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RODRIGUES ET AL, Nonlinear Dynamics in Science Education 118
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
Unravelling Student Learning: Exploring Nonlinear Dynamics in Science
Education
Humberto Rodrigues1, Antonio Jesús Molina-Fernandez2, Richard Lamb3, Ikeseon Choi4, &
Tosha Owens1
1.
East Carolina University, Neurocognition Science Laboratory; USA
2.
Department of Social, Work and Differential Psychology; Faculty of Psychology; Universidad Complutense de Madrid, Spain
3.
University of Georgia, USA
4.
Emery University, USA
Corresponding Author: Humberto Rodrigues, MSc
E-Mail Address: humberto.rodrigues@ua.pt
https://doi.org/10.56769/ijpn09311
Abstract
Background: Traditional education research often relies on static linear approaches to measure dynamic
systems involved in student information processing, overlooking the complexity of learning. Emerging
research in related fields acknowledges the highly dynamic and nonlinear nature of cognitive states and
information processing. Current educational research methods, predominantly based on quantitative and
qualitative "snapshot" examinations, inadequately capture the dynamic and nonlinear aspects of cognitive
processing during learning. Objective: This study aims to explore nonlinear dynamics as a means to
describe and understand student learning processes. Methods: This study analyzed actions of 158,000 high
school students in science-based immersive video games, specifically focusing on task completion within
a virtual setting. Students aged 14-18, enrolled in Earth Science, Biology, Chemistry, and Physics programs,
participated. Tasks, resembling Piagetian tasks, centered on volume conservation within a chemistry
classroom context, employing the Student Task and Cognition Model (STAC-M) to emphasize
computational cognition modeling. Results: The study tracked alterations in cognitive activations during
information processing using derivatives, modeled through parameters from the authors' cognitive dataset.
Employing the STAC-M model, achaotic attractors depicted convergence and sensitivity to initial
conditions, reflecting cognitive associations and stability. Random data lacked the observed dynamic
properties found in cognitive data, while bifurcation plots illustrated transitions from stability to chaos in
cognitive processing pathways, highlighting the system's intricate nature. Conclusions: Modern science
education explores beyond conventional assessments, acknowledging teaching methods' impact on
students' cognitive processing. Achaotic attractors depict shifts from stable to unstable mental activities,
highlighting the potential for diverse teaching approaches to minimize misconceptions and enable quicker
transitions to responsive, stable learning states, aligning with educational objectives.
Keyword: Cognition, Nonlinear Dynamics, Student Learning, Science Education.
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 119
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
Introduction
Science education practices lack
well-tailored teaching guidelines due to the
complexity of understanding science learning
and communicating that understanding to
teachers when using current assessments in
science education (Lamb et al., 2014;
Steenbeek et al., 2020). This complexity
arises from a variety of factors, including
political influences, business practices that
influence educational advice, legal and social
concerns, parental involvement issues, and
testing debates (Elliott et al, 2021; Hunkins et
al, 2022; Mark, 2023).
One approach that has been proposed
to overcome these challenges is to use sound
psychological theory based on biological and
neurological principles to develop effective
teaching guidelines. This approach does not
advocate a return to purely brain-based
learning but emphasizes understanding of
mechanisms of action related to
neuropsychological and physiological
aspects of science education. Indeed, while
behavioral studies are usually limited to
stimulus-response experiments,
neuroimaging tools offer the possibility of
visually representing and measuring the
neural mechanisms involved in various
processes between stimulus and response
(Elliott et al, 2021; Lamb et al., 2021; Mark,
2023). Combining evidence from behavioral
and neuroscience research increases the
strength of research compared to studies that
rely on only one source of evidence (Gabrieli,
2016). Where rapid progress is being made
and the ability to consider learning at
multiple levels and bring about an
interdisciplinary approach, it should consider
static externalizing behavior research and
small studies based on linear measures of
learning. For example, the integration of
neuroimaging into educational neuroscience
has revealed the neurological basis of
developmental fluctuations that impact
education. Brain differences have been
identified that predict differences in learning
performance between students exposed to
different curricula. Brain-based metrics often
outperform traditional metrics in predicting
effective educational strategies for
individuals. These results highlight the
potential of personalized education and
emphasize the need for educational
neuroscience to advance research and meet
the diverse needs of students and educators to
improve teaching methods (Gabrieli, 2016).
