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Mathematical Reasoning Presenters Leah Felcher

lfelcher_at_tcsg.edu Elaine Shapow

eshapow_at_comcast.net

Session Objectives

- Review standards for mathematical content for the

2014 GED Test and compare them to the 2002 GED

standards - Explore essential mathematical practices and

behaviors - Discuss beginning strategies for the classroom

Going the Next Step

- We should be educating all students according to

a common academic expectation, one that prepares

them for both postsecondary education and the

workforce. - (ACT, 2006)

Standards-Driven Curriculum

Design and Organization

Domain

Cluster

Standard

Assessment Target Standards

- Think, Pair, Share

Mathematical ReasoningNew Realities

What we know . . .

- People have a love-hate relationship with

mathematics - Twice as many people hated it as any other school

subject - It was also voted the most popular subject
- Associated Press Poll

Whats new in the Mathematical Reasoning domain?

- Identify absolute value of a rational number
- Determine when a numerical expression

is undefined - Factor polynomial expressions
- Solve linear inequalities

Whats new in the Mathematical Reasoning domain?

- Identify or graph the solution to a one variable

linear inequality - Solve real-world problems involving inequalities
- Write linear inequalities to

represent context - Represent or identify a

function in a table or graph

Whats not directly assessed on the 2014 GED

Math Reasoning Test?

- Select the appropriate operations to solve

problems - Relate basic arithmetic operations to one another
- Use estimation to solve problems and assess the

reasonableness of an answer - Identify and select appropriate units of metric

and customary measures - Read and interpret scales, meters, and gauges
- Compare and contrast different sets of data on

the basis of measures of central tendency - Recognize and use direct and indirect variation

New Mathematical Tools

TI-30XS MultiView Calculator

Its Your Turn!

Lets Have Some Fun!

http//education.ti.com/en/us/products/calculators

/scientific-calculators/ti-30xs-multiview/classroo

m-activities/activities-exchange

Teach Big Ideas!

What are the big ideas that I want students to

remember . . .

Problem Solving In Your Classroom

What opportunities do your students currently

have to grapple with non-routine complex tasks

and to

. reflect on their thinking and consolidate new

mathematical ideas and problem solving solutions?

Lets SOLVE a Math Problem

- Sure-Fire Steps to Becoming a Math Genius!

- Even Albert Einstein said
- Do not worry about your difficulties in

Mathematics. I can assure you mine are still

greater.

SOLVE a Problem

- S tudy the problem (What am I trying to find?)
- O rganize the facts (What do I know?)
- L ine up a plan (What steps will I take?)
- V erify your plan with action (How will I carry

out my plan?) - E xamine the results (Does my answer make

sense? If not, rework.) - Always double check!

S Study the problem

Each week, Bob gets paid 20 per hour for his

first 40 hours of work, plus 30 per hour for

every hour worked over 40 hours. Last month, Bob

made an additional 240 in overtime wages. If

Bob works 55 hours this week, how much will he

earn?

- What is the problem asking me to do?
- Find the question.

We are going to practice SOLVE with this one!

O Organize the Facts

- Identify each fact.
- Eliminate unnecessary facts.
- List all necessary facts.

Each week, Bob gets paid 20 per hour for his

first 40 hours of work, plus 30 per hour for

every hour worked over 40 hours. Last month, Bob

made an additional 240 in overtime wages. If

Bob works 55 hours this week, how much will he

earn?

L Line Up a Plan

- Select the operations to use.
- State the plan/strategy that you will use in

words.

I will use a multi-step approach. First, I will

multiply the number of regular work hours by the

regular hourly rate. Next, I will multiply the

number of hours of overtime by the overtime rate.

To obtain Bobs total weekly salary, I will add

the total amount earned for his regular salary

plus his overtime salary.

V Verify Your Plan

- 20.00
- x 40

Regular Salary

800.00 450.00

Regular Wages

Overtime Salary

800.00

1250.00

Hours Overtime

30.00x 15

Total Weekly Salary

450.00

Total Overtime Salary

E Examine the Results(Is it reasonable? Does

it make sense? Is it accurate?)

1250.00 IS reasonable because it is more than

Bobs average weekly salary. Also, the answer is

a whole number because all of the facts were

whole numbers ending in zeros. Therefore, Bob

made 1250.00 in salary for the week.

