Transcript Problems and Problem Solving

What is Literacy?
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Literacy is the ability to identify, understand,
interpret, create, communicate and compute,
using printed and written materials associated
with varying contexts. Literacy involves a
continuum of learning in enabling individuals
to achieve their goals, to develop their
knowledge and potential, and to participate
fully in their community and wider society.
United Nations Educational, Scientific, and Cultural Organization
(UNESCO)
Problems and Problem Solving
“Most, if not all, important mathematics
concepts and procedures can best be taught
through problem solving.”
--John Van de Walle
What is Problem Solving?
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“Problem solving means engaging in a
task for which the solution method is
not known in advance.”
--Principles and Standards for School
Mathematics
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It encompasses exploring, reasoning,
strategizing, estimating, conjecturing,
testing, explaining, and proving.
What is a Problem?
A problem is a task that requires the
learner to reason through a situation
that will be challenging but not
impossible.
 Most often, the learner is working with
a group of other students to meet the
challenge.
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Problem or Exercise?
An exercise is a set of number
sentences intended for practice in the
development of a skill.
 A problem is what we commonly refer
to as a “word problem.”
 But beware! Problems can become
exercises!!
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Common Characteristics of a
Good Problem
It should be challenging to the learner.
 It should hold the learner’s interest.
 The learner should be able to connect
the problem to her life and/or to other
math problems or subjects.
 It should contain a range of challenges.
 It should be able to be solved in several
ways.
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What Does It Mean to Be
Successful at Problem Solving?
Having success means that the child
has discovered a way of thinking about
mathematics that he had not
experienced before he came upon this
problem.
 Success will involve the process of
problem solving as well as
understanding the content presented.
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How many rectangles appear
in the figure below?
Success with “How Many
Rectangles”
Do the students resolve the question
about whether to include the squares in
their count of rectangles?
 Do they understand that squares meet
all the criteria to be considered a
rectangle?
 Do they recognize that there are many
different sizes of rectangles in the
drawing?
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Success with “How Many
Rectangles” (continued)
Have the students devised a way of
counting the rectangles they find?
 Do they find patterns in the number of
different-sized rectangles?
 Do they think about the concepts
embedded in the problem differently
than before?
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The Teacher’s Role in Problem
Solving
“The more regularly that teachers make it
part of the curriculum, the more opportunities
students will have to become successful
problem solvers.”
--Children Are Mathematical Problem Solvers
Choosing Problem-Solving
Tasks
The problem must be meaningful to the
students.
 The teacher must sometimes adapt the
problem to make it more meaningful.
 The teacher must work the problem to
anticipate mathematical ideas and
possible questions that problem might
raise.
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Presenting the Problem
It must be interesting and engaging.
 It must be presented so that all children
believe that it’s possible to solve the
problem, but that they will be
challenged.
 The teacher has to decide whether
students will work individually or in
groups.
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Group Work or Individual Work?
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In groups, students don’t give up as quickly.
Students have greater confidence in their abilities to solve
problems when working in groups.
When in a group, students hear a broad range of strategies
from others.
Kids enjoy working in groups!
Students remember what they learn better when they assist
each other.
If students are less productive, arrangements can be made for
them to work alone.
There will be a heightened noise level—but conversation is an
important part of the learning process.
Once the Kids Are Working…
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Allow students to “wrestle” with the problem
without just telling them the answer!
If we are just telling them what to do, the
students are not engaged in the process.
Finally, teachers have to determine how to
assess what the students are learning and
what they need to learn next.
 There are several ways to do this…
Assessing Understanding
Listen to and record the students’
conversations as they solve the
problem.
 Have students explain their solutions in
writing.
 Give them another problem that
requires them to use what they learned
in the first problem.
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Learning Mathematics through
Problem Solving
Students learn to apply the
mathematics as they are learning it.
 They can make connections within
mathematics and to other areas of the
curriculum.
 Students can understand what they
have learned.
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Expressions and Problem
Solving
“Math Expressions was developed to meet the
national need for a balanced program that
could expand the types of word problems to
those solved by other countries and use an
algebraic approach to word problem solving.”
In Kindergarten…
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Students act out family experiences
about meals they might eat at home.
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“Tom sets the table. He puts down 3 plates
and then 1 more. How many plates are on
the table?”
Using paper plates, each child can act
successfully solve a story problem.
In First Grade…
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Students might solve the following
problem and then explain their solutions
at the board:
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“I took 4 rides on the roller coaster. My
sister took 5 rides. How many roller coaster
rides did we take in all?”
Students could use any way that makes
sense to them to solve this problem.
In Second Grade…
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Using Solve and Discuss, students might
solve the following problem:
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“Last year our school had 5 computers in the
library. They bought some more over the summer.
Now there are 12. How many computers did they
buy over the summer?”
Two or three children might show their
solutions on the board. Students at their
desks should be encouraged to ask questions:
• How did you get 7
more?
5 + 7 = 12
C buy now
• Why did you start with
5?
5 + 5 + 2 = 12
C
buy
• How did you know 7 was
a partner?
now
xxxxx xxxxxxx 12
5c
7 buy
now
• How did you know 12
was the total?
• Where did you get 5 +
2?
In Third Grade…
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Students might solve the following
problem and record their answers in
several ways:
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“Chris picked 8 apples. His mother picked 6
more. How many apples do they have
now?”
Children might show their solutions in
several ways:
Math Mountain:
Count All:
now
14 now
T
14
8
6
P
P
Chris
Mom
Equation:
8 + 6 = 14
P P
xxxxxxxx xxxxxx
T
Chris
Mom
Count On:
8
xxxxxx
14
had
now
count on
6 more
In Fourth Grade…
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As the problem become more complex,
students may rely less on pictures and more
on ways to represent the steps in the
problem:
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“In the morning, 19 students were working on a
science project. In the afternoon, 3 students left
and 7 more students came to work on the project.
How many students were working on the project
at the end of the day?”
Manipulatives and drawing paper should still be
available for those students who would like to
use it. Following are abstract ways to represent
this problem:
Tommy’s Method
Write an equation for each step.
Find the total number of
students who worked on the
project.
19 + 7 = 26
Subtract the number of
students who left in the
afternoon.
26 – 3 = 23
Lucy’s Method
Write an equation for the whole problem.
Let n = the number of students
working on the project at the end of
the day.
Students who left
Students who arrived
in the afternoon.
in the afternoon.
19 – 3 + 7 = n
23 = n
In Fifth Grade…
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Students are still encouraged to solve
problems any way that works for them.
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“A right triangle has sides of 4 feet, 5 feet,
and 1 yard. What is its perimeter in
inches?”
In this case, students may well want to
draw a picture to assist them in solving
this problem.
Expressions: Inquiry + Fluency
Using Expressions, students balance
deep understanding with essential skills
and problem solving.
 Students invent, question, discover,
learn, and practice important math
strategies.
 Students explain their methods daily.
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Sources
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Children Are Mathematical Problem Solvers
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by Lynae E. Sakshaug, Melfried Olson, and Judith
Olson
Math Expressions
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developed by The Children’s Math Worlds
Research Project; Dr. Karen C. Fuson, Project
Director and Author