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Title: Instruction%20and%20Interventions%20within%20Response%20to%20Intervention%20Jim%20Wright%20www.interventioncentral.org


1
Instruction and Interventions within Response to
InterventionJim Wrightwww.interventioncentral.o
rg
2
Resources from This Workshop Available at
http//www.interventioncentral.org/ES_BOCES.php
3
Workshop Agenda
4
Intervention Research Development A Work in
Progress
5
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.
6
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.
7
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.
8
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
9
RTI Intervention Key Concepts
10
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.

11
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.

12
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).

13
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).

14
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

15
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.
16
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.
17
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.
18
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.
19
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.
20
Team Activity What Are Challenging Issues in
Your School Around the Topic of Academic
Interventions?
  • At your tables
  • Discuss the task of promoting the use of
    evidence-based academic interventions in your
    school.
  • What are enabling factors that should help you to
    promote the routine use of such interventions.
  • What are challenges or areas needing improvement
    to allow you to promote use of those
    interventions?

21
RTI Best Practicesin MathematicsInterventionsJ
im Wrightwww.interventioncentral.org
22
National Mathematics Advisory Panel Report13
March 2008
23
Math Advisory Panel Report athttp//www.ed.gov/
mathpanel
24
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
25
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).
26
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

27
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.
28
Mathematics is made of 50 percent formulas, 50
percent proofs, and 50 percent imagination.
Anonymous
29
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.
30
The Elements of Mathematical Proficiency What
the Experts Say
31
(No Transcript)
32
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.
33
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.
34
Five Strands of Mathematical Proficiency (NRC,
2002)
  • Table Activity Evaluate Your Schools Math
    Proficiency
  • As a group, review the National Research Council
    Strands of Math Proficiency.
  • Which strand do you feel that your school /
    curriculum does the best job of helping students
    to attain proficiency?
  • Which strand do you feel that your school /
    curriculum should put the greatest effort to
    figure out how to help students to attain
    proficiency?
  • Be prepared to share your results.
  • 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.
  • 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.

35
Three General Levels of Math Skill Development
(Kroesbergen Van Luit, 2003)
  • As students move from lower to higher grades,
    they move through levels of acquisition of math
    skills, to include
  • Number sense
  • Basic math operations (i.e., addition,
    subtraction, multiplication, division)
  • Problem-solving skills The solution of both
    verbal and nonverbal problems through the
    application of previously acquired information
    (Kroesbergen Van Luit, 2003, p. 98)

Source Kroesbergen, E., Van Luit, J. E. H.
(2003). Mathematics interventions for children
with special educational needs. Remedial and
Special Education, 24, 97-114..
36
Development of Number Sense
37
What is Number Sense? (Clarke Shinn, 2004)
  • the ability to understand the meaning of
    numbers and define different relationships among
    numbers. Children with number sense can
    recognize the relative size of numbers, use
    referents for measuring objects and events, and
    think and work with numbers in a flexible manner
    that treats numbers as a sensible system. p. 236

Source Clarke, B., Shinn, M. (2004). A
preliminary investigation into the identification
and development of early mathematics
curriculum-based measurement. School Psychology
Review, 33, 234248.
38
What Are Stages of Number Sense? (Berch, 2005,
p. 336)
  1. Innate Number Sense. Children appear to possess
    hard-wired ability (neurological foundation
    structures) to acquire number sense. Childrens
    innate capabilities appear also to include the
    ability to represent general amounts, not
    specific quantities. This innate number sense
    seems to be characterized by skills at estimation
    (approximate numerical judgments) and a
    counting system that can be described loosely as
    1, 2, 3, 4, a lot.
  2. Acquired Number Sense. Young students learn
    through indirect and direct instruction to count
    specific objects beyond four and to internalize a
    number line as a mental representation of those
    precise number values.

