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[1] Problem Solving HeuristicsThere is wide spread agreement on the importance of students developing effective problem solving skills. However, there is considerably less agreement on what these skills are and even less agreement on how to teach problem solving. The purpose of this article is to introduce or reintroduce readers to a few of George Polya's heuristics on problem solving and to encourage readers to pursue the subject further by reading his two volume set entitled Mathematical Discovery, On Understanding, Learning, and Teaching Problem Solving, (1962) published by John Wiley and Sons, New York.
Polya writes in the Preface to volume I "Solving problems is a practical art,
like swimming or skiing, or playing the piano: you can learn it only by
imitation and practice. If you wish to learn swimming you have to go into the
water, and if you wish to become a problem solver you have to solve
problems."
Reading a problem to clearly identify the three basic parts is an important,
although often overlooked heuristic. These parts are: Data---the information given in the problem, Unknown Variables---representing what the problem is asking for, and
Conditions or Constraints---that link the data to the
unknowns.
Example. The label on a half-pint carton of 2% milk claims 38% less fat than
whole milk. If a half-pint serving of 2% milk contains 5 grams of fat, how
many grams of fat are there in a half-pint serving of whole milk? Data: half-pint of 2% milk contains
5 grams of fat. Unknown Variable: x = the number of grams of fat in a half-pint serving
of whole milk. Condition: 2% milk has 38% less fat
than whole milk.
Model: Translate the condition into an equation by expressing the central idea
in two different ways and then setting the two ways equal to each other. In
this example, express the amount of fat in a half-pint of 2% milk in two
different ways--- 5 grams and
(
A useful insight to forming an equation model is given by noting that the two
sides of an equation represent the same thing. Thus to form an equation, find
two different ways of representing the same quantity and then set them equal
to each other.
Successive approximations is another useful heuristic. Although
sometimes called "guess and check" by students, forming a successive
approximation involves much more than guessing. The process involves making an
initial attempt, noting the resulting errors, modifying the attempt to reduce
the errors, and then trying again. This procedure is repeated until a
satisfactory solution is obtained. Graphically fitting a curve to a scatter
plot (see [6]) is a good example of the use of successive
approximations.
Polya'a Wishful Thinking heuristic is to assume the problem has been
solved and work backwards. The nature or form of a solution often provides
insights into what is needed to solve the problem.
Example. Determine the intersection of the two planes,
For more, read
Polya. [2] Read the Text
The following is a quotation from Isaac Newton describing how he learned
mathematics. The quote and its message about how to learn is worth sharing
with students.
"Took Descartes's Geometry in hand, thro he had been told it would be very
difficult, read some ten pages in it, then stopt, began again, went a little
further than the first time, stopt again, went back again to the beginning,
read on till by degrees he made himself master of the whole, to that degree
that he understood Descartes's Geometry better than he had done Euclid."
Source: The Mathematical Papers of Isaac Newton, vol 1,
p.5-6
One should remember that Newton was in the smart group. Everyone should expect
to have to read their text more than once, it is a proven way of learning. [3] Optimization Problems
The technology of the graphing calculator or a computer algebra system has
moved optimization problems from the realm of calculus to the realm of college
algebra. The following four optimization problems provide challenging problem
solving scenarios of increasing difficulty for college algebra
students.
Most shops involved in cutting smaller pieces of material from larger pieces,
such as carpentry or upholstery or metal working, have a scrap pile, a
collection of leftover pieces. Because the materials in these scrap piles have
already been "paid for," money can usually be saved by using a piece from
the scrap pile rather than cutting into new stock. Since many times the
craftsman starts with a rectangular shape piece of material, the value of a
scrap piece could be determined by the largest rectangle contained in the
piece. However, determining the size of the largest rectangle in a piece of
scrap can be a challenging problem as illustrated by the following
problems.
1. Consider a region bounded by the graphs of
2. Consider a region bounded by the graphs of
3. Consider a region bounded by the graphs of
4. Consider a region bounded by the graphs of
[4] Queries
1. (Submitted by Shane Brewer of Eastern Utah State College) Highway engineers
use the term grade to describe the slope or steepness of a road.
Highway signs alerting motorists of a steep hill ahead usually express the
grade in terms of percentages. For example, an 8% grade would indicate that
the road rises 8 feet, over a 100 foot horizontal distance. Clearly a 10%
grade indicates a steeper hill than does a 6% grade. If you saw a sign
"Warning: 100% Grade Ahead," what would you expect? 2. Think about blowing up a (spherical) beach ball. In particular, think about how the radius of the ball increases as the volume increases. Consider the three curves in the following multiplot. Do any of them approximate the shape of the radius as a function of the volume or is the radius curve completely different? Write a few sentences describing your thought process and how it led you to an answer.
[5] Recognizing Plots
In the following set of plots, A is the plot of
[6] The Wind Doth Blow
The cost of generating electricity from wind has decreased dramatically over
the past 20 years primarily due to (a) improved turbine design, (2) taller
towers to reach higher wind speeds (potential wind energy is proportional to
the cube of wind speed), (3) longer blade length (the swept area is
proportional to the square of the blade length). The American Wind Energy
Association predicts the proportion of U.S. electricity produced from wind
will grow from one percent today to 20 percent by 2020. Plot the following data and graphically fit a curve to the scatter plot.
[7] Notices
* Supported by the National Science Foundation and the U.S. Military Academy. |
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