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We all enjoy playing games and experience tells us that games can be very productive learning activities. However, before we commit ourselves to playing games in maths lessons, we need to answer the following questions:
- What is the justification for the use of games in mathematics lessons?
- Are some games better than others?
- What educational benefits are there to be gained from games?
This article considers the nature of games and their role in
supporting pupils in thinking creatively - particularly when trying
to identify winning strategies. Oldfield (1991) says that
mathematical games are activities which:
- involve a challenge, usually against one or more opponents;
- are governed by a set of rules and have a clear underlying structure;
- normally have a distinct finishing point;
- have specific mathematical cognitive objectives.
To illustrate some of the points I will focus on the game of
Nim its relatives. Nim-like games have been played since ancient
times. They are thought to have originated in China but the
earliest references to Nim in Europe were at the beginning of the
sixteenth century.
Nim is a strategy game in which players capture objects. The
player to take the last object wins. Nim can also be played so that
the person who makes the laast move (who takes the last object or
reaches a particular total) loses. All the examples used here can
be found on the NRICH website; several of them are
interactive.
Got It!
Got It! is an interactive verrsion of the classic game of Nim.
It involves players taking turns in adding numbers to reach a
chosen target (for example, any of the numbers 1, 2, 3 and 4 to
reach a target of 23).
Pupils are asked to work out a winning strategy and then:
- explore whether it matters who goes first
- identify how the strategy changes if there is a different number
- consider whether changing the target number influences who should go first
- identify how the strategy changes if you have more numbers to choose from (for example, 1-6)
- find a generalisation for who would win for any target number and any group of consecutive adding numbers starting from one
What is a mathematical game?
When considering the use of games for teaching mathematics, it
is useful to distinguish between an activity and a game. Gough (1999) says that:
A game needs to have two or more players, who take turns, each competing to achieve a winning situation of some kind, each able to exercise some choice about how to move at any time through the playing.
The key idea in this statement is that of choice. In this
sense, something like Snakes and Ladders is NOT a game because
winning relies totally on chance. The players make no decisions,
nor do they have to think further than counting. There is also no
interaction between players - nothing that one player does not
affect any other player's turn in any way.
There are many activities that we would describe as games,
which can be used to test mathematical understanding or practice
skills -for example, matching card games, games of odd one out and
so on. There are an enormous number of such games which support
work in number, algebra and geometry, and many can be found on the
NRICH site itself. However, these latter types of "games for
practice" are not of the sort Gough has in mind. Got It! is such a game because it
requires pupils to think about a strategy, not dependent on chance,
to acheive a winning position.
More noticeably, Got It! also offers a context in which pupils
can pose their own problems and investigate the environment created
by the game. They have to work out the mathematics and consider how
maths helps them win the game. Pupils will be engaging in an
investigation of a particular context, but they may be able to
apply their findings to other, similar games. Pupils should then be
encouraged to make their own Nim-like games based on similar
strategies and, through this, consider the underpinning
mathematical principles and think creatively about how to test out
their conclusions. They can also pose questions such as: "Is the
game fair and is there a certain strategy for winning?"
Benefits of Using Games
The advantages of using games in a mathematical programme have
been summarised in an article by Davies (1995) and include:
- Meaningful situations for the application of mathematical skills are created by games
- Motivation. Children freely choose to participate and enjoy playing
- Home and school. Games provide "hands-on" interactive tasks for both school and home.
- Positive attitude. Games provide opportunities for pupils to build positive self-concepts and develop positive attitudes towards mathematics, through reducing the fear of failure and error.
- Increased learning in comparison to more formal activities. Deeper learning can occur through games due to the increased interaction between children and opportunities to test intuitive ideas and problem solving strategies.
- Different levels of thinking. Games can allow children to operate at different levels of thinking and to learn from each other. In a group of children playing a game, one child might be encountering a concept for the first time, another may be developing his/her understanding of the concept, a third consolidating previously learned concepts.
- Assessment. Children's thinking often becomes apparent through the actions and decisions they make during a game, so the teacher has the opportunity to carry out diagnosis and assessment of learning in a non-threatening situation.
- Independence. Children can work independently of the teacher. The rules of the game and the children's motivation usually keep them on task.
- Make sure the game matches your mathematical objective.
- Use games for specific purposes, not just time-fillers.
- The game should have enough of an element of chance so that it allows weaker players to feel that they have a chance of winning.
- Keep the game completion time short.
- Use five or six 'basic' game structures so the children become familiar with the rules - vary the mathematics rather than the rules.