Over the past decade, researchers
outside of education have increasingly
focused on studying brain dynamics and
cognition in educational settings. However,
educational researchers have been reluctant
to implement tools to examine student
learning from this perspective (Elliott et al,
2021; Hunkins et al, 2022; Mark, 2023). This
lack of interdisciplinary inquiry isolates
education and hinders its progress and
collaboration with advances in science
teacher education and related fields. This gap
arises in part from traditional narratives in
science education research and policy that
treat the brain as a deterministic input-output
system (Hunkins et al, 2022). This is in stark
contrast to the contemporary view of
neuropsychology and neuroscience
researchers, who recognize the brain and
cognition as systems with varying outcomes
and provide a clear perspective on learning
mechanisms (Elliott et al, 2021; Hunkins et
al, 2022; Mark, 2023). In other words,
educational research typically relies on linear
models of cognitive processing (Lamb &
Firestone, 2022). Educational researchers
seek to standardize and control input by
expecting consistent results in student
performance and test scores (Corter et al.,
2011; Lee, 2013; Mark, 2023).
Educational theory and research often
cannot easily explain variation in student
outcomes other than by suggesting that this
variation is a product of individual
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 120
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
differences resulting from sociocultural
differences (Han et al., 2013). Although
labeling such variation as individual
differences acknowledges the underlying
variation in student outcomes, this label
explains little about how these individual
differences arise, especially as a function of
information processing or cognition. That is,
extensive explanations such as context and
culture do not complicate models of
cognitive processing, but rather increase the
complexity of the input while maintaining a
linear model of cognitive processing (Lamb
et al., 2015; Mark, 2023; Steenbeek et al.,
2020). Although the authors acknowledge
that this position was adopted due to the
limitations of measurement tools, linear
models of cognitive processing do not
support current neuropsychological and
neuroscience of the brain and cognition as
variable output systems (Janssen et al., 2021;
Lamb et al., 2018).
Therefore, unstable student
performance is a function of a nonlinear
processing system with emergent properties
(Chialvo, 2010). The application of nonlinear
dynamics to cognitive processing and
computational modeling of this cognitive
processing is relatively new with the
development of artificial neural networks in
psychology and is not routinely used in
educational research. If the education field
wants to achieve a more comprehensive
understanding of student learning and
continue to bridge the gap between
education, psychology, and neuroscience, a
closer look at nonlinear dynamics and its role
in cognition is necessary (Lamb et al., 2016).
In this context, it is important to consider that
further development of portable
neuroimaging tools offers promising
opportunities for more ecologically valid
research in education (Davidesco et al.,
2021). Wearable EEG has been used to
examine student engagement during real-
time instruction, as in the study of Landi et al.
it was shown (Landi et al., 2019).
Furthermore, functional near-infrared
spectroscopy (fNIRS) has been used to
observe neural patterns associated with
reading development in children at high risk
of illiteracy in rural Ivory Coast (Jasinska &
Guei, 2018).
In parallel, there has been a
significant increase in the use of
neuroscientific methods to study how
students learn. During this relatively short
period of time, important insights into the
neural mechanisms associated with
education-related skills have been gained.
Over the course of its development,
educational neuroscience has evolved into an
interdisciplinary field that integrates theories
and methods from education, psychology,
and cognitive neuroscience (Thomas &
Ansari, 2020).
Educational research often attributes
variations in student learning to individual
differences influenced by sociocultural
factors (Han et al., 2013). Despite this, it fails
to fully account for the complex, nonlinear
nature of how sociocultural inputs are
processed. Current approaches struggle to
explain how these differences manifest
within student outcomes based on
information processing or cognition.
Nonlinear dynamics play a crucial role in
cognition, yet their application in educational
research remains limited. Understanding
these dynamics is essential for a
comprehensive grasp of student learning in
science education (Lamb & Firestone, 2022).
Current research methods assume a static
linear approach to dynamic learning systems,
hindering a complete understanding. To
address this, educational researchers should
continually reassess parameters and errors as
the learning system evolves (Wolf et al.,
2013).
