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

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

3. 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. 4 How Many Triangles? Pair off with another person, count the number of triangles, explain the process, and record the number.

5. 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, 10.20.1997)

6. 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. 7 NCTM Standards: Content: Numbers and Operations Measurement Geometry Data Analysis and Probability Algebra Process: Problem Solving Reasoning and Proof Communication Connections Representation

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

9. 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. 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. 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. 12 Therefore, instructional and learning strategies should address: Memory Language and communication Processing Self-esteem Attention Organizational skills Math anxiety

13. 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. 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. 15 Best Practice in Teaching Strategies 1. Pretest 2. Describe 3. Model 4. Practice 5. Provide Feedback 6. Promote Generalization

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

17. 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. 18 Example for K-2 Add the robots!

19. 19 Example for K-2 Add the robots! + + = = 213

20. 20 Example for 3-5 Tilt or Balance the Equation! 3 * 4 = 2 * 6 ?

21. 21 Example 3-5 Represent the equation! 3 * 4 = 2 * 6 ?

22. 22 Example for 6-8 3 * + = 2 * - 4 Balance the Equation!

23. 23 Example for 6-8 Represent the Equation 3 * + = 2 * - 4

24. 24 Example for 6-8 3 * + = 2 * - 4 3 * 1 + 7 = 2 * 7 - 4 Solution

25. 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. 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. 27 3 Types of Mnemonics Keyword Strategy Pegword Strategy Letter Strategy

28. 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. 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. 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. 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. 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. 33 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.

34. 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. 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. 36 Activity: 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

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

38. 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. 39 ORDER Solve the problem (4 + 6) – 2 x 3 = ? (10) – 2 x 3 = ? (10) – 6 = 4

40. 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. 41 Please Excuse My Dear Aunt Sally (6 + 7) + 5 2 – 4 x 3 = ? 13 + 5 2 – 4 x 3 = ? 13 + 25 - 4 x 3 = ? 13 + 25 - 12 = ? 38 - 12 = ? = 26

42. 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. 43 GO Activity: Roles #1 works with the figures (1-16) #2 asks questions #3 records #4 reports out

44. 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. 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. 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. 47 Graphic Organizers: Assist students in organizing and retaining information when used consistently. Assist teachers by integrating into instruction through creative approaches.

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

49. 49 Coherent Graphic Organizers 1.Provide clearly labeled branch and sub branches. 2.Have numbers, arrows, or lines to show the connections or sequence of events. 3.Relate similarities. 4.Define accurately.

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

51. 51 Teacher-Directed Approach 1.Provide a partially incomplete GO for students 2.Have students read instructions or information 3.Fill out the GO with students 4.Review the completed GO 5.Assess students using an incomplete copy of the GO

52. 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. 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. 54 Types of Graphic Organizers Hierarchical diagramming Sequence charts Compare and contrast charts

55. 55 A Simple Hierarchical Graphic Organizer

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

57. 57 Another Hierarchical Graphic Organizer Category Subcategory List examples of each type

58. 58 Hierarchical Graphic Organizer – example Algebra Equations Inequalities 2x + 3 = 15 10y = 100 4x = 10x - 6 14 < 3x + 7 2x > y 6y ≠ 15

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

60. 60 Positive Integers Numbers What is it? Illustration/Example What are some examples? Properties/Attributes What is it like? Fractions Compare and Contrast - example Whole Numbers Negative Integers Zero -3, -8, -4000 6, 17, 25, 100 0

61. 61 Venn Diagram

62. 62 Venn Diagram - example Prime Numbers 57 11 13 Even Numbers 4 6 810 Multiples of 3 9 15 21 3 2 6

63. 63 Multiple Meanings

64. 64 Multiple Meanings – example TRI- ANGLES RightEquiangular AcuteObtuse 3 sides 3 angles 1 angle = 90° 3 sides 3 angles 3 angles < 90° 3 sides 3 angles 3 angles = 60° 3 sides 3 angles 1 angle > 90°

65. 65 Series of Definitions Word=Category +Attribute = + Definitions: ______________________ ________________________________

66. 66 Series of Definitions – example Word=Category +Attribute = + Definition: A four-sided figure with four equal sides and four right angles. Square Quadrilateral 4 equal sides & 4 equal angles (90°)

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

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

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

70. 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. 71 Problem Set 1 ParallelogramRhombus SquareQuadrilateral PolygonKite Irregular polygonTrapezoid Isosceles TrapezoidRectangle

72. 72 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

73. 73 Problem Set 3 AdditionMultiplication a + ba times b a plus ba x b sum of a and ba(b) ab SubtractionDivision a – ba/b a minus ba divided by b a less bb) a

74. 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. 75 Possible Solution to PS #1 POLYGON Parallelogram: has 2 pairs of parallel sides Kite Square, rectangle, rhombus Kite: has 0 sets of parallel sides Irregular: 4 sides w/irregular shape Quadrilateral Trapezoid: has 1 set of parallel sides Trapezoid, isosceles trapezoid

76. 76 Possible Solution to PS #2 REAL NUMBERS

77. 77 Possible Solution PS #3 Operations Subtraction Multiplication Division Addition ____a + b____ ___a plus b___ Sum of a and b ____a - b_____ __a minus b___ ___a less b____ ____a / b_____ _a divided by b_ _____a  b_____ ___a times b___ ____a x b_____ _____a(b)_____ _____ab______

78. 78 Possible Solution to PS #4 Numbers Operations RulesSymbols Geometric Figures Mathematics Triangle Quadrilateral Hexagon Integer Prime Rational Irrational Whole Composite Addition Subtraction Multiplication Division Corollary Postulatem║n √4 {1,2,3…}

79. 79 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.

80. 80 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 http://www.k8accescenter.org/training_resources/documents/MathGrap hicOrg.pdf Visual mapping software: Inspiration and Kidspiration (for lower grades) at http:/www.inspiration.com Math Matrix from the Center for Implementing Technology in Education. Available at http://www.citeducation.org/mathmatrix/

81. 81 Resources Hall, T., & Strangman, N. (2002).Graphic organizers. Wakefield, MA: National Center on Accessing the General Curriculum. Available at http://www.cast.org/publications/ncac/ncac_go.html 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 http://www.k8accesscenter.org/training_resources /udl/GraphicOrganizersHTML.asp

82. 82 How These Strategies Help Students Access Algebra Problem Representation Problem Solving (Reason) Self Monitoring Self Confidence

83. 83 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.

84. 84 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.

85. 85 Wrap-Up Questions

86. 86 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

87. 87 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