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RTI Framework for Success: Research-Based Math Instruction Intervention

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**1. **RTI Framework for Success: Research-Based Math Instruction & Intervention C. Lee Goss, Psy.D.
Attention & Learning Pathways
School Psychology Services

**2. **© C. Lee Goss, 2008 2 Acknowledgments
Rachel Brown-Chidsey, Ph.D.
Mark Steege, Ph.D.
Marcia Davidson, Ph.D.
RTI & research-based reading instruction is built on a foundation of years of research, professional expertise, and leadership:
Ed Daly, Ph.D., Stan Deno, Ph.D., Barbara Foorman, Ph.D., Doug Fuchs, Ph.D., Lynn Fuchs, Ph.D., Laurice Joseph, Ph.D., Dan Reschly, Ph.D., Ed Shapiro, Ph.D., Mark Shinn, Ph.D., David Tilly, Ph.D., Joseph Torgeson, Ph.D., & Sharon Vaughn, Ph.D., Amanda VanDerHeyden, Ph.D., Joseph Witt, Ph.D.

**3. **© C. Lee Goss, 2008 3 Status of Research-Based Math Instruction Research-Based Math Instruction is in it’s infancy compared to the current research available for reading & behavior
Current research in RTI for math is consistent with RTI in reading—the premise is the same
Focus on the importance and benefits of screening to detect students at-risk for difficulty in math achievement
Focus on formative assessment or progress monitoring data to inform instruction and intervention at all 3-Tiers

**4. **© C. Lee Goss, 2008 4 Benefits of Research-Based Math Instruction Provide all students with effective instruction and needed intervention that is supported by high quality peer-reviewed research studies
Early intervention and prevention methods to prevent poor math achievement or the need for identification for special education services.
Instruction is the one thing we can “control” on a daily basis to develop mastery of needed skills

**5. **© C. Lee Goss, 2008 5 Math Research Status U.S. National Research Council (NRC) conducted a review of math instruction research
A panel of researchers with expertise in math instruction reviewed the preceding 30 years of math instruction research and summarized findings in
Adding it Up: Helping Children Learn Mathematics (2001)

**6. **© C. Lee Goss, 2008 6 Adding It Up: 5 Essential Strands
1) Conceptual Understanding
2) Procedural Fluency
3) Strategic Competence
4) Adaptive Reasoning
5) Productive Disposition

**9. **© C. Lee Goss, 2008 9 Sample Skills Related to the Strands, Cont’d Note: Samples identified in the table are examples and do not represent all math skills
Key Point identified in Adding It Up:
All students need to be able to master all of the strands and skills identified in the table to develop proficiency in math

**10. **© C. Lee Goss, 2008 10 Adding It Up: 5 Essential Strands, Cont’d All 5 components or strands are interdependent
All 5 strands are identified as essential for all students
All 5 strands must be included in math instruction at all grade levels
The 5 strands provide a way to organize the math instruction
All strands can be matched with specific student instructional skills

**11. **© C. Lee Goss, 2008 11 National Mathematics Advisory Panel (NMAP) U.S. DOE appointed a National Mathematics Advisory Panel (NMAP) in 2006
Panel included researchers with strong expertise in mathematics and mathematics instruction
Panel charged with reviewing all available research about math instruction and summarizing findings
Final Report: Foundations for Success, 2008

**12. **© C. Lee Goss, 2008 12 NRC Report: Math Proficiency of U.S. Students International comparisons
Low fractions of proficiency on NAEP
Falling proficiency at higher grades
Heavy remedial demand upon entry into college
Achievement gap
Algebra as a gateway
(NMAP Report, 2008)

**13. **© C. Lee Goss, 2008 13 NRC Report: Math Proficiency of U.S. Students, Cont’d American students achievement in mathematics is mediocre compared to international peers
32% of our students are at or above the “proficient” level in Grade 8, but only 23% are proficient at Grade 12. Consistent with these findings is the vast and growing demand for remedial mathematics education among arriving students in 4-year colleges and community colleges across the nation.
On the TIMSS (Trends in International Mathematics and Science Study), U.S. students do less well in Grade 8 than grade 4. The performance is still poorer in Grade 12.
In the PISA (Programme for International Student Assessment), U.S. 15-year-olds ranked 25th among 30 developed nations in math literacy and problem solving.
Even in elementary school, only 7% of U.S. 4th-graders scored at the advanced level in TIMSS, compared to 38% of 4th-graders in Singapore, a world leader in mathematics achievement
(NMAP Report, 2008)

**14. **© C. Lee Goss, 2008 14 Basis of the Panel’s work
Review of 16,000 research studies and related documents.
Public testimony gathered from 110 individuals.
Review of written commentary from 160 organizations and individuals
12 public meetings held around the country
Analysis of survey results from 743 Algebra I teachers

**15. **© C. Lee Goss, 2008 15 NRC Effective Math Instruction Research Findings
Core Conclusions:
All U.S. children must develop math proficiency for successful academic achievement
Math skills must be viewed as important for all children to learn
Identified 5 Essential Components termed “Strands” of Effective Math Instruction

**16. **© C. Lee Goss, 2008 16 National Math Advisory Panel Report Fact Sheet Small Group Discussions by Grade Levels/Schools
Read and discuss NMAP Fact Sheet findings
Identify potential current math curricula strengths and weaknesses to share with the group
Any Surprises?

