# What is Problem Solving?

On this page we discuss "What is Problem Solving?" under the three headings:

Introduction Four Stages of Problem Solving Scientific Approach

Naturally enough, Problem Solving is about solving problems. And we’ll restrict ourselves to thinking about mathematical problems here even though Problem Solving in school has a wider goal. When you think about it, the whole aim of education is to equip children to solve problems. In the Mathematics Curriculum therefore, Problem Solving contributes to the generic skill of problem solving in the New Zealand Curriculum Framework.

But Problem Solving also contributes to mathematics itself. It is part of one whole area of the subject that, until fairly recently, has largely passed unnoticed in schools around the world. Mathematics consists of skills and processes. The skills are things that we are all familiar with. These include the basic arithmetical processes and the algorithms that go with them. They include algebra in all its levels as well as sophisticated areas such as the calculus. This is the side of the subject that is largely represented in the Strands of Number, Algebra, Statistics, Geometry and Measurement.

On the other hand, the processes of mathematics are the ways of using the skills creatively in new situations. Problem Solving is a mathematical process. As such it is to be found in the Strand of Mathematical Processes along with Logic and Reasoning, and Communication. This is the side of mathematics that enables us to use the skills in a wide variety of situations.

Before we get too far into the discussion of Problem Solving, it is worth pointing out that we find it useful to distinguish between the three words "method", "answer" and "solution". By "method" we mean the means used to get an answer. This will generally involve one or more problem solving strategies. On the other hand, we use "answer" to mean a number, quantity or some other entity that the problem is asking for. Finally, a "solution" is the whole process of solving a problem, including the method of obtaining an answer and the answer itself.

method + answer = solution

But
how
do
we
do
Problem
Solving?
There
appear
to
be
four
basic
steps.
Pólya
enunciated
these
in
1945
but
all
of
them
were
known
and
used
well
before
then.
And
we
mean
**well**
before
then.
The
Ancient
Greek
mathematicians
like
Euclid
and
Pythagoras
certainly
knew
how
it
was
done.

Pólya’s four stages of problem solving are listed below.

**Four
Stages
of
Problem
Solving
****Back
to
Top**

1.
Understand
and
explore
the
problem;

2.
Find
a
strategy;

3.
Use
the
strategy
to
solve
the
problem;

4.
Look
back
and
reflect
on
the
solution.

Although we have listed the Four Stages of Problem Solving in order, for difficult problems it may not be possible to simply move through them consecutively to produce an answer. It is frequently the case that children move backwards and forwards between and across the steps. In fact the diagram below is much more like what happens in practice

There
is
no
chance
of
being
able
to
solve
a
problem
unless
you
are
can
first
**understand**
it.
This
process
requires
not
only
knowing
what
you
have
to
find
but
also
the
key
pieces
of
information
that
somehow
need
to
be
put
together
to
obtain
the
answer.

Children (and adults too for that matter) will often not be able to absorb all the important information of a problem in one go. It will almost always be necessary to read a problem several times, both at the start and during working on it. During the solution process, children may find that they have to look back at the original question from time to time to make sure that they are on the right track. With younger children it is worth repeating the problem and then asking them to put the question in their own words. Older children might use a highlighter pen to mark and emphasise the most useful parts of the problem.

Pólya’s
second
stage
of
**finding
a
strategy**
tends
to
suggest
that
it
is
a
fairly
simple
matter
to
think
of
an
appropriate
strategy.
However,
there
are
certainly
problems
where
children
may
find
it
necessary
to
play
around
with
the
information
before
they
are
able
to
think
of
a
strategy
that
might
produce
a
solution.
This
exploratory
phase
will
also
help
them
to
understand
the
problem
better
and
may
make
them
aware
of
some
piece
of
information
that
they
had
neglected
after
the
first
reading.

Having
explored
the
problem
and
decided
on
a
plan
of
attack,
the
third
problem-solving
step,
**solve
the
problem**,
can
be
taken.
Hopefully
now
the
problem
will
be
solved
and
an
answer
obtained.
During
this
phase
it
is
important
for
the
children
to
keep
a
track
of
what
they
are
doing.
This
is
useful
to
show
others
what
they
have
done
and
it
is
also
helpful
in
finding
errors
should
the
right
answer
not
be
found.

At
this
point
many
children,
especially
mathematically
able
ones,
will
stop.
But
it
is
worth
getting
them
into
the
habit
of
**looking
back**
over
what
they
have
done.
There
are
several
good
reasons
for
this.
First
of
all
it
is
good
practice
for
them
to
check
their
working
and
make
sure
that
they
have
not
made
any
errors.
Second,
it
is
vital
to
make
sure
that
the
answer
they
obtained
is
in
fact
the
answer
to
the
problem
and
not
to
the
problem
that
they
thought
was
being
asked.
Third,
in
looking
back
and
thinking
a
little
more
about
the
problem,
children
are
often
able
to
see
another
way
of
solving
the
problem.
This
new
solution
may
be
a
nicer
solution
than
the
original
and
may
give
more
insight
into
what
is
really
going
on.
Finally,
the
better
students
especially,
may
be
able
to
generalise
or
extend
the
problem.

**Generalising**
a
problem
means
creating
a
problem
that
has
the
original
problem
as
a
special
case.
So
a
problem
about
three
pigs
may
be
changed
into
one
which
has
any
number
of
pigs.

