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DOI: 10.14738/aivp.92.9974
Publication Date: 25
th April, 2021
URL: http://dx.doi.org/10.14738/aivp.92.9974
Convergent and Divergent Thinking Skills in Electrical
Engineering Gaming Framework
1 Rani Deepika Balavendran Joseph, 2Jeanne Tunks, 3Gayatri Mehta
1Corresponding Author, Department of Electrical Engineering, University of North
Texas, Denton, Texas, United States.
2College of Education, University of North Texas, Denton, Texas, United States.
3Department of Electrical Engineering, University of North Texas, Denton, Texas,
United States.
ABSTRACT
This paper focuses on thinking competencies of the participants, who are
categorized based on the number of times they played the same level in an online
scientific game. An open-source scientific puzzle game UNTANGLED is used to
perform this research. Telemetry data of more than 700 players’ solutions were
considered for analysis and divided into two groups: single (players who played a
puzzle only one time) and multiple (players who played the same puzzle more than
one time). Analysis performed on these two groups helps to examine convergent
and divergent thinking skills. Statistical tests were performed to assist in learning
the significance of the two groups based on dependent variables score and type of
moves used in the gaming framework. The findings obtained show that
the single and multiple groups have no significant difference in their performance,
and multiple group players obtained top scores alongside single group players.
Results show that single group players could have convergent thinking that can
provide only one solution for a given problem. In contrast, multiple group players
could possess divergent thinking that can offer diverse solutions to the same
problem from different perspectives. Finally, this paper dispenses
recommendations for STEM educators and scientific game designers to develop
open-ended frameworks that reward both thinking competencies. The novelty of
this study is to show the significance of including features that enhance both
convergent and divergent thinking skills to solve open-ended problems in electrical
engineering.
Keywords: Electrical Engineering, UNTANGLED, Creative Problem Solving,
Convergent Thinking, Divergent Thinking
1. INTRODUCTION
The transparent description of the rules and procedures of a problem can make
problem-solving an easier task. However, this may not always be possible in a
rapidly changing world that has many surprises. An individual needs to conceive
creative problem-solving skills to face a capricious world. Especially, engineers
require such skills for solving unexpected problems, as engineering is the process of
developing novel and productive solutions to compete with advancing technologies
[1]. Fifty years ago, creative problem solving is treated as a skill related to fine arts
education and is underrated in the field of engineering. Apparently, researchers' and
educators' perspectives have changed after the launch of the first artificial satellite
“Sputnik” by the Soviet Union in the year 1957. In addition, previous studies [2]–[8]
estimates a problem-solving skill in an individual based on personality, cognitive
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style, and state of mind. Moreover, it determines the significance of including
creative problem-solving in engineering university education rather than
considering it as an innate ability. Proper training can help engineers to develop
thinking skills that prepare them to face the real world after their graduation. This
is because there is a huge difference in the types of problems students solve in the
classrooms to problems, they must solve in work areas. Convergent and divergent
thinking skills are considered prominent in developing design and creative skills in
an engineer to bridge the gap between the classroom and real-time problems [9].
According to studies [10]–[12], convergent thinking competencies are providing a
single solution for a given problem by following conventional problem-solving
procedures and judging decisions. Divergent thinking abilities are providing
multiple solutions for a given problem by thinking outside of the box and refining
the process. Charyton et al., [13], [14] developed an assessment mechanism that
includes personality attributes that influence the creative problem-solving process.
The attributes mentioned in their study are creative personality, creative
temperament, and cognitive risk tolerance. The creative process includes constraint
satisfaction (shapes and materials added), problem finding (coming up with new
uses), problem-solving (finding a novel solution), divergent thinking (more
solutions for the given problem), and convergent thinking (single solution for the
given problem). Pepler and Ross [15] examined the effect of play on divergent and
convergent problem-solving. Attributes for measuring divergent thinking are
fluency in problem defining, flexibility in providing ideas, and novelty of an idea. In
their research, convergent and divergent play material is provided to two groups of
children. The convergent group considers this as a puzzle, whereas the divergent
group considers this as a game. They observed that the convergent group spent two- thirds of the time in solving the puzzle, and the divergent was involved in a variety
of activities. Finally, the studies conclude that divergent group performance is better
than a convergent group in providing multiple and unique solutions. However,
convergent thinkers solve the given puzzles in a strategic way than divergent
thinkers. Similarly, in this paper, our focus is on these two thinking skills by using
UNTANGLED game data.
In this gaming framework, some gamers solved puzzles in one attempt, whereas the
remaining players solved the same puzzle in different ways in multiple attempts. For
analysis, the telemetry data is divided into two groups as single and multiple groups.
Gamers who played a given puzzle only one time are considered as in the single
group, whereas players who played a puzzle more than one time are represented as
the multiple groups. Different statistical tests are performed to analyze the
performance of these two groups. The rest of the paper is organized into six sections.
