1. 1 Strategies for Accessing Algebraic Concepts (K-8) Access Center April 25, 2007

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?

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 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)

11. 11 Therefore, instructional and learning strategies should address: Memory Language and communication Processing Self-esteem Attention Organizational skills Math anxiety

12. 12 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

13. 13 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

14. 14 Best Practice in Teaching Strategies 1. Pretest 2. Describe 3. Model 4. Practice 5. Provide Feedback 6. Promote Generalization

15. 15 Effective Strategies for Students with Disabilities Instructional Strategy: Concrete-Representational- Abstract (CRA) Instruction Learning Strategies: Mnemonics Graphic Organizers

16. 16 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.

17. 17 Example for 6-8 3 * + = 2 * - 4 Balance the Equation!

18. 18 Example for 6-8 3 * + = 2 * - 4 3 * 1 + 7 = 2 * 7 - 4 Solution

19. 19 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.

20. 20 3 Types of Mnemonics Keyword Strategy Pegword Strategy Letter Strategy

21. 21 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.

22. 22 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.

23. 23 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?

24. 24 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

25. 25 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

26. 26 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.

27. 27 Why are Graphic Organizers Important?

28. 28 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).

29. 29 Graphic Organizers: Assist students in organizing and retaining information when used consistently. Assist teachers by integrating into instruction through creative approaches.

30. 30 Graphic Organizers: Heighten student interest Should be coherent and consistently used Can be used with teacher- and student- directed approaches

31. 31 Key Features 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.

32. 32 How to Use Graphic Organizers in the Classroom Teacher-Directed Approach Student-Directed Approach

33. 33 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

34. 34 Teacher-Directed Approach - example 2. Example: 3. Non-example:4. Definition 1. Word: semicircle

35. 35 Teacher-Directed Approach – example 1. Word: semicircle 2. Example: 3. Non-example:4. Definition A semicircle is half of a circle.

36. 36 Student-Directed Approach Teacher uses a GO cover sheet with prompts Teacher acts as a facilitator Students check their answers with a teacher copy supplied on the overhead

37. 37 Student-Directed Approach - example From Word: To Category & Attribute Definitions: ______________________ ________________________________ Example

38. 38 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

39. 39 Types of Graphic Organizers Hierarchical diagramming Sequence charts Compare and contrast charts

40. 40 A Simple Hierarchical Graphic Organizer

41. 41 A Simple Hierarchical Graphic Organizer - example Algebra Calculus Trigonometry Geometry MATH

42. 42 Another Hierarchical Graphic Organizer Category Subcategory List examples of each type

43. 43 Hierarchical Graphic Organizer – example Algebra Equations Inequalities 2x + 3 = 15 10y = 100 4x = 10x - 6 14 < 3x + 7 2x > y 6y ≠ 15

44. 44 Category What is it? Illustration/Example What are some examples? Properties/Attributes What is it like? Subcategory Irregular set Compare and Contrast

45. 45 Integers Numbers What is it? Illustration/Example What are some examples? Properties/Attributes What is it like? Irrationals Compare and Contrast - example Rational Numbers Non-Integers Zero 6, 17, -25, 100 0

46. 46 Venn Diagram

47. 47 Venn Diagram - example Prime Numbers 57 11 13 Even Numbers 4 6 810 Multiples of 3 9 15 21 3 2 6

48. 48 Multiple Meanings

49. 49 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°

50. 50 Matching Activity Review Problem Set Respond to poll with type of graphic organizer that is a best fit

51. 51 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

52. 52 Possible Solution to PS #2 REAL NUMBERS

53. 53 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

54. 54 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______

55. 55 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.

56. 56 How These Strategies Help Students Access Algebra Problem Representation Problem Solving (Reason) Self Monitoring Self Confidence

57. 57 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.

58. 58 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.

59. 59 This module is available on our Web site: http://www.k8accesscenter.org/training_ resources/AlgebraicConceptsK-8.asp