ICOTS 2, 1986: George W. Bright
PROBABILITY GAMES
George W. Bright
University of Houston
John G. Harvey
University of Wisconsin
T h e r e is considerable, good evidence t h a t games can be effective tools i n
teaching mathematics a n d t h a t all games are n o t equally effective ( B r i g h t ,
Harvey, & Wheeler, 1985). One k e y t o effectiveness may b e t h e degree t o
which t h e mathematics content is involved i n t h e p l a y o f a game, since
t h e r e would seem t o be a corresponding involvement o f t h e game players
w i t h t h a t content.
I n o r d e r t o embed t h e mathematics content i n a game, it i s necessary t o
have a clear understanding o f both t h e game's instructional objectives and
t h e highest taxonomic level (Bloom, 1956) r e q u i r e d t o p l a y t h e game successfully T h i s can be combined w i t h an identification o f t h e instructional
level o f t h e game. (See B r i g h t e t al., 1983, 1985, f o r t h e relevant d e f i n i tions and a discussion o f t h e appropriate techniques.)
Previous Research
Although t h e r e has been some in-depth analysis o f y o u n g children's conceptions o f p r o b a b i l i t y concepts (e.g.,
Fischbein, 1975), t h e r e has been
almost no research on t h e effectiveness of p r o b a b i l i t y games w i t h these
children, i n spite o f t h e f a c t t h a t t h e r e are many suggestions f o r games
t h a t o u g h t t o p r o v i d e effective instruction f o r these c h i l d r e n . Extrapolation o f research w i t h older students seems t h e o n l y way c u r r e n t l y t o approach t h e t a s k o f t r y i n g t o plan effective uses o f p r o b a b i l i t y games w i t h
young children.
Four different p r o b a b i l i t y games have been used i n experimental studies.
The simplest i s a microcomputer game, T h e Jar Game (Kraus, 1982), designed f o r elementary school students. T h e p l a y e r is shown t w o "jars" on
t h e monitor screen, each containing green and gold cubes, and asked t o
select t h e jar t h a t gives t h e best chance of h a v i n g a g o l d cube randomly
selected. If a c o r r e c t choice is made, t h e p l a y e r receives bonus points.
Cubes are t h e n randomly selected from whichever j a r was chosen by t h e
player, and one p o i n t is awarded f o r each gold cube selected. Two studies
have r e p o r t e d treatments using t h i s game ( B r i g h t , 1985a, 198513). I n b o t h
cases, t h e subjects were preservice elementary school teachers, and t h e r e
was no e f f e c t on p r o b a b i l i t y learning a t t r i b u t a b l e t o t h e use o f t h e game.
Another game used i n research is a set o f f a i r / u n f a i r games (Romberg,
Harvey, Moser, & Montgomery, 1976). Students are asked t o p l a y each o f a
p a i r o f games several times, and then t o decide if t h e games are f a i r o r
unfair, i n t h e sense of whether one player has a b e t t e r chance o f winning.
T h e games involve dice, chips, o r spinners. B r i g h t , Harvey, and Wheeler
ICOTS 2, 1986: George W. Bright
(1980) demonstrated t h a t grade 7 students learned t o better i d e n t i f y f a i r
and unfair situations through use of these games, even without any i n struction from t h e teacher. I n a replication and extension of t h i s research,
B r i g h t e t al. (1985) showed t h a t f o r grades 6 and 8, t h e games were e f fective a t teaching the notion of fair/unfair, b u t only when the test situations included hypothetical data like that generated i n the play of t h e
game.
A t h i r d game used i n research is Number Golf (Bright, 1980). Students
r o l l dice and add o r subtract the sum generated t o a cumulative total, with
t h e objective of t r y i n g t o attain specified "goal numbers." With only one of
three groups of college-aged subjects d i d he f i n d a weak relationship between students' knowledge of probability concepts and the use of a f i r s t order strategy i n playing the game. B r i g h t e t al. (1985) found learning
effects only at a taxonomic level below that of t h e game with grade 7 and 9
students. However, there was no analysis of t h e strategies t h a t students
may have used i n play of the game.
The f o u r t h game is Capture (Schroeder, 1983). This game also involves
strategy, and it appears t o be t h e only game used with children aged 6 t o
11; his subjects were from grades 4, 5, and 6. Schroeder concluded t h a t
some subjects had l i t t l e d i f f i c u l t y applying probability concepts and e x plaining t h e strategies they used, while other students observed t h a t positions on t h e gameboard d i d not change hands equally often b u t were u n able t o relate t h i s observation t o probability. (For more details, see t h e
paper b y Schroeder i n this volume.)
I n sumiary, there is clear evidence t h a t probability can be taught t h r o u g h
games, b u t the role of students' strategy use may be important f o r understanding t h e effects of these games. Although only limited attention has i n
the past been given t o identification of students' strategies, techniques
have now been developed which may allow relating strategy use t o learning.
Measuring Game Playing Strategy
A game strategy is an algorithm that, when applied at any play of a game,
determines t h e move t o be made from among all of the possible moves at
t h a t play. This definition coincides with the one inherent i n the definition
both of an optimal strategy i n mathematical game theory and o f a winning
strategy i n artificial intelligence. The algorithms referred t o i n the defini tion are the ones used b y artificial intelligence game-playing programs
whenever it is possible t o specify. them. For example, i n NIM the optimal
strategy f o r the second player is "always pick u p enough objects so t h a t
the number of objects remaining is equal t o 1 (mod 4)."
