Copyright © University of Cambridge. All rights reserved.

This article is taken from the introduction to "Geometry Games", a photocopiable resource published by The Association of Teachers of Mathematics .

When I first started to work on the use of games in the classroom, I was amazed that so little justification existed for their use. The assumption seems to be that games are fun and so they are a good thing to do. Setting up games in the classroom is time consuming in terms of preparation of equipment and demanding from an organisational point of view.
Also, in some ways, it leads to a loss of teacher control of pupil learning, as well as the loss of the volume of recorded pupil activity produced by more formal activities. I felt therefore that a deeper analysis of what goes on when games are used as a pedagogic device was needed. Over three years I spent many hours watching groups of children playing games and tried to perceive the kind of
learning opportunities there were. Many of this set of games were invented, often with the help of the pupils, during this time. Others were tested out on seminar groups at the ATM conference.

I observed many interesting outcomes of playing games, some of them were fairly obvious, but others only struck me as time went on. I believe it is important to look at these because it is only too easy for critics to say that playing a game is not doing proper mathematics and that there is no record of what the pupils have done. The latter can to some extent be overcome, if overcome it must be,
by asking the pupils to write about what they did while playing the game.

The outcomes of this work are divided into three categories; learning, ways of working and pupil experience.

Learning

A game can generate an unreasonable amount of practice

By unreasonable I mean that the pupils carry out far more practice in geometric vocabulary and reasoning than they would be expected to do or ever manage to get through if they were faced with a conventional textbook exercise. This happens partly because they are working orally but also because their attention is on carrying out a 'move' in the game and not on how much work they are having to do.
It seems to give a sense of purpose to what is required of them.

Geometric games create a context for using geometric reasoning

Pupils do not necessarily find it natural to work with the properties of shapes, but many of these games require them to relate these properties to the shapes in a flexible way as they try to win the game.

A game will often result in the making of generalised statements

Not all games have this potential, but many do. For example when playing Quadrilateral Sets I was intrigued to be told by one of the players that he would not give this property to square because 'square has such a lot of properties'.

A game can allow the introduction of ideas that are difficult to develop in other ways.

In particular within these games it is the skill of understanding a diagram and visualising its properties and relationships. For example since shapes are shown in different orientations on the cards and the cards are inevitably being viewed from different angles, a lot of work can be done on recognising for example, a square whichever way up it is. Other forms of visualisation such as congruence
and the angle in the same segment are also included within the games.

Games seem to be able to lead pupils to work above their normal level

A game does not define the academic limits of the work in any way and since there is a natural wish to win, pupils will often devise ways of looking at the work they are doing which lead them way beyond what they are expected to achieve. The game situation appears to free the pupils from feeling a need to do something which the teacher wants and expects, thus allowing them to think freely about
the situation.

Ways of working

A game leads pupils to talk mathematics

I cannot prove that this is important but I came to believe that the need to talk about and justify one's moves leads pupils to talk about the properties of shapes far more than they would in completing a conventional exercise. One does not normally talk out loud about the properties of the shapes involved if they are in an exercise! This process was helped considerably by asking pairs of pupils
to work together as a single player. This I would also strongly recommend because...

A game can create discussion of all kinds

This is greatly enhanced by the pairing technique described above, because the partners have to verbalise their ideas about the next move to each other and justify their opinions. This not only helps them, but can be very informative for a passing teacher who can eavesdrop and assess where they have got to in their thinking. Co-operative games can be very useful in creating discussion too.

Games put pressure on players to work mentally

This is obvious in many ways but it is easy to overlook the fact that geometry almost always requires drawing and/or written work for the pupils. The provision of shapes and properties on cards means that they can be worked on mentally rather than on paper. This must be a great relief to those whose drawing skills are limited!

A game does not define the way in which a problem is to be solved or worked out

For example when a class is trying to guess which shape has been chosen in 'Shape Guess' the way in which they pose questions is for them to work out. In a property sorting game there is no laid down strategy for getting the properties and shapes to match.

