# Mathematical Games as an Aid to Teaching Mathematics

Mathematical games develop mathematical communication as pupils explain and justify their moves to one another. In addition, games can motivate students and engage them in thinking about, and applying, concepts and skills. Games give pupils an opportunity to communicate their ideas and justify their thinking.

In using games, the teacher plays an important role in encouraging pupils to explain their thinking and in keeping them focused on mathematical ideas. Asking them to explain and justify their moves during a trial round of the game played as a whole class demonstrates the type of thinking and communicating that is important for students to use later when they play the game in pairs.

Games contribute to the development of knowledge by having a positive affect on the atmosphere in the class which in turn produces a better mental attitude towards maths in the pupils. Educational games provide a unique opportunity for integrating the cognitive, affective and social aspects of learning.

### Using Games Successfully

The success of the games as learning tools depends on the teacher's talent in asking probing, open questions and ultimately how well the teacher establishes a classroom climate that encourages experimentation. Ultimately the focus must be on cognitive processes rather than on the correctness of final outcomes. The process by which 'wrong' answers are reached should be valued as much as processes producing 'right' answers.

Ernest (1986) claims that the success of mathematics teaching depends to a large extent on the active involvement of the learner and playing games demands involvement. Games cannot be played passively: players have to be actively involved. For this reason psychologists including Piaget, Bruner and Dienes suggest games have a very important part to play in learning, particularly in the learning of mathematics.

### Selecting a Game to Use

When considering what games to use it is vital that the context which they are to be used is considered. The thinking behind each game should be analysed and matched to the learning objectives that are to be met.

Looking at some of the questions which pupils should ask themselves when starting to play a game, and putting them under a mathematical heading gives a good idea to the higher order skills involved.

How do I play this?Interpretation
What is the best way of playing?Optimisation
How can I make sure of winning?Analysis
What happens if...?Variation
What are the chances of...?Probability
Form of statementMathematical idea
This game is the same as...Isomorphism
You can win by...A particular case
This works with all these games.Generalisation
Look, I can show you it does...Proving
I record the game like this...Symbolisation and Notation

Clearly the strategy to be used is the decision of the teacher, and is dependant upon various factors like the ability of the pupils, their motivation and sociability, the ethos of the school, and the degree of control that the teacher has.

### Methods of Playing Games

When games are used it is important that they are played properly for three reasons. First of all there is the intrinsic mathematics which is always present. Second, there is the high level of interest and motivation which playing games generates. Third, and perhaps most important, is the higher order understanding of the problem to be solved, which can only be gained by playing through different games. From this there are lines of attack which can be used when analysing games and trying to find a winning strategy. The teacher should demonstrate these and develop the skill in the pupils. For instance try making it simpler in some way, usually by making it smaller. If the full game is played on a grid of five by five cells, start the pupils playing and finding solutions on a three by three grid. If a solution cannot be found for the simpler version, it is very unlikely they will for the more complex case.

### Different Types of Games

It is important to remember that not all pupils like playing games, especially if they have weaker social skills. Others may not like playing games of any type because they do not like the competition. However, these pupils seem to be a minority. Children with weak number skills will not enjoy activities where this puts them at a disadvantage so using games which are non-competitive or involve an element of chance are best. Games in which chance plays a part, perhaps where a die, spinner, coin-tossing, or some other randomising device is used, can be helpful by giving weaker players a more level playing field. Such games are also a good introduction to the topic of probability.

Many 'patience' card games involve ordering the numbers forwards or backwards, whist 'beanie' and other games based on winning tricks, require pupils to decide which is the largest of a group of numbers.

#### Pontoon

Pontoon is an excellent game for practicing addition and subtraction of numbers up to 21. Because it contains an element of chance, pupils can play happily with the rest of the class without being at a disadvantage because they are weaker at maths. The gambling aspect can be left out of the game, but playing with matchsticks or other small objects adds more fun and extra counting practice.

