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Citation: Daniels, Jonathan S., David
Moreau, and Brooke N. Macnamara.
2022. Learning and Transfer in
Problem Solving Progressions.
Journal of Intelligence 10: 85.
https://doi.org/10.3390/
jintelligence10040085
Received: 29 August 2022
Accepted: 8 October 2022
Published: 12 October 2022
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Intelligence
Journal of
Article
Learning and Transfer in Problem Solving Progressions
Jonathan S. Daniels 1, * , David Moreau 2and Brooke N. Macnamara 1
1Department of Psychological Sciences, Case Western Reserve University, Cleveland, OH 44106, USA
2Department of Psychology, University of Auckland, Auckland 1010, New Zealand
*Correspondence: jonathan.daniels@cwru.edu
Abstract:
Do individuals learn more effectively when given progressive or variable problem-solving
experience, relative to consistent problem-solving experience? We investigated this question using
a Rubik’s Cube paradigm. Participants were randomly assigned to a progression-order condition,
where they practiced solving three progressively more difficult Rubik’s Cubes (i.e., 2
×
2
×
2 to
3×3×3
to 4
×
4
×
4), a variable-order condition, where they practiced solving three Rubik’s Cubes
of varying difficulty (e.g., 3
×
3
×
3 to 2
×
2
×
2 to 4
×
4
×
4), or a consistent-order condition, where
they consistently practiced on three
5×5×5
Rubik’s Cubes. All the participants then attempted
a
5×5×5
Rubik’s Cube test. We tested whether variable training is as effective as progressive
training for near transfer of spatial skills and whether progressive training is superior to consistent
training. We found no significant differences in performance across conditions. Participants’ fluid
reasoning predicted 5 ×5×5 Rubik’s Cube test performance regardless of training condition.
Keywords: learning; near-transfer; problem-solving; spatial reasoning; Rubik’s Cube
1. Introduction
People are often challenged when having to learn new skills in a limited amount of
time. In many cases, the most efficient way to learn a skill is to break it down to its core
components and gradually increase complexity. For instance, when learning mathematics,
we typically need to start with the basics, such as counting and addition before we move
to progressively more complex functions such as multiplication or division. Likewise,
when learning to read, we start by mapping sounds to written symbols before deciphering
high-level construction.
However, not all learning occurs from early, formal instruction like math and reading.
Without guided instruction, we typically apply various solutions to new situations often
making numerous errors (Ebbinghaus 1913). Slowly, patterns that lead to more directed
and efficient manners of problem-solving are discovered (Fitts and Posner 1967). In a world
where individuals are exposed to diverse and often unconstrained learning environments,
such as when learning to solve problems of a spatial nature, it is important to consider
how various learning progressions impacts one’s problem-solving abilities and overall
performance.
Learning, defined as our ability to encode new or modify old information and apply
it to future situations, has been studied for generations (Gross 2010). One important
component in the study of learning and memory is the transfer of learning—the idea that the
concepts learned from one situation can be applied to another (Woodworth and Thorndike
1901). Numerous studies have shown that transfer of learning through training can improve
one’s performance on more complex tasks (e.g., Broadbent et al. 2015;Meneghetti et al.
2017;Nayigizente and Richard 2000;Schiff and Vakil 2015;Vakil and Heled 2016).
The transfer of learning can be divided into two types: near and far transfer. Near
transfer—the application of learned situations to new, yet similar situations (as opposed
to far transfer, which is associated with new and relatively different situations; Schunk
J. Intell. 2022,10, 85. https://doi.org/10.3390/jintelligence10040085 https://www.mdpi.com/journal/jintelligence
J. Intell. 2022,10, 85 2 of 13
2004)—has often been studied for its application to spatial problems (e.g., Brunyéet al.
2019;Minear et al. 2016;Schiff and Vakil 2015;Trumbo et al. 2016;Vakil and Heled 2016).
Spatial problem-solving is considered a key component in a number of performance
domains from mathematics (Hegarty and Kozhevnikov 1999) to sports (Moreau et al.
2012). One of the tasks used to examine near transfer in spatial problem solving is the
Tower of Hanoi (e.g., Helfenstein 2005;Minsky et al. 1985;Nayigizente and Richard 2000;
Schiff and Vakil 2015;Tarasuik et al. 2017;Vakil and Heled 2016), a validated spatial
reasoning task (Ahonniska et al. 2000). Of this literature, only a small percentage of studies
has investigated training progressions, with designs systematically increasing levels of
difficulty. An exception is Vakil and Heled’s (2016) work investigating the schema theory
of discrete motor skill learning (Schmidt et al. 2011). The schema theory of discrete motor
learning posits that schemas form as rules and parameters that are compared to novel
situations. Based on this theory, Vakil and Heled (2016) hypothesized that participants
would naturally search for generalizable solutions to the Tower of Hanoi, rather than a
rigid and specific sequence of actions. To test this, they manipulated the order of training
using the Tower of Hanoi, where some participants received towers varying in difficulties
while others received the same level of difficulty. In support of the theory of discrete motor
learning, Vakil and Heled (2016) found that participants in the varied training condition
yielded better learning transfer than participants in the constant training condition.
However, in spatial problem-solving tasks, it has yet to be tested what type of varied
training leads to better transfer. In particular, the question remains whether varied training
needs to be progressive in nature, where participants gain experience with incrementally
more difficult versions of the task, or whether experiencing random variation is sufficient.
