College Career

Ready Ready

Mathematical Reasoning

Presenters

Leah Felcher – [email protected]

Elaine Shapow – [email protected]

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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 2

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) 3

Standards-Driven Curriculum

Standards/ Practices Student Achievement Classroom Instruction 4

Design and Organization

Domain Cluster Standard 5

Assessment Target Standards

• Think, Pair, Share

Mathematical Reasoning New Realities

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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 8

What’s 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 9

What’s 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 10

What’s 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 11

New Mathematical Tools

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TI-30XS MultiView Calculator

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It’s Your Turn!

"Calculators can only calculate - they cannot do mathematics." -- John A. Van de Walle http://education.ti.com/en/us/products/calcul ators/scientific-calculators/ti-30xs multiview/classroom-activities/activities exchange 14

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What are the big ideas that I want students to remember . . .

40 days (the test) 40 months (college) 40 years (life) 16

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?

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Let’s SOLVE a Math Problem

Even Albert Einstein said:

“Do not worry about your difficulties in Mathematics. I can assure you mine are still greater.”

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SOLVE a Problem

S O

tudy the problem (What am I trying to find?) 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!

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S = Study the problem

What is the problem asking me to do?

Find the question.

We are going to practice SOLVE with this one!

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?

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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?

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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 Bob’s total weekly salary, I will add the total amount earned for his regular salary plus his overtime salary.

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V = Verify Your Plan

$20.00

x 40 $800.00

Regular Wages $ 800.00

+ 450.00

$1250.00

$30.00

x 15 $450.00

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E = Examine the Results

(Is it reasonable? Does it make sense? Is it accurate?) $1250.00 IS reasonable because it is more than Bob’s 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.

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• • • • • • • • • • •

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 25

L V S O E Let’s Solve!

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L V S O E Let’s SOLVE!

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Quantitative Problem Solving Skills

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Geometric Reasoning Students 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.

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Focus on Geometric Reasoning

• • • • • Van Hiele Theory Level 1: Visualization Level 2: Analyze Level 3: Informal Deduction Level 4: Formal Deduction Level 5: Rigor 30

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) 31

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

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Visualization

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Visualization

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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 36

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?) 37

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. 38

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.

Description 2

Analysis

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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 41

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) 42

•

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 A A D B E C F 43

What do you think?

• Is it possible to draw a quadrilateral that has exactly 2 right angles and no parallel lines? • Try it. While you’re working, ask yourself . . . – What happens if…?

– What did that action tell me?

– What will be the next step?

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L V S O E Let’s SOLVE!

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Algebraic Reasoning Skills

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Algebraic Thinking in Adult Education by Myrna Manly and Lynda Ginsburg September 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 49

L V S O E Let’s 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 51

Symbolic Notation

A Few Examples

Sign

= (equal)

Arithmetic

. . . And the answer is + Addition operation Subtraction operation

Algebra

Equivalence between two quantities Positive number Negative number 52

Which Is Larger?

2 3 or 3 2 3 4 or 4 3 6 2 or 2 6 8 9 or 9 8 53

Patterns – Thinking Algebraically

• Finding patterns • Describing patterns • Explaining patterns • Predicting with patterns 54

Tiling Garden Beds

Research-Based Teaching Strategies

• Effective questioning • Teacher responses • Use of manipulatives • Conceptually-based teaching 56

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 . . .? 57

•

Teacher Responses

Phrases to Use – I’m 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. – That’s not correct. – I’m not sure you want to do that. 58

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 59

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.

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Mathematics is like a video game; If you just sit and watch, You’re wasting your time.

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Teaching Beyond the Facts “Trying to teach in the 21 st century without conceptual schema for knowledge is like trying to build a house without a blueprint.”

H. Lynn Erickson Concept-Based Curriculum and Instruction

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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 it’s not!

• Worksheets • Drill • Memorization of discrete facts 63

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 . . . 65

Instructional Element Curriculum Design Professional Development for Teachers

     o o o o 

Technology Manipulatives Instructional Strategies

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Assessment

             

Recommended Practices

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

Best Practices Review

• • • • • • Curriculum Design Professional Development Technology Manipulatives Instructional Strategies Assessment 66

Real-World Math

The Futures Channel http://www.thefutureschannel.com/algebra/algebra_real_world_movies.p

hp 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/MathInTheNews.asp

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“High achievement always occurs in the framework of high expectation.”

Charles F. Kettering (1876-1958)

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QUESTIONS, INSIGHTS, SUGGESTIONS

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Presenters Leah Felcher

Trainer/Consultant [email protected]

Elaine Shapow

Trainer/Consultant [email protected]

This workshop developed courtesy of GED Testing Service ® TCSG Adult Education office.

and the

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