ISBN 978-9915-698-39-7
General methodological proposals
for Computer Science research:
Prototyping, mathematical modeling,
and the scientific method
Camac Tiza, Maria Maura; Zevallos Vera,
Erika Juana; Castillo Paredes, Omar Tupac
Amaru; Ramírez Veliz, Juan Francisco;
Salazar Villavicencio, Ismael Edwin; Piedra
Isusqui, José César; Juscamayta Ramírez, Luis
Alberto
© Camac Tiza, Maria Maura; Zevallos Vera,
Erika Juana; Castillo Paredes, Omar Tupac
Amaru; Ramírez Veliz, Juan Francisco;
Salazar Villavicencio, Ismael Edwin; Piedra
Isusqui, José César; Juscamayta Ramírez, Luis
Alberto, 2025
First edition (1st ed.): October, 2025
Edited by:
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Uruguay.
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Editorial Mar Caribe
General methodological proposals for Computer
Science research: Prototyping, mathematical
modeling, and the scientific method
Colonia, Uruguay
2025
General methodological proposals for Computer
Science research: Prototyping, mathematical
modeling, and the scientific method
Index
Page
Introduction
6
Chapter I.
Methodological Pluralism in Computer Science: A
Comprehensive Framework for Research Paradigms,
Design Science, and Empirical Rigor
9
Chapter II.
Methodological Foundations of Computer Science: A
Comprehensive Analysis of Prototyping, Mathematical
Modeling, and the Scientic Method
32
Chapter III.
Mathematical Modeling and the Educational Research
Process: Theoretical Foundations, Methodological
Architectures, and Empirical Trajectories
58
Chapter IV.
Epistemological Convergence: The Integrated Role of
Prototyping, Mathematical Modeling, and the Scientic
Method in Complex Systems Engineering
81
Conclusion
106
Bibliography
108
Index of tables
Page
Table 1: Summary of Methodological Frameworks by
Research Goal
23
Table 2: Deterministic vs. Stochastic Modeling
37
Table 3: Research vs. Industrial Prototypes
40
Table 4: Comparing the Three Methodologies
45
Table 5: Traditional Problem Solving Research Vs.
Models and Modeling Perspective (MMP)
61
Table 6: Dimensions of the Quality Assurance Guide
(QAG)
70
6
Introduction
Research in Computer Science occupies, today, a unique place in the
academic landscape. Unlike traditional natural sciences, which study pre-
existing phenomena, or pure mathematics, which operates in a universe of logical
abstractions, computation is constantly moving between the articial and the
theoretical, between the construction of artifacts and the discovery of
fundamental laws. This hybrid nature has historically generated a certain
methodological fragmentation: Do we research to build more ecient systems?
To prove theorems about complexity? Or to empirically validate the behavior of
an algorithm in the real world?
The present book, "General methodological proposals for Computer Science
research: Prototyping, mathematical modeling, and the scientic method", was born
from the need to unify and formalize these approaches. Far from seeing these
methodologies as watertight compartments, this work proposes an integrative
vision where software engineering, mathematical rigor and empirical inquiry are
intertwined to give solidity to the advancement of computational knowledge.
The central structure of this text is based on three fundamental pillars that,
although dierent in their execution, are complementary in the search for
scientic answers:
- Prototyping: Often wrongly relegated to a mere development activity,
prototyping is claimed here as a legitimate and necessary research tool. In
a eld where systems reach levels of complexity unmanageable for purely
theoretical analysis, prototyping allows you to explore the design space,
identify hidden constraints, and validate the technical feasibility of an
idea. It is not just a maer of "making it work", but of using the constructed
7
artifact as a probe to interrogate technological reality and rene research
questions.
- Mathematical Modeling: If the prototype anchors us in practical reality, the
mathematical model elevates us towards the abstraction necessary to
generalize. This book examines how mathematical formalization allows
researchers to predict behaviors, ensure security properties, and optimize
resources before writing a single line of code. Modeling is not just an
exercise in rigor; It is the language that allows real-world problems to be
translated into computable structures, oering a framework for the
deductive reasoning that is the basis of computational theory.
