Problem Solving Finding the Mathematics Within the Task

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					    Problem Solving:
  Guiding the Experience
Professional Development Workshop
  KATM 2003 Annual Conference
          October 24, 2003

      David S. Allen, Ed.D.
                and
         Melisa Hancock
    Teacher in Residence, KSU
                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 it’s 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?
There is no royal road to critical thinking.
There’s not even a pauper’s paved path to
easy problem solving. Teaching today’s
children to become the thinking, caring
leaders who will be able to solve the world’s
increasingly complex and quantitative
problems requires a total commitment, not
just a Friday afternoon contribution.
                            (Willoughby, 1990)
            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)
“Problem Solving is more than obtaining answers. It is a tool,
a means of thinking, and a philosophy. It is a predisposition to
learn from every available opportunity the most that can be
gleaned from that experience.
                                        (NCTM, 2000)
             Math Standards (2001)

Content Standards         Process Standards
– Number and              – Problem Solving
  Operations              – Reasoning and
– Algebra                   Proof
– Geometry                – Communication
– Measurement             – Connections
– Data Analysis and       – Representation
  Probability
Three Problem Solving Approaches


1. Teaching for problem solving.

2. Teaching about problem solving.

3. Teaching via problem solving.
       Teaching for Problem Solving

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

1. Refers to instruction that focuses on
   strategies for solving problems
  a) Polya, 1954
  b) Four Step Method
  c) Heuristics
2. Process vs. Procedure
  a) Critical Thinking
  b) Examples
             Pedagogical Approach to
                Problem Solving
        Four Step               Bloom’s
         Process               Taxonomy

1. Read and understand        • Knowledge
   the problem                • Comprehension
2. Devise a plan              • Application

3. Carry out the plan         • Analysis
4. Check your answer          • Synthesis
                              • Evaluation
   Recommendations for Teaching About
           Problem Solving
1. Heuristics
2. Strategies taught in isolation are not
   meaningful to students.
3. Allow students to identify or create meaningful
   solution strategies.
4. Post strategies and refer to them often.
5. Demonstrate the need to draw upon a wide
   variety of solutions strategies.
6. THERE IS MORE THAN ONE WAY TO
   SKIN A CAT!
          Problem Solving Pitfalls

1. 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?
          Problem Solving Pitfalls
3. Unrealistic Problems
   • Mary’s 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 big—
     Subtract
   • If there was one large and one small—
     Divide
   • If it does not come out even—Multiply
      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?
      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?
       Teaching via Problem Solving


1. Uses a problem as a means of learning new
   ideas and for connecting new and already
   existing constructs.
2. Sample problems teaching via problem
   solving.
         Sample Problems (Via)
If you spill 6 counters and record how many
red sides and yellow sides come up each time,
do you think you’ll 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.

  Theoretical Probability vs. Experiments
    Problem Solving:
  Guiding the Experience
Professional Development Workshop
  KATM 2003 Annual Conference
          October 24, 2003

      David S. Allen, Ed.D.
                and
         Melisa Hancock
    Teacher in Residence, KSU

				
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