Title: ResearchBased Math Interventions for Middle School Students with Disabilities
1Research-Based Math Interventions for Middle
School Students with Disabilities
- Shanon D. Hardy, Ph.D.
- February 25, 2005
- Access Center
2Math is right up there with snakes, public
speaking, and heights. Burns, M. (1998). Math
Facing an American phobia. New York Math
Solutions Publications.
3Objectives
- Math Interventions for Students with
Disabilities
- Algebra
- Math Interventions for Algebra
- Effective Teaching Practices
- Accommodations
4NCTM (2000) Goals
- Becoming mathematical problem solvers
- Learning to communicate mathematically
- Learning to reason mathematically
- Becoming mathematical problem solvers through
representation
- Making connections
5Six 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.
6Math Difficulties
- Memory
- Language and communication disorders
- Processing Difficulties
- Poor self-esteem passive learners
- Attention
- Organizational Skills
- Math anxiety
7Curriculum Issues
- Spiraling curriculum
- Too rapid introduction of new concepts
- Insufficiently supported explanations and
activities
- Insufficient practice (Carnine, Jones, Dixon,
1994).
8Interventions 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
9Teacher 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
10Algebra
- 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).
11The 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
12Algebra 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
13Algebra 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)
14Algebra 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
15Empirically 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
16Concrete-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.
17Example (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.
18Representational 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).
19Conceptual 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.
20Other 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.
21Metacognitive 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.
22Structured 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
23Self-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.
24Example 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
25STAR (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
26STAR 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)
27Findings 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
28How 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
29Teacher 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.
30Corrective 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 .
31Accommodations
- 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
32Recommendations 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
33How 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).