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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:

Strategies for Accessing Algebraic Concepts (K-8)

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Strategies for Accessing Algebraic Concepts (K-8)

Access Center

September 20, 2006

- Introductions and Overview
- Objectives
- Background Information
- Challenges for Students with Disabilities
- Instructional and Learning Strategies
- Application of Strategies

- 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)

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

- 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)

- 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

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

-NCTM

- 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)

- 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)

- 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)

- Memory
- Language and communication
- Processing
- Self-esteem
- Attention
- Organizational skills
- Math anxiety

- 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 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

1. Pretest

2. Describe

3. Model

4. Practice

5. Provide Feedback

6. Promote Generalization

Instructional Strategy: Concrete-Representational- Abstract (CRA) Instruction

Learning Strategies: Mnemonics

Graphic Organizers

- 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

+

=

Tilt or Balance the Equation!

- 3 *4 =2* 6
- ?

3 * 4 = 2 * 6

?

Balance the Equation!

3 * +=2 * -4

Represent the Equation

3 * + = 2 * - 4

Solution

3 * + =2 * - 4

3 *1+7 =2 * 7-4

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?

- 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.

- Keyword Strategy
- Pegword Strategy
- Letter Strategy

- 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

- 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.

4x + 2x = 12

Represent the variable "x“ with circles.

+

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

6x = 12

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

The steps include:

- Search the word problem;
- Translate the words into an equation in picture form;
- Answer the problem; and
- Review the solution.

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?

- 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.

- 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

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

52

5 > 2

(Bernard, 1990)

- 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!

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.

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

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.

- #1 works with the figures (1-16)
- #2 asks questions
- #3 records
- #4 reports out

- Differentiate the figures that have like and unlike characteristics
- Create a definition for each set of figures.
- Report your results.

- 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?

- 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).

- Assist students in organizing and retaining information when used consistently.
- Assist teachers by integrating into instruction through creative
approaches.

- Heighten student interest
- Should be coherent and consistently used
- Can be used with teacher- and student- directed approaches

- Provide clearly labeled branch and sub branches.
- Have numbers, arrows, or lines to show the connections or sequence of events.
- Relate similarities.
- Define accurately.

- Teacher-Directed Approach
- Student-Directed Approach

- Provide a partially incomplete GO for students
- Have students read instructions or information
- Fill out the GO with students
- Review the completed GO
- Assess students using an incomplete copy of the GO

- 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

- 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

- Hierarchical diagramming
- Sequence charts
- Compare and contrast charts

Geometry

Algebra

MATH

Trigonometry

Calculus

Category

Subcategory

Subcategory

Subcategory

List examples of each type

Algebra

Equations

Inequalities

6y ≠15

14 < 3x + 7

2x > y

10y = 100

2x + 3 = 15

4x = 10x - 6

Compare and Contrast

Category

What is it?

Illustration/Example

Properties/Attributes

Subcategory

Irregular set

What are some examples?

What is it like?

Compare and Contrast - example

Numbers

What is it?

Illustration/Example

Properties/Attributes

6, 17, 25, 100

Positive Integers

Whole Numbers

-3, -8, -4000

Negative Integers

0

Zero

Fractions

What are some examples?

What is it like?

Prime Numbers

57

11 13

2

3

Even Numbers

4 6

810

Multiples of 3

9 15 21

6

3 sides

3 sides

3 angles

3 angles

3 angles = 60°

1 angle = 90°

3 sides

3 angles

3 angles < 90°

Right

Equiangular

TRI-

ANGLES

Acute

Obtuse

3 sides

3 angles

1 angle > 90°

Word=Category +Attribute

= +

Definitions: ______________________

________________________________

________________________________

Word=Category +Attribute

= +

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

4 equal sides &

4 equal angles (90°)

Square

Quadrilateral

1. Word:

2. Example:

4. Definition

3. Non-example:

1. Word: semicircle

2. Example:

4. Definition

3. Non-example:

A semicircle is half of a circle.

- Divide into groups
- Match the problem sets with the appropriate graphic organizer

- Which graphic organizer would be most suitable for showing these relationships?
- Why did you choose this type?
- Are there alternative choices?

ParallelogramRhombus

SquareQuadrilateral

PolygonKite

Irregular polygonTrapezoid

Isosceles TrapezoidRectangle

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

AdditionMultiplication

a + ba times b

a plus ba x b

sum of a and ba(b)

ab

SubtractionDivision

a – ba/b

a minus ba divided by b

a less bb) a

Use 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.

- 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.

- 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 at
http://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/

- 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

- Problem Representation
- Problem Solving (Reason)
- Self Monitoring
- Self Confidence

- 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.

- 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.

- Questions

Principles of an effective lesson:

Before the Lesson:

- Review
- Explain objectives, purpose, rationale for learning the strategy, and implementation of strategy
During 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

Grades K-2

- Use concrete materials to build an understanding of equality (same as) and inequality (more than and less than)
- Skip counting
Grades 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 variable
Grades 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