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Strategies for Accessing Algebraic Concepts (K-8)

Access Center

April 25, 2007

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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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How Many Triangles?

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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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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

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NCTM Standards:

Content:• Numbers and

Operations• Measurement• Geometry• Data Analysis and

Probability• Algebra

Process:• Problem Solving• Reasoning and

Proof• Communication• Connections• Representation

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

-NCTM

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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 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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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

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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

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Best Practice in Teaching Strategies

1. Pretest2. Describe3. Model4. Practice5. Provide Feedback6. Promote Generalization

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Effective Strategies for Students with Disabilities

Instructional Strategy: Concrete-Representational- Abstract (CRA)

Instruction

Learning Strategies: Mnemonics Graphic Organizers

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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 6-8

3 * + = 2 * - 4

Balance the Equation!

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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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3 Types of Mnemonics

• Keyword Strategy

• Pegword Strategy

• Letter Strategy

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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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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 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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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?

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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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Graphic Organizers:

• Assist students in organizing and retaining information when used consistently.

• Assist teachers by integrating into instruction through creative

approaches.

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Graphic Organizers:

• Heighten student interest

• Should be coherent and consistently used

• Can be used with teacher- and student- directed approaches

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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.

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How to Use Graphic Organizers in the Classroom

• Teacher-Directed Approach

• Student-Directed Approach

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Teacher-Directed Approach

1. Provide a partially incomplete GO for students

2. Have students read instructions or information

3. Fill out the GO with students4. Review the completed GO5. Assess students using an incomplete

copy of the GO

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Teacher-Directed Approach - example

2. Example:

3. Non-example:4. Definition

1. Word: semicircle

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Teacher-Directed Approach – example

1. Word: semicircle 2. Example:

3. Non-example:4. Definition

A semicircle is half of a circle.

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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

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Student-Directed Approach - example

From Word: To Category & Attribute

Definitions: ______________________

________________________________

________________________________

Example

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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

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Types of Graphic Organizers

• Hierarchical diagramming

• Sequence charts

• Compare and contrast charts

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A Simple Hierarchical Graphic Organizer

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A Simple Hierarchical Graphic Organizer - example

Algebra

Calculus Trigonometry

Geometry

MATH

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Another Hierarchical Graphic Organizer

Category

Subcategory Subcategory Subcategory

List examples of each type

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Hierarchical Graphic Organizer – example

Algebra

Equations Inequalities

2x +

3 =

15

10y

= 10

04x

= 1

0x -

6

14 < 3x + 7

2x > y

6y ≠ 15

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Category

What is it?Illustration/Example

What are some examples?

Properties/Attributes

What is it like?

Subcategory

Irregular set

Compare and Contrast

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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

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Venn Diagram

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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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Multiple Meanings

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Multiple Meanings – example

TRI-ANGLES

Right Equiangular

Acute Obtuse

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°

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Matching Activity

• Review Problem Set

• Respond to poll with type of graphic organizer that is a best fit

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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

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Possible Solution to PS #2REAL NUMBERS

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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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Possible Solution PS #3

Operations

Subtraction

MultiplicationDivision

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______

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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.

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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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This module is available on our Web site:

http://www.k8accesscenter.org/training_

resources/AlgebraicConceptsK-8.asp