- The Scientic Method: We address the application of the classical scientic
method—observation, hypothesis, experimentation, and analysis—in a
digital context. As computing moves into areas such as articial
intelligence, complex networks, and large-scale distributed systems,
software behavior often becomes stochastic or emergent. Here, empirical
validation becomes critical. We explore how to design controlled
experiments, how to collect meaningful data, and how to apply rigorous
statistical analysis to conrm that a proposed solution really represents a
signicant advance over the state of the art.
The purpose of these pages is not to prescribe a single "recipe" for research,
but to provide the reader—whether graduate student, academic, or industrial
researcher—with a versatile methodological toolbox. In the following chapters,
we will break down each approach, discussing its advantages, its limitations,
and, most importantly, its points of convergence.
It will be shown that the most impactful research usually occurs at the
intersection of these methods: when a mathematical model guides the creation of
a prototype, and this, in turn, is subjected to the scrutiny of the scientic method
to validate theoretical predictions. By mastering these three dimensions, the
8
Computer Science researcher not only becomes a beer system builder, but a
more complete scientist, capable of producing knowledge that is both
theoretically sound and practically relevant.
We invite the reader to go through these methodological proposals with
an open mind, willing to understand that in computing, theory and practice are
not opposites, but the two indispensable sides of the same scientic coin.
9
Chapter I.
Methodological Pluralism in Computer
Science: A Comprehensive Framework for
Research Paradigms, Design Science, and
Empirical Rigor
1. The Epistemological Crisis and the Search for
Identity
Computer Science (CS) stands as a unique anomaly in the history of academic
disciplines, vacillating between the abstract certainty of mathematics, the pragmatic utility
of engineering, and the empirical observation of the natural sciences. This tripartite identity
has engendered a rich, albeit often fragmented, landscape of methodological proposals.
Unlike physics or biology, where the "Scientific Method" provides a relatively unified
(though debated) foundation, computer science lacks a single, hegemonic research
paradigm. Instead, it operates under a "methodological eclecticism" where the validity of a
research contribution depends entirely on the ontological lens through which the computer
program is viewed.1
The discipline has matured from its mid-20th-century roots—where it was often
housed within departments of mathematics or electrical engineering—into a distinct field
with its own rigorous standards. However, this maturation has not resolved the
fundamental "ontological dispute" described by researchers at the University of South
Carolina and others: Is a computer program a mathematical expression governed by
deductive logic? Is it a technical artifact evaluated by its utility? Or is it a dynamic
phenomenon, akin to a biological process, to be observed and measured?.1
10
These questions are not merely philosophical musings; they dictate the day-to-day
mechanics of research. They determine whether a PhD student should spend their time
proving a theorem, conducting a controlled user study, or building a prototype system. As
the field expands into Artificial Intelligence (AI), Human-Computer Interaction (HCI), and
massive distributed systems, the definitions of "rigor" are shifting. The rise of the
"Replication Crisis" in AI and systems research has forced a re-evaluation of traditional
publication models, leading to the adoption of pre-registered reports, artifact badging, and
strict empirical standards.5
This report provides an exhaustive analysis of the methodological landscape of
computer science. It dissects the three dominant paradigms—Rationalist, Technocratic, and
Scientific—and explores the specific frameworks that operationalize these views, such as
Design Science Research (DSR), Action Research, and the emerging protocols for Digital
Twin validation and AI reproducibility. By synthesizing these diverse approaches, we aim
to provide a unified guide for constructing rigorous, high-impact research in the computing
disciplines.
2. The Tripartite Paradigms of Computing Research
To navigate the methodological options available, one must first locate their research
within the broader epistemological traditions of the field. The literature consistently
identifies three primary paradigms that govern how knowledge is acquired and validated
in computer science.
2.1 The Rationalist Paradigm: Computing as Mathematics
The Rationalist paradigm posits that computer science is a branch of mathematics.
In this view, the "computer program" is fundamentally a mathematical object—a formal
expression that exists independent of any physical machine.3 The ontology here is Platonist:
algorithms and data structures are discovered, not invented.
