Types of Problems
All types of mathematical problems serve a useful purpose in mathematics teaching, but different types of problem will achieve different learning objectives. Some problems require recall of facts and procedures, some stimulate different strategies, some depend on logic and reasoning, some have multiple solutions, and others demand decision making and creativity. In general, the more open-ended types of problems have greater potential for stimulating higher order mathematical thinking. This is partly because they usually involve a search for patterns and relationships between elements in the problem.
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.
Benefits of Open Problems and Investigations
In the context of classroom teaching one major advantage of using open problems and investigations is that, because there are multiple solutions, they cater for a wide range of mathematical abilities and stages of development in children. This is particularly evident with investigations that begin with the manipulation of some concrete material, because this allows `access' to the mathematics of the problem for less able children. The more able and experienced the child, the more sophisticated the investigation can become. Being `successful' can range from finding one possible solution in the form of a physical model, to a systematic presentation and explanation of every possible solution. This makes open activities valuable assessment opportunities, because the artificial limitations usually placed on children are removed and they have the opportunity to show what they are really capable of. Hidden talents are often revealed. Another benefit of multiple solutions and pathways is that this provides a rich source of material for mathematical discussion, which adds depth to the learning experience. Paragraph 252 of the Cockcroft report states:
``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.''
Opening Up Problems
Initially, it is easier to think of basic closed word problems than it is to devise open-problems, but luckily, a good source of open problems is actually these same basic closed problems.
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
||The final amount of money
||The number of coins
||All coins have the same value
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.
- 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?
All of these examples are more challenging than the original problem. Instead of recall of a single multiplication fact, a much greater range of mathematical skills is now required. It has become necessary to consider various combinations, maximums and minimums and methods of recording and checking for all possibilities. There is scope in several of these new problems (for example; 1, 2 and 5) for systematic recording of combinations of coins and for searching for patterns. While all of the examples of more open problems have multiple answers, numbers 2 and 5 require quite a lot of decision making to set some parameters and choose a pathway to explore. Therefore, these two `problems' could be considered to be investigations. Less experienced or less able children would benefit from working with real or toy coins. Having real objects to manipulate and create a visual display with, supports memory and increases the likelihood that they will look for similarities, differences and patterns in their solutions. Children who are reluctant to make written records may persist with an investigation longer and be more productive, if they can build a display of their thinking as they work. For example, if they were working with paper coins, these could be arranged and rearranged, then simply be glued into place.
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.