2:00-2:30



Tier 3: Intensive Individualized Interventions (Problem-Solving Model)

Tier 3 interventions are the most intensive offered in a school setting. About 5 % of a general-education student population may qualify for Tier 3 supports. Typically, the RTI Problem-Solving Team meets to develop intervention plans for Tier 3 students.

Students qualify for Tier 3 interventions because:– they are found to have a large skill gap when compared to their class or grade peers;

and/or– They did not respond to interventions provided previously at Tiers 1 & 2.

Tier 3 interventions are provided daily for sessions of 30 minutes. The student-teacher ratio is flexible but should allow the student to receive intensive, individualized instruction. The academic or behavioral progress of students in Tier 3 interventions is monitored at least weekly.

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



Tier 3: RTI Team• Decision Point: RTI Problem-Solving Team• Collaboration Opportunity: Weekly RTI Problem-Solving Team

meetings are scheduled to handle referrals of students that failed to respond to interventions from Tiers 1 & 2.

• Documentation: Teacher referral form; RTI Team minutes form; progress-monitoring data collected at least weekly.

• Decision Rules [Example]: If student has failed to respond adequately to 3 intervention trials of 6-8 weeks (from Tiers 2 and 3), the student may be referred to Special Education.





Advancing Through RTI: Flexibility in the TiersFor purposes of efficiency, students should be placed in small-group instruction at Tier 2.

However, group interventions may not always be possible because –due to scheduling or other issues—no group is available. (For example, students with RTI behavioral referrals may not have a group intervention available.)

In such a case, the student will go directly to the problem-solving process (Tier 3)—typically through a referral to the school RTI Team.

Nonetheless, the school must still document the same minimum number of interventions attempted for every student in RTI, whether or not a student first received interventions in a group setting.



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’)



RTI Teams: Improving Problem-Solving Through Effective Case Management Jim Wrightwww.interventioncentral.org





Case Manager: Role• Meets with the referring teacher(s) briefly prior to

the initial RTI Team meeting to review the teacher referral form, clarify teacher concerns, decide what additional data should be collected on the student.

• Touches base briefly with the referring teacher(s) after the RTI Team meeting to check that the intervention plan is running smoothly.

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Case Manager: Pre-Meeting• Prior to an initial RTI Problem-Solving Team

meeting, it is recommended that a case manager from the RTI Team schedule a brief (15-20 minute) ‘pre-meeting’ with the referring teacher. The purpose of this pre-meeting is for the case manager to share with the teacher the purpose of the upcoming full RTI Team meeting, to clarify student referral concerns, and to decide what data should be collected and brought to the RTI Team meeting.

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Case Manager: Pre-Meeting Steps

Here is a recommended agenda for the case manager-teacher pre-meeting:

1. Explain the purpose of the upcoming RTI Problem-Solving Team meeting: The case manager explains that the RTI Team meeting goals are to (a) fully understand the nature of the student’s academic and/or behavioral problems; (b) develop an evidence-based intervention plan for the student; and (c) set a goal for student improvement and select means to monitor the student’s response to the intervention plan.

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2. Define the student referral concern(s) in clear, specific terms.. The case manager reviews with the teacher the most important student referral concern(s), helping the teacher to define those concern(s) in clear, specific, observable terms. The teacher is also prompted to prioritize his or her top 1-2 student concerns.

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3. Decide what data should be brought to the RTI Team meeting. The case manager and teacher decide what student data should be collected and brought to the RTI Team meeting to provide insight into the nature of the student’s presenting concern(s).

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Case Manager: Tips• If you discover, when you meet with a referring

teacher prior to the RTI Team meeting, that his or her concern is vaguely worded, help the teacher to clarify the concern with the question “What does [teacher concern] look like in the classroom?”

• After the RTI Team meeting, consider sending periodic emails to the referring teacher(s) asking them how the intervention is going and inviting them to inform you if they require assistance.

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Academic Interventions ‘Critical Components’ Checklist




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This checklist summarizes the essential components of academic interventions. When preparing a student’s Tier 1, 2, or 3 academic intervention plan, use this document as a ‘pre-flight checklist’ to ensure that the academic intervention is of high quality, is sufficiently strong to address the identified student problem, is fully understood and supported by the teacher, and can be implemented with integrity. NOTE: While the checklist refers to the ‘teacher’ as the interventionist, it can also be used as a guide to ensure the quality of interventions implemented by non-instructional personnel, adult volunteers, parents, and peer (student) tutors.

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Allocating Sufficient Contact Time & Assuring Appropriate Student-Teacher RatioThe cumulative time set aside for an intervention and the amount of direct teacher contact are two factors that help to determine that intervention’s ‘strength’ (Yeaton & Sechrest, 1981).