11
Methodological Implications:
● Epistemology: Knowledge is a priori. It is derived through deductive reasoning from
axioms, independent of sensory experience or physical experimentation.1
● Core Method: The primary method is the Formal Proof. Research involves defining a
formal specification of a system and proving properties such as correctness,
termination, and complexity bounds.8
● Validation: A result is valid if the proof is logically sound. Empirical testing is viewed
with skepticism; as Dijkstra famously noted, testing can only show the presence of
bugs, not their absence. Only formal verification provides certainty.9
Research in Theoretical Computer Science (TCS), semantics, and type theory
operates almost exclusively within this paradigm. Here, the "methodology" section of a
paper describes the formal system (e.g., lambda calculus, temporal logic) and the proof
techniques (e.g., structural induction, diagonalization) employed.10
2.2 The Technocratic Paradigm: Computing as Engineering
The Technocratic paradigm views computer science as a design discipline, akin to
chemical or aeronautical engineering. Here, the program is a "technical artifact"—a tool
constructed to satisfy a specific set of requirements in a given context.1 The ontology is
functionalist: a program is defined by what it does and how well it solves a problem.
Methodological Implications:
● Epistemology: Knowledge is probabilistic and pragmatic. It is derived a posteriori through
the construction and evaluation of artifacts.
● Core Method: The primary method is Design Science Research (DSR). This involves
an iterative cycle of problem identification, artifact construction, and evaluation.12
● Validation: A result is valid if the artifact is shown to be useful, reliable, and efficient
relative to existing solutions. "Correctness" is less about mathematical truth and more
about specification compliance and user satisfaction.3
12
This paradigm dominates Software Engineering (SE), Database Systems, and
Network Architecture. The focus is on managing complexity and reliability in systems that
are too large to be formally proven correct.14
2.3 The Scientific Paradigm: Computing as a Natural Science
The Scientific paradigm, arguably the most rapidly expanding view, treats computer
systems as natural phenomena. This is particularly prevalent in Artificial Intelligence,
Artificial Life, and complex large-scale networks. Here, the program is an empirical entity
that exhibits behavior—often emergent and unpredictable—that must be observed.1
Methodological Implications:
● Epistemology: Knowledge is empirical. It is derived from observation, experimentation,
and statistical inference.
● Core Method: The primary method is the Controlled Experiment. Researchers
formulate hypotheses about system behavior and test them by manipulating
independent variables (e.g., dataset size, network load) and measuring dependent
variables (e.g., accuracy, latency).9
● Validation: A result is valid if it is statistically significant and reproducible. This
paradigm embraces the "falsifiability" criterion of the natural sciences.11
This paradigm acknowledges that modern software systems—especially those
involving deep learning or chaotic network dynamics—are often "black boxes" whose
internal states are analytically intractable. Therefore, we must study them in vivo (in
operation) rather than just in vitro (in static analysis).16
3. Design Science Research (DSR): The Engineering
Standard
For researchers in Software Engineering and Information Systems who aim to build
13
novel systems, Design Science Research (DSR) provides the codified methodological
framework. Unlike routine design (which applies known solutions to known problems),
DSR addresses "wicked problems" through the creation of innovative artifacts.12
3.1 The Peffers Process Model
The most widely cited framework for operationalizing DSR is the process model
developed by Peffers et al. (2007). This model provides a nominal process for conducting
and presenting design research, ensuring rigor in what might otherwise be ad-hoc
development.13 The process consists of six distinct steps:
1. Problem Identification and Motivation:
The researcher must define the specific research problem and justify the value of a
solution. This grounds the research in practical relevance. For example, "Current
distributed ledgers cannot scale to global retail transaction volumes".13 The output of
this phase is a definition of the problem space and the criteria for a successful solution.
2. Define Objectives for a Solution:
Based on the problem definition, the researcher infers the objectives of a potential
solution. These objectives can be quantitative (e.g., "Must process 10,000 transactions
per second") or qualitative (e.g., "Must preserve user anonymity"). This step bridges
the gap between the problem context and the technical architecture.13
3. Design and Development:
This is the core creative step where the Artifact is created. In CS, an artifact can be a
Construct (vocabulary/symbols), a Model (abstraction/representation), a Method
(algorithm/practice), or an Instantiation (working system/prototype).13 The researcher
determines the artifact's functionality and architecture.