Complexity arises from the number
and type of simultaneously interacting
variables (Steenbeek et al., 2020). The use of
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 121
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
nonlinear dynamics within neuroscience and
neuropsychology has provided means to
examine characteristics of the brain not
otherwise considered in clinical treatments
and educational processes (Rando et al.,
2011). Nonlinear dynamics and related
computational modeling provide tools to
examine cognitive processing at levels of
detail not obtained previously. Nonlinear
dynamics more popularly known as Chaos
Theory and the study of chaotic systems
suggests that these systems are highly
sensitive to initial starting conditions
resulting in the inability to predict long-term
outcomes despite deterministic properties
(Health & Hill, 2010). Lorenz (1969a,b,
1980) identified this occurrence and Merlees
(1972) names this occurrence the Butterfly
Effect while Lorenz prepared a talk for the
139th meeting of the Association for the
Advancement of Science. Researchers across
many disciplines, i.e., biology, physics,
psychology, and sociology; observe chaotic
systems in many natural systems such as
weather, population dynamics, human
interactions, and cognitive processing
(Melancon et al., 2000). These processes are
a part of the inherent complexity of all
nonlinear systems. Acceptance that learning
is a nonlinear dynamic process as evidence
by neuronal and structural activations in the
brain allows for examination of these
processes in a different context. A tacit
understanding while examining the brain and
related process is the presence of fractal
dimensions associated with the structures of
the brain and the presence of achaotic
(strange) attractors (Globus, 1992; Seoane &
Sanjuan, 2013). Fractal dimensions are an
inherent property of chaotic systems. It is the
presence of these fractal dimensions in
chaotic systems such as the brain that develop
the phase space for the achaotic attractors to
develop. Examination of fMRI and EEG data
suggests that there are achaotic attractors
associated with cognitive task present in
sufficiently large sets of cognitive data (Deco
et al., 2013; Hadriche et al., 2013). The
presence of achaotic attractors establishes the
underlying nature of neuronal interactions
occurring in the fractional dimensions of the
brain (Ivancevic & Ivancevic, 2010). Thus,
cognitive information processing is a self-
organizing system as a student engages in the
science learning process (Sha et al., 2012). It
is important to note this does not suggest that
outcomes of learning will spontaneously
materialize as engrams as sufficient and
necessary inputs are required to reduce
entropy and develop stable patterns (Savin et
al., 2010). Mathematicians characterize
fractal dimensions as self-similar systems
with non-integer dimensions in Euclidean
geometric space (phase space), (Calcagni,
2012; Doyne, 1982). For ideal fractals, this
self-similarity is scale independent of range
(Lesne & Lagues, 2012). Within biological
systems, fractals are not scale-independent.
Within this understanding, biological fractals
have characteristics known as essential self-
similarity meaning that each level of the
fractal is not identical but on average have the
same characteristics (Budd & Kisvarday,
2011).
Self-organizing systems and
specifically within non-linear dynamic
systems such as those found in student
science information processing, are areas of
stability and equilibrium that arise
spontaneously as an emergent property of the
system when sufficient energy is present and
reduces the entropy of the system (Kelso,
2014). Achaotic attractors popularly known
as strange attracters within the context of
non-linear system dynamics are these
spontaneous reductions in entropy and
represent a stable energy state for the system.
It is important to note the discussion is at the
system level. This property of stability in
dynamic systems sets the system into
equilibrium and provides resistance to
change despite the chaotic nature of the
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 122
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
system. These achaotic attractors are an
island of stability and represent the lowest
energy state within the dynamic system
suggesting this stability is inherently a
characteristic of the system. Within
neuroscience and neuropsychology, these
stabilized dynamic patterns are indicative of
cognitive patterns used in superimposition
when processing information (Fingelkurts et
al., 2011).
Descriptions of non-linear dynamic
systems such as student learning consist of
two components, the psychological state and
the dynamics (Nguyen & Zeng, 2012). As
educators, we often measure the state through
behavioral analysis of student learning via
tests, observations, and interviews. What
educators tend not to examine is the dynamic
components of the systems associated with
learning specifically within the brain. An
attractor is a set of properties that a system
tends to evolve toward despite initial starting
conditions, attractors in typical physical
systems dissipate over time resulting in loss
of the dynamic component of the system as
they evolve (Vallejo & Sanjuan, 2013).
Achaotic attractors differ in two important
ways from attractors. First, the properties of
the system in which achaotic attractors arise
are fractal in nature and second, due to the
self-organizational nature of the achaotic
attractor resulting from its fractal nature the
lowest energy point for the system is not that
of absolute dissipation but that of a dynamic
system orbiting given phase space (Negi et
al., 2014). Within the nonlinear dynamic
system of cognition, the achaotic attractor
formally represents the most efficient pattern
and contains the global patterns of thought
that process information (Mizraji & Lin,
2011; Kelso, 2014). The psychological state
of this system is represented by a vector
characterizing the movement of units of
quantized information from high entropy
areas to low entropy areas facilitated by
superimposition on an array of achaotic
attractors (Sulis & Trofimova, 2001). When
the unit of information is placed on the
appropriate achaotic attractor, it reaches the
lowest energy state in the system and is
retained as a part of that achaotic attractor
helping to establish a component of the
engram enriching the schema. In other words,
learning occurs.
