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Effective Math Instruction for Students with High Incidence Disabilities


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Title: Effective Math Instruction for Students with High Incidence Disabilities

Effective Math Instruction for Students with
High Incidence Disabilities
  • Joseph Calvin Gagnon, Ph.D.
  • George Mason University

Advanced Organizer Math Session
  1. Educational Reform
  2. The National Council of Teachers of Mathematics
  3. Characteristics of students with learning and
    emotional/behavioral disabilities
  4. Direct instruction (di)

Advanced Organizer Math Session
  1. Real world application and technology
  2. Student grouping
  3. Graduated instructional sequence
  4. Graphic Organizers
  5. Strategy instruction
  6. Instructional adaptations

Educational Reform
  • Standards-driven reform is the primary approach
    to assuring todays high school graduates are
    internationally competitive
  • Prompted by the public dissatisfaction and poor
    performance by U.S. students on international
    assessments (McLaughlin, Shepard, ODay, 1995),
    educators, curriculum specialists, and national
    organizations have focused on development of
    challenging standards for over a decade.

Educational Reform
  • Ensuring all students achieve in math is a
    national priority (IDEA, 1997 No Child Left
    Behind Act of 2001)
  • Success in math is considered a gateway to many
    educational and occupational opportunities
    (Jetter, 1993)

Educational Reform
  • Recent legislation has assisted these efforts and
    ensured that students with disabilities are
    included, to the maximum extent possible
  • Central to this notion of reform is the assertion
    that all students are, entitled to instruction
    that is grounded in a common set of challenging
    standards (McLaughlin, 1999, p. 10)

Educational Reform
  • Rigorous standards are especially crucial for
    students with learning disabilities (LD) and
    emotional disturbances (ED), who are commonly
    included in the general education environment.
  • These students have historically been provided a
    less rigorous curriculum with IEP goals that
  • Focus on computation (Shriner, Kim, Thurlow,
    Ysseldyke, 1993)
  • Have minimal linkage to long-term general
    education outcomes (Nolet McLaughlin, 2000
    Sands, Adams, Stout, 1995 Smith, 1990)

Educational Reform
  • The NCTM Standards are a critical component of
    the Standards-driven reform movement
  • At least 42 states have used the Standards as a
    guide to development of mathematics standards
    Blank and Dalkilic (1992) (as noted in Thurlow,

Characteristics of Students with LD
  • On average, adolescents with LD function 2.7
    grade levels below their nonlabeled peers
    (Wagner, 1995)
  • Secondary teachers have noted that many of their
    students experience difficulty in mathematics
    (McLeod Armstrong, 1982)

Characteristics of Students with LD
  • Adolescents with LD have difficulty with problem
    application and generally perform at the 5th
    grade level (Cawley Miller, 1989)
  • Secondary students with LD experience
    difficulties with a range of mathematics tasks,
  • Basic skills (Algozzine, OShea, Crews,
    Stoddard, 1987)
  • Higher-level skills/concepts and problem solving
    (Huntington, 1994 Hutchinson, 1993 Maccini
    Hughes, 2000 Maccini Ruhl, 2000)

Characteristics of Students with ED
  • Students with ED are typically 1.8 grade levels
    behind their nonlabeled peers (Wagner, 1995)
  • Adolescents with ED possess characteristics hat
    differentiate them from nonhandicapped peers
  • The academic success or failure of students
    labeled ED is greatly affected by the extent to
    which instruction is functional and recognized by
    students as relevant (Bos Vaughn, 1994)

Characteristics of Students with ED
  • These students often exhibit a general lack of
    persistence and concentration and have
    difficulties with independent class work
  • Secondary students with ED share a common set of
    learner characteristics that negatively affect
    their academic success motivational issues,
    anxiety, and impulse control
  • (Byrne, 1984 Dweck Elliot, 1983 Gottfied,
    1985 McNeil, 1998 Patten, 1983)

Characteristics of Students with ED
  • Students with ED obtain a percent correct rate
    between 20 and 76 on independent seatwork
    (Guntner Denny, 1998)
  • The ability to persist and work independently on
    open-ended mathematical tasks could greatly
    affect the level of success experienced in light
    of the more constructivist approach that guides
    the NCTM Standards

