Transcript Document

Response to Intervention
RTI for Math & Writing:
Grades K-8
Jim Wright
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Response to Intervention
Workshop PPTs and handout available at:
http://www.interventioncentral.org/gisd
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Response to Intervention
RTI: Key Concepts
Focus of Inquiry: What are central concepts of
RTI that are helpful in understanding how to
expand the model to include math and writing?
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Response to Intervention
RTI Assumption: Struggling Students Are ‘Typical’
Until Proven Otherwise…
RTI logic assumes that:
– A student who begins to struggle in general education is typical,
and that
– It is general education’s responsibility to find the instructional
strategies that will unlock the student’s learning potential
Only when the student shows through well-documented
interventions that he or she has ‘failed to respond to
intervention’ does RTI begin to investigate the possibility
that the student may have a learning disability or other
special education condition.
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Response to Intervention
RTI ‘Pyramid of
Interventions’
Tier 3
Tier 2
Tier 1
Tier 3: Intensive interventions.
Students who are ‘nonresponders’ to Tiers 1 & 2 are
referred to the RTI Team for more
intensive interventions.
Tier 2 Individualized
interventions. Subset of
students receive interventions
targeting specific needs.
Tier 1: Universal interventions.
Available to all students in a
classroom or school. Can consist
of whole-group or individual
strategies or supports.
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Response to Intervention
Avg Classroom Academic
Performance Level
Discrepancy 1: Skill Gap
Discrepancy 2:
(Current
Gap in Rate of
Performance Level)
Learning (‘Slope
Target
of Improvement’)
Student
‘Dual-Discrepancy’: RTI Model
of Learning Disability (Fuchs 2003)
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Response to Intervention
Evaluating a Student’s ‘NonResponder’ Status: An RTI
Checklist pp. 2-7
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Response to Intervention
RTI ‘Non-Responder’ Checklist: Purpose
The document Evaluating a Student’s ‘Non
Responder’ Status: An RTI Checklist was
created to help schools to:
• audit the quality of their current RTI efforts in any
academic area.
• create concrete guidelines for judging whether
RTI intervention efforts for a particular student
are of adequate quality.
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Response to Intervention
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Response to Intervention
Evaluating a Student’s ‘Non-Responder’
Status: An RTI Checklist
Interventions: Evidence-Based & Implemented With
Integrity
•
•
•
•
•
Tier 1: High-Quality Core Instruction
Tier 1: Classroom Intervention
Tier 2 & 3 Interventions: Minimum Number & Length
Tier 2 & 3 Interventions: Essential Elements
Tier 1, 2, & 3 Interventions: Intervention Integrity
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Response to Intervention
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Response to Intervention
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Response to Intervention
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Response to Intervention
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Response to Intervention
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Response to Intervention
Evaluating a Student’s ‘Non-Responder’
Status: An RTI Checklist
Academic Screenings: General Outcome Measures
and Skill-Based Measures
• Selection of Academic Screening Measures
• Local Norms Collected via Gradewide Academic
Screenings at Least 3 Times Per Year
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Response to Intervention
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Response to Intervention
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Response to Intervention
Evaluating a Student’s ‘Non-Responder’
Status: Activity
At your table: Discuss how your school or district may
use the document Evaluating a Student’s ‘NonResponder’ Status: An RTI Checklist to:
• help you to identify the ‘non-negotiable’ elements of
RTI as you extend it to cover math and writing.
Be prepared to share the main points of your
discussion with the large group.
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Response to Intervention
Core Instruction & Tier 1 Intervention
Focus of Inquiry: What are the indicators of
high-quality core instruction and classroom
(Tier 1) intervention for math?
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Response to Intervention
“
“
“Tier I of an RTI model involves quality core
instruction in general education and
benchmark assessments to screen students
and monitor progress in learning.” p. 9
“It is no accident that high-quality
intervention is listed first [in the RTI model],
because success in tiers 2 and 3 is quite
predicated on an effective tier 1. “ p. 65
”
”
Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools.
Routledge: New York.
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Response to Intervention
Common Core State
Standards Initiative
http://www.corestandards.org/
View the set of Common Core
Standards for reading and
mathematics being adopted by
states across America.
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Response to Intervention
Common Core
Standards, Curriculum, School Curriculum.
and Programs: How Do Outlines a uniform
sequence shared across
They Interrelate?
Common Core
Standards. Provide
external instructional
goals that guide the
development and
mapping of the
school’s curriculum.
However, the
sequence in which the
standards are taught
is up to the district and
school.
instructors for attaining the
Common Core Standards’
instructional goals. Scopeand-sequence charts bring
greater detail to the
general curriculum.
Curriculum mapping
ensures uniformity of
practice across
classrooms, eliminates
instructional gaps and
redundancy across grade
levels.
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Commercial
Instructional and
Intervention
Programs. Provide
materials for teaching
the curriculum.
Schools often piece
together materials
from multiple
programs to help
students to master the
curriculum. It should
be noted that specific
programs can change,
while the underlying
curriculum remains
unchanged.
Response to Intervention
An RTI Challenge: Limited Research to Support
Evidence-Based Math Interventions
“… in contrast to reading, core math programs that are
supported by research, or that have been constructed
according to clear research-based principles, are not
easy to identify. Not only have exemplary core
programs not been identified, but also there are no
tools available that we know of that will help schools
analyze core math programs to determine their
alignment with clear research-based principles.” p. 459
Source: Clarke, B., Baker, S., & Chard, D. (2008). Best practices in mathematics assessment and intervention with elementary
students. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 453-463).
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Response to Intervention
What Works
Clearinghouse Practice
Guide: Assisting Students
Struggling with
Mathematics: Response to
Intervention (RtI) for
Elementary and Middle
Schools
http://ies.ed.gov/ncee/wwc/
This publication provides 8
recommendations for effective
core instruction in mathematics
for K-8.
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Response to Intervention
Assisting Students Struggling with Mathematics: RtI for Elementary
& Middle Schools: 8 Recommendations
• Recommendation 1. Screen all
students to identify those at risk for
potential mathematics difficulties and
provide interventions to students
identified as at risk
• Recommendation 2. Instructional
materials for students receiving
interventions should focus intensely on
in-depth treatment of whole numbers in
kindergarten through grade 5 and on
rational numbers in grades 4 through 8.
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Response to Intervention
Assisting Students Struggling with Mathematics: RtI for Elementary
& Middle Schools: 8 Recommendations (Cont.)
• Recommendation 3. Instruction during
the intervention should be explicit and
systematic. This includes providing
models of proficient problem solving,
verbalization of thought processes,
guided practice, corrective feedback,
and frequent cumulative review
• Recommendation 4. Interventions
should include instruction on solving
word problems that is based on
common underlying structures.
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Response to Intervention
Assisting Students Struggling with Mathematics: RtI for Elementary
& Middle Schools: 8 Recommendations (Cont.)
• 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
• Recommendation 6. Interventions at
all grade levels should devote about 10
minutes in each session to building
fluent retrieval of basic arithmetic facts
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Response to Intervention
Assisting Students Struggling with Mathematics: RtI for Elementary
& Middle Schools: 8 Recommendations (Cont.)
• Recommendation 7. Monitor the
progress of students receiving
supplemental instruction and other
students who are at risk
• Recommendation 8. Include
motivational strategies in tier 2 and tier
3 interventions.
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Response to Intervention
How Do We Reach Low-Performing Math
Students?: Instructional Recommendations
Important elements of math instruction for low-performing
students:
–
–
–
–
“Providing teachers and students with data on student
performance”
“Using peers as tutors or instructional guides”
“Providing clear, specific feedback to parents on their children’s
mathematics success”
“Using principles of explicit instruction in teaching math
concepts and procedures.” p. 51
Source: Baker, S., Gersten, R., & Lee, D. (2002).A synthesis of empirical research on teaching mathematics to lowachieving students. The Elementary School Journal, 103(1), 51-73..
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Response to Intervention
Activity: How Do We Reach Low-Performing
Students? p.23
• Review the handout on p. 23
of your packet and consider
each of the elements found
to benefit low-performing
math students.
• For each element, brainstorm
ways that you could promote
this idea in your math
classroom.
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Response to Intervention
RTI at Tier 1: The Teacher as ‘First Responder’
Focus of Inquiry: What does Tier 1 intervention
look like for the general-education classroom
teacher who is supporting struggling students?
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Response to Intervention
RTI ‘Pyramid of
Interventions’
Tier 3
Tier 2
Tier 1
Tier 3: Intensive interventions.
Students who are ‘nonresponders’ to Tiers 1 & 2 are
referred to the RTI Team for more
intensive interventions.
Tier 2 Individualized
interventions. Subset of students
receive interventions targeting
specific needs.
Tier 1: Universal interventions.
Available to all students in a
classroom or school. Can consist
of whole-group or individual
strategies or supports.
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Response to Intervention
Tier 1 Core Instruction
Tier I core instruction:
• Is universal—available to all students.
• Can be delivered within classrooms or throughout the school.
• Is an ongoing process of developing strong classroom instructional
practices to reach the largest number of struggling learners.
All children have access to Tier 1 instruction/interventions. Teachers have
the capability to use those strategies without requiring outside assistance.
Tier 1 instruction encompasses:
• The school’s core curriculum.
• All published or teacher-made materials used to deliver that curriculum.
• Teacher use of ‘whole-group’ teaching & management strategies.
Tier I instruction addresses this question: Are strong classroom instructional
strategies sufficient to help the student to achieve academic success?
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Response to Intervention
Tier I (Classroom) Intervention
Tier 1 intervention:
• Targets ‘red flag’ students who are not successful with core
instruction alone.
• Uses ‘evidence-based’ strategies to address student academic
or behavioral concerns.
• Must be feasible to implement given the resources available in
the classroom.
Tier I intervention addresses the question: Does the student make
adequate progress when the instructor uses specific academic
or behavioral strategies matched to the presenting concern?
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Response to Intervention
The Key Role of Classroom Teachers as
‘Interventionists’ in RTI: 6 Steps
1. The teacher defines the student academic or
behavioral problem clearly.
2. The teacher decides on the best explanation for why the
problem is occurring.
3. The teacher selects ‘research-based’ interventions.
4. The teacher documents the student’s Tier 1 intervention plan.
5. The teacher monitors the student’s response (progress) to the
intervention plan.
6. The teacher knows what the next steps are when a student fails
to make adequate progress with Tier 1 interventions alone.
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Response to Intervention
Available on conference webpage
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Response to Intervention
RTI Interventions: What If There is No Commercial
Intervention Package or Program Available?
“Although commercially prepared programs and … manuals
and materials are inviting, they are not necessary. … A
recent review of research suggests that interventions are
research based and likely to be successful, if they are
correctly targeted and provide explicit instruction in the skill,
an appropriate level of challenge, sufficient opportunities to
respond to and practice the skill, and immediate feedback on
performance…Thus, these [elements] could be used as
criteria with which to judge potential …interventions.” p. 88
Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools.
Routledge: New York.
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Response to Intervention
Motivation Deficit 1: The student is
unmotivated because he or she cannot do
the assigned work.
• Profile of a Student with This Motivation Problem:
The student lacks essential skills required to do the task.
Handout pp.12-13
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work
• Profile of a Student with This Motivation Problem (Cont.):
Areas of deficit might include:
• Basic academic skills. Basic skills have straightforward criteria for correct
performance (e.g., the student defines vocabulary words or decodes text
or computes ‘math facts’) and comprise the building-blocks of more
complex academic tasks (Rupley, Blair, & Nichols, 2009).
• Cognitive strategies. Students employ specific cognitive strategies as
“guiding procedures” to complete more complex academic tasks such as
reading comprehension or writing (Rosenshine, 1995).
• Academic-enabling skills. Skills that are ‘academic enablers’ (DiPerna,
2006) are not tied to specific academic knowledge but rather aid student
learning across a wide range of settings and tasks (e.g., organizing work
materials, time management).
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• What the Research Says: When a student lacks the
capability to complete an academic task because of limited
or missing basic skills, cognitive strategies, or academicenabling skills, that student is still in the acquisition stage of
learning (Haring et al., 1978). That student cannot be
expected to be motivated or to be successful as a learner
unless he or she is first explicitly taught these weak or
absent essential skills (Daly, Witt, Martens & Dool, 1997).
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Verify the Presence of This Motivation Problem:
The teacher collects information (e.g., through observations
of the student engaging in academic tasks; interviews with
the student; examination of work products, quizzes, or
tests) demonstrating that the student lacks basic skills,
cognitive strategies, or academic-enabling skills essential to
the academic task.
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Fix This Motivation Problem: Students who are
not motivated because they lack essential skills need to be
taught those skills.
Direct-Instruction Format. Students learning new material,
concepts, or skills benefit from a ‘direct instruction’
approach. (Burns, VanDerHeyden & Boice, 2008;
Rosenshine, 1995; Rupley, Blair, & Nichols, 2009).
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Response to Intervention
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Fix This Motivation Problem: When following a
direct-instruction format, the teacher:
ensures that the lesson content is appropriately
matched to students’ abilities.
opens the lesson with a brief review of concepts or
material that were previously presented.
states the goals of the current day’s lesson.
breaks new material into small, manageable increments,
or steps.
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Fix This Motivation Problem: When following a
direct-instruction format, the teacher:
throughout the lesson, provides adequate explanations
and detailed instructions for all concepts and materials
being taught. NOTE: Verbal explanations can include
‘talk-alouds’ (e.g., the teacher describes and explains
each step of a cognitive strategy) and ‘think-alouds’
(e.g., the teacher applies a cognitive strategy to a
particular problem or task and verbalizes the steps in
applying the strategy).
regularly checks for student understanding by posing
frequent questions and eliciting group responses.
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Fix This Motivation Problem: When following a
direct-instruction format, the teacher:
verifies that students are experiencing sufficient success
in the lesson content to shape their learning in the
desired direction and to maintain student motivation and
engagement.
provides timely and regular performance feedback and
corrections throughout the lesson as needed to guide
student learning.
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Response to Intervention
Motivation Deficit 1: Cannot Do the Work (Cont.)
• How to Fix This Motivation Problem: When following a
direct-instruction format, the teacher:
allows students the chance to engage in practice
activities distributed throughout the lesson (e.g., through
teacher demonstration; then group practice with teacher
supervision and feedback; then independent, individual
student practice).
ensures that students have adequate support (e.g.,
clear and explicit instructions; teacher monitoring) to be
successful during independent seatwork practice
activities.
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Response to Intervention
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Response to Intervention
Activity: ‘Good Instruction is Research-Based’
• Review the elements of effective ‘direct
instruction’ that appear on page 13 of your
handout.
• Discuss how you can share this checklist with
others in your school to help them to realize that
teacher-delivered instruction that follows these
guidelines is ‘research-based’ and supports RTI,
e.g.:
– Whole-group: Tier 1 Core Instruction
– Small-group: Tier 1 Intervention; Tier 2/3 Intervention
– Individual student: Tier 3 Intervention
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Response to Intervention
Core Instruction & Tier 1 Intervention
Focus of Inquiry: How does a school select
‘research-based’ math instruction and
intervention ideas for use in the classroom?
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Response to Intervention
What Works
Clearinghouse
http://ies.ed.gov/ncee/wwc/
This website reviews core
instruction and intervention
programs in math and
reading/writing, as well as other
academic areas.
The site reviews existing
studies and draws conclusions
about whether specific
intervention programs show
evidence of effectiveness.
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Response to Intervention
Best Evidence
Encyclopedia
http://www.bestevidence.org/
This site provides reviews of
evidence-based reading and
math programs.
The website is sponsored by
the Johns Hopkins University
School of Education's Center for
Data-Driven Reform in
Education (CDDRE) .
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Response to Intervention
Intervention Central
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Response to Intervention
Math School-Wide Screenings
Focus of Inquiry: What math school-wide
screenings are available and how is that
information used in RTI?
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Response to Intervention
Building-Wide Screening: Assessing All Students
(Stewart & Silberglit, 2008)
Screening data in basic academic skills are collected at least 3
times per year (fall, winter, spring) from all students.
• Schools should consider using ‘curriculum-linked’ measures
such as Curriculum-Based Measurement that will show
generalized student growth in response to learning.
• If possible, schools should consider avoiding ‘curriculumlocked’ measures that are tied to a single commercial
instructional program.
Source: Stewart, L. H. & Silberglit, B. (2008). Best practices in developing academic local norms. In A. Thomas & J. Grimes
(Eds.), Best practices in school psychology V (pp. 225-242). Bethesda, MD: National Association of School Psychologists.
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Response to Intervention
Applications of Screening Data (Stewart & Silberglit, 2008)
Math screening data can be used to:
• Evaluate and improve the current core math instructional
program: How well are our children learning?
• Allocate resources to classrooms, grades, and buildings where
student academic needs are greatest: Where can we best put
our scarce resources to help struggling students?
• Guide the creation of targeted Tier 2/3 (supplemental
intervention) groups: What students need supplemental math
interventions—and what kinds of interventions do they need?
• Set academic goals for improvement for students on Tier 2
and Tier 3 interventions: Using local or research norms for
math, what progress do we expect for students on
intervention?
Source: Stewart, L. H. & Silberglit, B. (2008). Best practices in developing academic local norms. In A. Thomas & J. Grimes
(Eds.), Best practices in school psychology V (pp. 225-242). Bethesda, MD: National Association of School Psychologists.
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Response to Intervention
Clearinghouse for RTI Screening and ProgressMonitoring Tools
• The National Center on RTI (www.rti4success.org)
maintains pages rating the technical adequacy of RTI
screening and progress-monitoring tools.
• Schools should strongly consider selecting screening
tools that have national norms or benchmarks to help
them to assess the academic-risk level of their students.
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Response to Intervention
Methods of RTI Math
Screening/ProgressMonitoring
Description
Early Math Fluency
Kdg and Grade 1: One-minute
measures of numberline:
Quantity Discrimination, Missing
Number, Number Identification
Math Computation Fluency
Grades 1-8: Two-minute
assessments of math
computation skills.
Math Concepts & Applications Grades K-8: Mixed problems that
map to the Math Focal Points
from the NCTM.
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Response to Intervention
Early Math Fluency: Measuring ‘Number Sense’
• Early Math Fluency measures track primarygrade students’ acquisition of number sense
(defined as mastery of internal number line)
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Response to Intervention
Early Math Fluency: Measuring ‘Number Sense’
• Quantity Discrimination [1 minute]: The student is given a
worksheet with number pairs and, for each pair, identifies the
larger of the two numbers.
• Missing Number [1 minute]: The student is given a
worksheet with 4-digit number series with one digit randomly
left blank and, for each series, names the missing number.
• Number Identification [1 minute]: The student is given a
worksheet randomly generated numbers and reads off as
many as possible within the time limit.
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Response to Intervention
Numberfly Early Math
Fluency Generator
http://www.interventioncentral.org
Use this free online application to
design and create Early Math
Fluency Probes, including:
•Quantity Discrimination
•Missing Number
•Number Identification
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Response to Intervention
Math Computation Fluency: Computation Speed
and Accuracy
Math Computation Fluency [2 minutes]:
• The student is given a worksheet of computation
problems that either is a mix of different problem-types
(mixed-skill worksheet) or has problems all of the same
type (single-skill worksheet).
• The student has two minutes to answer as many
problems as possible.
• The computation probe is then scored, with the student
getting ‘credit’ for every correct digit.
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Response to Intervention
Computation Fluency: Benefits of Automaticity of
‘Arithmetic Combinations’ (Gersten, Jordan, & Flojo, 2005)
• There is a strong correlation between poor retrieval of
arithmetic combinations (‘math facts’) and global math
delays
• Automatic recall of arithmetic combinations frees up
student ‘cognitive capacity’ to allow for understanding of
higher-level problem-solving
• By internalizing numbers as mental constructs, students
can manipulate those numbers in their head, allowing for
the intuitive understanding of arithmetic properties, such
as associative property and commutative property
Source: Gersten, R., Jordan, N. C., & Flojo, J. R. (2005). Early identification and interventions for students with mathematics
difficulties. Journal of Learning Disabilities, 38, 293-304.
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Response to Intervention
Math Computation Fluency
Generator
http://www.interventioncentral.org
Use this free online application to
design and create curriculumbased measurement Math
Computation Probes, for the
basic math operations, including:
•Addition
•Subtraction
•Multiplication
•Division
NOTE: See pp. 20-22 for a listing
of math computation goals by
grade level.