Outside of spatial problem-solving, a working-memory training study (von Bastian and
Eschen 2016) suggested that random variation may be sufficient. von Bastian and Eschen
(2016) tested whether near transfer differed for participants randomly assigned to an
adaptive, progression-ordered training, a randomly-varied training, a self-selected task
difficulty training, or an active control. They found that training effects were not modulated
by condition. They concluded that training with varying levels of task difficulty, regardless
of progression order, produces training gains.
von Bastian and Eschen (2016) were not the first to suggest that random variation may
be beneficial for learning and perhaps more so than progressive sequences. Work by Shea
and Morgan (1979) showed evidence of this by introducing difficulty in training a motor
task via randomized order. Though participants in the randomized condition showed
slower acquisition during training, they showed greater transfer and generalizability on
later testing compared to their ordered-block counterparts. According to schema theory
(Schmidt 1975;Wulf and Schmidt 1988), variability in practice is beneficial because it
enhances the effectiveness of rules (schemata). Relatedly, Schmidt and Bjork (1992) argue
that across verbal and motor tasks, variable practice encourages additional information
processing leading to better transfer of learning. In this paper, we address the role of
variable practice further from the domain of spatial problem-solving.
Present Study
The goal of the present study was to examine ordering effects in problem-solving
progressions on a three-dimensional puzzle—the Rubik’s Cube. We chose to use three-
dimensional puzzles rather than a computerized task to best examine Schmidt et al.’s
(2011) theory of motor skill learning, and to increase the ecological validity of studying
three-dimensional problems (Schiff and Vakil 2015;Vakil and Heled 2016).
We specifically chose the Rubik’s Cube for its difficulty. Whereas the Tower of Hanoi is
fairly simple and training gains are easily found, only six percent of the global population
has solved a Rubik’s Cube (Vega 2018). We thus sought to test different types of varied
training in a task where participants would not be performing at ceiling with limited
training. If we find converging evidence, this offers strong support for previous findings
that used simpler tasks. In contrast, if we find countering evidence or no effects, this
J. Intell. 2022,10, 85 3 of 13
suggests that problem difficulty might be an important factor to consider when testing
hypotheses about training and learning transfer in spatial problem solving.
Further, while previous research in this area has been based on motor skill learning,
we were also interested in the pattern recognition and reasoning presumably active while
attempting to solve Rubik’s Cubes. We assume that pattern recognition and reasoning
are highly involved in Rubik’s Cube solving because Rubik’s Cubes have large problem
spaces, which are defined as the number of possible states in the problem and the number
of options at each stage of the solution (Newell and Simon 1972). As the problem space
increases, so does task difficulty, and the presumed recruitment of cognitive resources
(Frank and Macnamara 2021;Macnamara and Frank 2018).
Finally, examining transfer effects with a novel paradigm, rather than with a common
one from previous near-transfer literature, has the potential to give insight as to whether
any effects from varied training is paradigm-specific or widespread across spatial reasoning
problem-solving tasks. That is, the Rubik’s Cube not only differs from the Tower of Hanoi
in the size of its problem space, but on the type of actions needed at each stage. The Tower
of Hanoi involves moving independent objects (though with relational rules), whereas the
Rubik’s Cube involves changing the task object, where each move shifts other components
non-independently.
We tested several hypotheses. One hypothesis is that participants in the progression-
order group would perform significantly better on the final spatial problem-solving test
than those in the variable-order group. This would support findings from complex learning
studies where progression-order training is the norm (e.g., van Merriënboer and Kirschner
2017). Our competing hypothesis was that the variable-order group would have a similar
level of performance on the final spatial problem-solving test as the progression-order
group. This would support the schema theory of discrete motor skill learning and von
Bastian and Eschen’s (2016) finding that solely varying the levels of task difficulty without
adherence to a progression is as effective as progression-order training.
Second, we hypothesized that both the progressive-order and variable-order condi-
tions would outperform the consistent-order condition. We did not think it likely that
participants in the consistent-order condition would outperform the other groups, as
beginning training at a difficult level tends to slow learning (e.g., Ahissar et al. 2009).
Finally, we postulated that participants’ fluid reasoning—one’s capacity to think
logically and solve problems in novel situations, independent of acquired knowledge
(Cattell 1987)—would be positively correlated with performance on the final test, across
conditions. We also examined whether training condition moderated the relationship
between fluid reasoning and test performance.
2. Materials and Methods
Protocol and analyses were pre-registered, and materials and data can be found on
the Open Science Framework, osf.io/uk6w2.
2.1. Participants
An a priori power analysis indicated that a minimum of 128 participants was needed
to test our hypotheses, assuming a medium effect size (d= 0.5) with .80 power and alpha
set at .05. Our stopping rule was to collect data until the end of the week in which the 128th
participant was recruited. A total of 141 members of the Case Western Reserve University
community were recruited to participate in the study. Ninety-four participants were
recruited through various advertisements throughout campus and compensated fifteen
dollars at the completion of the study session. Forty-seven participants were recruited
through the Case Western Reserve University’s Psychology subject pool for partial course
credit in the Introduction to Psychology course. Eight participants who were recruited
had successfully solved a Rubik’s Cube in the past three years and, following our pre-
registered exclusion criteria, were excused from the study, and given partial course credit
or five dollars. The final sample was 133 (84 female). The mean age was 24 years old
J. Intell. 2022,10, 85 4 of 13
(
SD = 9.39 years
). This study was approved by the Institutional Review Board at Case
Western Reserve University.
