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Tiered Math Instruction

OrRTI ProjectSite Visit

April 22, 2010

ObjectivesObjectives

• Explore the latest research on Response to Intervention systems for Math

• Have dialogue about what would be necessary for a system to move toward RTI for Math

Do not worry about your problems with mathematics,

I assure you mine are far greater.

-Albert Einstein

The Process is Ongoing The Process is Ongoing and Long-Termand Long-Term

Adapted from

CONSENSUS

http://www.aea11.k12.ia.us/

The Math CaveatThe Math Caveat

• 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 (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) Moderate Assessment: Screening

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

Low Core/Tier 2/Tier 3

3. Systematic instruction Strong Core/Tier 2/Tier 3

4. Solving word problems Strong Core/Tier 2/Tier 3

5. Visual representations Moderate Core/Tier 2/Tier 3

6. Building fluency with basic arithmetic facts

Moderate Core/Tier 2/Tier 3

7. Progress monitoring Low Assessment: Progress Monitoring

8. Use of motivational strategies Low Core/Tier 2/Tier 3

Assessment Assessment RecommendationsRecommendations

• Recommendation 1: Universal Screening

• Recommendation 7: Progress Monitoring

Recommendation 1Recommendation 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 SystemsCoherent 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

General FeaturesGeneral 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

• The Math Measures:– K-1:

• Missing Number • Quantity Discrimination• Number Identification

– Grades 2-8: • Basic Facts • Concepts and Applications• Math Focal Points

– Secondary:• Prealgebra

Universal screenerUniversal screener

• K & 1 assessment• One minute measure• Individually administered

Early Numeracy CBM:Early Numeracy CBM:Missing NumberMissing Number

• K & 1 assessment• One minute

measure• Individually

administered

Early Numeracy CBM:Early Numeracy CBM:Quantity DiscriminationQuantity Discrimination

• Grades 1-8• Grows in complexity

through the grades• Two to four Minute

assessment (depending on grade)

• Scored on digits correct

• Group administered

Math-CBM:Math-CBM:ComputationComputation

5th grade example

• Grades 2-8• Grows in complexity

through the grades• Aligned with NCTM

Focal Points• Four to eight minutes

(depending on grade)• Scored on correct

answers (some have multiple answers)

• Group administered

Math-CBM:Math-CBM:Concepts and ApplicationsConcepts and Applications

• Items created according to NCTM Focal Points for grade level

• 48 items for screening (16 per focal point)

• Computer-based or pencil and paper administration

• Ongoing researchBen Clarke, 2009

easyCBMeasyCBM

CBM-AlgebraCBM-Algebra

• Designed by Foegen and colleagues (Iowa State University)

• Assess pre-algebra and basic algebra skills.

• Administered and scored similar to Math-CBM

Algebra Probe A-31 Page 1

Solve: 9 + a = 15 a =

Solve: 10 – 6 = g g =

Evaluate: 12 + (– 8) + 3

Simplify: 9 – 4d + 2 + 7d

Simplify: 2x + 4 + 3x + 5

Simplify: 5(b – 3) – b

Solve: 12 – e = 4 e =

Solve: q • 5 = 30 q =

Simplify: 4(3 + s) – 7

Evaluate: 8 – (– 6) – 4

Simplify: b + b + 2b

Simplify: 2 + w(w – 5)

Solve:

18

12

6

b

b =

Solve: 1 foot =12 inches 5 feet = ____ inches

Simplify: 7 – 3(f – 2)

Simplify: 4 – 7b + 5(b – 1)

Evaluate: – 5 + (– 4) – 1

Simplify: s + 2s – 4s

Solve: 63 c = 9 c =

Solve: x + 4 = 7 x =

Simplify: 2(s – 1) + 4 + 5s

Simplify: – 5(q + 3) + 9

Simplify: 8m – 9(m + 2)

Evaluate: 9 + (– 3) – 8

What was something that you already knew?

What was something new?

What is your district’s next step on this recommendation? Consensus? Infrastructure? Implementation?

Are there questions you still have?

Talk to a neighborTalk to a neighbor

Math Assessment Math Assessment ResourcesResources

• National Center on Student Progress Monitoring www.studentprogress.org

• National Center on RTI www.rti4success.org • Intervention Central’s Math Worksheet

Generator

www.interventioncentral.com• AIMSweb www.aimsweb.com• easyCBM easyCBM.com• Monitoring Basic Skills Progress (Fuchs,

Hamlet & Fuchs, 1998) • The ABC’s of CBM (Fuchs, Fuchs, & Howell,

2007)

SuggestionsSuggestions

• 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


• 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


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


• 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

• Growth rates for Math are much lower

– Grade 1, .3 digit per week growth



Instructional/Curricular Instructional/Curricular RecommendationsRecommendations

• 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


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

Core curriculum contentCore curriculum content

• Whole number• Rational number• Critical aspects of geometry and

measurement

Source: Ben Clarke & Scott Baker Pacific Institutes for Research


• 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).

Difficulty with fractions is pervasive and impedes further progress in

mathematics


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


• 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

SuggestionsSuggestions• 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.

Can you support this?Can you support this?

“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


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


• 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 Schema-based Strategy Instruction (Jitendra, Instruction (Jitendra,

2004)2004)• Teach students to represent quantitative

relationships graphically to solve problems.• Use explicit strategies:

1. Problem Identification2. Problem Representation3. Problem Solution

• Be systematic: Teach one type of problem at a time until students are proficient.

• Provide models of proficient problem solvingKathy Jungjohann


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


• 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.


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


• Provide 10 minutes per session of instructioninstruction 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 Basic” math facts are important!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)

Can you support this?Can you support this?

“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)


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


• 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 IES Math Instruction Big IdeasIdeas

• 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 National Mathematics Advisory Panel Final Advisory Panel Final

Report, 2008Report, 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.

• 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.


Report, 2008Report, 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


Report, 2008Report, 2008

Curriculum ReviewsCurriculum Reviews

• IES (What Works Clearinghouse)– http://ies.ed.gov/ncee/wwc/

• Best Evidence Encyclopedia– www.bestevidence.org

• Achievement Effects of Four Early Elementary School Math Curricula– http://ies.ed.gov/ncee/pubs/20094052/

pdf/20094052.pdf

Other ResourcesOther Resources

National Math Advisory Panel 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.orfocalpoints

PIR website (Best Practices/Articles) http://pacificir2.uoregon.edu:8100/

CA Intervention Standards http://www.cde.ca.gov/ci/ma/im/mathprogramnov2007.asp

What Works Clearinghouse http://ies.ed.gov/ncee/wwc/ Ben Clarke, 2009

Big IdeasBig 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 InfoContact Info

Dean Richards: [email protected], 503-431-4135

Jon Potter: [email protected], 503-431-4149

mailto:[email protected]

mailto:[email protected]

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Documents

math measures

math disability studies

minute assessment

reading disabilities

focal pointcomputer

rational number

nctm focal pointsfour

basic algebra skills