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Strategies for Accessing Algebraic Concepts (K-8) Access Center September 20, 2006 Agenda Introductions and Overview Objectives Background Information Challenges for Students with Disabilities Instructional and Learning Strategies Application of Strategies Objectives:

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Strategies for Accessing Algebraic Concepts (K-8)

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Presentation Transcript
  • Introductions and Overview
  • Objectives
  • Background Information
  • Challenges for Students with Disabilities
  • Instructional and Learning Strategies
  • Application of Strategies
  • To identify the National Council of Teachers of Mathematics (NCTM) content and process standards
  • To identify difficulties students with disabilities have with learning algebraic concepts
  • To identify and apply research-based instructional and learning strategies for accessing algebraic concepts (grades K-8)
how many triangles
How Many Triangles?

Pair off with another person, count the number of triangles, explain the process, and record the number.

why is algebra important
Why Is Algebra Important?
  • Language through which most of mathematics is communicated (NCTM, 1989)
  • Required course for high school graduation
  • Gateway course for higher math and science courses
  • Path to careers – math skills are critical in many professions (“Mathematics Equals Equality,” White Paper prepared for US Secretary of Education, 10.20.1997)
nctm goals for all students
NCTM Goals for All Students
  • Learn to value mathematics
  • Become confident in their ability to do mathematics
  • Become mathematical problem solvers
  • Learn to communicate mathematically
  • Learn to reason mathematically
nctm standards

Numbers and Operations



Data Analysis and Probability



Problem Solving

Reasoning and Proof




NCTM Standards:
“Teachers must be given the training and resources to provide the best mathematics for every child.”


challenges students experience with algebra
Challenges Students Experience with Algebra
  • Translate word problems into mathematical symbols (processing)
  • Distinguish between patterns or detailed information (visual)
  • Describe or paraphrase an explanation (auditory)
  • Link the concrete to a representation to the abstract (visual)
  • Remember vocabulary and processes (memory)
  • Show fluency with basic number operations (memory)
  • Maintain focus for a period of time (attention deficit)
  • Show written work (reversal of numbers and letters)
at the elementary level students with disabilities have difficulty with
At the Elementary Level, Students with Disabilities Have Difficulty with:
  • Solving problems (Montague, 1997; Xin Yan & Jitendra, 1999)
  • Visually representing problems (Montague, 2005)
  • Processing problem information (Montague, 2005)
  • Memory (Kroesbergen & Van Luit, 2003)
  • Self-monitoring (Montague, 2005)
at the middle school level students with disabilities have difficulty
At the Middle School Level, Students with Disabilities Have Difficulty:
  • Meeting content standards and passing state assessments(Thurlow, Albus, Spicuzza, & Thompson, 1998; Thurlow, Moen, & Wiley, 2005)
  • Mastering basic skills(Algozzine, O’Shea, Crews, & Stoddard, 1987; Cawley, Baker-Kroczynski, & Urban, 1992)
  • Reasoning algebraically(Maccini, McNaughton, & Ruhl, 1999)
  • Solving problems(Hutchinson, 1993; Montague, Bos, & Doucette, 1991)
therefore instructional and learning strategies should address
Therefore, instructional and learning strategies should address:
  • Memory
  • Language and communication
  • Processing
  • Self-esteem
  • Attention
  • Organizational skills
  • Math anxiety
instructional strategy
Instructional Strategy
  • Instructional Strategies are methods that can be used to deliver a variety of content objectives.
  • Examples: Concrete-Representational-Abstract (CRA) Instruction, Direct Instruction, Differentiated Instruction, Computer Assisted Instruction
learning strategy
Learning Strategy
  • Learning Strategies are techniques, principles, or rules that facilitate the acquisition, manipulation, integration, storage, and retrieval of information across situations and settings (Deshler, Ellis & Lenz, 1996)
  • Examples: Mnemonics, Graphic Organizers, Study Skills
best practice in teaching strategies
Best Practice in Teaching Strategies

1. Pretest

2. Describe

3. Model

4. Practice

5. Provide Feedback

6. Promote Generalization

effective strategies for students with disabilities
Effective Strategies for Students with Disabilities

