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 19962005 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
Level of
Scientific
Evidence
RTI Component
Moderate
Assessment:
Screening
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
IES Recommendation
1. Universal screening (Tier I)
2. Focus instruction on whole
number for grades k-5 and
rational number for grades 6-8
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
5th 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
4th 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 prealgebra 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
www.studentprogress.org
• National Center on RTI www.rti4success.org
• Intervention Central’s Math Worksheet Generator
www.interventioncentral.com
• AIMSweb www.aimsweb.com
• Monitoring Basic Skills Progress
(Fuchs, Hamlet & Fuchs, 1998)
• DIBELS Math (2nd 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
30th 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 25th 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 50th %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 problemsolving 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 indistrict 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 importantcontinue 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)
– http://ies.ed.gov/ncee/wwc/
• Best Evidence Encyclopedia
– www.bestevidence.org
Tier I in TTSD
•
•
•
•
•
45-90 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
 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.aspxlinkidentifier=id&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
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: drichards@ttsd.k12.or.us, 503-431-4135
Jon Potter: jpotter@ttsd.k12.or.us, 503-431-4149
Lisa Bates: lbates@ttsd.k12.or.us, 503-431-4079