4. Demonstration:
The efficacy of the artifact is demonstrated by solving the problem in a suitable context.
This could involve a simulation, a case study, a proof-of-concept implementation, or a
pilot project. The goal is to show that the artifact can solve the problem, not necessarily
14
that it is the optimal solution yet.13
5. Evaluation:
This is the critical differentiator between engineering and research. The researcher
must observe and measure how well the artifact supports the solution. Evaluation
methods vary by artifact type:
○ Observational: Case studies or field studies (e.g., deploying a new software
process in a company).
○ Analytical: Static analysis, complexity analysis, or architectural review.
○ Experimental: Controlled experiments comparing the artifact to state-of-the-art
baselines.
○ Testing: Functional (black-box) or structural (white-box) testing.
○ Descriptive: Using scenarios or walkthroughs to argue for utility.13
6. Communication:
The problem, the artifact, and the evaluation are communicated to relevant audiences
(researchers and practitioners). The structure of DSR papers typically follows this
process logic.13
3.2 The Three Cycles of DSR Relevance
Hevner et al. expanded on this by defining DSR as a set of three cycles that ensure
the research remains grounded 18:
● The Relevance Cycle: Connects the research to the "Environment" (people,
organizational systems, technical infrastructure). This provides the requirements and
the testing ground for field testing.
● The Design Cycle: The central iteration of building and evaluating the artifact. This is
where the hard technical work occurs.
● The Rigor Cycle: Connects the research to the "Knowledge Base" (scientific theories,
existing methods, experience). This ensures the researcher draws on past work and
contributes new knowledge back to the repository, preventing the "reinventing the
15
wheel" syndrome.18
3.3 Application in Sustainability and Requirements Engineering
Recent applications of DSR have demonstrated its utility in complex, value-driven
domains. For instance, the development of the Sustainability Awareness Framework
(SusAF) used DSR to create a tool that helps software engineers anticipate the long-term
sustainability effects of their systems.20 Similarly, researchers have used DSR cycles to
integrate the United Nations Sustainable Development Goals (SDGs) into the Requirements
Engineering process, showing that design artifacts can drive broad societal goals.20 This
highlights DSR's capacity to handle "wicked" sociotechnical problems where the
requirements are not fully known at the outset.
4. Empirical Methodologies: The Scientific Method in
Computing
As computer science moves away from pure theory, the adoption of the "Scientific
Paradigm" has necessitated the adaptation of empirical methods from the natural and social
sciences. This is most visible in the subfields of Human-Computer Interaction (HCI),
Empirical Software Engineering (ESE), and High-Performance Computing (HPC).9
4.1 The Taxonomy of Experiments
Not all "experiments" in CS are the same. A nuanced understanding of experimental
types is crucial for methodological clarity 9:
● Feasibility Experiments: Often called "existence proofs," these demonstrate that a new
tool or technique is possible. They are common in systems research (e.g., "We built a
compiler that optimizes for X").
● Trial Experiments: Evaluating a single system to characterize its performance (e.g.,
"How does this algorithm scale with N?").
16
● Comparison Experiments: The gold standard in algorithmic research. Two or more
systems are run under identical conditions to determine which performs better
regarding specific metrics (time, space, accuracy).
● Controlled Experiments: Borrowed from psychology/physics, where independent
variables are manipulated to measure their effect on dependent variables while holding
confounding factors constant.
4.2 Challenges in Testing Scientific Software
Testing software used for scientific discovery (e.g., climate models, physics
simulations) presents unique methodological challenges identified as the Oracle Problem.
In traditional business software, the "correct" answer is known (e.g., 2+2=4). In scientific
software, the correct answer is often the unknown value the software is being written to
discover.21
● Methodological Solutions: Researchers must rely on Metamorphic Testing (checking
if changes in input produce expected changes in output, even if the absolute output is
unknown) and Code Clone Detection to ensure consistency.
