Transcript Document

Response to Intervention
Math Skills: An Introduction
Jim Wright
www.interventioncentral.org
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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
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National
Mathematics
Advisory Panel
Report
13 March 2008
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Response to Intervention
Math Advisory Panel Report at:
http://www.ed.gov/about/bdscomm/list/
mathpanel/index.html
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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
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.
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.)
3. Applying: Being able to formulate problems
mathematically and to devise strategies for solving them
using concepts and procedures appropriately.
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.
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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.
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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.
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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.
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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
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
toderive 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.
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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
198543276
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
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Response to Intervention
Math Instruction: Unlock the Thoughts of Reluctant
Students Through Class Journaling
Students can effectively clarify their knowledge of math concepts and
problem-solving strategies through regular use of class ‘math journals’.
•
•
•
•
At the start of the year, the teacher introduces the journaling weekly assignment in
which students respond to teacher questions.
At first, the teacher presents ‘safe’ questions that tap into the students’ opinions and
attitudes about mathematics (e.g., ‘How important do you think it is nowadays for
cashiers in fast-food restaurants to be able to calculate in their head the amount of
change to give a customer?”). As students become comfortable with the journaling
activity, the teacher starts to pose questions about the students’ own mathematical
thinking relating to specific assignments. Students are encouraged to use numerals,
mathematical symbols, and diagrams in their journal entries to enhance their
explanations.
The teacher provides brief written comments on individual student entries, as well
as periodic oral feedback and encouragement to the entire class.
Teachers will find that journal entries are a concrete method for monitoring student
understanding of more abstract math concepts. To promote the quality of journal
entries, the teacher might also assign them an effort grade that will be calculated
into quarterly math report card grades.
Source: Baxter, J. A., Woodward, J., & Olson, D. (2005). Writing in mathematics: An alternative form of communication for
academically low-achieving students. Learning Disabilities Research & Practice, 20(2), 119–135.
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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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Response to Intervention
Math Review: Incremental Rehearsal of ‘Math Facts’
Step 3: The
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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
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9 x 2 =__
34 xx 85 =__
=__
42 xx 56 =__
=__
32 x 62 =__
3 x 62 =__
35 xx 63 =__
=__
85 x 43 =__
68 x 45 =__
64 xx 57 =__
=__
3 x 8 =__
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Response to Intervention
Applied Math: Helping Students to
Make Sense of ‘Story Problems’
Jim Wright
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Response to Intervention
‘Advanced Math’ Quotes from Yogi Berra—
• “Ninety percent of the game is half mental."
• “Pair up in threes."
• “You give 100 percent in the first half of the
game, and if that isn't enough in the second half
you give what's left.”
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Response to Intervention
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Response to Intervention
‘Mindful Math’: Applying a Simple Heuristic to Applied
Problems
By following an efficient 4-step plan, students can consistently
perform better on applied math problems.
•
•
•
•
UNDERSTAND THE PROBLEM. To fully grasp the problem, the
student may restate the problem in his or her own words, note key
information, and identify missing information.
DEVISE A PLAN. In mapping out a strategy to solve the problem, the
student may make a table, draw a diagram, or translate the verbal
problem into an equation.
CARRY OUT THE PLAN. The student implements the steps in the
plan, showing work and checking work for each step.
LOOK BACK. The student checks the results. If the answer is written
as an equation, the student puts the results in words and checks
whether the answer addresses the question posed in the original
word problem.
Source: Pólya, G. (1945). How to solve it. Princeton University Press: Princeton, N.J.
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Response to Intervention
Applied Problems: Timed Quiz
4-Step Problem-Solving:
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•
•
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UNDERSTAND
THE PROBLEM.
DEVISE A PLAN.
CARRY OUT THE
•
PLAN.
LOOK BACK.
There are six bananas. Suppose 6
monkeys require 6 minutes to eat
those 6 bananas. (Each monkey eats
at exactly the same rate.)
How many minutes would it take 3
monkeys to eat 3 bananas?
• How many monkeys would it take to
eat 48 bananas in 48 minutes?
Source: Puzzles & Brain Teasers: Monkeys & Bananas. (n.d.). Retrieved on October 22, 2007, from
http://www.syvum.com/cgi/online/serve.cgi/teasers/monkeys.tdf?0
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Response to Intervention
Applied Problems: Timed Quiz
4-Step Problem-Solving:
•
•
•
•
UNDERSTAND
THE PROBLEM.
DEVISE A PLAN.
CARRY OUT THE
PLAN.
LOOK BACK.
Mr. Brown has 12 black gloves and 6
brown gloves in his closet. He blindly
picks up some gloves from the closet.
What is the minimum number of
gloves Mr. Brown will have to pick to
be certain to find two gloves of the
same color?
Source: Puzzles & Brain Teasers: Monkeys & Bananas. (n.d.). Retrieved on October 22, 2007, from
http://www.syvum.com/cgi/online/serve.cgi/teasers/monkeys.tdf?0
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Response to Intervention
Applied Math Problems: Rationale
• Applied math problems (also known as ‘story’ or
‘word’ problems) are traditional tools for having
students apply math concepts and operations to
‘real-world’ settings.
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Response to Intervention
Sample Applied Problems
• Once upon a time, there were three little pigs - ages 2, 4, and 6.
Are their ages even or odd?
• Every day this past summer, Peter rode his bike to and from
work. Each round trip was 13 kilometers. His friend Marsha rode
her bike18 kilometers each day, but just for exercise. How much
further did Marsha ride her bike than Peter in one week?
• Suzy is ten years older than Billy, and next year she will be twice
as old as Billy. How old are they now?
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Response to Intervention
Applied Math Problems: Some Required
Competencies
For students to achieve success with applied problems, they must
be able to:
 Comprehend the text of written problems.
 Understand specialized math vocabulary (e.g., ‘quotient’).
 Understand specialized use of ‘common’ vocabulary (e.g.,
‘product’).
 Be able to translate verbal cues into a numeric equation.
 Ignore irrelevant information included in the problem.
 Interpret math graphics that may accompany the problem.
 Apply a plan to problem-solving.
 Check their work.
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Response to Intervention
Potential ‘Blockers’ of Higher-Level Math Problem-Solving:
A Sampler
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Limited reading skills
Failure to master--or develop automaticity in– basic math operations
Lack of knowledge of specialized math vocabulary (e.g., ‘quotient’)
Lack of familiarity with the specialized use of known words (e.g.,
‘product’)
• Inability to interpret specialized math symbols
(e.g., ‘4 < 2’)
• Difficulty ‘extracting’ underlying math operations from word/story
problems or identifying and ignoring extraneous information included in
word/story problems
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Response to Intervention
Comprehending Math Vocabulary: The Barrier of Abstraction
“…when it comes to abstract mathematical
concepts, words describe activities or relationships that
often lack a visual counterpart. Yet studies show that
children grasp the idea of quantity, as well as other
relational concepts, from a very early age…. As children
develop their capacity for understanding, language, and
its vocabulary, becomes a vital cognitive link between a
child’s natural sense of number and order and
conceptual learning. ”
-Chard, D. (n.d.)
Source: Chard, D. (n.d.. Vocabulary strategies for the mathematics classroom. Retrieved November 23, 2007, from
http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf.
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Response to Intervention
Math Vocabulary: Classroom (Tier I) Recommendations
• Preteach math vocabulary. Math vocabulary provides students with the
language tools to grasp abstract mathematical concepts and to explain
their own reasoning. Therefore, do not wait to teach that vocabulary
only at ‘point of use’. Instead, preview relevant math vocabulary as a
regular a part of the ‘background’ information that students receive in
preparation to learn new math concepts or operations.
• Model the relevant vocabulary when new concepts are taught.
Strengthen students’ grasp of new vocabulary by reviewing a number of
math problems with the class, each time consistently and explicitly
modeling the use of appropriate vocabulary to describe the concepts
being taught. Then have students engage in cooperative learning or
individual practice activities in which they too must successfully use the
new vocabulary—while the teacher provides targeted support to
students as needed.
• Ensure that students learn standard, widely accepted labels for
common math terms and operations and that they use them
consistently to describe their math problem-solving efforts.
Source: Chard, D. (n.d.. Vocabulary strategies for the mathematics classroom. Retrieved November 23, 2007, from
http://www.eduplace.com/state/pdf/author/chard_hmm05.pdf.
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Response to Intervention
Math Intervention: Tier I: High School: Peer Guided Pause
• Students are trained to work in pairs.
• At one or more appropriate review points in a math lecture, the
instructor directs students to pair up to work together for 4 minutes.
• During each Peer Guided Pause, students are given a worksheet that
contains one or more correctly completed word or number problems
illustrating the math concept(s) covered in the lecture. The sheet also
contains several additional, similar problems that pairs of students work
cooperatively to complete, along with an answer key.
• Student pairs are reminded to (a) monitor their understanding of the
lesson concepts; (b) review the correctly math model problem; (c) work
cooperatively on the additional problems, and (d) check their answers.
The teacher can direct student pairs to write their names on the
practice sheets and collect them to monitor student understanding.
Source: Hawkins, J., & Brady, M. P. (1994). The effects of independent and peer guided practice during instructional pauses on
the academic performance of students with mild handicaps. Education & Treatment of Children, 17 (1), 1-28.
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Response to Intervention
Applied Problems: Encourage Students to ‘Draw’ the Problem
Making a drawing of an applied, or ‘word’, problem is one easy
heuristic tool that students can use to help them to find the solution
and clarify misunderstandings.
• The teacher hands out a worksheet containing at least six word
problems. The teacher explains to students that making a picture of a
word problem sometimes makes that problem clearer and easier to
solve.
• The teacher and students then independently create drawings of
each of the problems on the worksheet. Next, the students show their
drawings for each problem, explaining each drawing and how it
relates to the word problem. The teacher also participates, explaining
his or her drawings to the class or group.
• Then students are directed independently to make drawings as an
intermediate problem-solving step when they are faced with
challenging word problems. NOTE: This strategy appears to be more
effective when used in later, rather than earlier, elementary grades.
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
Applied Problems: Individualized Self-Correction Checklists
Students can improve their accuracy on particular types of word and number
problems by using an ‘individualized self-instruction checklist’ that reminds them to
pay attention to their own specific error patterns.
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•
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The teacher meets with the student. Together they analyze common error patterns
that the student tends to commit on a particular problem type (e.g., ‘On addition
problems that require carrying, I don’t always remember to carry the number from
the previously added column.’).
For each type of error identified, the student and teacher together describe the
appropriate step to take to prevent the error from occurring (e.g., ‘When adding
each column, make sure to carry numbers when needed.’).
These self-check items are compiled into a single checklist. Students are then
encouraged to use their individualized self-instruction checklist whenever they work
independently on their number or word problems.
Source: Pólya, G. (1945). How to solve it. Princeton University Press: Princeton, N.J.
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Response to Intervention
Activity: Selecting the ‘Best of
the Best’ Tier I Intervention
Ideas
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Response to Intervention
Tier I Interventions List: Activity
• Scan the page of ‘Tier I’ (Classroom)
interventions on mathematics
from your packet
• Select 2-3 TOP ideas
from your reading that you
feel should be on
teachers’ ‘Tier I’ intervention
list at your school
• Be prepared to share your ideas
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Response to Intervention
Interpreting Math Graphics
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Response to Intervention
Housing
Bubble
Graphic:
New York Times
23 September 2007
Housing Price
Index = 171 in
2005
Housing Price
Index = 100 in
1987
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Response to Intervention
Classroom Challenges in Interpreting Math Graphics
When encountering math graphics, students may :
•
•
•
•
•
expect the answer to be easily accessible when in fact the
graphic may expect the reader to interpret and draw
conclusions
be inattentive to details of the graphic
treat irrelevant data as ‘relevant’
not pay close attention to questions before turning to
graphics to find the answer
fail to use their prior knowledge both to extend the
information on the graphic and to act as a possible ‘check’
on the information that it presents.
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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Response to Intervention
Using Question-Answer Relationships (QARs) to Interpret
Information from Math Graphics
Students can be more savvy interpreters of graphics in
applied math problems by applying the Question-Answer
Relationship (QAR) strategy. Four Kinds of QAR Questions:
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•
•
RIGHT THERE questions are fact-based and can be found in a single sentence, often
accompanied by 'clue' words that also appear in the question.
THINK AND SEARCH questions can be answered by information in the text but
require the scanning of text and making connections between different pieces of
factual information.
AUTHOR AND YOU questions require that students take information or opinions that
appear in the text and combine them with the reader's own experiences or opinions to
formulate an answer.
ON MY OWN questions are based on the students' own experiences and do not
require knowledge of the text to answer.
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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Response to Intervention
Using Question-Answer Relationships (QARs) to Interpret
Information from Math Graphics: 4-Step Teaching Sequence
1.
DISTINGUISHING DIFFERENT KINDS OF GRAPHICS. Students
are taught to differentiate between common types of graphics: e.g.,
table (grid with information contained in cells), chart (boxes with
possible connecting lines or arrows), picture (figure with labels), line
graph, bar graph.
Students note significant differences between the various graphics,
while the teacher records those observations on a wall chart. Next
students are given examples of graphics and asked to identify which
general kind of graphic each is.
Finally, students are assigned to go on a ‘graphics hunt’, locating
graphics in magazines and newspapers, labeling them, and bringing
to class to review.
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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Response to Intervention
Using Question-Answer Relationships (QARs) to Interpret
Information from Math Graphics: 4-Step Teaching Sequence
2.
INTERPRETING INFORMATION IN GRAPHICS. Students are paired
off, with stronger students matched with less strong ones. The
teacher spends at least one session presenting students with
examples from each of the graphics categories.
The presentation sequence is ordered so that students begin with
examples of the most concrete graphics and move toward the more
abstract: Pictures > tables > bar graphs > charts > line graphs.
At each session, student pairs examine graphics and discuss
questions such as: “What information does this graphic present?
What are strengths of this graphic for presenting data? What are
possible weaknesses?”
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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Response to Intervention
Using Question-Answer Relationships (QARs) to Interpret
Information from Math Graphics: 4-Step Teaching Sequence
3.
LINKING THE USE OF QARS TO GRAPHICS. Students are given a
series of data questions and correct answers, with each question
accompanied by a graphic that contains information needed to
formulate the answer.
Students are also each given index cards with titles and descriptions
of each of the 4 QAR questions: RIGHT THERE, THINK AND
SEARCH, AUTHOR AND YOU, ON MY OWN.
Working in small groups and then individually, students read the
questions, study the matching graphics, and ‘verify’ the answers as
correct. They then identify the type question being asked using their
QAR index cards.
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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Response to Intervention
Using Question-Answer Relationships (QARs) to Interpret
Information from Math Graphics: 4-Step Teaching Sequence
4.
USING QARS WITH GRAPHICS INDEPENDENTLY. When students
are ready to use the QAR strategy independently to read graphics,
they are given a laminated card as a reference with 6 steps to follow:
A.
B.
C.
D.
E.
F.
Read the question,
Review the graphic,
Reread the question,
Choose a QAR,
Answer the question, and
Locate the answer derived from the graphic in the answer choices offered.
Students are strongly encouraged NOT to read the answer choices
offered until they have first derived their own answer, so that those
choices don’t short-circuit their inquiry.
Source: Mesmer, H.A.E., & Hutchins, E.J. (2002). Using QARs with charts and graphs. The Reading Teacher, 56, 21–27.
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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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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: Phase 1: Math Interventions for Jared: ProgressMonitoring
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Response to Intervention
Tier 2: 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 2: Phase 2: Math Interventions for Jared: ProgressMonitoring
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Response to Intervention
Tier 2: 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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