Transcript Document

Response to Intervention
RTI Teams: Best Practices
in Elementary Mathematics
Interventions
Jim Wright
www.interventioncentral.org
www.interventioncentral.org
Response to Intervention
PowerPoints from this workshop available at:
http://www.interventioncentral.org/
math_workshop.php
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2
Response to Intervention
Workshop Agenda…
Response to Intervention & Math Interventions: Brief Introduction
Big Ideas in Learning
Foundations of Mathematical Skills
Math Computation: Strategies
Assessing Math Interventions: Curriculum-Based Measurement
Web Resources to Support Math Interventions
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Response to Intervention
RTI is a Model in Development
“Several proposals for operationalizing response
to intervention have been made…The field can
expect more efforts like these and, for a time at
least, different models to be tested…Therefore, it
is premature to advocate any single model.”
(Barnett, Daly, Jones, & Lentz, 2004 )
Source: Barnett, D. W., Daly, E. J., Jones, K. M., & Lentz, F.E. (2004). Response to intervention: Empirically based special
service decisions from single-case designs of increasing and decreasing intensity. Journal of Special Education, 38, 66-79.
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Response to Intervention
Georgia
‘Pyramid of
Intervention’
Source: Georgia Dept of Education: http://www.doe.k12.ga.us/
Retrieved 13 July 2007
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Response to Intervention
How can a school restructure to support RTI?
The school can organize its intervention efforts into 4 levels, or Tiers, that represent a
continuum of increasing intensity of support. (Kovaleski, 2003; Vaughn, 2003). In
Georgia, Tier 1 is the lowest level of intervention, Tier 4 is the most intensive
intervention level.
Tier 1
Standards-Based Classroom Learning: All students participate in general
education learning that includes implementation of the Georgia Performance
Standards through research-based practices, use of flexible groups for
differentiation of instruction, & frequent progress-monitoring.
Tier 2
Needs Based Learning: Targeted students participate in learning that is in
addition to Tier 1 and different by including formalized processes of intervention
& greater frequency of progress-monitoring.
Tier 3
SST Driven Learning: Targeted students participate in learning that is in
addition to Tier I & II and different by including individualized assessments,
interventions tailored to individual needs, referral for specially designed
instruction if needed.
Tier 4
Specially Designed Learning: Targeted students participate in learning that
includes specialized programs, adapted content, methodology, or instructional
delivery; Georgia Performance standards access/extension.
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Response to Intervention
The Purpose of RTI in Secondary Schools: What
Students Should It Serve?
Early Identification.
As students begin to
show need for
academic support, the
RTI model proactively
supports them with
early interventions to
close the skill or
performance gap with
peers.
Chronically At-Risk.
Students whose
school performance is
marginal across
school years but who
do not qualify for
special education
services are identified
by the RTI Team and
provided with ongoing
intervention support.
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Special Education.
Students who fail to
respond to
scientifically valid
general-education
interventions
implemented with
integrity are classified
as ‘non-responders’
and found eligible for
special education.
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Response to Intervention
Math Intervention Planning: Some Challenges for
Elementary RTI Teams
• There is no national consensus about what math
instruction should look like in elementary schools
• Schools may not have consistent expectations
for the ‘best practice’ math instruction strategies
that teachers should routinely use in the
classroom
• Schools may not have a full range of
assessment methods to collect baseline and
progress monitoring data on math difficulties
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Response to Intervention
Focus of This Math Interventions Workshop…
• Intervention and assessment strategies that
supplement the ‘core curriculum’
• NOTE: If greater than 20 percent of students in a
classroom or grade level experience significant
math difficulties, the focus should be on giving
the teacher skills for effective whole-group
instruction or on improving the ‘core curriculum’
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Response to Intervention
‘Big Ideas’ About Student
Learning
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Response to Intervention
Big Ideas: Student Social & Academic Behaviors
Are Strongly Influenced by the Instructional Setting
(Lentz & Shapiro, 1986)
• Students with learning problems do not exist in
isolation. Rather, their instructional environment
plays an enormously important role in these
students’ eventual success or failure
Source: Lentz, F. E. & Shapiro, E. S. (1986). Functional assessment of the academic environment. School Psychology Review,
15, 346-57.
