Published 2005 Revised 2017

There is debate amongst mathematics educators, as to what the difference is between open problems and investigations. Indeed, some say there is no difference. However, for the purposes of this paper, open problems are perceived as problems that have a clear goal yet have more than one solution. Investigations tend to have less specific goals, if any, and so have many possible pathways and outcomes open to the investigators own choice. There is a tendency for the investigator to pose his/her own problems to solve during the investigation.

``It is necessary to realise that much of the value of an investigation can be lost unless the outcome of the investigation is discussed. Such a discussion includes consideration not only of the method which has been used and the results that have been obtained but also of false trails which have been followed and mistakes which have been made in the course of the investigation.''

Moses, Bjork and Goldenberg (1990) suggest analysing the problem in terms of what is known (information given), what is unknown (what needs to be found out) and what restrictions are placed on its solutions, which makes it easier to then open up the problem. For example: ``How many 5p coins are needed to make 45p?''

Kind of Knowledge | Details | |

Known | The final amount of money | 45p |

Unknown | The number of coins | ? |

Restrictions | All coins have the same value | 5p |

Any of these variables can be altered to produce different problems. If the resulting problem has more than one solution, then it could be described as being more open. For example:

- Remove the restriction: How many coins does it take to make 45p?
- Remove known: I have a closed handful of 5p coins. How much do I have?
- Swap the known/unknown, change the restriction: I have 5 coins. Three are the same. How much money do I have?
- Swap the known/unknown, remove restriction: I have 5 coins. How much money could I have?
- Remove known and restriction, change unknown: I have some coins in my hand. How much money do I have?
- Change known, unknown, and restriction: What is the shortest/longest line that can be made with 5 coins?

Actually, specifying what component of the problem is known, unknown or restriction is not really important. The point is that once the variables have been identified, it then becomes easy to manipulate the problem and create new problems. Therefore, the invention of open problems is not dependent upon the creative mood of the teacher, but becomes a manageable procedure that can be applied to any problem. After a little practice, opening up a problem tends to become an automatic reaction to any basic word problem.

Cockcroft, W.H. (1982). Mathematics Counts. London: HMSO

Moses, Bjork & Goldenberg. (1990). Beyond Problem Solving: Problem Posing. Teaching and Learning Mathematics in the 1990's. NCTM Year Book. Reston, VA.