Transcript Section 1: Introduction and Rationale

Measurable Objectives
•Identify multiple problem solving strategies for a given
problem
•Identify skills needed as a problem solver
•Engage in problem solving
•Identify problem solving processes applied in personal and
student problem solving
•Identify curriculum resources associated with problem
solving instruction
“We can't solve problems by using the same kind
of thinking we used when we created them. ”
~Einstein, 1949
What skills are needed as a problem solver?
Reflecting on the Processes
Find the problem
card at your table.
 How did you solve this
problem?
 What skills did you need
Take a few
minutes on your
own to think about
the solution.
to solve this problem?
 How did you learn how to
solve problems like this?
 What factors influence
differences to approach
solution-finding?
~Ortiz, 2008
Why Problem Solving?
“The single best way to grow a better brain
is to engage in challenging problem
solving.”
~Jensen (1998)
“It doesn’t matter to our brains
whether we come up with the right
answer or not:
the neural growth happens because of
the process, not because we have found
the correct answer.”
~Jensen (1998)
What is in your textbook/curriculum
related to problem solving instruction?
Take five minutes to reflect in your journal about the resources
that you have available for problem solving instruction.
"Textbooks commonly…
 depict problem solving as a linear process.
 present problem solving as a series of steps.
 imply that solving mathematics problems is a
procedure to be memorized, practiced, and
habituated.
 lead to an emphasis on answer getting.”
~Polya, 1945
Instructional programs from Pre-K
through grade 12 should enable all
students to…
 build new mathematical knowledge through problem
solving;
 solve problems that arise in mathematics and in other
contexts;
 apply and adapt a variety of appropriate strategies to
solve problems;
 monitor and reflect on the process of mathematical
problem solving.
Taken from: NCTM, 2000. Principles & Standards for School Mathematics.
“If problem solving is treated as ‘apply the
procedure,’ then the students try to follow the
rules in subsequent problems. If you teach
problem solving as an approach, where you
must think and can apply anything that works,
then students are likely to be less rigid.”
~ Suydam, 1987
The Mangoes Problem
One night the King couldn't sleep, so he went down into the Royal kitchen,
where he found a bowl full of mangoes. Being hungry, he took 1/6 of the
mangoes. Later that same night, the Queen was hungry and couldn't sleep.
She, too, found the mangoes and took 1/5 of what the King had left.
Still later, the first Prince awoke, went to the kitchen, and ate 1/4 of the
remaining mangoes.
Even later, his brother, the second Prince, ate 1/3 of what was then left.
Finally, the third Prince ate 1/2 of what was left, leaving only three mangoes
for the servants.
How many mangoes were originally in the bowl?
~NCTM, Illuminations, 2008
Guess and Check
14 - 1/2 - 1/3 - 1/4 - 1/5 - 1/6
~NCTM, Illuminations, 2008
Draw a Picture
~NCTM, Illuminations, 2008
Work Backward
"Six represents two-thirds of something, so onethird must be three. So to get three-thirds, you
must add the six (for two-thirds) to three (for
one-third) and you have nine mangoes." Then,
going the next-backward step, he said, "Nine
needs one-fourth" (his words, meaning that
since nine is three-fourths of the previous
amount, it "needs" another fourth of this
amount added to it), "so nine is three-fourths:
divide by three (i.e., 9/3) and add this to nine,
obtaining twelve."
~NCTM, Illuminations, 2008
Write an Equation
 Since the King removed (1/6)x, then x - (1/6)x mangoes are left
after his removal. Thus, (5/6)x mangoes are left.
 The Queen removed one-fifth of (5/6)x, so (5/6)x - (1/5)(5/6)x, or
(4/6)x, mangoes are left after her removal.
 The first Prince removed one-fourth of (4/6)x mangoes, so (4/6x (1/4)(4/6)x, or (3/6)x, mangoes are left after the first Prince's
removal.
 The second Prince removed one-third of (3/6)x, so (3/6)x - (1/3)
(3/6)x, or (2/6)x, mangoes are left.
 Finally, the third Prince removed one-half of (2/6)x, leaving 3
mangoes, so (2/6)x - (1/2)(2/6)x = 1/6x = 3. Solving 1/6x = 3
results in x = 18.
