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Research-Based Math Interventions for Students with Disabilities Dr. Nedra Atwell Western Kentucky University
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.
Overview • Math Standards • Math Interventions for Students with Disabilities • Effective Teaching Practices • Algebra • Math Interventions for Algebra • Accommodations
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
Six NCTM General Principlesfor School Mathematics • Equity • Curriculum • Effective Teaching • Learning • Assessment • Importance of Technology
Math Difficulties • Memory • Language and communication disorders • Processing Difficulties • Poor self-esteem • Attention • Organizational Skills
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
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.
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
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.
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
Mnemonics STAR Strategy • Search the word problem • Translate the word into an equation in picture form • Answer the problem • Review the solution (Maccini & Gagnon’s article, “Preparing Students with Disabilities for Algebra”)
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. Using the STAR Strategy
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)
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.
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.
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
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
Computer Aided Instruction • Programs for remediation and instruction • Demonstration of concepts visually and with online manipulatives • Games • Spreadsheets
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.
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
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 .
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
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
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
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.
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 Authentic Context Cooperative Learning Peer Tutoring Planned Discovery Experiences Self-monitoring Practice Teach Big Ideas Structured Language Experiences 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.
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
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 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
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.
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.
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.
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.
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.
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)
STAR (for older students) • Answer the problem • Review the solution • Reread the problem • Ask question “Does the answer make sense? Why? • Check answer