Instructional Strategy: Concrete-Representational- Abstract (CRA) Instruction

Learning Strategies: Mnemonics

Graphic Organizers

concrete representational abstract instructional approach c r a
Concrete-Representational-Abstract Instructional Approach (C-R-A)
  • CONCRETE: Uses hands-on physical (concrete) models or manipulatives to represent numbers and unknowns.
  • REPRESENTATIONAL or semi-concrete: Draws or uses pictorial representations of the models.
  • ABSTRACT: Involves numbers as abstract symbols of pictorial displays.
example for 3 5
Example for 3-5

Tilt or Balance the Equation!

  • 3 *4 =2* 6
          • ?
example for 6 8
Example for 6-8

Balance the Equation!

3 * +=2 * -4

example for 6 823
Example for 6-8

Represent the Equation

3 * + = 2 * - 4

example for 6 824
Example for 6-8


3 * + =2 * - 4

3 *1+7 =2 * 7-4

case study
Case Study

Questions to Discuss:

  • How would you move these students along the instructional sequence?
  • How does CRA provide access to the curriculum for all of these students?
  • A set of strategies designed to help students improve their memory of new information.
  • Link new information to prior knowledge through the use of visual and/or acoustic cues.
3 types of mnemonics
3 Types of Mnemonics
  • Keyword Strategy
  • Pegword Strategy
  • Letter Strategy
why are mnemonics important
Why Are Mnemonics Important?
  • Mnemonics assist students with acquiring information in the least amount of time (Lenz, Ellis & Scanlon, 1996).
  • Mnemonics enhance student retention and learning through the systematic use of effective teaching variables.
draw letter strategy
Discover the sign

Read the problem

Answer or draw a representation of the problem using lines, tallies, or checks

Write the answer and check

DRAW: Letter Strategy
  • D iscover the variable
  • R ead the equation, identify operations, and think about the process to solve the equation.
  • A nswer the equation.
  • W rite the answer and check the equation.

4x + 2x = 12

Represent the variable "x“ with circles.


By combining like terms, there are six "x’s." 4x + 2x = 6x

6x = 12


Divide the total (12) equally among the circles.

6x = 12

The solution is the number of tallies represented in one circle – the variable ‘x." x = 2

star letter strategy
STAR: Letter Strategy

The steps include:

  • Search the word problem;
  • Translate the words into an equation in picture form;
  • Answer the problem; and
  • Review the solution.

The temperature changed by an average of -3° F per hour. The total temperature change was 15° F. How many hours did it take for the temperature to change?

  • Search: read the problem carefully, ask questions, and write down facts.
  • Translate: use manipulatives to express the temperature.
  • Answer the problem by using manipulatives.
  • Review the solution: reread and check for reasonableness.
  • Divide into groups
  • Read Preparing Students with Disabilities for Algebra (pg. 10-12; review examples pg.13-14)
  • Discuss examples from article of the integration of Mnemonics and CRA
example k 2 keyword strategy
Example K-2 Keyword Strategy

More than & less than (duck’s mouth open means more):


5 > 2

(Bernard, 1990)

example grade 3 5 letter strategy
Example Grade 3-5 Letter Strategy
  • O bserve the problem
  • Read the signs.
  • D ecide which operation to do first.
  • Execute the rule of order (Many Dogs Are Smelly!)
  • R elax, you're done!

Solve the problem

(4 + 6) – 2 x 3 = ?

(10) – 2 x 3 = ?

(10) – 6 = 4

example 6 8 letter strategy

Parentheses, brackets, and braces;

Exponents next;

Multiplication and Division, in order from left to right;

Addition and Subtraction, in order from left to right.

Example 6-8 Letter Strategy

Please Excuse My Dear Aunt Sally

p lease e xcuse m y d ear a unt s ally
Please Excuse My Dear Aunt Sally

(6 + 7) + 52 – 4 x 3 = ?

13 + 52 – 4 x 3 = ?

13 + 25 - 4 x 3 = ?

13 + 25 - 12 = ?

