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RTI - Mathematics: What do we know and where do we go from here?. Ben Clarke, Ph.D. Scott Baker. Ph.D. Pacific Institutes for Research . Increasing recognition of the importance of mathematical knowledge.

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Rti mathematics what do we know and where do we go from here l.jpg

RTI - Mathematics: What do we know and where do we go from here?

Ben Clarke, Ph.D.

Scott Baker. Ph.D.

Pacific Institutes for Research

Increasing recognition of the importance of mathematical knowledge l.jpg
Increasing recognition of the importance of mathematical knowledge

  • “For people to participate fully in society, they must know basic mathematics. Citizens who cannot reason mathematically are cut off from whole realms of human endeavor. Innumeracy deprives them not only of opportunity but also of competence in everyday tasks”. (Adding it Up, 2001)

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State of Mathematics knowledge

  • Achievement on the NAEP trending upward for 4th/8th grade and steady for 12th grade

    • Large numbers of students still lacking proficient skills

    • Persistent income and ethnicity gaps

    • Drop in achievement at the time algebra instruction begins

  • TIMS data indicate significant lower levels of achievement between US and other nations

    • Gap increase over time

  • Jobs requiring intensive mathematics and science knowledge will outpace job growth 3:1 (STEM) and everyday work will require greater mathematical understanding

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High Level of Interest in Mathematics Achievement knowledge

  • National Mathematics Advisory Report

  • National Council Teachers of Mathematics: Focal Points

  • National Research Council: Adding it Up

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Response to Intervention knowledge

  • Reauthorization of IDEA (2004) allowed for RTI to be included as a component in special education evaluations

  • Premised on the use of research based interventions and student response to intervention

    • Students who respond are not identified as learning disabled

    • Students who do not respond are referred for a complete evaluation and potential identification as learning disabled

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Response to Intervention knowledge

  • Linked closely to an early identification and prevention model of delivery

  • Provides for the delivery of tiered services across traditional boundaries (e.g.Special and General Education)

  • Most often implemented by schools using a schoolwide model of instruction…..

    • In Reading, but what about Math!

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Paucity of Research knowledge

  • A lit search for studies on reading disabilities studies and math disability studies from 1996-2005 found over 600 studies in the area of reading and less than 50 for mathematics (14:1)

  • Meta analysis conducted in the areas of low achievement and learning disabilties in mathematics consisted of 15, 38, and 58 studies - in comparison to the large number of studies that formed the basis of recommendations for the National Reading Panel

  • Specific RTI mathematics studies for a recent annotated bibliography totaled 9 studies

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Broad Issues to Consider (RTI) knowledge

  • Levels of Support / LD identification process

  • Standard prototcol / Problem Solving

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What is Needed for RTI knowledge

  • Primary:

    • Valid system for screening.

    • System for progress monitoring.

    • An array of evidence-base intervention or at least promising interventions for beginning Tier 2 students.

  • Secondary:

    • Diagnostic assessments

    • Core instructional program

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Tier 1: Components knowledge

  • Screening instruments

  • Core Curriculum based on expert judgment

  • Enhancements to the core curriculum

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Screening knowledge

  • Measures used in screening vary from short duration fluency measures to in-depth measures.

    • Short duration: Early Numeracy CBM; CBM computational and conceptual probes.

    • Number Knowledge Test (approx. 15 min).

    • Math specific tests such as TEMA, Key Math - also used in diagnostic testing.

  • Goal is to make accurate predictions about who needs and who does not need additional services.

  • Must balance efficiency of screening process with goal of accurate predictions.

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4 1 knowledge

5 11

9 4

Measures for Screening

  • Early Grades

    • Short duration fluency measures.

      • E.g. Early Magnitude Comparison

      • Missing Number in a series (strategic counting) e.g.. 7,_9…… X, 10,11

    • Robust Indicators….. But only for one year.

    • For long term prediction: working memory, PA measures show promise (E.g. reverse digit span)…

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Example: Number Knowledge Test knowledge

  • Level 1

    • If you had 4 chocolates and someone gave you 3 more, how many chocolates would you have?

    • Which is bigger: 5 or 4?

  • Level 3

    • What number comes 9 after 999?

    • Which difference is smaller: the difference between 48 and 36 or the difference between 84 and 73?

