ResearchBased Math Interventions for Students with Disabilities

1
Research-Based Math Interventions for Students
with Disabilities

• Dr. Nedra Atwell
• Western Kentucky University

2
For Some Students
Math is right up there with snakes, public
speaking, and heights. Burns, M. (1998). Math
Facing an American phobia. New York Math
Solutions Publications.
3
Overview

• Math Standards
• Math Interventions for Students with Disabilities
• Effective Teaching Practices
• Algebra
• Math Interventions for Algebra
• Accommodations

4
NCTM Goals (1989, 2000)

• Learning to value mathematics
• Becoming confident in their ability to do
  mathematics
• Becoming mathematical problem solvers
• Learning to communicate mathematically
• Learning to reason mathematically

5
Six NCTM General Principlesfor School Mathematics

• Equity
• Curriculum
• Effective Teaching
• Learning
• Assessment
• Importance of Technology

6
Math Difficulties

• Memory
• Language and communication disorders
• Processing Difficulties
• Poor self-esteem
• Attention
• Organizational Skills

7
Interventions Found Effective for Students with
Disabilities

• Manipulatives
• Concrete-Semi-concrete-Abstract Instruction
• Mnemonics
• Meta-cognitive strategies Self-monitoring,
  Self-Instruction
• Computer-Assisted Instruction
• Explicit Instruction

8
Research on Using Manipulatives

• The use of concrete materials
• Can produce meaningful use of notational systems
• Can increase student concept development
• Is positively related to increases in student
  mathematics achievement
• Is positively related to improved attitudes
  towards mathematics.

9
Issues with Manipulatives

• Teachers may not trust the usefulness or
  efficiency of manipulative objects for
  higher-level algebra.
• Classroom limitations Rigid schedules movement
  of students and teachers organization and supply
  of manipulatives.
• Dominance of textbook lessons

10
Issues with Manipulatives

• Confidence of teachers in their mathematics
  knowledge compared to confidence in the use of
  manipulatives
• One study (Howard Perry) secondary teachers
  used manipulatives once a month primary teachers
  used daily.

11
Concrete-Semi-concrete-Abstract (C-S-A) Phase of
Instruction

• C-S-A is an instructional sequence supporting
  students understanding of mathematical concepts.
• In the concrete phase, students represent the
  problem with concrete objects - manipulatives.
• In the semi-concrete or representational phase,
  students draw or use pictorial representations of
  the quantities
• During the abstract phase of instruction,
  students involve numeric representations, instead
  of pictorial displays. C-S-A is often integrated
  with meta-cognitive instruction, i.e. mnemonics

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Mnemonics STAR Strategy

• Search the word problem
• Translate the word into an equation in picture
  form
• Answer the problem
• Review the solution
• (Maccini Gagnons article, Preparing Students
  with Disabilities for Algebra)

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Using the STAR Strategy

• Search the word problem
• Students read the problem carefully,
• Regulate their thinking through self-questions,
  What facts do I know? What do I need to find?
  and,
• Write down facts.

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Using the STAR Strategy

• Translate the words into an equation in picture
  form
• Students choose a variable for the unknown
• Identify the operation (s)
• Represent the problem using CONCRETE APPLICATION
  of CSA.
• Draw a picture of the representation
  (SEMI-CONCRETE)
• Write an algebraic equation (ABSTRACT
  application)

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Using the STAR Strategy

• Answer the Problem
• Use the appropriate operations (, -, x or / )
• Use rules of solving simple equations
• Use rules to add/subtract positive and negative
  numbers
• Review the solution
• Reread the problem
• Check the reasonableness of the answer
• Check the answer.

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Metacognitive Strategies Self-Instruction

• Strategies include
• Advanced or Graphic Organizers
• Support from structured worksheets and strategy
  instruction
• General guidelines to direct themselves
• Re-read information for clarity
• Diagram representation of the problems before
  solving them
• Write algebraic equations for solving the
  problems.