Source Berch, D. B. (2005). Making sense of
number sense Implications for children with
mathematical disabilities. Journal of Learning
Disabilities, 38, 333-339...
39
Task Analysis of Number Sense Operations (Methe
Riley-Tillman, 2008)
  1. Counting
  2. Comparing and Ordering Ability to compare
    relative amounts e.g., more or less than ordinal
    numbers e.g., first, second, third)
  3. Equal partitioning Dividing larger set of
    objects into equal parts
  4. Composing and decomposing Able to create
    different subgroupings of larger sets (for
    example, stating that a group of 10 objects can
    be broken down into 6 objects and 4 objects or 3
    objects and 7 objects)
  5. Grouping and place value abstractly grouping
    objects into sets of 10 (p. 32) in base-10
    counting system.
  6. Adding to/taking away Ability to add and
    subtract amounts from sets by using accurate
    strategies that do not rely on laborious
    enumeration, counting, or equal partitioning. P.
    32

Source Methe, S. A., Riley-Tillman, T. C.
(2008). An informed approach to selecting and
designing early mathematics interventions. School
Psychology Forum Research into Practice, 2,
29-41.
40
Childrens Understanding of Counting Rules
  • The development of childrens counting ability
    depends upon the development of
  • One-to-one correspondence one and only one word
    tag, e.g., one, two, is assigned to each
    counted object.
  • Stable order the order of the word tags must be
    invariant across counted sets.
  • Cardinality the value of the final word tag
    represents the quantity of items in the counted
    set.
  • Abstraction objects of any kind can be
    collected together and counted.
  • Order irrelevance items within a given set can
    be tagged in any sequence.

Source Geary, D. C. (2004). Mathematics and
learning disabilities. Journal of Learning
Disabilities, 37, 4-15.
41
Math Computation Building FluencyJim
Wrightwww.interventioncentral.org
42
"Arithmetic is being able to count up to twenty
without taking off your shoes." Anonymous
43
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.
44
Internal Numberline
  • As students internalize the numberline, they are
    better able to perform mental arithmetic (the
    manipulation of numbers and math operations in
    their head).

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
16 17 18 1920 21 22 23 24 25 26 27 28 29
45
Associative Property
  • within an expression containing two or more of
    the same associative operators in a row, the
    order of operations does not matter as long as
    the sequence of the operands is not changed
  • Example
  • (23)510
  • 2(35)10

Source Associativity. Wikipedia. Retrieved
September 5, 2007, from http//en.wikipedia.org/wi
ki/Associative
46
Commutative Property
  • the ability to change the order of something
    without changing the end result.
  • Example
  • 23510
  • 25310

Source Associativity. Wikipedia. Retrieved
September 5, 2007, from http//en.wikipedia.org/wi
ki/Commutative
47
How much is 3 8? Strategies to Solve
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.
48
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.
49
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.
50
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..
51
Teaching Math Vocabulary
52
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.
53
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.
54
Vocabulary Why This Instructional Goal is
Important
  • As vocabulary terms become more specialized in
    content area courses, students are less able to
    derive the meaning of unfamiliar words from
    context alone.
  • Students must instead learn vocabulary through
    more direct means, including having opportunities
    to explicitly memorize words and their
    definitions.
  • Students may require 12 to 17 meaningful
    exposures to a word to learn it.

55
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.
56
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.
57
Teaching Math Symbols
58
Learning Math Symbols 3 Card Games
  1. The interventionist writes math symbols that the
    student is to learn on index cards. The names of
    those math symbols are written on separate cards.
    The cards can then be used for students to play
    matching games or to attempt to draw cards to get
    a pair.
  2. Create a card deck containing math symbols or
    their word equivalents. Students take turns
    drawing cards from the deck. If they can use the
    symbol/word on the selected card to generate a
    correct mathematical sentence, the student wins
    the card. For example, if the student draws a
    card with the term negative number and says
    that A negative number is a real number that is
    less than 0, the student wins the card.
  3. Create a deck containing math symbols and a
    series of numbers appropriate to the grade level.
    Students take turns drawing cards. The goral is
    for the student to lay down a series of cards to
    form a math expression. If the student correctly
    solves the expression, he or she earns a point
    for every card laid down.

