RTI: Academic Interventions for Difficult-to-Teach Students Jim Wright www.interventioncentral.org

1
RTI Academic Interventions for
Difficult-to-Teach StudentsJim
Wrightwww.interventioncentral.org
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Workshop Agenda
3
RTI Assumption Struggling Students Are Typical
Until Proven Otherwise

• RTI logic assumes that
• A student who begins to struggle in general
  education is typical, and that
• It is general educations responsibility to find
  the instructional strategies that will unlock the
  students learning potential
• Only when the student shows through
  well-documented interventions that he or she has
  failed to respond to intervention does RTI
  begin to investigate the possibility that the
  student may have a learning disability or other
  special education condition.

4
Essential Elements of RTI (Fairbanks, Sugai,
Guardino, Lathrop, 2007)

1. A continuum of evidence-based services available
   to all students" that range from universal to
   highly individualized intensive
2. Decision points to determine if students are
   performing significantly below the level of their
   peers in academic and social behavior domains"
3. Ongoing monitoring of student progress"
4. Employment of more intensive or different
   interventions when students do not improve in
   response" to lesser interventions
5. Evaluation for special education services if
   students do not respond to intervention
   instruction"

Source Fairbanks, S., Sugai, G., Guardino, S.,
Lathrop, M. (2007). Response to intervention
Examining classroom behavior support in second
grade. Exceptional Children, 73, p. 289.
5
Use Time Resources Efficiently By Collecting
Information Only on Things That Are Alterable

• Time should be spent thinking about things
  that the intervention team can influence through
  instruction, consultation, related services, or
  adjustments to the students program. These are
  things that are alterable.Beware of statements
  about cognitive processes that shift the focus
  from the curriculum and may even encourage
  questionable educational practice. They can also
  promote writing off a student because of the
  rationale that the students insufficient
  performance is due to a limited and fixed
  potential. p.359

Source Howell, K. W., Hosp, J. L., Kurns, S.
(2008). Best practices in curriculum-based
evaluation. In A. Thomas J. Grimes (Eds.), Best
practices in school psychology V (pp.349-362).
Bethesda, MD National Association of School
Psychologists.
6
School Instructional Time The Irreplaceable
Resource

• In the average school system, there are 330
  minutes in the instructional day, 1,650 minutes
  in the instructional week, and 56,700 minutes in
  the instructional year. Except in unusual
  circumstances, these are the only minutes we have
  to provide effective services for students. The
  number of years we have to apply these minutes is
  fixed. Therefore, each minute counts and schools
  cannot afford to support inefficient models of
  service delivery. p. 177

Source Batsche, G. M., Castillo, J. M., Dixon,
D. N., Forde, S. (2008). Best practices in
problem analysis. In A. Thomas J. Grimes
(Eds.), Best practices in school psychology V
(pp. 177-193).
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NYSED RTI Guidance Memo April 2008
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The Regents policy framework for RtIDefines
RtI to minimally include Appropriate
instruction delivered to all students in the
general education class by qualified personnel.
Appropriate instruction in reading means
scientific research-based reading programs that
include explicit and systematic instruction in
phonemic awareness, phonics, vocabulary
development, reading fluency (including oral
reading skills) and reading comprehension
strategies.Screenings applied to all students
in the class to identify those students who are
not making academic progress at expected rates.
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Instruction matched to student need with
increasingly intensive levels of targeted
intervention and instruction for students who do
not make satisfactory progress in their levels of
performance and/or in their rate of learning to
meet age or grade level standards.Repeated
assessments of student achievement which should
include curriculum based measures to determine if
interventions are resulting in student progress
toward age or grade level standards.The
application of information about the students
response to intervention to make educational
decisions about changes in goals, instruction
and/or services and the decision to make a
referral for special education programs and/or
services.
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Written notification to the parents when the
student requires an intervention beyond that
provided to all students in the general education
classroom that provides information about the
-amount and nature of student performance data
that will be collected and the general education
services that will be provided-strategies for
increasing the students rate of learning
and-parents right to request an evaluation for
special education programs and/or services.
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RTI Key Concepts
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Middle High School Lack of Consensus on an RTI
Model

• Because RTI has thus far been implemented
  primarily in early elementary grades, it is not
  clear precisely what RTI might look like at the
  high school level.

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.betterhighschool
s.org/pubs/ p. 3
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At the Federal Level A Hands-Off Approach to
RTI Implementation

• There are many RTI models and the regulations
  are written to accommodate the many different
  models that are currently in use. The Department
  does not mandate or endorse any particular model.
  Rather, the regulations provide States with the
  flexibility to adopt criteria that best meet
  local needs. Language that is more specific or
  prescriptive would not be appropriate. For
  example, while we recognize that rate of learning
  is often a key variable in assessing a childs
  response to intervention, it would not be
  appropriate for the regulations to set a standard
  for responsiveness or improvement in the rate of
  learning. p. 46653

Source U.S. Department of Education. (2006).
Assistance to States for the education of
children with disabilities and preschool grants
for children with disabilities final rule. 71
Fed. Reg. (August 14, 2006) 34 CFR Parts 300 and
301.
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The Purpose of RTI in Secondary Schools What
Students Should It Serve?
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RTI Pyramid of Interventions
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Complementary RTI Models Standard Treatment
Problem-Solving Protocols

• The two most commonly used RTI approaches are
  (1) standard treatment and (2) problem-solving
  protocol. While these two approaches to RTI are
  sometimes described as being very different from
  each other, they actually have several common
  elements, and both fit within a problem-solving
  framework. In practice, many schools and
  districts combine or blend aspects of the two
  approaches to fit their needs.

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.betterhighschool
s.org/pubs/ p. 5
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RTI Interventions Standard-Treatment vs.
Problem-Solving
There are two different vehicles that schools can
use to deliver RTI interventions Standard-Protoco
l (Standalone Intervention). Programs based on
scientifically valid instructional practices
(standard protocol) are created to address
frequent student referral concerns. These
services are provided outside of the classroom. A
middle school, for example, may set up a
structured math-tutoring program staffed by adult
volunteer tutors to provide assistance to
students with limited math skills. Students
referred for a Tier II math intervention would be
placed in this tutoring program. An advantage of
the standard-protocol approach is that it is
efficient and consistent large numbers of
students can be put into these group
interventions to receive a highly standardized
intervention. However, standard group
intervention protocols often cannot be
individualized easily to accommodate a specific
students unique needs. Problem-solving
(Classroom-Based Intervention). Individualized
research-based interventions match the profile of
a particular students strengths and limitations.
The classroom teacher often has a large role in
carrying out these interventions. A plus of the
problem-solving approach is that the intervention
can be customized to the students needs.
However, developing intervention plans for
individual students can be time-consuming.
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Tier I Instruction/Interventions

• Tier I instruction/interventions
• Are universalavailable to all students.
• Can be delivered within classrooms or throughout
  the school.
• Are likely to be put into place by the teacher at
  the first sign that a student is struggling.
• All children have access to Tier 1
  instruction/interventions. Teachers have the
  capability to use those strategies without
  requiring outside assistance.
• Tier 1 instruction/interventions encompass
• The schools core curriculum and all published or
  teacher-made materials used to deliver that
  curriculum.
• Teacher use of whole-group teaching
  management strategies.
• Teacher use of individualized strategies with
  specific students.
• Tier I instruction/interventions attempt to
  answer the question Are classroom instructional
  strategies supports sufficient to help the
  student to achieve academic success?

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Tier 1 Classroom-Level Interventions

• Decision Point Student is struggling and may
  face significant high-stakes negative outcome if
  situation does not improve.
• Collaboration Opportunity Teacher can refer the
  student to a grade-level, instruction team, or
  department meeting to brainstorm ideas OR
  teacher seeks out consultant in school to
  brainstorm intervention ideas.
• Documentation Teacher completes Classroom
  Intervention Form prior to carrying out
  intervention. Teacher collects classroom data.
• Decision Rule Example Teacher should refer
  student to the next level of RTI support if the
  intervention is not successful within 8
  instructional weeks.

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Tier 2 Supplemental (Standard-Protocol Model)
Interventions

• Tier 2 interventions are typically delivered in
  small-group format. About 15 of students in the
  typical school will require Tier 2/supplemental
  intervention support.
• Group size for Tier 2 interventions is limited
  to 4-6 students. Students placed in Tier 2
  interventions should have a shared profile of
  intervention need.
• The reading progress of students in Tier 2
  interventions are monitored at least 1-2 times
  per month.

Source Burns, M. K., Gibbons, K. A. (2008).
Implementing response-to-intervention in
elementary and secondary schools. Routledge New
York.
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Tier 2 Supplemental Interventions

• Decision Point Building-wide academic screenings
• Collaboration Opportunity After each
  building-wide academic screening, data teams
  meet (teachers at a grade level building
  principal reading teacher, etc.) At the meeting,
  the group considers how the assessment data
  should shape/inform core instruction.
  Additionally, the data team sets a cutpoint to
  determine which students should be recruited for
  Tier 2 group interventions. NOTE Team may
  continue to meet every 5 weeks to consider
  student progress in Tier 2 move students into
  and out of groups.
• Documentation Tier 2 instructor completes a Tier
  2 Group Assignment Sheet listing students and
  their corresponding interventions.
  Progress-monitoring occurs 1-2 times per month.
• Decision Rules Example Student is returned to
  Tier 1 support if they perform above the 25th
  percentile in the next school-wide screening.
  Student is referred to Tier 3 (RTI Team) if they
  fail to make expected progress despite two Tier 2
  (group-based) interventions.

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Scheduling Elementary Tier 2 Interventions
Option 3 Floating RTIGradewide Shared
Schedule. Each grade has a scheduled RTI time
across classrooms. No two grades share the same
RTI time. Advantages are that outside providers
can move from grade to grade providing push-in or
pull-out services and that students can be
grouped by need across different teachers within
the grade.
Anyplace Elementary School RTI Daily Schedule
Classroom 1
Classroom 2
Classroom 3
Grade K
900-930
Classroom 1
Classroom 2
Classroom 3
Grade 1
945-1015
Classroom 1
Classroom 2
Classroom 3
Grade 2
1030-1100
Classroom 1
Classroom 2
Classroom 3
Grade 3
1230-100
Classroom 1
Classroom 2
Classroom 3
Grade 4
115-145
Grade 5
Classroom 1
Classroom 2
Classroom 3
200-230
Source Burns, M. K., Gibbons, K. A. (2008).
Implementing response-to-intervention in
elementary and secondary schools Procedures to
assure scientific-based practices. New York
Routledge.
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Tier 3 Intensive Individualized Interventions
(Problem-Solving Model)

• Tier 3 interventions are the most intensive
  offered in a school setting. About 5 of a
  general-education student population may qualify
  for Tier 3 supports. Typically, the RTI
  Problem-Solving Team meets to develop
  intervention plans for Tier 3 students.
• Students qualify for Tier 3 interventions
  because
• they are found to have a large skill gap when
  compared to their class or grade peers and/or
• They did not respond to interventions provided
  previously at Tiers 1 2.
• Tier 3 interventions are provided daily for
  sessions of 30 minutes. The student-teacher ratio
  is flexible but should allow the student to
  receive intensive, individualized instruction.
  The academic or behavioral progress of students
  in Tier 3 interventions is monitored at least
  weekly.

Source Burns, M. K., Gibbons, K. A. (2008).
Implementing response-to-intervention in
elementary and secondary schools. Routledge New
York.
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Tier 3 RTI Team

• Decision Point RTI Problem-Solving Team
• Collaboration Opportunity Weekly RTI
  Problem-Solving Team meetings are scheduled to
  handle referrals of students that failed to
  respond to interventions from Tiers 1 2.
• Documentation Teacher referral form RTI Team
  minutes form progress-monitoring data collected
  at least weekly.
• Decision Rules Example If student has failed
  to respond adequately to 3 intervention trials of
  6-8 weeks (from Tiers 2 and 3), the student may
  be referred to Special Education.

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Advancing Through RTI Flexibility in the Tiers

• For purposes of efficiency, students should be
  placed in small-group instruction at Tier 2.
• However, group interventions may not always be
  possible because due to scheduling or other
  issuesno group is available. (For example,
  students with RTI behavioral referrals may not
  have a group intervention available.)
• In such a case, the student will go directly to
  the problem-solving process (Tier 3)typically
  through a referral to the school RTI Team.
• Nonetheless, the school must still document the
  same minimum number of interventions attempted
  for every student in RTI, whether or not a
  student first received interventions in a group
  setting.

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Target Student
Dual-Discrepancy RTI Model of Learning
Disability (Fuchs 2003)
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Intervention Research Development A Work in
Progress
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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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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.
34
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 processcriteria 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 shouldnt be afraid to ask)..
Learning Disabilities Research Practice,
22(2),129136.
35
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.betterhighschool
s.org/pubs/ p. 9
36
RTI Intervention Key Concepts
37
Essential Elements of Any Academic or Behavioral
Intervention (Treatment) Strategy

• Method of delivery (Who or what delivers the
  treatment?)Examples include teachers,
  paraprofessionals, parents, volunteers,
  computers.
• Treatment component (What makes the intervention
  effective?)Examples include activation of prior
  knowledge to help the student to make meaningful
  connections between known and new material
  guide practice (e.g., Paired Reading) to increase
  reading fluency periodic review of material to
  aid student retention.

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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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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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Core Instruction, Interventions, Accommodations
Modifications Sorting Them Out

• Accommodation. An accommodation is intended to
  help the student to fully access and participate
  in the general-education curriculum without
  changing the instructional content and without
  reducing the students 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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Teaching is giving it isnt taking away.
(Howell, Hosp Kurns, 2008 p. 356).


Source Howell, K. W., Hosp, J. L., Kurns, S.
(2008). Best practices in curriculum-based
evaluation. In A. Thomas J. Grimes (Eds.), Best
practices in school psychology V (pp.349-362).
Bethesda, MD National Association of School
Psychologists..
42
Core Instruction, Interventions, Accommodations
Modifications Sorting Them Out

• Modification. A modification changes the
  expectations of what a student is expected to
  know or dotypically 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

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Big Ideas The Four Stages of Learning Can Be
Summed Up in the Instructional Hierarchy pp.
2-3(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.
44
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.
45
RTI Interventions What If There is No Commercial
Intervention Package or Program Available?

• Although commercially prepared programs and the
  subsequent manuals and materials are inviting,
  they are not necessary. A recent review of
  research suggests that interventions are research
  based and likely to be successful, if they are
  correctly targeted and provide explicit
  instruction in the skill, an appropriate level of
  challenge, sufficient opportunities to respond to
  and practice the skill, and immediate feedback on
  performanceThus, these elements could be used
  as criteria with which to judge potential tier 2
  interventions. p. 88

Source Burns, M. K., Gibbons, K. A. (2008).
Implementing response-to-intervention in
elementary and secondary schools. Routledge New
York.
46
Research-Based Elements of Effective Academic
Interventions

• Correctly targeted The intervention is
  appropriately matched to the students 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.
47
Interventions Potential Fatal Flaws

• 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 students response to the intervention
• Clearly defined problem. The students 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 students 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.
48
RTI Best Practicesin MathematicsInterventionsJ
im Wrightwww.interventioncentral.org
49
National Mathematics Advisory Panel Report13
March 2008
50
Math Advisory Panel Report athttp//www.ed.gov/
mathpanel
51
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/rep
ort/final-factsheet.html
52
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).
53
Math Intervention Planning Some Challenges for
Elementary RTI Teams

• There is no national consensus about what math
  instruction should look like in elementary
  schools
• Schools may not have consistent expectations for
  the best practice math instruction strategies
  that teachers should routinely use in the
  classroom
• Schools may not have a full range of assessment
  methods to collect baseline and progress
  monitoring data on math difficulties

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Profile of Students With Significant Math
Difficulties

• 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 numberminuendfrom 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 students 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.
55
Mathematics is made of 50 percent formulas, 50
percent proofs, and 50 percent imagination.
Anonymous
56
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 averageUsing 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.
57
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..
58
The Elements of Mathematical Proficiency What
the Experts Say
59
(No Transcript)
60
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.
61
Five Strands of Mathematical Proficiency (Cont.)

1. Reasoning Using logic to explain and justify a
   solution to a problem or to extend from something
   known to something less known.
2. Engaging Seeing mathematics as sensible, useful,
   and doableif you work at itand 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.
62
Math Computation Building FluencyJim
Wrightwww.interventioncentral.org
63
"Arithmetic is being able to count up to twenty
without taking off your shoes." Anonymous
64
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.
65
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 childrens
  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.
66
Big Ideas Learn Unit (Heward, 1996)

• The three essential elements of effective student
  learning include
• 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.
• Active Student Response. The student answers the
  item, solves the problem presented, or completes
  the academic task. Answering the teachers
  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.
• Performance Feedback. The student receives timely
  feedback about whether his or her response is
  correctoften 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, CABrooks/Cole.
67
Math Intervention Tier I or II Elementary
Secondary Self-Administered Arithmetic
Combination Drills With Performance
Self-Monitoring Incentives

1. The student is given a math computation worksheet
   of a specific problem type, along with an answer
   key Academic Opportunity to Respond.
2. 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.
3. 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
4. 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
5. The student records the days score of TOTAL
   number of correct digits on his or her personal
   performance chart.
6. 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, CABrooks/Cole.
68
Self-Administered Arithmetic Combination Drills
69
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.
70
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
71
Math Computation Problem Interspersal Technique

• The teacher first identifies the range of
  challenging problem-types (number problems
  appropriately matched to the students 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 11 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 fifth-grade students
mathematics accuracy. School Psychology Review,
34, 543-555..
72
Teaching Math Vocabulary
73
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 childs 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/a
uthor/chard_hmm05.pdf.
74
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
  vocabularywhile 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/a
uthor/chard_hmm05.pdf.
75
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 dont
  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 assessmentrather than treating
  as an afterthought.

Source Adams, T. L. (2003). Reading mathematics
More than words can say. The Reading Teacher,
56(8), 786-795.
76
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), 119135.
77
Building Student Skills inApplied Math
ProblemsJim Wrightwww.interventioncentral.org
78
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 childrens 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 low-achieving students.
The Elementary School Journal, 103(1), 51-73..
79
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 lt 2)
• Difficulty extracting underlying math
  operations from word/story problems
• Difficulty identifying and ignoring extraneous
  information included in word/story problems

80
Math Intervention Ideas for Higher-Level Math
ProblemsJim Wrightwww.interventioncentral.org
81
Applied Problems
82
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.

83
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 fifth-grade students
mathematics accuracy. School Psychology Review,
34, 543-555..
84
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.
• 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 dont 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.
85
Interpreting Math Graphics A Reading
Comprehension Intervention
86
Housing Bubble GraphicNew York Times23
September 2007
87
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, 2127.
88
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
• 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, 2127.
89
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, 2127.
90
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence

1. 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 gt tables gt bar graphs gt charts gt 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, 2127.
91
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence

1. 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, 2127.
92
Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence

• 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
• 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
  dont short-circuit their inquiry.

Source Mesmer, H.A.E., Hutchins, E.J. (2002).
Using QARs with charts and graphs. The Reading
Teacher, 56, 2127.
93
Developing Student Metacognitive Abilities
94
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 solvingThat 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.
95
Elements of Metacognitive Processes

• Self-instruction helps students to identify and
  direct the problem-solving strategies prior to
  execution. Self-questioning promot