Tiered Math Instruction OrRTI Project Site Visit December 9, 2009

Do not worry about your problems with mathematics, I assure you mine are far greater. -Albert Einstein

The Math Caveat A lit search for studies on reading disabilities studies and math disability studies from found over 600 studies in the area of reading and less than 50 for mathematics (12:1) Specific RTI mathematics studies for a recent annotated bibliography totaled 9 studies

IES Recommendation Level of Scientific Evidence RTI Component 1.Universal screening (Tier I)ModerateAssessment: Screening 2.Focus instruction on whole number for grades k-5 and rational number for grades 6-8 LowCore/Tier 2/Tier 3 3.Systematic instructionStrongCore/Tier 2/Tier 3 4.Solving word problemsStrongCore/Tier 2/Tier 3 5.Visual representationsModerateCore/Tier 2/Tier 3 6.Building fluency with basic arithmetic facts ModerateCore/Tier 2/Tier 3 7.Progress monitoringLowAssessment: Progress Monitoring 8.Use of motivational strategiesLowCore/Tier 2/Tier 3

Assessment Recommendations Recommendation 1: Universal Screening Recommendation 7: Progress Monitoring

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

Coherent Assessment Systems Each type of assessment has a purpose The design of the tool should match the purpose –What are the implications for screening tools used with all students? Think purpose not tool How do each of these purposes fit together? Ben Clarke, 2009

Features Short duration measures (1 to 5 minute(s) fluency measures) –Note many measures that are short duration also used in progress monitoring. Longer duration measures (untimed up to 20 minutes) often examine multiple aspects of number sense –Issue of purpose is critical to examine Most research examines predictive validity from Fall to Spring. Ben Clarke, 2009

Universal Screening The Math Measures: –K-1: Missing Number Quantity Discrimination Number Identification VanDerheyden: K-CBM –Grades 2-5: Basic Facts Concepts and Applications Math Focal Points –-Secondary: Prealgebra

Universal screener Missing Number K & 1 assessment One minute assessment Individually administered

Universal screener Quantity Discrimination K & 1 assessment One Minute assessment Individually administered

Universal screener Computation 5 th grade example 1-5 grade Grows in complexity through the grades Two to four Minute assessment (depending on grade) Scored on digits correct Group administered

Universal screener Monitoring Basic Skills 4 th grade example 2-5 grade Grows in complexity through the grades Four to eight minutes (depending on grade) Scored on correct answers (some have multiple answers) Group administered Fuchs, Fuchs and Hamlett

Example: Reflecting critical math content easy-CBM Items created according to NCTM Focal Points for grade level 48 items for screening (16 per focal point) Ongoing research (not reviewed in practice guide) Ben Clarke, 2009

easy-CBM: Number and Operations Ben Clarke, 2009

Middle School 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 Ben Clarke, 2009

Math Screening & Monitoring National Center on Student Progress Monitoring National Center on RTI Intervention Central’s Math Worksheet Generator AIMSweb Monitoring Basic Skills Progress (Fuchs, Hamlet & Fuchs, 1998) DIBELS Math (2 nd year Beta) Easy CBM

Suggestions Have a district level team select measures based on critical criteria such as reliability, validity and efficiency. Use the same screening tool across a district to enable analyzing results across schools Ben Clarke, 2009

Suggestions Select screening measures based on the content they cover with a emphasis on critical instructional objectives for each grade level. –Lower elementary: Whole Number –Upper elementary: Rational Number –Across grades: Computational Fluency (hallmark of MLD) In grades 4-8, use screening measures in combination with state testing data. Ben Clarke, 2009

Universal Screening TTSD Decision Rules –K: Students receiving only “o” and/or “/” in the “Progression of Mathematics Stages” on the Progress Report are screened using CBM. –1-2: Students receiving only “1” and/or “/” in “math” on the Progress Report are screened using CBM. –3-5: Students receiving only “1,” “2,” and/or “/” in “math” on the Progress Report AND scoring below the 30 th percentile on the OAKS, are screened using CBM. –Students who meet the above criteria are assessed using Curriculum Based Measurements (CBM: Missing Number for K/1 and Basic Facts for 2-5). Students scoring below the 25 th percentile on CBMs are placed in Second Tier Interventions.

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

Suggestions Monitor the progress of tier 2, tier 3 and borderline tier 1 students at least once a month using grade appropriate general outcome measures. Use curriculum-embedded assessments in intervention materials –Will provide a more accurate index of whether or not the student is obtaining instructional objectives –Combined with progress monitoring provides a proximal and distal measure of performance Ben Clarke, 2009

TTSD Progress Monitoring CBMs are given every other week –Trained instructional assistants will complete progress monitoring Review trend lines every 12 weeks –We need a longer intervention period because growth on math CBMs happens in small increments –Look at rates of growth published by AIMSWeb

Growth trajectories for responders/non responders can be based on local and class or grade performance OR Use projected rate of growth from national norms—e.g. AIMSweb 50 th %tile –Grade 1,.30 digit per week growth –Grade 3,.40 digit per week growth –Grade 5,.70 digit per week growth

Instructional/Curricular Recommendations Recommendation 2: whole numbers/rational numbers Recommendation 3: systematic instruction Recommendation 4: solving word problems Recommendation 5: visual representation Recommendation 6: fluent retrieval of facts Recommendation 8: motivational strategies

Recommendation 2 Instructional materials for students receiving interventions should focus intensely on in-depth treatment of whole numbers in K-3 and on rational numbers in grades 4-8. Evidence: Low

Suggestions For tier 2 and 3 students in grades K-3, interventions should focus on the properties of whole number and operations. Some older students would also benefit from this approach. For tier 2 and 3 students in grades 4-8, interventions should focus on in depth coverage of rational number and advanced topics in whole number (e.g. long division).

