Playing (Mathematics) Games - When is a Game Not a Game (Gough)

7/26/2019 Playing (Mathematics) Games - When is a Game Not a Game (Gough)

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Playing (Mathematics) Games: When is a Game not a Game?

Dr John Gough - Deakin University (Burwood Campus) - [email protected]

[FromAustralian Primary Mathematics Classroom (APMC), vol. 4, no. 2,

1999, pp 12-17.]

When is a mathematics 'game' not a game? Imagine we have a 3x3 grid, as for

Noughts-&-Crosses, and nine bottle tops, numbered 1 to 9. One person can be

asked to place one bottle top in each cell of the 3x3 grid, so that every row of

three cells, every column of three cells, and the two diagonals add up to a total

of 15. Yes, this is the familiar Magic Square task. I call this kind of thing a

puzzle. It can be made to feel like a game, if the 'player' is told not to move

any bottle top that has already been placed, until all have been placed; then, if

one of the correct 'winning' patterns has not yet been made, thereafter the

player is allowed to swap any two of the bottle tops. But even though there isnow one 'player', and defined 'turns' or 'rounds' of play, and a 'winning'

position, it is still a puzzle. Tetris, the computer game, is a puzzle-race, a

puzzle with a randomising element and a time-limit. If you think of it as a

game, who are you playing against? How does the opponent win the game?

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. Snakes and

Ladders is not a 'game': even though the players take 'turns', and the first to the

end is the 'winner', the players have no choice about what they can do in their

turnthey just follow the dice, plus the possible consequences of ending on a

ladder or snake. Pure luck. No choice. No interaction between players, so that

what I do in my turn can effect what you can do in your turn.

A genuine game can have an element of luck, from a dice, spinner, coin, or

draw of card or domino. But no 'game' is 100% luck. Players must be able to

choose how to use their luck, as happens frequently in the dice-driven games

Monopoly and Backgammon. But Backgammon, no more than a glorified

version of Ludo, itself a version of Snakes and Ladders, is only played by

adults because of the extra interest that comes from the gambling related to

who will win, and at what agreed odds. I believe any game that relies ongambling to make it interesting (poker, 21, and Two-Up spring to mind) has

no place in a socially responsible school curriculum. The exception is the

study of gambling within the probability (or Chance and Data) curriculum,

showing that the odds of the casino or house controlling the game are rigged

against the players to make a profit!

Is it clear what I have in mind by the word 'game'?


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The Magic Square puzzle becomes a real 'game' when two players take it in

turns to place one of the unused bottletops in any of the empty cells of the 3x3

grid. The loser is the first player who is forced to place one of the bottle tops

so that the numbers in one or more of the columns, rows or diagonals cannot

add up to the required total, or, alternatively, both players win if the numbers

are all satisfactorily placed and the magic square pattern is jointly achieved.(Versions of this game are called Fifteens and Fifteens All, in Hill and Gough

1992 p 34).

Such games can be reversed, in which case the playing strategy is altered, and

the game-thinking must change. What would be a 'win' in one version of a

game is defined to be a way of losing. This is the 'misere' version of the

original game. Misere versions can be devised for almost any game, with a

dramatic change in the playing strategies and thinking. For example, Misere

Draughts is a version of Draughts in which the winner is the first player to

lose all of his or her pieces: a player must jump if it is possible to do so.Misere Noughts and Crosses makes the winner be the first player who does

not complete a row or column or diagonal of all Xs or all Os.

Bingo, although it shares many features of a game, is not a 'game', by this

definition. Players have no choice about what to do when a number is called:

they either have or do not have the numberpure luck. Clearly skill, and

choice are important aspects of a real game, although of course some games

combine this with some amount of luck. Similarly 'races', where two or more

players each try to be the first to correctly answer a mathematics question, are

not 'games': what one player does in a turn does not effect what another playercan do in the next turn, except, perhaps, to eliminate one or more unsuccessful

player. Likewise Buzz is not a game. Can you see why not?

One exception to these general game characteristics is the guess-my-secret

game, where logic and successive rounds of cunning question-and-elimination

aim to discover a 'secret'. Mastermind is the classic commercially published

logic game which fits this pattern. Dictionary, a little like Hangman, is

another, less well known, but easier to play. One player chooses a word to be

the secret 'target', and other players take turns to guess the secret wordby

suggesting a particular guessed word, and being told by the seceret-keeper

whether or not each guessed word comes alphabetically before or after the

secret word, just like the better known Guess-My-Number game, where the

person with a secret chooses a secret number between 0 and 10 000 (Gough,

1992 p 25, Hill and Gough, 1993, p. 81).

A danger in using mathematics games to introduce a concept (such as using

Battleships to start children thinking about coordinates), or to practise a

concept formally treated by ordinary classroom instruction: students may be

so distracted by their natural interest in playing to win, that they fail to focus


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on the mathematics. For example, they may use coordinates, without really

learning about them. Worse: with Battleships or street-directory letter-digit

coordinates, the letter-part can never be confused for the number-part,

however much fun, which is poor conceptual preparation for using number-

number or (x, y) Cartesian coordinates where the order of the two coordinate-

parts really matters.

A related danger is learning incorrect lessons. For example, children playing

Ludo need to roll a 6 before bringing a counter into play from their Home

Base. Given seemingly interminable waits for a 6, children can mis-learn that

a 6 is harder to roll, using an ordinary dice, than any of the other numbers

(which are not so critically necessary in Ludo).

I am not suggesting that non-'games', such as Buzz, should not be used as

learning or practice activities, although with Buzz the process of elimination

means that the weaker students who would benefit from Buzz-experience arethe ones soonest left wanly watching. So change the rules so that they have,

say, 'nine lives' before being eliminated. Similarly Bingo can provide good

practice, and students can enjoy it. But there are more important issues.