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Response to Intervention
Math Computation Fluency: Computation Speed
and Accuracy
• Strength: Computation Fluency provides good
information about a student’s proficiency with math
facts, a strong indicator of his or her ability to do mental
arithmetic.
• Drawback: Computation Fluency taps only a narrow set
of math competencies and is not a good ‘general
outcome measure’ or predictor of more global math
performance.
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Response to Intervention
Math Concepts & Applications
Math Concepts & Applications [www.easycbm.com
online administration]: The student goes online to
complete a mixed-skills series of ‘concepts &
applications’ in mathematics.
.
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Response to Intervention
EasyCBM Math Concepts &
Applications
http://www.easycbm.com
This website provides two levels of
support:
• Teacher Version [free]: Any
teacher can create a free account
and use easycbm tools to monitor
student progress on interventions.
NOTE: There are 16 items on the
C&A Teacher Version probes.
• District Version [pay]: Allows
schools to screen student
populations 3 times per year.
NOTE: There are 45 items on the
C&A District Version probes.
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Response to Intervention
Math Concepts & Applications
• Strengths: Concepts & Applications measures sample
a broad array of math skills and concepts.
They also tap into a student’s conceptual knowledge of
mathematics, not just procedural knowledge.
• Drawback: While Concepts & Applications measures
sample a broader array of math skills and concepts,
they do not provide deeper information about the
student’s performance on any one skill. (Nor were they
designed to!)
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Response to Intervention
Using Research Math Norms to Estimate Risk:
Example Using EasyCBM
• Low Risk: At or above the 20th percentile: Core instruction alone is
sufficient for the student.
• Some Risk: 10th to 20th percentile: The student will benefit from
additional intervention, which may be provided by the classroom teacher
or other provider.
• At Risk: Below 10th percentile : The student requires intensive
intervention, which may be provided by the classroom teacher or other
provider.
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Response to Intervention
Creating a School-Wide RTI Math Screening
Plan: Recommendations
1. Analyze your student demographics and academic
performance and select math (or other) academic
screeners matched to those demographics.
2. Consider piloting new screening tools (e.g., at single
grade levels or in selected classrooms) before rolling out
through all grade levels.
3. Ensure that any discussion about grade- or school- or
district-wide adoption of RTI screening tools includes
general education and special education input.
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Response to Intervention
Creating a School-Wide RTI Math Screening
Plan: Recommendations (Cont.)
4. When adopting a screening tool, inventory all formal
assessments administered in your school. Discuss
whether any EXISTING assessments can be made
optional or dropped whenever new screening tools are
being added.
5. If possible, use screening tools found by the National
Center on RTI to have ‘technical adequacy’.
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Response to Intervention
Team Activity: Creating an RTI Math Screening Plan for
Your School
• Review the recommendations just presented on
school-wide screening tools in math, including
Early Math Fluency, Math Computation Fluency,
and Concepts & Applications.
• If your school is currently using a set of math
school-wide screeners, discuss how you might
evaluate them to ensure that they are adequate
and meet your needs.
• If your school does NOT yet have a set of
school-wide screeners, discuss how you might
begin to select and pilot these screeners.
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Response to Intervention
Math Interventions
Focus of Inquiry: What are sample intervention
programs or ideas to address math delays?
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Response to Intervention
Recommended Instruction and Intervention
Programs for ‘Number Sense’ (Clements & Sarama, 2011)
•
•
•
•
RightStart Math
Building Blocks Math
Big Math for Little Kids
Number Worlds
Source: Clements, D. H. & Sarama, J. (2011). Early childhood math intervention. Science, 333, 968-970.
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Response to Intervention
Profile of Students With Significant Math Difficulties p. 15
Spatial organization. The student commits errors such as misaligning numbers in columns in a
multiplication problem or confusing directionality in a subtraction problem (and subtracting
the original number—minuend—from the figure to be subtracted (subtrahend).
Visual detail. The student misreads a mathematical sign or leaves out a decimal or dollar sign in
the answer.
Procedural errors. The student skips or adds a step in a computation sequence. Or the student
misapplies a learned rule from one arithmetic procedure when completing another, different
arithmetic procedure.
Inability to ‘shift psychological set’. The student does not shift from one operation type (e.g.,
addition) to another (e.g., multiplication) when warranted.
Graphomotor. The student’s poor handwriting can cause him or her to misread handwritten
numbers, leading to errors in computation.
Memory. The student fails to remember a specific math fact needed to solve a problem. (The
student may KNOW the math fact but not be able to recall it at ‘point of performance’.)
Judgment and reasoning. The student comes up with solutions to problems that are clearly
unreasonable. However, the student is not able adequately to evaluate those responses to
gauge whether they actually make sense in context.
Source: Rourke, B. P. (1993). Arithmetic disabilities, specific & otherwise: A neuropsychological perspective. Journal of Learning
Disabilities, 26, 214-226.
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Response to Intervention
Activity: Profile of Math Difficulties p. 15
• Review the profile of
students with significant
math difficulties that
appears on p. 15 of your
handout.
• For each item in the
profile, discuss what
methods you might use to
discover whether a
particular student
experiences this difficulty.
Jot your ideas in the
‘NOTES’ column.
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Response to Intervention
Math Challenge: The student
has not yet acquired math facts.