Section 2 provides an introduction on the framework used in this paper, section 3
gives detailed description on the experimental set-up, section 4 explains analytical
results performed on single and multiple attempt players in different aspects,
section 5 elaborates the results obtained, section 6 made a few recommendations
for STEM educators, and the last section includes the conclusion of this paper.
1.1 Proposition
The purpose of this study is to perceive whether there is any difference in the
performance of single and multiple groups. The analysis is performed to answer
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
questions like which group has a greater number of players, which group players
score best, is there any significant difference in performance between the groups,
and how do these two groups differ when solving complex puzzles.
2. BACKGROUND
Recently, games with scientific purpose have gained attention as researchers are
developing gaming frameworks to use human intelligence to solve complex
problems in various fields [16]–[18]. A scientific puzzle game, UNTANGLED, which
is developed by Dr. Mehta and her team is used for this study [19]–[22]. The
empirical purpose of this framework is to develop efficient mapping algorithms for
low power portable devices by using human problem-solving strategies. This game
has been online since 2012, and there are thousands of players’ solutions in the
gaming database. In the year 2012, UNTANGLED received the People Choice Award
in the games & apps category of the International Science Engineering Visualization
Challenge conducted by the Science [23] and National Science Foundation (NSF)
[24]. The game has developed in four different versions over a period, whereas in
this paper first version of the game is analyzed. Personal information such as
demographic, geographic, and psychographic data of a player is not saved in the
game database. There are in-depth tutorials included in this framework, which helps
to elucidate gaming rules. Bronze, silver, and gold badges or medals will pop-up on
the screen as incentives to motivate players. In addition, there is a leaderboard
feature that helps to view scores of players all over the world. IRB protocols of the
University were followed for conducting this study. This game is designed with 13
subgames, and each subgame has easy, medium, and hard levels. In total, there are
172 different puzzles in this game. The next section shows how the gaming
framework looks, what are the various constraints in each puzzle, what are the
features included in this game, what are the rules and regulations of the subgames,
and how players can solve puzzles.
3. EXPLORATORY MODEL
This section describes the scientific puzzle game that is used in this paper. There are
two kinds of puzzles, regular and constrained, in this game. Regular puzzles consist
of only red rectangular blocks and constrained puzzles consist of red rectangular
and blue circular blocks. In this framework, a data flow graph is disclosed as a
puzzle/graph to the players by hiding details of computational elements. Players
need to position these blocks as per rules provided to overcome violations that, in
turn, increases the score. Based on the type and number of blocks, there are 13
different subgames. However, each subgame is different based on the connectivity
between the blocks, a variety of blocks, and constraints in placing blocks. Based on
these constraints, each subgame has a different number of levels. These levels
represent digital signal and image processing applications benchmarks. The number
of blocks and connections between the blocks in these seven levels are specified in
table 1. Columns in the table represent the seven levels that are most played, and
the two rows indicate a number of blocks and number of connections between those
blocks. The number of blocks can be the total number of red and blue blocks present
in the puzzle, and the connections are the total degree of connectivity for each block.
Based on these blocks and connections, the level L1, L2, and L3 are considered as
easy levels in this gaming framework. Similarly, the levels L4 and L5 are represented
as medium levels, and L6 and L7 as hard levels. Table 1 determines that for
succeeding levels, either there is an increase in a number of nodes or in a number of
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connections. Though there is an equal number of nodes in levels L2, L3, and L4, there
is an increase in a number of connections from L2-L4. Hence, the total number of
nodes and connections defines the complexity of levels.
TABLE 1. BLOCKS AND CONNECTION INFORMATION OF L1-L7 LEVELS
Levels L1 L2 L3 L4 L5 L6 L7
Number of Blocks 24 29 29 29 36 52 61
Number of
Connections
29 29 34 36 53 63 72
Figure 1 (a) and (b) shows the connection between the blocks in two subgames. For
this study, analysis is performed on two subgames, in which one is regular, and
another one is a constrained subgame. These two subgames are termed as
4Way2Hop and 4Way2Hop-IO in the game. For this paper, these two subgames are
named as SG1(4Way2Hop) and SG2 (4Way2Hop-IO), where SG stands for subgame.
Figure 1 is a model used to explain the connectivity, whose appearance is different
from the original game. Figure 1(a) is the connectivity of blocks in SG1, and figure
1(b) is the connectivity in the SG2 subgame. SG1 consists of only red rectangular
blocks, whereas SG2 contains red rectangular blocks in addition to blue circular
blocks. Players must place these blue blocks around the periphery of a puzzle/graph.
Because of this constraint of placing the blue blocks, SG2 could be complex
compared to SG1. In these two subgames, the connectivity between the blocks is the
same. That is, a node can connect to four immediate neighbor nodes (horizontally
and vertically), and by hopping two nodes in horizontal and vertical directions.