2.2. Measures and Procedure
Participants were brought into the lab and seated at one of six computer stations. In
front of each participant were four Rubik’s Cubes, covered completely with the same-sized
occluder such that participants did not know what was underneath and could not ascertain
size differences. The occluders were bottomless boxes designed to cover Rubik’s cubes.
The Rubik’s cubes remained covered until it was time to complete that puzzle. Thus,
participants were unaware of an upcoming Rubik’s Cube complexity.
Before revealing the Rubik’s cubes, participants were asked to complete the Raven’s
Advances Progressive Matrices (RAPM; Raven 2003). The RAPM is a psychometrically-
validated measure of fluid reasoning designed for adults with above-average reasoning.
The RAPM is a multiple-choice test in which participants are tasked with completing
matrix patterns by choosing the correct missing item. Odd-numbered items (18 total) of
progressive difficulty were used and participants were given a ten-minute time limit to
solve as many problems as they could.
Following the RAPM, instructions appeared on the computer screen in front of the
participant. Computer stimuli were created and presented using E-Prime 2 (Psychology
Software Tools 2002) and displayed on a 23-inch OptiPlex 9030 computer screens. Stimuli
were displayed at a resolution of 1920 ×1080 pixels at a refresh rate of 59 Hz.
The instructions on the computer screen read as follows: “In front of you, under
each box is a puzzle. Your job is to solve each in order from left to right as best as you
can. To solve each puzzle, you must successfully match all colors until there is only one
distinct color on each side.” All participants had four minutes to independently work
on each Rubik’s Cube before being prompted to move to the next, with thirty, twenty,
and ten second warnings on the computer screen prior to the four-minute deadline. The
four-minute timeline was based on pilot feedback. Greater amounts of time resulted in
frustration and pilot participants indicating that they felt they would “undo” their progress
by continuing beyond this amount of time.
Participants were assigned to one of three conditions in counterbalanced order. In the
progression-order condition, participants were tasked with solving increasingly difficult
standardized Rubik’s Cubes under time restrictions (2
×
2
×
2
→
3
×
3
×
3
→
4
×
4
×
4). In the variable-order condition, participants were given the same time restrictions and
were tasked with solving Rubik’s Cubes of varying difficulty in pseudo-random order
(e.g., 3
×
3
×
3
→
2
×
2
×
2
→
4
×
4
×
4). (An order of 2
×
2
×
2
→
3
×
3
×
3
→
4
×
4
×
4 was excluded as it would be equivalent to the progression-order condition). In the
consistent-order condition, participants had the same time restrictions to solve three 5
×
5
×5 Rubik’s Cubes. Figures 1–3show the initial Rubik’s Cubes in each condition.
J. Intell. 2022, 10, x FOR PEER REVIEW 5 of 14
Figure 1. Consistent-order condition. Cube sizes are identical across the learning phase and at test
(5 × 5 × 5).
Figure 2. Progression-order condition. Cubes increase in size during the learning phase (2 × 2 × 2; 3
× 3 × 3; 4 × 4 × 4) before completing the test (5 × 5 × 5).
Figure 3. Variable-order condition example. Cubes vary in size during the learning phase (here, 3 ×
3 × 3; 4 × 4 × 4, 2 × 2 × 2) before the test (5 × 5 × 5).
2.3. Rubik’s Cube Scoring
To measure performance on the Rubik’s Cubes, we recorded the maximum propor-
tion of matching colors (green, blue, red, orange, white and yellow) on a single Rubik’s
Cube side. For each color, we recorded the side with the highest number of that color. For
example, a 3 × 3 × 3 Rubik’s Cube has nine squares on each of its six sides. If the participant
had successfully matched a maximum of five red, two orange, four green, three blue, four
white, and three yellow square pieces on any side, their completion would be recorded as
21 56
⁄ or 38%.
∗
(1)
Percent completion does not always indicate better progress toward solving a Ru-
bik’s Cube. That is, when solving a Rubik’s Cube, sometimes like colors need to be moved
apart to make progress toward the solution. However, given that the Rubik’s Cubes could
not be completely solved by novices in the timeframe given, participants were given the
goal of moving as many like colors on the same side in the time allotted. Again, based on
our piloting, the time chosen represented the time in which most pilot participants made
the most progress toward this goal.
Figure 1.
Consistent-order condition. Cube sizes are identical across the learning phase and at test
(5 ×5×5).
J. Intell. 2022,10, 85 5 of 13
J. Intell. 2022, 10, x FOR PEER REVIEW 5 of 14
Figure 1. Consistent-order condition. Cube sizes are identical across the learning phase and at test
(5 × 5 × 5).
Figure 2. Progression-order condition. Cubes increase in size during the learning phase (2 × 2 × 2; 3
× 3 × 3; 4 × 4 × 4) before completing the test (5 × 5 × 5).
Figure 3. Variable-order condition example. Cubes vary in size during the learning phase (here, 3 ×
3 × 3; 4 × 4 × 4, 2 × 2 × 2) before the test (5 × 5 × 5).
2.3. Rubik’s Cube Scoring
To measure performance on the Rubik’s Cubes, we recorded the maximum propor-
tion of matching colors (green, blue, red, orange, white and yellow) on a single Rubik’s
Cube side. For each color, we recorded the side with the highest number of that color. For
example, a 3 × 3 × 3 Rubik’s Cube has nine squares on each of its six sides. If the participant
had successfully matched a maximum of five red, two orange, four green, three blue, four
white, and three yellow square pieces on any side, their completion would be recorded as
21 56
⁄ or 38%.