Cognitive processing of science
information is characterized via modes of
thinking associated with processing of
knowledge and skills and develops as
resonance in the brain (Fox & Christoff,
2014). Several research studies establish
considerable evidence that general resonance
within the brain occurs as it functions as an
information processor (Mori & Kai, 2002;
Spreng et al., 2013). Resonance is the
tendency of a system to move and oscillate
between values during attempts to equate the
values. In this context, cognitive processing
resonance patterns show up as background
“noise” on electroencephalography readings
(Dominguez et al., 2013). Resonance is an
inherent property of nonlinear dynamic
systems and is ubiquitous. It follows that
resonance patterns found, as a function of
information processing, would be present in
data sets with sufficiently large numbers of
data points to capture this occurrence.
Sufficient computational models using such
data would also contain evidence of this
resonance and the model would be usable to
identify this property of the system (Lamb et
al., 2016). Derivatives of the computational
model equations built with cognitive data
related to information processing would
provide a view of the patterns of neuronal
activation (thinking) as an achaotic attractor.
Computational psychologists understand
achaotic attractors within the quantitative
psychology literature to be graphical
representation of schema (Hirsh et al., 2012;
Enewyk, 1991). Stimulus inputs triggering
cognition related to science information
processing requires students to provide fit to
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 123
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
existing schema or create new schema in
order to process information. This “fitting”
occurs via superimposition of the achaotic
attractor at a rate of approximately 89
superimpositions per second (Toft-Nielsen et
al., 2014). Within the context of nonlinear
dynamic systems, the patterns of neuronal
excitation over time would form resonance
patterns manifested as achaotic attractors.
Thus, these achaotic attractors are the global
pattern for the schema. The achaotic
attractors act as the framework for stimulus
data to be processed against. The existence of
nonlinear dynamics is well evidenced in
cortical neurodynamics literature (Noack,
2012). Learning new concepts or
incorporating new knowledge creates new
achaotic attractors that are variable with
sufficient energy inputs. The numbers of new
achaotic attractors are exemplified as
bifurcations (new pathways) within an
environmentally supplied input (Steenbeek et
al., 2020).
In summary the brain, seem to operate
as a nonlinear dynamic system with fractal
properties with a limited range of scale. The
fractal components of the brain provide for
development of achaotic attractors. Each
achaotic attractor acts as an engram for
particular stimuli allowing for
superimposition within the memory. Through
achaotic attractor properties such as entropy
reducing when orbiting specific phase space
input stimulus data can become
superimposed and incorporated into the
achaotic attractor and learning can develop.
Purpose, Research Question, and
Hypothesis
The purpose of this study is to
establish explanatory mechanisms for
understanding the variable outputs seen
within student cognitive processing of
science-based tasks. The secondary purpose
of this paper is to examine underlying
nonlinear dynamic structures of student
science based cognitive data through
examination of achaotic attractors and
bifurcation. The third purpose of this paper is
to take a step toward that goal by
characterizing the information processing
aspects of cognition and brain function in
these ways - through attractor and chaotic
properties. The research questions addressed
in this study are:
1. Does student science-based
cognitive processing data
examined using computational
models exhibit chaotic properties
in the form bifurcation and
achaotic attractors?
2. Does the development of
bifurcation and achaotic attractors
arise as an artificial construct of
the computational model, or are
the chaotic properties
independent of the computational
model?
Consideration of the research questions
results in the following hypotheses:
Hypothesis 1 (H1): Using suitable
data and theoretical framing data
derived from cognitive tasks will
exhibit chaotic properties with
bifurcation when compared to
random data inputs into the
computational model.
Hypothesis 2 (H2): Development of
the attractor and bifurcation will not
derive from inherent properties of the
computational model.
