First Year Seminar
Problem Solving Through Recreational Mathematics
Polya Study Questions
pp. 1-22
What are the four phases of solving a problem? Briefly describe each phase.
pp. 37-59
What does analogy mean?
Polya claims that "two questions are better than one." What does he mean?
What is inference by analogy?
What is an auxiliary element? Give an example.
Polya writes: "...there is nothing to learn about reasoning and invention
if the motive and purpose of the most conspicuous step remain incomprehensible." (p. 50)
In regards to presenting mathematics, what can we learn from this?
What is an auxiliary problem? What is its role?
What are bilateral and unilateral reduction?
What is a bright idea? Give an example, either from the book or from your own experience.
pp. 59-85
According to Polya, why is it useful to check your argument in a new way?
If the proof of a result brings certainty, then why is it better to have two proofs than one?
How do devising a plan and carrying out the plan differ? Give an example.
What is a "Euclidean argument"? What purpose does it serve? What is Polya's critique of this
form of argument?
Compare "intuitive insight" and "formal proof". How do they differ? How do they work together?
pp. 85-98
Polya says that a mathematical definition "creates" the mathematical meaning of a term. What does he mean? What is an example?
What does it mean to "deflate a problem"?
What does Polya mean by "going back to the definitions"?
Polya claims that "determination and emotions" play an important
role in problem solving. What is his point?
In the section Did You Use All the Data, Polya makes a distinction between "practical" problems and "perfectly stated" problems. What is their difference and what bearing does it have on solving
problems?
pp. 98-121
Polya warns us not to let our suspicions "grow without examination until it becomes ineradicable."
What does this mean? Give an example.
Polya quotes Pappus of Alexandria: "Assume what is required to be done as already done."
Explain this.
Why does Polya say that sometimes the "more general problem may be easier to solve"? Give
an example, either from the book or from your own experience, mathematical or otherwise.
What is heuristic reasoning? What role does it play in mathematics?
What is the role of induction in mathematics? How is induction different from "mathematical induction"?
pp. 121-141
What is a lemma?
What is "modern heuristic"?
Polya asserts that "notation is not only easily recognizable but particularly helpful
in shaping our conception when the order and connection of the signs suggest the order
and connection of the objects." Explain this by way of an example.
pp. 141-160
According to Pappus, what are the processes of "analysis" and "synthesis"? Illustrate these processes with an example, mathematical or otherwise.
What is meant by the phrase, "assume the problem as solved"?
What are some of the key differences between practical and mathematical problems? Give an example.
How are "problems to find" different from "problems to prove". How do they differ in their
principle parts? Give some examples.
pp. 160-177
What are reductio ad absurdum and the indirect proof? Give examples of each.
What is Polya's criticism of "routine problems"?
What are the rules of discovery the rules of style and the rules of teaching?
Polya claims that setting up equations is like a problem in "translation". What does he mean
by this?
Explain the method of translation in the following problem: find the dimensions of a rectangle
whose perimeter is 33 and whose area is 38.
pp. 178-197
What is a "heuristic suggestion"? What is its role in problem solving?
Compare and contrast "plausible reasoning" and "demonstrative reasoning". How
are they alike? How are they different?
What is "specialization" in mathematics. Give an example.
What is a "counterexample"? Give of an example of how a counterexample is used.
pp. 197-215
What is the meaning of the saying, "take counsel of your pillow?"
What is a "test by dimension"? How is a test by dimension useful?
Find the surface area of a cylinder with height h and radius r
and apply a test by dimension.
What does Polya mean by "variation of the problem"? Why is this important?
Give an example from the text or from your own experience.