Critical Item? Intervention Element Notes

Time Allocated. The time set aside for the intervention is appropriate for the type and level of student problem (Burns & Gibbons, 2008; Kratochwill, Clements & Kalymon, 2007). When evaluating whether the amount of time allocated is adequate, consider:Length of each intervention session.Frequency of sessions (e.g.., daily, 3 times per week)Duration of intervention period (e.g., 6 instructional weeks)

Student-Teacher Ratio. The student receives sufficient contact from the teacher or other person delivering the intervention to make that intervention effective. NOTE: Generally, supplemental intervention groups should be limited to 6-7 students (Burns & Gibbons, 2008).



Matching the Intervention to the Student ProblemAcademic interventions are not selected at random. First, the student academic problem(s) is defined clearly and in detail. Then, the likely explanations for the academic problem(s) are identified to understand which intervention(s) are likely to help—and which should be avoided.Critical Item? Intervention Element Notes

Problem Definition. The student academic problem(s) to be addressed in the intervention are defined in clear, specific, measureable terms (Bergan, 1995; Witt, VanDerHeyden & Gilbertson, 2004). The full problem definition describes:Conditions. Describe the environmental conditions or task demands in place when the academic problem is observed. Problem Description. Describe the actual observable academic behavior in which the student is engaged. Include rate, accuracy, or other quantitative information of student performance.Typical or Expected Level of Performance. Provide a typical or expected performance criterion for this skill or behavior. Typical or expected academic performance can be calculated using a variety of sources,



Sample Math Problem Identification Statements

ConditionsWhen shown random number pairs 0-20 during a 1-minute assessment…

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Problem Description

…Charlie can correctly identify 6 number pairs…

Typical/Expected Level of Performance

…while the median rate in the 1st grade classroom is 22 number pairs.

When shown 20 key vocabulary terms required for grade 4 math performance…

…Haley can provide correct definitions for 10 terms…

…while the curriculum expectation is that students will have 100 percent mastery of those terms.



Matching the Intervention to the Student Problem (Cont.)Critical Item? Intervention Element Notes

Appropriate Target. Selected intervention(s) are appropriate for the identified student problem(s) (Burns, VanDerHeyden & Boice, 2008). TIP: Use the Instructional Hierarchy (Haring et al., 1978) to select academic interventions according to the four stages of learning:Acquisition. The student has begun to learn how to complete the target skill correctly but is not yet accurate in the skill. Interventions should improve accuracy.Fluency. The student is able to complete the target skill accurately but works slowly. Interventions should increase the student’s speed of responding (fluency) as well as to maintain accuracy.Generalization. The student may have acquired the target skill but does not typically use it in the full range of appropriate situations or settings. Or the student may confuse the target skill with ‘similar’ skills. Interventions should get the student to use the skill in the widest possible range of settings and situations, or to accurately discriminate between the target skill and ‘similar’ skills.Adaptation. The student is not yet able to modify or adapt an existing skill to fit novel task-demands or situations. Interventions should help the student to identify key concepts or elements from previously learned skills that can be adapted to the new demands or situations.



Matching the Intervention to the Student Problem (Cont.)Critical Item? Intervention Element Notes

‘Can’t Do/Won’t Do’ Check. The teacher has determined whether the student problem is primarily a skill or knowledge deficit (‘can’t do’) or whether student motivation plays a main or supporting role in academic underperformance (‘wont do’). If motivation appears to be a significant factor contributing to the problem, the intervention plan includes strategies to engage the student (e.g., high interest learning activities; rewards/incentives; increased student choice in academic assignments, etc.) (Skinner, Pappas & Davis, 2005; Witt, VanDerHeyden & Gilbertson, 2004).



Activity: Matching the Intervention to the Student ProblemIn your teams:

• Consider these critical aspects of academic intervention: Clear and specific problem-identification statement (Conditions,

Problem Description, Typical/Expected Level of Performance).

Appropriate intervention target (e.g., selected intervention is appropriately matched to Acquisition, Fluency, Generalization, or Adaptation phase of Instructional Hierarchy).

Can’t Do/Won’t Do Check (Clarification of whether motivation plays a significant role in student academic underperformance).

• What steps can your RTI Team and school take to ensure that each of these intervention elements is taken into consideration?



Incorporating Effective Instructional ElementsThese effective ‘building blocks’ of instruction are well-known and well-supported by the research. They should be considered when selecting or creating any academic intervention.Critical Item? Intervention Element Notes

Explicit Instruction. Student skills have been broken down “into manageable and deliberately sequenced steps” and the teacher provided“ overt strategies for students to learn and practice new skills” (Burns, VanDerHeyden & Boice, 2008, p.1153).