Source Adams, T. L. (2003). Reading mathematics
More than words can say. The Reading Teacher,
56(8), 786-795.
59
Developing Student Metacognitive Abilities
60
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.
61
Elements of Metacognitive Processes
  • Self-instruction helps students to identify and
    direct the problem-solving strategies prior to
    execution. Self-questioning promotes internal
    dialogue for systematically analyzing problem
    information and regulating execution of cognitive
    strategies. Self-monitoring promotes appropriate
    use of specific strategies and encourages
    students to monitor general performance.
    Emphasis added. p. 231

Source Montague, M. (1992). The effects of
cognitive and metacognitive strategy instruction
on the mathematical problem solving of middle
school students with learning disabilities.
Journal of Learning Disabilities, 25, 230-248.
62
Combining Cognitive Metacognitive Strategies to
Assist Students With Mathematical Problem Solving
  • Solving an advanced math problem independently
    requires the coordination of a number of complex
    skills. The following strategies combine both
    cognitive and metacognitive elements (Montague,
    1992 Montague Dietz, 2009). First, the student
    is taught a 7-step process for attacking a math
    word problem (cognitive strategy). Second, the
    instructor trains the student to use a three-part
    self-coaching routine for each of the seven
    problem-solving steps (metacognitive strategy).

63
Cognitive Portion of Combined Problem Solving
Approach
  • In the cognitive part of this multi-strategy
    intervention, the student learns an explicit
    series of steps to analyze and solve a math
    problem. Those steps include
  • Reading the problem. The student reads the
    problem carefully, noting and attempting to clear
    up any areas of uncertainly or confusion (e.g.,
    unknown vocabulary terms).
  • Paraphrasing the problem. The student restates
    the problem in his or her own words.
  • Drawing the problem. The student creates a
    drawing of the problem, creating a visual
    representation of the word problem.
  • Creating a plan to solve the problem. The student
    decides on the best way to solve the problem and
    develops a plan to do so.
  • Predicting/Estimating the answer. The student
    estimates or predicts what the answer to the
    problem will be. The student may compute a quick
    approximation of the answer, using rounding or
    other shortcuts.
  • Computing the answer. The student follows the
    plan developed earlier to compute the answer to
    the problem.
  • Checking the answer. The student methodically
    checks the calculations for each step of the
    problem. The student also compares the actual
    answer to the estimated answer calculated in a
    previous step to ensure that there is general
    agreement between the two values.

64
Metacognitive Portion of Combined Problem Solving
Approach
  • The metacognitive component of the intervention
    is a three-part routine that follows a sequence
    of Say, Ask, Check. For each of the 7
    problem-solving steps reviewed above
  • The student first self-instructs by stating, or
    saying, the purpose of the step (Say).
  • The student next self-questions by asking what
    he or she intends to do to complete the step
    (Ask).
  • The student concludes the step by
    self-monitoring, or checking, the successful
    completion of the step (Check).

65
Combined Cognitive Metacognitive Elements of
Strategy
66
Combined Cognitive Metacognitive Elements of
Strategy
67
Combined Cognitive Metacognitive Elements of
Strategy
68
Combined Cognitive Metacognitive Elements of
Strategy
69
Combined Cognitive Metacognitive Elements of
Strategy
70
Combined Cognitive Metacognitive Elements of
Strategy
71
Combined Cognitive Metacognitive Elements of
Strategy
72
Applied Problems Pop Quiz
  • Q To move their armies, the Romans built over
    50,000 miles of roads. Imagine driving all those
    miles! Now imagine driving those miles in the
    first gasoline-driven car that has only three
    wheels and could reach a top speed of about 10
    miles per hour.
  • For safety's sake, let's bring along a spare
    tire. As you drive the 50,000 miles, you rotate
    the spare with the other tires so that all four
    tires get the same amount of wear. Can you figure
    out how many miles of wear each tire accumulates?

Directions As a team, read the following
problem. At your tables, apply the 7-step
problem-solving (cognitive) strategy to complete
the problem. As you complete each step of the
problem, apply the Say-Ask-Check metacognitive
sequence. Try to complete the entire 7 steps
within the time allocated for this exercise.
  • 7-Step Problem-SolvingProcess
  • Reading the problem.
  • Paraphrasing the problem.
  • Drawing the problem.
  • Creating a plan to solve the problem.
  • Predicting/Estimat-ing the answer.
  • Computing the answer.
  • Checking the answer.

A Since the four wheels of the three-wheeled
car share the journey equally, simply take
three-fourths of the total distance (50,000
miles) and you'll get 37,500 miles for each
tire.
Source The Math Forum _at_ Drexel Critical
Thinking Puzzles/Spare My Brain. Retrieved from
http//mathforum.org/k12/k12puzzles/critical.think
ing/puzz2.html
73
Finding a Way Out of the Research-Based Maze
A Guide for SchoolsJim Wrightwww.intervention
central.org
74
Innovations in Education Efficacy vs.
Effectiveness
  • A useful distinction has recently emerged
    between efficacy and effectiveness (Schoenwald
    Hoagwood, 2001). Efficacy refers to intervention
    outcomes that are produced by researchers and
    program developers under ideal conditions of
    implementation (i.e., adequate resources, close
    supervision ). In contrast, effectiveness refers
    to demonstration(s) of socially valid outcomes
    under normal conditions of usage in the target
    setting(s) for which the intervention was
    developed. Demonstrations of effectiveness are
    far more difficult than demonstrations of
    efficacy. In fact, numerous promising
    interventions and approaches fail to bridge the
    gap between efficacy and effectiveness.
    Emphasis added

Source Walker, H. M. (2004). Use of
evidence-based interventions in schools Where
we've been, where we are, and where we need to
go. School Psychology Review, 33, 398-407. p. 400
75
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Define the Academic or Behavioral Needs Requiring
    Intervention in Detail and Using Standard
    Terminology. Effective interventions cannot be
    reliably identified and matched to student needs
    if those needs are loosely or vaguely defined.
  • Overly broad academic goal statement a student
    will know her letters.
  • More focused goal statement When shown any
    letter in uppercase or lowercase form, the
    student will accurately identify the letter name
    and its corresponding sound without assistance.
  • When possible, describe academic behaviors
    selected as intervention target using standard
    terminology to make it easier to locate
    appropriate evidence-based intervention ideas.

76
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Develop Consensus in Your School About What is
    Meant by Evidence-Based.
  • Compile a list of trusted professional
    organizations and journals. Continue to add to
    this list of trusted organizations and journals
    over time.

77
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Develop Consensus in Your School About What is
    Meant by Evidence-Based.
  • Draft a definition of evidence-based. Example
    The International Reading Association (2002)
    provides these guidelines Produce
    objective dataso that different evaluators
    should be able to draw similar conclusions when
    reviewing the data from the studies. Have
    valid research results that can reasonably be
    applied to the kinds of real-world reading tasks
    that children must master in actual classrooms.
    Yield reliable and replicable findings that would
    not be expected to change significantly based on
    such arbitrary factors as the day or time that
    data on the interventions were collected or who
    collected them.
  • Employ current best-practice methods in
    observation or experimentation to reduce the
    probability that other sources of potential bias
    crept into the studies and compromised the
    results.
  • Checked before publication by independent
    experts, who review the methods, data, and
    conclusions of the studies.

78
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Develop Consensus in Your School About What is
    Meant by Evidence-Based.
  • Adopting a research continuum. It can be useful
    for schools to use a research continuum that
    establishes categories for interventions in
    descending levels of research quality. The
    continuum would be used as an aid to judge
    whether specific instructional practices or
    interventions are supported by research of
    sufficient quantity and quality for use in
    schools.

79
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Use Impartial On-Line Rating Sites to Evaluate
    Commercial Intervention Products. Cautions to
    keep in mind when using these sites
  • They typically rely on existing research only.
  • There can potential delays / lag time between the
    publication of new research and these sites
    evaluation of that research.

80
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82
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Know the Research-Based Components That Are
    Building Blocks of Effective Interventions.
    Research indicates (Burns, VanDerHeyden, Boice,
    2008) that, to be maximally effective,
    interventions should
  • be matched to the students academic needs
  • be delivered using explicit instruction
  • provide the student with adequate success in the
    instructional task
  • give the student a high opportunity to respond
  • provide timely performance feedback.

83
Finding a Way Out of the Research-Based Maze A
Guide for Schools
  • Keep Up With Emerging Intervention Research
    Through Knowledge Brokers.
  • Districts first define manageable and sensible
    intervention topic areas, such as alphabetics
    and reading fluency.
  • Then district or school staff members are
    selected to serve as knowledge brokers based on
    their training, experience, and/or interest.
  • Knowledge brokers regularly read educational
    research journals and other publications from
    reputable organizations or government agencies to
    keep up with emerging research in their
    intervention topic area.
  • They periodically share their expertise with
    other district RTI planners to ensure that the
    schools are using the best available intervention
    strategies.

