Transcript Problem Solving

PROBLEM SOLVING
Chris Watts
DEFENSE OF THE
THESIS 2K8
Chris Watts
Acknowledgement
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
mathematician.
Motivation
• 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
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What is the unknown?
What are the data?
What is the condition?
Draw a figure.
Introduce suitable notation.
 Devising a Plan
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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
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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?
http://www.geocities.com/polyapower/TheList.html
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.
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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
Problems
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
Problems
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
retrievable.
AMTNYS, Jan. 2007
Problems
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
Problems
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
Problems
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….
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…explicitly teach the tools and tactics and model the selfquestioning techniques learned from my research.
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…develop a repertoire of creative problems to assign for
extra credit, in addition to more innovative homework.
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…encourage students to investigate problems genuine to
them.
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…foster an environment of creativity and risk-taking.
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…incorporate the mathematical history of content into
lessons.
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…continue to solve problems and work independently.