Transcript Document

Response to Intervention
Math Reasoning: Assisting the
Struggling Middle and High
School Learner
Jim Wright
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Response to Intervention
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Response to Intervention
Intervention Research &
Development: A Work in
Progress
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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
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
Tier 1: What Are the Recommended Elements of ‘Core
Curriculum’?: More Research Needed
“In essence, we now have a good beginning on the
evaluation of Tier 2 and 3 interventions, but no idea
about what it will take to get the core curriculum to
work at Tier 1. A complicating issue with this potential
line of research is that many schools use multiple
materials as their core program.” p. 640
Source: Kovaleski, J. F. (2007). Response to intervention: Considerations for research and systems change. School
Psychology Review, 36, 638-646.
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Response to Intervention
Limitations of Intervention Research…
“…the list of evidence-based interventions is quite
small relative to the need [of RTI]…. Thus, limited
dissemination of interventions is likely to be a
practical problem as individuals move forward in the
application of RTI models in applied settings.” p. 33
Source: Kratochwill, T. R., Clements, M. A., & Kalymon, K. M. (2007). Response to intervention: Conceptual and
methodological issues in implementation. In Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (Eds.), Handbook of
response to intervention: The science and practice of assessment and intervention. New York: Springer.
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Response to Intervention
Schools Need to Review Tier 1 (Classroom)
Interventions to Ensure That They Are Supported
By Research
There is a lack of agreement about what is meant by
‘scientifically validated’ classroom (Tier I) interventions. Districts
should establish a ‘vetting’ process—criteria for judging whether
a particular instructional or intervention approach should be
considered empirically based.
Source: Fuchs, D., & Deshler, D. D. (2007). What we need to know about responsiveness to intervention (and shouldn’t be
afraid to ask).. Learning Disabilities Research & Practice, 22(2),129–136.
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Response to Intervention
What Are Appropriate Content-Area Tier 1
Universal Interventions for Secondary Schools?
“High schools need to determine what
constitutes high-quality universal instruction
across content areas. In addition, high school
teachers need professional development in, for
example, differentiated instructional techniques
that will help ensure student access to
instruction interventions that are effectively
implemented.”
Source: Duffy, H. (August 2007). Meeting the needs of significantly struggling learners in high school. Washington, DC: National High
School Center. Retrieved from http://www.betterhighschools.org/pubs/ p. 9
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Response to Intervention
Intervention: Key Concepts
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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
Scripting Interventions to Promote Better Compliance
Interventions should be written up in a ‘scripted’
format to ensure that:
– Teachers have sufficient information about the
intervention to implement it correctly; and
– External observers can view the teacher implementing the
intervention strategy and—using the script as a
checklist—verify that each step of the intervention was
implemented correctly (‘treatment integrity’).
Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools.
Routledge: New York.
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Response to Intervention
Intervention Script
Builder Form
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Response to Intervention
Increasing the Intensity of an Intervention: Key Dimensions
Interventions can move up the RTI Tiers through being
intensified across several dimensions, including:
•
•
•
•
•
Type of intervention strategy or materials used
Student-teacher ratio
Length of intervention sessions
Frequency of intervention sessions
Duration of the intervention period (e.g., extending an intervention
from 5 weeks to 10 weeks)
• Motivation strategies
Source: Burns, M. K., & Gibbons, K. A. (2008). Implementing response-to-intervention in elementary and secondary schools.
Routledge: New York.
Kratochwill, T. R., Clements, M. A., & Kalymon, K. M. (2007). Response to intervention: Conceptual and methodological issues
in implementation. In Jimerson, S. R., Burns, M. K., & VanDerHeyden, A. M. (Eds.), Handbook of response to intervention: The
science and practice of assessment and intervention. New York: Springer.
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Response to Intervention
Research-Based Elements of
Effective Academic Interventions
• ‘Correctly targeted’: The intervention is appropriately matched
to the student’s academic or behavioral needs.
• ‘Explicit instruction’: Student skills have been broken down
“into manageable and deliberately sequenced steps and
providing overt strategies for students to learn and practice new
skills” p.1153
• ‘Appropriate level of challenge’: The student experiences
adequate success with the instructional task.
• ‘High opportunity to respond’: The student actively responds
at a rate frequent enough to promote effective learning.
• ‘Feedback’: The student receives prompt performance feedback
about the work completed.
Source: Burns, M. K., VanDerHeyden, A. M., & Boice, C. H. (2008). Best practices in intensive academic interventions. In A.
Thomas & J. Grimes (Eds.), Best practices in school psychology V (pp.1151-1162). Bethesda, MD: National Association of
School Psychologists.
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Response to Intervention
Core Instruction, Interventions, Accommodations
& Modifications: Sorting Them Out
• Core Instruction. Those instructional strategies
that are used routinely with all students in a
general-education setting are considered ‘core
instruction’. High-quality instruction is essential
and forms the foundation of RTI academic
support. NOTE: While it is important to verify that
good core instructional practices are in place for
a struggling student, those routine practices do
not ‘count’ as individual student interventions.
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Response to Intervention
Core Instruction, Interventions, Accommodations
& Modifications: Sorting Them Out
• Intervention. 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 can be thought of as “a set of
actions that, when taken, have demonstrated
ability to change a fixed educational trajectory”
(Methe & Riley-Tillman, 2008; p. 37).
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Response to Intervention
Core Instruction, Interventions, Accommodations
& Modifications: Sorting Them Out
• Accommodation. An accommodation is intended to help
the student to fully access and participate in the generaleducation curriculum without changing the instructional
content and without reducing the student’s rate of learning
(Skinner, Pappas & Davis, 2005). An accommodation is
intended to remove barriers to learning while still expecting
that students will master the same instructional content as
their typical peers.
– Accommodation example 1: Students are allowed to supplement
silent reading of a novel by listening to the book on tape.
– Accommodation example 2: For unmotivated students, the
instructor breaks larger assignments into smaller ‘chunks’ and
providing students with performance feedback and praise for each
completed ‘chunk’ of assigned work (Skinner, Pappas & Davis,
2005).
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Response to Intervention
Core Instruction, Interventions, Accommodations &
Modifications: Sorting Them Out
• Modification. A modification changes the expectations of
what a student is expected to know or do—typically by
lowering the academic standards against which the student
is to be evaluated.
Examples of modifications:
– Giving a student five math computation problems for practice
instead of the 20 problems assigned to the rest of the class
– Letting the student consult course notes during a test when peers
are not permitted to do so
– Allowing a student to select a much easier book for a book report
than would be allowed to his or her classmates.
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Response to Intervention
‘Intervention Footprint’: 7-Step Lifecycle of an Intervention Plan…
1.
2.
3.
4.
5.
6.
7.
Information about the student’s academic or
behavioral concerns is collected.
The intervention plan is developed to match
student presenting concerns.
Preparations are made to implement the plan.
The plan begins.
The integrity of the plan’s implementation is
measured.
Formative data is collected to evaluate the plan’s
effectiveness.
The plan is discontinued, modified, or replaced.
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Response to Intervention
Interventions: Potential ‘Fatal Flaws’
1.
2.
3.
4.
Any intervention must include 4 essential elements. The absence of any one of the
elements would be considered a ‘fatal flaw’ (Witt, VanDerHeyden & Gilbertson, 2004) that
blocks the school from drawing meaningful conclusions from the student’s response to the
intervention:
Clearly defined problem. The student’s target concern is stated in specific, observable,
measureable terms. This ‘problem identification statement’ is the most important step of the
problem-solving model (Bergan, 1995), as a clearly defined problem allows the teacher or
RTI Team to select a well-matched intervention to address it.
Baseline data. The teacher or RTI Team measures the student’s academic skills in the
target concern (e.g., reading fluency, math computation) prior to beginning the intervention.
Baseline data becomes the point of comparison throughout the intervention to help the
school to determine whether that intervention is effective.
Performance goal. The teacher or RTI Team sets a specific, data-based goal for student
improvement during the intervention and a checkpoint date by which the goal should be
attained.
Progress-monitoring plan. The teacher or RTI Team collects student data regularly to
determine whether the student is on-track to reach the performance goal.
Source: Witt, J. C., VanDerHeyden, A. M., & Gilbertson, D. (2004). Troubleshooting behavioral interventions. A systematic
process for finding and eliminating problems. School Psychology Review, 33, 363-383.
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Response to Intervention
How Do We Reach Low-Performing Math
Students?: Instructional Recommendations
Important elements of math instruction for low-performing
students:
–
–
–
–
“Providing teachers and students with data on student
performance”
“Using peers as tutors or instructional guides”
“Providing clear, specific feedback to parents on their children’s
mathematics success”
“Using principles of explicit instruction in teaching math
concepts and procedures.” p. 51
Source: Baker, S., Gersten, R., & Lee, D. (2002).A synthesis of empirical research on teaching mathematics to lowachieving students. The Elementary School Journal, 103(1), 51-73..
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Response to Intervention
Profile of Students With Significant Math Difficulties
1.
2.
3.
4.
5.
6.
7.
Spatial organization. The student commits errors such as misaligning numbers in columns
in a multiplication problem or confusing directionality in a subtraction problem (and
subtracting the original number—minuend—from the figure to be subtracted (subtrahend).
Visual detail. The student misreads a mathematical sign or leaves out a decimal or dollar
sign in the answer.
Procedural errors. The student skips or adds a step in a computation sequence. Or the
student misapplies a learned rule from one arithmetic procedure when completing another,
different arithmetic procedure.
Inability to ‘shift psychological set’. The student does not shift from one operation type
(e.g., addition) to another (e.g., multiplication) when warranted.
Graphomotor. The student’s poor handwriting can cause him or her to misread
handwritten numbers, leading to errors in computation.
Memory. The student fails to remember a specific math fact needed to solve a problem.
(The student may KNOW the math fact but not be able to recall it at ‘point of performance’.)
Judgment and reasoning. The student comes up with solutions to problems that are
clearly unreasonable. However, the student is not able adequately to evaluate those
responses to gauge whether they actually make sense in context.
Source: Rourke, B. P. (1993). Arithmetic disabilities, specific & otherwise: A neuropsychological perspective. Journal of Learning
Disabilities, 26, 214-226.
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Response to Intervention
Team Activity: Define ‘Math Reasoning’…
At your table:
1. Appoint a recorder/spokesperson.
2. Discuss the term ‘math reasoning’ at the
secondary level. Task-analyze the term
and break it down into the essential
subskills.
3. Be prepared to report out on your work.
4. What is the role of the Student Support
Team in assisting teachers to promote
‘math reasoning’?
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Response to Intervention
Assisting Students in Accessing Contextual, Conceptual, &
Procedural Knowledge When Solving Math Problems
“Well-structured, organized knowledge allows people to solve novel
problems and to remember more information than do memorized facts
or procedures... Such well-structured knowledge requires that people
integrate their contextual, conceptual and procedural knowledge in a
domain. Unfortunately, U.S. students rarely have such integrated and
robust knowledge in mathematics or science. Designing learning
environments that support integrated knowledge is a key challenge for
the field, especially given the low number of established tools for
guiding this design process.” p. 313
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Types of Knowledge: Definitions
Conceptual Knowledge: “…integrated knowledge of
important principles (e.g., knowledge of number magnitudes)
that can be flexibly applied to new tasks. Conceptual
knowledge can be used to guide comprehension of problems
and to generate new problem-solving strategies or to adapt
existing strategies to solve novel problems.” p. 317
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Types of Knowledge: Definitions
Procedural Knowledge: “…knowledge of
subcomponents of a correct procedure. Procedures
are a type of strategy that involve step-by-step actions
for solving problems, and most procedures require
integration of multiple skills. For example, the
conventional procedure for adding fractions with
unlike denominators requires knowing how to find a
common denominator, how to find equivalent
fractions, and how to add fractions with like
denominators.” 318
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Types of Knowledge: Definitions
Contextual Knowledge: “…our knowledge of how things
work in specific, real-world situations, which develops from
our everyday, informal interactions with the world. Students’
contextual knowledge can be elicited by situating problems
in story contexts.” p. 316
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Math Problem Scaffolding Examples
(Modeled after Rittle-Johnson & Koedinger, 2005)
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Math Problem Scaffolding Examples
(Modeled after Rittle-Johnson & Koedinger, 2005)
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Leveraging the Power of Contextual Knowledge in Story Problems:
Use Familiar Student Contexts
“Past research on fraction learning
indicates that food contexts are
particularly meaningful contexts
for students (Mack, 1990, 1993).”
p. 319
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Math Problem Scaffolding Examples
(Modeled after Rittle-Johnson & Koedinger, 2005)
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
www.interventioncentral.org
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Response to Intervention
Math Problem Scaffolding Examples
(Modeled after Rittle-Johnson & Koedinger, 2005)
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Research is Unclear Whether Math Problems in Story or Symbolic
Format Are More Difficult
The reason for contradictory findings about the relative difficulty of
math problems in story or symbolic format may be explained by gradespecific challenges in math.
“First, young children have pervasive exposure to single-digit
numerals, but some words and syntactic forms are still unknown or
unfamiliar [explaining why younger students may find story problems
more challenging]. In comparison, older children have less exposure to
large, multidigit numerals and algebraic symbols and have much better
reading and comprehension skills [explaining why older students may
find symbolic problems more challenging].” p. 317
Source: Rittle-Johnson, B., & Koedinger, K. R. (2005). Designing knowledge scaffolds to support mathematical problemsolving. Cognition and Instruction, 23(3), 313–349.
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Response to Intervention
Strands of Math Proficiency
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Response to Intervention
5 Strands of Mathematical Proficiency
1. Understanding
2. Computing
3. Applying
4. Reasoning
5. Engagement
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
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
Five Strands of Mathematical Proficiency (NRC, 2002)
Conceptual
Knowledge
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.
Procedural
Knowledge
3.
Applying: Being able to formulate problems
mathematically and to devise strategies for solving
them using concepts and procedures appropriately.
Metacognition
4.
Reasoning: Using logic to explain and justify a
solution to a problem or to extend from something
known to something less known.
Synthesis
5.
Engaging: Seeing mathematics as sensible, useful,
and doable—if you work at it—and being willing to do
the work.
Motivation
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Response to Intervention
Table Activity: Evaluate Your
School’s Math
Proficiency…
•
•
•
•
As a group, review the
National Research
Council ‘Strands of Math
Proficiency’.
Which strand do you feel
that your school /
curriculum does the best
job of helping students to
attain proficiency?
Which strand do you feel
that your school /
curriculum should put the
greatest effort to figure
out how to help students
to attain proficiency?
Be prepared to share
your results.
Five Strands of Mathematical
Proficiency (NRC, 2002)
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.
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.
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Response to Intervention
RTI & Secondary Literacy:
Explicit Vocabulary Instruction
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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
Promoting Math Vocabulary: Other Guidelines
–
–
–
–
–
–
Create a standard list of math vocabulary for each grade level (elementary)
or course/subject area (for example, geometry).
Periodically check students’ mastery of math vocabulary (e.g., through
quizzes, math journals, guided discussion, etc.).
Assist students in learning new math vocabulary by first assessing their
previous knowledge of vocabulary terms (e.g., protractor; product) and
then using that past knowledge to build an understanding of the term.
For particular assignments, have students identify math vocabulary that
they don’t understand. In a cooperative learning activity, have students
discuss the terms. Then review any remaining vocabulary questions with
the entire class.
Encourage students to use a math dictionary in their vocabulary work.
Make vocabulary a central part of instruction, curriculum, and
assessment—rather than treating as an afterthought.
Source: Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher, 56(8), 786-795.
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Response to Intervention
Vocabulary: Why This Instructional Goal is
Important
As vocabulary terms become more specialized in
content area courses, students are less able to derive
the meaning of unfamiliar words from context alone.
Students must instead learn vocabulary through more
direct means, including having opportunities to explicitly
memorize words and their definitions.
Students may require 12 to 17 meaningful exposures to
a word to learn it.
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Response to Intervention
Enhance Vocabulary Instruction Through Use of
Graphic Organizers or Displays: A Sampling
Teachers can use graphic displays to structure
their vocabulary discussions and activities
(Boardman et al., 2008; Fisher, 2007; Texas
Reading Initiative, 2002).
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Response to Intervention
4-Square Graphic Display
The student divides a page into four quadrants.
In the upper left section, the student writes the
target word. In the lower left section, the student
writes the word definition. In the upper right
section, the student generates a list of examples
that illustrate the term, and in the lower right
section, the student writes ‘non-examples’ (e.g.,
terms that are the opposite of the target
vocabulary word).
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Response to Intervention
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Response to Intervention
Semantic Word Definition Map
The graphic display contains sections in which
the student writes the word, its definition (‘what
is this?’), additional details that extend its
meaning (‘What is it like?’), as well as a listing of
examples and ‘non-examples’ (e.g., terms that
are the opposite of the target vocabulary word).
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Response to Intervention
Word Definition Map Example
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Response to Intervention
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Response to Intervention
Semantic Feature Analysis
A target vocabulary term is selected for analysis
in this grid-like graphic display. Possible features
or properties of the term appear along the top
margin, while examples of the term are listed ion
the left margin. The student considers the
vocabulary term and its definition. Then the
student evaluates each example of the term to
determine whether it does or does not match
each possible term property or element.
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Response to Intervention
Semantic Feature Analysis Example
• VOCABULARY TERM: TRANSPORTATION
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Response to Intervention
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Response to Intervention
Comparison/Contrast (Venn) Diagram
Two terms are listed and defined. For each
term, the student brainstorms qualities or
properties or examples that illustrate the term’s
meaning. Then the student groups those
qualities, properties, and examples into 3
sections:
A. items unique to Term 1
B. items unique to Term 2
C. items shared by both terms
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Response to Intervention
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Response to Intervention
Provide Regular In-Class Instruction and
Review of Vocabulary Terms, Definitions
Present important new vocabulary terms in class,
along with student-friendly definitions. Provide
‘example sentences’/contextual sentences to
illustrate the use of the term. Assign students to
write example sentences employing new vocabulary
to illustrate their mastery of the terms.
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Response to Intervention
Generate ‘Possible Sentences’
The teacher selects 6 to 8 challenging new vocabulary
terms and 4 to 6 easier, more familiar vocabulary items
relevant to the lesson. Introduce the vocabulary terms to
the class. Have students write sentences that contain at
least two words from the posted vocabulary list. Then write
examples of student sentences on the board until all words
from the list have been used. After the assigned reading,
review the ‘possible sentences’ that were previously
generated. Evaluate as a group whether, based on the
passage, the sentence is ‘possible’ (true) in its current form.
If needed, have the group recommend how to change the
sentence to make it ‘possible’.
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Response to Intervention
Provide Dictionary Training
The student is trained to use an Internet lookup strategy
to better understand dictionary or glossary definitions of
key vocabulary items.
– The student first looks up the word and its meaning(s) in the
dictionary/glossary.
– If necessary, the student isolates the specific word meaning
that appears to be the appropriate match for the term as it
appears in course texts and discussion.
– The student goes to an Internet search engine (e.g., Google)
and locates at least five text samples in which the term is
used in context and appears to match the selected dictionary
definition.
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Response to Intervention
RTI & Secondary Literacy:
Extended Discussion
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Response to Intervention
Extended Discussions: Why This Instructional Goal
is Important
Extended, guided group discussion is a powerful means to help
students to learn vocabulary and advanced concepts. Discussion
can also model for students various ‘thinking processes’ and
cognitive strategies (Kamil et al. 2008, p. 22). To be effective,
guided discussion should go beyond students answering a series
of factual questions posed by the teacher: Quality discussions are
typically open-ended and exploratory in nature, allowing for
multiple points of view (Kamil et al., 2008).
When group discussion is used regularly and well in instruction,
students show increased growth in literacy skills. Content-area
teachers can use it to demonstrate the ‘habits of mind’ and
patterns of thinking of experts in various their discipline: e.g.,
historians, mathematicians, chemists, engineers, literacy critics,
etc.
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Response to Intervention
Use a ‘Standard Protocol’ to Structure Extended
Discussions
Good extended classwide discussions elicit a
wide range of student opinions, subject
individual viewpoints to critical scrutiny in a
supportive manner, put forth alternative views,
and bring closure by summarizing the main
points of the discussion. Teachers can use a
simple structure to effectively and reliably
organize their discussions…
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Response to Intervention
‘Standard Protocol’ Discussion Format
A. Pose questions to the class that require students to explain their
positions and their reasoning .
B. When needed, ‘think aloud’ as the discussion leader to model
good reasoning practices (e.g., taking a clear stand on a topic).
C. Supportively challenge student views by offering possible
counter arguments.
D. Single out and mention examples of effective student reasoning.
E. Avoid being overly directive; the purpose of extended
discussions is to more fully investigate and think about complex
topics.
F. Sum up the general ground covered in the discussion and
highlight the main ideas covered.
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Response to Intervention
RTI & Secondary Literacy:
Reading Comprehension
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Response to Intervention
Reading Comprehension: Why This Instructional
Goal is Important
Students require strong reading comprehension skills to
succeed in challenging content-area classes.
At present, there is no clear evidence that any one
reading comprehension instructional technique is clearly
superior to others. In fact, it appears that students
benefit from being taught any self-directed practice that
prompts them to engage more actively in understanding
the meaning of text (Kamil et al., 2008).
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Response to Intervention
Assist Students in Setting ‘Content Goals’ for
Reading
Students are more likely to be motivated to read--and to
read more closely—if they have specific content-related
reading goals in mind. At the start of a reading
assignment, for example, the instructor has students
state what questions they might seek to answer or what
topics they would like to learn more about in their
reading. The student or teacher writes down these
questions. After students have completed the assignee
reading, they review their original questions and share
what they have learned (e.g., through discussion in
large group or cooperative learning group, or even as a
written assignment).
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Response to Intervention
Teach Students to Monitor Their Own
Comprehension and Apply ‘Fix-Up’ Skills
•
•
•
•
Teachers can teach students specific strategies to
monitor their understanding of text and independently
use ‘fix-up’ skills as needed. Examples of student
monitoring and repair skills for reading comprehension
include encouraging them to:
Stop after every paragraph to summarize its main idea
Reread the sentence or paragraph again if necessary
Generate and write down questions that arise during
reading
Restate challenging or confusing ideas or concepts
from the text in the student’s own words
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Response to Intervention
Teach Students to Identify
Underlying Structures of Math
Problems
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Response to Intervention
Algebra Word Problems: Predictable Elements
Most algebra problems contain predictable elements:
1.
2.
3.
4.
Assignment statements: Assign a “particular numerical value to some variable.”
Relational statements: “Specify a single relationship between two variables.”
Questions: “Involve the requested solution (e.g.,’ What is X?’).”
Relevant facts: “Any other type of information that might be useful for solving the
problem.”
Problem translation: The process of “taking each of these forms of information
and using them to develop corresponding algebraic equations.”
The most common problems were noted:
•
Students may fail to discriminate relevant from irrelevant information.
•
Students may commit translation errors when processing relational
statements.
Source: National Mathematics Advisory Panel. (2008). Foundations for success: The final Report of the National Mathematics
Advisory Panel: Chapter 4: Report of the Task Group on Learning Processes. U.S. Department of Education: Washington, DC.
Retrieved from http://www.ed.gov/about/bdscomm/list/mathpanel/reports.html
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Response to Intervention
Developing Student
Metacognitive Abilities
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Response to Intervention
Importance of Metacognitive Strategy Use…
“Metacognitive processes focus on self-awareness
of cognitive knowledge that is presumed to be
necessary for effective problem solving, and they
direct and regulate cognitive processes and
strategies during problem solving…That is,
successful problem solvers, consciously or
unconsciously (depending on task demands), use
self-instruction, self-questioning, and self-monitoring
to gain access to strategic knowledge, guide
execution of strategies, and regulate use of
strategies and problem-solving performance.” p. 231
Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem
solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
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Response to Intervention
Elements of Metacognitive Processes
“Self-instruction helps students to identify and
direct the problem-solving strategies prior to
execution. Self-questioning promotes internal
dialogue for systematically analyzing problem
information and regulating execution of cognitive
strategies. Self-monitoring promotes appropriate
use of specific strategies and encourages students
to monitor general performance. [Emphasis added].”
p. 231
Source: Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem
solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.
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Response to Intervention
Combining Cognitive & Metacognitive Strategies to Assist
Students With Mathematical Problem Solving
Solving an advanced math problem independently
requires the coordination of a number of complex skills.
The following strategies combine both cognitive and
metacognitive elements (Montague, 1992; Montague &
Dietz, 2009). First, the student is taught a 7-step
process for attacking a math word problem (cognitive
strategy). Second, the instructor trains the student to
use a three-part self-coaching routine for each of the
seven problem-solving steps (metacognitive strategy).
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Response to Intervention
Cognitive Portion of Combined Problem Solving Approach
In the cognitive part of this multi-strategy intervention, the student learns an explicit series
of steps to analyze and solve a math problem. Those steps include:
1. Reading the problem. The student reads the problem carefully, noting and attempting
to clear up any areas of uncertainly or confusion (e.g., unknown vocabulary terms).
2. Paraphrasing the problem. The student restates the problem in his or her own words.
3. ‘Drawing’ the problem. The student creates a drawing of the problem, creating a
visual representation of the word problem.
4. Creating a plan to solve the problem. The student decides on the best way to solve
the problem and develops a plan to do so.
5. Predicting/Estimating the answer. The student estimates or predicts what the answer
to the problem will be. The student may compute a quick approximation of the answer,
using rounding or other shortcuts.
6. Computing the answer. The student follows the plan developed earlier to compute the
answer to the problem.
7. Checking the answer. The student methodically checks the calculations for each step
of the problem. The student also compares the actual answer to the estimated answer
calculated in a previous step to ensure that there is general agreement between the two
values.
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Response to Intervention
Metacognitive Portion of Combined Problem Solving Approach
The metacognitive component of the intervention is a threepart routine that follows a sequence of ‘Say’, ‘Ask, ‘Check’. For
each of the 7 problem-solving steps reviewed above:
• The student first self-instructs by stating, or ‘saying’, the
purpose of the step (‘Say’).
• The student next self-questions by ‘asking’ what he or she
intends to do to complete the step (‘Ask’).
• The student concludes the step by self-monitoring, or
‘checking’, the successful completion of the step (‘Check’).
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Combined Cognitive & Metacognitive Elements of
Strategy
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Response to Intervention
Applied Problems: Pop Quiz
7-Step Problem-Solving:Process
1.
2.
3.
4.
5.
6.
7.
Reading the
problem.
Paraphrasing the
problem.
‘Drawing’ the
problem.
Creating a plan to
solve the problem.
Predicting/Estimating the answer.
Computing the
answer.
Checking the
answer.
Q:
“To move theirAs
armies,
the Romans
over
Directions:
a team,
read thebuilt
following
50,000 miles of roads. Imagine driving all those miles!
problem.
Atdriving
your tables,
apply
thefirst
7-step
Now
imagine
those miles
in the
gasolineproblem-solving
(cognitive)
strategy
to
driven car that has only three wheels and could reach
problem.
As per
youhour.
complete each
acomplete
top speedthe
of about
10 miles
stepsafety's
of thesake,
problem,
apply
thea‘Say-AskFor
let's bring
along
spare tire. As
Check’
Tryspare
to with
you
drivemetacognitive
the 50,000 miles,sequence.
you rotate the
the
other tires
that all7four
tireswithin
get thethe
same
complete
thesoentire
steps
time
amount
of wear.
Canexercise.
you figure out how many miles
allocated
for this
of wear each tire accumulates?”
A: “Since the four wheels of the three-wheeled car
share the journey equally, simply take
three-fourths of the total distance (50,000
miles) and you'll get 37,500 miles for each tire.”
Source: The Math Forum @ Drexel: Critical Thinking Puzzles/Spare My Brain. Retrieved from
http://mathforum.org/k12/k12puzzles/critical.thinking/puzz2.html
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Response to Intervention
RTI & Math Reasoning:
Key Next Steps…
1.
Define ‘math reasoning’ and task-analyze to create
a checklist of subskills that make up that term. (This
checklist can be framed as student ‘look-for’
behaviors and adjusted to each grade level).
2.
Develop school-wide screening measures to
identify students at-risk for math computation and
math reasoning skills. Also develop the capacity to
complete diagnostic math assessments for students
with more severe math deficits.
3.
Set up ‘knowledge brokers’ in your school who will
monitor math instructional and intervention
programs and research findings by attending
workshops, visiting websites, reading professional
journals, etc.—and give them opportunities to share
these updates with school staff.
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At your tables:,
Discuss these key
‘next steps’ for
moving forward
with RTI & Math
Reasoning in your
school or district.
Begin to draft an
action plan to
implement each of
these steps.
Be prepared to
report out on your
work.
84
Response to Intervention
Team Activity: Favorite Math Websites…
At your table:
1. Discuss math websites that you have
used and have found to be helpful.
2. Be prepared to report out on your
favorite math websites.
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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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