Solution: Use these strategies:
• Incremental Rehearsal
• Cover-Copy-Compare
• Peer Tutoring in Math Computation with
Constant Time Delay
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Response to Intervention
Acquisition Stage: Math Review: Incremental
Rehearsal of ‘Math Facts’
Step 1: The tutor writes
down on a series of index
cards the math facts that the
student needs to learn. The
problems are written without
the answers.
4 x 5 =__
2 x 6 =__
5 x 5 =__
3 x 2 =__
3 x 8 =__
5 x 3 =__
6 x 5 =__
9 x 2 =__
3 x 6 =__
8 x 2 =__
4 x 7 =__
8 x 4 =__
9 x 7 =__
7 x 6 =__
3 x 5 =__
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Response to Intervention
Math Review: Incremental Rehearsal of ‘Math Facts’
Step 2: The tutor reviews
the ‘math fact’ cards with
the student. Any card
that the student can
answer within 2 seconds
is sorted into the
‘KNOWN’ pile. Any card
that the student cannot
answer within two
seconds—or answers
incorrectly—is sorted into
the ‘UNKNOWN’ pile.
‘KNOWN’ Facts
‘UNKNOWN’ Facts
4 x 5 =__
2 x 6 =__
3 x 8 =__
3 x 2 =__
5 x 3 =__
9 x 2 =__
3 x 6 =__
8 x 4 =__
5 x 5 =__
6 x 5 =__
4 x 7 =__
8 x 2 =__
9 x 7 =__
7 x 6 =__
3 x 5 =__
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Response to Intervention
Math Review: Incremental Rehearsal of ‘Math Facts’
Step 3: The
Nexttutor
the tutor
is now
then
takes
repeats
ready
a math
the
to follow
sequence--adding
fact afrom
nine-step
the ‘known’
incremental-rehearsal
yet another
pile and known
pairs it with
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the tutor
When
deck
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shown
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student
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student
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an ‘unknown’
off
student
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whole
tutor
it.series
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gives
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review
deck then
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and provide
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percent
answer.
‘known’ to 10 percent ‘unknown’ material )
3 x 8 =__
4 x 5 =__
2 x 6 =__
3 x 2 =__
3 x 6 =__
5 x 3 =__
8 x 4 =__
6 x 5 =__
4 x 7 =__
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Response to Intervention
Math Review: Incremental Rehearsal of ‘Math Facts’
Step 4: At
Thethis
student
point, isthethen
lastpresented
‘known’ math
with fact
a new
that‘unknown’
had beenmath
added
fact
to to
the
answer--and
student’s
review
the deck
reviewis sequence
discarded is(placed
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back into
repeated
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pile of
until
‘known’
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and the
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‘known’
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arestudent
discontinued
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deck when
for future
timedrills.
runs out or when the
student answers an ‘unknown’ math fact incorrectly three times.
9 x 2 =__
34 xx 85 =__
=__
42 xx 56 =__
=__
32 x 26 =__
3 x 62 =__
35 xx 63 =__
=__
85 x 43 =__
68 x 54 =__
64 xx 57 =__
=__
3 x 8 =__
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83
Response to Intervention
Cover-Copy-Compare:
Math Computational Fluency-Building Intervention
The student is given sheet with correctly completed
math problems in left column and index card.
For each problem, the student:
–
–
–
–
–
studies the model
covers the model with index card
copies the problem from memory
solves the problem
uncovers the correctly completed model to check answer
Source: Skinner, C.H., Turco, T.L., Beatty, K.L., & Rasavage, C. (1989). Cover, copy, and compare: A method for increasing
multiplication performance. School Psychology Review, 18, 412-420.
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Response to Intervention
Peer Tutoring in Math
Computation with
Constant Time Delay
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Response to Intervention
Peer Tutoring in Math Computation with Constant
Time Delay
• DESCRIPTION: This intervention employs students as reciprocal
peer tutors to target acquisition of basic math facts (math
computation) using constant time delay (Menesses & Gresham,
2009; Telecsan, Slaton, & Stevens, 1999). Each tutoring
‘session’ is brief and includes its own progress-monitoring
component--making this a convenient and time-efficient math
intervention for busy classrooms.
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86
Response to Intervention
Peer Tutoring in Math Computation with Constant
Time Delay
MATERIALS:
Student Packet: A work folder is created for each tutor pair. The
folder contains:
 10 math fact cards with equations written on the front and correct
answer appearing on the back. NOTE: The set of cards is
replenished and updated regularly as tutoring pairs master their
math facts.
 Progress-monitoring form for each student.
 Pencils.
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Response to Intervention
Peer Tutoring in Math Computation with Constant Time Delay
PREPARATION: To prepare for the tutoring program, the teacher
selects students to participate and trains them to serve as tutors.
Select Student Participants. Students being considered for the
reciprocal peer tutor program should at minimum meet these
criteria (Telecsan, Slaton, & Stevens, 1999, Menesses &
Gresham, 2009):
Is able and willing to follow directions;
Shows generally appropriate classroom behavior;
Can attend to a lesson or learning activity for at least 20
minutes.