Core curriculum content Whole number: understand place value, compose/decompose numbers, leaning 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 Source: Ben Clarke & Scott Baker Pacific Institutes for Research

Difficulty with fractions is pervasive and impedes further progress in mathematics

Recommendation 3 Instruction provided in math interventions should be explicit and systematic, incorporating modeling of proficient problem-solving, verbalization of thought processes, guided practice, corrective feedback and frequent cumulative review. Evidence: Strong

Suggestions Districts should appoint committees with experts in mathematics instruction and mathematicians to ensure specific criteria are covered in-depth in adopted curriculums. –Integrate computation with problem solving and pictorial representations –Stress reasoning underlying calculation methods –Build algorithmic proficiency –Contain frequent review of mathematical principles –Contain assessments to appropriately place students in the program

Suggestions Ensure that intervention materials are systematic and explicit and include numerous models of easy and difficult problems with accompanying teacher think-alouds. Provide students with opportunities to solve problems in a group and communicate problem- solving strategies. Ensure that instructional materials include cumulative review in each session. –May need to supplement curriculum with more modeling, think-alouds, practice and cumulative review.

“Explicit instruction with students who have mathematical difficulties has shown consistently positive effects on performance with word problems and computations. Results are consistent for students with learning disabilities, as well as other student who perform in the lowest third of a typical class.” National Mathematics Advisory Panel Final Report p. xxiii

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

Suggestions Teach students about the structure of various problem types, how to categorize problems, and how to determine appropriate solutions. –Math curriculum material might not classify the problems in the lessons into problem types, so in- district math experts may need to do this Teach students to recognize the common underlying structure between familiar and unfamiliar problems and to transfer known solution methods from familiar to unfamiliar problems.

Schema-based Strategy Instruction (Jitendra, 2004) Teach students to represent quantitative relationships graphically to solve problems. Use explicit strategies: 1.Problem Identification 2.Problem Representation 3.Problem Solution Be systematic: Teach one type of problem at a time until students are proficient. Provide models of proficient problem solving Kathy Jungjohann

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

Suggestions Use visual representations such as number lines, arrays, and strip diagrams. If necessary consider expeditious use of concrete manipulatives before visual representations. The goal should be to move toward abstract understanding. –Because many curricular materials do not include sufficient examples of visual representations, the interventionist may need the help of the mathematics coach or other teachers in developing the visuals.

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

Suggestions Provide 10 minutes per session of instruction to build quick retrieval of basic facts. Consider the use of technology, flash cards, and other materials to support extensive practice to facilitate automatic retrieval. For student in K-2 grade explicitly teach strategies for efficient counting to improve the retrieval of math facts. Teach students in grades 2-8 how to use their knowledge of math properties to derive facts in their heads.

“Basic” math facts are important! Basic math facts knowledge –Difficulty in automatic retrieval of basic math facts impedes more advanced math operations Fluency in math operations –Distinguishes between students with poor math skills to those with good skills (Landerl, Bevan, & Butterworth, 2004; Passolunghi & Siegel, 2004)

“the general concept of automaticity... is that, with extended practice, specific skills can read a level of proficiency where skill execution is rapid and accurate with little or no conscious monitoring … attentional resources can be allocated to other tasks or processes, including higher-level executive or control function” (Goldman & Pellegrino, 1987, p. 145 as quoted in Journal of Learning Disabilities, “Early Identification of Students with Math Disabilities,” July/August 2005 p 294

Recommendation 8 Include motivational strategies in Tier 2 and Tier 3 interventions. Evidence: Low

Suggestions Reinforce or praise students for their effort and for attending to and being engaged in the lesson. Consider rewarding student accomplishment. Allow students to chart their progress and to set goals for improvement.

IES Math Instruction Big Ideas Provide explicit and systematic instruction in problem solving. Teach common underlying structures of word problems. Use visual representations Verbalize your thought process Model proficient problem solving, providing guided practice, corrective feedback and frequent cumulative review Reinforce effort

National Mathematics Advisory Panel Final Report, 2008 Conceptual understanding, computational fluency and problem-solving skills are each essential and mutually reinforcing. Effort-based learning has greater impact than the notion of inherent ability The notion of “developmentally appropriate practices” based on age or grade level has consistently been proven to be wrong. Instead, learning is contingent on prior opportunities to learn.

National Mathematics Advisory Panel Final Report, 2008 Professional development is important- continue to build content knowledge as well as learning strategies. Teachers who know the math content they are teaching, including the content before and beyond, have the most impact on student achievement.

National Mathematics Advisory Panel Final Report, 2008 Use formative assessments Low achievers need explicit instruction in addition to daily core instruction Technology supports drill practice and automaticity Gifted students should accelerate and receive enrichment

Curriculum Reviews IES (What Works Clearinghouse) – Best Evidence Encyclopedia –

Tier I in TTSD minutes core instruction K-12 curriculum alignment Systematic instruction and feedback Teach content to mastery Focus on fractions!

Tier II Interventions for Math in TTSD (Within the Core) Kindergarten –Increased teacher attention during math Grades 1-5 –10 minutes of additional guided practice per day OR –10 minutes of Computer Assisted Instruction (CAI) per day

Other Resources  NMAP   Center On Instruction - Mathematics   NCTM focal points   PIR website (Best Practices/Articles)   National Center Progress Monitoring   CA Intervention Standards  Ben Clarke, 2009

Big Ideas Choose valid and reliable Screening and Progress Monitoring assessments that are linked to district standards Focus on Core instruction first Supplement existing curriculum with effective instructional strategies Build early number sense and fluency with basic skills

Contact Info Dean Richards: Jon Potter: Lisa Bates: