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Title: 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
2
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).
7
NYSED RTI Guidance Memo April 2008
8
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9
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.
10
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.
11
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.
12
RTI Key Concepts
13
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
14
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.
15
The Purpose of RTI in Secondary Schools What
Students Should It Serve?
16
RTI Pyramid of Interventions
17
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
18
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.
19
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?

20
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.

21
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22
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.
23
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.

24
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25
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.
26
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.
27
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.

28
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29
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.

30
Target Student
Dual-Discrepancy RTI Model of Learning
Disability (Fuchs 2003)
31
Intervention Research Development A Work in
Progress
32
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.
33
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.

38
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.

39
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).

40
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).

41
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

43
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

54
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
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