● Cultural Friction: There is often a disconnect between domain scientists (who view the
code and the model as inseparable) and software engineers (who view them as
distinct). Effective methodology requires bridging this gap through rigorous
documentation and unit testing of sub-components where the answer is known.21
4.3 Simulation as a Methodological Proxy
When physical experimentation is impossible—due to cost, scale (e.g., the Internet),
or danger (e.g., nuclear simulations)—Simulation becomes the primary method. However,
a simulation is only as good as its correspondence to reality.22
● Verification: The process of ensuring the computer program (the simulator) correctly
implements the conceptual model. "Did we build the model right?" This involves
standard software testing techniques.
17
● Validation: The process of ensuring the model accurately represents the real-world
system. "Did we build the right model?"
○ Retrodiction: Validating the model by feeding it historical input data and checking
if it reproduces the historical output data.24
○ Prediction: Validating by comparing model forecasts with future real-world
events.
● Sensitivity Analysis: A critical methodological step where input parameters are
systematically varied to determine how sensitive the output is to uncertainty. If small
changes in input cause massive changes in output, the model may be too unstable for
reliable conclusions.25
5. Methodologies in Artificial Intelligence and
Machine Learning
The rapid ascent of AI has birthed its own set of methodological norms and crises.
The complexity of modern deep neural networks (DNNs) means they act as "black boxes,"
requiring new investigative techniques to understand why they work.26
5.1 The Ablation Study: Isolating Causality
One of the most important methodological contributions from the AI community is
the Ablation Study. In complex learning systems with many components (e.g., a new loss
function, a specific layer type, a data augmentation technique), it is insufficient to show that
the "whole package" works.
● Mechanism: An ablation study systematically removes one component at a time (e.g.,
"ablating" the attention mechanism) to measure its specific contribution to the overall
performance.26
● Purpose: This serves as a "sensitivity analysis" for architectural decisions. It prevents
"feature creep" where researchers add unnecessary complexity that does not actually
18
contribute to the result. In modern AI conferences (NeurIPS, ICML, ICLR), an ablation
study is often a mandatory requirement for acceptance.27
5.2 The Crisis of Reproducibility and Benchmarking
AI research faces a severe reproducibility crisis. Models often perform well on
specific benchmarks but fail to generalize.
● Benchmarking Methodologies: While benchmarks (like ImageNet or GLUE) provide
a standard yardstick, over-reliance leads to overfitting the benchmark.
Methodologically sound research now requires Out-of-Distribution (OOD) testing,
where models are evaluated on data that differs statistically from the training set.28
● The AI Scientist Concept: New research is exploring the use of "AI Scientists"—
automated agents powered by LLMs that can generate hypotheses, write code, run
experiments, and even draft papers. While this promises to accelerate discovery, it
introduces methodological risks regarding the verification of these automated findings.
Benchmarks like A2D are being developed to test these agents on their ability to
conduct reproducible science.28
5.3 Benchmarking for Reproducibility Agents
Recent initiatives are attempting to automate the reproducibility check itself.
Benchmarks are being created to test whether AI agents can take a published paper's
codebase and data and successfully reproduce the reported results. This represents a "meta-
methodology"—using AI to verify the methodology of AI research.27
6. Action Research and Qualitative Inquiry
Not all computer science is quantitative. In fields like Information Systems and
industrial Software Engineering, the human element is paramount. Here, Action Research
(AR) and Case Study methodologies are essential.16
19
6.1 Action Research: Intervening to Learn
Action Research differs from traditional observational science in that the researcher
actively participates in the system to improve it. It is highly relevant for introducing new
software methodologies (e.g., Agile, DevOps) into an organization.31
● The Cycle: The process involves Diagnosing (identifying the problem in the
organization), Action Planning (designing the intervention), Action Taking
(implementing the change), Evaluating (measuring the impact), and Specifying
Learning (deriving generalizable knowledge).31
● Distinction from Consulting: Unlike consulting, which focuses solely on solving the
client's problem, AR focuses on generating scientific knowledge. The researcher must
have a theoretical framework before the intervention and must reflect on the theoretical
implications after.33
6.2 Ethnography and Grounded Theory
In Human-Computer Interaction (HCI), understanding the "user" requires more
than surveys.
