Problem Solving: Finding the Mathematics Within the Task

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Problem Solving Finding the Mathematics Within
the Task

• Professional Development Workshop
• Junction City, Kansas
• August 14, 2002
• David S. Allen, Ed.D.
• and
• Emily R. Finney

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Introduction

• Welcome
• Introduction and Personal History
• Problem Solving Defined (Group Activity)
• What are your classroom goals with respect to
  problem solving?
• What are the characteristics of a problem solving
  environment?
• What are the characteristics of a problem solving
  task?

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Sample Problems
Acrobats, Grandmas, and Ivan Round 1 On one side
are four acrobats, each of equal strength. On the
other side are five neighborhood grandmas, each
of equal strength. The result is dead even. Round
2 On one side is Ivan, a dog. Ivan is pitted
against two of the grandmas and one acrobat.
Again its a draw. Round 3 Ivan and three
grandmas are on one side, and the four acrobats
are on the other. Who will win the third round?
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There is no royal road to critical thinking.
Theres not even a paupers paved path to easy
problem solving. Teaching todays children to
become the thinking, caring leaders who will be
able to solve the worlds increasingly complex
and quantitative problems requires a total
commitment, not just a Friday afternoon
contribution. (Willoughby, 1990)
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Problem Solving Defined!

• Problem Solving means engaging in a task for
  which the solution method is not known in
  advance.
• (NCTM, 2000)
• A problem is a situation in which a person
  is seeking some goal and for which a suitable
  course of action is not immediately apparent.
• (Marilyn Burns, 2001)
• Solving problems takes place when students
  think flexibly, creatively, and analytically to
  define, examine, diagnose, and unravel
  complicated problems. There must be some blockage
  on the part of the potential problem solver. That
  is a mathematical task is a problem only if the
  problem solver reaches a point where he or she
  does not know how to proceed.

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Math Standards

• Content Standards
• Number and Operations
• Algebra
• Geometry
• Measurement
• Data Analysis and Probability

• Process Standards
• Problem Solving
• Reasoning and Proof
• Communication
• Connections
• Representation

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Criteria for Mathematical Problems

• There is a perplexing situation that the student
  understands.
• The student is interested in finding a solution.
• The student is unable to proceed directly toward
  a solution.
• The solution requires the use of mathematical
  ideas.

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Dealing In Horses!
A man bought a horse for 50 and sold it for 60.
He then bought the horse back for 70 and sold it
again for 80. What do you think was the
financial outcome of these transactions?
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Criteria for Mathematical Problems

• Was the Dealing in Horses problem a problem for
  you? Why or why not? Answer in terms of the four
  criteria.
• How will your students answer these questions?
• Consider the implications of asking your students
  these questions?

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Additional Problem Solving Experience!
A man owned a fox, a rabbit, and a bag
of corn. One day he was on the bank of a river,
where there was a boat only large enough for him
to cross with one of his possessions. If he left
the fox and the rabbit alone, the fox would eat
the rabbit. If he left the rabbit and the corn
alone the rabbit would eat the corn. How did he
get safely across the river with all three of his
possessions?
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Use of Nontraditional Problem-Solving Experiences

• Does the river crossing problem constitute a true
  problem-solving experience?
• What benefits do you see in using this problem
  with your students?
• Do traditional algorithmic problem-solving
  strategies lend themselves to helping to solve
  this type of problem?

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Similar Crossing Problems
Danielle is lost in the jungle with her
pet crocodile, her monkey, and her pet parrot
with a broken wing. She comes across a rickety
rope bridge that goes over a huge river with
rapids. Danielle realizes that she will have to
hold on to the side of the bridge with one hand
while holding an animal friend with the other
hand, as they cross the bridge. She cant leave
the crocodile with the monkey and she cant leave
the parrot with the crocodile. What is the fewest
trips Danielle will have to make across the
rickety bridge to get all her friends over the
river?
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Three Problem Solving Approaches

• Teaching for problem solving.
• Teaching about problem solving.
• Teaching via problem solving.