∗
(1)
Percent completion does not always indicate better progress toward solving a Ru-
bik’s Cube. That is, when solving a Rubik’s Cube, sometimes like colors need to be moved
apart to make progress toward the solution. However, given that the Rubik’s Cubes could
not be completely solved by novices in the timeframe given, participants were given the
goal of moving as many like colors on the same side in the time allotted. Again, based on
our piloting, the time chosen represented the time in which most pilot participants made
the most progress toward this goal.
Figure 2.
Progression-order condition. Cubes increase in size during the learning phase (2
×
2
×
2;
3×3×3; 4 ×4×4) before completing the test (5 ×5×5).
J. Intell. 2022, 10, x FOR PEER REVIEW 5 of 14
Figure 1. Consistent-order condition. Cube sizes are identical across the learning phase and at test
(5 × 5 × 5).
Figure 2. Progression-order condition. Cubes increase in size during the learning phase (2 × 2 × 2; 3
× 3 × 3; 4 × 4 × 4) before completing the test (5 × 5 × 5).
Figure 3. Variable-order condition example. Cubes vary in size during the learning phase (here, 3 ×
3 × 3; 4 × 4 × 4, 2 × 2 × 2) before the test (5 × 5 × 5).
2.3. Rubik’s Cube Scoring
To measure performance on the Rubik’s Cubes, we recorded the maximum propor-
tion of matching colors (green, blue, red, orange, white and yellow) on a single Rubik’s
Cube side. For each color, we recorded the side with the highest number of that color. For
example, a 3 × 3 × 3 Rubik’s Cube has nine squares on each of its six sides. If the participant
had successfully matched a maximum of five red, two orange, four green, three blue, four
white, and three yellow square pieces on any side, their completion would be recorded as
21 56
⁄ or 38%.
∗
(1)
Percent completion does not always indicate better progress toward solving a Ru-
bik’s Cube. That is, when solving a Rubik’s Cube, sometimes like colors need to be moved
apart to make progress toward the solution. However, given that the Rubik’s Cubes could
not be completely solved by novices in the timeframe given, participants were given the
goal of moving as many like colors on the same side in the time allotted. Again, based on
our piloting, the time chosen represented the time in which most pilot participants made
the most progress toward this goal.
Figure 3.
Variable-order condition example. Cubes vary in size during the learning phase (here,
3×3×3; 4 ×4×4, 2 ×2×2) before the test (5 ×5×5).
All participants were then given four minutes to solve a 5
×
5
×
5 Rubik’s Cubes as
best as they could with thirty, twenty, and ten second warnings prior to the four-minute
deadline. Scrambling of each Rubik’s Cube was based on algorithms generated from
TNoodle-WCA-0.13.5, the official scrambling program for the World Cube Association
(Fleischman et al. n.d.). Each participant received the same Rubik’s Cube configuration
for each respective puzzle (i.e., all 2
×
2
×
2 puzzles were the same initial configuration,
all 3
×
3
×
3 puzzles were the same configuration, etc.). Configurations can be found on
Open Science Framework, osf.io/uk6w2. Finally, participants were asked to describe their
thoughts and strategies during the tasks in an open-ended response using the computer
keyboard, as well as indicate their gender and age on a computerized questionnaire.
2.3. Rubik’s Cube Scoring
To measure performance on the Rubik’s Cubes, we recorded the maximum proportion
of matching colors (green, blue, red, orange, white and yellow) on a single Rubik’s Cube
side. For each color, we recorded the side with the highest number of that color. For
example, a 3
×
3
×
3 Rubik’s Cube has nine squares on each of its six sides. If the
participant had successfully matched a maximum of five red, two orange, four green, three
blue, four white, and three yellow square pieces on any side, their completion would be
recorded as 21/56 or 38%.
Σ(maximum number of like colors on a side)
total pieces on puzzle =5[red]+2[orange]+4[green]+3[blue]+4[white]+3[yellow]
56 [6∗9]=21
56 (1)
Percent completion does not always indicate better progress toward solving a Rubik’s
Cube. That is, when solving a Rubik’s Cube, sometimes like colors need to be moved apart
to make progress toward the solution. However, given that the Rubik’s Cubes could not
be completely solved by novices in the timeframe given, participants were given the goal
of moving as many like colors on the same side in the time allotted. Again, based on our
piloting, the time chosen represented the time in which most pilot participants made the
most progress toward this goal.
J. Intell. 2022,10, 85 6 of 13
3. Results
Pre-registered analyses were as follows: (1) an independent samples t-test, comparing
participants’ 5
×
5
×
5 Rubik’s Cube test completion between the progression-order
and variable-order groups, (2) an equivalence test (Lakens 2017) to test the competing
hypothesis that there was no difference on test performance between the progression-order
and the consistent-order conditions, (3) an independent samples t-test comparing the
progression-order and consistent-order groups, and (4) a Pearson’s correlation to test for a
relationship between participants’ RAPM performance and their 5
×
5
×
5 Rubik’s Cube
Completion test. We also pre-registered an exploratory moderator analysis and conducted
an exploratory ANCOVA. Finally, we conducted additional Bayesian analyses to evaluate
the evidence in favor of the null and alternate hypotheses.
All priors used in the reported analyses are default prior scales (Morey et al. 2016).
