Kilpatrick Algebra Game
Kilpatrick Algebra Game
Kilpatrick Algebra Game
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THE
CONSISTENCY
LEARNING
OF
OF
STRATEGIES
IN
THE
MATHEMATICAL STRUCTURES
NICHOLASA. BRANCA
Assistant Professor of Education
Stanford University
Stanford, California
JEREMYKILPATRICK
Associate Professor of Mathematics
Teachers College, Columbia University
New York, New York
The term strategy abounds these days in our educational literature. Children are alleged to have strategies for playing games and for solving problems; teachers are encouraged to develop strategies of instruction. But
like its cousin style, strategy is often applied loosely and inappropriately
to any sort of regularity in behavior.
Researchers in mathematics education should be cautious in using the
term. One assumes, for example, that if a person has a strategy for playing a game, he will be aware that he is acting according to some plan,
however vague or rudimentary. If the pattern of questions a person asks
or moves he makes is to be called a strategy, moreover, then one ought to
expect that when faced with a similar task on another occasion, the person's behavior will show a similar pattern. On the other hand, if the pattern of behavior is neither conscious nor consistent across tasks and occasions, it hardly deserves to be considered a strategy.
A recent series of investigations by Dienes and Jeeves (1965, 1970)
has rekindled enthusiasm in the search for strategies. Dienes and Jeeves
identified three so-called strategies based on the moves subjects made in
learning a game embodying the structure of a mathematical group. They
presented evidence that a subject's retrospective account (termed an evaluation) of how the game worked reflected the moves he had made in playing it. Such issues as whether there is a hierarchy of strategies and evaluations, whether there are sex and age differences in strategies and evaluations, and how the learning of one group structure affects the learning of
another group structure have been Dienes and Jeeves's major concerns in
This report is based upon the senior author's dissertation conducted
in partial fulfillment of the requirementsfor the Ed.D. degree at Teachers
College, Columbia University.
132
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their research. The question arises, however, as to how consistent subjects' strategies and evaluations are across different embodiments of the
same group structure and between one mathematical structure and another.
The purpose of the present study is to investigate this question.
A Game Embodying the Klein Group
As an aid in understanding the present study, consider first one of the
tasks used by Dienes and Jeeves (1965). The subject and the experimenter had identical sets of four cards, each of a different color. One of
the experimenter's cards was placed in the window of an apparatus. The
subject played a card, and the two cards together (the card in the window and the subject's card) determined which card appeared next in the
window-a binary operation. The rules for the operation, which in this
case were those of the Klein group, are given in Table 1.
After the subject played a card, he was to predict which card would
appear next in the window. Then he was to play another card, make another prediction, and so on, until his predictions were consistently correct.
The object was to learn the rules of the game.
When asked afterwards how the game worked, subjects gave three types
of evaluations: memory, indicating that each of the 16 combinations was
memorized separately; pattern, indicating that the subject divided the
game into parts in which similar combinations of cards were seen as giving
similar outcomes (for example, any two members of the set {Orange, Blue,
Green) give the third member as an outcome); and operator, indicating
that the subject regarded the card he played as operating on the card in
the window (for example, Green interchanges Yellow and Green, and
it interchanges Orange and Blue). All evaluations fell into one or a
combination of these three types.
Dienes and Jeeves argued that one could identify, in the sequences of
moves, three strategies-memory (or random), pattern, and operatorthat corresponded to the three types of evaluations. They devised two
scores-a pattern score and an operator score. The pattern score reflected
the proportion of the subject's plays that were within a given region of the
group table. The operator score reflected the proportion of the subject's
plays in which the same card was played repeatedly in succession.
TABLE 1
OUTCOMESOF PLAYING A CARD AGAINSTA CARD IN THE WINDOW
Card in the window
Card
played
Yellow
Orange
Blue
Green
Yellow
Orange
Yellow
Orange
Blue
Green
Orange
Yellow
Green
Blue
Blue
Blue
Green
Yellow
Orange
Green
Green
Blue
Orange
Yellow
May 1972
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133
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labeled Saturn, Mars, Venus, and Jupiter. One bulb was lit initially. By
throwing a switch, the subject caused that bulb to go out and then one of
the four bulbs to light. A play consisted of throwing a switch and predicting
which bulb would light. The object was to learn the rules of the game. The
Light Game was isomorphic to the Color Game, with the bulb that was
lit playing the part of the card in the window and the switch that was
thrown playing the part of the card the subject played. Saturn, Mars,
Venus, and Jupiter corresponded to Yellow, Orange, Blue, and Green,
respectively.
Map Game. The apparatus for this task consisted of a miniature car
and a map of the United States on which were marked five cities (Birmingham, Chicago, Denver, San Francisco, and Washington) connected by
eight fictitious highways. (A highway was defined as linking two and only
two cities.) The car was placed on the map at one of the cities. The subject
chose a city as her first destination and then was told which city she would
encounter first on her journey if she were to traverse the least number
of passable highways enroute (the condition of the highways was not
initially known by the subject-they could be closed, open in one direction only, or open in both directions). After the subject had moved to the
first city on her route, she was then free to choose any destination again,
predicting which city she would then visit first. The object was to learn
the rules of the game, that is, the condition of each highway. The task
has a network structure, not a group structure, as can be seen in Table 2,
but like the other two tasks, 16 outcomes of a binary operation must be
found. The operation was traveling from the departure city to the destination city, the outcome was the city first visited, and the 16 starred outcomes
in Table 2 allowed the subject to infer the condition of each direction of
the eight highways.