**17. **© C. Lee Goss, 2008 17 NMAP Final Report Findings NMAP Report identified 6 main steps needed to improve math achievement:
Pre-K to 8th grade math curriculum should be streamlined to emphasize a narrower set of the most critical skills and topics
Implementation of best practice instruction methods and knowledge of how children learn with a focus on the benefits and importance of:
Early Intervention
Conceptual Understanding
Fluency
Automaticity
Effort

**18. **© C. Lee Goss, 2008 18 NMAP Final Report Findings, Cont’d
3) Elementary grade teachers must have strong math skills in order to teach math well
4) Math instruction should not be purely ‘student-centered’ or “teacher centered” but must be an integration of both perspectives based on the findings of research
5) National assessments such as National Assessment of Educational Progress (NAEP) should be strengthened to include emphasis on the most critical math knowledge and skills
6) There is a need for more rigorous research about math instruction and the findings of such research must be used to improve teaching practices

**19. **© C. Lee Goss, 2008 19 NMAP Final Report Findings, Cont’d
Key NMAP Conclusion
Importance of early student mastery of basic math skills such as computation fluency
NMAP Conclusion coincides with other recent research findings about math instruction (Baker, Gerston, & Lee, 2002)

**20. **© C. Lee Goss, 2008 20 Research Findings on Effective Math Instruction Effective math instruction includes a focus on:
Basic Skills
Teaching explicitly and systematically
Frequent progress monitoring data to inform instruction and student progress
(Baker, Gerston, & Lee, 2002)

**21. **© C. Lee Goss, 2008 21 Research Informs Federal Policies NCLB, IDEIA 2004, NMAP Report, ME Regs & ME RTI Guidelines highlight the importance of research-based instruction and research based interventions before a referral to special education is made.
CBMs meet extensive research criteria for benchmark and progress assessments to measure student response to research-based instruction

**22. **© C. Lee Goss, 2008 22 RTI: Prevention & Early Intervention Scientifically-based math curriculum that incorporates effective instruction techniques
Importance of early mastery of number values and math calculation fluency
Not unusual for students to develop some math skills better than others (e.g. struggle w/ subtraction but master multiplication easily).

**23. **© C. Lee Goss, 2008 23 RTI: Prevention & Early Intervention, Cont’d RTI methods of prevention are a very helpful and effective way to reduce the number of students needing remediation
Research demonstrates that many children experiencing difficulty respond to early intervention and explicit research-based instruction techniques

**24. **© C. Lee Goss, 2008 24 RTI is a Well-Child Prevention Model Prevention efforts to foster educational success for all students in school
Incorporates regular screening of ALL children to determine who is at risk for developing academic and/or behavioral problems
Provides strategic interventions for children determined at-risk in the early stages of a problem
Acknowledges that some children will have serious and persistent problems that will require intensive support

**25. **© C. Lee Goss, 2008 25 RTI and Identification of LD Copious studies have shown the inaccuracy of IQ score-achievement test discrepancy scores for documenting the presence of a specific learning disability (SLD)
Evidence of high risk for both false positive (over identification) and false negative (under identification) results
Research evidence that math calculation fluency is highly correlated with future math achievement
Response to Intervention/Early Intervention Models are recommended instead of IQ Discrepancy “Wait to Fail” Model
Identification of the need for research based math interventions can be identified with CBM Benchmarks

**26. **© C. Lee Goss, 2008 26 Locating Research-Based Math Curricula Information about the efficacy of specific curricula can be found in peer-reviewed research journals and specific internet resources:
ERIC is a federally-sponsored database of research in education
http://www.eric.ed.gov/
National Math Advisory Panel Report
www.nationalmathpanel.org
What Works Clearing House is a new initiative of the Institute of Education Sciences developed to share information about effective programs
http://ies.ed.gov/ncee/wwc/

**27. **© C. Lee Goss, 2008 27 FAQ: Where is the List for RBIs? There is not a list of research-based instruction or interventions
There are credible research resources
RBI Goal: research studies (scientific evidence) that document efficacy of instruction or intervention in multiple peer-reviewed journals
Although more research is needed to inform effective math instruction, current research is consistent with NMAP findings

**28. **© C. Lee Goss, 2008 28 Caveat Emptor Buyer beware!
Not all programs or curricula advertised as scientifically-based will actually have adequate data to support their use
There is no “list” of approved programs so teachers and other educational personnel must be informed consumers and determine whether materials meet the standards of effective research-based curricula

