Teaching Math to Students with Disabilities

Present Perspectives

“Math is hard” (Barbie, 1994)

US 15 year olds ranked 24th (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 1st generation who are worse off economically than their parents

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 NCTM’s standards

Debate over math difficulties vs math disabilities

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

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?

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

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

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

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

Promising approaches to teaching

mathematics to students with disabilities

Math Expressions Saxon Strategic math Series Touch Math Number Worlds Curriculum Montessori methods and materials What works clearing house

Resources for teaching math

Illuminations

MathVids

Teaching Math to Students with Disabilities

Strategies

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

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.

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.

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.

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.

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

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

What is explicit modeling?

Student

Teacher

Mathematical concept

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

Instructional techniques….(con’t)

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

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

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

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

Now it’s 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 group’s focus student

Think about the student’s strengths & needs Review the student’s IEP and corresponding

curricular framework Be prepared to share your ideas with the

class