creative problems

CREATIVE PROBLEMS
CREATIVE PROBLEMS
(LOGICAL vTHINKING)
PROBLEM SOLVING
IS:
WHAT YOU DO, WHEN YOU DON'T KNOW WHAT TO DO!
This assumes you do something. You must recognize that there is a problem and
be motivated to solve it. A good problem solver ~ght take a break, but seldom
qm~.
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Problem solving is a LIFE SKILL. It should be discussed and applied across the
curriculum, and developed as an essential life skill.
Life is filled with problems which are becoming very sophisticated. They could be
moral, physical, focus on environmentalissues, etc. Limiting problem solving to
mathematics in the decades ahead is unacceptable.
.
For a more extensive development of the teaching of problem solving, and over
200 problems/puzzles categorized by useful strategies, see: "Problem Solving,
What YouDo When YouDon't Know What to Do", Catalogue #0049 Exclusive
Educational Produc~.
The followmg pages contain a brief introduction to the process developed in that
book, as well as some thoughts on evaluating problem solving.
That is followed by some favourite problems from that book as well as many new
ones.
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SOLVING A PROBLEM'
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/:>
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When solving a problem we usually follow 4 very logical steps, based on the work
of George Polya(2). These steps do not need to be memorized. They may be
.
-.
posted for older children, but probing questions from peers, teachers or parents
will maintain the natural flow of the steps.
1. Understand the Problem"
2. Choose a Strategy
3. Solve the Problem
4. Think -About It
For young students:
1. Tell me about your problem.
2. What could we do?
3. Let's try it.
4. What do you think?
SUGGESTIONS FOR EACH STEP: .
1. Understand the Problem'
.Say or write the problem in your own words.
.Pretend YOU are in that problem.
. Look up words. in the dictionary.
.Underline important words.
.Discuss the problem with a friend.
.Ask questions.
.Sometimes making a drawing helps.
Hint: If a student leaves the page blank because he/she does not understand the
problem, INSIST that the reason it was not understood be written down. Ask
them to write what they DO know as well. This helps both students arid teachers.
This is a difficult task at first, but it will prove to be beneficial.
(2)
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G. Polya; How To Solve It; Princeton, N.J. Princeton University Press, 1945.
WORLD'SMOSTPOPULARPUZZLES&.PROBLEMS
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CREATIVEPROBLEMS
2. Choose a Strategy
.
. Good problem solvers know a variety of strategies.
.Take a risk! Try sometlllng.
.If it doesn't work try another strategy.
.Keep a list of strategies you can use,
. Brainstorm
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ideas in a group.
3. Solve. the Problem,
.Work through the problem using the strategy chosen.
'. If you get stuck, try another strategy.
. If you are really stuck you may want to pause and come back to
it another time.
.If you have an answer, think about it.
.'
4. Think About It
. .
.'
.Is it a reasonable answer?
.Are there any other answers?
. Is there a different method to try?
.Have you ever solved a similar problem?
.Make up a similar question. (to reinforce the strategy)
.Change the problem slightly, e.g. "What would happen if...??"
. Discuss why you enjoyed (or disliked) the problem.
)
WORLD'S MOST POPULAR PUZZLES& PROBLEMS
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CREATIVE PROBLEMS
USEFUL STRATEGIES
. COITectanswers are important.
. The process is equally important.
Is the cOITectanswer or the process more likely to be used tomoITow,next week or
next year?
THE PROCESS OR STRATEGY!
Students must understand the value of the strategy. Too often textbooks tell
students which strategy to usetather than having them decide, e.g. "Problems
Using Trial and Error" is sometimesused as a title, or instructions sometimes say,
"Make a drawing to show how to get your answer." It is possible that the student
did not need the drawing to solve the problem, or may have used another strategy
altogether.
A student must eventuallyunderstand that a strategy has helped solve a problem in
order to understand its value, and therefore, use it again.
Trial and error is probably the most common and intuitive strategy for a child, (e.g.
learning to walk, talk, etc...). School programs should hone rather than discourage
trial and error because it encourages risk taking.
