trol = Teacher Resources on Line = trol
Games in the Classroom
|What research has been done into the use of games in mathematics seems to have been concerned with games that were devised for the teaching, practising and/or reinforcement of specific skills such as negative numbers, coordinates, fractions etc., which are referred to (here) as 'educational games'. Generally, that research has reported good results both in terms of motivation and in the measured progress in learning that took place. The interest here is in games of a type that could well be played by choice to occupy moments of leisure which, to distinguish them from 'educational games', are here referred to as 'mathematical games'.|
First of all there is a need to clarify what we mean by 'mathematical games'. Games is an all embracing word covering such diverse items as boxing, chess, cards, football, hop-scotch, polo, skittles and many hundreds more. Here we shall confine ourselves (more or less) to games which
* have only 2 players
* involve only thinking skills
[No physical activity other than moving a counter, making a mark etc. is required.]
* offer full information at all times
[The state of play is clearly visible to both players and there are no hidden elements as in playing cards.]
* do not, in general, involve luck
[Exceptions to this will be made.]
* people might well play for pleasure
[This excludes 'educational games'.]
* are usually finished within a reasonable span of time
[This is merely to exclude such games as chess, draughts etc. which, though acceptable in other respects, are impracticable for our purposes. Some games can go on without end since they allow repeatable positions. These are not necessarily excluded, but an end needs to be defined, perhaps by limiting time or the number of moves allowed, before a draw is declared.]
* require a minimum of special equipment
[This allows multiple copies of a single game to be made or reproduced as cheaply as possible, and rules out proprietory games needing special boards, pieces etc. - as well as (usually) having an extensive set of rules.]
Two simple well-known games that embody all of the above are
Noughts and Crosses (Tic-Tac-Toe) and Nim.
How can games be used to further mathematical education? This question has to be addressed and answered if we are to devote time and resources to playing games in the classroom. Let us first look at some questions which players might pose to themselves on settling down to play a game, and the mathematical heading under which we might class such a question
|Form of question:||Mathematical heading:|
|1. "How do I play this?"||Interpretation|
|2. "What is the best way of playing?"||Optimisation|
|3. "How can I make sure of winning?"||Analysis|
|4. "What happens if . . . ?"||Variation|
|5. "What are the chances of . . . ?"||Probability|
|Given a chance to develop answers to questions like that could lead to statements commencing as listed below, together with the mathematical idea being covered in such a statement|
|Form of statememt:||Mathematical idea:|
|6. "This game is the same as . . ."||Isomorphism|
|7. "You can win by . . ."||A particular case|
|8. "This works with all these games . . ."||Generalisation|
|9. "Look, I can show you it does . . ."||Proving|
|10. "I record the game like this . . ."||Symbolisation and Notation|
|Of course, none of it is as clear cut as that tabulation seems to suggest. Broadly speaking though, the first five do cover the implicit mathematics that can go on when these games are played, while the second five suggest the opportunities offered in seeking responses and making them explicit. Not all games offer all these possibilities, just as not all pupils are capable of dealing with them, but the potential is enormous if you know what to look for.
Making the Link
The above rough, but workable, connections made between games and mathematics indicates two basic approaches to the use of games in the classroom. The first five connections indicate that a covert approach is possible, where the games are played and the mathematics is (assumed to be) done intrinsically. The second five need demands to be made for some extrinsic mathematics to be done, perhaps via discussion, but more likely requiring some written output. There is also the possibility of an in-between position. Structured opportunities for playing could be provided, together with suggestions as to how an investigation might be developed that would be acceptable for assessment purposes for any who were inclined to adopt it.
Clearly the strategy to be used will be the decision of the teacher, and would be formed in response to all sorts of factors - the ability of the pupils, their motivation and sociability, the ethos of the school, the degree of autonomy that the teacher has, and so on. In order to integrate the idea of games into the mathematics curriculum, the scheme adopted by one school was as follows.
Year 7 pupils (11/12 year olds) had one period (30 minutes) for games-playing each week. It was limited only to playing the games (the covert or intrinsic model) using the same game for the entire class, and a different game every week. Inter-class games were organised from time to time to provide variety. It was a very popular activity.|
Year 8 pupils (12/13 year olds) played intermittently as a leisure activity, not only to fill in some odd moments but also to keep them aware of games.
Year 9 pupils (13/14 year olds) in some groups played games on a semi-regular basis, usually with written work being required at the end. This would be of a fairly simple nature, being structured, and usually required to be done as a homework.
