Response to Intervention
RTI: Best Practices
in Mathematics
Interventions for Elementary
Students
Jim Wright
www.interventioncentral.org
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Response to Intervention
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National
Mathematics
Advisory Panel
Report
13 March 2008
2
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
“Mathematics is made of 50 percent formulas,
50 percent proofs, and 50 percent
imagination.”
–Anonymous
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Response to Intervention
An RTI Challenge: Limited Research to Support
Evidence-Based Math Interventions
“… in contrast to reading, core math programs that are
supported by research, or that have been constructed
according to clear research-based principles, are not
easy to identify. Not only have exemplary core
programs not been identified, but also there are no
tools available that we know of that will help schools
analyze core math programs to determine their
alignment with clear research-based principles.” p. 459
Source: Clarke, B., Baker, S., & Chard, D. (2008). Best practices in mathematics assessment and intervention with elementary
students. In A. Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp. 453-463).
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Response to Intervention
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
Math Computation: Building
Fluency
Jim Wright
www.interventioncentral.org
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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.
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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
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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 p. 67
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 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
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,
which represented the 25th
percentile performance level on
the grade level’s universal
screening for math computation.
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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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Response to Intervention
Math Intervention Ideas for
Secondary Classrooms
Jim Wright
www.interventioncentral.org
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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
Potential ‘Blockers’ of Higher-Level Math Problem-Solving:
A Sampler
 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
 Difficulty identifying and ignoring extraneous information included in
word/story problems
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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:
•
•
•
•
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
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
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
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Response to Intervention
Secondary GroupBased Math
Intervention
Example
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Response to Intervention
1.
2.
3.
Math Mentors: Training Students to Independently Use
On-Line Math-Help Resources
Math mentors are recruited (school personnel, adult volunteers,
student teachers, peer tutors) who have a good working knowledge of
algebra.
The school meets with each math mentor to verify mentor’s algebra
knowledge.
The school trains math mentors in 30-minute tutoring protocol, to
include:
A.
B.
C.
4.
Requiring that students keep a math journal detailing questions from notes and
homework.
Holding the student accountable to bring journal, questions to tutoring session.
Ensuring that a minimum of 25 minutes of 30 minute session are spent on
tutoring.
Mentors are introduced to online algebra resources (e.g.,
www.algebrahelp.com, www.math.com) and encouraged to browse
them and become familiar with the site content and navigation.
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Response to Intervention
Math Mentors: Training Students to Independently Use
On-Line Math-Help Resources
5.
Mentors are trained during ‘math mentor’ sessions to:
A.
B.
C.
6.
Examine student math journal
Answer student algebra questions
Direct the student to go online to algebra tutorial websites while mentor
supervises. Student is to find the section(s) of the websites that answer their
questions.
As the student shows increased confidence with algebra and with
navigation of the math-help websites, the mentor directs the student
to:
A.
B.
C.
D.
Note math homework questions in the math journal
Attempt to find answers independently on math-help websites
Note in the journal any successful or unsuccessful attempts to independently get
answers online
Bring journal and remaining questions to next mentoring meeting.
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Response to Intervention
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