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Examples ofSelf-Monitoring Strategies

• Cue cards to ask themselves while representing
  problems (card is eventually withdrawn)
• Structured worksheet to help organize their
  problem-solving activities that contained spaces
  for goals, unknowns, knowns, and visual
  representations.
• Questions as prompts for students while solving
  problems

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Structured Worksheet
Strategy questions Write
a check after completing each task Search the
word problem Read the problem carefully
___________________ Ask yourself
questions What facts do I know?
___________________________ What do
I need to find? _______________________
_____ Write down facts
_________________
Adapted from Maccinni Hughes, 2000
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Computer Aided Instruction

• Programs for remediation and instruction
• Demonstration of concepts visually and with
  online manipulatives
• Games
• Spreadsheets

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Use Explicit Instruction

• Begin lesson by
• Tapping prior knowledge
• Modeling how to solve problems while thinking
  aloud
• Prompting students when they needed assistance in
  the activity.

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Empirically Validated Components of Effective
Instruction

• Teacher-based activities
• C-S-A (Manipulatives)
• Direct/Explicit instruction
• Teaching Prerequisite Skills
  
  
  
• Computer Assisted Instruction
• Strategy Instruction
• Structured Worksheets Diagramming
• Graphic organizers

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Reinforce strategy application through corrective
positive feedback

• Examine students math work noting patterns and
  evidence of strategy.
• Meet with students individually or in small
  groups.
• Makes one positive statement about students work
  or thinking.
• Specify error patterns.
• Demonstrate how to complete the problem using one
  of the strategies.
• Provide an opportunity to practice the strategy
  on a similar problem type (guided practice).
• End with a positive comment .

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Recommendations and Conclusions

• Provide instruction in basic arithmetic.
• Use think-aloud techniques
• Allot time to teach specific strategies.
• Provide guided practice before independent
  practice
• Provide a physical and pictorial model
• Relate to real-life events
• Let students practice, practice, practice

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Algebra I and Students with Disabilities

• Algebra I can and should be taught to all
  students, including students with disabilities
• May need more than one class
• May need practical ways of demonstrating skills
  and competencies
• May need supplementary materials

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NCTM (2000) Goals

• Becoming mathematical problem solvers
• Learning to communicate mathematically
• Learning to reason mathematically
• Becoming mathematical problem solvers through
  representation
• Making connections

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Six General Principles

• Equity math is for all students, regardless of
  personal characteristics, background, or physical
  challenges
• Curriculum math should be viewed as an
  integrated whole, as opposed to isolated facts to
  be learned or memorized
• Effective Teaching teachers display 3
  attributes deep understanding of math,
  understanding of individual student development
  and how children learn math ability to select
  strategies and tasks that promote student learning

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Six General Principles

• Problem Solving - Students will gain an
  understanding of math through classes that
  promote problem-solving, thinking, and reasoning
• Continual Assessment of student performance,
  growth and understanding via varied techniques
  (portfolios, math assessments embedded in
  real-world problems
• Importance of Technology use of these tools may
  enhance learning by providing opportunities for
  exploration and concept representation.
  Supplement traditional.

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Math Difficulties

• Memory
• Language and communication disorders
• Processing Difficulties
• Poor self-esteem passive learners
• Attention
• Organizational Skills
• Math anxiety

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Curriculum Issues

• Spiraling curriculum
• Too rapid introduction of new concepts
• Insufficiently supported explanations and
  activities
• Insufficient practice (Carnine, Jones, Dixon,
  1994).

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Interventions Found Effective for Students with
Disabilities

• Reinforcement and corrective feedback for fluency
• Concrete-Representational-Abstract Instruction
• Direct/Explicit Instruction
• Demonstration Plus Permanent Model
• Verbalization while problem solving
• Big Ideas
• Metacognitive strategies Self-monitoring,
  Self-Instruction
• Computer-Assisted Instruction
• Monitoring student progress
• Teaching skills to mastery

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Teacher Directed/Explicit Instruction
Student Directed/Implicit Instruction
Explicit Teacher Modeling Building Meaningful
Student Connections C-R-A Sequence of
Instruction Manipulatives Strategy
Learning Scaffolding Instruction
Authentic Context Cooperative Learning Peer
Tutoring Planned Discovery Experiences Self-monito
ring Practice
Teach Big Ideas Structured Language Experiences
Allsopp, Kyger, 2000
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Algebra

• Language through which most of mathematics is
  communicated (NCTM, 1989).
• Completion of Algebra for high school graduation
• Gateway course for higher math and science
  courses postsecondary education
• Jobs math skills critical for success in 100
  professions, basic algebra skills essential in
  70 of them (Saunders, 1980).