The hypothesis that chaotic properties
and bifurcation will occur will result in the
support for the non-linear dynamic
framework of cognitive processing within the
human brain and aid educational researchers
in the development of methods and
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 124
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
frameworks to address this property. This
understanding can provide further evidence
that structures in the brain naturally develop
differential outputs despite uniform inputs
because of sensitivity to initial starting
conditions. This sensitivity to initial starting
conditions is popularly known as the
Butterfly Effect. This effect is most
commonly discussed in terms of weather in
which slight variations in the initial starting
parameters of weather models yields
radically different forecasts. This sensitivity
is an inherent property of non-linear dynamic
systems. This characteristic of cognitive
processing provides explanatory mechanisms
for understanding student learning as an
individualistic property tied to physiological
structures and system properties within the
brain. It also sets the conditions for science
education researchers to examine science
information processing in light of individual
differences with possible mechanisms of
action and include aspects physiological
considerations.
Methods
Samples
The unit of analysis for this study is
student actions toward task completion while
playing science based immersive video
games (n=158,000); Serious Educational
Games (SEG) as conceptualized by Annetta
(2008), It is not the students themselves that
are of interest, only the data garnered while
the student completed tasks within a virtual
environment. The authors added information
related to the sample for contextual purposes.
Table 1 provides the demographic data for
the study. The targeted population is fulltime
high school students enrolled in science
programs for grades 9-12 taking Earth
Science, Biology, Chemistry, and Physics.
Student’s ages ranged from 14 to 18.
Participant selection criteria is (1) taking
their current science class for the first time;
(2) taking the course as a member of a class
and not in an online or virtual capacity; (3)
admitted into the class within the first two
weeks of class. The data set for this study
consisted of derived data from preexisting
data using the target population descriptors to
screen data points.
Table 1.
Demographic composition of the study sample (N=606)
Note. Numbers in parentheses represent national statistics from the most recent Census figures.
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 125
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
Task Presentation
The task was presented to the students
via a SEG Environment. The game
environment used a first-person view of the
task with students selecting responses and
actions in an open-ended play environment.
The tasks were similar to typical Piagetian
tasks used in educational psychology
specifically this study focused on the volume
conservation task. In this particular
computational model, the volume
conservation task was modeled in the context
of a chemistry classroom using an artificial
neural network. The computational model is
known as the Student Task and Cognition
Model (STAC-M) developed by Lamb in
2013. The underlying concepts examined in
the context of science class were
conservation of volume. With a
computational model, the original sample
becomes less important as the model and its
underlying processing nodes (representing
cognition) now act as the subject of the study
(Lamb et al., 2019; Lamb, 2019).
Analysis and Modeling
Initial parameterization of the subject
response to the task sets of presented in the
SEG occurred through a combination of the
two parameter logistic model – item response
model (2PLM-IRT) parameterization (a and
b), IRT-True Score Analysis (Lamb et al.,
2012; Lamb et al., 2014) and Artificial
Neural Network modeling, under a model of
cognitive diagnostics known as the
deterministic-input, noisy “and” gate model
(DINA-N), (Status et al., 2020; Steenbeek et
al., 2020; Basokcu, 2014; Lamb, 2013; Lamb
et al., 2014). Population parameters for the
science tasks were established by the authors
using IRT-True Score analysis resulting in
the initial probabilities and assisting in the
identification of cognitive components
known as cognitive attributes used in task
processing through a Q-matrix. These
identified tasks then provided inputs into the
Student Task and Cognition Model (STAC-
M). The STAC-M is a computational model
developed by Lamb in 2013 and 2014. The
use of a computational model allows for the
development of a system of equations
processing science task information the way
student would. The basis of the artificial
neural network (ANN) information
processing or propagation is the Hebb’s
Equation Synaptic Propagation Rule (Lamb,
2013; Lamb et al., 2014)). This particular
computational model consists of seven
interconnected processing elements or
artificial neurons resulting in the production
probabilistic outcomes associated with the
task. The STAC-M was designed using SAS
JMP 11.0 and Rx64 3.0.2.
Achaotic Attractor Development
Several neural algorithms exist for the
estimation of neural processing such as those
found in the STAC-M (Lamb & Firestone,
2017). Specifically the authors use Hebbian
dynamics with modification via the Ojasian
rules for examinations of learning via
increasing the learning functions associated
with the network. Within this framework
weighting adaptation are randomly varied
using Ω+/-2 to ensure they do not approach
+/-∞. Approaching +/-∞ would create
uninterpretable results. In addition, the action
potentials seen within biological nervous
systems are limited in their production as
well. Thus, the addition of limits is not
arbitrary. The authors of the study consider
the following equations starting with
Equation 1.
