Mathematics problems can be serious and widespread for students with disabilities.  Evidence-based or research-based instructions provide teachers with tried-and-tested strategies to improve student learning.  Below you will find a list of some of those strategies.      

Math Interventions Found Effective for Students with Disabilities

1.)  Reinforcement and corrective feedback for fluency

2.)  Concrete-Representational-Abstract Instruction (Teacher Directed/Explicit Instruction)

3.)  Direct/Explicit Instruction/Modeling (Teacher Directed/Explicit Instruction)

4.)  Demonstration Plus Permanent Model

5.)  Verbalization while problem solving

6.)  Big Ideas (Strategy Learning)

7.)  Strategy Instruction (Student Directed/Implicit Instruction)

                        a.)  Metacognitive strategies: Self-monitoring, Self-Instruction

b.)  Structured Worksheets; Diagramming

c.)  Mnemonics (PEMDAS)

d.)  Graphic organizers

8.)  Computer-Assisted Instruction

9.)  Monitoring student progress

10.)  Teaching skills to mastery

Source 1: Shanon D. Hardy, Ph.D Powerpoint Slides, February 25, 2005 Access Center, Accessed from: http://www.k8accesscenter.org/index.php/category/math/

Source 2: Seifert, Kathy. (2010).  University of Minnesota Powerpoint Lecture, EPSY 5615 accessed 3/10/2010

Four Methods of Instruction Show the Most Promise:

1.)  Systematic and explicit instruction, a detailed instructional approach in which teachers guide students through a defined instructional sequence. Within systematic and explicit instruction students learn to regularly apply strategies that effective learners use as a fundamental part of mastering concepts.

2.)  Self-instruction, through which students learn to manage their own learning with specific prompting or solution-oriented questions.

3.)  Peer tutoring, an approach that involves pairing students together to learn or practice an academic task.

4.)  Visual representation, which uses manipulatives, pictures, number lines, and graphs of functions and relationships to teach mathematical concepts.

Source: Steedly, Kathlyn,. Dragoo, Kyrie,. Arafeh, Sousan & Luke, Stephen D. (2008).  Effective Mathematics Instruction,  Evidence for Education, 3, (1).  accessed 3/13/2010 from:

http://www.nichcy.org/Research/EvidenceForEducation/Documents/NICHCY_EE_Math.pdf

Evidence Based Instruction for Math (list):

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

a.)  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.

b.)  Students first represent the problem with objects - manipulatives.

c.)  Then advance to semi-concrete or representational phase and draw or use pictorial representations of the quantities

d.)  Abstract phase of instruction involves numeric representations, instead of pictorial displays.  C-R-A is often integrated with metacognitive instruction, i.e. STAR strategy.

Source: Shanon D. Hardy, Ph.D Powerpoint Slides, February 25, 2005 Access Center, Accessed from: http://www.k8accesscenter.org/index.php/category/math/

Source: Maccini, P., & Hughes, C. A. (2000).  Effects of a problem-solving strategy on the introductory algebra performance of secondary students with learning disabilities.  Learning Disabilities Research & Practice, 15, 10-21.

            ***National Library of Virtual Manipulative http://nlvm.usu.edu/en/nav/topic_t_3.html   ******

2.)  Cognitive Strategy Instruction:

a.)  Cognitive Strategy Instruction:  Students are taught and memorize explicit steps for approaching and solving problems, and they apply these steps by verbalizing them, first overtly and gradually fading their overtly use over time.  Students are taught cognitive steps (read the problem, paraphrase, visualize with a picture or diagram, hypothesize a plan to solve the problem, estimate the answer, compute, and check. 

1.)  Following intervention of strategy instruction and structured worksheets, students used the general guidelines to direct themselves to:

a.)  re-read information for clarity;

b.) diagram representation of the problems before solving them;

c.) write algebraic equations for solving the problems.

b.)  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_________________

Source: Shanon D. Hardy, Ph.D Powerpoint Slides, February 25, 2005 Access Center, Accessed from: http://www.k8accesscenter.org/index.php/category/math/

4.)  Task Analysis/Problem Solving Strategies:

*Source: Montague (1992)

1.      Read for understanding

2.      Paraphrase in your own words

3.      Visualize a picture or diagram

4.      Hypothesize a plan to solve the problem

5.      Estimate or predict the answer

6.      Compute the answer

7.      Check to be sure everything is correct

*Source: Miller, Strawser, & Mercer,1996

1.      Read the problem

2.      What is the question the problem asks?

3.      To answer the question, do I have to:

4.      ___ Add             ___ Subtract       ___ Multiply   ___ Divide

5.      What information is not needed?

6.      Write out the problem using numbers

7.      Solve the problem

8.      Check the answer

*Source: Fleischner, Nuzum, & Marzola (1987)

1.  Read            

a. What is the question?

2.  Reread  

a. What is the necessary information?