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Upper Grades Screening knowledge

  • Algebra measures

    • Designed by Foegen and colleagues assess pre-algebra and basic algebra skills. Administered and scored similar to Math-CBM

  • Math CBM Computation and Concepts and Applications

    • Concepts and Applications showed greater valdity in 6th, 7th, and 8th grade

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Core curriculum knowledge

  • National Math Panel

    • Need to develop understanding and mastery of

      • Whole number: understand place value, compose/decompose numbers, meaning of operations, algorithms and automaticity with facts, apply to problem solving, use/knowledge of commutative, associative, and distributive properties,

      • Rational number: locate +/- fractions on number line, represent/compare fractions, decimals percents, sums, differences products and quotients of fractions are fractions, understand relationship between fractions, decimals, and percents, understand fractions as rates, proportionality, and probability, computational facility

      • Critical aspects of geometry and measurement:similar triangles, slope of straight line/linear functions, analyze properties of two and three dimensional shapes and determine perimeter, area, volume, and surface area

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Less is More knowledge

  • US curricula tend to cover more topics with less depth resulting in persistent review across grades versus closure after exposure, development and refinement.

  • NCTM Focal Points

    • Emphasize critical topics at each grade level

      • (e.g. 2nd grade)

    • Still contains more non-arithmetic coverage than TIMSS

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Example Tier 1 intervention: VanDerHayden et al. (STEEP) knowledge

  • Entire class is screened on a computation probe

  • If class is below criterion established by Deno, then entire class receives Tier 1 intervention (i.e. practice in computation and facts for 30 min daily)

  • Students who do not respond to the Tier 1 intervention are provided a similar Tier 2 intervention consisting of peer tutoring on computation problems

  • Limited scope and duration of the Tier 1 intervention

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Tier 2 and 3 Components knowledge

  • Progress monitoring and diagnostic assessments

  • Standard protocol interventions

  • Instructional design considerations

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Progress Monitoring Assessments knowledge

  • Progress monitoring measures:

    • Some screening measures have promise as General Outcome Measures but need more research to also be used as progress monitoring measures.

    • Well researched progress monitoring measures are available for grades one and up.

    • These possess weaker criterion-related validity than reading measures. (Foegen et al, in press)

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Diagnostic Assessments knowledge

  • Deciding when to use

    • Prior to intervention or

    • If the intervention is not successful

  • Sources

    • Compilation of data from progress monitoring

    • Curriculum-Based Assessment (e.g. Howell)

    • In-depth math specific measure (e.g. TEMA)

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Tier 2/3 (Interventions) knowledge

  • Standard protocol approach

  • Students maintain in the program for a set duration of time

    • I.e. Progress monitoring data collected but not used for educational decision-making

  • Limit scope to critical topics

    • Avoid low grade dose of the same material and same approach

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Intervention Content knowledge

  • Wu and Milgram (CA standards) for at-risk 4th to 7th grade students

  • Recommend 2 hours of instruction per day

  • Taught by teacher with content knowledge expertise

  • Topics

    • Place Value and Basic Number Skills (1st-3rd grade skills)

    • Fractions and Decimals

    • Ratios, Rates, Percents, and Proportions

    • The Core Processes of Mathematics

    • Functions and Equations

    • Measurement

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Example: Fuchs 1st Grade Small Group Tutoring knowledge

  • 41 1st-grade teachers in 6 Title 1 and 4 non-Title 1 schools (92% consented students)

  • Conducted weekly CBM Computation

  • AR: Using Week 4 CBM Computation, 139 lowest performing (21% of 667 consented students); randomly assigned to control or tutoring

  • NAR: 528 remaining students with consent

  • Of 528 NAR:

    • All weekly CBM Computation

    • 180 sampled for individual and group pre/posttesting

    • 348 group pre/posttested

  • With attrition, samples sizes of:

    • 127 AR: 63 control + 64 tutored

    • 437 NAR: 145 individually/group tested + 292 group tested

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Tutoring knowledge

  • Small groups (11 groups of two students and 16 groups of three students)

  • 3 times per week outside classrooms

  • Each session:

    • 30 min of tutor-led instruction

    • 10 min of student use of practice to improve automatic retrieval of math facts

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Tutor-Led Instruction knowledge

  • Concrete-representational-abstract model, which relies on concrete objects to promote conceptual understanding (e.g., base-10 blocks for place value instruction)

  • 17 scripted topics addressing number concepts, numeration, computation, story problems (e.g., not geometry, measurement, charts/figures, money)

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Flash Card Activity knowledge

  • Final 10 minutes of each session devoted to practice to improve retrieval of math facts

  • Students individually presented basic addition and subtraction problems and earn points for correct answers

  • Students receive additional practice for incorrect answers

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17 Scripted Topics knowledge

  • Identifying and Writing Numbers

  • Identifying More & Less Objects

  • Sequencing Numbers

  • Using <, >, and =

  • Skip Counting by 10s, 5s, and 2s

  • Introduction to Place Value

  • Place Value

  • Identifying Operations

  • Writing Addition and Subtraction Sentences

  • Place Value

  • Addition Facts

  • Subtraction Facts

  • Addition and Subtraction Facts Review

  • Place Value Review

  • 2-Digit Addition

  • 2-Digit Subtraction

  • Missing Addends

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Tutoring Efficacy knowledge

  • Improvement

  • Weekly CBM Computation Slope

    • AR tutored = NAR > AR control

  • WJ III Calculation

    • AR tutored > NAR and AR

  • Grade 1 Concepts/Applications

    • AR tutored > NAR and AR control

  • Story Problems

    • NAR > AR tutored > AR control

  • First-grade tutoring enhances outcomes.