(a) (B)
FIGURE 1. CONNECTIVITY BETWEEN THE BLOCKS IN THE SUBGAMES (A) SG1, AND (B) SG2
(a) (B)
FIGURE 2. AN EXAMPLE OF AN INITIAL GRAPH (L1 LEVEL) (A) SG1, AND (B) SG2
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
Figure 2 depicts an original puzzle as it appears on the players’ screen with red
rectangular blocks in a grid, and connections between the blocks are in red and
black. Players must arrange these red and blue blocks based on constraints provided
to overcome violations. Being an open-ended problem, each puzzle can be solved in
multiple ways. As they move blocks, the grid size varies based on the blocks in the
row and column position of the grid. If they solve a puzzle in a more compact way
with zero violations, then that solution is considered as one of the feasible solutions.
To obtain a feasible solution, players can use different types of moves, which are
available in a gaming framework. These different move types are single, multi, swap,
add pass gate, and remove pass gate.
Apart from these, there are undo, flip, rotate, and paint options in the framework. In
the top left corner of fig 2, there are rubrics like score, violations, and time, which
will keep track of time taken by a player to solve a puzzle. In the center of the game
window, there are undo/redo, paint, flip/rotate, add/remove pass gate, and save
options. There are three stars on the top left, which is found above the score. These
three stars can be considered as three stages of performance, which are activated
based on the performance of players. The top right corner of the screen shows
symbols for reloading, pause, help (?), mute/unmute, and window close options.
Reload helps the player to restart the game from the beginning, a pause is for taking
a break in between, and help gives information on features in the game.
Figure 3(a) and (b) show screenshots of two feasible solutions to the original graph
in fig 2(a). Players can use any move types that are mentioned earlier. There is a
chance a player may solve the whole puzzle using only single move type or a
combination of any two move types. From the players’ solutions, a single move is
considered as a move in demand that is used by all players. In addition, most used
moves are multi, swap, rotate, add pass, and remove pass. A single move is used to
take one step at a time, a multi move is used to select a group of nodes together to
move them from one position to another, swap move is used to interchange the
position of two nodes, add pass gate is used to add pink blocks to connect one block
to another, and remove pass gate is used to remove the added pink blocks.
(a) (B)
FIGURE 3. TWO FEASIBLE SOLUTIONS PROVIDED BY TWO DIFFERENT PLAYERS FOR
SG1– L1 (A) SOLUTION 1 WITH GRID SIZE 4X6 (B) SOLUTION 2 WITH GRID SIZE 5X6
From these two solutions, it is evident that though there are no violations, while the
solution 2 score is less when compared to the solution 1 score. In solution 1, the
player has placed 24 blocks in four rows and six columns without any empty cells.
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In solution 2, the player has salved the same puzzle by placing 24 blocks in five rows
and six columns with four empty cells, and the player used two pink blocks.
Apparently, figure 3 concludes that by arranging blocks without any violations in a
compact way made solution 1 a better solution compared to solution 2. Pink color
blocks in solution 2 are called pass gates and are used to make a connection between
the blocks to overcome violations. The next section presents the analytical results of
players who played SG1. The performance of single and multiple group players,
separated based on the number of attempts made, are compared. In other words,
analysis performed shows whether strategies used by players who played the same
puzzle multiple times differed from players who played only one time. In addition,
analysis is performed on a number of attempts to help to solve advanced/complex
games (SG2).
4. DATA ANALYSIS
This study performed an analysis on the telemetry data of players who played the
subgames SG1 and SG2. The extracted data is categorized into two groups based on
the number of attempts made. Players who obtained feasible solutions in their first
attempt are treated as one group (single), and players who reached feasible
solutions in either their first or consecutive attempts by playing the same puzzle
repeatedly are considered as another group (multiple). In these results, the
categorical variable that represents these two groups is named as an attempt made.
Attributes considered to analyze the performance of the two groups are score and
type of moves (single, multi, swap, rotate, add pass, and remove pass). Statistical
tests are performed to see if there is any significant difference in the performance of
the two groups, and observations were made on the performance of these two
groups in the more complex subgame SG2 in terms of score.
4.1 Percentage of players
Figure 4 is a stacked bar graph that shows the percentage of players who played
each level a single or multiple time. This figure is obtained by extracting the
subgame SG1 players’ data for seven levels (L1-L7). The light blue indicates a
percentage of players who made a single attempt, and dark blue indicates a
percentage of players who made multiple attempts. In figure 4, the x-axis shows
levels L1-L7, and the y-axis represents the percentage of players. The legend at the
top right illustrates the nominal variable attempt made, which has two categories:
single and multiple. Level L1 determines that from the total number of players,
53.05% of players made a single attempt, and 46.95% made multiple attempts. In
level L2, 61.54% of players made a single attempt, and 38.46% made multiple
attempts. These multiple attempts can be any number as there is no restriction
provided to players on a number of attempts. In this paper, multiple attempt players
who reached feasible solutions are considered for analysis.