38 - 12 = ?

= 26

graphic organizers gos
Graphic Organizers (GOs)

A graphic organizer is a tool or process to build word knowledge by relating similarities of meaning to the definition of a word. This can relate to any subject—math, history, literature, etc.

go activity roles
GO Activity: Roles
  • #1 works with the figures (1-16)
  • #2 asks questions
  • #3 records
  • #4 reports out
go activity directions
GO Activity: Directions
  • Differentiate the figures that have like and unlike characteristics
  • Create a definition for each set of figures.
  • Report your results.
go activity discussion
GO Activity: Discussion
  • Use chart paper to show visual grouping
  • How many groups of figures?
  • What are the similarities and differences that defined each group?
  • How did you define each group?
why are graphic organizers important
Why are Graphic Organizers Important?
  • GOs connect content in a meaningful way to help students gain a clearer understanding of the material (Fountas & Pinnell, 2001, as cited in Baxendrall, 2003).
  • GOs help students maintain the information over time (Fountas & Pinnell, 2001, as cited in Baxendrall, 2003).
graphic organizers
Graphic Organizers:
  • Assist students in organizing and retaining information when used consistently.
  • Assist teachers by integrating into instruction through creative


graphic organizers48
Graphic Organizers:
  • Heighten student interest
  • Should be coherent and consistently used
  • Can be used with teacher- and student- directed approaches
coherent graphic organizers
Coherent Graphic Organizers
  • Provide clearly labeled branch and sub branches.
  • Have numbers, arrows, or lines to show the connections or sequence of events.
  • Relate similarities.
  • Define accurately.
how to use graphic organizers in the classroom
How to Use Graphic Organizers in the Classroom
  • Teacher-Directed Approach
  • Student-Directed Approach
teacher directed approach
Teacher-Directed Approach
  • Provide a partially incomplete GO for students
  • Have students read instructions or information
  • Fill out the GO with students
  • Review the completed GO
  • Assess students using an incomplete copy of the GO
student directed approach
Student-Directed Approach
  • Teacher uses a GO cover sheet with prompts
    • Example: Teacher provides a cover sheet that includes page numbers and paragraph numbers to locate information needed to fill out GO
  • Teacher acts as a facilitator
  • Students check their answers with a teacher copy supplied on the overhead
strategies to teach graphic organizers
Strategies to Teach Graphic Organizers
  • Framing the lesson
  • Previewing
  • Modeling with a think aloud
  • Guided practice
  • Independent practice
  • Check for understanding
  • Peer mediated instruction
  • Simplifying the content or structure of the GO
types of graphic organizers
Types of Graphic Organizers
  • Hierarchical diagramming
  • Sequence charts
  • Compare and contrast charts
a simple hierarchical graphic organizer example
A Simple Hierarchical Graphic Organizer - example






another hierarchical graphic organizer
Another Hierarchical Graphic Organizer





List examples of each type

hierarchical graphic organizer example
Hierarchical Graphic Organizer – example




6y ≠15

14 < 3x + 7

2x > y

10y = 100

2x + 3 = 15

4x = 10x - 6


Compare and Contrast


What is it?




Irregular set

What are some examples?

What is it like?


Compare and Contrast - example


What is it?



6, 17, 25, 100

Positive Integers

Whole Numbers

-3, -8, -4000

Negative Integers




What are some examples?

What is it like?

venn diagram example

Prime Numbers

5 7

11 13



Even Numbers

4 6

8 10

Multiples of 3

9 15 21


Venn Diagram - example
multiple meanings example

3 sides

3 sides

3 angles

3 angles

3 angles = 60°

1 angle = 90°

3 sides

3 angles

3 angles < 90°

Multiple Meanings – example







3 sides

3 angles

1 angle > 90°

series of definitions
Series of Definitions

Word = Category + Attribute

= +

Definitions: ______________________



series of definitions example
Series of Definitions – example

Word = Category + Attribute

= +

Definition: A four-sided figure with four equal sides and four right angles.