● Ethnography: Involves deep immersion in the user's environment to understand the
social and contextual factors of technology use. This is crucial for "Contextual Design".34
● Grounded Theory: A bottom-up methodology where the researcher does not start with
a hypothesis. Instead, they collect data (interviews, observations) and code it to allow
a theory to "emerge" from the data. This is particularly useful for exploring new
phenomena where no existing theory exists.30
7. Digital Twins: The Convergence of Modeling and
Engineering
The concept of the Digital Twin (DT) represents a convergence of simulation, data
science, and systems engineering. A Digital Twin is a dynamic virtual replica of a physical
20
asset (e.g., a jet engine, a factory) that is continuously updated with real-time data.37
7.1 Validation methodologies for Digital Twins
Validating a DT is more complex than validating a static simulation because the
physical system changes over time (degradation, maintenance).
● TEVV (Testing, Evaluation, Verification, and Validation): A comprehensive
framework for DTs. It involves Unit Testing of individual models, Integration Testing
of the data pipelines, and Performance Evaluation of the predictive accuracy.37
● Data Provenance and Security: Since DTs rely on real-time IoT data, the validation
methodology must include checks for data integrity and anomaly detection. A
"poisoned" data stream can invalidate the twin's predictions.37
● Lifecycle Validation: The validation is not a one-time event but a continuous loop. As
the physical asset ages, the digital model must be re-calibrated and re-validated (a
process known as "Model Updating" or "Continuous Validation") to ensure it remains
a faithful representation.38
8. The Crisis of Rigor: Artifact Review and Empirical
Standards
The increasing complexity of software and data has led to a recognition that a text-
based research paper is an insufficient record of scientific work. "An article about
computational science... is not the scholarship itself, it's merely scholarship advertisement.
The actual scholarship is the complete software development environment and the complete
set of instructions which generated the figures".40
8.1 ACM Artifact Review and Badging
To address this, the ACM and other organizations have instituted Artifact Review
processes. Authors submit their code, data, and scripts alongside the paper.41
21
● Badges:
○ Artifacts Available: Code is permanently archived (e.g., on Zenodo or ACM DL).
○ Artifacts Evaluated – Functional: The artifact runs and produces the expected
results.
○ Artifacts Evaluated – Reusable: The artifact is well-documented and can be easily
reused by others.
○ Results Reproduced: An independent team used the author's artifacts to obtain the
same results.
○ Results Replicated: An independent team obtained the same results using their
own implementation.42
● Impact: This methodology forces researchers to prioritize "Practical Reproducibility"
(e.g., using Docker containers) from day one, rather than trying to clean up code after
acceptance.44
8.2 Registered Reports
To combat publication bias (where only positive results are published) and p-
hacking (manipulating analysis to find significance), the field is adopting Registered
Reports (RR).6
● The Workflow:
1. Stage 1: Researchers submit the Introduction, Methodology, and Analysis Plan
before collecting data.
2. In-Principle Acceptance (IPA): If the methodology is sound, the journal commits
to publishing the paper regardless of the results.46
3. Stage 2: After data collection, the final paper is reviewed to ensure the protocol
was followed.
● Adoption: First introduced in Software Engineering at the MSR 2020 conference, this
format is now available in major journals like Empirical Software Engineering (EMSE)
and ACM Transactions on Information Systems (TOIS).6 It is particularly effective for
22
hypothesis-driven empirical studies.
8.3 ACM SIGSOFT Empirical Standards
To standardize the quality of reviews, the ACM SIGSOFT community has released
Empirical Standards for common methodologies (Experiments, Case Studies,
Benchmarking, etc.).7
● Usage: These standards provide checklists of "Essential," "Desirable," and
"Extraordinary" attributes. Researchers use them to design studies, and reviewers use
them to evaluate papers.