For the Bayesian ANCOVA, the prior scale on fixed effects was set to 0.5, the prior scale
on random effects to 1, and the prior scale on the covariate to 0.354. The latter was also
used in the Bayesian Linear Regression. All Bayesian t-tests used a Cauchy prior with a
width of
√
2/2 (~0.707), meaning that half of parameter values are set to lie within the
interquartile range [
−
0.707; 0.707]. In the next sections, we report these with robustness
checks to quantify the influence of our choice of priors on the results.
3.1. Manipulation Check
To provide evidence of learning, we examined whether participants performed better
on a given cube size if they received it later in their training sequence compared to partici-
pants who received the same sized cube earlier in their training. We found no significant
differences in order of exposure for the 2
×
2
×
2 or the 3
×
3
×
3 cubes. However, partici-
pants who encountered the 4
×
4
×
4 cube at the end of the learning phase (whether in the
progression-order or variable-order condition) outperformed participants who encountered
the 4 ×4×4 cube first, p= .004, or second, p= .033.
3.1.1. Progression-Order v. Variable-Order
To test the hypothesis that the progression-order group would outperform the variable-
order group on the 5
×
5
×
5 Rubik’s Cube test, an independent samples t-test was
conducted using JASP (JASP Team 2018). This method was used instead of a one-way
between-groups ANOVA because this hypothesis focused on a difference between these two
conditions. There was no significant difference between progression-order test performance
(M= 0.318, SD = 0.041) and variable-order test performance (M= 0.313, SD = 0.044),
p= .558;
d= 0.11 (see Figure 4).
J. Intell. 2022, 10, x FOR PEER REVIEW 7 of 14
Figure 4. The 5 × 5 × 5 Rubik’s Cube mean completion in the progression-order (n = 44), consistent-
order (n = 45), and variable-order (n = 44) groups. Error bars represent standard errors of the mean.
To test the competing hypothesis that there was no difference in test performance
between the progression-order and the variable-order conditions, an equivalence test was
conducted. Equivalence tests are conducted to test whether an observed effect is statisti-
cally smaller than the smallest effect size one would be concerned with, assuming there
exists effects in a given population (Lakens et al. 2018). An Independent Groups Student’s
Equivalence test was run in R-studio (Lakens 2017; RStudio Team 2016); forty-four partic-
ipants in each condition, 0.80 power, and alpha set at .05. A mean difference of 0.006 sup-
ported statistical equivalence between the effects, suggesting that the conditions made a
trivial difference in overall performance (see Figure 5).
Figure 5. Results of the Independent Groups Student’s Equivalence test. The horizontal line indi-
cates the confidence interval of the two one-sided t-test procedure, dotted vertical lines indicate
equivalence bounds.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Progression-order Variable-order Consistent-order
Mean 5x5x5 Test Proportion
Completed
Figure 4.
The 5
×
5
×
5 Rubik’s Cube mean completion in the progression-order (n= 44), consistent-order
(n= 45), and variable-order (n= 44) groups. Error bars represent standard errors of the mean.
J. Intell. 2022,10, 85 7 of 13
To test the competing hypothesis that there was no difference in test performance
between the progression-order and the variable-order conditions, an equivalence test
was conducted. Equivalence tests are conducted to test whether an observed effect is
statistically smaller than the smallest effect size one would be concerned with, assuming
there exists effects in a given population (Lakens et al. 2018). An Independent Groups
Student’s Equivalence test was run in R-studio (Lakens 2017;RStudio Team 2016); forty-
four participants in each condition, 0.80 power, and alpha set at .05. A mean difference of
0.006 supported statistical equivalence between the effects, suggesting that the conditions
made a trivial difference in overall performance (see Figure 5).
J. Intell. 2022, 10, x FOR PEER REVIEW 7 of 14
Figure 4. The 5 × 5 × 5 Rubik’s Cube mean completion in the progression-order (n = 44), consistent-
order (n = 45), and variable-order (n = 44) groups. Error bars represent standard errors of the mean.
To test the competing hypothesis that there was no difference in test performance
between the progression-order and the variable-order conditions, an equivalence test was
conducted. Equivalence tests are conducted to test whether an observed effect is statisti-
cally smaller than the smallest effect size one would be concerned with, assuming there
exists effects in a given population (Lakens et al. 2018). An Independent Groups Student’s
Equivalence test was run in R-studio (Lakens 2017; RStudio Team 2016); forty-four partic-
ipants in each condition, 0.80 power, and alpha set at .05. A mean difference of 0.006 sup-
ported statistical equivalence between the effects, suggesting that the conditions made a
trivial difference in overall performance (see Figure 5).
Figure 5. Results of the Independent Groups Student’s Equivalence test. The horizontal line indi-
cates the confidence interval of the two one-sided t-test procedure, dotted vertical lines indicate
equivalence bounds.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
Progression-order Variable-order Consistent-order
Mean 5x5x5 Test Proportion
Completed
Figure 5.
Results of the Independent Groups Student’s Equivalence test. The horizontal line indicates
the confidence interval of the two one-sided t-test procedure, dotted vertical lines indicate equivalence
bounds.
We also conducted a Bayesian independent samples t-test using JASP (JASP Team
2018). A Bayes factor of 3.77 with an error percentage of .033 was found in favor of the null
hypothesis (see Figure 6).