A game often can be played at more then one level

As a result of the previous property of games, it is possible for the players to play the game with more or less skill or perception depending on their own competence. They will often learn to develop their level of play by watching and listening to the other players.

Pupil experience

This last set of outcomes were the ones of which I only gradually became aware and which I see as of particular interest because so many of them are concerned with the players' feelings about what is happening.

It is acceptable to learn the rules of a game gradually

No-one expects to be able to learn all the rules of a game at once. Pupils do not seem to distinguish between the procedural rules and the mathematical rules. This allows pupils to query the mathematical rules without loss of face. For example it frees them to continue to query the definition of a rhombus until they feel confident about it.

Games are played in a context in which there is usually unthreatening help available

When a game is played by pairs then a pupil's partner is always there to help make decisions about what to do or to explain something which her partner does not understand. In addition in most groups other players will make suggestions if a player is stuck, if only to keep the game moving on. For example when a pairs game is played there is usually ample advice from other players as to whether
they 'match' or not!

The pieces used in a game are concrete objects

When a pupil is faced by a set of examples to work he feels pressure to answer them in the given order and there is a sense of failure if one proves too hard. Very often, in a game, there is a choice of which 'piece' one uses. In playing a property card or a shape one can often choose to use first the one which one feels most confident about. I have had cards held out to me for help - much more
expressive than pointing to a problem on a page! I have even found cards under the table and wondered whether they had been dropped to get rid of them! There are perhaps implications here about setting formal exercises which do not have numbers and are scattered randomly on the page and inviting the pupils to solve them in the order that they prefer. Or even perhaps presenting problems on a set
of cards, but that would be hard to organise!

A game allows a pupil to hide until he feels confident

In watching groups of pupils playing games I noticed that there were those who played quietly and rather mechanically for a while and then suddenly started to join in and make suggestions. It seemed that in a game situation they had the freedom to assess the situation without pressure until they felt they had things sorted out and then to contribute in a more positive fashion.

Clearly there are other types of activity which produce many of these outcomes, only some of them are in any way unique to the game playing situation. However what seems clear to me is that games are a way of reducing teacher domination and control of the situation in a way which has positive outcomes for the learners.

Introducing games in the classroom

If a game is to be played by the whole class at the same time, working in groups of 6 or so, then clearly some thought has to be given to how the pupils are taught the rules and procedures of the game. What follows here are some thoughts about how to make this as efficient as possible.

It will be noted that although the games in this booklet are organised by the area of geometry to which they relate, there are in each section games of a similar type, e.g. pairs games. The re-use of different types of game for different mathematical purposes simplifies their introduction greatly as the class have only to be told that it is a pairs game or a happy family game and they know how to
play it. At most it may be necessary to look at the particular type of card to be used.

In teaching games to large groups I have found three different methods that work well depending on the game and the situation.

- Introduce the game to one group of pupils while the others are completing some individual work and then divide the whole class into groups, putting one of the first group into each group to teach the game to the group.

- Play the game with the class divided into the groups in which they will subsequently play and play the game with the whole class, each group acting as a single player.

- Choose a set of pupils to come to the front of the class and play the game as a demonstration, possibly with assistance in decision making from the whole class. If this is done it may be useful to have large-size cards, which can be seen by the whole class or cards made from OHT film and cut up so that they can be projected.

The use of pairs working as a single player has already been discussed as a way of encouraging discussion and indeed concept development. It has other advantages in that a larger number of pupils can use the same set of materials, so that there are fewer groups in the classroom to set-up and manage and less materials to produce and store!

Storage is inevitably a problem and I found myself losing and remaking sets of games until I adopted the system of keeping a class set of any particular game in a plastic box clearly marked with its name! Plastic bags of games do not work at all well. If the pupils can manage the size of card shown in this book, then it is quite easy to store class sets in a small box. I found that they were
big enough, but the cards can, of course, be enlarged if this is felt desirable.

ATM have also published What Kind of Game is Algebra? , a photocopiable resource with more than 50 algebra games also produced by Gillian Hatch.