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Response to Intervention
Big Ideas: Learn Unit (Heward, 1996)
The three essential elements of effective student learning include:
1. Academic Opportunity to Respond. The student is presented with
a meaningful opportunity to respond to an academic task. A question posed by
the teacher, a math word problem, and a spelling item on an educational computer
‘Word Gobbler’ game could all be considered academic opportunities to respond.
2. Active Student Response. The student answers the item, solves the problem
presented, or completes the academic task. Answering the teacher’s question,
computing the answer to a math word problem (and showing all work), and typing
in the correct spelling of an item when playing an educational computer game are
all examples of active student responding.
3. Performance Feedback. The student receives timely feedback about whether his
or her response is correct—often with praise and encouragement. A teacher
exclaiming ‘Right! Good job!’ when a student gives an response in class, a student
using an answer key to check her answer to a math word problem, and a
computer message that says ‘Congratulations! You get 2 points for correctly
spelling this word!” are all examples of performance feedback.
Source: 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.
www.interventioncentral.org
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Response to Intervention
Big Ideas: The Four Stages of Learning Can Be
Summed Up in the ‘Instructional Hierarchy’
(Haring et al., 1978)
Student learning can be thought of as a multi-stage process. The
universal stages of learning include:
• Acquisition: The student is just acquiring the skill.
• Fluency: The student can perform the skill but
must make that skill ‘automatic’.
• Generalization: The student must perform the skill
across situations or settings.
• Adaptation: The student confronts novel task
demands that require that the student adapt a
current skill to meet new requirements.
Source: Haring, N.G., Lovitt, T.C., Eaton, M.D., & Hansen, C.L. (1978). The fourth R: Research in the classroom. Columbus,
OH: Charles E. Merrill Publishing Co.
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Response to Intervention
www.interventioncentral.org
National
Mathematics
Advisory Panel
Report
13 March 2008
14
Response to Intervention
Math Advisory Panel Report at:
http://www.ed.gov/mathpanel
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Response to Intervention
2008 National Math Advisory Panel Report: Recommendations
• “The areas to be studied in mathematics from pre-kindergarten through
eighth grade should be streamlined and a well-defined set of the most
important topics should be emphasized in the early grades. Any approach
that revisits topics year after year without bringing them to closure should
be avoided.”
• “Proficiency with whole numbers, fractions, and certain aspects of geometry
and measurement are the foundations for algebra. Of these, knowledge of
fractions is the most important foundational skill not developed among
American students.”
• “Conceptual understanding, computational and procedural fluency, and
problem solving skills are equally important and mutually reinforce each
other. Debates regarding the relative importance of each of these
components of mathematics are misguided.”
• “Students should develop immediate recall of arithmetic facts to free the
“working memory” for solving more complex problems.”
Source: National Math Panel Fact Sheet. (March 2008). Retrieved on March 14, 2008, from
http://www.ed.gov/about/bdscomm/list/mathpanel/report/final-factsheet.html
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Response to Intervention
‘Elbow Group’ Activity: What are common student
mathematics concerns in your school?
In your ‘elbow groups’:
•
Discuss the most common
student mathematics
problems that you encounter
in your school(s). At what
grade level do you typically
encounter these problems?
•
Be prepared to share your
discussion points with the
larger group.
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Response to Intervention
“Mathematics is made of 50 percent formulas,
50 percent proofs, and 50 percent
imagination.”