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Teaching for Problem Solving

• Uses real-life problems as a setting in which
  students can apply and practice recently taught
  concepts and skills.
• Janalea has 2 dogs. Landree has 5 dogs. How many
  more dogs does Landree have than Janalea?
• Traditional problem-solving experiences familiar
  to most adults.

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Teaching About Problem Solving

• Refers to instruction that focuses on strategies
  for solving problems
• Polya, 1954
• Four Step Method
• Heuristics
• Process vs. Procedure
• Critical Thinking
• Examples

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Pedagogical Approach to Problem Solving

• Four Step
• Process
• Read and understand the problem
• Devise a plan
• 3. Carry out the plan
• 4. Check your answer

• Blooms
• Taxonomy
• Knowledge
• Comprehension
• Application
• Analysis
• Synthesis
• Evaluation

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Recommendations for Teaching About Problem Solving

• Heuristics
• Strategies taught in isolation are not meaningful
  to students.
• Allow students to identify or create meaningful
  solution strategies.
• Post strategies and refer to them often.
• Demonstrate the need to draw upon a wide variety
  of solutions strategies.
• THERE IS MORE THAN ONE WAY TO SKIN A CAT!

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Problem Solving Pitfalls

• Rules often provide the thinking for the
  children.
• If 1 man can jump a stream that is 3 meters wide,
  how wide a stream can 5 men jump?

• 2. Key Words often encourage students to avoid
  thinking about the problem.
• Mary walked 11 meters north. She then turned and
  walked 7 meters west. Did she turn right or left?

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Problem Solving Pitfalls

• 3. Unrealistic Problems
• Marys mother needs three hours to do the
  laundry. If Mary helps her, they can do the
  laundry in only two hours. How long would it take
  Mary to do the laundry by herself?

• 4. Non-pertinent Clues
• If there are two numbers that are bigSubtract
• If there was one large and one smallDivide
• If it does not come out evenMultiply

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Sample problems (About)
Jennifer wants to buy 12 new baseball
cards. The Collector Store sells two cards for 25
cents. The Cards and Book Store has three cards
for 33 cents. Where should Jennifer buy the
cards? Why?

• The center region on a dart board is worth
  100 points the next ring is worth 50 points the
  next, 25 points and the outermost, 10 points.
  Betty throws six darts and earns a score of 150.
  Where might her darts have landed?

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Sample problems (About)
Rebecca has a pocketful of change. She
would like to buy a soda, which costs 0.55. How
could she pay for the soda so that she would
eliminate the most change from her pocket?

• I counted 22 legs in my house. All the legs
  were on cats, people, and spiders. How many of
  each creature--cats, people, and spiders--might
  be in the house? See how many different ways you
  can answer this riddle. How many can you find?

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Teaching via Problem Solving

• Uses a problem as a means of learning new
  mathematical ideas and for connecting new and
  already constructed mathematical notions.
• Sample problems teaching via problem solving.

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The Rectangle Problem
What percent of the 4 x 10 rectangle is shaded?
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The Rectangle Problem
Possible solutions identified by elementary
teachers.

• Change 6/40 to an equivalent fraction
• (6/40) X (2.5/2.5) 15/100
• Some teachers wondered where the 2.5 came from.