Appropriate Level of Challenge. The student experienced sufficient success in the academic task(s) to shape learning in the desired direction as well as to maintain student motivation (Burns, VanDerHeyden & Boice, 2008).

Active Engagement. The intervention ensures that the student is engaged in ‘active accurate responding’ (Skinner, Pappas & Davis, 2005).at a rate frequent enough to capture student attention and to optimize effective learning.

Performance Feedback. The student receives prompt performance feedback about the work completed (Burns, VanDerHeyden & Boice, 2008).

Maintenance of Academic Standards. If the intervention includes any accommodations to better support the struggling learner (e.g., preferential seating, breaking a longer assignment into smaller chunks), those accommodations do not substantially lower the academic standards against which the student is to be evaluated and are not likely to reduce the student’s rate of learning (Skinner, Pappas & Davis, 2005).



Activity: Incorporating Effective Instructional Elements

In your teams:

• Think about the effective instructional elements reviewed in this workshop.

• How can your school assist teachers to ensure that effective instructional elements are included in math interventions?

Incorporating Effective Instructional ElementsCritical Item?

Intervention Element Notes

Explicit Instruction. Appropriate Level of Challenge. Active Engagement.. Performance Feedback.

Maintenance of Academic Standards.



Verifying Teacher Understanding & Providing Teacher SupportThe teacher is an active agent in the intervention, with primary responsibility for putting it into practice in a busy classroom. It is important, then, that the teacher fully understands how to do the intervention, believes that he or she can do it, and knows whom to seek out if there are problems with the intervention.Critical Item? Intervention Element Notes

Teacher Responsibility. The teacher understands his or her responsibility to implement the academic intervention(s) with integrity.

Teacher Acceptability. The teacher states that he or she finds the academic intervention feasible and acceptable for the identified student problem.

Step-by-Step Intervention Script. The essential steps of the intervention are written as an ‘intervention script’--a series of clearly described steps—to ensure teacher understanding and make implementation easier (Hawkins, Morrison, Musti-Rao & Hawkins, 2008).

Intervention Training. If the teacher requires training to carry out the intervention, that training has been arranged.

Intervention Elements: Negotiable vs. Non-Negotiable. The teacher knows all of the steps of the intervention. Additionally, the teacher knows which of the intervention steps are ‘non-negotiable’ (they must be completed exactly as designed) and which are ‘negotiable’ (the teacher has some latitude in how to carry out those steps) (Hawkins, Morrison, Musti-Rao & Hawkins, 2008).

Assistance With the Intervention. If the intervention cannot be implemented as designed for any reason (e.g., student absence, lack of materials, etc.), the teacher knows how to get assistance quickly to either fix the problem(s) to the current intervention or to change the intervention.



Activity: Verifying Teacher Understanding & Providing Teacher Support

In your teams:

• Review the checklist for verifying that teachers understand all elements of the intervention and actively support its use.

• How will your school ensure that teachers in Tier 1 will understand and support the math interventions such as the example selected by your team?

Verifying Teacher Understanding & Providing Teacher SupportCritical Item? Intervention Element

Teacher Responsibility Teacher Acceptability. Step-by-Step Intervention Script. Intervention Training. Intervention Elements: Negotiable

vs. Non-Negotiable Assistance With the Intervention



Documenting the Intervention & Collecting DataInterventions only have meaning if they are done within a larger data-based context. For example, interventions that lack baseline data, goal(s) for improvement, and a progress-monitoring plan are ‘fatally flawed’ (Witt, VanDerHeyden & Gilbertson, 2004).Critical Item? Intervention Element Notes

Intervention Documentation. The teacher understands and can manage all documentation required for this intervention (e.g., maintaining a log of intervention sessions, etc.).

Checkup Date. Before the intervention begins, a future checkup date is selected to review the intervention to determine if it is successful. Time elapsing between the start of the intervention and the checkup date should be short enough to allow a timely review of the intervention but long enough to give the school sufficient time to judge with confidence whether the intervention worked.

Baseline. Before the intervention begins, the teacher has collected information about the student’s baseline level of performance in the identified area(s) of academic concern (Witt, VanDerHeyden & Gilbertson, 2004).

Goal. Before the intervention begins, the teacher has set a specific goal for predicted student improvement to use as a minimum standard for success (Witt, VanDerHeyden & Gilbertson, 2004). The goal is the expected student outcome by the checkup date if the intervention is successful.

Progress-Monitoring. During the intervention, the teacher collects progress-monitoring data of sufficient quality and at a sufficient frequency to determine at the checkup date whether that intervention is successful (Witt, VanDerHeyden & Gilbertson, 2004).



Activity: Putting Math Interventions into a ‘Data Context’…

In your teams:

• Appoint a recorder.