Six Math Instructional Recommendations
  1. Incorporate components of direct instruction
  2. Teach strategies
  3. Embed math in real-world activities and include
    the use of technology into instruction

Math Instructional Recommendations cont.
  • Group for instruction
  • Incorporate a graduated (i.e., Concrete-Semiconcre
    te-Abstract) instructional Sequence
  • Use instructional adaptations

Direct Instruction (di)
  • Effective Teaching
  • Researchers note that incorporating efficient and
    effective teaching components into the teaching
    routine promotes student learning and retention
    (Rosenshine Stevens, 1986).
  • These include
  • Daily review
  • Presentation (provide overview of lesson, teach
    new skills at a fast rate and in small
    increments, model procedures, check for
    understanding, teach to mastery)

Direct Instruction (di)
  • guided practice
  • corrective and positive feedback
  • independent practice
  • frequent reviews (cumulative weekly and monthly

Direct Instruction (di)
  • Nationally, close to 70 or more of general and
    special education teachers reported being
    prepared to use di
  • Teachers reported using di variables frequently
    (i.e., 2 4 times per week to daily) (Gagnon
    Maccini, 2004)
  • This is promising given that use of techniques
    consistent with teacher-directed instruction has
    been empirically validated for teaching math to
    secondary students with LD (Kelly, Gersten,
    Carnine, 1990 Moore Carnine, 1989)

Direct Instruction
  • Researchers (Gagnon Maccini, 2005) recommend
    providing direct instruction on a daily/weekly
    basis and providing weekly and monthly cumulative

Direct Instruction
  • Example Gagnon, J. C., Maccini, P. (2005).
    Direct instruction in mathematics for youth with
    learning disabilities in middle school.
    Washington, DC The Access Center Improving
    Outcomes for all Students K-8.
  • http//www.k8accesscenter.org/training_resources/

Real World Problem Solving and Technology
  • Technology-based instructional approaches can
    significantly affect student learning and
    acquisition of higher-level math concepts
    particularly when embedded within real-world
    problem solving tasks (Maccini Gagnon, 2005)
  • This approach relies on the use of a computer,
    calculator, or other specialized systems as the
    mode of instruction (Vergason Anderegg, 1997)

Real World Problem Solving and Technology
  • Technology-based instruction can
  • Assist teachers in moving away from a focus on
    memorization and routine manipulation of numbers
    in formulas and toward instruction and activities
    embedded in real-world problems (Bottge
    Hasselbring, 1993)
  • Promote active student learning (Kelly, Gersten,
    Carnine, 1990)

Real World Problem Solving and Technology
  • Embedding problem solving information within a
    real world context helps
  • Activate student conceptual knowledge when
    presented with a real-life problem solving
    situation (Gagne, Yekovich, Yekovich, 1993)
  • Improve student motivation, participation, and
    generalization (Palloway Patton, 1997)

Real World Problem Solving and Technology
  • Rather than capitalizing on the insights and
    motivation that students bring to the classroom,
    schools may actually be wasting valuable time by
    withholding more authentic and motivating
    problems until prerequisite skills are
    acquired (Bottge et al., 2001, p. 312)
  • It is effective to use videodisc-based
    interventions that embed interesting and
    age-appropriate problem-solving situations
    (Bottge, 1999 Bottge Hasselbring, 1993 Bottge
    et al., 2001 Bottge et al., 2002)

Real World Problem Solving and Technology
  • Recommendations
  • Incorporate di (e.g., model, guided practice,
    review, feedback) within technology-based
  • Incorporate effective instructional design
    variables (examples follow) within
    technology-based instruction to reduce student
    confusion and mathematical errors

Real World Problem Solving and Technology
  • Discrimination Skills are introduced, practiced,
    and mixed with other types of problems. Specific
    instruction and remediation provide for
  • Range of Examples Students introduced to
    fractions less than one, improper fractions, and
    provided strategies for reading and writing both

Real World Problem Solving and Technology
  • Explicit Strategy Teaching Students provided
    explicit problem solving strategies
  • Computer software should incorporate a wide range
    of examples and nonexamples into instruction for
    discrimination practice and generalization