**29. **© C. Lee Goss, 2008 29 Systematic & Explicit Math Instruction Strategic Skill Lesson
Review (Activate Prior Knowledge)
Introduce New Skill
Best Practice Instruction
I do it, We do it, You do it, Immediate Feedback
Application to Controlled Skill Practice
Teach to Mastery
Generalization of Skill

**30. **© C. Lee Goss, 2008 30

**31. **© C. Lee Goss, 2008 31
Calculators: However, the Panel’s survey of the nation’s algebra teachers indicated that the use of calculators in prior grades was one of their concerns. The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected.
Calculators: However, the Panel’s survey of the nation’s algebra teachers indicated that the use of calculators in prior grades was one of their concerns. The Panel cautions that to the degree that calculators impede the development of automaticity, fluency in computation will be adversely affected.

**32. **© C. Lee Goss, 2008 32 RTI: Math @ Secondary Level RTI: Mathematics is not as advanced as reading
Challenges at secondary level:
Math becomes much more specialized in terms of content
Graduation requirements and state assessments are aligned to particular math courses (e.g., Algebra 1, Geometry 1, etc.)
- Application of RTI in mathematics at the secondary level is in the development stage

**33. **© C. Lee Goss, 2008 33 RTI: Math @ Secondary Level, Cont’d RTI IS happening at the secondary level
Exs: Minnesota, Colorado
Systematically applied and evaluated models are in the emerging stage
Exs: Minnesota Example
The Good News:
There is a research base to apply the RTI tiered model of instruction in math

**34. **© C. Lee Goss, 2008 34 Core Components of Math Instruction Core components essential to mathematics skill development
Number/number sense appears to be most critical for K-12 math success
Math Calculation Fluency
Math Calculation Fluency is highly correlated with mathematics achievement and success in Algebra
Lack of number sense and math calculation fluency seems to be a consistent issue for students who fail Algebra at the secondary level

**35. **© C. Lee Goss, 2008 35 How Math Calculation Fluency Impacts Comprehension Fluency allows the mind to concentrate on comprehension
Automatic math calculation fluency leaves working memory free to think about application and problem solving strategies

**36. **© C. Lee Goss, 2008 36 Research-based Instructional Practices Explicit systematic instruction within authentic contexts
Teaching strategies for learning and doing mathematics including use of graphic organizers
Grounding abstract concepts within concrete experiences (concrete-representational-abstract sequence of instruction)

**37. **© C. Lee Goss, 2008 37 Research-Based Instructional Practices, Cont’d 4) Providing multiple opportunities for students to apply their mathematical understanding with immediate corrective feedback
Newly learned concepts and
Maintenance/Judicious Review of previously learned concepts
5) Continuous progress monitoring/instructional decision-making
Key Point: These practices can be applied in the general education classroom in whole class, small group, and one-to-one situations.

**38. **© C. Lee Goss, 2008 38 Research on Effective Mathematics Instruction Research synthesis on effective mathematics instruction for students with mathematics difficulties: Gersten, Baker, & Chard (2006). Effective instructional practices for students with difficulties in mathematics: Findings from a research synthesis.
Center on Instruction www.centeroninstruction.org

**39. **© C. Lee Goss, 2008 39 Best Practice Instruction Process Explicit, Sequential, Scaffolded Instruction
Model: I do it
Rehearsal: We do it together
Practice: You do it (class/peer support)
w/ multiple opportunities for repetition & practice when learning skills w/ immediate corrective feedback
Mastery: Evidence that you can perform the skill w/ automaticity on your own
Generalization: Evidence that you can perform the skill w/ automaticity in different contexts/environments
Judicious review of prior skills taught to mastery

**40. **© C. Lee Goss, 2008 40 Importance of Immediate Corrective Feedback Error correction is a key feature of effective instruction
Error correction is crucial to effective and efficient learning
If errors are identified and corrected immediately in the learning sequence, students master skills more quickly with less frustration

**41. **© C. Lee Goss, 2008 41 Tier 1: Research-Based Math Instruction What Works Clearinghouse (WWC) http://ies.ed.gov/ncee/wwc/
Elementary School Math
Middle School Math
Notice the nonexistent research reviews available to date at the high school level
Goal of website to locate and summarize information about instructional methods and materials for teachers
Includes Summary Tables of findings about curricula and instruction for many subject areas
A common table legend indicates the trend of evidence for each program
Click on the title of the program for detailed information about each program reviewed

**42. **© C. Lee Goss, 2008 42 Tier 1: Small Group Research Activity Break into small groups by grade level/current math curricula
Log on to What Works Clearinghouse (WWC)
http://ies.ed.gov/ncee/wwc/
Identify your current math curricula or pick a curricula to review
Review & Summarize research evidence to share w/ the group
Click on the title of the program to summarize the research for specific grade levels and current effectiveness ratings