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88
Response to Intervention
Peer Tutoring in Math Computation with Constant Time Delay
Select Student Participants (Cont.). Students being considered for the
reciprocal peer tutor program should at minimum meet these criteria
(Telecsan, Slaton, & Stevens, 1999, Menesses & Gresham, 2009):
 Is able to name all numbers from 0 to 18 (if tutoring in addition or
subtraction math facts) and name all numbers from 0 to 81 (if tutoring in
multiplication or division math facts).
• Can correctly read aloud a sampling of 10 math-facts (equation plus
answer) that will be used in the tutoring sessions. (NOTE: The student
does not need to have memorized or otherwise mastered these math
facts to participate—just be able to read them aloud from cards without
errors).
• [To document a deficit in math computation] When given a two-minute
math computation probe to complete independently, computes fewer
than 20 correct digits (Grades 1-3) or fewer than 40 correct digits
(Grades 4 and up) (Deno & Mirkin, 1977).
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Response to Intervention
Peer Tutoring in Math
Computation: Teacher
Nomination Form
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Response to Intervention
Peer Tutoring in Math Computation with Constant Time Delay
Tutoring Activity. Each tutoring ‘session’ last for 3 minutes. The tutor:
– Presents Cards. The tutor presents each card to the tutee for 3
seconds.
– Provides Tutor Feedback. [When the tutee responds correctly] The
tutor acknowledges the correct answer and presents the next card.
[When the tutee does not respond within 3 seconds or responds
incorrectly] The tutor states the correct answer and has the tutee
repeat the correct answer. The tutor then presents the next card.
– Provides Praise. The tutor praises the tutee immediately following
correct answers.
– Shuffles Cards. When the tutor and tutee have reviewed all of the
math-fact carts, the tutor shuffles them before again presenting
cards.
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Response to Intervention
Peer Tutoring in Math Computation with Constant Time Delay
Progress-Monitoring Activity. The tutor concludes each 3-minute tutoring
session by assessing the number of math facts mastered by the tutee.
The tutor follows this sequence:
– Presents Cards. The tutor presents each card to the tutee for 3
seconds.
– Remains Silent. The tutor does not provide performance feedback
or praise to the tutee, or otherwise talk during the assessment
phase.
– Sorts Cards. Based on the tutee’s responses, the tutor sorts the
math-fact cards into ‘correct’ and ‘incorrect’ piles.
– Counts Cards and Records Totals. The tutor counts the number of
cards in the ‘correct’ and ‘incorrect’ piles and records the totals on
the tutee’s progress-monitoring chart.
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Response to Intervention
Peer Tutoring in Math Computation with Constant Time Delay
Tutoring Integrity Checks. As the student pairs complete the tutoring
activities, the supervising adult monitors the integrity with which the
intervention is carried out. At the conclusion of the tutoring session, the
adult gives feedback to the student pairs, praising successful
implementation and providing corrective feedback to students as
needed. NOTE: Teachers can use the attached form Peer Tutoring in
Math Computation with Constant Time Delay: Integrity Checklist to
conduct integrity checks of the intervention and student progressmonitoring components of the math peer tutoring.
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Response to Intervention
Peer Tutoring in
Math
Computation:
Intervention
Integrity Sheet:
(Part 1:
Tutoring
Activity)
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Response to Intervention
Peer Tutoring in
Math
Computation:
Intervention
Integrity Sheet
(Part 2:
ProgressMonitoring)
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Response to Intervention
Peer Tutoring in
Math
Computation:
Score Sheet
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Response to Intervention
Math Challenge: The student
has acquired math computation
skills but is not yet fluent.
Solution: Use these strategies:
• Explicit Time Drills
• Self-Administered Arithmetic Combination Drills
With Performance Self-Monitoring & Incentives
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97
Response to Intervention
Explicit Time Drills:
Math Computational Fluency-Building Intervention
Explicit time-drills are a method to boost students’ rate of
responding on math-fact worksheets.
The teacher hands out the worksheet. Students are told that they
will have 3 minutes to work on problems on the sheet. The
teacher starts the stop watch and tells the students to start work.
At the end of the first minute in the 3-minute span, the teacher
‘calls time’, stops the stopwatch, and tells the students to
underline the last number written and to put their pencils in the
air. Then students are told to resume work and the teacher
restarts the stopwatch. This process is repeated at the end of
minutes 2 and 3. At the conclusion of the 3 minutes, the teacher
collects the student worksheets.
Source: Rhymer, K. N., Skinner, C. H., Jackson, S., McNeill, S., Smith, T., & Jackson, B. (2002). The 1-minute explicit timing
intervention: The influence of mathematics problem difficulty. Journal of Instructional Psychology, 29(4), 305-311.
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98
Response to Intervention
Fluency Stage: Math Computation
Self-Administered Arithmetic Combination Drills With Performance
Self-Monitoring & Incentives
1.
2.
3.
4.
5.
6.
The student is given a math computation worksheet of a specific problem type, along with
an answer key [Academic Opportunity to Respond].
The student consults his or her performance chart and notes previous performance. The
student is encouraged to try to ‘beat’ his or her most recent score.
The student is given a pre-selected amount of time (e.g., 5 minutes) to complete as many
problems as possible. The student sets a timer and works on the computation sheet until
the timer rings. [Active Student Responding]
The student checks his or her work, giving credit for each correct digit (digit of correct
value appearing in the correct place-position in the answer). [Performance Feedback]
The student records the day’s score of TOTAL number of correct digits on his or her
personal performance chart.
