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Topic: Re: Mathematics in Everyday Situations. LESH (1985)
Replies: 9   Last Post: Aug 23, 1994 3:12 PM

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Sarah Seastone

Posts: 21
Registered: 12/3/04
MATHEMATICS AND LEARNING: POLYA
Posted: Aug 16, 1994 9:06 AM
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This is the third in a series of discussions intended to provide an
informative and sometimes provocative overview of the thinking of some key
researchers in mathematics education and educational psychology.

COMMON-SENSE QUESTIONS

POLYA (1957)

Polya describes a four-stage approach to mathematical problem-solving. He
bases his approach on common-sense questions that would naturally occur to
an experienced problem-solver. Polya claims teachers should pose these
questions to students in as natural and unobtrusive a way as possible, the
goal being to encourage independence and internalization of this
framework.

Chapter:

Polya, G. (1957). In the classroom. In Polya, G., How to solve it (pp.
1-32). Garden City, NY: Doubleday and Company, Inc.

Overview:

o Effective problem-solving consists of four main phases: understanding
the problem, devising a plan, carrying out the plan, and looking back.
Understanding the problem includes labeling and identifying unknowns,
condition(s), and data, and determining the solubility of the problem.
Devising a plan means drawing on prior knowledge to frame an appropriate
technique, restating the problem if necessary. The chosen technique is
then carried out in the third stage. The final phase consists of checking
the correctness of the solution and adding the problem to one's store of
knowledge for use in solving future problems.

o The teacher prompts the students in applying these phases with the use
of key suggestions and questions.

o An important characteristic of Polya's framework is generality: it can
and should be applied to many different types of problems.

o Another key is practice. This framework should be applied often and in
many different situations so that students can gain experience and
confidence in problem-solving.


Direct Quotes (and some comments):

o "There are two aims which the teacher may have in view when addressing
to his students a question or a suggestion...: First, to help the student
to solve the problem at hand. Second, to develop the student's ability so
that he may solve future problems by himself" (p. 3-4).

o "If the same question is repeatedly helpful, the student will scarcely
fail to notice it and he will be induced to ask the question by himself in
a similar situation. Asking the question repeatedly, he may succeed once
in elicting the right idea. By such a success, he discovers the right way
of using the question, and then he has really assimilated it" (p. 4).

o "[Appropriate questions and suggestions] have two common
characteristics, common sense and generality. As they proceed from plain
common sense they very often come naturally; they could have occurred to
the student himself. As they are general, they help unobtrusively; they
just indicate a general direction and leave plenty for the student to do"
(p. 4).

o Regarding the inadvisability of such specific questions as "Could you
use the Pythagorean Theorem?" -- a possible question when trying to find
the length of the diagonal of a rectangular parallelepiped -- Polya says:


"The intention may be the best, but the question is about the worst....
[T]here is a long sequence of objections against that sort of help:

(1) If the student is near to the solution, he may understand the
suggestion implied by the question; but if he is not, he quite possibly
will not see at all the point at which the question is driving. Thus the
question fails to help where help is most needed.

(2) If the suggestion is understood, it gives the whole secret away,
very little remains for the student to do.

(3) The suggestion is of too special a nature. Even if the student can
make use of it in solving the present problem, nothing is learned for
future problems. The question is not instructive.

(4) Even if he understands the suggestion, the student can scarcely
understand how the teacher came to the idea of putting such a question.
And how could...the student...find such a question by himself? It appears
as an unnatural surprise, as a rabbit pulled out of a hat; it is really
not instructive" (p. 22).

o "Begin with a general question or suggestion..., and, if necessary, come
down gradually to more specific and concrete questions or suggestions
till you reach one which elicits a response in the student's mind" (p.
20).

o "A good teacher should understand and impress on his students the view
that no problem whatever is completely exhausted. There remains always
something to do; with sufficient study and penetration, we could improve
any solution, and, in any case, we can always improve our understanding of
the solution" (p. 15).

o "One of the first and foremost duties of the teacher is not to give his
students the impression that mathematical problems have little connection
with each other, and no connection at all with anything else....The
teacher should encourage the students to imagine cases in which they could
utilize again the procedure used, or apply the result obtained" (p.
15-16). [Polya uses the example of finding the measurement of the
diagonal of a parallelepiped. From that problem he poses the related,
real-life problem of determining the length of several wires needed to
secure an erect flagpole].

-- Summarized by Andrea Hall





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