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## Strategies for Accessing Algebraic Concepts (K-8)

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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:**• 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?**Pair off with another person, count the number of triangles, explain the process, and record the number.**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**• Learn to value mathematics • Become confident in their ability to do mathematics • Become mathematical problem solvers • Learn to communicate mathematically • Learn to reason mathematically**Content:**Numbers and Operations Measurement Geometry Data Analysis and Probability Algebra Process: Problem Solving Reasoning and Proof Communication Connections Representation NCTM Standards:**“Teachers must be given the training and resources to**provide the best mathematics for every child.” -NCTM**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: • 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: • 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: • Memory • Language and communication • Processing • Self-esteem • Attention • Organizational skills • Math anxiety**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 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**1. Pretest 2. Describe 3. Model 4. Practice 5. Provide Feedback 6. Promote Generalization**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: 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 K-2Add the robots!**+ = 2 1 3 + =**Example for 3-5**Tilt or Balance the Equation! • 3 *4 =2* 6 • ?**Example 3-5Represent the equation!**3 * 4 = 2 * 6 ?**Example for 6-8**Balance the Equation! 3 * +=2 * -4**Example for 6-8**Represent the Equation 3 * + = 2 * - 4**Example for 6-8**Solution 3 * + =2 * - 4 3 *1+7 =2 * 7-4**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?**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.**3 Types of Mnemonics**• Keyword Strategy • Pegword Strategy • Letter Strategy**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.**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**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.**DRAW**4x + 2x = 12 Represent the variable "x“ with circles. + By combining like terms, there are six "x’s." 4x + 2x = 6x 6x = 12**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**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.**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?**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.**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**Example K-2 Keyword Strategy**More than & less than (duck’s mouth open means more): 52 5 > 2 (Bernard, 1990)**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!**ORDER**Solve the problem (4 + 6) – 2 x 3 = ? (10) – 2 x 3 = ? (10) – 6 = 4**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. Example 6-8 Letter Strategy Please Excuse My Dear Aunt Sally**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)**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**• #1 works with the figures (1-16) • #2 asks questions • #3 records • #4 reports out**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**• 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?**• 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:**• Assist students in organizing and retaining information when used consistently. • Assist teachers by integrating into instruction through creative approaches.**Graphic Organizers:**• Heighten student interest • Should be coherent and consistently used • Can be used with teacher- and student- directed approaches**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**• Teacher-Directed Approach • Student-Directed Approach