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