3.   Think

a. Putting together means addition

b. Taking apart means subtraction

4. Solve; write the equation

5. Check; recalculate, label, & compare

Generic

1. Read problem

                        a. circle unknown words (ask)

                        b. underline cue words

2. Choose operation

3. Write down necessary numbers

4. Cross out unnecessary information

5. Write equation

6. Solve problem

7. Check answer; “Ask yourself, does this make sense?”

Source: Seifert, Kathy. (2010).  University of Minnesota Powerpoint Lecture, EPSY 5615 accessed 3/10/2010

3.)  Self-Monitoring Strategy

a.)  Students were provided with a cue card listing four questions to ask themselves while representing problems; card was eventually withdrawn

b.)  Results = students’ representation of the algebraic word problems were similar to those of experts (Hutchinson, 1993).

c.)  Students also given a structured worksheet to help organize their problem-solving activities that contained spaces for goals, unknowns, knowns, visual representations.

d.)  Questions served as prompts for students use while solving problems

1.)  Have I read and understood each sentence.  Any words whose meaning I have to ask

2.)  Have I got the whole picture, a representation of the problem?

3.)  Have I written down my representation on the work sheet – goal, unknowns, known, type of problem, equation?

4.)  What should I look for in a new problem to see if it is the same type of problem?

4.)  Strategy Instruction Mnemonics (PEMDAS)

a.)  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

b.)  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

Source: Shanon D. Hardy, Ph.D Powerpoint Slides, February 25, 2005 Access Center, Accessed from: http://www.k8accesscenter.org/index.php/category/math/

Other Useful Strategies:                                                                                                          

1.)  Teaching Multiplication:

a.)  Notice how the second digit of the answers go down from (9) to 1(8) to 2(7) to 3(6) to 4(5) to 5(4) to 6(3) to 7(2) to 8(1). 

            9 X 1 = 9

            9 X 2 = 18

            9 X 3 = 27

            9 X 4 = 36

9 X 5 = 45

            9 X 6 = 54

            9 X 7 = 63

            9 X 8 = 72

            9 X 9 = 81

2.)  Teaching Multiplication: Bent Finger Strategy:

Using this strategy, students hold their two hands, palms down, in front of them.  They then count from left to right on their fingers by the number of the fact, and bend down only the relevant finger.  That is, for 9 X 5, students count to 5 starting with their left little finger to their left thumb, and only bend down that thumb.   Then, the fingers to the left of the bent finger represent the 10s and the fingers to the right of the bent finger represent the 1s of the product.  In the case of 9 X 5, there are 4 fingers to the left and 5 fingers to the right of the bent finger, so the answer is 45.

Source: Mastropieri, M.A. & Scruggs, T.E. (2007). The inclusive classroom: Strategies for effective instruction (4^th ed.).Upper Saddle River, NJ:  Prentice-Hall. pg 340, 2007).

3.)  Touch Math for Subtraction and Addition Computation

Strategy represents quantity by dots on each of the numbers 1-9.  Numbers 1-5 have solid dots.  After 5, Touch Math uses circled dots, or “double touch points,” each of which represents the quantity of 2.  Students learn to touch each of the touch points once, and to touch each double touch point twice, with their pencil when counting.     

Source: Mastropieri, M.A. & Scruggs, T.E. (2007). The inclusive classroom: Strategies for effective instruction (4^th ed.).Upper Saddle River, NJ:  Prentice-Hall. pg. 345). 

4.)  Constructed number line

Used to help students simplify fractions to simplest form.  The teacher used a sequence chart GO (i.e., a constructed number line) to help students visualize how to represent equivalent fractions and how to simplify fractions to simplest terms. The teacher then models, using examples, how to use the constructed number line to reduce fractions to simplest form: 8/16 = 4/8 = 2/4 = 1/2 (see graphic below).  The teacher uses different colored pens to denote each fractional part.  The teacher provides guided practice and monitors students as they construct a number line with their rulers.

To check for accuracy, the teacher has students compare their number lines with the markings on their individual rulers.

*Source: Constructed Number Line, Access Center: Improving Outcomes for All AStudents, Retrieved 5/2/2010 from http://www.k8accesscenter.org

Accommodations for math

a.)  Use vertical lines or graph paper in math to help the student keep math problems in correct order

b.)  Highlight symbols, different colors

c.)  Use different colors for rules, relationships

References:                                                                                                                                                            

Shanon D. Hardy, Ph.D Powerpoint Slides, February 25, 2005 Access Center, Accessed from: http://www.k8accesscenter.org/index.php/category/math/

Mastropieri, M.A. & Scruggs, T.E. (2007). The inclusive classroom: Strategies for effective instruction (4^th ed.).Upper Saddle River, NJ:  Prentice-Hall. pg. 345). 

Source: Seifert, Kathy. (2010).  University of Minnesota Powerpoint Lecture, EPSY 5615 accessed 3/10/2010

Steedly, Kathlyn,. Dragoo, Kyrie,. Arafeh, Sousan & Luke, Stephen D. (2008).  Effective Mathematics Instruction,  Evidence for Education, 3, (1).  accessed 3/13/2010 from: http://www.nichcy.org/Research/EvidenceForEducation/Documents/NICHCY_EE_Math.pdf

Mercer, C. D., & Miller, S. P. (1992).  teaching students with learning problems in math to acquire, understand and apply basic math facts.  Remedial and Special education, 13, 19-35, 61.