J. Intell. 2022, 10, x FOR PEER REVIEW 8 of 14
We also conducted a Bayesian independent samples t-test using JASP (JASP Team
2018). A Bayes factor of 3.77 with an error percentage of .033 was found in favor of the
null hypothesis (see Figure 6).
Figure 6. Results of Bayesian t-test for the progression-order and variable-order conditions. The top-
left pane shows prior and posterior densities. The top-right pane shows the descriptive plot. The
bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the se-
quential analysis.
3.1.2. Progression-Order v. Consistent-Order
To test whether the progression-order group outperformed the consistent-order
group on the 5 × 5 × 5 Rubik’s Cube test, an independent samples t-test was conducted
(JASP Team 2018). Again, these methods were used instead of a one-way between-groups
ANOVA because this hypothesis focused on a difference between two conditions. There
was no significant difference between progression-order test performance (M = 0.318, SD
= 0.041) and consistent-order test performance (M = 0.323, SD = 0.042), p = .530, d = −0.13,
(see Figure 5). A Bayes factor of 3.87 with an error percentage of .033 was found in favor
of the null hypothesis (see Figure 7).
Figure 6.
Results of Bayesian t-test for the progression-order and variable-order conditions. The
top-left pane shows prior and posterior densities. The top-right pane shows the descriptive plot.
The bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the
sequential analysis.
J. Intell. 2022,10, 85 8 of 13
3.1.2. Progression-Order v. Consistent-Order
To test whether the progression-order group outperformed the consistent-order group
on the 5
×
5
×
5 Rubik’s Cube test, an independent samples t-test was conducted (JASP
Team 2018). Again, these methods were used instead of a one-way between-groups ANOVA
because this hypothesis focused on a difference between two conditions. There was no
significant difference between progression-order test performance (M= 0.318, SD = 0.041)
and consistent-order test performance (M= 0.323, SD = 0.042), p= .530, d=
−
0.13, (see
Figure 5). A Bayes factor of 3.87 with an error percentage of .033 was found in favor of the
null hypothesis (see Figure 7).
J. Intell. 2022, 10, x FOR PEER REVIEW 8 of 14
We also conducted a Bayesian independent samples t-test using JASP (JASP Team
2018). A Bayes factor of 3.77 with an error percentage of .033 was found in favor of the
null hypothesis (see Figure 6).
Figure 6. Results of Bayesian t-test for the progression-order and variable-order conditions. The top-
left pane shows prior and posterior densities. The top-right pane shows the descriptive plot. The
bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the se-
quential analysis.
3.1.2. Progression-Order v. Consistent-Order
To test whether the progression-order group outperformed the consistent-order
group on the 5 × 5 × 5 Rubik’s Cube test, an independent samples t-test was conducted
(JASP Team 2018). Again, these methods were used instead of a one-way between-groups
ANOVA because this hypothesis focused on a difference between two conditions. There
was no significant difference between progression-order test performance (M = 0.318, SD
= 0.041) and consistent-order test performance (M = 0.323, SD = 0.042), p = .530, d = −0.13,
(see Figure 5). A Bayes factor of 3.87 with an error percentage of .033 was found in favor
of the null hypothesis (see Figure 7).
J. Intell. 2022, 10, x FOR PEER REVIEW 9 of 14
Figure 7. Results of Bayesian t-test for the progression-order and consistent-order groups. The top-
left pane shows prior and posteriors densities. The top-right pane shows the descriptive plot. The
bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the se-
quential analysis.
3.1.3. Fluid Reasoning as a Predictor of Test Performance
We tested the strength of the correlation between participants’ fluid reasoning as
measured by the Raven’s Advance Progressive Matrices and their 5 × 5 × 5 Rubik’s Cube
performance across the full sample. There was a weak, positive correlation, r = .24, p =.005,
(see Figure 8) and a Bayes factor of 5.27 in favor of the alternative hypothesis—that fluid
reasoning predicts Rubik’s Cube performance.
Figure 8. The 5 × 5 × 5 Rubik’s Cube Completion test and RAPM accuracy percentage across the full
sample.
3.2. Exploratory Analyses
In addition to the pre-registered analyses and the Bayesian analyses, we conducted
two exploratory analyses. It could have been the case that variability in fluid reasoning
was suppressing the effect of training order condition. To test whether 5 × 5 × 5 Rubik’s
Cube completion depended on condition after controlling for fluid reasoning, we con-
ducted with an ANCOVA where fluid reasoning served as the covariate. Condition did
not emerge as a significant predictor after controlling for fluid reasoning F(2,129) = 0.62, p
= .540. A Bayesian ANCOVA verified this result, producing a Bayes factor of 7.409 in favor
of the null hypothesis.
We also tested whether fluid reasoning moderated the effectiveness of condition.
Condition was dummy-coded with the consistent condition serving as the reference
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
RAPM Percentage Correct
Test 5×5×5 Completion
Figure 7.
Results of Bayesian t-test for the progression-order and consistent-order groups. The
top-left pane shows prior and posteriors densities. The top-right pane shows the descriptive plot.
The bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the
sequential analysis.
3.1.3. Fluid Reasoning as a Predictor of Test Performance
We tested the strength of the correlation between participants’ fluid reasoning as
measured by the Raven’s Advance Progressive Matrices and their 5
×
5
×
5 Rubik’s Cube
performance across the full sample. There was a weak, positive correlation, r= .24, p=.005,
(see Figure 8) and a Bayes factor of 5.27 in favor of the alternative hypothesis—that fluid
reasoning predicts Rubik’s Cube performance.