Figure 4 depicts that there are a smaller number of multiple attempt players on easy
levels compared to single attempt players. That is, more than 50% of players in the
levels L1-L4 are single attempt players, and more than 60% of players in L5-L7 are
multiple attempt players. Finally, this can conclude that players play the same level
multiple times if the level complexity is high and more often obtained a feasible
solution in a single attempt when the level complexity is less. Table 2 shows which
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
group obtained the maximum score on each level. Columns in the table indicate the
levels and rows depict the single and multiple groups.
FIGURE 4. PERCENTAGE OF SINGLE AND MULTIPLE ATTEMPT PLAYERS
TABLE 2. MAXIMUM SCORE OBTAINED BY TWO GROUPS IN EACH LEVEL
Levels
L1 L2 L3 L4 L5 L6 L7
Single 184,500 126,340 67,420 101,390 248,930 383,230 274,740
Multiple 184,700 126,330 67,400 101,680 246,290 383,740 275,110
For instance, in fig 5 for level L1, the maximum score obtained by a single group
player is 184,500, and a multiple group player is 184,700. This explains that the top
player is the player who made multiple attempts in level L1. Next, in levels L2, L3,
and L5, the maximum score is obtained by a player who made a single attempt. In
addition, in level L4, L6, and L7, the maximum score is obtained by a player who
made multiple attempts. Finally, figure 4 and table 2 conclude that among levels L5,
L6, and L7, in which there are more than 60% of players who made multiple
attempts, in L6 and L7, the top player is a multiple group player.
4.2. Single and Multiple players in constraint game
Table 3 depicts the analysis of the type of moves used by the two groups of players
to measure the difference in the performance of players to obtain a feasible
solution. For this analysis, the type of moves is considered as a nominal dependent
variable, and a variable attempt made is considered as an independent nominal
variable. The type of moves considered for this analysis are single, multi, swap,
rotate, add pass gate, and remove pass gate. Based on sample characteristics such as
size of each group and variable types, a Kruskal-Wallis (KW) test is performed to
find statistically significant differences between the two groups. This test is
considered as a non-parametric test and suitable for extracted data from the
UNTANGLED database. Asymptotic significance (asymp.sig) or p-value helps to
determine whether there is a statistically significance between the two groups or
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not. If p-value is ≤0.05, then there is a significance between single and multiple
groups.
TABLE 3. SIGNIFICANT LEVEL OF TWO GROUPS OF PLAYERS IN SG1
Level Single
Move
Swap
Move
Multi
Move
Rotate
Move
Add
Move
Remove
Move
L1 Chi-Square 4.745 .317 4.055 .885 6.712 .069
Df 1 1 1 1 1 1
Asymp. Sig. .029 .573 .044 .347 .010 .793
L2 Chi-Square .066 1.985 2.784 .000 3.019 .168
Df 1 1 1 1 1 1
Asymp. Sig. .797 .159 .095 1.000 .082 .682
L3 Chi-Square 1.979 .600 6.678 1.057 4.649 2.743
Df 1 1 1 1 1 1
Asymp. Sig. .159 .438 .010 .304 .031 .098
L4 Chi-Square .006 .652 2.465 .000 .528 .214
Df 1 1 1 1 1 1
Asymp. Sig. .936 .420 .116 1.000 .467 .644
L5 Chi-Square 4.124 2.191 6.612 .000 1.827 .216
Df 1 1 1 1 1 1
Asymp. Sig. .042 .139 .010 1.000 .176 .642
L6 Chi-Square 1.303 .889 1.045 .000 4.304 1.641
Df 1 1 1 1 1 1
Asymp. Sig. .254 .346 .307 1.000 .038 .200
L7 Chi-Square 6.582 .872 1.724 4.187 .588 .632
Df 1 1 1 1 1 1
Asymp. Sig. .010 .350 .189 .041 .443 .427
In table 3, the significance value for single move type of levels L1, L5, and L7 is ≤0.05.
That is, for these levels there is a significant difference between the mean number
of single moves used by the two groups. Similarly, there is no significance in mean
number of multi moves used by two groups in any levels. There is no such pattern
observed in the statistical results obtained. For some levels there is statistically
significant difference in the means of move type used and for some there is no such
difference. But, when the overall cases are observed, in most of the cases the
asymp.sig value in table 3 depicts that there is no statistically significant difference
between means of the two groups. Mean ranks in table 4 provide additional
information on which group used a greater number of moves. Mean ranks are
calculated between the two groups obtained after performing KW test as shown in
table 4. For instance, in level L1, for multi move type the single group has a greater
mean rank than the multiple group. This determines that single group players used
more multi moves than the multiple group players.