4 equal sides &

4 equal angles (90°)



four square graphic organizer
Four-Square Graphic Organizer

1. Word:

2. Example:

4. Definition

3. Non-example:

four square graphic organizer example
Four-Square Graphic Organizer – example

1. Word: semicircle

2. Example:

4. Definition

3. Non-example:

A semicircle is half of a circle.

matching activity
Matching Activity
  • Divide into groups
  • Match the problem sets with the appropriate graphic organizer
matching activity70
Matching Activity
  • Which graphic organizer would be most suitable for showing these relationships?
  • Why did you choose this type?
  • Are there alternative choices?
problem set 1
Problem Set 1

Parallelogram Rhombus

Square Quadrilateral

Polygon Kite

Irregular polygon Trapezoid

Isosceles Trapezoid Rectangle

problem set 2
Problem Set 2

Counting Numbers: 1, 2, 3, 4, 5, 6, . . .

Whole Numbers: 0, 1, 2, 3, 4, . . .

Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .

Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1

Reals: all numbers

Irrationals: π, non-repeating decimal

problem set 3
Problem Set 3

Addition Multiplication

a + b a times b

a plus b a x b

sum of a and b a(b)


Subtraction Division

a – b a/b

a minus b a divided by b

a less b b) a

problem set 4
Problem Set 4

Use the following words to organize into categories and subcategories of


NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division.

graphic organizer summary
Graphic Organizer Summary
  • GOs are a valuable tool for assisting students with LD in basic mathematical procedures and problem solving.
  • Teachers should:
    • Consistently, coherently, and creatively use GOs.
    • Employ teacher-directed and student-directed approaches.
    • Address individual needs via curricular adaptations.
  • Maccini, P., & Gagnon, J. C. (2005). Math graphic organizers for students with disabilities. Washington, DC: The Access Center: Improving Outcomes for all Students K-8. Available at

  • Visual mapping software: Inspiration and Kidspiration (for lower grades) at http:/
  • Math Matrix from the Center for Implementing Technology in Education. Available at
  • Hall, T., & Strangman, N. (2002).Graphic organizers. Wakefield, MA: National Center on Accessing the General Curriculum. Available at
  • Strangman, N., Hall, T., Meyer, A. (2003) Graphic Organizers and Implications for Universal Design for Learning: Curriculum Enhancement Report. Wakefield, MA: National Center on Accessing the General Curriculum. Available at
how these strategies help students access algebra
How These Strategies Help Students Access Algebra
  • Problem Representation
  • Problem Solving (Reason)
  • Self Monitoring
  • Self Confidence
  • Provide a physical and pictorial model, such as diagrams or hands-on materials, to aid the process for solving equations/problems.
  • Use think-aloud techniques when modeling steps to solve equations/problems. Demonstrate the steps to the strategy while verbalizing the related thinking.
  • Provide guided practice before independent practice so that students can first understand what to do for each step and then understand why.
additional recommendations
Additional Recommendations:
  • Continue to instruct secondary math students with mild disabilities in basic arithmetic. Poor arithmetic background will make some algebraic questions cumbersome and difficult.
  • Allot time to teach specific strategies. Students will need time to learn and practice the strategy on a regular basis.
wrap up
  • Questions
closing activity
Closing Activity

Principles of an effective lesson:

Before the Lesson:

  • Review
  • Explain objectives, purpose, rationale for learning the strategy, and implementation of strategy

During the Lesson:

  • Model the task
  • Prompt students in dialogue to promote the development of problem-solving strategies and reflective thinking
  • Provide guided and independent practice
  • Use corrective and positive feedback
concepts for developing a lesson
Concepts for Developing a Lesson

Grades K-2

  • Use concrete materials to build an understanding of equality (same as) and inequality (more than and less than)
  • Skip counting

Grades 3- 5

  • Explore properties of equality in number sentences (e.g., when equals are added to equals the sums are equal)
  • Use physical models to investigate and describe how a change in one variable affects a second variable

Grades 6-8

  • Positive and negative numbers (e.g., general concept, addition, subtraction, multiplication, division)
  • Investigate the use of systems of equations, tables, and graphs to represent mathematical relationships