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Strategies for Accessing Algebraic Concepts (K-8)

Access Center

September 20, 2006

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Agenda

• Introductions and Overview

• Objectives

• Background Information

• Challenges for Students with Disabilities

• Instructional and Learning Strategies

• Application of Strategies

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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)

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Make ‘Em Laugh with Math

Question: What did one math book say to the other?

Answer: Don’t bother me. I’ve got my own problems!

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NYS Mathematics StandardStudents will:• Understand the concepts of and become proficient

with the skills of mathematics• Communicate and reason mathematically• Become problem solvers by using appropriate tools

and strategies

This is achieved through the integrated study ofnumber sense and operations, algebra, geometry,measurement, and statistics and probability.

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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The Goals of Mathematics Instruction

Help students to:• Internalize mathematical relationships so that they

can connect these relationships to their preexisting ideas (conceptual understanding)

• Gain the skills that would enable them to be fluent in carrying out accurate procedures (procedural fluency)

• Formulate, represent and solve mathematical problems (problem solving)

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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Synopsis of New York State Mathematics Core Curriculum

• Process strands• Content Strands• Bands within the Content Strands• Grade-by-Grade Performance

Indicators

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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The Process Strands

• The process strands highlight ways of acquiring and using content knowledge.

• They help to give meaning to mathematics and help students to see mathematics as a discipline rather than a set of isolated skills.

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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The Process Strands

• Problem Solving• Reasoning and Proof• Communication• Connections• Representation

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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The Content Strands

• The content strands explicitly describe the content students should learn.

• This broad range of content, taught in an integrated fashion, allows students to see how various mathematics knowledge is related, not only within mathematics, but also to other disciplines and the real world as well.

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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The Content Strands

• Number Sense and Operations• Algebra• Geometry• Measurement• Statistics and Probability

Adapted from NYS Mathematics Core Curriculum MST Standard 3 Pre-K – Grade 12, 2005

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How Many Triangles?

Pair off with another person, count the number of triangles, explain the process, and record the number.

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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)

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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)

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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)

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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)

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Therefore, instructional and learning strategies should

address:• Memory

• Language and communication

• Processing

• Self-esteem

• Attention

• Organizational skills

• Math anxiety

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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.

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Example for K-2

Add the robots!

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Example for K-2

Add the robots!

+

+

=

=2 1 3

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Example for 3-5

Tilt or Balance the Equation!

3 * 4 = 2 * 6 ?

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Example 3-5

Represent the equation!

3 * 4 = 2 * 6 ?

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Example for 6-8

3 * + = 2 * - 4

Balance the Equation!

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Example for 6-8Represent the Equation

3 * + = 2 * - 4

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Example for 6-8

3 * + = 2 * - 4

3 * 1 + 7 = 2 * 7 - 4

Solution

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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.

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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.

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2 Types of Mnemonics

• Keyword Strategy

• Letter Strategy

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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.

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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?

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Example K-2 Keyword Strategy

More than & less than (duck’s mouth open means more):

5 2

5 > 2

(Bernard, 1990)

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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

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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

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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!

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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.

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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).

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Venn Diagram - example

Prime Numbers

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7 11 13

Even Numbers

4 6 8 10

Multiples of 3

9 15 21

32

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Series of Definitions

Word = Category + Attribute

= +

Definitions: ______________________

________________________________

________________________________

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Four-Square Graphic Organizer

1. Word: 2. Example:

3. Non-example:4. Definition

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Four-Square Graphic Organizer – example

1. Word: semicircle 2. Example:

3. Non-example:4. Definition

A semicircle is half of a circle.

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Problem Set 3Addition Multiplication a + b a times b a plus b a x b sum of a and b a(b)

ab

Subtraction Divisiona – b a/ba minus b a divided by ba less b b) a

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How These Strategies Help Students Access Algebra

• Problem Representation

• Problem Solving (Reason)

• Self Monitoring

• Self Confidence

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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.

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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.

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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 athttp://www.k8accescenter.org/training_resources/documents/MathGraphicOrg.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/

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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