Playing Games to Learn Mathematics (Gough)

7/26/2019 Playing Games to Learn Mathematics (Gough)

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Playing Games To Learn Mathematics

John Gough Deakin University (retired, 2012)

[email protected]

[From Judy Mousely & Mary Rice (ed.s)Mathematics: of Primary Importance. Brunswick:Mathematical Association of Victoria, 1993: Proceedings of the 30th Annual Conference of

the MAV, La Trobe University, 2, 3 December 1993; pp. 218-221.]

Abstract

The word game, applied to mathematics games, is often poorly understood or defined. It

gets confused with puzzles, and worksheets or other activities that are recreational.

However, it can be useful to adopt a clear definition: a (mathematics) game involves

mathematics or mathematical thinking, has two or more players who take turns, and

requires some player choice and interaction. This is distinct from one-player game-like

activities that are more clearly labelled as puzzles, and different from luck races and

dressed-up worksheets.

Introduction

We take for granted the ideas that mathematics games are fun, a painless way of learning

number facts without drilling, and that anything apparently active is better than a page of a

textbook, a blackboard full of "sums", or a morning of chanting tables and mental

arithmetic. But everything we take for granted should be carefully, critically considered.

Just because a book or article calls some mathematics activity a "game" doesn't necessarily

make it so. Not every so called "mathematics game" actually involves real mathematics.

How can we tell the difference?

What makes a "game" really be a "game"? I have discussed this before in several articles in

Prime Number(with, surprisingly, no reader feedback! - is there anyone out there who

reads and uses these things?), as well as my book with Tom Hill, and will only summarise

here.

A "game" is a playing and thinking activity which satisfies the following conditions:

(a) it involves more than one "player";

(b) players take turns to play;

(c) players interact with each other, that is, what one player does in a turn will effect

in some way what the subsequent player or players can do in following

turns; and

(d) although there may be some amount of luck (such as a dice roll, or a dealing of

cards), there should also be some room for a player to choose how to play.

What does this mean? Rubik's Cube, jigsaw puzzles, peg solitaire and card patience, for

example, are not"games" because they involve only one player, and hence lack player


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interaction. They can be called "activities" or "puzzles". Bingo and Tattslotto are not

games, because there is no player interaction, and, in the case of Bingo, nothing but luck in

the numbers the players have on their Bingo cards, and no player choice about what to do

as the numbers are called.

Other "matching" activities, such as Coin Equivalents Bingo, or Analog and Digital Clock

Dominos are (usually) not games, because they depend on the luck of the call, or of the

domino draw. (However the standard game of dominoes, with scoring, is a superb game

with considerable scope for player choice and skill! Try it!) In fact they are virtually no

more than a "dressed up worksheet" (match each item in the first row with an item in the

second row), a collection of "sums", first to finish "wins"but not a game. Snakes and

Ladders is not a game because it is total luck, has no player choice and no player

interaction, unlike, say, Backgammon, which is an elaboration of Snakes and Ladders.

Tables Footy and Pacman Arithmetic, and the like, are not games because players do not

take turns and do not interact, they merely race to answer each question first.

I am not suggesting any of these non-games are not worthwhile activities for occasional

challenge, recreation or practice, simply that they do not deserve to be called "games", and

should be recognised as "luck races", "dressed up worksheets", and the like. Also they do

not have the appeal of a real game, such as the classic games (which include Monopoly,

Scrabble, chess, draughts, dominoes, backgammon, or Mastermind). Children will not

choose to play them for fun outside of a classroom. We should hope that, when we offer

children the chance to play a game in a mathematics class, it is sufficiently good as a game

that children would choose to play it at home or at lunchtime. And also hope that it has

some identifiable mathematics content that could be fitted comfortably into the Scope and

Sequence chart at the back of The Mathematics Framework: P-10 (1988, 96-101) or intothe "nutshell" descriptors of theMathematics Profiles Handbook: Number and Space

(1992). Chess, for example, certainly is a game, and is also mathematical because it

includes problem solving, strategic thinking, and spatial thinking, albeit very generally.

