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Looking at the whole issue of problem solving I find that there are
many views as to what it really means. It seems to go from
something that is not just a written calculation such as, "5 + 7 =
?", but instead words are used, "Jane had 5 pencils and her brother
gave her 7, how many did she now have?", to something far more
complicated in which the problem is multifaceted: "Design a system for traffic lights to
operate in, when there is a junction of three major roads and
For the purposes of this article I would like to use the idea that
problem solving is something that a pupil becomes involved in when
(s)he has to do some extra exploration in order to come up with a
solution. This means that a problem-solving activity in a class may
not always involve problem solving for the most able, who may
instantly produce a solution.
Most of the experiences that I have had in which I have observed
problem solving have been when pupils have been involved in
investigations. Along the way they come across parts that require
problem solving and other parts that just flow, as it were,
naturally from them.
Let's look at an example, Buying a
Here is what one group of pupils did (they sent their
solution in to NRICH and you can see it in full here
"We made a list of all the
possible coins that Lolla might have used, 1p, 2p, 5p, 10p, 20p,
50p, £1, and £2."
So for some this was problem solving as they did not know straight
away and had to get some coins and see what was available. Others
knew, without cause for extra thinking, but by just
"We think that if she used just
1p coins she would have paid 6p altogether, which we think is too
little for a balloon."
This shows a bit of simple calculation that had an instant answer
and then some opinions shared about the cost of balloons.
"We think if she had used £2
coins she would have spent £12 and this was far too much for a
"We then looked at combining two
different coins together. We discovered that with a 1p and 2p she
could have paid 7p, 8p, 9p, 10p, or 11p. With the 1p and the 5p she
could have paid 10p, 14p, 18p, 22p or 26p."
The above involved problem solving for some of the group as they
worked out how, when you replace one of the 1p coins with the 5p
coin, you were taking away 1 but adding 5 and soon they realised
that this was just a matter of adding 4. So, what had started as
problem solving at this stage was overcome and became a known
"We moved on to using 1p and 10p
and realised that the smallest amount would be using five 1p and
one 10p making 15p altogether and then replacing a 1p with a 10p
meant they would have 9p more! We said that this is a pattern. We
explained this as 5 (1ps) and the larger coin and then add 1 less
than the larger coin each time."
The fact that they acquired in the previous stage becomes used over
and over again, and so generates a pattern. Having seen that there
was a pattern, they can use it to move on.
The teacher then divided the group up into smaller groups to look
at particular combinations of used coins.
"We recorded our results. With
the 1p, the 2p and the 5p, she could have paid 11p, 12p, 13p, 14p,
15p, 16p, 17p, 19p, 20p or 23p.
With the 5p, the 10p and the 50p,
she could have paid 80p, 85p, 90p, 95p, £1.25, £1.30,
£1.35, £1.70, £1.75 or £2.15."
To me, this is a good example of a problem-solving exercise. It
gave opportunities for all the pupils to explore something that
they did not instantly know the answer to or how to go about
However, I'd like to stress that the activity became a
problem-solving one partly due to the way the teacher behaved.
There were very few interventions by the teacher except to organise
the pupils into groups and to supply the coins. The teacher
- Asking a question that seemed to ask the pupils to guess what
was in her head.
- Speaking in such a way so as to bring about a state of anxiety
or having to rush in order to get an answer.
This helps in allowing pupils to realise that their thoughts
are valued and that they are encouraged to think for themselves,
and not just remember what the teacher said on previous days.
Some older, more experienced pupils might have instantly known
a way of proceeding. They still may have enjoyed the challenge and
be pleased to know that they found all the mathematical solutions
possible. But maybe for them it was not problem solving. There are
different kinds of learners and thinkers in our classroom and some
like to get at the answer as quickly as possible and others
thoroughly enjoy the processes involved along the way.
Let's now look at another example. Sort the
"I sorted the houses into three
groups because they were different styles: one group of three had 3
windows on two floors and a door on the right hand side; another
group of three houses had 5 windows on two floors and a door in the
middle; another group of three houses had 5 windows on three floors
and a door on the right hand side. I sorted the houses by the
position of the doors: one group of six houses had the door on the
right hand side and another group of three houses had the door in
the middle making it symmetrical."
A group of pupils from a First school said:
"We found 10 groups:
window numbers 2
number of doors 1
number of floors
A Key Stage 3 student, Gabby, also noticed that you could sort
the houses according to the steepness of their roofs. For her it
was probably not problem solving.
Now for some pupils the idea of being asked to group some
houses together when they will have to decide on the criteria is
quite a challenge, in fact a problem-solving situation. I would
imagine that there would be a lot of conversation between
youngsters and an adult in order for some steps to be taken. For it
to be problem solving the adult has to be non-directive and just
help the youngster to understand what they are aiming
A further challenge for a still wider range of pupils in Key
Stages 1, 2 and 3 comes from Teddy Town
When I've worked with pupils on a similar activity (see below)
I've found that the 9 and 10 year olds do not find this much of a
problem. The challenge is fairly easily done and they are happy to
achieve a solution. Some pupils are worried that they find
solutions which are different and yet still correct. Younger pupils
find it to be quite a problem-solving activity. Some try a strategy
of placing the houses first so that they comply to the rule of each
being different, and then place the teddies. They try lots of times
and most will achieve it and a few miss it because they produce a
Teddy with a House that gets repeated.