TABLE 4. MEAN RANK OF TWO GROUPS OF PLAYERS IN SG1 ON SCORE
Type of
Moves
Attempt
Made
Mean Rank
L1 L2 L3 L4 L5 L6 L7
Single
Move Single 74.9
0 53.10 59.14 43.67 36.82 40.61 35.94
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
Multiple 91.0
8
51.54 50.62 43.23 26.45 34.57 24.14
Swap
Move
Single 80.5
5
55.76 57.25 45.21 34.68 39.89 30.89
Multiple 84.7
1
47.29 52.62 40.76 27.15 34.91 26.59
Multi Move
Single 89.3
9 56.21 62.53 46.80 38.75 40.15 32.03
Multiple 74.7
2
46.56 47.05 38.20 25.83 34.79 26.04
Rotate
Move
Single 82.9
4 52.50 54.50 43.50 29.00 36.50 30.06
Multiple 82.0
0
52.50 55.53 43.50 29.00 36.50 27.00
Add Move
Single 91.3
9 47.86 60.76 45.01 34.18 43.89 25.64
Multiple 72.4
5
55.40 48.92 41.08 27.31 33.03 29.15
Remove
Move
Single 83.2
9 51.65 58.53 42.65 30.61 40.89 30.25
Multiple 81.6
1
53.03 51.27 44.86 28.48 34.44 26.91
TABLE 5. SIGNIFICANT LEVEL OF TWO GROUPS OF PLAYERS IN SG1 WITH RESPECT TO SCORE
Score
L1 L2 L3 L4 L5 L6 L7
Chi- Square
2.938 11.268 1.287 .542 11.632 34.759 15.371
df 1 1 1 1 1 1 1
Asymp.
Sig.
.086 .001 .257 .461 .001 .000 .000
TABLE 6. MEAN RANK OF TWO GROUPS OF PLAYERS IN SG1 WITH RESPECT TO SCORE
Scor
e
Attem
pt
Made
Mean Rank
L1 L2 L3 L4 L5 L6 L7
Single 1261.
91
837.0
7
732.8
1
651.0
1
426.5
3
355.1
3
310.7
4
Multip
le
1212.
52
759.4
5
707.6
0
635.5
6
487.1
7
451.9
3
370.2
1
For this level L1, the p-value in table 3 is <0.05, which shows there is significant
difference in the mean number of moves used by the two groups of players. In other
words, as the p-value in this case is 0.04, this concludes that there is a slightly
significant difference between the means of these two groups. The p-value of the
rotate move is 1.00 for levels L2, L4, L5, and L6. The corresponding mean ranks for
these levels is same for the two groups, which concludes both groups used almost
the same number of rotate moves to solve a puzzle. The next statistical test is
performed on continuous dependent variable Score. The asymp. sig value in table 5
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is ≤0.05 for all levels except L1, L3, and L4, which says there is statistically significant
difference in the mean scores obtained by the two groups. For levels L1, L3 and L4
there is no difference in mean scores obtained by the two groups of players. The p- value for levels L2, L5, and L6 is 0.001, and for L7 is 0.00. These values conclude that
there is highly significant difference between mean scores obtained by the two
groups of players.
In table 6, the mean ranks for the two groups of players are shown by considering
score as dependent variable. Highest mean rank shows which group has the highest
scores. Table 6 strengthens the results shown in figure 4. That is, mean ranks for
levels L5, L6, and L7 show that the highest scores are obtained by multiple group
players. In other words, these findings depict that a larger number of players made
multiple attempts in complex levels and maximum scores are also obtained by the
players who played those levels multiple times. The next analysis is performed on
the constrained game (SG2). Statistically significant difference between the two
groups of players on the dependent variable score is considered for this analysis.
The main purpose of performing analysis on the constrained puzzle is to observe
whether there is any difference in the scores obtained by these two groups.
4.3. Single and Multiple players in constrained game
Among 647 players who played SG1, 315 players solved puzzles in a single attempt,
whereas the remaining 332 players solved in multiple attempts. In the same way,
among 704 players who played SG2, 420 players solved the game in a single attempt
and 284 players made multiple attempts. This depicts that players are making
multiple attempts in the advanced puzzles.
TABLE 7. SIGNIFICANT LEVEL OF TWO GROUPS OF PLAYERS IN SG2 WITH RESPECT TO SCORE
Scor
e
L1 L2 L3 L4 L5 L6 L7
Chi- Square
1.13
3
7.65
4
6.640 2.42
0
3.24
1
7.85
0
6.57
0
df 1 1 1 1 1 1 1
Asymp.
Sig.