Real World Problem Solving and Technology
  • Recommendations
  • Incorporate technology-based tutorial programs
    that embed basic math skills and higher order
    thinking within problem-solving situations
  • This allows students to practice remedial skills
    within context
  • For example, it is recommended that computers be
    available to students with LD for tutorial

Real World Problem Solving and Technology
  • Limitations to the use of technology
  • The review was limited to 11 published articles
    that met all criteria
  • Although 73 (n 8) of the studies determined
    significant treatment effects, three of the
    studies noted that the proficiency levels of
    students with disabilities fell below the
    established criterion for learning of 80

Real World Problem Solving and Technology
  • Further, of the articles that obtained
    significant findings, only 45 (n 5) of the
    interventions directly programmed for maintenance
    and 55
  • (n 6) programmed for generalization
  • The generalizability of the findings may also be
    of concern because no information was available
    on new technologies (e.g., DVD and streaming

Technology and Real-World Activities
  • Research Recommendations
  • It is recommended (Gagnon Maccini, 2005) to
    provide technology-based learning activities
    real-world activities that incorporate effective
    teaching variables on a daily/weekly basis

Real World Problem Solving and Technology
  • Calculators
  • In one study, calculator use was the most
    prevalent adaptation noted by teachers Maccini
    Gagnon, 2002)
  • Consistent with Etlinger and Ogletree (1982),
    teacher responses involved two primary

Real World Problem Solving and Technology
  • The "practical" function The use of calculators
    to complete tedious calculations, save time,
    increase student motivation, and to decrease math
  • The "pedagogical" function Relates to
    similarities between calculators, textbooks, and
    manipulatives in that each enhances student
    understanding and competence in mathematics

Real World Problem Solving and Technology
  • These classifications are consistent with the
    five primary functions of calculators as stated
    by the NCTM
  • Within the practical classification, NCTM
    identifies the use of calculators to
  • Perform tedious computations that arise when
    working with real data in problem solving
  • Concentrate on the problem-solving process rather
    than calculations associated with problems
  • Gain access to mathematics beyond their level of
    computational skill

Real World Problem Solving and Technology
  • The pedagogical function coincides with two other
    uses identified by NCTM
  • To explore, develop, and reinforce concepts
    including estimation, computation, approximation,
    and properties
  • To experiment with math ideas and discover

Real World Problem Solving and Technology
  • Salend and Hoffstetter (1996) asserted the
    importance of
  • Training students to use calculators
  • Using an overhead projector to teach this skill
  • Locating and describing the function of each key
    to students
  • Providing examples of calculator use

Real World Problem Solving and Technology
  • Students should be provided opportunities to
    practice calculations, including estimation
    skills and reviewing answers obtained through
    calculator use

Real World Problem Solving and Technology
  • The following recommendations for teachers are
  • Model calculator application
  • Use calculators in computation, problem solving,
    concept development, pattern recognition, data
    analysis, and graphing
  • Integrate calculator use in assessment and

Real World Problem Solving and Technology
  • Remain current with state-of-the-art technology
  • Explore and develop new ways to use calculators
    to support instruction and assessment

  • Maccini, P., Gagnon, J. C. (2005). Mathematics
    and technology-based interventions for secondary
    students with learning disabilities. In D.
    Edyburn, K. Higgins, R. Boone, The handbook of
    special education technology research and
    practice (pp. 599-622). Winston-Salem, NC
    Knowledge By Design, Inc.

Grouping for Instruction
  • Grouping for instruction involves cooperative
    group activities and peer tutoring.
  • Cooperative learning refers to an instructional
    arrangement for teaching academic and
    collaborative skills to a small, heterogeneous
    group of students (Rivera, 1996, p. 1)
  • Peer tutoring is a systematic tutoring
    arrangement in which peers rotate assisting one
    another, where one acts as the tutor (coach) and
    the other as the tutee (learner)

