1. Mathematics for Students with Learning Disabilities Background Information Number Experiences Quantifying the World Math Anxiety and Myths about math The Concepts (How they are formed) Connected Teaching

2. Background 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.

3. Overall concerns for students 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

4. Number 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.

5. Number Sense As important to math as phonological awareness is to reading (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 numeral Note: Use frames to teach number patterns: 2 4 6 8 12 14 16 20

6. Anxiety 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 11 th grade they share a positive attitude about math in the 12 th grade. However, why do more men take advanced high school courses and score high in math achievement tests by the 12 th grade?

7. The Concepts (How they are 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)

8. Connected Teaching CRA instruction Fluency Direct Instruction Applications Use of strategies Best Practices 1.Advance Organizer 2.Model 3.Guided Practice 4.Independent Practice 5.Feedback 6.Maintenance and Generalization

9. CRA 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.

12. Implement CRA instruction in 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

13. Algorithms 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

14. Direct 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?

15. Word 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.

16. 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

17. Sum 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?