The student receives praise or a reward if he or she exceeds the most recently posted
number of correct digits.
Application of ‘Learn Unit’ framework from : Heward, W.L. (1996). Three low-tech strategies for increasing the frequency of active student
response during group instruction. In R. Gardner, D. M.S ainato, J. O. Cooper, T. E. Heron, W. L. Heward, J. W. Eshleman,& T. A. Grossi
(Eds.), Behavior analysis in education: Focus on measurably superior instruction (pp.283-320). Pacific Grove, CA:Brooks/Cole.
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Response to Intervention
Self-Administered Arithmetic Combination Drills:
Examples of Student Worksheet and Answer Key
Worksheets created using Math Worksheet Generator. Available online at:
http://www.interventioncentral.org/htmdocs/tools/mathprobe/addsing.php
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Response to Intervention
Self-Administered Arithmetic Combination Drills…
Reward Given
Reward Given
Reward Given
Reward Given
No Reward
No Reward
No Reward
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Response to Intervention
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Response to Intervention
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Response to Intervention
Math Computation Fluency:
RTI Case Study
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Response to Intervention
RTI: Individual Case Study: Math Computation
• Jared is a fourth-grade student. His teacher, Mrs.
Rogers, became concerned because Jared is
much slower in completing math computation
problems than are his classmates.
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105
Response to Intervention
Tier 1: Math Interventions for Jared
• Jared’s school uses the Everyday Math curriculum
(McGraw Hill/University of Chicago). In addition to the
basic curriculum the series contains intervention
exercises for students who need additional practice or
remediation.
The instructor, Mrs. Rogers, works with a small group of
children in her room—including Jared—having them
complete these practice exercises to boost their math
computation fluency.
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Response to Intervention
Tier 2: Standard Protocol (Group): Math
Interventions for Jared
• Jared did not make sufficient progress in his Tier 1 intervention. So his
teacher brought the student up at a grade-level ‘data meeting’ held
once per month to consider students for Tier 2 interventions. The team
and teacher decided that Jared would be placed on the school’s
educational math software, AMATH Building Blocks, a ‘self-paced,
individualized mathematics tutorial covering the math traditionally
taught in grades K-4’.
Jared worked on the software in 20-minute daily sessions to increase
computation fluency in basic multiplication problems.
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Response to Intervention
Tier 2: Math Interventions for Jared (Cont.)
• During this ‘standardtreatment protocol’ Tier 2
intervention, Jared was
assessed using
Curriculum-Based
Measurement (CBM)
Math probes. The goal
was to bring Jared up to
at least 40 correct digits
per 2 minutes.
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Response to Intervention
Tier 2: Math Interventions for Jared (Cont.)
• Progress-monitoring worksheets were created using
the Math Computation Probe Generator on Intervention
Central (www.interventioncentral.org).
Example of Math
Computation
Probe: Answer
Key
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Response to Intervention
Tier 2: Math Interventions for Jared: Progress-Monitoring
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Response to Intervention
Tier 3: Individualized Plan: Math Interventions for Jared
• Progress-monitoring data showed that Jared did not make
expected progress in his Tier 2 intervention. So the school’s
RTI Problem-Solving Team met on the student. The team and
teacher noted that Jared counted on his fingers when
completing multiplication problems. This greatly slowed down
his computation fluency. The team decided to use a researchbased strategy, Explicit Time Drills, to increase Jared’s
computation speed and eliminate his dependence on fingercounting.
During this individualized intervention, Jared continued to be
assessed using Curriculum-Based Measurement (CBM) Math
probes. The goal was to bring Jared up to at least 40 correct
digits per 2 minutes.
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111
Response to Intervention
Explicit Time Drills:
Math Computational Fluency-Building Intervention
Explicit time-drills are a method to boost students’ rate of
responding on math-fact worksheets.
The teacher hands out the worksheet. Students are told that they
will have 3 minutes to work on problems on the sheet. The
teacher starts the stop watch and tells the students to start work.
At the end of the first minute in the 3-minute span, the teacher
‘calls time’, stops the stopwatch, and tells the students to
underline the last number written and to put their pencils in the
air. Then students are told to resume work and the teacher
restarts the stopwatch. This process is repeated at the end of
minutes 2 and 3. At the conclusion of the 3 minutes, the teacher
collects the student worksheets.
Source: Rhymer, K. N., Skinner, C. H., Jackson, S., McNeill, S., Smith, T., & Jackson, B. (2002). The 1-minute explicit timing
intervention: The influence of mathematics problem difficulty. Journal of Instructional Psychology, 29(4), 305-311.
www.interventioncentral.org
112
Response to Intervention
Tier 3: Math Interventions for Jared: Progress-Monitoring
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113
Response to Intervention
Tier 3: Math Interventions for Jared
Explicit Timed Drill Intervention: Outcome
• The progress-monitoring data showed that Jared was well
on track to meet his computation goal. At the RTI Team
follow-up meeting, the team and teacher agreed to
continue the fluency-building intervention for at least 3
more weeks. It was also noted that Jared no longer relied
on finger-counting when completing number problems, a
good sign that he had overcome an obstacle to math
computation.
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