Problem Solving - Get Now PowerPoint

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                  Chris Watts
         Chris Watts

 To Kathleen Lewis, Lynn Carlson, Magdalena
    Mosbo, and Mary Harrell for consistently
  listening to and encouraging me through my
            mathematical insecurities.

  To the whole Oswego math department, for
       helping me mature into a growing

• Unchallenged

• Bored

• Misguided

• Terrified
Road to Discovery

• Dissatisfaction with High School

• Abstract Algebra

• Probability / Statistics

• Elementary Problem Solving
The Paper

• 5 Solved Problems

• Research
  • How to Solve It (George Pólya)
  • The Art and Craft of Problem Solving (Paul Zeitz)
Research: How to Solve It

• Written for teachers and students

• Structure

• Two types of problems
  • Problems to find
  • Problems to prove
Research: How to Solve It

• Luck

• Four Steps
  • Understanding the Problem
  • Devising a Plan
  • Executing the Plan
  • Looking Back
Research: How to Solve It

• “The List”
  • “What is the unknown?”
  • “Have you seen a similar problem?”
  • Teachers’ role
Research: “The List”
 Understanding the Problem
      What is the unknown?
      What are the data?
      What is the condition?
      Draw a figure.
      Introduce suitable notation.
 Devising a Plan
      Find a connection between the data and unknown.
      Do you know a related problem?
      Could you restate the problem?
      Solve a related problem.
 Executing the Plan and Looking Back
      Have you checked each step?
      Is it evident that each step is correct?
      Can you prove that each step is correct?
      Can you check the result?
      Can you derive the solution differently?
      Can you use the result or method for some other problem?
Research: The Art and Craft of
Problem Solving (ACPS)
• Types of Problems
  • Recreational
  • Contest
  • Journal
  • Open-Ended

• Exercises and Problems
ACPS: An Analogy
The average (non-problem-solver) math student is like someone
  who goes to a gym three times a week to do lots of repetitions
  with low weights on various exercise machines. In contrast, the
  problem solver goes on a long, hard backpacking trip. Both
  people get stronger. The problem solver gets hot, cold, wet,
  tired, and hungry. The problem solver gets lost, and has to find
  his or her way. The problem solver gets blisters. The problem
  solver climbs to the top of mountains, sees hitherto undreamed
  of vistas. The problem solver arrives at places of amazing
  beauty, and experiences ecstasy that is amplified by the effort
  expended to get there. When the problem solver returns home,
  he or she is energized by the adventure, and cannot stop
  gushing about the wonderful experience. Meanwhile, the gym
  rat has gotten steadily stronger, but has not had much fun, and
  has little to share with others (page x).
Research: ACPS

• Problem solving is learned.
      • History

• There is no wrong path.

• It has a definite structure.
      • Strategies
      • Tactics
      • Tools
      Strategies                Tactics                 Tools
   Bend the Rules*         Extreme Principle*          Factor*
  Penultimate Step        Pigeonhole Principle   Add Zero Creatively
Get Your Hands Dirty*          Invariants           Invent a Font
 Restate the Problem          Symmetry*               AM-GM
Obtain Partial Solution      Substitution*       Massage Inequalities
Change Point of View      Modular Arithmetic*     Telescoping Series
Research: ACPS (Uniqueness)
• Emphasis should be placed more on exploration
  than presentation.

• Problem solving involves more than intelligence.
  •   There is always some luck involved.
  •   There must be a genuine, deep-rooted interest.
  •   Great thinkers must have mental toughness.
  •   Positive thinking is necessary for clear thinking.
  •   Fostering a constructive atmosphere is critical.
  •   Education is good iff it promotes exploration.

• Problem solving is fun.
The Paper

• Research
  • How to Solve It (George Pólya)
  • The Art and Craft of Problem Solving (Paul Zeitz)

• 5 Solved Problems
  • Contest and Journal Problems
  • Domestic and International Contests
  • Investigation and Reflections

1. Let k ≥ 1 be an integer. Show there are
   exactly 3k-1 integers n such that:
      n has k digits,
      all of the digits are odd,
      n is divisible by 5, and
      m = n/5 has k odd digits.

     Austrian-Polish Mathematics Competition 1996

2. We call an integer m “retrievable” if for
  some integers x and y, m = 3x2 + 4y2.
  Show that if m is retrievable, then 13m is

                            AMTNYS, Jan. 2007
3. At ABC University, the mascot does as many
   pushups after each ABCU score as the team has
   accumulated. The team always makes extra points
   after touchdowns, so it scores only in increments
   of 3 and 7. For each sequence a1, a2, …, an where
   each ak = 3 or 7, let P(a1, a2, …, an) denote the total
   number of pushups the mascot does for the
   scoring sequence a1, a2, …, an. For example, P(3,7,3)
   = 3 + (3 + 7) + (3 + 7 + 3) = 26. Call a positive
   integer k accessible if there is a scoring sequence
   a1, a2, …, an such that P(a1, a2, …, an) = k. Is there a
   number K such that for all t ≥ K, t is accessible? If
   not, prove it, and if so, find K.
                                Pi Mu Epsilon, Spring 2007

4. Players 1, 2, 3, …, n are seated around a table
   and each has a single penny. Player 1 passes a
   penny to Player 2, who then passes two pennies
   to Player 3. Player 3 then passes one penny to
   Player 4, who passes two pennies to Player 5,
   and so on, players alternately passing one penny
   or two to the next player who still has some
   pennies. A player who runs out of pennies drops
   out of the game and leaves the table. Find an
   infinite set of numbers n for which some player
   ends up with all n pennies.
                                      Putnam, 1997

5. What is the expected length of a
  standard NHL shootout where the
  probability of each shooter scoring a goal
  is 1/3?

                              AMTNYS: Jan ’07
Looking Back

As a student…
  • Math is about exploring problems.
  • “School math” is necessary for “real math.”
  • Problem solving is not random.
  • Problem solving is developable.
  • A positive attitude promotes clear thinking.
Reflections (Looking Back)

As a prospective teacher…
  • Finding problems genuine to students is key.
  • Teachers must model effective strategies.
  • Students should learn how mathematics’ great
    thinkers approached problems historically.
  • Struggling through problems helps teachers
    empathize with struggling students.
  • A positive attitude promotes clear thinking.
So, what now? I will….
•   …explicitly teach the tools and tactics and model the self-
    questioning techniques learned from my research.

•   …develop a repertoire of creative problems to assign for
    extra credit, in addition to more innovative homework.

•   …encourage students to investigate problems genuine to

•   …foster an environment of creativity and risk-taking.

•   …incorporate the mathematical history of content into

•   …continue to solve problems and work independently.