**43. **© C. Lee Goss, 2008 43 Tier 1: Core Class Instruction

**44. **© C. Lee Goss, 2008 44 Tier 2: Supplemental Math Interventions Tier 2 = Tier 1 + Tier 2
Small Group Additional Instruction
3-5 students (homogeneous skill groups) identifed for targeted skill instruction matched to student needs
2-5 days/week based on individual student needs and progress monitoring data

**45. **© C. Lee Goss, 2008 45 Tier 2: Math Intervention Researh Evidence Baker, S., Gerston, R., & Lee, D.S. (2002). Elementary School Journal.
Baker, et. al. reviewed all available studies (15) which investigated the effects of math interventions for low-achieving students that were NOT receiving special education services.
Key feature of this research is that it included students in need of Tier 2 interventions.
On-going math research is still needed; however, results from previous research offer a great entry point for planning Tier 2 interventions.

**46. **© C. Lee Goss, 2008 46 Tier 2: Math Intervention 6 Key Principles Math interventions at Tier 2 level must incorporate 6 instructional principles:
Instructional Explicitness
Instructional design that eases the learning challenge
A strong conceptual basis for procedures that are taught
An emphasis on drill and practice
Cumulative review as part of drill and practice
Motivators to help students regulate their attention and behavior and to work hard
(Lynn Fuchs, Vanderbilt University)

**47. **© C. Lee Goss, 2008 47 1) Instructional Explicitness Typicially developing children profit from Tier 1 instruction even though it may require inductive reasoning.
Students at-risk for serious mathematics difficulties, however, fail to profit from inductive instructional methods in a way that produces understanding of the structure, meaning, and operational requirements of mathematics.
(Fuchs & Fuchs, Vanderbilt University)

**48. **© C. Lee Goss, 2008 48 Instructional Explicitness, Cont’d A meta-analysis of 58 math studies (Kroesbergen & Van Luit, 2003) revealed that students with math disability benefit more from explicit instruction than from discovery-oriented methods.
Tier 2 application: Effective intervention in Tier 2 requires an explicit, didactic form of instruction where the teacher directly shares the information the students need to learn.
(Fuchs & Fuchs, 2008)

**49. **© C. Lee Goss, 2008 49 Explicit Instruction Research shows that when additional highly explicit and sequenced math instruction was provided for Tier 2 at-risk students, they made significant gains in math skills (Baker et. al., 2002)
Ex: Student who has struggled to master the steps in solving word problems needs additional instruction and practice in how to “translate” words into numerical equations

**50. **© C. Lee Goss, 2008 50 2) Instructional Design Explicitness is critical but not enough
Effective instruction also requires careful organization and design
Goal: To anticipate and eliminate misunderstandings through precise explanations and carefully sequenced and integrated instruction
Tier 2 application: To close the achievement gap as quickly as possible

**51. **© C. Lee Goss, 2008 51 Instructional Design, Cont’d Tier 2 application: To close the achievement gap as quickly as possible
Key for Mathematics due to the many branches and strands with distinct conceptual and procedural demands
Instructional efficiency is critical

**52. **© C. Lee Goss, 2008 52 Instructional Design, Cont’d Careful instructional design begins by teaching a set of foundational skills the student can apply across the entire program:
Multi-digit calculations
Solving algebraic equations
Checking Work
Such foundational skills can be taught as intact instructional targets and then applied efficiently across subsequent units along with word problem instruction.
Overall goal: To purposely conceptualize, organize and teach students to recognize problem types so student can recognize novel problems as familiar vs. random

**53. **© C. Lee Goss, 2008 53 Instructional Design Goals To purposely conceptualize, organize and teach students to recognize problem types so student can recognize novel problems as familiar vs. random
Generalization within problem type structure so the student can dismiss irrelevant information, find missing information (e.g., solve equations), and find relevant information in charts and graphs with efficiency and predictability

**54. **© C. Lee Goss, 2008 54 3) Conceptual Basis for Procedures Taught Effective instruction requires a strong conceptual basis for the procedures that are taught—Why?
Neglect in emphasis on the concepts for mathematical procedures can result in confusion, learning gaps, and a failure to maintain and integrate previously mastered content

**55. **© C. Lee Goss, 2008 55 4) Emphasis on Drill and Practice Drill and Practice should include:
Practice in sorting problems into problem types
Mixing problem types within the daily lesson
Daily Review

**56. **© C. Lee Goss, 2008 56 5) Cumulative Review Practice should be rich in cumulative review
Cumulative Review includes:
Continual reliance and practice of foundational math skills
Use of mixed problem types
Sorting practice
Paper & Pencil Review