● Benefit: This reduces the subjectivity of peer review and provides clear guidance on
what constitutes a "methodologically sound" contribution.36
9. Ethics as a Methodological Component
Ethics is no longer an afterthought; it is a core component of research methodology.
The integration of "Society, Ethics, and Professionalism" (SEP) into the ACM/IEEE CS2023
curriculum highlights this shift.51
9.1 Institutional Review Boards (IRB)
For any research involving human subjects—which includes user studies, surveys
of developers, and even analyzing data from public repositories if it contains PII (Personally
Identifiable Information)—IRB approval is a mandatory methodological step.53
● Risk Minimization: The protocol must detail how risks to participants (including
privacy risks) are minimized.
● Informed Consent: Researchers must document how participants are informed of the
study's purpose and their rights.
● Snowball Sampling: Special care is needed for "snowball sampling" (participants
recruiting others), as this can introduce coercion or reveal social network structures,
requiring specific ethical safeguards.54
23
9.2 Value Sensitive Design (VSD)
VSD is a methodology that accounts for human values (privacy, autonomy, trust,
environmental sustainability) throughout the design process.55
● Methodology: It involves Conceptual Investigations (identifying values), Empirical
Investigations (understanding how stakeholders prioritize values), and Technical
Investigations (designing the system to support these values).
● Green Computing: With the massive energy footprint of AI, "Green Technologies" and
sustainability impact assessments are becoming standard methodological
requirements for systems research.20
10. Structuring the Rigorous Research Proposal
The methodology of computer science is a composite discipline that draws on the
rigor of mathematics, the creativity of engineering, and the empiricism of the natural
sciences. A strong research proposal must clearly articulate its paradigm and adhere to the
specific standards of that tradition (see Table 1).
Table 1: Summary of Methodological Frameworks by Research Goal
Research Goal
Validation Standard
Proving a Theorem
Mathematical
Correctness
Building a System
Utility, Efficiency,
Relevance
Understanding a
Phenomenon
Statistical
Significance,
Reproducibility
24
Improving a Process
Organizational
Learning, Adoption
Creating an AI Model
State-of-the-Art
Performance,
Generalizability
Whether one is proving the properties of a cryptographic protocol, designing a
sustainable software architecture, or conducting a large-scale randomized control trial on
user interface design, the unifying theme is rigor. By adopting formal frameworks like
Peffers' DSR model, adhering to Artifact Review standards, and pre-registering empirical
hypotheses, computer scientists can ensure their work stands on a solid methodological
foundation, capable of weathering the scrutiny of peer review and contributing lasting value
to the body of knowledge. The future of the field belongs to the "methodological polyglot"—
the researcher who can fluidly move between the whiteboard proof, the git repository, and
the statistical analysis package.
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32
Chapter II.
Methodological Foundations of Computer
Science: A Comprehensive Analysis of
Prototyping, Mathematical Modeling, and
the Scientific Method
1. The Epistemological Pluralism of Computer Science
Computer Science (CS) occupies a unique and often contested epistemological space,
straddling the boundaries between the abstract precision of mathematics, the pragmatic
constructivism of engineering, and the observational rigor of the natural sciences.1 Unlike
traditional disciplines defined by a singular mode of inquiry—such as the deductive proofs
of mathematics or the inductive experiments of biology—computer science is intrinsically
pluralistic. It investigates phenomena that are at once artificial (created by humans) and
mathematical (governed by formal logic), necessitating a research methodology that is
robust enough to handle this duality.2
The discipline has historically struggled with an "identity crisis," oscillating between
being viewed as a field deeply rooted in strong theories—such as computational complexity,
Turing machines, and formal semantics—and an engineering discipline focused on the
creation of artifacts that transform society, such as the Von Neumann architecture, the
internet, and distributed systems.1 This duality suggests that computer science inherits its
research methods from both ancestors: the mathematical approach utilizing axioms,
postulates, and proofs; and the engineering approach employing quantification,
measurements, and comparison.1 Furthermore, as computer systems have grown to levels
of complexity where their behavior cannot always be predicted solely through analysis,
researchers increasingly treat them as natural objects to be observed, measured, and
33
subjected to hypothesis testing, introducing the empirical or scientific paradigm.2