- being systematic
- transforming information (e.g. inventing a method for recording moves)
- searching for patterns
- applying mathematics (e. g. doing calculations and algebra)
- manipulating variables
- working backwards or simplifying the problem
- hypothesising and testing
- generalising (perhaps even producing a formula)
Variations of the Same Game
This small group of activities is taken from the Mathematical
Games Archive on the NRICH site. They all have a related structure
that can be used to develop the skills of strategic planning and
reasoning as well as ideas of analogy and, through this,
generalising. Students at all levels of ability and age can access
them. At the most basic level, they offer opportunities for
practising arithmetical skills. At a higher level, they can be used
to promote mathematical discussion by demanding detailed and
reasoned explanations for a winning strategy, or an explanation of
the mathematics that links the games. Once pupils identify
generalisations by introducing some variables (number of counters
to start with and number of counters that can be removed) further
variables can be explored and new generalisations will emerge such
as the number of piles.
Make a pile of seven counters. Two
players each take turns to remove either one or two counters from
the pile. The player who takes the last counter is the
winner.
Play the game several times to begin searching for winning
strategies. Ask whether it matters who goes first and encourage
pupils to record their moves. Encourage opponents to become
partners in investigation as they test their theories. The aim for
each player is not to win but to find out how to win. To do this,
it is often worth trying ideas out by suggesting that each pupil
play a particular role (always taking two counters, for
instance).
In the classroom the teacher's role is critical. Getting
pupilsto try out and test ideas without leading them along a
particular path is an important skill. Ways to help with this
include listening in on pupils' conversations and spending time
with the class listing some of their conjectures on the board.
Establishing some of the theories pupils have been working on helps
them identify what they have noticed so far. Another approach is to
use a "conjecture board" and ask pupils to write their hypotheses
on it as they work. Pupils come to see the board as their own tool
for thinking aloud and sharing ideas. A nudge from you to a group
that has a good idea will result in a conjecture appearing on the
board.
Encourage pupilsto explain and justify their strategies and
suggest that they try to convince others of its validity. Then,
once a strategy has been identified, it is time for a "What if?"
question.
- What happens if you start the game with a different number of counters? (A series of key numbers will emerge, as well as some interesting observations about odds and evens and multiples).
- What if you can take a different number of counters away?
- What if the player to remove the last counter(s) is the loser?
Make a row of 3 counters, a row of
4 and a row of 5. Two players each take turns to remove any number
of counters from a particular row. The player left with the last
counter is the loser (or winner, as agreed at the start).
Place a counter on each of the
four coloured squares. Two players take turns to move any counter
one, two or three spaces, until they reach the end of the track and
are removed. No jumping is allowed. The winner (or loser as agreed)
is the person left with sliding the last counter of the
track.
This is a game for two players.
You will need a game board and four counters (or coins). If you are
drawing it yourself, count the spots carefully. Place a counter on
each star. Players take turns to move any counter, moving out
towards the snail's tail. A counter can only be moved by sliding it
ahead 1, 2 or 3 spots. Counters cannot jump on or pass each other.
When a counter reaches the tail, it slides of and is out of the
game. The winner is the player who slides the last counter off the
snail.
Does it matter who goes first? What happens if there are more
or less spots to start with? Does it matter if the stars are in
different places? Make up your own Slippery Snail game. What is the
connection between this and NIM?
Links to the Framework
The following objectives are to be found in the Frameworks for
teaching mathematics for Key Stages 1, 2 and 3, and apply to the
activities in this theme.
- Solve mathematical problems or puzzles, recognise simple patterns and relationships, generalise and predict.
- Understand addition and subtraction mental calculation strategies.
- Use letter symbols to represent unknown numbers or variables.
- Represent problems mathematically.
- Explain and justify methods and conclusions.
- Use logical argument to establish the truth of a statement.
- Solve increasingly demanding problems and evaluate solutions.
- Present a concise, reasoned argument, using symbols, diagrams, graphs and related explanatory text.
- Suggest extensions to problems, conjecture and generalise.
And finally. Having
mastered some of the fine detail concerning the strategies
associated with some of these games the next step is to encourage
pupils to think of Nim like games of their own.
This article is based on a number of articles on the NRICH
site produced by NRICH team members. For more information search
for "
games " on the NRICH website.
References
- Aldridge, S. and Badham, V. (1993). "Beyond just a game". Pamphlet Number 21 . Primary Mathematics Association.
- Davies, B. (1995). "The role of games in mathematics". Square One . Vol.5. No. 2
- Gough, J. (1999). "Playing mathematical games: When is a game not a game?" Australian Primary Mathematics Classroom . Vol 4. No.2
- Oldfield, B. (1991). "Games in the learning of mathematics." Mathematics in Schools . January