Doing Math Critical Thinking: Fostering Students Mathematical Thinking

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Doing Math Critical Thinking Fostering
Students Mathematical Thinking

• Dana Pomykal Franz
• Associate Professor

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Focus in High School MathematicsReasoning and
Sense Making "Reasoning and sense making
should be a part of the mathematicsclassroom
every day. NCTM, 2009
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NCTM Problem Solving Principle

• Instructional programs from prekindergarten
  through grade 12 should enable all students to
• build new mathematical knowledge through problem
  solving
• solve problems that arise in mathematics and in
  other contexts
• apply and adapt a variety of appropriate
  strategies to solve problems
• monitor and reflect on the process of
  mathematical problem solving.

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NCTM Communication Principle

• Instructional programs from prekindergarten
  through grade 12 should enable all students to
• organize and consolidate their mathematical
  thinking through communication
• communicate their mathematical thinking
  coherently and clearly to peers, teachers, and
  others
• analyze and evaluate the mathematical thinking
  and strategies of others
• use the language of mathematics to express
  mathematical ideas precisely.

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Types of Mathematics TasksStein, Smith,
Henningsen, Silver, 2009
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Depth of Knowledge

• Level 1-Recall
• Level 2-Basic Application of Skill/Concept
• Level 3-Strategic Thinking
• Level 4-Extended Thinking

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Do They Match?
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Traditional Problem

• Nicoles Carpeting Task
• Nicole was redecorating her house. She has
  decided to recarpet her bedroom, which is 15
  feet long and 10 feet wide. How many square feet
  of carpeting will she need to purchase?

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Advanced Problem

• Nicoles Carpeting Problem
• Nicole wants to redecorate her bedroom. She
  decides to recarpet. If her room is 5 feet
  longer than it is wide, write an equation to
  represent the area of her room. If you know her
  room is 10 feet wide, how many square feet of
  carpet will she need? If the carpet is sold by
  the square yard, how many square yards will she
  need?

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Doing Mathematics

• The Fencing Problem
• Coach Tates class will raise rabbits for use in
  their science fair project (no rabbits will be
  hurt). They have 32 feet of fencing with which to
  build a rectangular rabbit pen to keep the
  rabbits.
• If Coach Tates students want their rabbits to
  have as much room as possible, how long would
  each of the sides of the pen be?
• How long would each of the sides of the pen be if
  they had only 16 feet?
• How would you go about determining the pen with
  the most room for any amount of fencing?
  Organize your work so that someone else who reads
  it will understand it.
• Stein, et. al.

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Lets Compare

• Traditional/
• Advanced

• Doing Mathematics

• DOK Level(s)?
• Mathematical Tasks
• Mathematical Requirements
• Challenges
• Student Characteristics

• DOK Level(s)?
• Mathematical Tasks
• Mathematical Requirements
• Challenges
• Student Characteristics

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Challenging Students to Do Math

• Model thinking doing for students
• Set High Expectations for completing problems
• Expect all students to participate
• Create a climate where risk-taking is encouraged
  honored
• Honor ALL student thinking
• Anticipate multiple solutions understand the
  mathematics you are asking for
• Become a facilitator not director

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Another Problem

• Beams are designed as supports for various types
  of bridges. The beams are constructed using
  rods. The number of rods used to construct the
  bottom of the beam determines the length of the
  beam. A beam of length 4

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Cont.

• How many rods are needed to make a beam of length
  5? Of length 8? Of length 20?
• How many rods are needed for a beam of length
  223?
• Write a rule or formula for finding the number of
  rods needed to make a beam of any length.
  Explain your rule or formula.
• Adapted from Townsend, Lannin, Barker

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Your Turn

• Traditional Form In a game, a players scores
  on five successive turns were 8, -11, 7, -7,
  6. After which two turns was the players total
  score the same? How many points were scored
  altogether during those five turns?
• (Houghton Mifflin 2002, pg. 220).

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One Suggested Problem

• You have a set of integer cards from -9 to 9 in
  a bag, a six-sided die, and a set of plus and
  minus (/-) operation cards. Shake the bag and
  draw out 8 integer cards. Roll the die. This is
  your target number. Use the 8 drawn integer cards
  and any of the operation cards to make integer
  sentence that use addition and subtraction and
  which total the number on your die. Record your
  sentence. Repeat.
• Kabiri Smith 2003

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Function Problems

• Determine whether the following relationships
  could be descriptions of functions. For each
  relationship that you decide is a function, state
  its input and output variables. If the
  relationship is not a function, explain why not.
• 1. The volume of a cube varies as the length of
  one of its edges is changed.
• 2. The surface area of a (spherical) balloon
  changes as you blow air into it.
• 3. To each student enrolled in the course, a
  grade (AF) is assigned.
• 4. The time required for a certain chemical
  reaction is related to the amount of catalyst
  present during the reaction.
• 5. Ms. Spencer decides to record her students
  birthdays, so she asks students whose birthdays
  are in January to raise their hands, then those
  whose birthdays are in February, and so on.
• 6. The measurement of one wall of your classroom
  taken in inches is related to that measurement
• taken in meters.
• 7. Each of your classes is assigned to meet on
  specific days of the week.
• 8. At the end of the course, David recorded the
  names of students who earned As and the
• names of students who earned Bs.
• 9. Given a temperature in degrees Fahrenheit, you
  can determine the temperature in degrees
• Celsius.
• 10. For any popular musician or music group
  today, you can list the albums attributed to
• that group.
• Hartter, 2009

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Questions or Discussion?