Separately from any mathematical justification, chess could also be educationally justified

in terms of life skills, personal development, and recreation studies.

The following mathematics games are offered as examples of activities that are genuinely

games, and that genuinely involve overtly mathematical thinking as part of their essential

play, including logical reasoning, arithmetic, probability, spatial thinking and, all of them,

problem solving and strategic thinking.

COLOR COUNTRIES

adapted from J.J. del Grande (1973, p 89 ) and Gough (1978, p 122).

This is a nim-type gameof successive elimination of possibilities, combined with spatial

thinking. It was developed from the classic Four-Color Map Coloring Problem - given a


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map of countries, can it be colored using only four colors, so that no countries which share

adjacent boundaries have the same color? Try it.

Number of players: 2

Equipment: pencil and paper, three pencils of different colors, or counters of three

different colors.Setting up: draw a 4 x 4 or 5 x 5 square grid.

Playing: Players take turns to color an empty cell in the grid according to the rules:

1. no squares side by side may be the same color.

2. squares which are diagonally adjacent may have the same color.

The winner is the player who is last able to move.

Scoring: The winner scores 1 for each cell which can not be colored. The first player to

score a total of 20 wins the whole match.

Variations: Of course this can be played in black and white, writing A, B or C instead of

coloring with three different colors. Try using different playing boards, such as a triangular,

hexagonal, or brickwall pattern. Alternatively, instead of coloring grid cells, color celledges, with the rule that no two joined edges can be the same color. Alternatively, color the

vertices of the grid, with the rule that no two vertices joined by a common edge may have

the same color. Alternatively, use colored cubes, stacking them face against face on a 3 x 3

base grid, with the rule that no two adjacent cubes may have the same color, and no more

than three cubes can be stacked on top of each other. For all of these variations, there is also

a misere version, in which the first player who is unable to move does not lose (as the rule

stands) but instead is the winner.

GUESS WHICH

adapted from D 1-1 (LL) in Meirovitz and Jacobs (1987, p 24).

This is a very simple attribute and logic game, similar to the popular game Guess Who,commercially published by Milton Braddley. Notice that in such logic games, where

players take turns to try to deduce another player's secret (compare this with Battleships or

Mastermind) there is a special kind of question-and-answer player interaction that is

different from chess-like games.

Number of Players: 2, one called the Hider, and the other called the Finder.

Equipment: a pack of cards, possibly simplified (e.g. Aces to 6s in four suits).

Setting Up: The Hider shuffles the cards and deals them face up, neatly,

for example, in a 6 x 4 array.

The Hider then secretly chooses one of these cards, and secretly notes this.

Playing: The Finder asks questions which can only be answered "Yes" or "No", trying tofind out which is the Hider's secretly chosen card.

Scoring: The Hider scores the number of questions needed for the Finder to

correctly identify the secretly chosen card. Then players exchange roles.

After both players have had an equal number of turns being Hider and Finder,

the player with the highest score wins the whole match.


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Variations: Instead of a pack of cards, use a set of dominoes, or a set of attribute blocks, or

any set of objects (farm animals, zoo animals, dinosaurs, dog biscuits, pasta) whose

attributes unambiguously vary - color, number, shape, size, texture, thickness, material.

Instead of using only one pack of cards, use two packs of cards; then players act as Finder

for themselves as well as Hider for the other player: first to correctly identify the otherplayer's secretly "hidden" card is the winner.

SPLITTER

adapted from "Acey, Twoesy" inMathematics Course Advice(Primary) (1992) Upper

Chance and Data, Unit One, page 5, itself an adaptation of the popular gambling card game

variously known, with slightly differing house rules, as "Acey Deucey", "In Between",

"Ace-Deuce" or "Yablon" (e.g. The Way to Play Bantam Books, 1975, p 250).