**57. **© C. Lee Goss, 2008 57 6) Motivation Motivators help students regulate their attention and behavior
Motivators help students continue to work hard on difficult tasks
Students at-risk for poor math achievement often display attention, motivation, and self-regulation difficulties that may adversely affect their behavior and learning (Fuchs et. al., 2006; Montague, 2007)

**58. **© C. Lee Goss, 2008 58 Motivation, Cont’d Tier 2 application:
At-Risk students have often experienced failure in mathematics achievement, particularly at the secondary level, which may contribute to a lack of motivation or avoidance behaviors
Learning may be adversely impacted for fear of failure
For effective instruction, Tier 2 interventions must incorporate systematic self-regulation and motivators, and for many students tangible reinforces are required
(Fuchs et. al., 2003)

**59. **© C. Lee Goss, 2008 59 Intervention Central Math Practice Worksheet Generator Go to the following website: www.interventioncentral.org
In the section labeled “Online Tools” click on “Math Worksheet Generator” on the right hand side of the page.
When the new math page loads, click on the button labeled “mixed skills”.
Click on the specific skill(s) you want included in your practice worksheet.

**60. **© C. Lee Goss, 2008 60 Intervention Central: Math Practice Worksheet Generator, Cont’d At the bottom of the page choose how many columns, rows, and what size font you want students to use.
Click on “Single Skill Computation Probe.”
Once you click on the button to create the probe, your browser should create and load a new tab or page with the teacher (answer) key first. Once this page is loaded, click on the text at the very top of the page which says “click for student worksheet.”
Another new tab or page will load. This is the one you want to print for the students. Either print as many copies as you need for the students, or print one copy and then make enough photocopies.
Print a copy of the answer sheet onto an overhead transparency.
(Brown-Chidsey, Bronaugh, & McGraw, 2008)

**61. **© C. Lee Goss, 2008 61 “Interspersing Easy and Hard Math Problems” Purpose: To provide practice opportunities solving known
and new problems
Research: McDonald & Ardoin, 2007
Ingredients: Easy and difficult math problems
Prep Time: 5-10 minutes
Activity time: 10 minutes per worksheet

**62. **© C. Lee Goss, 2008 62 Intervention Central: Mixed Skill Math Practice Worksheet Generator Identify what computation skills students have mastered and what ones the students need to practice in order to master (e.g., addition with re-grouping, subtraction without re-grouping).
Go to the following website: www.interventioncentral.org
In the section labeled “Online Tools” click on “Math Worksheet Generator” on the right hand side of the page.
When the new math page loads, click on the button labeled “mixed skills”.
Click on the specific skill(s) you want included in your practice worksheet.

**63. **© C. Lee Goss, 2008 63 Intervention Central: Mixed Skill Math Practice Worksheet Generator, Cont’d At the bottom of the page choose how many columns, rows, and what size font you want students to use.
Click on “Single Skill Computation Probe.”
Once you click on the button to create the probe, your browser should create and load a new tab or page with the teacher (answer) key first. Once this page is loaded, click on the text at the very top of the page which says “click for student worksheet.”
Another new tab or page will load. This is the one you want to print for the students. Either print as many copies as you need for the students, or print one copy and then make enough photocopies.
Print a copy of the answer sheet onto an overhead transparency

**64. **© C. Lee Goss, 2008 64 Timed Practice, Practice, Practice Purpose: To help students build fluency with basic computation skills
Research: The National Mathematics Advisory Panel, 2008
Ingredients: Computation problems matched to students’ independent practice level
Timer or stopwatch
Graph on which students can enter the number of correct answers (attached)
Prep Time: 5-10 minutes
Activity Time: 2 minutes per worksheet

**65. **© C. Lee Goss, 2008 65 Timed Practice, Practice, Practice Preparation:
Identify what computation skill students most need to practice for the purpose of developing fluency (i.e., accuracy plus speed).
Go to the following website: www.interventioncentral.org
In the section labeled “Online Tools” click on “Math Worksheet Generator” on the right hand side of the page.
When the new math page loads, click on the computation skill(s) you want students to practice (e.g., addition, subtraction, multiplication, division, mixed skills).
Click on the specific skill(s) you want included in your practice worksheet.

**66. **© C. Lee Goss, 2008 66 Timed Practice, Practice, Practice, Cont’d At the bottom of the page choose 4 columns, 5 rows, and medium size font. For the timed version of this activity, it is important that the same number of items be used for each timed session.
Click on “Single Skill Computation Probe.”
Once you click on the button to create the probe, your browser should create and load a new tab or page with the teacher (answer) key first. Once this page is loaded, click on the text at the very top of the page which says “click for student worksheet.”
Another new tab or page will load. This is the one you want to print for the students. Either print as many copies as you need for the students, or print one copy and then make enough photocopies.
Print a copy of the answer sheet onto an overhead transparency (optional).

