1. Strategies for Accessing Algebraic Concepts (K-8)
Access Center
September 20, 2006

2. Agenda Introductions and Overview Objectives Background Information
Challenges for Students with Disabilities
Instructional and Learning Strategies
Application of Strategies

3. Objectives:
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)

4. How Many Triangles?
Pair off with another person, count the number of triangles, explain the process, and record the number.

5. 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, )
“Mathematics Equals Opportunity” Evaluative Report for the US Department of Education

6. 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

7. NCTM Standards: Content: Process: Numbers and Operations
Measurement
Geometry
Data Analysis and Probability
Algebra
Process:
Problem Solving
Reasoning and Proof
Communication
Connections
Representation

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

9. 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)

10. 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)

11. 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)

12. Therefore, instructional and learning strategies should address:
Memory
Language and communication
Processing
Self-esteem
Attention
Organizational skills
Math anxiety

13. 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

14. 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

15. Best Practice in Teaching Strategies
1. Pretest
2. Describe
3. Model
4. Practice
5. Provide Feedback
6. Promote Generalization

16. Effective Strategies for Students with Disabilities
Instructional Strategy: Concrete-Representational Abstract (CRA) Instruction
Learning Strategies: Mnemonics
Graphic Organizers

17. 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.

18. Example for K-2 Add the robots!

19. Example for K-2 Add the robots!+ = 2 1 3 + =

20. Example for 3-5
Tilt or Balance the Equation!
3 * = * 6 ?

21. Example 3-5 Represent the equation!
3 * = * 6 ?

22. Example for 6-8
Balance the Equation!
3 * = *

23. Example for 6-8
Represent the Equation
3 * = 2 *

24. Example for 6-8
Solution
3 * = *
3 * = *

25. 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?

26. Mnemonics
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.

27. 3 Types of Mnemonics
Keyword Strategy
Pegword Strategy
Letter Strategy

28. 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.

29. 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

30. DRAW 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.

31. DRAW 4x + 2x = 12 Represent the variable "x“ with circles. +
By combining like terms, there are six "x’s." 4x + 2x = 6x
6x = 12

32. DRAW 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

33. STAR: Letter Strategy Search the word problem;
The steps include:
Search the word problem;
Translate the words into an equation in picture form;
Answer the problem; and
Review the solution.

34. STAR
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?

35. STAR:
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.

36. Activity: Divide into groups
Read Preparing Students with Disabilities for Algebra (pg ; review examples pg.13-14)
Discuss examples from article of the integration of Mnemonics and CRA

37. Example K-2 Keyword Strategy
More than & less than (duck’s mouth open means more):
5 > 2
(Bernard, 1990)

38. Example Grade 3-5 Letter Strategy
O bserve the problem
R ead the signs.
D ecide which operation to do first.
E xecute the rule of order (Many Dogs Are Smelly!)
R elax, you're done!

39. ORDER Solve the problem (4 + 6) – 2 x 3 = ? (10) – 2 x 3 = ?
(10) – = 4

40. Example 6-8 Letter Strategy
PRE-ALGEBRA: ORDER OF OPERATIONS
Parentheses, brackets, and braces;
Exponents next;
Multiplication and Division, in order from left to right;
Addition and Subtraction, in order from left to right.
Please Excuse My Dear Aunt Sally

41. Please Excuse My Dear Aunt Sally
= ?
= ?
= 26

42. 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.

43. GO Activity: Roles #1 works with the figures (1-16) #2 asks questions
#3 records
#4 reports out

44. GO Activity: Directions
Differentiate the figures that have like and unlike characteristics
Create a definition for each set of figures.
Report your results.

45. 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?

46. 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).

47. Graphic Organizers:
Assist students in organizing and retaining information when used consistently.
Assist teachers by integrating into instruction through creative
approaches.

48. Graphic Organizers: Heighten student interest
Should be coherent and consistently used
Can be used with teacher- and student- directed approaches

49. 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.

50. How to Use Graphic Organizers in the Classroom
Teacher-Directed Approach
Student-Directed Approach

51. 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

52. 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

53. 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

54. Types of Graphic Organizers
Hierarchical diagramming
Sequence charts
Compare and contrast charts

55. A Simple Hierarchical Graphic Organizer

56. A Simple Hierarchical Graphic Organizer - example
Geometry
Algebra
MATH
Trigonometry
Calculus

57. Another Hierarchical Graphic Organizer
Category
Subcategory
Subcategory
Subcategory
This example relates a category, subdivided into subcategories then lists or shows examples of each type.
List examples of each type

58. Hierarchical Graphic Organizer – example
Algebra
Equations
Inequalities
This example relates a category, subdivided into subcategories then lists or shows examples of each type.
6y ≠ 15
2x + 3 = 15
10y = 100
14 < 3x + 7
2x > y
4x = 10x - 6

59. Compare and Contrast Category What is it? Illustration/Example
Properties/Attributes
Subcategory
Irregular set
What are some examples?
What is it like?

60. Compare and Contrast - example
Numbers
What is it?
Illustration/Example
Properties/Attributes
6, 17, 25, 100
Positive Integers
Whole Numbers
-3, -8, -4000
Negative Integers
Zero
Fractions
What are some examples?
What is it like?

61. Venn Diagram

62. Venn Diagram - example Prime Numbers 5 7 11 13 2 3 Even Numbers 4 6
5 7
Even Numbers
8 10
Multiples of 3 3 2 6

63. Multiple Meanings

64. Multiple Meanings – example
Right
Equiangular
3 sides
3 angles
1 angle = 90°
3 sides
3 angles
3 angles = 60°
TRI-
ANGLES
Acute
Obtuse
3 sides
3 angles
3 angles < 90°
3 sides
3 angles
1 angle > 90°

65. Series of Definitions Word = Category + Attribute = +=
Definitions: ______________________
________________________________
Example 4 can be considered a sequential organizer showing the process of defining a word by combining the category and attribute.

66. 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°)
Square
Quadrilateral
Example 4 can be considered a sequential organizer showing the process of defining a word by combining the category and attribute.

67. Four-Square Graphic Organizer
1. Word:
2. Example:
4. Definition
3. Non-example:

68. Four-Square Graphic Organizer – example
1. Word: semicircle
2. Example:
4. Definition
3. Non-example:
A semicircle is half of a circle.

69. Matching Activity Divide into groups
Match the problem sets with the appropriate graphic organizer

70. Matching Activity
Which graphic organizer would be most suitable for showing these relationships?
Why did you choose this type?
Are there alternative choices?

71. Problem Set 1 Parallelogram Rhombus Square Quadrilateral Polygon Kite
Irregular polygon Trapezoid
Isosceles Trapezoid Rectangle

72. Problem Set 2 Counting Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: , -2, -1, 0, 1, 2, 3,
Rationals: 0, …1/10, …1/5, …1/4, , …1/2, …1
Reals: all numbers
Irrationals: π, non-repeating decimal

73. Problem Set 3 Addition Multiplication a + b a times b a plus b a x b
sum of a and b a(b) ab
Subtraction Division
a – b a/b
a minus b a divided by b
a less b b) a

74. Problem Set 4
Use the following words to organize into categories and subcategories of
Mathematics:
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.

75. 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.

76. Resources
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
Math Matrix from the Center for Implementing Technology in Education. Available at

77. Resources
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

78. How These Strategies Help Students Access Algebra
Problem Representation
Problem Solving (Reason)
Self Monitoring
Self Confidence

79. Recommendations:
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.

80. 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.

81. Wrap-Up
Questions

82. 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

83. 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