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Break TimeBreak Time
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
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
SuggestionsSuggestions
• 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
Recommendation 7Recommendation 7
Monitor the progress of students receiving supplemental instruction and other students who are at risk.Evidence: Low
SuggestionsSuggestions
• 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
– Grade 3, .4 digit per week growth
– Grade 5, .7 digit per week growth
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
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
Recommendation 2Recommendation 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
Core curriculum contentCore curriculum content
• Whole number• Rational number• Critical aspects of geometry and
measurement
Source: Ben Clarke & Scott Baker Pacific Institutes for Research
SuggestionsSuggestions
• 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
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
Recommendation 3Recommendation 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
SuggestionsSuggestions
• 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
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
Recommendation 4Recommendation 4
Interventions should include instruction on solving word problems that is based on common underlying structures.Evidence: Strong
SuggestionsSuggestions
• 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
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
Recommendation 5Recommendation 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
SuggestionsSuggestions
• 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 6Recommendation 6
Interventions at all grade levels should devote about 10 minutes in each session to building fluent retrieval of basic arithmetic facts.Evidence: Moderate
SuggestionsSuggestions
• 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)
Recommendation 8Recommendation 8
Include motivational strategies in Tier 2 and Tier 3 interventions.Evidence: Low
SuggestionsSuggestions
• 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.
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
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.
National Mathematics National Mathematics Advisory Panel Final Advisory Panel Final
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
National Mathematics National Mathematics Advisory Panel Final Advisory Panel Final
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: drichards@ttsd.k12.or.us, 503-431-4135
Jon Potter: jpotter@ttsd.k12.or.us, 503-431-4149
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