#### Dice and Counters

Any game which involves throwing dice and moving counters helps build confidence with numbers. These games can be made more difficult by using two dice and working out the move by adding the two numbers or finding the difference between them. The numbers can also be multiplied together but this often means that the games are completed too quickly unless, like Monopoly, it involves travelling around a board many times.

#### Score-Keeping

Score keeping comes into many games. Mostly this just means adding numbers together but some games are more complicated. Scrabble and darts both involve multiplying by 2 and 3. The darts game 301 provides excellent practice at subtraction and this can be developed to work with negative numbers.

#### Pairs

Games involving pairing cards can be very flexible. For instance the pairs of cards can form the two halves of an equation, marked with two equivalent fractions or a percentage and its decimal equivalent.

#### Computer Games

When choosing computers and electronic games, they should contain the following features:

• Several levels of difficulty so that differentiation is allowed.

• The ability to practice one skill at a time.

• More than one attempt allowed before the correct answer is given. This allows time for rethinking and means that accidentally pressing the wrong key isn't a disaster.

• A response to a wrong answer which is less interesting than the response to a correct one. Many games fail on this point.

### Personal Experience

When choosing any game it is important to remember that they stop being fun if they are used all the time. From this Researcher's own experience, having used games whilst teaching a higher set year seven class and a bottom set year eight class, it can be seen that games can play an important part in teaching maths:

On three occasions with year eight, whilst working on a unit about variables which involved expanding brackets, I used a multiplication bingo game. The idea initially was to increase pupil's quick recall of basic multiplication facts. The first time proved to be a very difficult lesson. The pupils were very keen to play but did not abide by the rules calling out answers to the multiplications drawn from the bucket. Although this defeated the object of the game I decided that they should continue as it was good practice for the next time.
One week later when we played the game again they pupils took it much more seriously and did not call out answers. There was a new atmosphere of competition in the class and I was very happy with the response to the game. In the next lesson I was able to try a few variations on the theme to test the pupils' knowledge further. To start I discussed with the class what was actually involved in playing the game. Once we agreed that we were marking off numbers that were the product of two others I then asked the question 'Which numbers would you not want on your bingo card?' I was very pleased that the correct response came from a pupil and in this way I was able to introduce the idea of 'prime numbers' which was new to a lot of the class. On pre-prepared blank bingo cards I instructed the pupils to write ten numbers of their own choice between zero and fifty. By keeping the cards after the game I was able to check through their choice of numbers to see if they understood the concept of prime numbers. A further dimension to playing the game was added when I decided that instead of pulling multiplication cards out of a bucket, I would point to a number on a pupil's card and they would have to shout out the requisite multiplication pair. This not only gave the game a new feel but more importantly gave the pupils practice at the reverse operation of factorising.

In this case, playing the games was beneficial to the pupils. In the bottom set class there were a few pupils who were very disillusioned with maths. However, even if their number skills had not improved, it became very clear when on numerous occasions it was asked if the game could be played again that their confidence and attitude towards maths lessons had.

When using games with the year seven class it became immediately evident that they were more receptive to playing / working in this manner. This could be because of the difference in top and bottom sets or because the pupils were used to playing games in primary school and had not lost the skill yet.

### Conclusion

From the research that this Researcher has made and the - all be it limited - experience at using games in teaching, it seems that they are very beneficial to the pupils learning under the right conditions. The games must be appropriate for their use in class and the teacher must be clear about the objectives of the game. Certain games such as Monopoly and the variations of this are best used for out of class activities such as math clubs. Games such as classroom darts and fraction pairs have a clear link to a mathematical topic and are a very useful classroom activity.

Most importantly the pupils need to be practiced at using games and solving problems analytically so that they do not waste valuable time. This method of learning is common in primary school but is not carried on through secondary. If time is dedicated to these types of activities in the schemes of work throughout Key Stages 3 and 4, the pupils' problem solving skills become finely tuned, the teacher is freed from the feeling that they are 'losing time' by playing games and methods and strategies for future course work can be developed at an early age.