Problem Solving: Developing Proficiency in the Content Area

1
Problem Solving Developing Proficiency in the
Content Area

• Professional Development Workshop
• Ft. Riley Middle School
• September 29, 2002
• David S. Allen, Ed.D.
• and
• Melisa Hancock
• Teacher in Residence, KSU

2
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?

3
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?
4
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)
5
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. (Van Dewalle,
  2002)

6
Math Standards (2001)

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

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

7
Criteria for 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 previously
  acquired knowledge, skills, and understanding.

8
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?
9
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?

10
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?
11
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?

12
Three Problem Solving Approaches

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

13
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.

14
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

15
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

16
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!

17
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?

18
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

19
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?

20
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?

21
Teaching via Problem Solving

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

22
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.
23
Problem-Based Learning in the Content Area

All education involves either
problem solving or preparation for problem
solving. 1. Mathematics (What does this
equal?) 2. Literary analysis (What does this
mean?) 3. Scientific experiments (Why and how
does this happen?) 4. Historical
investigation (What took place, and why
did it occur that way?)
24
Problem-Based Learning in the Content Area
Through problem-based learning
teachers show students how to answer questions
and solve problems. When teachers and schools
skip the problem-formulating stage -- handing
facts and procedures to students without giving
them a chance to develop their own questions and
investigate by themselves -- students may
memorize material but will not fully understand
or be able to use it. Problem-based learning
provides a structure for discover that helps
students internalize learning and leads to
greater comprehension. (Delisle 1997)

25
Problem-Based Learning in the Content Area

• 1. PBL deals with problems that are as close to
  real-life situations as possible.
• Students make a greater attempt to understand and
  remember when they see connections between the
  material they study and their own lives.
• Students occasionally ask why they need to study
  a subject or what use the information will be
  will be to them.
• Students acquire new knowledge or skills to solve
  a problem or complete a task that is highly
  relevant to their lives.

26
Example (High School ESL Class)

• It has come to the attention of several
  community leaders that the working conditions of
  immigrants in many communities are substandard.
  You are a member of a community group that has
  been asked to investigate these working
  conditions. You are collaborating with several
  labor unions who wish to organize the workers.
  You will present a report containing
  recommendations to the State Labor Commission. In
  that report you will push for enforcement of
  current laws -- and enactment of new laws -- to
  protect the workers.

27
Problem-Based Learning in the Content Area

• 2. PBL promotes students active engagement with
  learning.
• Passive learning -- students take notes during a
  teachers lecture and repeat the same information
  back on tests.
• Student responses to questions are dictated by
  textbooks and are already known by the teacher.
• Even in math and science classes teachers rarely
  allow students to discover principles for
  themselves but instead present the mathematical
  techniques or scientific laws then students
  practice what they have been taught.

28
Example (7th Grade)

• A number of children and their parents have
  complained about the food that is served in the
  school cafeteria. Their complaints range from the
  nutritional value of the food to the fact that
  the food does not reflect the cultures from which
  the children come. Your class has been asked by
  the principal to look into these complaints and
  to prepare a set of recommendations regarding the
  foods served in the school cafeteria.

29
Problem-Based Learning in the Content Area

• 3. PBL promotes and interdisciplinary approach
• Students who are switching subjects throughout
  the day act surprised when their science teacher
  corrects spelling on lab reports.
• Students see each subject as totally isolated
  from any others.
• Real world work rarely fits the narrow boundaries
  of a single academic discipline. A newspaper
  reporter needs needs a knowledge of English and
  history as well as science and statistics.

30
Example (Art, Math, Language)

• A new school is to be built for our
  neighborhood at a cost of 8 million. By law, 1.5
  percent of this cost is to be devoted to the
  installation of art in the school. You are
  members of a committee that must do two things.
  The first is to set up rules for the artists who
  wish to submit proposals, and the second is to
  determine how the school will select the art.

31
Problem-Based Learning in the Content Area

• 4. PBL requires students to make choices about
  how and what they will learn.
• With the continuing explosion of knowledge and
  the rapid pace of technological change, schools
  can no longer present students with all the
  information they need for their entire lives.
• The most important skill schools can teach
  students is how to learn on their own.

32
Example (4th Grade)

• The state legislature is considering changes
  to the Kansas State Constitution. An advocacy
  group is proposing a Bill of Rights for Children
  and wishes you and your classmates to make some
  suggestions for items to be included in the bill.
  You will propose items to the Bill of Rights and
  include reasons why these rights are needed.

33
Problem-Based Learning in the Content Area

• 5. PBL promotes collaborative learning.
• Students using PBL build teamwork skills as they
  learn from each other and work together to solve
  the problem.
• Ideal for classes with a range of academic
  abilities. Students in each group can work on
  different aspects of the problem.
• Students from diverse backgrounds will see
  different aspects of the problem and have varying
  ideas that could lead to solutions.

34
Example (8th Grade Social Studies)


There has been a great deal of discussion
about violence in the schools. Some schools have
adopted dress codes as part of the solution and
your school, although peaceful, is now
considering a dress code as a preventive measure.
You and your classmates have been asked to write
a report on the effect dress codes have on the
behavior of students.
35
Problem-Based Learning in the Content Area

• 6. PBL helps raise the quality of education.
• The technique requires students to put forth more
  thought and effort than assignments requiring
  rote memorization.
• A well-written problem forces students to learn
  from a variety of different sources and to make
  decisions based on their research.
• This process enables students to meet standards
  calling for the development of advanced cognitive
  skills, research skills, and problem-solving
  skills.

36
Problem-Based Learning in the Content Area

• Teachers Role
• The PBL Teacher as a Curriculum Designer
• Frequency
• Problem Identification
• Write problem statement
• The PBL Teacher as Guide
• Teacher acts as a facilitator
• Sets the climate and helps students connect to
  the problem

37
Problem-Based Learning in the Content Area

• Teachers Role
• The PBL Teacher as Evaluator
• Effectiveness of the Problem
• Ongoing evaluation process
• Student Performance
• Not just for a grade but also as a diagnostic
  tool for identification of student difficulties
• Teacher Performance
• Evaluation of support and guidance

38
Problem-Based Learning in the Content Area

• Developing a Problem
• A large part of the art of instruction lies
  in making the difficulty of new problems large
  enough to challenge thought, and small enough so
  that, in addition to the confusion naturally
  attending the novel elements, their shall be
  luminous familiar spots from which helpful
  suggestions may spring.
• (Dewey, 1916)

39
Problem-Based Learning in the Content Area

• Developing a Problem
• Selecting Content and Skills
• Determining Availability of Resources
• Writing a Problem Statement
• Developmentally appropriate
• Grounded in student experience
• Curriculum based
• Attention to teaching and learning styles
• Ill-structured

40
Problem Solving Developing Proficiency in the
Content Area
Professional Development Workshop Ft. Riley
Middle School September 29, 2002 David S. Allen,
Ed.D. and Melisa Hancock Teacher in Residence,
KSU http//coe.ksu.edu/allen/index.html