J. Intell. 2022, 10, x FOR PEER REVIEW 9 of 14
Figure 7. Results of Bayesian t-test for the progression-order and consistent-order groups. The top-
left pane shows prior and posteriors densities. The top-right pane shows the descriptive plot. The
bottom-left pane shows the Bayes Factor Robustness check. The bottom-right pane shows the se-
quential analysis.
3.1.3. Fluid Reasoning as a Predictor of Test Performance
We tested the strength of the correlation between participants’ fluid reasoning as
measured by the Raven’s Advance Progressive Matrices and their 5 × 5 × 5 Rubik’s Cube
performance across the full sample. There was a weak, positive correlation, r = .24, p =.005,
(see Figure 8) and a Bayes factor of 5.27 in favor of the alternative hypothesis—that fluid
reasoning predicts Rubik’s Cube performance.
Figure 8. The 5 × 5 × 5 Rubik’s Cube Completion test and RAPM accuracy percentage across the full
sample.
3.2. Exploratory Analyses
In addition to the pre-registered analyses and the Bayesian analyses, we conducted
two exploratory analyses. It could have been the case that variability in fluid reasoning
was suppressing the effect of training order condition. To test whether 5 × 5 × 5 Rubik’s
Cube completion depended on condition after controlling for fluid reasoning, we con-
ducted with an ANCOVA where fluid reasoning served as the covariate. Condition did
not emerge as a significant predictor after controlling for fluid reasoning F(2,129) = 0.62, p
= .540. A Bayesian ANCOVA verified this result, producing a Bayes factor of 7.409 in favor
of the null hypothesis.
We also tested whether fluid reasoning moderated the effectiveness of condition.
Condition was dummy-coded with the consistent condition serving as the reference
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
RAPM Percentage Correct
Test 5×5×5 Completion
Figure 8.
The 5
×
5
×
5 Rubik’s Cube Completion test and RAPM accuracy percentage across the full sample.
J. Intell. 2022,10, 85 9 of 13
3.2. Exploratory Analyses
In addition to the pre-registered analyses and the Bayesian analyses, we conducted
two exploratory analyses. It could have been the case that variability in fluid reasoning was
suppressing the effect of training order condition. To test whether 5
×
5
×
5 Rubik’s Cube
completion depended on condition after controlling for fluid reasoning, we conducted with
an ANCOVA where fluid reasoning served as the covariate. Condition did not emerge
as a significant predictor after controlling for fluid reasoning F(2,129) = 0.62, p= .540. A
Bayesian ANCOVA verified this result, producing a Bayes factor of 7.409 in favor of the
null hypothesis.
We also tested whether fluid reasoning moderated the effectiveness of condition. Con-
dition was dummy-coded with the consistent condition serving as the reference group,
RAPM scores were centered, and dummy code
×
centered RAPM were created. Centered
RAPM scores and dummy-coded conditions were entered into the model first and inter-
actions were entered in the second step. Ravens and condition together explained 7% of
the variance, R
2
= .067, p= .029. R
2
did not significantly change with the addition of the
interactions (R2change = .004, p= .776), indicating no significant moderation. A Bayesian
Linear Regression verified this result, producing a Bayes factor of 1.597, R
2
= .067, in favor
of the null hypothesis.
4. Discussion
We sought to test the effects of the order of difficulty during training in a test of
spatial problem solving and to test the influence of fluid intelligence on spatial problem
solving. We first examined whether progression-order training yielded superior results to a
variable-order condition on the final Rubik’s Cube performance, which would be in line
with the methods of many progression-order training paradigms (e.g., van Merriënboer and
Kirschner 2017) or whether there was no difference between progression-order training and
variable-order training. Our results demonstrated no difference between the progression-
and variable-order conditions. However, we also did not find that participants in these
conditions outperformed participants in the consistent-difficulty condition, limiting the
support our results offer for suggesting any variability is important (von Bastian and
Eschen 2016) or for the schema theory of discrete motor skill learning, which also suggests
participants should yield better transfer from varied versus consistent training.
Our results support our hypothesis that participants’ fluid reasoning would be pos-
itively correlated with performance on the spatial reasoning test, across conditions. The
positive correlation found between fluid reasoning scores and 5
×
5
×
5 Rubik’s Cube com-
pletion suggests that cognitive abilities are important considerations for predicting overall
performance on spatial problem-solving tasks, regardless of the training progression.
Indeed, our results are in line with other studies that have found that spatial reasoning
predicts performance but does not interact with the type of training. For example, Keehner
et al. (2006,2008), found that spatial reasoning ability predicted performance on a spatial
rotation task regardless of the type of familiarization participants had with the stimuli (e.g.,
only seeing rotations of models versus interacting with the models, experienced versus
novice laparoscopic surgeons).
Considering the results of Vakil and Heled (2016), in which varied training on the
Tower of Hanoi led to better schematic representation of the problem, we believe the lack
of a superior method of learning may arise from the difficulty of the Rubik’s Cube. The
Rubik’s Cube is substantially more complex than the Tower of Hanoi, and takes a large
span of time to become familiar with; its complexities even evaded its inventor, Erno Rubik,
who took nearly a month to solve it (Great Big Story 2017).
