1. Teaching Math to Students with Disabilities Present Perspectives

2. Math is hard (Barbie, 1994) US 15 year olds ranked 24 th (among 29 developed nations) in the 2003 International Student Assessment in math literacy and problem solving 7% of US students scored in the advanced level in the 2004 Trends in Math and Science Study Almost half of America's 17 year olds did not pass The National Assessment of Educational Progress math test 2006 Hart/Winston Poll found that 76% of Americans believe that if the next generation does not work to improve its skills it risks becoming the 1 st generation who are worse off economically than their parents

3. How did we get here? Math skills have received less attention than reading skills because of the perception that they are not as important in real life Ongoing debate over how explicitly children must be taught skills based on formulas or algorithms vs a more inquiry-based approach Teacher preparation – general concern about elementary preservice training programs Little reference to students with disabilities in NCTMs standards Debate over math difficulties vs math disabilities

4. Developmental dyscalculia developmental difficulties or disabilities involving quantitative concepts, information, or processes Dyscalculia is where dyslexia was 20 years ago it needs to be brought into the public domain Jess Blackburn, Dyscalculia & Dyslexia Interest Group

5. What defines mathematical learning disabilities? Genetic basis Presently only determined by behavior (which behaviors: knowledge of facts? procedures? conceptual understanding? Speed and accuracy?) Depending on the criteria incidence can include from 4 to 48% of students Mathematical difficulties vs. mathematical disabilities: different degrees of the same problem or different problems?

6. National Mathematics Advisory Panel Established in 2006 To examine: Critical skills & skill progressions Role & appropriate design of standards & assessment Process by which students of various abilities and backgrounds learn mathematics Effective instructional practices, programs & materials Training (pre and post service) Research in support of mathematics education

7. NCMT final Report (2008) Curricular content Focused: must include the most important topics underlying success in school algebra (whole numbers, fractions, and particular aspects of geometry and measurement) Coherent: effective, logical progressions Proficiency: students should understand key concepts, achieve automaticity as appropriate; develop flexible, accurate, and automatic execution of the standard algorithms, and use these competencies to solve problems

8. What is the structure of mathematical learning disabilities? Issues with retrieval of arithmetic facts Difficulties understanding mathematical concepts and executing relevant procedures Difficulties choosing among alternate strategies Trouble understanding the language of story problems, teacher instructions and textbooks

10. Math instruction issues that impact students who have math learning problems Spiraling curriculum Teaching understanding/algorithm driven instruction Teaching to mastery Reforms that are cyclical in nature

11. Promising approaches to teaching mathematics to students with disabilities Math Expressions Saxon Strategic math Series Touch Math Number Worlds Curriculum Touch Math Number Worlds Curriculum Montessori methods and materials What works clearing house

12. Resources for teaching math Illuminations MathVids

13. Teaching Math to Students with Disabilities Strategies

14. Application of effective teaching practices for students who have learning problems Concrete-to-representational-to-abstract instruction (C-R-A Instruction) Explicitly model mathematics concepts/skills and problem solving strategies Creating authentic mathematics learning contexts

15. Concrete-to-Representational-to-Abstract Instruction (C-R-A Instruction) Concrete: each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, pattern blocks) Concrete Representational: the math concept is next modeled at the representational (semi-concrete) level (e.g. tallies, dots, circles) Abstract: The math concept is finally modeled at the abstract level (numbers & mathematical symbols) should be used in conjunction with the concrete materials and representational drawings.

16. Concrete-to-Representational-to-Abstract Instruction (C-R-A Instruction) Concrete: each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, pattern blocks) Representational: the math concept is next modeled at the representational (semi-concrete) level (e.g. tallies, dots, circles) Abstract: The math concept is finally modeled at the abstract level (numbers & mathematical symbols) should be used in conjunction with the concrete materials and representational drawings.

17. Concrete-to-Representational-to-Abstract Instruction (C-R-A Instruction) Concrete: each math concept/skill is first modeled with concrete materials (e.g. chips, unifix cubes, base ten blocks, pattern blocks) Representational: the math concept is next modeled at the representational (semi-concrete) level (e.g. tallies, dots, circles) Abstract: The math concept is finally modeled at the abstract level (numbers & mathematical symbols) should be used in conjunction with the concrete materials and representational drawings.

18. Important Considerations Use appropriate concrete objects After students demonstrate mastery at the concrete level, then teach appropriate drawing techniques when students problem solve by drawing simple representations After students demonstrate mastery at the representational level use appropriate strategies for assisting students to move to the abstract level.

19. How to implement C-R-A instruction When initially teaching a math concept/skill, describe and model it using concrete objects Provide students multiple opportunities using concrete objects Provide multiple practice opportunities where students draw their solutions or use pictures to problem solve When students demonstrate mastery by drawing solutions, describe and model how to perform the skills using only numbers and math symbols Provide multiple opportunities for students to practice performing the skill using only numbers and symbols After students master performing the skill at the abstract level, ensure students maintain their skill level by providing periodic practice Example

20. Explicit Modeling Provides a clear and accessible format for initially acquiring an understanding of the mathematics concept/skill Provides a process for becoming independent learners and problem solvers

21. What is explicit modeling? Student Teacher Mathematical concept

22. Instructional techniques…. Identify what students will learn (visually and auditorily) Link what they already know (e.g. prerequisite concepts/skills, prior real life experiences, areas of interest) Discuss the relevance/meaning of the skill/concept

23. Instructional techniques….(cont) Break math concept/skill into 3 – 4 learnable features or parts Describe each using visual examples Provide both examples and non-examples of the mathematics concept/skill Explicitly cue students to essential attributes of the mathematic concept/skill you model (e.g. color coding) Example

24. Implementing Explicit Modeling Select appropriate level to model the concept or skill (concrete, representational, abstract) Break concept/skills into logical/learnable parts Provide a meaningful context for the concept/skill (e.g. word problem) Provide visual, auditory, kinesthetic and tactile means for illustrating important aspects of the concept/skill Think aloud as you illustrate each feature or step of the concept/skill Link each step of the process (e.g. restate what you did in the previous step, what you are going to do in the next step) Periodically check for understanding with questions Maintain a lively pace while being conscious of student information processing difficulties Model a concept/skill at least three times

25. Authentic Mathematics Learning Contexts Explicitly connects the target math concept/skill to a relevant and meaningful context, therefore promoting a deeper level of understanding for students Requires teachers to think about ways the concept skill occurs in naturally occurring contexts The authentic context must be explicitly connected to the targeted concept/skill Example

26. Implementation Choose appropriate context Activate students prior knowledge of authentic context, identify the math concept/skill students will learn and explicitly relate it to the context Involve students by prompting thinking about how the math concept/skill is relevant Check for understanding Provide opportunities for students to apply math concept/skill within authentic context Provide review and closure, continuing to explicitly link target concept/skill to authentic context Provide multiple opportunities for student practice

27. Now its your turn… Using your case study information apply at least one of the three selected teaching strategy (C-R-A, Explicit Modeling or Authentic Concepts) to your groups focus student Think about the students strengths & needs Review the students IEP and corresponding curricular framework Be prepared to share your ideas with the class