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Title: ResearchBased Math Interventions for Middle School Students with Disabilities


1
Research-Based Math Interventions for Middle
School Students with Disabilities
  • Shanon D. Hardy, Ph.D.
  • February 25, 2005
  • Access Center

2
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
Objectives
  • Math Interventions for Students with
    Disabilities
  • Algebra
  • Math Interventions for Algebra
  • Effective Teaching Practices
  • Accommodations

4
NCTM (2000) Goals
  • Becoming mathematical problem solvers
  • Learning to communicate mathematically
  • Learning to reason mathematically
  • Becoming mathematical problem solvers through
    representation
  • Making connections

5
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
  • 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.

6
Math Difficulties
  • Memory
  • Language and communication disorders
  • Processing Difficulties
  • Poor self-esteem passive learners
  • Attention
  • Organizational Skills
  • Math anxiety

7
Curriculum Issues
  • Spiraling curriculum
  • Too rapid introduction of new concepts
  • Insufficiently supported explanations and
    activities
  • Insufficient practice (Carnine, Jones, Dixon,
    1994).

8
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

9
Teacher Directed/Explicit Instruction
Student Directed/Implicit Instruction
Explicit Teacher Modeling Building Meaningful Stu
dent Connections C-R-A Sequence of Instruction M
anipulatives
Strategy Learning Scaffolding Instruction Teach
Big Ideas
Structured Language Experiences
Authentic Context Cooperative Learning Peer Tuto
ring Planned Discovery Experiences Self-monitori
ng
Practice
Allsopp, Kyger, 2000
10
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).

11
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.
  • 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

12
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

13
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)

14
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 subgoals 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

15
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

16
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.

17
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.
  • 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
    workmat 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.

18
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).

19
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.

20
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.

21
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.

22
Structured Worksheet
Strategy questions Write
a check after completing each task
Search the word problem Read the problem ca
refully ___________________
Ask yourself questions What fa
cts do I know? ______________________
_____ What do I need to find?
____________________________ Write down
facts I know I have two
rates_________________
Adapted from Maccinni Hughes, 2000
23
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.
  • 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.

24
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

25
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)
  • Answer the problem
  • Review the solution
  • Reread the problem
  • Ask question Does the answer make sense? Why?
  • Check answer

26
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)

27
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

28
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

29
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
    prerequisitie 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.

30
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 .

31
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

32
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.
  • 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

33
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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