A Few Problem-Solving Strategies

- Look for patterns
- Consider all possibilities
- Make an organized list
- Draw a picture
- Guess and check
- Write an equation
- Construct a table or graph
- Act it out
- Use objects
- Work backward
- Solve a simpler (or similar) problem

Lets Solve!

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Lets SOLVE!

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Quantitative Problem

Solving Skills

Geometric ReasoningStudents will need

- proficiency in basic measurement and geometric

thinking. - to know the basic formulas for calculating the

area of a square or perimeter of a circle. - to know how to apply higher level formulas, such

as those associated with surface area and volume. - to be able to geometrically reason.

Focus on Geometric Reasoning

- Van Hiele Theory
- Level 1 Visualization
- Level 2 Analyze
- Level 3 Informal Deduction
- Level 4 Formal Deduction
- Level 5 Rigor

Visualization

- Recognize and name shapes by appearance
- Do not recognize properties or if they do, do not

use them for sorting or recognition - May not recognize shape in different orientation

(e.g., shape at right not recognized as square)

Can You See It?

- Object will be shown for 3 seconds.
- For each image, what did you notice the first

time you saw the shape? - What features were in your first pictures?
- What did you miss when you first saw each shape?
- How did you revise your pictures?

Visualization

Visualization

Visualization

Implications for Instruction - Visualization

- Provide activities that have students sort

shapes, identify and describe shapes (e.g., Venn

diagrams) - Have students use manipulatives
- Build and draw shapes
- Put together and take apart shapes
- Make sure students see shapes in different

orientations - Make sure students see different sizes of each

shape

Analysis

- Can identify some properties of shapes
- Use appropriate vocabulary
- Cannot explain relationship between shape and

properties (e.g., why is second shape not a

rectangle?)

Analysis

- Description 1
- The design looks like a bird with
- a hexagon body
- a square for the head
- triangles for the beak and tail and
- triangles for the feet.

Analysis Activity

- Work in pairs to construct the figure with the

provided colored shapes. - One person is given the picture and the other

person is given the actual colored shapes. - The person with the picture must describe to the

person with the shape how to construct the

figure. - Time limit will be 5 minutes.

Analysis

- Description 2

Implications for Instruction - Analysis

- Work with manipulatives
- Define properties, make measurements, and look

for patterns - Explore what happens if a measurement or property

is changed - Discuss what defines a shape
- Use activities that emphasize classes of shapes

and their properties - Classify shapes based on lists of properties

Informal Deduction

- Can see relationships of properties within shapes
- Can recognize interrelationships among shapes or

classes of shapes (e.g., where does a rhombus fit

among all quadrilaterals?) - Can follow informal proofs (e.g., every square is

a rhombus because all sides are congruent)

Deduction

- Usually not reached before high school maybe not

until college - Can construct proofs
- Understand the importance of deduction
- Understand how postulates, axioms, and

definitions are used in proofs

What do you think?

- Is it possible to draw a quadrilateral that has

exactly 2 right angles and no parallel lines? - Try it. While youre working, ask yourself . . .
- What happens if?
- What did that action tell me?
- What will be the next step?

(No Transcript)

Lets SOLVE!

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Algebraic Reasoning Skills

Algebraic Thinking in Adult Educationby Myrna

Manly and Lynda GinsburgSeptember 2010

Algebraic Thinking in Adult Education

- Create opportunities for algebraic thinking as a

part of regular instruction - Integrate elements of algebraic thinking into

arithmetic instruction - Acquiring symbolic language
- Recognizing patterns and making generalizations
- Reorganize formal algebra instruction to

emphasize its applications

Adapted from National Institute for Literacy,

Algebraic Thinking in Adult Education,

Washington, DC 20006

Lets SOLVE One More Time!

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Some Big Ideas in Algebra

- Variable
- Symbolic Notation
- Equality
- Ratio and Proportion
- Pattern Generalization
- Equations and Inequalities
- Multiple Representations of Functions

Symbolic Notation

- A Few Examples

Sign Arithmetic Algebra

(equal) . . . And the answer is Equivalence between two quantities

Addition operation Positive number

- Subtraction operation Negative number

Which Is Larger?