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The Trouble with Algebra

• Students have difficulty with Algebra for one of
  the same reasons they have difficulty with
  arithmetic an inability to translate word
  problems into mathematical symbols (equations)
  that they can solve.
• Students with mild disabilities are unable to
  distinguish between relevant and irrelevant
  information difficulty paraphrasing and imaging
  problem situation
• Algebraic translation involves assigning
  variables, noting constants, and representing
  relationships among variables.

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The Trouble with Algebra

• Abstract using symbols to represent numbers and
  other values. Hard to use manipulatives
  (concrete) to show linear equations
• Erroneous assumption that many students are
  familiar with basic vocabulary and operations
  many still are not fluent in number sense
• Attention to detail is crucial
• All work must be shown

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Algebra Textbooks

• Of the math curricula taught by teachers, 75 to
  95 is derived directly from district supplied
  textbooks (Tyson Woodward, 1989).
• Covers wide range of topics
• Not usually aligned with C-S-A sequence.
• http//www.mathematicallycorrect.com/a1foerst.htm

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Algebra and Students with Disabilities

• 17 year old students with mild disabilities
  performed at levels typically observed in 10 year
  old non-disabled students (Cawley Miller,
  1989).
• Students with mild disabilities did not perform
  as well in basic operations as peers without
  disabilities and the discrepancy between
  achievement scores increased with age (Cawley,
  Parmar, Yan, Miller, 1996)
• Performance tends to plateau at the
  fifth-or-sixth grade level (Cawley Miller, 1989)

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Algebra Terminology

• Problem representation students mentally
  construct the problem-solving situation and
  integrate information from the word problem into
  an algebraic representation using symbols to
  replace unknown quantities (ask for explanations)
• Problem solution value of unknown variables is
  derived by applying appropriate arithmetic or
  algebraic operations divide the solution into
  sequential steps within the problem to solve
  the sub goals and goals of problem. Must divide
  the solution into sequential steps.
• Self-monitoring students monitor their own
  thinking and strategies to represent and solve
  word problems failure to self-monitor may result
  in incorrect solutions

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Empirically Validated Components of Effective
Instruction for Algebra

• Teacher-based activities
• C-R-A (Manipulatives)
• Direct/Explicit instruction - modeling
• Instructional Variables LIP, teach
  prerequisites
  
  
  
• Computer Assisted Instruction
• Strategy Instruction
• Metacognitive Strategy
• Structured Worksheets Diagramming
• Mnemonics (PEMDAS)
• Graphic organizers

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Concrete-Representational-Abstract (C-R-A) Phase
of Instruction

• Instructional method incorporates hands-on
  materials and pictorial representations. For
  algebra, must also include aids to represent
  arithmetic processes, as well as physical and
  pictorial materials to represent unknowns.
• Students first represent the problem with objects
  - manipulatives.
• Then advance to semi-concrete or representational
  phase and draw or use pictorial representations
  of the quantities
• Abstract phase of instruction involves numeric
  representations, instead of pictorial displays.
  C-R-A is often integrated with metacognitive
  instruction, i.e. STAR strategy.

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Example (Concrete Stage)

• In state college, Pennsylvania, the temperature
  on a certain days was -2F. The temperature rose
  by 9ºF by the afternoon. What was the
  temperature in the afternoon?
• Students first search the word problem (read the
  problem carefully, regulate their thinking
  through self-questions, and write down facts.