In contrast to the Rubik’s Cube, the Tower of Hanoi is more easily solved because
progress is easier to recognize (i.e., number of ascending size disks on the furthest right
spoke). The Tower of Hanoi can be solved in 2n
−
1 moves, where nequals the number of
disks (Petkovi´c 2009). People can usually solve the Tower of Hanoi the first time they see it
within a few minutes, and many can optimally solve it. In contrast, a person is unlikely to
J. Intell. 2022,10, 85 10 of 13
solve a scrambled Rubik’s Cube the first time they try, and optimally solving the Rubik’s
cube requires the Hamiltonian path problem, a mathematical problem with a million-dollar
prize for anyone able to solve it (Demaine et al. 2018;Ramachandran n.d.). While our
lack of clear differences by condition might be due to needing longer training sessions on
the Rubik’s Cube due to its difficulty, it may also be the case that varied training may be
paradigm specific rather than widespread across spatial reasoning problems.
Despite limited findings, this paradigm has the potential to offer insight into the role
of cognitive abilities on complex spatial problem-solving following training. Many studies
investigating cognitive predictors of spatial task performance either use relatively simple
laboratory tasks (e.g., the Tower of Hanoi) or use more complex real-world tasks where
domain-specific knowledge and other confounds may be difficult to control (Keehner
et al. 2006;Keehner et al. 2004;Minsky et al. 1985;Nayigizente and Richard 2000;Schiff
and Vakil 2015;Tarasuik et al. 2017;Vakil and Heled 2016). These nuisance factors might
be why findings in the literature are mixed. For example, Keehner et al. (2006) trained
participants on using an angled laparoscope and found that spatial reasoning ability and
general reasoning ability predicted performance initially, but only spatial reasoning ability
continued to predict performance following twelve sessions of practice. However, another
study by Keehner et al. (2004) found that spatial reasoning ability predicted performance
following introductory laparoscopic training, but not following advanced laparoscopic
training. In this study, participants differed in their surgical knowledge, with those taking
the introductory training having limited surgical experience and those taking the advanced
training having considerable surgical experience. Indeed, interactions between general
cognitive ability and domain-specific knowledge have been found in other spatial tasks
such as geological bedrock mapping (Hambrick et al. 2012), where visuospatial ability
predicted performance among individuals with low geology knowledge, but not high
geology knowledge.
5. Limitations and Future Directions
We designed our training paradigm to resemble those used in Tower of Hanoi training
tasks and in von Bastian and Eschen’s (2016) working memory training tasks. That is, par-
ticipants were given experience with variants of the task and then tested on a more difficult
version of the task to examine near transfer. The trade-off with being consistent with these
training paradigms is that we could not include a measure of baseline performance without
interfering with the order conditions. Likewise, we did not provide strategy training on
the Rubik’s Cube, which might have produced significant effects, but would not have been
comparable to either Tower of Hanoi and in von Bastian and Eschen’s (2016) working
memory training tasks. Strategy training may be a fruitful direction for future research.
The brief, i.e., four-minute time limits per puzzle, were chosen based on pilot per-
formance and feedback from pilot participants (i.e., frustration levels). However, longer
training sessions may be needed to observe improvements. Analogous studies of the Tower
of Hanoi typically have no time restrictions for each training trial (Schiff and Vakil 2015;
Vakil and Heled 2016). When time restrictions were imposed they were typically short, e.g.,
five-minute limits for puzzles that required only five moves to solve (Tarasuik et al. 2017).
To study problem-solving progression with the Rubik’s Cube while avoiding participant
frustration might therefore require a long-term study conducted over multiple sessions to
provide insight into learning progressions using this paradigm. This approach may yield
evidence for the schema theory of discrete motor skill learning as additional time may
allow for participants to explore various possible solutions.
Longer training studies using the Rubik’s Cube paradigm could elucidate the role of
the acquisition of domain-specific knowledge, which is difficult to control when observing
real-world trainees. Likewise, domain-specific knowledge could be manipulated in a longer
training paradigm. Domain-specific knowledge manipulations would allow researchers
to examine interactions with cognitive abilities and types of training progressions, as well
as how fluid reasoning might predict domain-specific knowledge acquisition rates, which
J. Intell. 2022,10, 85 11 of 13
in turn predict final performance. Further, longer training studies using this paradigm
could shed light on the indirect role of cognitive abilities on spatial problem solving via
influence on the rate of domain-specific knowledge acquisition. Understanding the role of
cognitive abilities and various training conditions on a spatial problem-solving task could
lead to better-informed training paradigms in domains such as sports, medicine, science,
or engineering.
Author Contributions:
Conceptualization J.S.D., B.N.M. and D.M.; methodology, software, valida-
tion, formal analysis, investigation, resources, J.S.D. and B.N.M., data curation, writing—original
draft preparation, writing—review and editing, J.S.D. and D.M. and B.N.M.; visualization, J.S.D.;
supervision, B.N.M.; project administration, funding acquisition, J.S.D. All authors have read and
agreed to the published version of the manuscript.
Funding:
This research was funded by Case Western Reserve University’s Support of Undergraduate
Research and Creative Endeavors.
Institutional Review Board Statement:
The study was conducted in accordance with the Declaration
of Helsinki, and approved by the Institutional Review Board (or Ethics Committee) of Case Western
Reserve University (protocol code: 2169 and date of approval: 21 December 2017) for studies involving
humans.
Informed Consent Statement:
Informed consent was obtained from all subjects involved in the
study.
Data Availability Statement:
Protocol and analyses were pre-registered, and materials and data can
be found on the Open Science Framework, osf.io/uk6w2.
Conflicts of Interest: The authors declare no conflict of interest.
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