–Anonymous
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Response to Intervention
Who is At Risk for Poor Math Performance?: A
Proactive Stance
“…we use the term mathematics difficulties rather than
mathematics disabilities. Children who exhibit
mathematics difficulties include those performing in the
low average range (e.g., at or below the 35th percentile)
as well as those performing well below average…Using
higher percentile cutoffs increases the likelihood that
young children who go on to have serious math
problems will be picked up in the screening.” p. 295
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
Profile of Students with Math Difficulties
(Kroesbergen & Van Luit, 2003)
[Although the group of students with
difficulties in learning math is very
heterogeneous], in general, these students
have memory deficits leading to difficulties
in the acquisition and remembering of math knowledge.
Moreover, they often show inadequate use of strategies
for solving math tasks, caused by problems with the
acquisition and the application of both cognitive and
metacognitive strategies.
Because of these problems, they also show deficits in
generalization and transfer of learned knowledge to new
and unknown tasks.
Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs.
Remedial and Special Education, 24, 97-114..
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Response to Intervention
The Elements of Mathematical
Proficiency: What the Experts Say…
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Response to Intervention
5 Strands of Mathematical
Proficiency
5 Big Ideas in Beginning
Reading
1. Understanding
1. Phonemic Awareness
2. Computing
2. Alphabetic Principle
3. Applying
3. Fluency with Text
4. Reasoning
4. Vocabulary
5. Engagement
5. Comprehension
Source: National Research Council. (2002). Helping
children learn mathematics. Mathematics Learning Study
Committee, J. Kilpatrick & J. Swafford, Editors, Center for
Education, Division of Behavioral and Social Sciences
and Education. Washington, DC: National Academy
Press.
Source: Big ideas in beginning reading.
University of Oregon. Retrieved September 23,
2007, from http://reading.uoregon.edu/index.php
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Response to Intervention
Five Strands of Mathematical Proficiency
1.
Understanding: Comprehending mathematical concepts,
operations, and relations--knowing what mathematical
symbols, diagrams, and procedures mean.
2.
Computing: Carrying out mathematical procedures, such
as adding, subtracting, multiplying, and dividing numbers
flexibly, accurately, efficiently, and appropriately.
3.
Applying: Being able to formulate problems
mathematically and to devise strategies for solving them
using concepts and procedures appropriately.
Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study
Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and
Education. Washington, DC: National Academy Press.
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Response to Intervention
Five Strands of Mathematical Proficiency (Cont.)
4.
Reasoning: Using logic to explain and justify a solution to
a problem or to extend from something known to
something less known.
5.
Engaging: Seeing mathematics as sensible, useful, and
doable—if you work at it—and being willing to do the
work.
Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study
Committee, J. Kilpatrick & J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and
Education. Washington, DC: National Academy Press.
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Response to Intervention
Three General Levels of Math Skill Development
(Kroesbergen & Van Luit, 2003)
As students move from lower to higher grades, they move through
levels of acquisition of math skills, to include:
• Number sense
• Basic math operations (i.e., addition, subtraction,
multiplication, division)
• Problem-solving skills: “The solution of both verbal
and nonverbal problems through the application of previously
acquired information” (Kroesbergen & Van Luit, 2003, p. 98)
Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs.
Remedial and Special Education, 24, 97-114..
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Response to Intervention
What is ‘Number Sense’?
(Clarke & Shinn, 2004)
“… the ability to understand the meaning of
numbers and define different relationships among
numbers.
Children with number sense can recognize the
relative size of numbers, use referents for
measuring objects and events, and think and work
with numbers in a flexible manner that treats
numbers as a sensible system.” p. 236
Source: Clarke, B., & Shinn, M. (2004). A preliminary investigation into the identification and development of early mathematics
curriculum-based measurement. School Psychology Review, 33, 234–248.
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Response to Intervention
What Are Stages of ‘Number
Sense’?
(Berch, 2005, p. 336)
1.
2.
Innate Number Sense. Children appear to possess ‘hardwired’ ability (neurological ‘foundation structures’) to acquire
number sense. Children’s innate capabilities appear also to be
to ‘represent general amounts’, not specific quantities. This
innate number sense seems to be characterized by skills at
estimation (‘approximate numerical judgments’) and a counting
system that can be described loosely as ‘1, 2, 3, 4, … a lot’.