.287 .006 .010 .120 .072 .005 .010
TABLE 8. MEAN RANK OF TWO GROUPS OF PLAYERS IN SG2 WITH RESPECT TO SCORE
Scor
e
Attemp
t Made
Mean Rank
L1 L2 L3 L4 L5 L6 L7
Single 105.3
7
84.8
0
67.6
8
55.2
3
23.9
1
17.3
1
10.4
1
Multipl
e
115.0
4
65.1
5
51.3
6
46.1
4
31.7
1
28.9
1
18.5
0
Tables 7 and 8 are the results of a non-parametric Kruskal-Wallis test performed on
the SG2 sample for finding statistically significant difference between the two
groups of players based on score. Like SG1, the mean rank is high for L1, L5, L6 and
L7 levels, which depicts that multiple group players for these levels obtained high
scores. For levels L1 and L4, asymp.sig value is >0.05, which says there is no
significant difference between the score obtained by the two groups of players,
whereas for the remaining levels there is a significant difference. In other words,
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
there is difference between the score obtained by the two groups of players in level
L2, L3, L5, L6, and L7. The above analytical results conclude that single and multiple
attempt group players equally contribute to solve the given puzzle. In fact, multiple
attempt players are trying to come up with multiple solutions for the same puzzle
even though they obtained the feasible solution in their first attempt. Finally, results
obtained illustrate that both groups of players performed equally to provide feasible
solutions in the UNTANGLED framework. The next sections discuss on results
obtained and provide suggestions for STEM education.
5. DISCUSSION
This paper examined how the players’ thinking skills help to solve the puzzles when
they solve open ended problems. Inductive approach in our research helps to
examine convergent and divergent thinking that are essential skills for problem
solving. Researchers [9], [25]–[27] explain the importance of these skills in
engineering and describe that convergent thinking is the qualitative process of
providing a single and fixed solution for a given problem, while divergent thinking
is a quantitative process of providing multiple solutions for a given problem in
different perspectives. For this study, the type of moves and score obtained is used
to measure the performance of these two thinking skills. This paper analyzed the
solutions of random players who played these seven puzzles available in the game.
The findings demonstrate that a few numbers of players solved initial puzzles with
a smaller number of nodes and connections in a single attempt, whereas for puzzles
like L5, L6, and L7 that have a greater number of nodes and connections, more than
60% of the players made multiple attempts. Also, noticed that among these levels
only in L5 and L7 do players who made multiple attempts scored higher than the
single attempt group. These results obtained cannot conclude that divergent
thinking strategy can be used to solve complex puzzles. But, based on the p-value
>0.05 in almost all cases emphasizes that there is no statistically significant
difference in the scores obtained and type of moves used by the two groups of
players. Finally, results conclude that either in regular or constrained puzzles, there
is no significant difference in number of moves used by the two groups. However, in
some levels players who made multiple attempts scored the highest score whereas
in some other cases it is vice versa.
From these observations one can determine that players in the single group reached
a feasible solution in a first attempt and they did not try in multiple ways with
different perspective. In contrast, players in the multiple group played the same
puzzle repeatedly in different ways. These players made multiple attempts though
they reached a feasible solution in their initial attempts. This study observed that
the average number of attempts made by the multiple group’s players is five. Hence,
this research emphasizes that one needs to provide scope for both kinds of thinking
skills to solve open-ended problems, which are like puzzles shown in this game. An
individual should be able to come with multiple possible perspectives of solving a
problem and should know when players must stop the game with proper analysis.
They should be able to judge their solution, evaluate it, and if required need to play
it again to get to feasible solutions.
The study supports the findings of Lin [28] on creative problem solving on math
abilities which suggest that attributes like divergent thinking, convergent thinking,
motivation, general knowledge, and environment have equal importance in
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developing creative problem solving. The author’s cluster analysis results show that
correlations between divergent and convergent thinking is high compared to any
other attributes in creative problem solving in education. In conclusion to the
discussion, this paper observes that a few factors or features embedded in gaming
framework might have helped players with convergent and divergent thinking to
provide feasible solutions for the given open-ended problems. Some of these
features are having no restriction in choosing a puzzle to play, freedom on making
number of attempts, no curtailment on time for solving a puzzle, and so on. From
these findings obtained, the next section made few recommendations to educators
and the scientific game designers.
6. IMPLICATIONS
Researchers [29]–[32] explain the relationship between creativity, problem solving,
and individual learning styles. They demonstrated how different learning styles or
processes can enhance creative problem solving through learning processes and
they suggest considering individual learning style in pedagogy. In support to this,
our study suggests that Science, Technology, Engineering, and Mathematics (STEM)
educators can design course structure or training methodologies for students to
prepare them to face novel situations and solve unforeseen problems creatively.
Educators can use techniques in their courses that encourage both convergent and
divergent thinking. Like the engineering game used in this paper, educators can
develop or use such online frameworks that include features that enhance both
convergent and divergent thinking skills. They can provide some entry level
problems to introduce to conventional procedures, use mock quizzes/exams with
zero penalty on grades to give freedom for expressing thoughts, offer problems or
projects related to real world engineering problems and create a highly interactive
environment that can enhance students’ performance. Additionally, scientific game
designers can develop games that provide flexibility for multiple solutions, concise
instructions on how to analyze or evaluate their solutions, and opportunities to
work in groups. That is, designers can include leaderboard feature like the one in
the gaming framework that is used in this study can motivate players to boost their
performance. Designers can also include multiplayer option to encourage
collaborative game play to further improve the performance of participants.
