MAKING PROBLEM
SOLVING LESS
PROBLEMATIC
Prepared by SER Literacy & Numeracy Lead Coaches - 2013
What is problem solving?
The term ‘problem’ and ‘problem solving’ occur in
many subject areas, however, is most commonly
associated with mathematics.
‘Solving a problem is finding the unknown means to a
distinctly conceived end…to find a way where no way
is known off-hand. For a question to be a problem, it
must represent a challenge that cannot be resolved by
some routine procedure. Problem solving is a process
of accepting a challenge and striving to resolve it.’
(George Polya – ‘the father of problem solving’ 1945)
Why teach problem solving?
• An interesting and enjoyable way
to learn mathematics.
• An opportunity to learn
mathematics with greater
understanding.
• Produces positive attitudes
towards mathematics.
• Teaches varied ways of thinking,
flexibility and creativity.
• Teaches general problem solving
skills generally applicable in other
KLAs and life.
• Encourages cooperative skills.
3
The Importance of Problem Solving
‘With exposure, experience, and shared
learning, children will develop a repertoire
of problem-solving strategies that they can
use flexibly when faced with new problemsolving situations.’
Burris, Anita (2005). Understanding the Math You Teach. Merrill Prentice Hall, ISBN 0-13110737-2
.
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Australian Curriculum
Why do children have difficulties
with PROBLEM SOLVING?
• When solving problems students will need to know general
strategies and techniques that will guide the choice of which
skills or knowledge to use at each stage in problem solving.
• When a problem has ‘a new twist’ students cannot recall how to
go about it – this is when general strategies are useful in
providing possible approaches that may lead to a solution.
• Some children find it difficult to think of ideas and strategies. A
brain-storming session might help students reflect on problems
they have previously solved.

The misconceptions that students have, are a problem with the
understanding proficiency strand rather than with fluency. When
students don't understand something we can't fix it by just doing
more of the same.
Approaches to teaching
problem solving …
The approaches teachers use:
Teaching for problem solving - knowledge, skills and
understanding (the mathematics)
Teaching about problem solving - experience-based techniques for
problem solving, learning, and discovery that give a solution which is not guaranteed
to be optimal and behaviours (the
strategies and processes)
Teaching through problem solving - posing questions and
investigations as key to learning new mathematics
(beginning a unit of work with a problem the students
cannot do yet)
Procedural ( Closed) and
Open-ended Problems
Procedural problems:
One- or two-step simple word problems
Open-ended
problems:
Problems that require mathematical
analysis and reasoning;
Open-ended problems
can be solved in more than one way,
and can have more than one solution.
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Procedural or Open-ended?
A typical conventional classroom task is:
a
Find the perimeter of this rectangle.
10 cm
4cm
a
A corresponding open-ended task is:
a
The perimeter of a rectangle is 28cm. What
might be the length and width of the
rectangle?
Newman’s analysis of children’s
problem solving errors
1. Reading
Can students read the words of the problem?
2. Comprehension
Can students understand the meaning?
3. Transformation
Can students determine a way to solve the problem?
4. Process Skills
Can students do the mathematics?
5. Encoding
Can students record and interpret their answer?
(cited in NAPLAN Numeracy 2009 by Bob Wellham K-12
Mathematics Consultant, Swansea)
So what do we do?
Help students to ………
- read the questions
- comprehend what they read
-provide strategies that aid understanding
-teach students to check their answers are
reasonable.
In multiple choice questions – many students
guess or are mislead.
How can we do this?
• Teach a problem solving model explicitly
• Increase the mathematical metalanguage used
• Train students to explain how they get their
answers
• Scaffold the teaching of the problem solving
strategies
• Assess for differentiation of learning
• Explicit teaching of concepts
Polya’s Model – Problem Sovling
George Polya has had an important
influence on problem solving in
mathematics education. He stated
that good problem solvers tend to
forget the details and focus on the
structure of the problem, while poor
problem solvers do the opposite.
Four-Step Process:
1. Understand the problem (See)
2. Devise a plan (Plan)
3. Carry out the plan (Do)
4. Look back (Check)
Four Step Process
Newman’s Analysis aligned
with Polya’s model
•See
•Check
•Plan
1.Reading the
problem
2.Comprehending
what is read
3. Carrying out a
transformation
from the words of
the problem:
the selection of an
appropriate
mathematical
strategy
5. Encoding
the answer in
an acceptable
written form
4. Applying the
process skills
demanded by the
selected strategy
•Do
Step 1 – SEE:
Understand the problem
Students can be overwhelmed just by
reading a problem. At this point group
discussion is beneficial.
Questions that may help you lead students
to an understanding of the problem:
•What are you asked to find or show?
•What information is given? (valuable/useless?)
•Has anyone seen a problem like this before?
•Can you restate the problem in your own
words?
•What conditions/operations apply?
•What type of answer do you expect/Can you
give an estimate?
•What units will be used in the answer?
Visualisation
Is an important way to read
and understand the
problem.
Do I see pictures in my
mind?
How do they help me
understand the situation?
Imagine the situation
What’s going on here?
Drawing pictures/models: Using models to
visualise/understand the problem
Problem:
Sally had some stickers. After she gave her best friend 280
stickers she had 754 stickers left. How many stickers did Sally
have in the first place?
Model
drawing
Sally had 1034 stickers in the first place.
Step 2- PLAN:
Devise a Plan
1. Is this problem similar to a
problem you’ve done before?
(Can you use the same method
or part of the previous plan?)
2. What strategy would help you
solve it? (Students can best learn
skills at choosing an appropriate
strategy by exposure to many
different problems.)