23 or 32 34 or 43 62 or 26 89 or 98

Patterns Thinking Algebraically

- Finding patterns
- Describing patterns
- Explaining patterns
- Predicting with patterns

Tiling Garden Beds

Research-Based Teaching Strategies

- Effective questioning
- Teacher responses
- Use of manipulatives
- Conceptually-based teaching

Effective Questions Techniques

- Ask challenging, well-crafted,

open-ended questions, such as - What would happen if . . . ?
- What would have to happen for . . .?
- What happens when . . . ?
- How could you . . . ?
- Can you explain why you decided . . .?

Teacher Responses

- Phrases to Use
- Im not sure I understand, could you show me an

example of ... ? - What do you think the next step should be?
- Where would you use ... ?
- Could ____ be an answer?
- How do you know you are correct?
- Phrases to Avoid
- Let me show you how to do this.
- Thats not correct.
- Im not sure you want to do that.

Math journals help students to . . .

- Be aware of what they do and do

not know - Make use of prior knowledge
- Identify their mathematical questions
- Develop their ability to problem solve
- Monitor their own progress
- Make connections
- Communicate more precisely

Algebra Manipulatives (the C of CRA)

- Students with access to virtual manipulatives

achieved higher gains than those students taught

without manipulatives. - Students using hands-on and manipulatives were

able to explain the how and why of algebraic

problem solving.

Make It Real!

Mathematics is like a video game If you just

sit and watch, Youre wasting your time.

Teaching Beyond the Facts

- Trying to teach in the 21st century without

conceptual schema for knowledge is like trying to

build a house without a blueprint. - H. Lynn Erickson
- Concept-Based Curriculum and Instruction

Conceptual Teaching

- What is conceptual teaching?
- Using schema to organize new knowledge
- Developing units around concepts to help students

learn - Providing schema based on students prior

knowledge or experiences - Teaching knowledge/skill/concept in context
- What its not!
- Worksheets
- Drill
- Memorization of discrete facts

MICROLAB protocol review

My Teaching Reflections

- One secret I have about teaching algebra is . .

. - My worst experience with teaching algebra was

when . . . - My best experience with teaching algebra was when

. . .

Best Practices Review

Instructional Element Recommended Practices

Curriculum Design Ensure mathematics curriculum is based on challenging content Ensure curriculum is standards based Clearly identify skills, concepts and knowledge to be mastered Ensure that the mathematics curriculum is vertically and horizontally articulated

Professional Development for Teachers Provide professional development which focuses on Knowing/understanding standards Using standards as a basis for instructional planning Teaching using best practices Multiple approaches to assessment Develop/provide instructional support materials such as curriculum maps and pacing guides and provide math coaches

Technology Provide professional development on the use of instructional technology tools Provide student access to a variety of technology tools Integrate the use of technology across all mathematics curricula and courses

Manipulatives Use manipulatives to develop understanding of mathematical concepts Use manipulatives to demonstrate word problems Ensure use of manipulatives is aligned with underlying math concepts

Instructional Strategies Focus lessons on specific concept/skills that are standards based Differentiate instruction through flexible grouping, individualizing lessons, compacting, using tiered assignments, and varying question levels Ensure that instructional activities are learner-centered and emphasize inquiry/problem-solving Use experience and prior knowledge as a basis for building new knowledge Use cooperative learning strategies and make real life connections Use scaffolding to make connections to concepts, procedures and understanding Ask probing questions which require students to justify their responses Emphasize the development of basic computational skills

Assessment Ensure assessment strategies are aligned with standards/concepts being taught Evaluate both student progress/performance and teacher effectiveness Utilize student self-monitoring techniques Provide guided practice with feedback Conduct error analyses of student work Utilize both traditional and alternative assessment strategies Ensure the inclusion of diagnostic, formative and summative strategies Increase use of open-ended assessment techniques

- Curriculum Design
- Professional Development
- Technology
- Manipulatives
- Instructional Strategies
- Assessment

Real-World Math

- The Futures Channel
- http//www.thefutureschannel.com/algebra/algebra_r

eal_world_movies.php - Real-World Math
- http//www.realworldmath.org/
- Get the Math
- http//www.thirteen.org/get-the-math/
- Math in the News http//www.media4math.com/MathInT

heNews.asp

- High achievement always occurs in the framework

of high expectation. - Charles F. Kettering (1876-1958)

Questions, Insights, Suggestions

Presenters Leah Felcher Trainer/Consultant lfelch

er_at_tcsg.edu Elaine Shapow Trainer/Consultant esh

apow_at_comcast.net

This workshop developed courtesy of GED Testing

Service and the TCSG Adult Education office.