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Example (Concrete Stage)

• Second step Translate the words into an
  equation in picture form prompts students to
  identify the operation(s) and represent the
  problem using concrete manipulatives. Students
  first put two tiles in the negative area of the
  work mat to represent -2 and 9 tiles in the
  positive area to represent 9 and then cancel
  opposites. 2 and -2
• Third step, Answer the Problem involves counting
  the remaining tiles 7 and the fourth step
  Review the solution involves rereading the
  problem and checking the reasonableness of the
  answer. Need 80 mastery on two probes before
  going to semi-concrete.

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Representational to Abstract

• Structured worksheet provided to cue students to
  use the first two steps of STAR. However,
  instead of manipulatives, students represent word
  problems using drawings of the algebra tiles.
• Third phase of instruction students represent and
  solve math problems using numerical symbols,
  answer the problem using a rule, and review the
  solution. The problem described would be -2F
  (9F) x, apply the rule for adding integers,
  solve the problem (x 7).

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Conceptual Problems with Manipulatives in Algebra

• Some researchers found that in Concrete steps,
  the materials (manipulatives) did not adequately
  represent algebraic variables and coefficients.
  For example, equation X35 and 5X 15 are
  easily represented but representations did not
  differentiate coefficients from exponents.
• May lead to confusion. By asking students to
  represent X with a cube, the coefficient is
  misrepresented. Instead of thinking five cubes
  is 5X, mathematically, five cubes should be X5
  when working with exponents.

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Other Issues with Manipulatives in Algebra

• Teachers may not trust the usefulness or
  efficiency of manipulative objects for
  higher-level algebra.
• Rigid timetables, movement of students and
  teachers make it difficult to organize the supply
  of manipulatives in classes.
• Dominance of textbook lessons in secondary math
  classrooms and ease with which the use of such
  texts can be arranged, could also effect the
  regular use of manipulatives.
• Teachers feel confident in their use but they
  also know that they dont know everything they
  need to know about manipulatives.
• One study (Howard Perry) secondary teachers
  used manipulatives once a month primary teachers
  used daily.

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Metacognitive Strategies

• Many studies found that prior to instruction many
  students bypassed problem representation and
  started with trying to solve the problems.
• Advance or Graphic Organizers
• Following intervention of strategy instruction
  and structured worksheets, students used the
  general guidelines to direct themselves to
• 1. re-read information for clarity
• 2. diagram representation of the problems before
  solving them
• 3. write algebraic equations for solving the
  problems.

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Self-Monitoring Strategy

• Students were provided with a cue card listing
  four questions to ask themselves while
  representing problems card was eventually
  withdrawn
• Results students representation of the
  algebraic word problems were similar to those of
  experts (Hutchinson, 1993).
• Students also given a structured worksheet to
  help organize their problem-solving activities
  that contained spaces for goals, unknowns,
  knowns, visual representations.

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Self-Monitoring Strategy

• Questions served as prompts for students use
  while solving problems
• Have I read and understood each sentence. Any
  words whose meaning I have to ask
• Have I got the whole picture, a representation of
  the problem
• Have I written down my representation on the work
  sheet goal, unknowns, known, type of problem,
  equation
• What should I look for in a new problem to see if
  it is the same type of problem.

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Example Strategy Instruction - DRAW

• Discover the sign
• Read the problem
• Answer or DRAW a conceptual representation of the
  problem using lines and tallies, and check
• Write the answer and check.
• First three steps address problem representation,
  last problem solution

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STAR (for older students)

• Search the word problem
• Read the problem carefully
• Ask yourself questions What facts do I know?
  What do I need to find?
• Translate the words into an equation in picture
  form
• Choose a variable
• Identify the operation(s)
• Represent the problem with the Algebra Lab Gear
  (concrete application)
• Draw a picture of the representation
  (semi-concrete application)
• Write an algebraic equation (abstract application)

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STAR (for older students)

• Answer the problem
• Review the solution
• Reread the problem
• Ask question Does the answer make sense? Why?
• Check answer

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STAR adapted from
Strategic Math Series by Mercer and Miller, 1991.

