Effective Math Instruction for Students with High Incidence Disabilities

1
Effective Math Instruction for Students with
High Incidence Disabilities

• Joseph Calvin Gagnon, Ph.D.
• George Mason University

2
Advanced Organizer Math Session

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

3
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

4
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.

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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)

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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)

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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)

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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,
  2000)

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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)

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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,
  including
• Basic skills (Algozzine, OShea, Crews,
  Stoddard, 1987)
• Higher-level skills/concepts and problem solving
  (Huntington, 1994 Hutchinson, 1993 Maccini
  Hughes, 2000 Maccini Ruhl, 2000)

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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)

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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)

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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

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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

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Math Instructional Recommendations cont.

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

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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)

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Direct Instruction (di)

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

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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)

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Direct Instruction

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

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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/
  directinstructionmath.asp

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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)

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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)

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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)

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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)

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Real World Problem Solving and Technology

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

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Real World Problem Solving and Technology

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

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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

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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
  assistance

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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

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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
  video)

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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

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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
  categories

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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
  anxiety
• The "pedagogical" function Relates to
  similarities between calculators, textbooks, and
  manipulatives in that each enhances student
  understanding and competence in mathematics

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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
  situations
• Concentrate on the problem-solving process rather
  than calculations associated with problems
• Gain access to mathematics beyond their level of
  computational skill

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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
  patterns

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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

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Real World Problem Solving and Technology

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

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Real World Problem Solving and Technology

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

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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

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Resource

• 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.

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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)

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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,
  2000)
• Peer-assisted learning promotes computational
  skills (Calhoon Fuchs, 2003)
• Classwide peer tutoring is effective in
  strengthening basic math skills (Allsopp, 1997)

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Grouping for Instruction

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

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Grouping for Instruction

• Classwide peer tutoring (CWPT)
• Typically, the entire class participates in CWPT
  simultaneously
• 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

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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

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Grouping for Instruction

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

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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
  activity

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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

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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

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Grouping for Instruction

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

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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,
  367-379.

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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
  representations)
• Iconic mode (e.g., the seeing phase visualizing
  representations of the problem - semiconcrete
  representations)
• Symbolic mode (e.g., using abstract symbols to
  represent the problem - abstract representations)

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C-S-A

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

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C-S-A

• 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.

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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

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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
  solution)

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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)

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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
  solution
• 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

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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

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Instructional Adaptations

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

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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)

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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)

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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
  g_resources/documents/MathGraphicOrg.pdf

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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

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Instructional Adaptations

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

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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

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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)

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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.

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Resources

• 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/