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### Problem Solving

### 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
- 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?
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 There is no wrong path. It has a definite structure.

- Problem solving is learned.
- History

- Strategies
- Tactics
- Tools

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

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….

- …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 them.
- …foster an environment of creativity and risk-taking.
- …incorporate the mathematical history of content into lessons.
- …continue to solve problems and work independently.

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