84
Defining Academic Problems Get It Right and
Interventions Are More Likely to Be
EffectiveJim Wrightwww.interventioncentral.org
85
Defining Academic Problems Recommended Steps
  1. Be knowledgeable of the school academic
    curriculum and key student academic skills that
    are taught. The teacher should have a good
    survey-level knowledge of the key academic skills
    outlined in the schools curriculumfor the grade
    level of their classroom as well as earlier grade
    levels. If the curriculum alone is not adequate
    for describing a students academic deficit, the
    instructor can make use of research-based
    definitions or complete a task analysis to
    further define the academic problem area. Here
    are guidelines for consulting curriculum and
    research-based definitions and for conducting a
    task analysis for more global skills.

86
Defining Academic Problems Recommended Steps
  • Curriculum. The teacher can review the schools
    curriculum and related documents (e.g.,
    score-and-sequence charts curriculum maps) to
    select specific academic skill or performance
    goals. First, determine the approximate grade or
    level in the curriculum that matches the
    students skills. Then, review the curriculum at
    that alternate grade level to find appropriate
    descriptions of the students relevant academic
    deficit. For example, a second-grade student
    had limited phonemic awareness. The student was
    not able accurately to deconstruct a spoken word
    into its component sound-units, or phonemes. In
    the schools curriculum, children were expected
    to attain proficiency in phonemic awareness by
    the close of grade 1. The teacher went off
    level to review the grade 1 curriculum and found
    a specific description of phonemic awareness that
    she could use as a starting point in defining the
    students skill deficit.

87
Defining Academic Problems Recommended Steps
  • Research-Based Skill Definitions. Even when a
    schools curriculum identifies key skills,
    schools may find it useful to corroborate or
    elaborate those skill definitions by reviewing
    alternative definitions published in research
    journals or other trusted sources. For example,
    a student had delays in solving quadratic
    equations. The math instructor found that the
    schools math curriculum did not provide a
    detailed description of the skills required to
    successfully complete quadratic equations. So the
    teacher reviewed the National Mathematics
    Advisory Panel report (Fennell et al., 2008) and
    found a detailed description of component skills
    for solving quadratic equations. By combining the
    skill definitions from the school curriculum with
    the more detailed descriptions taken from the
    research-based document, the teacher could better
    pinpoint the students academic deficit in
    specific terms.

88
Defining Academic Problems Recommended Steps
  • Task Analysis. Students may possess deficits in
    more global academic enabling skills that are
    essential for academic success. Teachers can
    complete an task analysis of the relevant skill
    by breaking it down into a checklist of
    constituent subskills. An instructor can use the
    resulting checklist to verify that the student
    can or cannot perform each of the subskills that
    make up the global academic enabling
    skill.For example, teachers at a middle school
    noted that many of their students seemed to have
    poor organization skills. Those instructors
    conducted a task analysis and determined that--in
    their classrooms--the essential subskills of
    student organization included (a) arriving to
    class on time (b) bringing work materials to
    class (c) following teacher directions in a
    timely manner (d) knowing how to request teacher
    assistance when needed and (e) having an
    uncluttered desk with only essential work
    materials.

89
Defining Academic Problems Recommended Steps
  • Describe the academic problem in specific,
    skill-based terms (Batsche et al., 2008 Upah,
    2008). Write a clear, brief description of the
    academic skill or performance deficit that
    focuses on a specific skill or performance area.
    Here are sample problem-identification
    statements
  • John reads aloud from grade-appropriate text much
    more slowly than his classmates.
  • Ann lacks proficiency with multiplication math
    problems (double-digit times double-digit with no
    regrouping).
  • Tye does not turn in homework assignments.
  • Angela produces limited text on in-class writing
    assignments.

