Mathematics for Students Mathematics for Students with Learning Disabilitieswith Learning Disabilities

• Background Information• Number Experiences• Quantifying the World• Math Anxiety and Myths about math• The Concepts (How they are formed)• Connected Teaching

Background InformationBackground Information

• For every ___ years of schools, students with disabilities gain 1 year of math achievement.

• What grade level of math do most high school students with learning disabilities top out at?

• Most students with disabilities are not knowledgeable of needed consumer math skills (Algozzine et al., 1987).

• Students with learning disabilities learn arithmetic through hills and valleys (Cawley, Parmar, & Miller, 1997).

• Many elementary school preservice teachers show a distaste for mathematics.

Overall concerns for Overall concerns for students with learning disabilitiesstudents with learning disabilities

• Abstractness of numbers• Low number sense• Poorly formed ideas and algorithms

– (requiring systematic instruction over constructivism)

• Overgeneralization or incorrect use of algorithms

• Poor recall of facts and procedures• Generalization and maintenance

– (increases with difficulty of problems)– different presentation confuses

Number ExperiencesNumber Experiences

• A concept is an idea or mental image. Children develop concepts from physical objects through mental abstractions.

• How can we help young children experience the importance of numbers?

• Arithmetic is used to refer to manipulations with numbers and computations while mathematics is concerned with thinking about quantities and relationships among them (Polloway & Patton, 1993). To learn mathematics children must be taught the relationships between quantities and shown relevance behind arithmetic.

Number SenseNumber SenseAs important to math as phonological awareness is to reading As important to math as phonological awareness is to reading

(Gersten and Chard, 1999).(Gersten and Chard, 1999).

• Numerals to objects• Which is larger? 8 or 18• Which is closer to ___?• Counting• Counting on• Backwards on• Place Value• Writing numerals to match oral word and writing

number words to match numeralNote: Use frames to teach number patterns: 2 4 6 8 12 14 16 20

Anxiety and Myths about mathAnxiety and Myths about math

• Most elementary school children have positive experiences with mathematics and arithmetic.

• Do adults have positive experiences with math?• Females score on average with males in math

until secondary school (Xin, 2000). When males and females take the same math course in 11th grade they share a positive attitude about math in the 12th grade. However, why do more men take advanced high school courses and score high in math achievement tests by the 12th grade?

The Concepts The Concepts (How they are (How they are formed)formed)

• Sensorimotor- objects exist out of sight (0-2)• Preoperational- ability to think in symbols (2-7)• Concrete- manipulatives offer medium for

instruction(7-11) conservation of objects • Formal operations- abstract problem solving

(11+)

• Much research challenges Piaget’s theory• The order of development and the age of

onset may be incorrect (Demby; Miller)

Connected TeachingConnected Teaching• CRA instruction• Fluency • Direct

Instruction• Applications• Use of strategies

Best Practices1. Advance Organizer2. Model3. Guided Practice4. Independent

Practice5. Feedback6. Maintenance and

Generalization

CRA instructionCRA instruction• 62% of primary teachers use

manipulatives while only 8% of secondary teachers use hands-on materials (Howard, Perry, & Lindsay, 1996). Why?

• Concrete - from fingers to objects• Representational - from objects to

pictures• Abstract - from pictures to numerals

• What programs cover some of these components? Touch math, etc.

Implement CRA instruction in Implement CRA instruction in your classroom. Here’s how:your classroom. Here’s how:

• Choose the math topic to be taught• Review abstract steps to solve the problem• Adjust the steps to eliminate notation or

calculation tricks • Match the abstract steps with an

appropriate concrete manipulative• Arrange concrete and representational

lessons• Teach each concrete, representational, and

abstract lesson to student mastery (accuracy without hesitation)

• Help students generalize learning through word problems and problem solving events

Algorithms and FluencyAlgorithms and Fluency

• Students apply algorithms properly after they learn the concept.

• Why do shortcuts and algorithms not work as well when learning is new or the concept is difficult?

• Fluency measures can only be used with instruction after students show mastery.

• Fluency programs– Great Leaps Math– Precision Teaching– Teacher Made probes

Direct InstructionDirect Instruction

• Explain how you can apply direct instruction to teaching the 6 times multiplication table.

• What other mathematics areas would be appropriate for direct instruction?

Word ProblemsWord Problems

Students with disabilities do not paraphrase or visualize word problems. There is a connection between reading comprehension difficulties and poor performance solving for word problems.

(Montague, Bos, & Doucette, 1991)

How can we help? Examples

7 cars 6 groups of- 3 cars x 3 apples___ cars ___ apples

• after seeing this pattern, leave some blanks for students to fill in. Then list needed information to solve, followed by extraneous info. Once students show mastery, have them write their own word problems.

Word Problems (cont)Word Problems (cont)• Teach word problems as problem solving

situations that need to be interpreted• Teach strategies for recognizing types of

problems• i.e., focus on reading comprehension

strategies– KWL– RAPQ– Word walls for vocabulary

Sum it UpSum it Up

• What activities can we do in the classroom to help children prepare to think in symbols and numbers?

• What is the difference between elementary or middle school boys and girls?

• What is CRA instruction?• When should a teacher use fluency or

algorithms to solve math problems?