Acquired Number Sense. Young students learn through
indirect and direct instruction to count specific objects beyond
four and to internalize a number line as a mental representation
of those precise number values.
Source: Berch, D. B. (2005). Making sense of number sense: Implications for children with mathematical disabilities. Journal of
Learning Disabilities, 38, 333-339...
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Response to Intervention
The Basic Number Line is as Familiar as a Well-Known
Place to People Who Have Mastered Arithmetic
Combinations
Moravia, NY
Number Line: 0-144
0 1 2 3 4
11 12 13 14
21 22 23 24
31 32 33 34
41 42 43 44
51 52 53 54
61 62 63 64
71 72 73 74
81 82 83 84
91 92 93 94
101 102 103
111 112 113
121 122 123
131 132 133
141 142 143
5 6 7 8
15 16 17 18
25 26 27 28
35 36 37 38
45 46 47 48
55 56 57 58
65 66 67 68
75 76 77 78
85 86 87 88
95 96 97 98
104 105 106
114 115 116
124 125 126
134 135 136
144
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9 10
19 20
29 30
39 40
49 50
59 60
69 70
79 80
89 90
99 100
107 108
117 118
127 128
137 138
109
119
129
139
110
120
130
140
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Response to Intervention
Internal Numberline
As students internalize the numberline, they are better
able to perform ‘mental arithmetic’ (the manipulation of
numbers and math operations in their head).
0
1
2
3
4
5
6
7
8
9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
328
÷774===21
7
9X
–
2
2+4=6
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Response to Intervention
Mental Arithmetic: A Demonstration
332 x 420 = ?
Directions: As you watch this video of a person using
mental arithmetic to solve a computation problem, note
the strategies and ‘shortcuts’ that he employs to make
the task more manageable.
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Response to Intervention
\Mental Arithmetic Demonstration: What Tools Were Used?
Solving
for…
6. Use
Mnemonic
Strategy
5. Continue
with
Next
‘Chunk’
of Problem: Math
4. Use Mnemonic
Strategy
to Remember
Intermediate
7. AddShortcut
Intermediate
Products:
Chunk
2.Remember
Break
Problem
intointo Smaller
Intermediate
1.toEstimate
Answer
Computation Product
Tasks
Manageable Chunks
Product
3. Apply Math Shortcut:
332
332
Add Zeros in One’s300
Place
6,640
332
132,800
132,800
for Each
of Ten
X Multiple
420
x
20
’66
is
a
x
400
1,328
‘1=3-2’
xfamous
x&46,640
‘800 is a
332national
x
10
=
3320
road’
132,800
+
6000
=
138,800
120,000
x1,328
100
toll-free
& ’40 is speed
number
’
132,800
+
640
=
139,440
3320
x
2
=
6640
limit
in
front
132,800
of house’
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Response to Intervention
Math Computation: Building
Fluency
Jim Wright
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Response to Intervention
"Arithmetic is being able to count up to twenty
without taking off your shoes."
–Anonymous
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Response to Intervention
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.
www.interventioncentral.org
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Response to Intervention
How much is 3 + 8?: Strategies to Solve…
Least efficient strategy: Count out and group 3 objects; count out and
group 8 objects; count all objects:
=11
+
More efficient strategy: Begin at the number 3 and ‘count up’ 8 more
digits (often using fingers for counting):
3+8
More efficient strategy: Begin at the number 8 (larger number) and
‘count up’ 3 more digits:
8+ 3
Most efficient strategy: ‘3 + 8’ arithmetic combination is stored in
memory and automatically retrieved: Answer = 11
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.
www.interventioncentral.org
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Response to Intervention
Math Skills: Importance of Fluency in Basic Math
Operations
“[A key step in math education is] to learn the four basic
mathematical operations (i.e., addition, subtraction,
multiplication, and division). Knowledge of these
operations and a capacity to perform mental arithmetic
play an important role in the development of children’s
later math skills. Most children with math learning
difficulties are unable to master the four basic
operations before leaving elementary school and, thus,
need special attention to acquire the skills. A …
category of interventions is therefore aimed at the
acquisition and automatization of basic math skills.”
