Tiered Math Instruction
OrRTI Project
November 20, 2009
Do not worry about your problems with
mathematics,
I assure you mine are far greater.
-Albert Einstein
Objectives
• Look at IES
recommendations for
assessment and instruction
in Mathematics
• Understand the major
findings of the National
Math Advisory Panel report
and it’s implications to core
curriculum
• Look at possible
interventions to support
struggling mathematicians
The 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
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.
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
Ben Clarke, 2009
Fall to Spring.
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
•
•
•
•
Number Identification
K & 1 assessment
One Minute assessment
Individually administered
VanDerheyden: K-CBM
Ben Clarke, 2009
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
easy-CBM: Number and Operations
Ben Clarke, 2009
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
Ben Clarke, 2009
guide)
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
Basic Skills (in Algebra)
• 60 items; 5 minutes
• Problems include:
–
–
–
–
–
–
Solving basic fact equations;
Applying the distributive property;
Working with integers;
Combining like terms;
Simplifying expressions;
Applying proportional reasoning
• Scoring: # of problems correct
Ben Clarke, 2009
Basic Pre-algebra skills
A lgebra Probe A-31
Page 1
Solve:
9 + a = 15
a=
Evaluate:
12 + (– 8) + 3
Solve:
10 – 6 = g
g=
Simplify:
9 – 4d + 2 + 7d
Simplify:
2x + 4 + 3x + 5
Simplify:
5(b – 3) – b
Solve:
12 – e = 4
e=
Simplify:
4(3 + s) – 7
Solve:
q • 5 = 30
q=
Evaluate:
8 – (– 6) – 4
Simplify:
b + b + 2b
Simplify:
2 + w(w – 5)
Solve:
Solve:
1 foot =12 inches
5 feet = ____ inches
b
12

6
18
b=
Simplify:
7 – 3(f – 2)
Simplify:
4 – 7b + 5(b – 1)
Evaluate:
– 5 + (– 4) – 1
Simplify:
s + 2s – 4s
Solve:
63  c = 9
c=
Simplify:
2(s – 1) + 4 + 5s
Solve:
x+4=7
x=
Simplify:
– 5(q + 3) + 9
Simplify:
8m – 9(m + 2)
Evaluate:
9 + (– 3) – 8
Ben Clarke, 2009
Math Screening & Monitoring
• National Center on Student Progress Monitoring
(www.studentprogress.org)
• Intervention Central’s Math Worksheet Generator
(www.interventioncentral.com)
• AIMSweb
(www.aimsweb.com)
• Monitoring Basic Skills Progress
(Fuchs, Hamlet & Fuchs, 1998)
• The ABC’s of CBM (Hosp, Hosp,& Howell, 2007)
• DIBELS Math (2nd year Beta)
• Easy CBM
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.
Suggestions
• Have a district level team select measures
based on critical criteria such as reliability,
validity and efficiency.
– Team should have measurement expertise (e.g. school
psychologist) and mathematics (e.g. math specialist)
– Set up a screening to occur twice a year (Fall and
Winter)
– Be aware of students who fall near the cut scores
Ben Clarke, 2009
Suggestions
• Use the same screening tool across a district
to enable analyzing results across schools
– Districts may use results to determine the
effectiveness of district initiatives.
– May also be used to determine systematic areas
of weakness and provide support in that area
(e.g. fractions)
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)
Ben Clarke, 2009
Suggestions
• In grades 4-8, use screening measures in
combination with state testing data.
– Use state testing data from the previous year as the first
cut in a screening system.
– Can then use a screening measure with a reduced pool
of students or a more diagnostic measure linked to the
intervention program for a second cut.
Ben Clarke, 2009
Roadblocks
• Resistance may be encountered in
allocating time and resources to the
collection of screening data.
• Suggested Approach: Use data collection
teams to streamline the data collection and
analysis process.
Ben Clarke, 2009
Roadblocks
• Questions may arise about testing students
who are “doing fine”.
• Suggested Approach: Screening all students
allows the school or district to evaluate the
impact of instructional approaches
– Screening all students creates a distribution of
performance allowing the identification of atrisk students
Ben Clarke, 2009
Roadblocks
• Screening may identify students as at-risk
who do not need services and miss students
who do.
• Suggested Approach: Schools should
frequently examine the sensitivity and
specificity of screening measures to ensure
a proper balance and accurate decisions
about student risk status.
Ben Clarke, 2009
Roadblocks
• Screening may identify large numbers of students
who need support beyond the current resources of
the school or district.
• Suggested Approach: Schools and districts should
– Allocate resources to the students with the most risk
and at critical grade levels
and
– Implement school wide interventions to all students in
areas of school wide low performance (e.g. Fractions)
Ben Clarke, 2009
Recommendation 7
Monitor the progress of students
receiving supplemental instruction and
other students who are at risk.
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.
– Same team that worked on screening can also
work on progress monitoring
– Need to carefully consider capacity to model
growth in the context of instructional decision
making
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
Suggestions
• Use curriculum-embedded assessments in
intervention materials
– Frequency of measures can vary - every day to
once every week.
– 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 measuue of performance
Ben Clarke, 2009
Roadblocks
• Students within classes are at very different
levels.
• Suggested Approach: Group students across
classes to create groups with similar needs.
Ben Clarke, 2009
Roadblocks
• Insufficient time for teachers to implement
progress monitoring.
• Suggested Approach: Train
paraprofessionals or other school staff to
administer progress monitoring measures.
Ben Clarke, 2009
Math
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.
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.