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Tutoring Efficacy knowledge

  • Did tutoring decrease MD prevalence at end of 1st grade?

    Yes, across identification options, tutoring substantially decreased prevalence.

  • Example

    Final Low Achievement (<10th percentile) on Gr 1 Concepts/Applications, prevalence went from 9.75% without tutoring to 5.14% with tutoring.

    • ~ 2.5 million fewer children identified MD

    • At end of 2nd grade, MD prevalence was still twice as high in the untutored group.

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Tier 2/3 Instructional Design knowledge

  • Previous Syntheses on mathematics interventions

    • Not LD specific (Xin & Jitendra, 1999; Kroesbergen & Van Luit, 2003; Baker, Gersten, & Lee, 2002)

    • Organized on basis of dependent measure (Swanson, Hoskyn, & Lee, 1999)

  • Fuchs, Fuchs, Mathes, and Lipsey (2002) meta-analysis in reading: LD vs. Low Achieving students

    • LD lower performing than low achieving (d = .61)

    • Discrepancy between LD and low achieving greater on timed vs. untimed measures -- (automaticity / speed of processing implications)

    • LD vs. low achieving differences were greater when LD determinations based on objective measures vs. more subjective approaches (e.g., team decision making)

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Purpose knowledge

  • To synthesize experimental and quasi-experimental research on instructional approaches that enhance the mathematics performance of students with learning disabilities

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Method knowledge

  • Review of all published studies and dissertations between 1970 and 2003

  • Conduct meta-analysis to identify trends in the literature

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Findings knowledge

  • N of studies in meta-analysis: 38

  • N Intervention Categories: 8

    • 3 clusters of studies

  • N of Effect Sizes: 59

  • Range of Effect Sizes per category: 4 to 10

  • Effect size range: –0.43 to 2.96

  • Average Unweighted ES: –0.11 to 1.62

  • Average Weighted ES: –0.04 to 1.53

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Intervention Categories knowledge

  • Feedback to Teachers: N ES = 10

  • Feedback to students: N ES = 9

  • Visuals N ES = 9

    • (including graphic organizers and pictures)

  • Explicit Instruction: N ES = 9

  • Student Verbalizations: N ES = 5

  • Range and sequence of examples N ES = 5

  • Peer Assisted Instruction: N ES = 7

  • Computer Assisted Instruction: N ES = 5

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Providing Feedback to Teachers knowledge

  • Feedback provided to teachers on the status of student learning resulted in small effects

    • (10 ESs; d = .31*; range = –0.06 to 0.71)

  • Feedback provided includes specific information (e.g., instructional recommendations) along with feedback on student performance

    • (5 ESs; d = .29*; range = –0.06 to 0.71)

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Providing Feedback to Students about their Performance knowledge

  • Overall: Providing students with feedback about their performance

    • 9 ESs; d = 0.21*; range = –0.43 to 0.64

  • Without specific goals

    • 5 ESs; d = 0.33*; range = 0.04 to 0.64

  • With specific goals

    • 4 ESs; d = –0.18; range = –0.43 to 0.11

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Use of Visuals knowledge

  • Student use graphic representations to clarify or solve problems resulted in moderate effect sizes

    • 5 ESs; d = .56*; range = 0.32 to1.5

  • Teachers using pictures or diagrams to explain how to solve a problem resulted in moderate effects

    • 4 ESs; d = .55*; range = –0.38 to 1.15)

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Use of Visuals knowledge

  • Teacher had to use the visual representation during her initial teaching/demonstration OR

  • Students had to use visuals while solving the problem

    • Could not be an optional step

    • Visuals were used in solving 1-2 step arithmetic story problems

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Explicit Instruction knowledge

  • Explicit instruction used to teach a single skill resulted in large effects

    • 4 ESs; d = 1.72*; range = 0.88 to 2.49

  • Explicit instruction used to teach multiple related skills resulted in equally large effects

    • 5 ESs; d = 1.53*; range = 0.61 to 2.96

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Explicit Instruction knowledge

  • Teacher demonstrated a step-by-step plan (strategy) for solving the problem

  • This step-by-step plan had to be problem-specific and not just a generic, heuristic guide for solving problems

  • Students had to use the same procedure/steps shown by the teacher to solve the problem

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Student Verbalizations knowledge