7. CONCLUSION
This paper showcased the performance of two groups of players who played an
open-ended scientific puzzle game, which are single and multiple groups based on
the number of attempts made by players. Statistical tests elucidate that there is no
significant difference in performance between these two groups in the SG1game.
Additionally, in SG2, though there are fewer players who are interested in multiple
attempts, findings illustrate that this group is equally competitive with the single
group. Results show that divergent thinking helps to solve more complex puzzles at
higher scores. This concludes that divergent thinking is a unique and important
thinking competency in solving scientific puzzles. Study determines that both
groups can contribute to feasible solutions in the gaming framework, it is clearly
important for educators and scientific game designers to develop an environment
that engages the long-term interest of players with both convergent and divergent
thinking strategies. Although divergent thinking is more common in art education,
STEM educators can increase the interest of students with divergent thinking by
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Joseph, R. D. B., Tunks, J., & Mehta, G. (2021). Convergent and Divergent Thinking Skills in Electrical Engineering
Gaming Framework. European Journal of Applied Sciences, 9(2). 190-203.
URL: http://dx.doi.org/10.14738/aivp.92.9974
providing online pedagogical tools which allow for freedom in making errors and
reward multiple ideas. Finally, this paper suggests that to serve the purpose of
scientific games or to encourage the students' creative problem-solving skills in the
field of engineering, educators and designers need to develop a framework that can
enhance both convergent and divergent thinking competencies.
8. ACKNOWLEDGEMENTS
We would like to thank National Science Foundation (NSF) for supporting our work
with a grant NSF CCF-1617475.
References
[1] OECD, PISA 2012 Results : Creative Problem Solving, vol. V. 2012.
[2] J. P. Adams, S. Kaczmarczyk, P. Picton, and P. Demian, “Problem Solving and Creativity in
Engineering: Perceptions of Novices and Professionals,” Lect. Notes Eng. Comput. Sci., vol. I,
2009.
[3] D. H. Cropley and A. J. Cropley, “Fostering Creativity in Engineering Undergraduates,” High
Abil. Stud., vol. 11, no. 2, pp. 207–219, 2000, doi: 10.1080/13598130020001223.
[4] D. H. Cropley, A. J. Cropley, and B. L. Sandwith, “Creativity in the engineering domain,”
Cambridge Handb. Creat. across Domains, pp. 261–275, 2017, doi:
10.1017/9781316274385.015.
[5] D. L. Dekker, “Engineering design processes, problem solving & creativity,” Proc. - Front.
Educ. Conf., vol. 1, pp. 445–448, 1995.
[6] M. Karyotaki and A. Drigas, “Online and other ICT-based training tools for problem-solving
skills,” Int. J. Emerg. Technol. Learn., vol. 11, no. 6, pp. 35–39, 2016, doi:
10.3991/ijet.v11i06.5340.
[7] B. D. Nielsen, C. L. Pickett, and D. K. Simonton, “Conceptual versus experimental creativity:
Which works best on convergent and divergent thinking tasks?,” Psychol. Aesthetics, Creat.
Arts, vol. 2, no. 3, pp. 131–138, Aug. 2008, doi: 10.1037/1931-3896.2.3.131.
[8] D. K. Simonton, “Creativity: Cognitive, personal, developmental, and social aspects.,” Am.
Psychol., vol. 55, no. 1, pp. 151–158, 2000, doi: 10.1037/0003-066X.55.1.151.
[9] R. Pathan, U. Khwaja, D. Reddy, and V. V. Kamat, “Teaching and Learning of Divergent &
Convergent Thinking Skills using DCT,” Proc. - IEEE 8th Int. Conf. Technol. Educ. T4E 2016,
no. September, pp. 54–61, 2017, doi: 10.1109/T4E.2016.020.
[10] A. Y. K. and D. A. Kolb, “Learning Styles and Learning Spaces: Enhancing Experiential
Learning in Higher Education,” vol. 4, no. 2, pp. 193–212, 2005, doi: 10.31219/osf.io/rdq97.
[11] A. Y. Kolb and D. A. Kolb, “The Kolb Learning Style Inventory - Version 4.0,” Exp. Based
Learn. Syst. Inc., p. 234, 2013.
[12] Saul McLeod, “Kolb Learning Style,” 2017. http://www.simplypsychology.org/learning- kolb.html.
[13] C. Charyton, R. J. Jagacinski, and J. A. Merrill, “CEDA: A research instrument for creative
engineering design assessment.,” Psychol. Aesthetics, Creat. Arts, vol. 2, no. 3, pp. 147–154,
Aug. 2008, doi: 10.1037/1931-3896.2.3.147.