(1)
where χ is inputs to the system in the form of
a matrix of data collected from the task
completions in the virtual environment, y is
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 126
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
the outputs and ω is the potentiation weight
of each neural pathway developed from the
data set as vector through the matrix data
points. Modification of the Hebbian synaptic
(artificial and biological) potential occurs to
account for change in the connection strength
over time. This modification of Hebbian
system by Oja (Apairin, 2012) generates
stability within the dynamic system over time
allowing for examination of these dynamic
properties via perturbations and the return to
stability. Equation 2 provides the relationship
as examined in this way.
(2)
Within equation 2,3 and 4,
account
for the rate of change in the synaptic weights
for the modeled system. Rewriting the
equation as a derivative as the limit of the
weighting parameter approaches +/- 2.
(3)
(4)
Using the Hebb-Oja model of signal
propagation the authors of the study consider
a network of N=7 interconnected neurons
with binary states ( . When
active , the activated neuron produces
one action potential for propagation.
Activation in the case of biological neurons
can occur though stimulation from external
or internal sources through one’s senses or
through interaction of memory (prior
knowledge). The synaptic potentiation of an
individual neuron (i) at a specific time
(t=1…∞) is illustrated in equation 5. For the
purposes of this study the limit as it
approaches +2 is used as negative
potentiation associated with the
nuerodynamics are considered inhibitory and
beyond the scope of this study.
(5)
The authors rearranged the equations
to include a constant activation threshold (θi)
due to the continuous nature of inputs
associated with biological systems. The
constant input associated with individual
neuron in the system allows for the creation
of a step function illustrating the threshold of
activation and a specific time point. Thus,
equation 5 is rewritten to include
:
Up to this point within the modeling
of educational processes specifically,
information processing threshold values, are
treated as constant reducing the complexity
of the system dynamics and creating
linearity. In practical terms, the creation of
linearity reduces analytical complexity and
results in researchers expecting uniform
outcome in student learning when measured
by instruments such as cognitive tests.
Reexamination of learning system using a
nonlinear dynamic approach provides a more
realistic understanding of how student
process and apply information and
knowledge in science classes.
Results
Derivatives of the equation 6 provide
point values for changes in activations
associated with information processing
which can be modeled as a series over time
using parameters developed from the
cognitive data set collected by the authors.
Graphical representation in the form of an
attractor is possible using activations seem
within the STAC-M computational model.
The activations are modeled using random
points from the cognitive data collected from
high school students engaging in science
information processing. The data is passed
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 127
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
through a seven-neuron artificial neural
network simulating the student cognition.
Figure 1 is the achaotic attractor generated
using Rx64 3.0.2. This formation is
characterized as an achaotic attractor using
mathematical criteria developed by (Wei &
Yang, 2012). Within this attractor, set A in
that the basin of the achaotic attractor consist
of all points convergent to orbits at A
Rn in
fractal Euclidean space (Rn) with a non-
integer dimension also understood as phase
space. The second criterion is that the
achaotic attractor be sensitive to initial
starting conditions, as is the case in this
attractor. Interpretation of this set condition A
Rn also suggests that the attractor basin be
tangible in that the achoatic attractor cannot
be a single point or discontinuous such as in
a dissipative system such as a pendulum.
More to this point, if a change occurs in the
attractor basin, then the measure of the basin
attraction would change generating a new
achaotic attractor represented a new
cognitive association and thus a new or
modified schema is generated and stabilized.
A second sustentative and necessary
condition is that the achaotic attractor set
must be contained within a dense orbit as
seen in the green orbital tracks in Figure 1.
The existence of these tracks is suggestive of
an orbit that passed infinitely close to the
points of the achaotic attractor ensuring that
this achaotic attractor is not two or more
separate achaotic attractors superimposed
upon one another confounding the separation
of cognitive processing elements. Since the
value of w oscillates between the values for
Ω, curve changes in θ, the maximum slope, is
represented as a function between any two
random points on the attractor creating
movement to the attractor orbits and stability
as a cognitive processing element.
Figure 1.
Achaotic attractor plot for student cognitive processing derived from the STAC-M
Note. This output illustrates a coherent attracter with well-defined edges.
Figure 2 is an example of the use of
random data developed using runif(250000,
min=-2, max=2) in Rx64 3.0.2. Examination
of the “attractor” in Figure 2 illustrates a lack
of these characteristic requirements found for
Figure 1. Parameterization of the random
data set occurred in the same way as the
cognitive data using the DINA-N model
(Lamb 2013; Lamb et al., 2014). The
visualization of the data reveals a random
data structure without dynamic properties as
outlined above. Within this visualization,
there is no underlying pattern associated with
the nonlinear dynamic system.
