Research-Based Math Interventions for Middle School

Students with Disabilities

Shanon D. Hardy, Ph.D.

February 25, 2005

Access Center

Math is right up there with snakes, public speaking, and heights.

Burns, M. (1998). Math: Facing an American phobia. New York: Math Solutions Publications.

Objectives

Math Interventions for Students with Disabilities

Algebra Math Interventions for Algebra Effective Teaching Practices Accommodations

NCTM (2000) Goals

Becoming mathematical problem solvers Learning to communicate mathematically Learning to reason mathematically Becoming mathematical problem solvers

through representation Making connections

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.

Math Difficulties

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

Curriculum Issues

Spiraling curriculum Too rapid introduction of new concepts Insufficiently supported explanations and

activities Insufficient practice (Carnine, Jones, &

Dixon, 1994).

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

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

Teach Big Ideas

Structured Language Experiences

Authentic Context

Cooperative Learning

Peer Tutoring

Planned Discovery Experiences

Self-monitoring

Practice

Allsopp, & Kyger, 2000

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).

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

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

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)

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

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

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.

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.

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).

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 X+3=5 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.

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 don’t know everything they need to know about manipulatives.

One study (Howard & Perry) secondary teachers used manipulatives once a month; primary teachers used daily.

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.

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 I know I have two rates_________________

Adapted from Maccinni & Hughes, 2000

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.

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

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

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)

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 problem=solving skills. Includes meaningful activities

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

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 1:00. 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 (11+5)/2 before do 2X-5 = 11.

• Explicitly instruct students in specific skills using think aloud techniques when modeling.

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 .

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

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

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).