In
Problem
4
of
What
Is
A
Problem?,
there
is
a
problem
on
towers.
The
last
part
of
that
problem
asks
how
many
towers
can
be
built
for
*any*
particular
height.
The
answer
to
this
problem
will
contain
the
answer
to
the
previous
three
questions.
There
we
were
asked
for
the
number
of
towers
of
height
one,
two
and
three.
If
we
have
some
sort
of
formula,
or
expression,
for
any
height,
then
we
can
substitute
into
that
formula
to
get
the
answer
for
height
three,
for
instance.
So
the
"any"
height
formula
is
a
generalisation
of
the
height
three
case.
It
contains
the
height
three
case
as
a
special
example.

**Extending**
a
problem
is
a
related
idea.
Here
though,
we
are
looking
at
a
new
problem
that
is
somehow
related
to
the
first
one.
For
instance,
a
problem
that
involves
addition
might
be
looked
at
to
see
if
it
makes
any
sense
with
multiplication.
A
rather
nice
problem
is
to
take
any
whole
number
and
divide
it
by
two
if
it’s
even
and
multiply
it
by
three
and
add
one
if
it’s
odd.
Keep
repeating
this
manipulation.
Is
the
answer
you
get
eventually
1?
We’ll
do
an
example.
Let’s
start
with
34.
Then
we
get

34 17 52 26 13 40 20 10 5 16 8 4 2 1

We
certainly
got
to
1
then.
Now
it
turns
out
that
no
one
in
the
world
knows
if
you
will
always
get
to
1
this
way,
no
matter
where
you
start.
That’s
something
for
you
to
worry
about.
But
where
does
the
extension
come
in?
Well
we
can
extend
this
problem,
make
another
problem
that’s
a
bit
like
it,
by
just
changing
the
3
to
5.
So
this
time
instead
of
dividing
by
2
if
the
number
is
even
and
multiplying
it
by
three
and
adding
one
if
it’s
odd,
try
dividing
by
2
if
the
number
is
even
and
multiplying
it
by
5
and
adding
one
if
it’s
odd.
This
new
problem
doesn’t
contain
the
first
one
as
a
special
case,
so
it’s
not
a
generalisation.
It
**is**
an
extension
though
–
it’s
a
problem
that
is
closely
related
to
the
original.
You
might
like
to
see
if
this
new
problem
always
ends
up
at
1.
Or
is
that
easy?

It
is
by
this
method
of
generalisation
and
extension
that
mathematics
makes
great
strides
forward.
Up
until
Pythagoras’
time,
many
right-angled
triangles
were
known.
For
instance,
it
was
known
that
a
triangle
with
sides
3,
4
and
5
was
a
right-angled
triangle.
Similarly
people
knew
that
triangles
with
sides
5,
12
and
13,
and
7,
24
and
25
were
right
angled.
Pythagoras’
generalisation
was
to
show
that
EVERY
triangle
with
sides
a,
b,
c
was
a
right-angled
triangle
if
and
only
if
a^{2}
+
b^{2}
=
c^{2}.

This
brings
us
to
an
aspect
of
problem
solving
that
we
haven’t
mentioned
so
far.
That
is
**justification**
(or
proof).
Your
students
may
often
be
able
to
guess
what
the
answer
to
a
problem
is
but
their
solution
is
not
complete
until
they
can
justify
their
answer.

Now in some problems it is hard to find a justification. Indeed you may believe that it is not something that any of the class can do. So you may be happy that the children can guess the answer. However, bear in mind that this justification is what sets mathematics apart from every other discipline. Consequently the justification step is an important one that shouldn’t be missed too often.

**Scientific
Approach
****Back
to
Top**

Another way of looking at the Problem Solving process is what might be called the scientific approach. We show this in the diagram below.

Here the problem is given and initially the idea is to experiment with it or explore it in order to get some feeling as to how to proceed. After a while it is hoped that the solver is able to make a conjecture or guess what the answer might be. If the conjecture is true it might be possible to prove or justify it. In that case the looking back process sets in and an effort is made to generalise or extend the problem. In this case you have essentially chosen a new problem and so the whole process starts over again.

Sometimes, however, the conjecture is wrong and so a counter-example is found. This is an example that contradicts the conjecture. In that case another conjecture is sought and you have to look for a proof or another counterexample.

Some
problems
are
too
hard
so
it
is
necessary
to
give
up.
Now
you
may
give
up
so
that
you
can
take
a
rest,
in
which
case
it
is
a
‘for
now’
giving
up.
Actually
this
is
a
good
problem
solving
strategy.
Often
when
you
give
up
for
a
while
your
subconscious
takes
over
and
comes
up
with
a
good
idea
that
you
can
follow.
On
the
other
hand,
some
problems
are
so
hard
that
you
eventually
have
to
give
up
‘for
ever’.
There
have
been
many
difficult
problems
throughout
history
that
mathematicians
have
had
to
give
up
on.
(More
can
be
found
on
this
in
__What
is
Mathematics?__)

That then is a rough overview of what Problem Solving is all about. For simple problems the four stage Pólya method and the scientific method can be followed through without any difficulty. But when the problem is hard it often takes a lot of to-ing and fro-ing before the problem is finally solved – if it ever is!