90
Defining Academic Problems Recommended Steps
  • Develop a fuller description of the academic
    problem to provide a meaningful instructional
    context. When the teacher has described the
    students academic problem, the next step is to
    expand the problem definition to put it into a
    meaningful context. This expanded definition
    includes information about the conditions under
    which the academic problem is observed and
    typical or expected level of performance.
  • Conditions. Describe the environmental conditions
    or task demands in place when the academic
    problem is observed.
  • Problem Description. Describe the actual
    observable academic behavior in which the student
    is engaged. Include rate, accuracy, or other
    quantitative information of student performance.
  • Typical or Expected Level of Performance. Provide
    a typical or expected performance criterion for
    this skill or behavior. Typical or expected
    academic performance can be calculated using a
    variety of sources,

91
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92
Defining Academic Problems Recommended Steps
  1. Develop a hypothesis statement to explain the
    academic skill or performance problem. The
    hypothesis states the assumed reason(s) or
    cause(s) for the students academic problems.
    Once it has been developed, the hypothesis
    statement acts as a compass needle, pointing
    toward interventions that most logically address
    the student academic problems.

93
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94
Activity Defining Academic Interventions
  • Consider the structured format that was reviewed
    for defining academic interventions.
  • How can you use this framework to support RTI in
    your school?

95
Tier 1 Case Example Patricia Reading
Comprehension
96
Case Example Reading Comprehension
  • The Problem
  • A student, Patricia, struggled in her social
    studies class, particularly in understanding the
    course readings. Her teacher, Ms. Cardamone,
    decided that the problem was significant enough
    that the student required some individualized
    support.

97
Case Example Reading Comprehension
  • The Evidence
  • Student Interview. Ms. Cardamone met with
    Patricia to ask her questions about her
    difficulties with social studies content and
    assignments. Patricia said that when she reads
    the course text and other assigned readings, she
    doesnt have difficulty with the vocabulary but
    often realizes after reading half a page that she
    hasnt really understood what she has read.
    Sometimes she has to reread a page several times
    and that can be frustrating.

98
Case Example Reading Comprehension
  • The Evidence (Cont.)
  • Review of Records. Past teacher report card
    comments suggest that Patricia has had difficulty
    with reading comprehension tasks in earlier
    grades. She had received help in middle school in
    the reading lab, although there was no record of
    what specific interventions were tried in that
    setting.
  • Input from Other Teachers. Ms. Cardamone checked
    with other teachers who have Patricia in their
    classes. All expressed concern about Patricias
    reading comprehension skills. The English
    teacher noted that Patricia appears to have
    difficulty pulling the main idea from a passage,
    which limits her ability to extract key
    information from texts and to review that
    information for tests.
  •  

99
Case Example Reading Comprehension
  • The Intervention
  • Ms. Cardamone decided, based on the evidence
    collected, that Patricia would benefit from
    training in identifying the main idea from a
    passage, rather than trying to retain all the
    information presented in the text. She selected
    two simple interventions Question Generation and
    Text Lookback. She arranged to have Patricia meet
    with the Reading Lab teacher to learn these two
    strategies. Then Ms. Cardamone scheduled time to
    meet with Patricia to demonstrate how to use
    these strategies effectively with the social
    studies course text and other assigned readings.

100
  • Students are taught to boost their comprehension
    of expository passages by (1) locating the main
    idea or key ideas in the passage and (2)
    generating questions based on that information.

QuestionGeneration
http//www.interventioncentral.org/htmdocs/interve
ntions/rdngcompr/qgen.php
101
  • Text lookback is a simple strategy that students
    can use to boost their recall of expository prose
    by identifying questions that require information
    from the text and then looking back in the text
    in a methodical manner to locate that
    information.

Text Lookback
http//www.interventioncentral.org/htmdocs/interve
ntions/rdngcompr/txtlkbk.php
102
Case Example Reading Comprehension
  • The Outcome
  • When the intervention had been in place for 4
    weeks, Ms. Cardamone noted that Patricia appeared
    to have a somewhat better grasp of course content
    and expressed a greater grasp of material from
    the text. She shared her intervention ideas with
    other teachers working with Patricia. While
    Patricias grades did improve in Social Studies,
    they were still only borderline passing. Ms.
    Cardamone decided to continue her classroom
    interventions with Patricia but also to refer the
    student to the RTI Team to see if the student
    could receive any additional support to build her
    reading comprehension skills.

103
Tier 1 Case Example Justin Non-Compliance
104
Case Example Non-Compliance
  • The Problem
  • Justin showed a pattern from the start of the
    school year of not complying with teacher
    requests in his English class. His teacher, Mr.
    Steubin, noted
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