Source: Kroesbergen, E., & Van Luit, J. E. H. (2003). Mathematics interventions for children with special educational needs.
Remedial and Special Education, 24, 97-114.
www.interventioncentral.org
38
Response to Intervention
Big Ideas: Learn Unit (Heward, 1996)
The three essential elements of effective student learning include:
1. Academic Opportunity to Respond. The student is presented with
a meaningful opportunity to respond to an academic task. A question posed by
the teacher, a math word problem, and a spelling item on an educational computer
‘Word Gobbler’ game could all be considered academic opportunities to respond.
2. Active Student Response. The student answers the item, solves the problem
presented, or completes the academic task. Answering the teacher’s question,
computing the answer to a math word problem (and showing all work), and typing
in the correct spelling of an item when playing an educational computer game are
all examples of active student responding.
3. Performance Feedback. The student receives timely feedback about whether his
or her response is correct—often with praise and encouragement. A teacher
exclaiming ‘Right! Good job!’ when a student gives an response in class, a student
using an answer key to check her answer to a math word problem, and a
computer message that says ‘Congratulations! You get 2 points for correctly
spelling this word!” are all examples of performance feedback.
Source: 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.
www.interventioncentral.org
39
Response to Intervention
Math Intervention: Tier I or II: Elementary & Secondary:
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
How to… Use PPT Group Timers in the Classroom
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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
Math Shortcuts: Cognitive Energy- and Time-Savers
“Recently, some researchers…have argued that
children can derive answers quickly and with minimal
cognitive effort by employing calculation principles or
“shortcuts,” such as using a known number combination
to derive an answer (2 + 2 = 4, so 2 + 3 =5), relations
among operations (6 + 4 =10, so 10 −4 = 6) … and so
forth. This approach to instruction is consonant with
recommendations by the National Research Council
(2001). Instruction along these lines may be much more
productive than rote drill without linkage to counting
strategy use.” p. 301
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.
www.interventioncentral.org
45
Response to Intervention
Math Multiplication Shortcut: ‘The 9 Times Quickie’
• The student uses fingers as markers to find the product of singledigit multiplication arithmetic combinations with 9.
• Fingers to the left of the lowered finger stands for the ’10’s place
value.
• Fingers to the right stand for the ‘1’s place value.
99xx10
198765432
Source: Russell, D. (n.d.). Math facts to learn the facts. Retrieved November 9, 2007, from http://math.about.com/bltricks.htm
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Response to Intervention
Students Who ‘Understand’ Mathematical
Concepts Can Discover Their Own ‘Shortcuts’
“Students who learn with understanding have less to
learn because they see common patterns in
superficially different situations. If they understand the
general principle that the order in which two numbers
are multiplied doesn’t matter—3 x 5 is the same as 5 x
3, for example—they have about half as many ‘number
facts’ to learn.” p. 10
Source: National Research Council. (2002). Helping children learn mathematics. Mathematics Learning Study Committee, J. Kilpatrick &
J. Swafford, Editors, Center for Education, Division of Behavioral and Social Sciences and Education. Washington, DC: National
Academy Press.
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Response to Intervention
Application of Math Shortcuts to Intervention Plans
• Students who struggle with math may find
computational ‘shortcuts’ to be motivating.
• Teaching and modeling of shortcuts provides students
with strategies to make computation less ‘cognitively
demanding’.
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Response to Intervention
Math Computation: Motivate With ‘Errorless
Learning’ Worksheets
In this version of an ‘errorless learning’ approach, the
student is directed to complete math facts as quickly as
possible. If the student comes to a number problem that
he or she cannot solve, the student is encouraged to locate
the problem and its correct answer in the key at the top of the page
and write it in.