Grouping for Instruction
  • Grouping for instruction involves the use of
    small group instruction, one-on-one support,
    cooperative group activities, individualized
    instruction, and peer tutoring
  • Research
  • Grouping adaptations reduce occurrences of
    behavioral problems (Penno, Frank, Wacker,
  • Peer-assisted learning promotes computational
    skills (Calhoon Fuchs, 2003)
  • Classwide peer tutoring is effective in
    strengthening basic math skills (Allsopp, 1997)

Grouping for Instruction
  • Peer tutoring has several benefits including
  • Promoting active student responding
  • Providing students opportunities to correct
  • Providing students with immediate feedback
  • Teaching self-management
  • Providing a structured, task-focused opportunity
    for positive social interaction

Grouping for Instruction
  • Classwide peer tutoring (CWPT)
  • Typically, the entire class participates in CWPT
  • First, students are paired and the pairs are
    separated into two groups within the classroom
  • Each session last approximately 30 minutes and
    can be implemented from two to five days per week

Grouping for Instruction
  • Within a session, each student
  • Spends 10 minutes acting as the tutor
  • Spends 10 minutes as the tutee
  • Students are then provided approximately 5
    minutes to record their individual points

Grouping for Instruction
  • Points may be earned individually for
  • Correct responses
  • Error correction
  • Following the tutoring procedures

Grouping for Instruction
  • To increase self-management and positive student
    interactions, teachers may designate certain
    instances where the tutor provides the points to
    the tutee
  • At the end of the week, teams meeting a certain
    criteria level may earn a special reinforcing

Grouping for Instruction
  • To implement CWPT
  • The teacher must identify the specific procedural
    steps and expectations for students to follow
  • Students should be taught the exact procedures
    through a simple three-step process
  • First, the teacher explains and posts the exact
    list of procedures

Grouping for Instruction
  1. Then, the teacher models the peer tutoring
    process and allows students to participate in
    role playing
  2. Lastly, the teacher provides an opportunity for
    the students to use the process and receive
    feedback on their correct use of the format

Grouping for Instruction
  • Recommendations for Practice
  • Cooperative learning strategies
  • Classwide peer-tutoring
  • Use of teaching assistants to support learning

Grouping for Instruction Research
  • Research Recommendations
  • It is recommended (Allsopp, 1997 OMelia
    Rosenberg, 1994) to provide cooperative learning
    and peer tutoring arrangements 2-4 times per week
  • Resource
  • Allsopp, D. H. (1997). Using classwide peer
    tutoring to teach beginning algebra
    problem-solving skills in heterogeneous
    classrooms. Remedial and Special Education, 18,

Concrete-Semiconcrete-Abstract Instructional
Sequence (C-S-A)
  • Bruners structure-oriented theory of learning
  • Enactive mode (e.g., the doing phase - using
    concrete objects to represent problems - concrete
  • Iconic mode (e.g., the seeing phase visualizing
    representations of the problem - semiconcrete
  • Symbolic mode (e.g., using abstract symbols to
    represent the problem - abstract representations)

  • Empirical studies have validated CSA use with
    students with high incidence disabilities for
  • Whole number operations
  • Word problems
  • Place value
  • Introductory algebra skills

  • Implications
  • It is recommended (Gagnon Maccini, 2001) to
    provide instruction using the graduated
    instructional sequence on a daily/weekly basis
    when introducing new math concepts and advancing
    to more abstract ideas.
  • Resources (Handout)
  • Gagnon, J. C., Maccini, P. (2001). Preparing
    students with disabilities for algebra
    Kindergarten through secondary school. TEACHING
    Exceptional Children, 33(2), 8-15.

Teach Strategies
  • A strategy refers to, a plan that not only
    specifies the sequence of needed actions but also
    consists of critical guidelines and rules related
    to making effective decisions during a problem
    solving process (Ellis Lenz, 1996, p. 24). A
    number of features help to make strategies
    effective for students, including

Teach Strategies
  • Use strategy steps that include familiar words,
    are stated simply and concisely, and begin with
    action verbs to facilitate student involvement
    (e.g., Read the problem carefully), and that are
    sequenced appropriately
  • Use strategy steps that remind students to read
    the problem carefully, to obtain a whole picture
    of the problem (problem representation), to solve
    the problem, and to check their answers (problem