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 128
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
Figure 2.
Plot of cognitive processing using STAC-M derivatives with random data
Note. This output illustrates a chaotic system with undefined edges.
Figure 3 is a bifurcation plot of
cognitive processing. Instead of using
individual diagrams to show the behavior of
science information processing neurons with
different processing pathways, the
bifurcation plot combines the pathways into
one diagram illustrating stability and
instability. The bifurcation diagram in Figure
3 is used to assemble activations into a single
coherent picture. The diagram show changes
in one parameter energy input (stimulus
creating activation) would change the
number of possible outcomes of system and
move the system from stability (the lines) to
instability as seen at input 30 through 50.
Values of the parameter are represented from
left to right with the final parameter value
driving the system to increasing nonlinearity
and chaotic action. When the parameter is
low (left), the number of pathways is low, and
cognition is stable. As the parameter changes
the equilibrium, level associated with the
cognitive pathways reduces splitting into two
stable possibilities. As stimulation of the
neurons continues, the number of possible
pathways oscillates between differing
numbers of pathways ultimately the
bifurcations occur more frequently and the
system becomes chaotic approaching an
infinite number of possible pathways.
Through careful quantification of the
stimulus and neuronal patterns of excitation,
it becomes possible to examine the
development and alternative conceptions
related to information until the load is such
that the number of possible pathways is
chaotic, and the system is unstable and
unable to process.
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 129
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
Figure 3.
Bifurcation plot of student information processing during science task completion
Discussion
The use of a conceptual framework of
nonlinear dynamics in learning can have a
significant impact on the field of education,
science education, educational measurement,
and educational psychology. In particular,
examination of cognitive processing of
science-based information opens education
to greater use of transdisciplinary tools such
as fNIRS, electroencephalography (EEG),
and other psychophysiological tools in
understanding how student learning occurs
(Lamb et al., 2022). In turn this deeper
understanding allows greater translation into
practice, though it is important to note there
is often a distance between understanding of
a mechanism of action and the ability to
manipulate and make use of it. Drawing upon
the tools of mathematics, neuroscience, and
neuropsychology it is possible for
educational researchers and by extension
teachers to understand the interplay of
multiple processing units used by science
students to understand the world around
them, computationally model the dynamic
processing, and develop mechanisms of
action for examination. To answer research
question 1, does student science-based
cognitive processing data examined using
computational models exhibit chaotic
properties in the form bifurcation and
achaotic attractors the authors used
computational model data to develop the
achaotic attractors for examination. To
answer research question 2; does the
development of bifurcation and achaotic
attractors arise as an artifact construct of the
computational model or are the chaotic
properties independent of the computational
model; the authors used applied
mathematical modeling of random data.
Examination of the achaotic attractor and the
bifurcation plot in conjunction with their
underlying mathematical properties in Figure
1 provides evidence that science information
processing generalized to cognitive
processing do exhibit the properties of
nonlinear dynamic systems to include fractal
dimensions when engaged in science
RODRIGUES ET AL, Nonlinear Dynamics in Science Education 130
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
learning. This affirms research question 1 and
supports hypothesis 1. Examination of the
random data set visualization shown in
Figure 2 provides evidence that the outcomes
do not arise from an artificial artifact from the
computational model used to develop the
achaotic attractor and bifurcation plot and
supports hypothesis 2.
Of particular interest to science
educators are the possible long-term
behaviors of these cognitive systems as they
move from unstable to stable within the
achaotic attractor. This is akin to examination
of the change in currently held conceptions to
new conceptions and the necessary energy
inputs to change them. The stability of the
attractor directly examines the stability of
cognitive processing patterns ties to science
information. Meaning that stable patterns of
cognitive processing will be difficult to
change, in this light, the achaotic attractor
developed upon faulty processing will not
readily change and requires a significant
change in the entropy of the system to affect
a new attractor Lamb et al., 2021). This
phenomenon has long been exhibited within
the misconception literature (Chi, 1993;
Ebenezer et al., 2010; Elliott et al, 2021;
Hunkins et al, 2022; Lamb et al., 2014; Mark,
2023; Posner et al., 1982; Steenbeek et al.,
2020). An incorrectly developed achaotic
attractor will require significant stimulus
perturbations (neuronal propagation) in order
to create bifurcation moving from stable to an
unstable condition allowing for
redevelopment of the achaotic attractor from
the chaotic system. It is now possible to
compare directly the changes in one stable
pattern of cognition in science to another
stable pattern of cognition in science. The
bifurcation plot provides a visualization of
the mechanism of action related to change of
knowledge states (Lamb et al., 2014;
Steenbeek et al., 2020).
