Mathematical Reasoning

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Mathematical Reasoning Presenters Leah Felcher
lfelcher_at_tcsg.edu Elaine Shapow
eshapow_at_comcast.net
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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

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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)

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Standards-Driven Curriculum
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Design and Organization
Domain
Cluster
Standard
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Assessment Target Standards

• Think, Pair, Share

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Mathematical ReasoningNew 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

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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

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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

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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

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New Mathematical Tools
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TI-30XS MultiView Calculator
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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
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Teach Big Ideas!
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What are the big ideas that I want students to
remember . . .
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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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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.

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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!

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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!
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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 Bobs 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

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
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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
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.
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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

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Lets Solve!

• S
• O
• L
• V
• E

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Lets SOLVE!

• S
• O
• L
• V
• E

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Quantitative Problem
Solving Skills
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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.

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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

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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)

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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?

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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

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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?)

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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.

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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.

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Analysis

• Description 2

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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

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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)

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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

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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?

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Lets SOLVE!

• S
• O
• L
• V
• E

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Algebraic Reasoning Skills
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Algebraic Thinking in Adult Educationby Myrna
Manly and Lynda GinsburgSeptember 2010
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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
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Lets SOLVE One More Time!

• S
• O
• L
• V
• E

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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

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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
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Which Is Larger?
23 or 32 34 or 43 62 or 26 89 or 98
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Patterns Thinking Algebraically

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

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Tiling Garden Beds
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Research-Based Teaching Strategies

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

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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 . . .?

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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.

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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

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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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Make It Real!
Mathematics is like a video game If you just
sit and watch, Youre wasting your time.
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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

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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 its not!
• Worksheets
• Drill
• Memorization of discrete facts

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MICROLAB protocol review
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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
  . . .

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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

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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

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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 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.