3. Are the units consistent?
Problem-solving strategies
include:
• Draw a picture or diagram
• Act it out
• Use concrete materials/make a model
• Look for a pattern
• Make an orderly list or table
• Work backwards
• Use direct/logical reasoning
• Solve a simpler problem
• Predict and test (guess & check)
• Write an equation/use a formula
• Eliminate possibilities
Make a Drawing or Diagram
• Stress that there is no need to draw
detailed pictures.
• Encourage children to draw only what is
essential to tell about the problem.
Act It Out
• Stress that other objects may be used in
the place of the real thing.
• The value of acting it out becomes
clearer when the problems are more
challenging.
Use concrete materials/
make a model
• Students use materials to help them
discover the relationship they need to
see, to lead them to a solution.
• Using concrete materials may make the
problem easier to see.
Look for a Pattern
• This involves identifying a pattern and
predicting what will come next.
• Often students will construct a table, then
use it to look for a pattern.
• Explain to the students that:
o They can look at a series of shapes, colours or
numbers to see if you can find a pattern
o The pattern should repeat
o The pattern is not always obvious
•
Example: Look for a pattern
using a table
PROBLEM: Restaurants often
use small square tables to
seat customers. One chair is
placed on each side of the
table. Four chairs fit around
one square table.
Restaurants handle larger
groups of customers by
pushing together tables.
Two tables pushed together
will seat six customers.
How many people will 4
tables pushed together
seat?
Number
of
tables
Number
of
people
1
4
2
6
3
8
4
?
Construct a Table or
Organised List
• This is an efficient way to
classify or order a large
amount of data.
• An organised list provides
a systematic way to
record computations.
The letters ABCD, can
be put into a different
order: DCBA or BADC.
How many different
combinations of the
letters ABCD can you
make? To answer this
question students may
choose to make a list.
By making a
SYSTEMATIC list,
students will be able to
see every possible
combination.
Work Backwards
• This strategy seems to have limited
application opportunities, however, is a
powerful tool when it can be used.
• Some problems are posed in such a way
that students are given the final conditions
of an action and are asked about
something that occurred earlier.
Example of Working Backwards
Direct /Logical Reasoning
Explicitly teach students:
•To tackle the problem step by step
•That each piece of information is a piece of
the puzzle, put all pieces together to find the
solution
•To read each clue thoroughly and work by a
process of elimination
•When the first plan (strategy) is unsuccessful,
to try another one
Solve a Simpler or Similar
Problem
• Some solutions are
difficult because the
problem contains large
numbers or
complicated patterns.
• Sometimes a simpler
representation will
show a pattern which
can help solve a
problem.
Teach the students to:
· Set aside the original
problem and work through a
simpler related problem
· Replace larger numbers
with smaller numbers to
make calculations easier,
then apply same method of
solving it to the original
problem
· Look for a pattern that may
be emerging
Predict & Test (Guess and
Check)
• This strategy does not include “wild” or “blind”
guesses.
• Students should be encouraged to incorporate
what they know into their guesses - a logical
guess.
• The “Check” portion of this strategy must be
stressed.
• When repeated guesses are necessary, using
what has been learned from earlier guesses
should help make each subsequent guess better
and better.
Write an equation/
use a formula
• Students can sometimes make sense of
a problem by changing the written
problem to a number sentence.
Eliminate possibilities
• Eliminating possibilities is a strategy
where students use a process of
elimination until they find the correct
answer.
• This is a problem-solving strategy that
can be used in basic math problems or
to help solve logic problems.
Step 3 – DO:
Carry out the plan
• At this stage students need to follow
the relevant steps to solve the
problem.
• They need to learn how to check
each step of the solution and ensure
the accuracy of the computation as
they implement their chosen
problem solving strategy. They
need to be explicitly taught how to
keep an accurate record of each
step and how to avoid making
careless mistakes and
computational errors.
Step 4 – CHECK:
Look Back
1. Does my answer make
sense?
2. Check the mathematics by
working backwards,
estimating, showing that the
answer is reasonable, or
doing the problem another
way.
3. Have I answered the
question?
4. Have I learned anything new
from solving this problem?
Polya’s Model in action
Maddison, Bella, Tia and Talia all
exchanged Valentine’s Day cards. How
many Valentine’s Day cards were
exchanged?
See
Understanding the Problem
Do I understand all of the words?
Exchange = swap
Valentine = loved one
Many = more than one
What am I asked to do?
I am asked to find out how many cards were
exchanged between the 4 friends.
Can I restate the problem in my
own words?
Four friends want to give each other a card.
How many cards would be swapped if they
were each to get a card from one another?
Can I think of a picture or
diagram
that might help me
understand the problem?
Four
Friends
Is there enough
information to enable
me to solve the
problem?
Yes!
Plan
Devise a plan
List
Maddison
Bella-Tia-Talia
Bella
Tia-Talia-Maddison
Tia
Talia-Maddison-Bella
Talia
Maddison-Bella-Tia
Diagram
Maddison
Bella
Tia
Talia
Act it out
Three friends and I could stand in a
circle. Each of us would hold
enough pieces of paper to give
one another one each. Then I
could count how many pieces of
paper we held altogether.
Look for a Pattern
Maddison would need ***
Bella would need ***
Tia would need ***
Talia would need ***
SO… ***+***+***+*** = 12
Solve a Simpler Problem
If Maddison was to give a card to each of her
three friends, She would need 3 cards.
SO…
There are four friends, each needing 4 cards.
SO…
4 X 3 = 12
Write an Equation
4 friends each need to give out 3
cards.
SO…
4 X 3 = 12
OR…
3 + 3 + 3 + 3 = 12
Problem Solving Think board
What is the question asking?
What information is important?
Write your answer as a complete
sentence.
What strategy will you use to
solve the problem?
Attack the problem!
Do the maths!
DO