The best games are those that students spontaneously choose to play when

time is available. I doubt that any children in their right mind would choose to

play, say, Tables Races, or Fraction Bingo, or Buzz, or many other standard

mathematics-practice games, except in a please-the-teacher school context.

Yet many students willingly play Chess, Draughts, Ludo, Scrabble,

Buccanneer, 500, Monopoly and other games. Apart from the arithmetic

involved in scoring, and problem solving in developing good playing

strategies, students are rarely aware of mathematics in these games, and regard

them as non-mathematical. (Where is the 'mathematics' in, say, Scrabble? Far

more than just scoring!)

Better still, although they involve a great deal of repetitive practice, these

games are not merely good for practice. Students typically learn to play these

games by actually playing them. Good games are in themselves learning

experiences, not merely practice experiences. Very young students gain

invaluable experience with alphabetical order by playing Dictionary, without

first requiring them to learn their alphabet. The game becomes a context inwhich students are keen to learn something new. Playing includes a lot of

practising. Learning can be concurrent, and need not be done before playing,

or learning to play the game.

Now for an example. Mancalais a strategy board game for two players. It is

also known as Wari, Owari, Swahili, Pitfall, and many other names. The game

may go back as far as Ancient Egypt. Versions are now played almost

everywhere that Africans or Muslims have spread across the world, from


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Africa to Hawaii, West Indies, Indonesia and the Philippines. Amazingly, the

game is little known in the English-speaking world, despite the high

reputation it has with such game enthusiasts as R.C. Bell, Sid Sackson, and

Herbert Kohl. The version I give here is a simplification Herb Kohl made,

from a Philippine version, for his classes of Afro-American students in New

York ghetto schools. 'None of the children had played Wari before and so[unlike with chess] everyone started out as equals' (Kohl 1974 p 156).

Interestingly Howard Gardner (in Frames of Mind, 1983 pp 161-162),

discussing an aspect of number sense, says 'If one is looking for instances

of highly developed numerical ability in Africa, the best place to look is in a

game such as kala [mancala] a pit-and-pebbles game considered the most

arithmetical game with a mass following anywhere in the world'.

Where is the mathematics in Mancala?Any board game poses spatial

thinking problems, and tactical (how to do best in this move?) and strategic

(how to do best through a sequence of moves?) problems. Can we evaluate themathematics learning that occurs as children develop their playing skill? We

can start by asking children to reflect on what they think they are learning, and

then to try to identify what kinds of mathematics they are aware of using as

they play. Are children beginning to plan ahead, to anticipate the responses of

their opponents, and less often struck by 'I wish I could have that move back

again' reactions? If the game is genuinely successful, children will choose to

play it in their genuinely spare time, making their own set to play the game at

home, and teaching the game to parents and friends.

Why play a game, such as Mancala, instead of something like FractionDominoes, Time Bingo, or Tables Races, or Fifteen All, where arithmetic and

number facts are far more prominent? This depends. If you want dedicated

practice, play a more mathematically dedicated game. We can also justify

teaching games because students need to use recreation time effectively

now, and as adults. We need a Recreation curriculum which, for reasons of

cultural continuity and social cohesion, includes the best of the world's great

games, such as chess, draughts, go, dominoes, 500, nine men's morris, and

mancala, for example. These are very enjoyable to play, in their own right, of

courseotherwise they would not be great world games. They also entail lots

of (sometimes subconsciously) intuitive mathematics.

Players 2

Board Mini-Mancala playing board, or half an egg carton and two larger

cups.


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Equipment Beads, marbles, shells, beans, or suitable counters.

Aim The player who ends with the most beads by moving around

the board according to the rules, placing and capturing, is the winner.

Setting Up At the start, place three beads in each small cup.

Playing Players take turns. In each turn a player picks up all the

counters in any one of the small cups on his or her side of the board. Then,

moving clockwise around the board, the player 'sows' one of the set of picked

up beads in each successive cup, until all the picked up beads have been

'sown', like 'sowing' seeds in small holes in the ground. If a piece lands in one

of the player's Home Cups that then stays there as a point for that player. If,

also, at the end of a move, the last piece lands in what had up until then been

an empty cup of that player's, then all of the opponent's beads in the cup

immediately opposite (above or below, on the board) are 'captured', that is

they are removed from that cup and placed in the moving player's Home Cup,

adding to the final score for that player.

Scoring The game ends when one player has no more pieces on his or

her side. This player scores the number of beads in his or her Home Cup. The

other player, who has some beads left on his or her side of the board, scores

the number of beads in his or her Home Cup, minus the number of beads left

in the other cups.

Variations Play with six small cups on each side, and start with 3, or 4, or

5 beads in each small cup. For an even simpler game, start play with three

small cups on each side, and only 2 beads in each small cup.

References and Further Reading

Bell, R.C. (1960).Board and Table Games, Oxford University Press, London.

Diagram Group, (1975) The Way To Play, Paddington Press, London.

Gardner, H. (1983). Frames of Mind: The Theory of Multiple Intelligences,

Basic Books: Harper Collins, New York.Gough, J. (1992). 'Dictionary', Prime Number, vol. 7 no. 4, p. 25.

Hill, T., Gough, J. (1992). Work it Out With Maths Games, Oxford University

Press, Melbourne.

Kohl, H.R. (1974).Math, Writing & Games in the Open Classroom, Random

House, New York.

Sackson, S. (1969).A Gamut of Games, Castle Books, New York.

31 Mathematical Games for the Classroom, AAMT, undated.

Playing (Mathematics) Games - When is a Game Not a Game (Gough)

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