Incidentally, because of the seriousness of addictive gambling behaviour, the gambling

origins of this and other games should be concealed, and gambling of any sort should be

actively discouraged in schools. Where gambling-type games are used in mathematics

classes this should only be for the study of probability and strategy. Games such as poker,

or blackjack (also known as Twenty One and Pontoon), whose only playing interest derives

from gambling may be subjected to probabalistic analysis but should not be played.

Number of Players: 2 or more.

Equipment: 1,2 or 3 dice; pencil and paper to score.

Playing: Players take turns.

In each turn a player rolls two dice (or rolls one die, twice), and notes the result. The

player scores the difference between the two numbers (e.g. rolling a 2 and a 5, the player

scores 3 points), or 1 if the numbers are the same.The player may also choose to roll a third die, and will score a bonus of 10 points if this

number "splits" or is in between the first two numbers. However the player scores nothing

in this round if the third die does not "split" between the first two dice rolls. The first player

to reach a total score of 100 wins.

Variations: Notice that there is no player interaction with these dice rolls, except in

deciding whether or not to risk getting the bonus points for "splitting" in order to beat an

opponent's score. Compare this with the card version "Acey, Twoesy", where seeing which

cards have been drawn by an opponent can effect what a player chooses to do. Try the

variation of a player losing 10 points if an attempted"split" fails. Investigate the frequency

with which the scoring differences between the first two dice occur. Try adapting the same

game to use a set of double-six dominoes.

WHOLLY FRACTIONS

Number of players: 2 or more.

Equipment: a number pack of cards (an ordinary pack, with no royalty cards)

pencil and paper to score, a fraction-wall chart to check, and a calculator


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to settle disputes.

Setting up: shuffle the pack, and deal each player four cards face down, and deal four cards

face up between the players.

Playing: Players take turns.

In each turn a player tries to make pairs of cards so that each pair represents a fraction, andthe set of fractions adds up to 1. The player may use any card in his or her hand, in

combination with any card that is face up on the table.

When a player makes a fraction or set of fractions totalling 1, the player scores as many

points as the number of cards the player has used.

E.g. any matched pair of cards represents 1, but will only score 2 points.

Alternatively, 2, 3, 5, 10 can represent 2/4 and 5/10 which adds to 1 and will score 4 points

because it uses four cards.

Then the player is dealt four new cards, and play resumes.

If a player is unable to make a fraction or set of fractions that totals 1, the player

draws one more card from the remaining stock of undealt cards,

and the player's turn ends.

When no further play can take place, or all the cards in the middle have been used, or all

the remaining cards have been dealt, and no one can make a set totalling 1, all the cards are

shuffled, and a new deal begins.

The first player to reach a total score of 50 wins the whole game.

Variations: Allow subtraction as well as addition to reach a total of 1. For example,1, 4, 8

and 10, can make 10/8 - 1/4 which totals 1. To avoid players continually resorting to trivial

play, forbid any scoring for matched pairs. To encourage adventurous thinking, add one

bonus point for each different denominator in the set of fractions used to score. To extend

the range of fractions, let J = 11, Q = 12, K = 13.

References

del Grande, J. J., (1973, p 89 ).Math, Book 2 . Windsor, Melbourne: Lloyd O'Neill.

Gough, J. (1978) Games and Puzzles Vinculum 15 (4) 121-122.

Gough, J., (1990). Mathematics Games.Prime Number, 5 (4) 22-23.

Gough, J., (1991). Mathematics Games That Really Teach Mathematics. Prime Number, 6

(1) 3-6.

Gough, J., (1991). Some Mathematics Games and Comments About Using Games. Prime

Number, 6 (2) 17-22.

Hill, T., and Gough, J., (1992). Work it Out With Mathematics Games. Melbourne: Oxford

University Press.

Meirovitz, M., Jacobs, P.I., (1987).Brain Muscle Builders: Games to Increase Your

Natural Intelligence. Princeton, N.J.: Hawker Brownlow/ Trillium.

Playing Games to Learn Mathematics (Gough)

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