• For the math intervention that your team selected, brainstorm methods to measure student progress.

• Discuss how teachers would collect baseline data, set a goal for improvement.

• How frequently should progress-monitoring data be collected?



References

• Bergan, J. R. (1995). Evolution of a problem-solving model of consultation. Journal of Educational and Psychological Consultation, 6(2), 111-123.

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

• Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in intensive academic interventions. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp.1151-1162). Bethesda, MD: National Association of School Psychologists.

• 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.

• Hawkins, R. O., Morrison, J. Q., Musti-Rao, S., & Hawkins, J. A. (2008). Treatment integrity for academic interventions in real- world settings. School Psychology Forum, 2(3), 1-15.

• Kratochwill, T. R., Clements, M. A., & Kalymon, K. M. (2007). Response to intervention: Conceptual and methodological issues in implementation. In Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (Eds.), Handbook of response to intervention: The science and practice of assessment and intervention. New York: Springer.

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

• Witt, J. C., VanDerHeyden, A. M., & Gilbertson, D. (2004). Troubleshooting behavioral interventions. A systematic process for finding and eliminating problems. School Psychology Review, 33, 363-383.

• Yeaton, W. M. & Sechrest, L. (1981). Critical dimensions in the choice and maintenance of successful treatments: Strength, integrity, and effectiveness. Journal of Consulting and Clinical Psychology, 49, 156-167.

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RTI: Selected Math InterventionsJim Wrightwww.interventioncentral.org



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

Self-Monitoring & Incentives p. 28

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 Intervention: Teacher Feedback

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Cover-Copy-Compare: Math Computational Fluency-Building Intervention p. 19

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

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

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



Math Computation: Problem Interspersal Technique p. 23• The teacher first identifies the range of ‘challenging’ problem-types

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

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

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

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

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



Developing Student Metacognitive Abilities



Importance of Metacognitive Strategy Use…“Metacognitive processes focus on self-awareness of cognitive knowledge that is presumed to be necessary for effective problem solving, and they direct and regulate cognitive processes and strategies during problem solving…That is, successful problem solvers, consciously or unconsciously (depending on task demands), use self-instruction, self-questioning, and self-monitoring to gain access to strategic knowledge, guide execution of strategies, and regulate use of strategies and problem-solving performance.” p. 231

Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.



Elements of Metacognitive Processes

“Self-instruction helps students to identify and direct the problem-solving strategies prior to execution. Self-questioning promotes internal dialogue for systematically analyzing problem information and regulating execution of cognitive strategies. Self-monitoring promotes appropriate use of specific strategies and encourages students to monitor general performance. [Emphasis added].” p. 231

Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.



Combining Cognitive & Metacognitive Strategies to Assist Students With Mathematical Problem Solving p.

Solving an advanced math problem independently requires the coordination of a number of complex skills. The following strategies combine both cognitive and metacognitive elements (Montague, 1992; Montague & Dietz, 2009). First, the student is taught a 7-step process for attacking a math word problem (cognitive strategy). Second, the instructor trains the student to use a three-part self-coaching routine for each of the seven problem-solving steps (metacognitive strategy).



Cognitive Portion of Combined Problem Solving Approach

In the cognitive part of this multi-strategy intervention, the student learns an explicit series of steps to analyze and solve a math problem. Those steps include:

1. Reading the problem. The student reads the problem carefully, noting and attempting to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).

2. Paraphrasing the problem. The student restates the problem in his or her own words.3. ‘Drawing’ the problem. The student creates a drawing of the problem, creating a

visual representation of the word problem.4. Creating a plan to solve the problem. The student decides on the best way to solve

the problem and develops a plan to do so.5. Predicting/Estimating the answer. The student estimates or predicts what the answer

to the problem will be. The student may compute a quick approximation of the answer, using rounding or other shortcuts.

6. Computing the answer. The student follows the plan developed earlier to compute the answer to the problem.

7. Checking the answer. The student methodically checks the calculations for each step of the problem. The student also compares the actual answer to the estimated answer calculated in a previous step to ensure that there is general agreement between the two values.



Metacognitive Portion of Combined Problem Solving Approach

The metacognitive component of the intervention is a three-part routine that follows a sequence of ‘Say’, ‘Ask, ‘Check’. For each of the 7 problem-solving steps reviewed above:

• The student first self-instructs by stating, or ‘saying’, the purpose of the step (‘Say’).

• The student next self-questions by ‘asking’ what he or she intends to do to complete the step (‘Ask’).

• The student concludes the step by self-monitoring, or ‘checking’, the successful completion of the step (‘Check’).



Combined Cognitive & Metacognitive Elements of Strategy



















Documents

Response to Intervention RTI for Math: Process Check Jim Wright