Title: RTI: Academic Interventions for Difficult-to-Teach Students Jim Wright www.interventioncentral.org
1RTI Academic Interventions for
Difficult-to-Teach StudentsJim
Wrightwww.interventioncentral.org
2Workshop Agenda
3RTI 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.
4Essential Elements of RTI (Fairbanks, Sugai,
Guardino, Lathrop, 2007)
- A continuum of evidence-based services available
to all students" that range from universal to
highly individualized intensive - Decision points to determine if students are
performing significantly below the level of their
peers in academic and social behavior domains" - Ongoing monitoring of student progress"
- Employment of more intensive or different
interventions when students do not improve in
response" to lesser interventions - 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.
5Use 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.
6School 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).
7NYSED RTI Guidance Memo April 2008
8(No Transcript)
9The 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.
10Instruction 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.
11Written 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.
12RTI Key Concepts
13Middle 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
14At 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.
15The Purpose of RTI in Secondary Schools What
Students Should It Serve?
16RTI Pyramid of Interventions
17Complementary 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
18RTI 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.
19Tier 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?
20Tier 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(No Transcript)
22Tier 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.
23Tier 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(No Transcript)
25Scheduling 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.
26Tier 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.
27Tier 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(No Transcript)
29Advancing 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.
30Target Student
Dual-Discrepancy RTI Model of Learning
Disability (Fuchs 2003)
31Intervention Research Development A Work in
Progress
32Tier 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.
33Limitations 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.
34Schools 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.
35What 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
36RTI Intervention Key Concepts
37Essential 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.
38Core 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.
39Core 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).
40Core 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).
41Teaching 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..
42Core 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
43Big 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.
44Increasing 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.
45RTI 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.
46Research-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.
47Interventions 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.
48RTI Best Practicesin MathematicsInterventionsJ
im Wrightwww.interventioncentral.org
49National Mathematics Advisory Panel Report13
March 2008
50Math Advisory Panel Report athttp//www.ed.gov/
mathpanel
512008 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
52An 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).
53Math 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
54Profile 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.
55Mathematics is made of 50 percent formulas, 50
percent proofs, and 50 percent imagination.
Anonymous
56Who 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.
57Profile 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..
58The Elements of Mathematical Proficiency What
the Experts Say
59(No Transcript)
60Five Strands of Mathematical Proficiency
- Understanding Comprehending mathematical
concepts, operations, and relations--knowing what
mathematical symbols, diagrams, and procedures
mean. - Computing Carrying out mathematical procedures,
such as adding, subtracting, multiplying, and
dividing numbers flexibly, accurately,
efficiently, and appropriately. - 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.
61Five Strands of Mathematical Proficiency (Cont.)
- Reasoning Using logic to explain and justify a
solution to a problem or to extend from something
known to something less known. - 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.
62Math Computation Building FluencyJim
Wrightwww.interventioncentral.org
63"Arithmetic is being able to count up to twenty
without taking off your shoes." Anonymous
64Benefits 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.
65Math 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.
66Big 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.
67Math Intervention Tier I or II Elementary
Secondary Self-Administered Arithmetic
Combination Drills With Performance
Self-Monitoring Incentives
- The student is given a math computation worksheet
of a specific problem type, along with an answer
key Academic Opportunity to Respond. - The student consults his or her performance chart
and notes previous performance. The student is
encouraged to try to beat his or her most
recent score. - The student is given a pre-selected amount of
time (e.g., 5 minutes) to complete as many
problems as possible. The student sets a timer
and works on the computation sheet until the
timer rings. Active Student Responding - The student checks his or her work, giving credit
for each correct digit (digit of correct value
appearing in the correct place-position in the
answer). Performance Feedback - The student records the days score of TOTAL
number of correct digits on his or her personal
performance chart. - The student receives praise or a reward if he or
she exceeds the most recently posted number of
correct digits.
Application of Learn Unit framework from
Heward, W.L. (1996). Three low-tech strategies
for increasing the frequency of active student
response during group instruction. In R. Gardner,
D. M.S ainato, J. O. Cooper, T. E. Heron, W. L.
Heward, J. W. Eshleman, T. A. Grossi (Eds.),
Behavior analysis in education Focus on
measurably superior instruction (pp.283-320).
Pacific Grove, CABrooks/Cole.
68Self-Administered Arithmetic Combination Drills
69Cover-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.
70Math 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
71Math 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..
72Teaching Math Vocabulary
73Comprehending 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.
74Math 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.
75Promoting 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.
76Math 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.
77Building Student Skills inApplied Math
ProblemsJim Wrightwww.interventioncentral.org
78How 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..
79Potential 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
80Math Intervention Ideas for Higher-Level Math
ProblemsJim Wrightwww.interventioncentral.org
81Applied Problems
82Applied 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.
83Applied 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..
84Applied 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.
85Interpreting Math Graphics A Reading
Comprehension Intervention
86Housing Bubble GraphicNew York Times23
September 2007
87Classroom 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.
88Using 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.
89Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
- 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.
90Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
- 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.
91Using Question-Answer Relationships (QARs) to
Interpret Information from Math Graphics 4-Step
Teaching Sequence
- 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.
92Using 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.
93Developing Student Metacognitive Abilities
94Importance 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.
95Elements of Metacognitive Processes
- Self-instruction helps students to identify and
direct the problem-solving strategies prior to
execution. Self-questioning promot