  • Student use of verbalizations while solving problems resulted in large effects

    • 5 ESs; d = 1.25*; range = 0.23 to 2.49

  • In all studies students verbalized the solution while solving problems

  • In all but 1 study, focus was narrow -- e.g., single digit addition/subtraction; 1-2 step arithmetic story problems involving addition/subtraction

    • Students did not have to verbalize a range of solutions

    • In most complex verbalization study -- ES = 0.23

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Range and Sequence of Examples knowledge

  • Controlled range and sequence of instructional examples (e.g., Concrete – Representational – Abstract) resulted in moderate effects

    • 5 ESs; d = .53*; range = .12 to 1.15

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Peer Assisted Learning knowledge

  • Use of same age and cross age peers for learning and extended practice of math content resulted in moderate effects

    • 7 ESs; d = 0.42*; range = –0.27 to 1.19

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Computer Assisted Instruction knowledge

  • Student use of computers for activities, games, and practice in mathematics resulted in no effect

    • 5 ESs (3 studies); d = –0.04; range = –0.60 to 0.15)

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LD Vs. Low Achieving knowledgeBaker, Gersten, & Lee (2002)

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Summary knowledge

  • Results of this meta-analysis suggest that students with learning disabilities benefit from:

    • Curricula that employ explicit instruction, include a range of examples that are sequenced from concrete to abstract;

    • Student Verbalization and use of visuals for problem solving;

    • Teacher instruction that uses visuals to demonstrate problem solving

    • Peer assisted learning particularly for extended practice

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Assisting Students Struggling with Mathematics: Response to Intervention for Elementary and Middle Schools

The report is available on the IES website:

http://ies.ed.gov/ncee &


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Search for Coherence Intervention for Elementary and Middle Schools

Panel works to develop 5 to 10 assertions that are:

  • Forceful and useful


  • Do not encompass all things for all people

  • Do not read like a book chapter or article

    Challenges for the panel:

  • State of math research

  • Distinguishing between tiers of support

    Jump start the process by using individuals with topical expertise and complementary views

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The Topics Intervention for Elementary and Middle Schools

  • Tier 1

    • Universal Screening

  • Tier 2 and Tier 3

    • Focus instruction on whole number for grades k-5 and rational number for grades 6-8.

    • Explicit and systematic instruction

    • Solving word problems

    • Use of Visual Representations

    • Building fluency with basic arithmetic facts

    • Progress monitoring

    • Use of motivational strategies

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Recommendation 1 Intervention for Elementary and Middle Schools

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

  • Level of Evidence: Moderate

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Evidence Intervention for Elementary and Middle Schools

  • Technical evidence

    • Reliability and validity with a focus on predictive validity

    • Greater evidence in the earlier grades

  • Content of Measures

    • Single aspect of number sense (e.g. strategic counting)

    • Or Broad measures incorporating multiple aspects of number sense

    • Measures reflecting the computation and concepts and applications objectives for a specific grade level

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Suggestions Intervention for Elementary and Middle Schools

  • Have a building level team select measures based on critical criteria such as reliability, validity and efficiency.

  • Select screening measures based on the content they cover with a emphasis on critical instructional objectives for each grade level.

  • In grades 4-8, use screening measures in combination with state testing data.

  • Use the same screening tool across a district to enable analyzing results across schools

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Roadblocks Intervention for Elementary and Middle Schools

  • Resistance may be encountered in allocating time resources to the collection of screening data

  • Questions may arise about testing students who are “doing fine”.

  • Screening may identify students as at-risk who do not need services and miss students who do.

  • Screening may identify large numbers of students who need support beyond the current resources of the school or district.

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Roadblocks Intervention for Elementary and Middle Schools

  • Resistance may be encountered in allocating time resources to the collection of screening data.

  • Suggested Approach: Use data collection teams to streamline the data collection and analysis process.

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Final thoughts Intervention for Elementary and Middle Schools

  • RTI for identification is only possible if tiered support and corresponding elements are in place

  • Professional Development is critical in enhancing both the teaching of mathematics and data-based instructional decision-making

  • Districts and schools should think of developing math specialists similar to reading specialists

  • Our understanding of how best to teach and assess mathematics is rapidly expanding - Stay connected and be flexible in your approach to supporting mathematics achievement

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Resources Intervention for Elementary and Middle Schools

  • NMAP

    • http://www.ed.gov/about/bdscomm/list/mathpanel/index.html

  • Center On Instruction - Mathematics

    • http://www.centeroninstruction.org/resources.cfm?category=math

  • NCTM focal points

    • http://www.nctm.org/focalpoints.aspx?linkidentifier=id&amp;itemid=270

  • PIR website (Best Practices/Articles)

    • http://pacificir2.uoregon.edu:8100/

  • National Center Progress Monitoring

    • http://www.studentprogress.org/

  • CA Intervention Standards

    • http://www.cde.ca.gov/ci/ma/im/mathprogramnov2007.asp