A typical solution may look like:
In order for problem solving to take place the teacher has to
restrain from intervening and allow the exploration to take place.
The challenge then goes further for the older pupils in using four
Teddies and Houses of each colour to place on a 4 by 4 grid.
Some pupils will take a piece of paper and try to draw a
solution by placing the Houses first and then the Teddies. Much
rubbing out occurs because they find themselves repeating some of
the combinations of a Teddy and a House. But having not been
successful on about the fifth attempt the idea of using something
physical to represent the Teddies and the Houses shows a good move
in problem solving. I have witnessed others deciding to write down
the 16 possible combinations and when drawing the solution will
cross out each combination as they use it. This is a
problem-solving strategy that they come up with that for them may
Other pupils will have some inkling that they want to use
coloured cubes to represent the Teddies and the Houses. They may
even go so far as to sort out the 16 different combinations before
trying to arrange them in the grid. All sorts of systems are tested
out. A row might be put down and then the next, following the rules
carefully. Then they find that they can only half-complete the
third row and nothing else can be placed. In the problem-solving
world they have to decide how far back they have to go, before
trying some different combinations. Eventually they get there and
come up with a solution, maybe like this one (the smaller square
representing the Teddy, the larger one, the House):
I have always considered it a privilege to sit with children
when they are going through a problem-solving exercise like this
and see the work that has to be done. In challenges like this it is
good to be able to observe the very many different problem-solving
skills that are used. It also sometimes shows a pupil that certain
known approaches do not always work. For example putting the houses
in an array like this:
seems to have worked in some previous problems. But in this
challenge does not work. The way in which the pupil deals with this
can prove to be a valuable learning situation.
I have used a similar challenge, Tea Cups
, over one
hundred times and youngsters like it. Some have remarked that they
liked it being wrapped up in a story because if the challenge were
simply put with no story attached they would not be keen to solve
This next example is to do with numbers but is not
straightforward and so becomes a problem-solving situation for Key
Stage 1. The problem is called Birthday Cakes
was published on NRICH in May 2003.
A learner called Alice, who sent her solution into NRICH,
"First I got 24 lollipop
sticks then I fiddled around and sorted out Jack's birthdays and
put in Kate's birthdays where they fitted. I had three tries to get
She sent in this picture:
I believe this is a particularly good challenge for problem
solving as the answer is very unlikely to be readily available for
pupils at Key Stage 1, and many at Key Stage 2. Many pupils will be
able to access the solution by a variety of routes. But as well as
the teacher not intervening, the pupils need to have some feeling
that they could explore this and get at least somewhere near an
answer. I feel also, that many pupils will take to the question as
it may be something they can relate to. They must also know that
they have the freedom to think for themselves and are free to get
whatever apparatus may be of use.
Here is a final Key Stage 2 activity, Brush Loads
published on NRICH in July 2006.
Lan explored the pattern for greater numbers of cubes, as did
Rohaan. Lan describes the method used:
"From counting cubes, we
With 5 cubes, the least
number of BLs is 15, and the largest is 21.
With 6 cubes, the least
number of BLs is 16, and the largest is 25.
With 7 cubes, the least
number of BLs is 19, and the largest is 29.
With 8 cubes, the least
number of BLs is 20, and the largest is 33.
With 9 cubes, the least
number of BLs is 23, and the largest is 37.
We remark the
With 5 cubes, the least
number is 15, and the largest is 21.
With 6 cubes, the least
number is 15+1 = 16, the largest is 21+4 = 25.
With 7 cubes, the least
number is 16+3 = 19, the largest is 25+4 = 29.
With 8 cubes, the least
number is 19+1 = 20, the largest is 29+4 = 33.
With 9 cubes, the least
number is 20+3 = 23, the largest is 33+4 = 37."
Rohaan then made some more observations:
"The largest number is
always (4 x number of cubes) + 1, and the smallest depends on
whether the number of cubes is odd or even.
Odd: (n-5) x 2 + 15 and
Even: (n-6) x 2 + 16"
So, having got some results by problem solving they are able
to explore further by noticing relationships. Other pupils may take
it further still by trying to understand WHY these relationships
One of the difficulties that have to be overcome with this
challenge is in the counting of the faces that are visible. I have
observed many children missing ones out or counting some faces
twice. It's a problem that needs solving and so they have to invent
a method for getting the counting accurate. It will not be a
problem for pupils who have come across similar situations in which
they had missed or counted again, and learnt a way round it.
To sum up, here are some final thoughts about "doing" problem
solving in the classroom:
- The teacher is the creator of the environment that allows
problem solving to take place.
- Investigations are a very good source of challenges to present
- Adults involved need to be more of a listener than a
- Pupils need to know that they are respected and their thoughts
and ideas are valued.
- The teacher should not push the pupils down their own route but
allow the problem-solving pupil to problem solve.