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 Balloon:  


 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 remembering.
 
"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 balloon!"  
 
"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 fact.

"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 solving it.

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 avoided:

• 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.

Let's now look at another example. Sort the Street.

 
 Lucy said:
 
"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:

• 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 to pupils.
• Adults involved need to be more of a listener than a speaker.
• 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.