The modeled dynamic system will
possess multiple limit sets (bifurcations) with
each attractor (pattern of neuronal
interactions) converging to stability over time
as knowledge related to tasks becomes stable
within the basins. Each attractor represents
different stable patterns within the brain.
Through examination of the achaotic
attractor and the bifurcation plot, it is
possible to get an understanding of the
requirements to change a stable achaotic
attractor to an unstable attractor and back
again. Given the bifurcation plots,
independent axis is energy generalized to
input one can argue that an alternative
measure of energy in a system is entropy.
Thus, the authors speculates that it may be
possible to use a system of equations related
to the entropy of a state to provide resolution
on the requirements energetically to change a
stable knowledge state. This proposition is
speculative and requires extensive
experimental testing. The author proposes to
use electroencephalography and functional
near-infrared spectroscopy to accomplish
this. The use of this technology as a
measurement tool in conjunction with current
educational measures will allow users to
triangulate physiological measures and
changes with psychological states. This will
allow greater development of computational
models of student cognition, increases the
quality of data, and in turn increase the
quality of the derived attractor for
examination of student learning (Elliott et al,
2021; Hunkins et al, 2022; Mark, 2023;
Steenbeek et al., 2020).
Limitations
The use of nonlinear dynamics as a
modeling approach in education faces a
significant challenge related to its sensitivity
to initial conditions. From a statistical
perspective, gathering a substantial amount
of data is necessary for developing
simulations and analyzing outcomes.
Another issue lies in measuring and
INTERNATIONAL JOURNAL OF PSYCHOLOGY AND NEUROSCIENCE, 9(3), 118-137 131
ORIGINAL
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023). Unraveling Student Learning: Exploring Nonlinear Dynamics in
Science Education. International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi: https://doi.org/10.56769/ijpn09311
describing learning through nonlinear
dynamics, as the time neurons spend within a
specific fractal dimension phase space is
limited and rapidly changing. Therefore, data
collected at one point in time and phase space
may not be applicable to other points. One
potential solution involves employing
computational models that simulate time at
varying rates. However, the accuracy of such
models, especially in capturing fractal
dimensionality, emergent properties, and
interactions within multiple systems, remains
a serious limitation. This challenge is
comparable to the complexity faced in
solving three-body problems in physics and
weather prediction during the 1960s, which
often required approximations without
providing specific answers.
Conclusion
The landscape of science education is
undergoing a transformation in assessment,
moving beyond traditional content
knowledge tests (Steenbeek et al., 2020;
Hunkins et al., 2022; Elliott et al., 2021;
Mark, 2023; Lamb et al., 2014). In the realm
of science instruction, teaching strategies
wield the power to elicit both positive and
negative stimuli. These stimuli can act as
precursors, influencing students' cognitive
processing, akin to achaotic attractors
transitioning from stability to instability and,
in some cases, maladaptive states. These
shifts in mental activity (cognitive
processing), are exemplified through
bifurcations, offering opportunities to choose
alternative pathways.
An appreciation for nonlinear
dynamic systems, such as the learning
environment within a classroom, encourages
a perspective that values diverse teaching
styles. This approach diminishes the
prevalence of stable attractors linked to
misconceptions, paving the way for less
stable transitional phases. These transitional
periods create conditions conducive to swift
shifts toward stable attractors, presenting
opportunities for effective interventions
aligned with the learning objectives set by
teachers in the classroom.
Acknowledgments: The authors want to
thank all participants for their contribution to
the present study.
Declaration ethical approval: All
procedures performed in studies involving
human participants were in accordance with
the ethical standards of the institutional
and/or national research committee and with
the 1964 Helsinki Declaration and its later
amendments or comparable ethical standards
and received approval from appropriate
ethical review boards.
Competing interests: The authors declare no
conflict of interest.
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How to cite this Article:
Rodrigues, H., Molina-Fernandez, A. J., Lamb, L., Choi, I., & Owens, T. (2023).
Unraveling Student Learning: Exploring Nonlinear Dynamics in Science Education.
International Journal of Psychology and Neuroscience, 9(3), 118-137. Doi:
https://doi.org/10.56769/ijpn09311
Received: 11/10/2023 Revised:16/11/2023 Accepted: 12/12/2023
Published online: 31/12/2023 ISSN: 2183-5829
