Effective Math Instruction for Students with

High Incidence Disabilities

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

Advanced Organizer Math Session

- Educational Reform
- The National Council of Teachers of Mathematics

Standards - Characteristics of students with learning and

emotional/behavioral disabilities - Direct instruction (di)

Advanced Organizer Math Session

- Real world application and technology
- Student grouping
- Graduated instructional sequence
- Graphic Organizers
- Strategy instruction
- 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,

2000)

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,

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)

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

- Incorporate components of direct instruction
- Teach strategies
- 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

reviews)

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

review.

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

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

interventions - 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

discrimination - 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

assistance

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

video)

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

categories

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

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

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

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

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

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

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.

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,

2000) - 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

errors - 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

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

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

activity

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

- Then, the teacher models the peer tutoring

process and allows students to participate in

role playing - 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,

367-379.

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)

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

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.

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

solution)

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

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

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

g_resources/documents/MathGraphicOrg.pdf

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.

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/