Such speed drills build computational fluency while promoting
students’ ability to visualize and to use a mental number line.
TIP: Consider turning this activity into a ‘speed drill’. The student is
given a kitchen timer and instructed to set the timer for a
predetermined span of time (e.g., 2 minutes) for each drill. The
student completes as many problems as possible before the timer
rings. The student then graphs the number of problems correctly
computed each day on a time-series graph, attempting to better his or
her previous score.
Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282
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Response to Intervention
‘Errorless
Learning’ Worksheet Sample
Source: Caron, T. A. (2007). Learning multiplication the easy way. The Clearing House, 80, 278-282
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Response to Intervention
Math Computation: Two Ideas to Jump-Start
Active Academic Responding
Here are two ideas to accomplish increased academic
responding on math tasks.
• Break longer assignments into shorter assignments with
performance feedback given after each shorter ‘chunk’ (e.g., break a
20-minute math computation worksheet task into 3 seven-minute
assignments). Breaking longer assignments into briefer segments
also allows the teacher to praise struggling students more frequently
for work completion and effort, providing an additional ‘natural’
reinforcer.
• Allow students to respond to easier practice items orally rather than in
written form to speed up the rate of correct responses.
Source: Skinner, C. H., Pappas, D. N., & Davis, K. A. (2005). Enhancing academic engagement: Providing opportunities for
responding and influencing students to choose to respond. Psychology in the Schools, 42, 389-403.
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Response to Intervention
Math Computation: Problem Interspersal Technique
• The teacher first identifies the range of ‘challenging’ problem-types
(number problems appropriately matched to the student’s current
instructional level) that are to appear on the worksheet.
• Then the teacher creates a series of ‘easy’ problems that the students
can complete very quickly (e.g., adding or subtracting two 1-digit
numbers). The teacher next prepares a series of student math
computation worksheets with ‘easy’ computation problems
interspersed at a fixed rate among the ‘challenging’ problems.
• If the student is expected to complete the worksheet independently,
‘challenging’ and ‘easy’ problems should be interspersed at a 1:1 ratio
(that is, every ‘challenging’ problem in the worksheet is preceded
and/or followed by an ‘easy’ problem).
• If the student is to have the problems read aloud and then asked to
solve the problems mentally and write down only the answer, the
items should appear on the worksheet at a ratio of 3 ‘challenging’
problems for every ‘easy’ one (that is, every 3 ‘challenging’ problems
are preceded and/or followed by an ‘easy’ one).
Source: Hawkins, J., Skinner, C. H., & Oliver, R. (2005). The effects of task demands and additive interspersal ratios on fifthgrade students’ mathematics accuracy. School Psychology Review, 34, 543-555..
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Response to Intervention
How to… Create an Interspersal-Problems Worksheet
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Response to Intervention
Additional Math Interventions
Jim Wright
www.interventioncentral.org
www.interventioncentral.org
Response to Intervention
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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56
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
the
sequence:
problem
unknown
to First,
theproblem.
growing
the tutor
When
deck
presents
of
shown
index
the
each
cards
student
ofbeing
thewith
two
reviewed
aproblems,
single and
index
the
each
card
student
time is
asked
containing
prompting
to read
the
an ‘unknown’
off
student
the problem
to math
answer
and
fact.
the
answer
The
whole
tutor
it.series
readsofthe
math
problem
facts—until
aloud,the
gives
the answer,
review
deck then
contains
prompts
a total
theofstudent
one ‘unknown’
to read off
math
thefact
same
andunknown
nine ‘known’
problem
math
and provide
facts
(a ratiothe
of 90
correct
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
once again
back into
repeated
the original
each time
pile of
until
‘known’
the
‘unknown’ math
problems)
and the
factpreviously
is grouped
‘unknown’
with ninemath
‘known’
factmath
is now
facts—and
treated asonthe
and
firston.