**67. **© C. Lee Goss, 2008 67 Timed Practice Activity Instructions
Say to students: “TODAY WE ARE GOING TO PRACTICE _____________. I WANT YOU TO DO THE PROBLEMS AS ACCURATELY AND AS FAST AS YOU CAN. YOU WILL HAVE 2 MINUTES TO FINISH THE PAPER I AM HANDING OUT.
Hand out worksheet and say to students: “DO NOT START UNTILI SAY BEGIN.”
Once all students have their worksheets, say: “BEGIN.”
Begin your timer and give students 2 minutes to work on the problems. At the end of 2 minutes, say to students: “STOP. PASS YOU WORKSHEET TO A CLASSMATE. I WILL PUT THE CORRECT ANSWERS ON THE OVERHEAD – or – READ THEM OUT LOUD. MARK YOUR CLASSMATES’ ANSWER CORRECT OR INCORRECT. WRITE THE NUMBER OF CORRECT ANSWERS ON THE TOP OF THE PAGE.”

**68. **© C. Lee Goss, 2008 68 Timed Practice Activity Instructions, Cont’d When finished correcting answers say to students: “PASS THE WORK SHEET BACK TO ITS OWNER. TAKE OUT YOUR MATH GRAPH AND FILL IN THE NUMBER CORRECT FOR TODAY. BE SURE TO WRITE TODAY’S DATE AT THE BOTTOM OF THE GRAPH. WHEN YOU ARE DONE, PUT YOUR GRAPH AWAY.”
Say: “NOW, LOOK AT YOUR CORRECT AND INCORRECT ANSWERS. FIX THE ONES YOU GOT WRONG.”
Collect student worksheets and graphs.

**69. **© C. Lee Goss, 2008 69 Tier 2: Small Group Instruction/Intervention

**70. **© C. Lee Goss, 2008 70 Tier 3: Individualized Math Interventions
Flashcards
Flashcards w/ Folding in New Items
Cover, Copy, Compare
Great Leaps/Computer Practice
More research is needed
Presently, Tier 3 interventions typically involve more intensive versions of methods used at Tier 2.

**71. **© C. Lee Goss, 2008 71 Flashcards Purpose: To build student fluency with computation facts
Research: Cates, 2005; Shapiro, 2004
Ingredients: Blank 3” x 5” flashcards
Sets of math computation problems
Marker
Prep Time: 20 minutes
Activity Time: 10 minutes

**72. **© C. Lee Goss, 2008 72 Flashcards, Cont’d Preparation:
Identify sets of 10 math computation problems that the student needs to master
Write one math problem on one side of each card.
Write the answer to each problem on the other side of the cards.

**73. **© C. Lee Goss, 2008 73 Flashcards w/ Folding in New Items Purpose: To build and maintain students’ fluency with computation facts
Research Base: Shapiro, 2004
Ingredients: Blank 3” x 5” flashcards
Sets of known and new math computation problems
Marker
Prep Time: 20 minutes
Activity time: 10 minutes

**74. **© C. Lee Goss, 2008 74 Flashcards w/ Folding in New Items, Cont’d Preparation:
See Flashcards
Once the student has mastered all of the items in a set of 10 problems, create an additional set of flashcards with new math problems. Write one new math problem on one side of each card; write the answer to each new problem on the other side of the cards.
Shuffle the set of mastered problems and remove 2 randomly selected cards; replace the removed cards with 2 of the new (unknown) problem cards.

**75. **© C. Lee Goss, 2008 75 Cover, Copy, Compare Purpose: To provide students with errorless practice of computation skills
Research base: Shapiro, 2004
Ingredients: Computation problems matched to students’ instructional learning level
Pencil
Prep Time: 5-10 minutes
Activity Time: 5 minutes per worksheet

**76. **© C. Lee Goss, 2008 76 Cover, Copy, Compare, Cont’d Preparation:
Identify what computation skill students most need to learn (e.g., addition with grouping, multiplication, etc.)
Go to the following website: www.interventioncentral.org
In the section labeled “Online Tools” click on “Math Worksheet Generator” on the right hand side of the page.
When the new math page loads, click on the computation skill you want students to practice.
Click on the specific skill you want included in your cover-copy-compare worksheet.
At the bottom of the page select the number of columns, rows, and font size appropriate for the student.
Click on “Cover-Copy-Compare Worksheet.”
Once you click on the button to create the probe, your browser should create and load a new tab or page with the student worksheet.
Either print as many copies as you need for students, or print one copy and then make enough photocopies.
Print a copy of the answer sheet.

**77. **© C. Lee Goss, 2008 77 Tier 3: Intensive Instruction/Intervention

**78. **© C. Lee Goss, 2008 78 Research-Based Math: RTI 3-Tier Model RTI model allows for use of standard assessment practices as well as individual problem-solving.
RTI helps to build a bridge between general and special education
Focus is on exit as much as entrance.
Matches our belief about education for all children.