Seldom is one strategy used alone. For example, when we use trial and eITor,it
may be helpful to make a list of the guesses.
The following list of strategies are not sequenced, although some may be more
appropriate for young students than others.
'Trial and Error
This is sometimes called "guess and check". As a life skill it may be the most
frequently used strategy.
Making a Drawing/Sketch it
This strategy frequently helps in understanding a problem, as well as leading to a
solution.
",
Logicalrhinking
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It is important to provide many opportunities for students to consider:
"If I did this, then
would be true."
This may have been a strategy which has been avoided or overlooked with young
students.
,.
Find
"
APattern.
>, ,',',;
,,":,'
Patterns are so common in designs and numbers that we sometimes overlook it as
a problem solving strategy.
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WORLD'S MOST POPULAR PUZZLES &.PROBLEMS
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CREATIVE PROBLEMS
Work Backwards
A valuable strategy ftequently used in life situations-.
Elimination
By ruling out the least obvious choices or answers, one is able to focus on the best
alternatives. Another useful life skill!
.:Make A List. .
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This strategy helps to organize work. Making a table or chart can be considered. a
similar strategy.
Act It OuVMake A Model
Three or four p~ople can often underst~d a problem better and also see possible
solutions if they act it out or make a model of it. Puppets may be used.
Using concrete materialsor models can serve as an imitation of acting it out.
Brainstorming
Brainstorming should become part of a child's vocabulary at an early age. It can
be used to help understand a problem. It can be used to help choose a strategy.
Rules for Brainstorming:
1. Record ALL suggestions.
2. Do not discuss or evaluate suggestions when they are given.
3. Encourage everyone to participate.
4. Limit the time for receiving suggestions to a short period.
Estimation
Make the Number Smaller
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,..
.
'"
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These two strategies are useful for the more routine arithmetic problems/skill
applications.
It is surprising sometimes when doing arithmeticapplications that a child is
stumped when determiningthe cost of one orange, if a crate of 144 cost $56.16.
But that samechilddividesquicklyif thenumbersare rounded off or made
smaller.
. Writea'n'
Equatidn
'
This is an important organizer,particularly since the use of a variable is essential
to algebraic concepts. However, it is of little value to students below Grade 7
because a variable is a difficultconcept to grasp. It also has limited value as a life
skill. Have you ever said to yourself, "I must let 'x' represent my change..", as
you are leaving a check-out counter? Quite unlikely!
1would recommend this strategy beginningwith numerical problems such as,
Twice a number plus 6 equals 20. What is the number? 2x + 6 = 20.
WORLD'S MOST POPULAR PUZZLES & PROBLEMS
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{
CREATIVE
pa~BLEMS)
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EVALUATING PROBLEM SOLVING
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There are a number of ways to evaluate problem solving: OBSERVING,
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QUESTIONING, LISTENING, READING, etc. Evaluation is too frequently,
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done on the end product which often measures only the perfOlmance(or lack of it) ,
If your objectives incorporate problem solving,then the actions of the student
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when stuck,' must be evaluated. How does.the student respond when he/she does
not know what to do?
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There are a number of qualities we should be observing when evaluating problem
solving.
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1. willing to take risks
I
2. chooses a good strategy
3. perseveres
4. finds a useful solution
5. works well in a small group
6. uses organization skills
7.
8.
You may wish to add other characteristics.
Keeping records of pupils' work is always a formidable task. Perhaps a check list
for each student would be helpful which includes space for anecdotes. One
recommendation follows and may be photocopied for use. It is derived from an
article in the Arithmetic Teacher,January 1983. (3)
(3)Randall. 1. Charles; "Teaching: Evaluation and Problem Solving", Arithmetic
Teacher; January 1983; The National Council of Teachers of Mathematics Inc.;
Reston Virginia
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WORLD'S MOsr POPULAR PUZZLES &.PROBLEMS"
1996 EXCWSIVE EDUCATIONAL PRODUcrs