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The Rectangle Problem
Possible solutions identified by elementary
teachers.
2. Each of the ten columns represents 10 percent,
so I took four of the shaded boxes and filled the
first column to make 10 percent and the last two
boxes fill half of the second column to make 5
percent, so the total shaded area is 15 percent.
10 5
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The Rectangle Problem
Possible solutions identified by elementary
teachers.
3. Draw the picture twice. In the first picture,
I can see that I have 6 out of 40. In the second
picture, I have 6 more out of 40 more. If I just
draw half of the picture again, I pick up 3 out
of 20, so that gives a total of 15 our of 100.
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Sample Problems (Via)
Toss two coins together 25 times. After each
toss, record what comes up -- two heads, two
tails, or one head and one tail. What do you
think will be the result? Record your
predictions for comparison with your actual
results. Prediction Outcome two heads
____ two heads ____ two tails _____ two tails
_____ one head/one tail____ one head/one
tail____
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Sample Problems (Via)
If you spill 6 counters and record
how many red sides and yellow sides come up each
time, do you think youll get one result more
often than the others? If so, what will it be?
Why do you think that? Try it, spilling the
counters at least 25 times. Record your
prediction and your actual results.
Extension Try the experiment with other numbers
of counters.
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Pizzas Small, Medium, and Large In the
thousands of pizza restaurants across the U.S.,
pizzas are sold in small, medium, and large
sizesusually measured by the diameter of the
circular pie. Of course, the prices are different
for the three sizes. Do you think a large pizza
is usually the best buy? The Sole D
Italia Pizzeria sells small, medium, and large
pies. The small pie is nine inches in diameter,
the medium pie is twelve inches in diameter, and
the large pie is fifteen inches in diameter. For
a plain cheese, small pizza, Sole D Italia
charges 6 for a medium pizza, it charges 9
and for a large pizza, it charges 12. Are these
fair prices? Which measures should be
most closely related to the prices
chargedcircumference or area or radius or
diameter? Why?
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The Prison Problem
There was a jail with 100 cells in
it, all in a long row. The warden was feeling
very jolly one night and told his assistant that
he wanted the assistant to unlock all the cells.
This should be done, he told the assistant, by
putting the key in each lock and turning it once.
Following the order, the
assistant unlocked all the cells and then came
back to report that the job was done. Meanwhile,
however, the warden had second thoughts. Maybe
I shouldnt let all the prisoners go free, he
said. Go back and leave the first cell open,
but lock the second one, by putting the key in
and turning it once. The leave the third open,
but lock the fourth, and continued on this way
for the entire row. The
assistant wasnt very surprised at this request.
The warden often changed his mind. After
finishing this task, the assistant returned, and
again the warden had other thoughts. Heres
what I really want you to do, he said. Go back
down the row. Leave the first two cells as they
are, and put your key in the third cell and turn
it once. Then leave the fourth and fifth cells
alone and turn the key in the sixth. Continue
down the row this way. The
assistant again did as instructed. Fortunately,
the prisoners were still asleep. As a mater of
fact the assistant was getting pretty sleepy, but
there was no chance for rest yet. The warden
changed his mind again, and the assistant had to
go back again, and turn the lock in the fourth
cell and in every fourth cell down the row.
This continued all through the night,
next turning the lock in every fifth cell, and
then in every sixth, and on and on, until on the
last trip, the assistant just had to turn the key
in the hundredth cell. When the prisoner finally
woke up, which ones could walk out of their
cells?
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Sure-Fire Rules For Problem Solving

• Most problems are addition.

2. If more than two numbers are given, it has
to be addition.

• When only two numbers are given and they are
  about the same
• subtract.

4. Consider subtraction when money is involved,
particularly if one amount is a round
figure like 50 or 10.00.

• If two numbers are given and one is much larger
  than the other,
• try division.

• Very few problems involve division with a
  remainder. When you
• get a remainder, cross out the division
  and multiply instead.

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Sure-Fire Rules For Problem Solving
7. If you see a fraction, invert it.
8. If you see a decimal, move it.
9. If you see a negative or positive sign, change
it.
10. If the Rules 1-9 do not seem to work, make
one last desperate attempt. Take the set of
numbers in the problem and perform about
two pages of random operations using these
numbers. You should circle about five or
six answers on each page just in case one
of them happens to be the answer. You might get
some partial credit for trying hard.

• Never, never spend too much time solving
  problems. This set of
• rules will get you through even the longest
  assignment in no
• more than 10 minutes with very little
  thinking!

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Problem Solving Finding the Mathematics Within
the Task
Professional Development Workshop Junction City,
Kansas August 14, 2002 David S. Allen, Ed.D. and
Emily R. Finney