Strategy Instruction
  • Strategy Instruction Can Include
  • Structured worksheets/cue cards to help students
    remember problem solving steps or strategies for
    solving problems
  • Mnemonics to help students recall problem solving
    steps or important facts
  • Research
  • Strategy instruction that incorporated a
    first-letter mnemonic and structured worksheets
    helped students with LD learn prealgebra skills
    and concepts (Maccini Hughes, 2000)

Strategy Instruction Implications
  • Need to use
  • On a daily/weekly basis, use strategies that
    incorporate memory devices, sequenced strategy
    steps, and both problem representation and
  • Resource
  • Maccini, P., Gagnon, J. C. (in press). Math
    strategy instruction for middle school students
    with learning disabilities. Washington, DC The
    Access Center Improving Outcomes for all
    Students K-8

Instructional Adaptations
  • Instructional adaptations include structured
    worksheets/graphic organizers, self-monitoring
    devices, and advance organizers.
  • Provide graphic organizer/structured worksheets
    to help students remember and recall information
    (e.g., steps to a strategy).
  • Incorporate self-monitoring to help students
    monitor their problem solving behavior

Instructional Adaptations
  • Use advance organizers to help students identify,
    organize, understand, and retain information
    (Lenz, Bulgren, Hudson, 1990).

Instructional Adaptations Organizers
  • Students with disabilities have difficulties
  • Remembering and recalling information (Olson
    Platt, 1996)
  • Identifying relevant information
  • Organizing information
  • Using visual organizers, such as structured
    worksheets, prompt cards, or graphic organizers
    helps students analyze and solve math problems
    (Gagnon Maccini, 2001)

Instructional Adaptations Organizers
  • Graphic organizers should be taught to students
    using di, used when introducing new material, and
    used during instruction to help students organize
    the information (Maccini Gagnon, 2005)
  • Self-monitoring or individualized
    self-instruction checklists should be used to
    help prompt students to use the correct
    steps/procedures (Dunlap Dunlap, 1989)

Instructional Adaptations Organizers
  • For examples of organizes, key components, ways
    to develop them and instruct students in using
    organizers, see
  • Maccini, P., Gagnon, J. C. (2005). Math
    graphic organizers for students with
    disabilities. Washington, DC The Access Center
    Improving Outcomes for all Students K-8.
    Available at http//www.k8accesscenter.org/trainin

Instructional Adaptations
  • Recommendations for Practice
  • Include assignment adaptations to maintain
    student attention
  • Examples
  • Salend (1990) supports the adaptation of
    assignments through
  • A decrease in the number of problems assigned and
    includes three related suggestions
  • Reviewing previously mastered skills
  • Dividing a task or worksheet in to smaller tasks
    or sections

Instructional Adaptations
  • Inappropriate student behavior decreases when
    students are presented with a sequence of
    shortened assignments versus one long assignment
    (Dunlap et al., 1993)

Instructional Adaptations
  • Meese (1994) identifies several effective
    assignment modifications
  • Divide assignments into chunks and have timelines
    for each chunk
  • Extend time for completing assignments
  • Encourage the use of calculators and computers
  • Allow groups to complete some written assignments

Instructional Adaptations
  • 5. Reduce the amount of copying needed throughout
    the assignment (e.g., from board, notetaking)
  • 6. Require students to paraphrase an assignment's
    tasks (p. 350-351)
  • 7. A reduction in the number of problems assigned
    to students (Salend, 1994)

Instructional Adaptations Research
  • It is recommended to use these instructional
    adaptations daily (advance organizer), or on an
    as needed basis (graphic organizer,
    self-monitoring devices).
  • For example, it is recommended to provide an
    advance organizer to help orient students to the
    lesson-of-the-day or the new topic.

  • For more information on teaching reading and math
    to secondary students with emotional and
    behavioral disorders, see
  • Gagnon, J. C., Wehby, J., Strong, A., Falk, K.
    B. (2005). Research-based reading and math
    interventions for youth with emotional
    disturbance. In L. M. Bullock, R. A. Gable, K.
    J. Melloy (Eds.), Sixth CCBD mini-library series.
    Arlington, VA Council for Children with
    Behavioral Disorders.
  • Available http//www.cec.sped.org/
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