~NCTM, Illuminations, 2008
George Polya (1887-1985)
George Polya was one of the most famous mathematics
educators of the 20th century. He strongly believed that
the skill of problem solving could and should be
taught—it is not something that you are born with. He
identified four principles that form the basis for
problem solving:
 Understand the problem
 Devise a plan
 Carry out the plan
 Look back (reflect)
~Ortiz, 2008
Getting Acquainted With A
Problem
 Where should I start?
 Start from the statement of the problem.
 What can I do?
 Visualize the problem as a whole as clearly and as vividly as you
can. Do not concern yourself with details for the moment.
 What can I gain by doing so?
 You should understand the problem, familiarize yourself with it,
impress its purpose on your mind. The attention bestowed on
the problem may also stimulate your memory and prepare for the
recollection of relevant points.
~Polya, 1945
Introduce New Problem:
Applying the Process
The houses on Main Street are numbered
consecutively from 1 to 150. How many house
numbers contain at least one digit 7?
~Reardon, 2001
Area 1:
Understand the Problem
 What are you asked to find out or show?
 Can you draw a picture or diagram to help you understand the
problem?
 Can you restate the problem in your own words?
 Can you work out some numerical examples that would help make
the problem more clear?
Critical Questions
 Do you understand all the words used in stating
the problem?
 What are you asked to find or show?
 Can you restate the problem in your own
words?
 Can you think of a picture or diagram that
would help you understand the problem?
 Is there enough information to enable you to
find a solution?
 What if students cannot understand the problem?
 What can you do?
Area 2:
Devise a Plan to Solve the Problem
A partial list of problem solving strategies include:
•Solve the simpler problem
•Guess and check
•Experiment
•Make an organized list
•Act it out
•Draw a picture or diagram
•Work backwards
•Look for a pattern
•Use deduction
•Make a table
•Change your point of view
•Use a variable
Critical Questions
 Have you seen it before? Or, have you seen the
same problem in a slightly different form?
 Do you know a related problem? Do you know
a theorem that could be useful?
 Look a the unknown, and try to think of a
familiar problem having the same or a similar
unknown.
 Could you restate the problem? Go back to
definitions
 What if students cannot understand the problem?
 What can you do?
Now that you have a plan,
what will you do with it?
Area 3:
Implementing a Solution Plan
 Carrying out the plan is usually easier than devising the plan
 Be patient—most problems are not solved quickly nor on the first
attempt
 If a plan does not work immediately, be persistent
 Do not let yourself get discouraged
 If one strategy isn’t working, try a different one
 Experiment with different plans
 Allow for mistakes
 Work collaboratively
 Check
How do students implement?
 Draw it
 Use calculator
 Use computation skills
Area 4:
Reflecting on the Problem:
Looking Back
 Does your answer make sense? Did you answer all of the
questions?
 What did you learn by doing this?
 Could you have done this problem another way—maybe even an
easier way?

~Reardon Problem Solving Gifts, Inc. (2001)
“Students need to view themselves as capable
of using their growing mathematical
knowledge to make sense of new problem
situations in the world around them.”
~Newell & Simon, 1972
Reflection
 What problem solving processes do you most
commonly use or encourage students to use?
 What strategy have you learned or been reminded to
use more often?
Welcome Back !
To Day 2 of Keys of Problem Solving (KoPS)
Review of Day 1
 Legislation steering mathematics
 Recommendations from research
 Next Generation Standards
 RtI
 Problem solving process
Day 2
Focus will be on instruction and intervention
based on assessment results
1-accurate
problem
identification
2-problem
analysis
3-design a plan
and implement
4-evaluate effect
1&2
Understanding
the Problem
3
Devising a Plan
to Solve the
Problem
4
Reflecting on
the Problem:
Looking Back
3
Implementing
a Solution Plan
Polya hypothesized that problem solving is not
an innate skill, but rather something that can
be developed. He explains, “Solving problems is
a practical skill, let us say, like swimming….
Trying to solve problems, you have to observe
and imitate what other people do when solving
problems, and, finally, you learn to solve
problems by doing them.”
~Polya, 1945, p. 5
Example
Three darts hit this dart
board and each scores a 1,
5, or 10. The total score
is the sum of the scores
for the three darts. There
could be three 1’s, two 1’s
and a 5, one 5 and two
10’s, and so on. How
many different possible
total scores could a
person get with three
darts?