Daily review
‘known’
mathsessions
fact in new
arestudent
discontinued
revieweither
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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Response to Intervention
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Response to Intervention
Measuring the ‘Intervention
Footprint’: Issues of
Planning, Documentation, &
Follow-Through
Jim Wright
www.interventioncentral.org
www.interventioncentral.org
Response to Intervention
Essential Elements of Any Academic or Behavioral
Intervention (‘Treatment’) Strategy:
•
Method of delivery (‘Who or what delivers the treatment?’)
Examples include teachers, paraprofessionals, parents, volunteers,
computers.
•
Treatment component (‘What makes the intervention effective?’)
Examples include activation of prior knowledge to help the student to
make meaningful connections between ‘known’ and new material;
guide practice (e.g., Paired Reading) to increase reading fluency;
periodic review of material to aid student retention. As an example of a
research-based commercial program, Read Naturally ‘combines
teacher modeling, repeated reading and progress monitoring to
remediate fluency problems’.
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61
Response to Intervention
Interventions, Accommodations & Modifications:
Sorting Them Out
• Interventions. An academic intervention is a strategy
used to teach a new skill, build fluency in a skill, or
encourage a child to apply an existing skill to new
situations or settings.
An intervention is said to be research-based when it has
been demonstrated to be effective in one or more
articles published in peer–reviewed scientific journals.
Interventions might be based on commercial programs
such as Read Naturally. The school may also develop
and implement an intervention that is based on
guidelines provided in research articles—such as
Paired Reading (Topping, 1987).
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Response to Intervention
Interventions, Accommodations & Modifications:
Sorting Them Out
• Accommodations. An accommodation is intended to help
the student to fully access the general-education curriculum
without changing the instructional content. An
accommodation for students who are slow readers, for
example, may include having them supplement their silent
reading of a novel by listening to the book on tape.
An accommodation is intended to remove barriers to
learning while still expecting that students will master the
same instructional content as their typical peers. Informal
accommodations may be used at the classroom level or be
incorporated into a more intensive, individualized
intervention plan.
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Response to Intervention
Interventions, Accommodations & Modifications:
Sorting Them Out
• Modifications. A modification changes the expectations of
what a student is expected to know or do—typically by
lowering the academic expectations against which the
student is to be evaluated.
Examples of modifications are reducing the number of
multiple-choice items in a test from five to four or shortening
a spelling list. Under RTI, modifications are generally not
included in a student’s intervention plan, because the
working assumption is that the student can be successful in
the curriculum with appropriate interventions and
accommodations alone.
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Response to Intervention
Writing Quality ‘Problem Identification’
Statements
www.interventioncentral.org
Response to Intervention
Writing Quality ‘Problem Identification’ Statements
• A frequent problem at RTI Team meetings is that
teacher referral concerns are written in vague terms. If
the referral concern is not written in explicit, observable,
measurable terms, it will be very difficult to write clear
goals for improvement or select appropriate
interventions.
• Use this ‘test’ for evaluating the quality of a problemidentification (‘teacher-concern’) statement: Can a third
party enter a classroom with the problem definition
in hand and know when they see the behavior and
when they don’t?
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Response to Intervention
Writing Quality ‘Teacher Referral Concern’ Statements:
Examples
• Needs Work: The student is disruptive.
• Better: During independent seatwork , the
student is out of her seat frequently and talking
with other students.
• Needs Work: The student doesn’t do his math.
• Better: When math homework is assigned, the
student turns in math homework only about 20
percent of the time. Assignments turned in are
often not fully completed.
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Response to Intervention
Math Computation Fluency:
RTI Case Study
www.interventioncentral.org
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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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 referred the student to the RTI Intervention Team. 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 group-based Tier 2
intervention, Jared was
assessed using CurriculumBased 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 the first phase of his Tier 2 intervention.
So the RTI Intervention Team met again 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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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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Response to Intervention
Tier 3: Math Interventions for Jared: Progress-Monitoring
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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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