**79. **© C. Lee Goss, 2008 79

**80. **© C. Lee Goss, 2008 80 RTI 3-Tier Model Process

**81. **Example of Tier Level Interventions

**82. **© C. Lee Goss, 2008 82 Minnesota Example of RTI: Mathematics Break into small groups by schools/grade levels
Review Example of RTI: Mathematics at the high school level in Minnesota
Identify RTI methods for screening, instruction/intervention, and continuous progress monitoring
Identify existing or similar implementation opportunities at your school
Identify challenges for implementation at your school
Summarize Lessons Learned

**83. **© C. Lee Goss, 2008 83 Tier 1: Core Research-Based Class Instruction Critical Elements
Universal Benchmark Screening and identification of skill level and at-risk students
Direct and Explicit instruction and in-class support matched to student skill levels
Incorporates current research findings (e.g., Adding it Up: 5 strands and NMAP, 2008 research-based instruction components)
Treatment Integrity
On-going professional development

**84. **© C. Lee Goss, 2008 84 Strategic Intervention Highly strategic/targeted interventions vs. comprehensive (everything)
“All paths lead to nowhere”
Time is critical
Focus on identified priority skills

**85. **© C. Lee Goss, 2008 85 Strategic Intervention Outcomes At-risk students have heterogeneous skills
Screening and frequent progress monitoring benefits and informs skill groups and instruction
Gains are specific to skill match to instruction
Evidence of increased outcomes when students monitor their academic progress, set goals, and chart results toward benchmark goals

**86. **© C. Lee Goss, 2008 86 Treatment Integrity Treatment Integrity is critical to determination of response to intervention
Fidelity means adherence to the parts of the curriculum/program and procedures
Fidelity increases the likelihood the curriculum/program will be successful
Fidelity and Creativity are NOT mutually exclusive.
Treatment Integrity should be documented
Teacher Self-Report Check Lists & Teacher Logs of Instruction/Intervention Components and Student Response

**87. **© C. Lee Goss, 2008 87 What Math CBMs Offer CBMs in Math provide data about student proficiency with math calculation skills that have been shown to be predictive of later math achievement
Other math instruction elements are also important and should also be monitored

**88. **© C. Lee Goss, 2008 88 Math CBM Resources @ Secondary Level System to Enhance Educational Performance (K-12): www.isteep.com
Intervention Central (Free): www.interventioncentral.org
AIMSweb (K-8): www.aimsweb.com
www.studentprogress.org

**89. **© C. Lee Goss, 2008 89 Interpreting CBM Scores Each CBM activity generates a score specific to an important math skill
Allows teachers to identify the instructional area(s) needed
Progress monitoring allows review of whether the instruction is working

**90. **© C. Lee Goss, 2008 90 What Fluency Measures Fluency = Timed
Fluency = Speed + Accuracy
Research has shown that BOTH rate of speed and accuracy of math calculations are important for successful math achievement
Each CBM task is a fluency measure because it allows for comparisons between students over time of skill automaticity
Requires standardized administrations for each student for meaningful data and progress monitoring

**91. **© C. Lee Goss, 2008 91 Advantages of Charting CBM Progress Data Small gains are visible
Steady growth over time is visible
Student is motivated by visible progress
If progress is not occurring teacher can respond quickly to modify instruction
A clear benchmark goal is in sight

**92. **© C. Lee Goss, 2008 92 Benefits of CBMs Reliability and validity are established.
Tests are efficient and economical.
Scoring interpretation and record-keeping can be done by computer.
Repeated assessments do not spoil the results.
Subtest content is research-supported.
End-of-year achievement predicted by CBM score.
Instructional goals for each grade are established.
Decision making for individual children is facilitated.
Progress Monitoring possible w/ programs and curriculum

**93. **© C. Lee Goss, 2008 93 Progress Monitoring All interventions should be evaluated with regular progress monitoring
Weekly assessments should be done with CBMs related to the skill of greatest concern to check progress
Performance should be graphed to show whether student is improving
If no progress, change instruction

**94. **© C. Lee Goss, 2008 94 Components of Instruction The instruction should match the student’s area(s) of learning needs
Some interventions will cover more than one skill, but others are more targeted
If a student moves forward in one area but not another, provide instruction in the area of greatest need

**95. **© C. Lee Goss, 2008 95 Changing Instruction/Intervention If a student does not show progress, a different instructional method or intervention is needed
Changes can include changing the intensity, duration, or frequency of instruction
An entirely different intervention may be needed

**96. **© C. Lee Goss, 2008 96 Potential Benefits of RTI Research has shown that students performing below grade level in reading, writing or mathematics benefit from the increased attention to instructional interventions and progress monitoring
Students who have LD that have gone undetected in elementary school stand a better chance of being identified for special education services.
RTI takes the focus off individual student deficits and refocuses attention on the interaction between teaching and learning.
RTI has the potential to reduce the number of students incorrectly identified as having LD when they may be struggling due to cultural differences or poor instruction (Cortiella, 2005).