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
Schema-based strategy
instruction (Jitendra, 2004)
• Teach student 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 Jungjahann
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.
Point of Discussion
“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
Roadblocks
• Interventionists might not be familiar with using
explicit instruction and might not realize how
much practice is needed for students in tier 2 and
tier 3 to master the material being taught.
• Suggested Approach: Have interventionists
observe lessons, practice with instructional
materials, and provide them with corrective
feedback on implementation
Roadblocks
• Those teaching in the intervention might not
be experts or feel comfortable with the math
content.
• Suggested Approach: Train interventionists to
explain math content (including math concepts,
vocabulary, procedures, reasoning and methods)
using clear, student-friendly language.
Roadblocks
• The intervention materials might not incorporate enough
modeling, think-alouds, practice or cumulative review to
improve students’ math performance.
• Suggested Approach: Consider having a math specialist
develop an instructional template which contains the
elements of instruction identified above and which can be
applied to various lessons.
– If possible, have a math specialist coach new
interventionists on how to use materials most
effectively.
Recommendation 4
•
Interventions should include instruction on
solving word problems that is based on
common underlying structures.
Suggestions
• Teach students about the structure of various
problem types, how to categorize problems, and
how to determine appropriate solutions.
• Teach students to recognize the common
underlying structure between familiar and
unfamiliar problems and to transfer known
solution methods from familiar to unfamiliar
problems.
Roadblocks
 Math curriculum material might not classify the
problems in the lessons into problem types
 Suggested Approach: Use a math specialist or a state or
district curriculum guide to help identify the problem types
covered in the curriculum at each level and the
recommended strategies for solving them.
• Students must be taught to understand a set of problem
types and a reliable strategy for solving each type.
Roadblocks
 As problems get more complex, so will the
problem types and the task of
discriminating among them.
 Suggested Approach: Explicitly and
systematically teach teachers and interventionists
to identify problem types and how to teach
students to differentiate one problem type from
another.
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.
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.
Roadblocks
• 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.
• If interventionists do not fully understand the
mathematical ideas behind the (representations),
they are unlikely to be able to teach it to struggling
students
Recommendation 6
Interventions at all grade levels should
devote about 10 minutes in each session to
building fluent retrieval of basic arithmetic
facts.
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)
Point of Discussion
“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
Recommendation 8
Include motivational strategies in Tier 2
and Tier 3 interventions.
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.
Mindset
• Incorporate social and intellectual support
from peers and teachers
• Teach students that effort has a huge impact
on math achievement
Big Ideas from IES
• 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, provide
guided practice, corrective feedback, and
frequent cumulative review.
Putting it all Together for Multitiered Instruction
• National Math Panel
• Process in TTSD
Core curriculum and instruction
National Mathematics Advisory Panel Final
Report, 2008
• Curricular Content moving toward algebra
• Teacher Proficiency
• Conceptual Understanding Interdependent
and
• Fluency and Automaticity
mutually
• Problem Solving
reinforcing
Core curriculum and instruction
Curricular Content
Focus + Coherence =
Depth
Breadth
Linear proficiency
vs.
Spiraling
(Closure after Exposure)
Learning Processes
• 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
• Teacher induction programs have positive
effects on all teachers.
• 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.
Practices That Work
• Using 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
So What? Now What?
• What information coincided with your
understanding of effective math instruction,
or practices in your district?
• What surprised you?
• What implications does the report have for
this school year? Future years?
Tier I in TTSD
•
•
•
•
•
45-90 minutes core instruction
K-12 curriculum alignment
Systematic instruction and feedback
Teach content to mastery
Focus on fractions!
What about interventions?
• Emphasis on research-based instructional
strategies (not “programs”)
• Increase opportunities to practice a skill
correctly
– Guided practice (“I do, We do, You do”)
– Correction routine
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
Tier II & III:
Research on Best Practices
Baker, Gersten, and Lee, 2002
• Demonstrated, significant effects for:
– Progress monitoring feedback, especially when
accompanied by instructional recommendations
– Peer Assisted Learning
– Explicit teacher led and contextualized teacher
facilitated approaches
– Concrete feedback to Parents
Interventions
• Emphasis on research-based
instructional strategies (not programs)
• Increase opportunities to practice a
skill correctly
– Guided practice (“I do, We do, You do”)
– Correction routine
• There are few, but an increasing number
of research based curricula available
Intervention lists
• IES
– http://ies.ed.gov/ncee/wwc/reports/Topic.aspx?tid=04#s=13
• Best Evidence
– http://www.bestevidence.org/math/elem/elem_math.htm
How to start and Next steps
• As you get started consider
– Focus on one grade or grade bands
• Long term trajectories suggest end of K critical benchmark
• May have more expertise/comfort with whole number
approach
– Screening before progress monitoring
– Strategies for collecting data
Ben Clarke, 2009
Resources
 NMAP
 http://www.ed.gov/about/bdscomm/list/mathpanel/index.html
 Center On Instruction - Mathematics
 http://www.centeroninstruction.org/resources.cfm?category=m
ath
 NCTM focal points
 http://www.nctm.orfocalpoints.aspxlinkidentifier=id&ite
mid=270
 PIR website (Best Practices/Articles)
 http://pacificir2.uoregon.edu:8100/
 National Center Progress Monitoring
 http://www.studentprogress.org/
 CA Intervention Standards
Ben Clarke, 2009
Discussion
From where you sit in your current job, was
the presentation consistent with how you
think about RtI in Math?
Why? Why not?
Contacts
• 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
Break Time