Although t h i s definition b y default applies t o expert strategies; t h a t is,
winning strategies; many games have less-than-expert strategies t h a t s t u dents might use. These are called novice strategies. One of the goals o f
using probability games seems t o be t o help students develop better strategies; t h a t is, t o develop strategies t h a t become more like expert/winning
strategies.
ICOTS 2, 1986: George W. Bright
A s t r a t e g y i n s t r u c t i o n a l game, then,
i s an i n s t r u c t i o n a l game ( B r i g h t e t
al., 1985) t h a t has a t least one e x p e r t s t r a t e g y w h i c h (a) involves t h e cont e n t o f t h e p r i m a r y i n s t r u c t i o n a l o b j e c t i v e o f t h a t game a n d (b) r e q u i r e s
use of t h e c o n t e n t a t t h e taxonomic level o f t h e game. C a p t u r e a n d Number
Golf a r e examples o f s t r a t e g y i n s t r u c t i o n a l games, b u t T h e J a r Game i s
not, since t h e r e i s n o s t r a t e g y f o r w i n n i n g t h a t game.
T h e analysis o f t h e s t r a t e g y used by a s t u d e n t can b e made by c o n s i d e r i n g
t h e p l a y s o f t h e game as successive solutions t o e q u i v a l e n t sets o f p r o b lems. P r i o r t o t h e analysis, t h e possible e x p e r t a n d novice strategies t h a t
m i g h t b e applied i n p l a y i n g t h e game m u s t b e specified. T h e analysis uses
techniques similar t o those used t o analyze t h e c o n t e n t o f i n s t r u c t i o n a l sequences.
T h e analysis begins by t h e development o f a move m a t r i x . One column o f
t h e m a t r i x i s generated f o r each t u r n in a game. T h e number o f rows o f
t h e m a t r i x i s one more t h a n t h e number o f i d e n t i f i e d strategies. Each row,
save t h e l a s t one, i s labeled w i t h one o f t h e strategies; these rows a r e c a l l e d s t r a t e g y rows o f t h e m a t r i x . T h e e n t r y i n r o w i and column j i s
1 if t h e j t h move was consistent w i t h s t r a t e g y i , o r 0 if t h e j t h
move was inconsistent w i t h s t r a t e g y i. T h e l a s t r o w i s used t o r e c o r d
instances when none o f t h e i d e n t i f i e d strategies i s used; 1 is e n t e r e d i n
column j if t h e j t h move was inconsistent w i t h a l l i d e n t i f i e d s t r a t e gies, a n d 0 i s e n t e r e d if t h e j t h move was consistent w i t h a t least one
o f t h e i d e n t i f i e d strategies.
T h e move m a t r i x i s t h e n used t o generate t w o numeric values. T h e f i r s t i s
a 0, 1, o r 2, d e p e n d i n g o n whether (a) n o discernable category o f s t r a t e gies was used, (b) novice strategies w e r e most consistently used, o r (c)
e x p e r t strategies were most consistently used. T h e second value i s a r a t i o
o f (a) moves consistent w i t h t h e i d e n t i f i e d s t r a t e g y category t o (b) t h e
t o t a l number o f times t h a t category c o u l d have been observed. (Since it i s
possible, f o r example, t h a t e x p e r t strategies m i g h t n o t b e applicable t o
each move, it i s necessary t o r e s t r i c t t h e denominator o f t h i s r a t i o t o t h e
l a r g e s t s u b s e t o f moves i n w h i c h t h e i d e n t i f i e d c a t e g o r y o f s t r a t e g y use
could have been observed.)
From t h e se t w o generated values f o r each game, a player's use o f s t r a t e gies can b e t r a c k e d across multiple p l a y s o f a game. T h i s t r a c k i n g i s b e g u n w i t h t h e l a s t f e w p l a y s o f a game, since use o f strategies i s more l i k e Iy t o b e o b s e r v e d as s t u d e n t s become familiar w i t h t h e game. If no categ o r y o f s t r a t e g y use can b e observed i n t h i s analysis, t h e n a weighted
average across a l l p l a y s can be computed as an approximation o f t h e use o f
strategies.
Conclusions
These research techniques can b e used t o determine (a) relationships b e tween t h e development a n d use o f game strategies a n d t h e acquisition o f
c o n t e n t knowledge, (b) whether r e p l a y s o f games a f f e c t t h e development
a n d use o f strategies o r t h e acquisition o f c o n t e n t knowledge, (c) w h e t h e r
t h e use o f p a r t i c u l a r strategies by an opponent affects e i t h e r t h e develop-
ICOTS 2, 1986: George W. Bright
ment and use o f strategies or t h e acquisition o f content knowledge, and
(d) whether t h e development and use o f strategies in one game transfers
t o t h e development a n d use o f strategies i n o t h e r games o r affects t h e acquisition o f content knowledge i n o t h e r games. It is, o f course, hypothesized t h a t improvement i n s t r a t e g y o v e r many plays o f a game would b e
associated w i t h increased knowledge, b u t t h i s needs t o be v e r i f i e d empirically.
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