**97. **© C. Lee Goss, 2008 97 Longitudinal Intervention Research

**98. **© C. Lee Goss, 2008 98 Longitudinal Intervention Research: Best-Practice Grouping students for instruction based on student skill
Monitoring their progress over small periods of time
Adjusting instruction based on the data
And providing students feedback on their performance is one of the most powerful sets of educational practices that exists.
(Reschly & Tilly, 2006)

**99. **© C. Lee Goss, 2008 99 Big Ideas to Take Home Although there is less research about math interventions, there are studies which indicate specific interventions can be effective for most students.
Tier 1, direct instruction has been found to be very effective. For some students, Tier 1 alone is not enough, and many students can benefit from Tier 2 interventions.
Three main Tier 2 math intervention components have been shown to be most effective:
1) peer-teaching
2) explicit instruction
3) formative assessment and feedback for students and teachers

**100. **© C. Lee Goss, 2008 100 Big Ideas to Take Home, Cont’d Current research evidence that computation fluency is an important aspect of overall math skill development.
For students who still struggle after participating in Tier 2 instruction, diagnostic teaching and evaluation at Tier 3 can be a way to identify the specific nature of their math difficulties.
Match specific interventions to student skill levels and needs

**101. **© C. Lee Goss, 2008 101 Big Ideas to Take Home, Cont’d CBM benchmarks can be used to identify which students are at risk for math problems
CBMs are fast and effective for identifying student’s math skills
Instruction/Intervention should be matched to student current skill level
Early intervention for at-risk population is critical

**102. **© C. Lee Goss, 2008 102 Big Ideas to Take Home, Cont’d Select interventions on the basis of student’s CBM performance confirmed with other data
CBMs can be used to monitor a student’s progress with implemented math intervention
Progress data that shows little or no improvement means that a new, or more intensive, intervention is needed

**103. **© C. Lee Goss, 2008 103 Website Resources for Math Instruction
Designing Effective Mathematics Instruction: A Direct Instruction Approach by: Marcy Stein, Diane Kinder, Jerry Silbert, and Douglas W. Carnine. Website: http://www.prenhall.com/stein/
Intervention Central Website: http://www.interventioncentral.org
National Center on Student Progress Monitoring:
http://www.studentprogress.org/
National Math Advisory Panel Website: http://www.ed.gov/about/bdscomm/list/mathpanel/index.html
RTI Action Network Website: http://www.rtinetwork.org/
System to Enhance Educational Performance Website: http://www.isteep.com/
What Works Clearinghouse (WWC) Website: http://ies.ed.gov/ncee/wwc/

**104. **© C. Lee Goss, 2008 104 References Baker, Scott, Gerston, Russ, & Lee, D. S. A synthesis of empirical research on teaching mathematics to low-achieving students. Elementary School Journal, 103, 51-73, 2002.
Brown-Chidsey, Rachel, Bronaugh, Louise, & McGraw, Kelly. RTI in the classroom: Guidelines and recipes for success. New York: Guilford Press, 2008.
Carnine, Doug. Instructional design in mathematics for students with learning disabilities. Journal of Learning Disabilities, 30, 130-141,1997.
Cates, Gary L. Effects of peer versus computer-assisted drill on mathematics response rates. Psychology in the Schools, 42, 637-646, 2005.
Intervention Central. Math worksheet generator. Retrieved 4 August 2008 from: http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php, 2008.
Kilpatrick, Jeremy, Swafford, Jane, & Findell, Bradford (Eds.). Adding it Up: Helping Children Learn Mathematics. Washington, DC: National Research Council, 2001.

**105. **© C. Lee Goss, 2008 105 References, Cont’d McDonald, Elizabeth, & Ardoin, Scott. Interspersing easy math problems among challenging problems: detection of interspersal effects in whole-class applications. Journal of Behavioral Education, 16, 342-354, 2007.
National Mathematics Advisory Panel. Foundations for success [Final report]. Washington, DC: U.S. Department of Education, 2008.
Peer Assisted Learning Strategies. [Website]. Retrieved 4 August 2008 from: http://kc.vanderbilt.edu/pals/, 2008.
Shapiro, Edward S. Academic skills problems: Direct assessment and intervention (3rd. Ed.). New York: Guilford, 2004.
Simon, Rebecca, & Hanrahan, James. An evaluation of the Touch Math method for teaching addition to students with learning disabilities in mathematics. European Journal of Special Needs Education, 19, 191-209, 2004.
Stein, Marcy, Kinder, Diane, Silbert, Jerry, Carnine, Doug. W. Designing Effective Mathematics Instruction: A Direct Instruction Approach (4th Ed.). Upper Saddle River, NJ: Pearson-Prentice Hall, 2006.