10 points
5 points
1 point
There are 10 different
possibilities…
# of 1’s
# of 5’s
# of 10’s
Score
3
0
0
3
2
1
0
7
2
0
1
12
1
2
0
11
1
1
1
16
1
0
2
21
0
3
0
15
0
2
1
20
0
1
2
25
0
0
3
30
~Ortiz, 2008
Additional ways to reflect on
the problem…
 Reflect on the plan
 Justify and explain your answers
How Does the Process Relate to
Student Learning?
My Instruction?
Holistic Rubrics
~ Florida Department of Education. FCAT
Mathematics Sunshine State Standards
Test Book, Released: Fall 2007.
Problem Solving
Process/Holistic Rubric
Using this one source
of student data, what
does this student
sample reflect?
Florida Department of Education. FCAT
Mathematics Sunshine State Standards
Test Book, Released: Fall 2007.
Mathematical Thinking
“Providing a challenging investigation to small
groups of students facilitates ongoing
reasoning, argument, and assessment
throughout the problem solving process.”
~NCTM, (2005)
Problem Solving is a dynamic
process
1-accurate
problem
identification
2-problem
analysis
3-design a plan
and implement
4-evaluate effect
1&2
Understanding
the Problem
3
Devising a Plan
to Solve the
Problem
4
Reflecting on
the Problem:
Looking Back
3
Implementing
a Solution Plan
Let’s Look at a Problem…
Growing Giant Sequoias
Goals:
•Use given data to decide which
growing condition fosters better
growth of tree seedlings
•Develop a decision-making
procedure based on the data for
use in future experiments
~NCTM (2005)
Advance Organizer
1. Read about sequoias from
newspaper article
2. Answer readiness questions
3. Learn about professor’s work
regarding growing conditions
4. Analyze professor’s data
~NCTM (2005)
Connections to Problem Solving
•Build new mathematical
knowledge through problem solving
•Solve problems that arise in
mathematics and other contexts
•Apply and adapt a variety of
appropriate strategies to solve
problems
•Monitor and reflect on the process
of mathematical problem solving
~NCTM (2005)
~NCTM (2005)
~NCTM (2005)
You will be analyzing data for several purposes:
• To compare two growing conditions
• To compensate for missing data
• To make a decision about the data set
• To develop a procedure, based on the data, for
making decisions about future experiments
~NCTM, 2005
~Ortiz, 2008
Applying What We Learned
In table groups, use the student data that was just
reviewed to answer the following questions:
 What are my students’ strengths related to problem





solving?
What are my students’ areas of need related to problem
solving?
What is an instructional goal(s) for improvement?
What can I implement to make a learning difference?
How can I teach differently to assure learning?
What assessments will be used to monitor student
learning?
Looking Back at the Goals
Goals:
•Use given data to decide which
growing condition fosters better growth
of tree seedlings
•Develop a decision-making procedure
based on the data for use in future
experiments
~NCTM (2005)
Assessment
Reevaluate and revise procedures in response
to self-assessment, peer assessment, and
comments made by the teacher during informal
assessment.
Reflection
 Think and reason about how
you extrapolated information
from incomplete data sets.
 Take two minutes to journal
about how thinking and
reasoning capabilities may be
developed through a lesson
such as this one.
 What question, task, or activity
should I pose next?
References
Jensen, E. (1998). Teaching with the brain in mind. Alexandria, VA: Association for
Supervision and Curriculum Development.
National Council of Teachers of Mathematics. (2000). Principles & Standards for School
Mathematics.
National Council of Teachers of Mathematics (2008). Illuminations. Located at:
http://illuminations.nctm.org/LessonDetail.aspx?ID=L264
National Council of Teachers of Mathematics. (2005). Navigating through problem
solving and reasoning in grade 4.
Newell, A., and H. A. Simon. 1972. Human problem solving. Englewood Cliffs, NJ: Prentice
Hall.
References
Ortiz, E. (2008). Problem-solving process map. Adapted from G. Polya’s
model.
Pólya, George (1945). How to solve it. Princeton University Press.
Reardon, T. (2001). Teaching problem solving strategies in the 5-12
curriculum: Thank you George Polya. Located at:
http://www.austintown.k12.oh.us/~aust_tr/PSS%20Teaching%20Problem%20
olving%20Strategies.pdf
Suydam, M. (1987). Indications from research on problem solving. In F. R. Curcio
(Ed.), Teaching and learning: A problem solving focus. Reston, VA: National
Council of Teachers of Mathematics.