In Years 10 and 11 (the examination groups), organised games-playing was done only on appropriate occasions and usually there would be one of two possible follow-up modes: either, everyone was required to submit a piece of work in response to an open-ended question or, guidance was given as to how those who cared could undertake an investigation arising from the game just experienced. Just once in a while a classroom lesson might follow to discuss and/or demonstrate some possible developments. Of course, in all years, games were much used by teachers as an end-of-term activity.
Is there any need to actually play the games if the object is merely to provide a setting for doing some mathematics? After all the game could be explained very quickly and the follow-up work started immediately. However, it is best that the games are played properly for three reasons. First of all there is the intrinsic mathematics which always present. Second, there is the high level of interest and motivation which games-playing generates. Third, and perhaps most important, is the deeper understanding of the situation to be worked on which can be gained only by playing through several games. Far too often students are expected to try and analyse a situation of which they usually have little or no previous experience. Here at least there is a chance to remedy that defect. To provide everyone with experience of a common activity, a system needs to be put in place which affords playing opportunities to all.
There are many different ways of setting up a system for the playing of games. These vary from a no-system free-for-all (anybody plays anybody at any game) to the sort of framework encountered in competitions (ladders, knock-outs, leagues and other variations). For classroom purposes the best is probably the progressive model where everyone is playing the same game but the players have some routine of changing around after each game (or set of games) so that, as far possible, no one plays against the same opponent twice. This is very simple to organise and, once everyone is familiar with it, just about runs itself.Progressive Games-playing.
Assume all the playing positions (table or desks) are numbered 1 to n (actually or notionally) with two players in each position. Designate which player of each pair is to be the 'mover' (say the left- or right-hand player). This designation then stays the same throughout the playing session. Every time a game (or a set of 3 or 5 or . . . games) is completed then the designated player moves up one numbered position: 1 to 2, 2 to 3 etc. and n moves to 1. A practical point is that no moving may be done until, every pair having finished their game (or set of games), the controller (=teacher) calls "Move!" If this is not implemented then queuing will inevitably result and chaos will swiftly follow. A plus for this mode of play is that by the time the third or fourth move is made everyone should be playing to the same rules!
A little more interest can be added by supplying score cards on which results can be recorded. If there is an odd number of players then, either the teacher can become a participant or, the moving players all have a turn at 'sitting-out' and this happens naturally. If scoring is being used then sitting-out should count as a 'win', or perhaps score half-a-point, as appropriate.
|The Analysis of Games|
There are four main lines of attack when analysing games and trying to find a winning strategy.
* Make a tree diagram of all possible positions that can develop from the starting position. This method is very laborious and takes up a lot of space but must lead to a solution eventually. Usually it requires a notation to be devised in order to keep an economical record of any particular position, and that by itself can provide insights into the game. But very often, once the tree has been started, shortcuts become apparent and it is not necessary to draw the complete tree.
* Work backwards. In this case, if a winning position is set up you can then consider the possible positions that must have existed before that one, and then before that and so on. It is rather like a tree-diagram in reverse. However, what can happen in this case is that, certain positions which always lead to a winning position become apparent, whilst others can be discarded. So the backwards development need only proceed from the 'good' positions which obviously are the better ones to aim for.
* Make it simpler in some way, usually by making it smaller. Thus if the full game is played on a grid of five by five cells, try playing and analysing it on a three by three grid. If an analysis cannot be found for the simpler version, it is very unlikely to be possible for the more complex case.
* The special case of NIM. The analysis of this well-known mathematical game was devised nearly 100 years ago and should be known to all students of this subject. Thetype of analysis used can be applied to many other games.
It is as well to remember that not all pupils like playing games - especially if they are full-time losers! Apart from that obvious reason, there are some people who just do not like games at all, at least, not of the variety we are considering here; fortunately they seem to be a small minority. Games in which chance plays a part, perhaps where a die or spinner or coin-tossing or some other randomising device is used, can be helpful in giving weaker players a better chance. Such games also serve to introduce the topic of probability.
Some games involve arithmetic. Triangle Sum (to be found in Games 1) is one. In order to retain the game-like feel it is essential that the arithmetic is kept simple. It may be tempting to insert larger numbers into the game or set higher final target totals, but this should be resisted. The arithmetic should be easy enough to be done without any major effort or else a "real game" becomes an "educational game".
Another factor to be considered is whether the whole business of equating games with doing mathematics is a good thing or not, in terms of pupils' general development, but that is a broader matter which only needs bearing in mind before going too far. After all,playing games is meant to be a social activity, a hobby and a relaxation!