• Six elements used in each lesson
• Provide an advance organizer identify the new
  skill and provide a rationale for learning
• Describe and model
• Conduct guided practice
• Conduct independent practice
• Give posttest
• Provide feedback (positive and corrective)

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Findings on Algebra Interventions

• Results students with mild disabilities can
  successfully learn to represent and solve
  algebraic word problems when appropriate
  instruction is provided. However, given the
  small number of studies currently available, it
  is unlikely that a classroom educator can
  implement any of the interventions described here
  without substantial modifications to meet
  particular classroom needs.
• One finding from all research is that a
  comprehensive instructional program is necessary
  to ensure that instruction does not lead to
  splintered understanding that slows acquisition
  of sophistical problemsolving skills. Includes
  meaningful activities

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How Teachers Can Make a Change Principles of
Effective Instruction

• (BEFORE LESSON)
• Review
• Explanation of objectives or informed teaching
  precise statements of the goal, rationale for
  learning the strategy, and information on when
  the strategy should be implemented (LIP).
• DURING LESSON)
• Modeling the task
• Prompting - engage students in dialogue that
  promotes the development of student-generated
  problem-solving strategies and reflective
  thinking (students self-evaluate while they are
  solving problems).
• Guided and independent practice wide range of
  examples
• Corrective and positive feedbacks

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Teacher Variables Arithmetic to Algebra Gap

• Teachers need to attend to the following
  instructional techniques to help students make
  connections between arithmetic and algebra and
  understand algebraic notation. Three principles
• Teach through stories that connect math
  instruction to students lives. Example- you
  live in Tampa, Florida and want to go to a UF
  football game which is 120 miles away. You know
  you can travel 50 miles an hour from Tampa to
  Gainesville. The game starts in the afternoon,
  so you want to arrive in Gainesville at 100.
  How many hours will it take you to travel to
  Gainesville? Use the formula - Distance (miles)
  Speed (m/h) X Time (hours).
• Prepare students for more difficult concepts by
  making sure students have the necessary
  prerequisite knowledge for learning a new math
  strategy. Students should know how to do
  (115)/2 before do 2X-5 11.
• Explicitly instruct students in specific skills
  using think aloud techniques when modeling.

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Corrective and Positive Feedback

• Reinforce strategy application through feedback
• first examine students math work. While noting
  error patterns, the teacher looks for evidence
  related to the presence or absence of strategy
  use.
• Once this is completed, the teach meets with
  students individually or in small groups.
• Makes one positive statement about students work
  or thinking.
• Next, specify error patterns. Then demonstrate
  how to complete the problem using one of the
  strategies.
• Student then given an opportunity to practice the
  strategy on a similar problem type (guided
  practice).
• Ends with teacher responding with another
  positive comment .

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Accommodations

• Use vertical lines or graph paper in math to help
  the student keep math problems in correct order
• Highlight symbols, different colors
• Use different colors for rules, relationships

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Recommendations and Conclusions

• Continue to instruct secondary math students with
  mild disabilities in basic arithmetic. Poor
  arithmetic background will make some algebraic
  questions cumbersome and difficult.
• Use think-aloud techniques when modeling steps to
  solve equations. Demonstrate the steps to the
  strategy while verbalizing the related thinking.
  
• Must allot time to teach specific strategies.
  Students will need time to learn and practice the
  strategy on a regular basis.

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Recommendations and Conclusions

• Provide guided practice before independent
  practice so that students can first understand
  what to do for each step and then understand why.
  When constructing their interpretation of steps
  under teacher guidance, students need to
  understand why they are solving equations. When
  students build their own proper understanding of
  how to solve equations, it is less likely that
  they will forget the steps.
• Provide a physical and pictorial model, such as
  diagrams or hands-on materials, to aid the
  process for solving equations.
• Relate to real life events
• Let students practice, practice, practice

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How Students and Teachers Interact During Learning

• When students cannot construct knowledge for
  themselves, they need some instruction.
• People are sometimes better at remembering
  information that they create for themselves than
  information they receive passively, but in other
  cases they remember as well or better information
  that is provided than information they create.
• Real competence only comes with extensive
  practice. Anderson, Reder, Simon (1995).

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• Thank you for your time and attention.