Procedure
The tasks were presented in individual interviews, one task per interview,
at intervals of approximately two weeks. The order of the tasks was the
same for all subjects: Color Game, Map Game, Light Game. The interviewer kept track of the subject's moves and predictions in learning each
game and the evaluation she gave at the end of the interview of how the
game worked.
Each subject played each game until she thought she knew all of the
rules. She was asked to give the rules and was encouraged to continue
playing if she overlooked or forgot some of them. Because the Color Game
proved more difficult and discouraging for the subjects than had been
anticipated, in the Map Game each subject was allowed to stop whenever
she thought she knew the rules. As a consequence, many subjects stopped
playing the Map Game prematurely, and the game appeared much easier
to them than their performances indicated.
May 1972
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135
TABLE 2
OFCHOOSING
A DESTINATION
OUTCOMES
CITYFROMA DEPARTURE
CITY
Destination city
Departure city
Birmingham
Chicago
Denver
San Francisco
Washington
Denver*
Chicago*
* An outcome that can be used to infer the condition of the eight highways.
Results
A subject was classified as successful on a task if she gave an evaluation
of the game that accounted for 14 or more of the 16 binary combinations.
By this criterion, 49, 47, and 54 subjects were successful on the Color
Game, the Light Game, and the Map Game, respectively. Comparisons
with Dienes and Jeeves's data on successes and failures are not possible
since they did not report these data. Only the successful subjects were included in their analyses.
The Relationship between Evaluations and Strategies
In direct contrast to the findings of Dienes and Jeeves, successful subjects who gave an operator evaluation on the Color Game did not tend to
have higher operator scores than successful subjects who gave other evaluations. Of the 13 successful subjects who gave an operator or partial-operator evaluation, only 6 had operator scores that were above the median for
the successful subjects. Similarly, of the 42 successful subjects who gave
a pattern or partial-pattern evaluation on the Color Game, only 22 had
pattern scores that were above the median for the successful subjects.
The results for the Light Game were strikingly similar. Only 6 of the
14 successful subjects who gave an operator or partial-operator evaluation
had operator scores above the median. Only 22 of the 42 successful subjects who gave a pattern or partial-pattern evaluation had pattern scores
above the median.
There was some evidence of a relationship between evaluations and
strategies on the Map Game, at least for one kind of evaluation and one
kind of strategy. Subjects who, in explaining the Map Game, described the
condition of each of the eight highways separately were credited as giving
an individual-roads evaluation; the other subjects, who described the game
in a more global fashion by explaining how one traveled from one city
to another, were credited with a detour-routes evaluation. Each subject
was given a back-and-forth strategy score, which reflected the proportion
of moves in which she returned to the city she had just left, and a detour
strategy score, which reflected the proportion of moves in which the same
city was chosen repeatedly as the destination. Of the 39 successful subjects
136
Journal for Research in Mathematics Education
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Operatoror
partial-operator
Pure pattern
Memoryor
pattern-memory
Other
Operator or
partialoperator
Pure
pattern
Memory or
patternmemory
8
4
2
12
4
7
0
0
9
1
5
1
31
6
7
3
May 1972
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Other
137
TABLE 4
SUCCESSFULSUBJECTS'EVALUATIONSOF THE COLOR GAME AND THE LIGHT GAME
Light Game evaluation
Color Game
evaluation
Operator or
partialoperator
Operator or
partial-operator
Pure pattern
Memory or
pattern-memory
Memory or
pattemnmemory
Pure
pattern
7
2
2
10
3
4
Individual-roads
Detour-routes
Consistent
Not consistent
Consistent
success
Consistent
failure
Not
consistent
29
25
31
15
33
5
16
26
11
9
138
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TABLE6
RELATIONSHIPOF BACK-AND-FORTH STRATEGY SCORES ON THE MAP GAME
TO CONSISTENCY OF PERFORMANCEON THE GROUP STRUCTURE TASKS
Back-and-forth strategy scores
High
Low
X2
Consistent failure
28
10
12
30
Not consistent
10
10
sis 4 was not investigated as stated. Instead, several links between behavior
on the group structure tasks and behavior on the network task were examined separately.
Subjects who gave the same evaluation on both group structure tasks
showed no greater tendency than the other subjects to give a particular
evaluation on the Map Game; however, subjects who were successful on
both group structure tasks tended to give an individual-roads evaluation on
the Map Game (see Table 5). Subjects who were successful on both group
structure tasks tended to have high back-and-forth strategy scores on the
Map Game; subjects who failed both group structure tasks tended to have
low back-and-forth strategy scores (see Table 6). Of the 32 subjects who
consistently scored above the median operator score on both group structure tasks, 20 (or 62.5% ) gave an individual-roads evaluation on the Map
Game, a percent that differed little from the corresponding percent for the
entire sample (60%).
Relationships with Age and Intelligence
The two age groups did not differ significantly in evaluations or strategy
scores on any of the three tasks. There were no age differences in number of
plays, number of errors (incorrect predictions), and percent of successful
subjects.
Intelligence was significantly related to performance on the two group
structure tasks, but not the network task: the product-moment correlation
coefficients between IQ and number of errors on the Color Game, the
Light Game, and the Map Game, respectively, were -.53, -.55, and -.04.
In contrast, Dienes and Jeeves reported a zero correlation between IQ and
total errors on their Klein group task.
Discussion
The considerably greater difficulty of the group structure tasks in the
present study, as compared with Dienes and Jeeves's study, was apparently
due chiefly to unfamiliarity with the tasks. According to Dienes,' the children in his study had been participating in a special mathematics program
1 Z. P. Dienes, personal communication, 7 February 1970.
May 1972
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139
TeachersCollege,ColumbiaUniversity)Ann Arbor,Mich.:UniversityMicrofilms,
140
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