[14] C. Charyton and J. A. Merrill, “Assessing general Creativity and Creative engineering Design
in first year engineering students,” J. Eng. Educ., vol. 98, no. 2, pp. 145–156, 2009, doi:
10.1002/j.2168-9830.2009.tb01013.x.
[15] D. J. Pepler and H. S. Ross, “The Effects of Play on Convergent and Divergent Problem
Solving,” Child Dev., vol. 52, no. 4, p. 1202, 1981, doi: 10.2307/1129507.
[16] L. Von Ahn, R. Liu, and M. Blum, “Peekaboom: A Game for Locating Objects in Images,” 2006.
[17] S. Cooper et al., “Predicting protein structures with a multiplayer online game,” Nature, vol.
Page 14 of 14
203
European Journal of Applied Sciences, Volume 9 No. 2, April 2021
Services for Science and Education, United Kingdom
466, no. 7307, pp. 756–760, 2010, doi: 10.1038/nature09304.
[18] L. Von Ahn and L. Dabbish, “Labeling Images with a Computer Game,” 2004. Accessed: Sep.
19, 2019. [Online]. Available: http://www.espgame.org.
[19] G. Mehta et al., “Untangled: A game environment for discovery of creative mapping
strategies,” ACM Trans. Reconfigurable Technol. Syst., vol. 6, no. 3, 2013, doi:
10.1145/2517325.
[20] G. Mehta, K. K. Patel, N. Parde, and N. S. Pollard, “Data-Driven Mapping Using Local
Patterns,” IEEE Trans. Comput. Des. Integr. Circuits Syst., vol. 32, no. 11, pp. 1668–1681, Nov.
2013, doi: 10.1109/TCAD.2013.2272541.
[21] G. Mehta, K. Patel, and N. S. Pollard, “On fast iterative mapping algorithms for stripe based
coarse-grained reconfigurable architectures,” Int. J. Electron., vol. 102, no. 1, pp. 3–17, Jan.
2015, doi: 10.1080/00207217.2014.938310.
[22] A. K. Sistla, K. Patel, and G. Mehta, “Crowdsourcing the mapping problem for design space
exploration of custom reconfigurable architecture designs,” Hum. Comput., vol. 2, no. 1, pp.
69–103, 2015, doi: 10.15346/hc.v2i1.5.
[23] Science, “Science/AAAS | Special Issue: 2012 International Science & Engineering
Visualization Challenge,” 2012. https://www.sciencemag.org/site/special/vis2012/.
[24] NSF, “Winners of 10th Annual International Science & Technology Visualization Challenge
Announced,” 2013.
https://www.nsf.gov/news/news_summ.jsp?cntn_id=126758&WT.mc_id=USNSF_51&WT.
mc_ev=click.
[25] D. A. Kolb, “Experiential Learning: Experience as The Source of Learning and Development,”
Prentice Hall, Inc., no. 1984, pp. 20–38, 1984, doi: 10.1016/B978-0-7506-7223-8.50017-4.
[26] D. A. Kolb and A. Y. Kolb, “Research on Validity and Educational Applications,” no. May 2016,
pp. 0–233, 2013, doi: 10.1016/S0020-7519(02)00196-0.
[27] P. D. Reddy, S. Iyer, and M. Sasikumar, “Teaching and learning of divergent and convergent
thinking through open-problem solving in a data structures course,” Proc. - 2016 Int. Conf.
Learn. Teach. Comput. Eng. LaTiCE 2016, pp. 178–185, 2016, doi: 10.1109/LaTiCE.2016.13.
[28] C.-Y. Lin, “Threshold Effects of Creative Problem-Solving Attributes on Creativity in the Math
Abilities of Taiwanese Upper Elementary Students,” Educ. Res. Int., vol. 2017, pp. 1–9, 2017,
doi: 10.1155/2017/4571383.
[29] S.-C. Wang and J.-Y. Chern, “The ‘Night Owl’ Learning Style of Art Students: Creativity and
Daily Rhythm,” Int. J. Art Des. Educ., vol. 27, no. 2, pp. 202–209, Jun. 2008, doi:
10.1111/j.1476-8070.2008.00575.x.
[30] I. Y. Kazu, “The Effect of Learning Styles on Education and the Teaching Process,” J. Soc. Sci.,
vol. 5, no. 2, pp. 85–94, 2009, doi: 10.3844/jssp.2009.85.94.
[31] M. Mimi, Muhaffyza and M. Heong, Yee, “Identifying Relationship Involving Learning Styles
And Problem Solving Skills Among Vocational Students Faculty of Technical Education
University Tun Hussein Onn Malaysia Muhammad Rashid Rajuddin Faculty of Education
University Technology Malaysia Email :,” J. Tech. Educ. Train., vol. 3, no. 1, pp. 37–46, 2011.
[32] K. C. Tsai, “Taiwanese elementary student’s creativity, creative personality, and learning
styles: An exploratory study,” Alberta J. Educ. Res., vol. 60, no. 3, pp. 464–473, 2014.