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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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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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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 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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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 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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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?

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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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DRAW: Letter Strategy

• Discover the sign• Read the problem• Answer or draw a

representation of the problem using lines, tallies, or checks

• Write the answer and check

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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.

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DRAW

4x + 2x = 12

Represent the variable "x“ with circles.

+

By combining like terms, there are six "x’s." 4x + 2x = 6x

6x = 12

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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

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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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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.

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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

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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 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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ORDER

Solve the problem

(4 + 6) – 2 x 3 = ?

(10) – 2 x 3 = ?

(10) – 6 = 4

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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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GO Activity: Roles

• #1 works with the figures (1-16)

• #2 asks questions

• #3 records

• #4 reports out

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GO Activity: Directions

• Differentiate the figures that have like and unlike characteristics

• Create a definition for each set of figures.

• Report your results.

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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?

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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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Coherent Graphic Organizers

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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Student-Directed Approach• Teacher uses a GO cover sheet with

prompts– Example: Teacher provides a cover sheet

that includes page numbers and paragraph numbers to locate information needed to fill out GO

• Teacher acts as a facilitator• Students check their answers with a

teacher copy supplied on the overhead

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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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Positive Integers

Numbers

What is it?Illustration/Example

What are some examples?

Properties/Attributes

What is it like?

Fractions

Compare and Contrast - example

Whole Numbers Negative Integers

Zero

-3, -8, -4000

6, 17, 25, 100

0

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Venn Diagram

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Venn Diagram - example

Prime Numbers

5

7 11 13

Even Numbers

4 6 8 10

Multiples of 3

9 15 21

32

6

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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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Series of Definitions

Word = Category + Attribute

= +

Definitions: ______________________

________________________________

________________________________

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Series of Definitions – example

Word = Category + Attribute

= +

Definition: A four-sided figure with four equal sides and four right angles.

Square Quadrilateral 4 equal sides & 4 equal angles (90°)

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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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Matching Activity

• Divide into groups

• Match the problem sets with the appropriate graphic organizer

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Matching Activity

• Which graphic organizer would be most suitable for showing these relationships?

• Why did you choose this type?

• Are there alternative choices?

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Problem Set 1

Parallelogram Rhombus

Square Quadrilateral

Polygon Kite

Irregular polygon Trapezoid

Isosceles Trapezoid Rectangle

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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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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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Problem Set 4Use the following words to organize into categories and subcategories of

Mathematics:NUMBERS, OPERATIONS, Postulates, RULE, Triangles, GEOMETRIC FIGURES, SYMBOLS, corollaries, squares, rational, prime, Integers, addition, hexagon, irrational, {1, 2, 3…}, multiplication, composite, m || n, whole, quadrilateral, subtraction, division.

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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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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

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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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Wrap-Up

• Questions

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Closing ActivityPrinciples of an effective lesson:Before the Lesson:• Review • Explain objectives, purpose, rationale for learning the

strategy, and implementation of strategyDuring the Lesson:• Model the task• Prompt students in dialogue to promote the

development of problem-solving strategies and reflective thinking

• Provide guided and independent practice• Use corrective and positive feedback

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Concepts for Developing a Lesson

Grades K-2• Use concrete materials to build an understanding of equality

(same as) and inequality (more than and less than)• Skip countingGrades 3- 5• Explore properties of equality in number sentences (e.g., when

equals are added to equals the sums are equal)• Use physical models to investigate and describe how a change

in one variable affects a second variableGrades 6-8• Positive and negative numbers (e.g., general concept, addition,

subtraction, multiplication, division)• Investigate the use of systems of equations, tables, and graphs

to represent mathematical relationships