Consequently, the methodological landscape of modern computer science can be
broadly categorized into three primaries, yet interconnected, proposals: Mathematical
Modeling, Prototyping (often framed within Constructive Research or Design Science), and
the Scientific Method.5 Each methodology offers unique advantages and is suited for
different types of research inquiries. Mathematical modeling provides a framework for
abstract reasoning, enabling researchers to simulate systems and predict their behaviors
through formal logic.5 Prototyping enables researchers to create tangible representations of
their ideas, facilitating iterative testing, refinement, and the demonstration of feasibility.5
Meanwhile, the scientific method provides a structured approach to inquiry, emphasizing
the systematic testing of hypotheses and the validation of empirical findings through
controlled experimentation.5
This report provides an exhaustive analysis of these three methodological pillars. It
explores not only their individual mechanics and validity criteria but also the "interplay"
among them that enriches the research landscape.5 By synthesizing theoretical foundations
with practical applications, this analysis elucidates how computer science generates
knowledge, bridging the gap between the abstract "universe" of the computer—an ever-
developing artifact—and the concrete reality of its application in society.2
1.1 The "Sciences of the Artificial"
To understand the methodologies of computer science, one must first define the
object of investigation. In physics or biology, the object of study usually pre-exists the
observer. In computer science, the universe of study is the computer itself—an artifact
created by human intelligence.2 This aligns computer science with Herbert Simon's concept
of the "Sciences of the Artificial," where the goal is not merely to describe the world as it is,
but to design courses of action to change existing situations into preferred ones.7
34
This artificial nature complicates the application of traditional scientific methods. If
the object of study is a program written by the researcher, "experimentation" takes on a
different meaning than in the natural sciences. It may refer to "demonstration" (showing
that the artifact functions as intended) rather than "hypothesis testing" (proving a universal
theory true or false).3 This distinction is critical in separating constructive research, which
aims to build a solution to a specific persisting problem, from empirical research, which
aims to test the feasibility or performance of that solution using empirical evidence.8
2. Mathematical Modeling: The Theoretical Pillar
Mathematical modeling serves as the theoretical backbone of computer science,
providing the necessary tools for abstraction, specification, and verification. In this
paradigm, computer science is treated as a branch of applied mathematics, where systems
are specified using logic, probability, and algebra, and properties are verified through
deductive reasoning rather than observation.2
2.1 The Role of Abstraction and Formalism
Abstraction is the process of developing a conceptual veneer that hides the
complexity of internals, allowing researchers to focus on the essence of a problem without
being bogged down by implementation details.9 Mathematical modeling makes this
abstraction precise. By stripping away extraneous variables, researchers can create a model
that captures the fundamental behavior of a system, whether it is an algorithm, a network
protocol, or a database transaction system.2
This reliance on abstraction aligns computer science with the "Analytic Paradigm,"
where researchers propose formal theories or sets of axioms and derive results that are then
compared with empirical observations.10 The primary tool here is Formal Methods, which
rely on abstract models to represent system properties mathematically and precise
semantics to define the meaning of these models without ambiguity.11
35
2.1.1 Taxonomy of Formal Approaches
Formal methods are broadly categorized into two approaches based on their focus:
● Model-Oriented Formalisms: These focus on constructing explicit mathematical
models of the system's state and operations. They allow for detailed simulation and
refinement, making them suitable for constructive design. Examples include Z, VDM,
and state transition systems.12
● Property-Oriented Formalisms: These emphasize axiomatic descriptions of desired
behaviors and invariants. They use logical predicates to assert what the system must
satisfy without prescribing how. This distinction influences their applicability: model-
oriented suits constructive design, while property-oriented excels in abstract
validation.12
2.2 The Research Cycle in Theoretical Computer Science (TCS)
Research in Theoretical Computer Science (TCS) follows a distinct lifecycle that
mirrors the mathematical tradition rather than the empirical scientific method. It is
characterized by a "Formal Methodology," where the basic idea is to abstract away as many
details as possible to leave behind only the essence of the problem.13
The TCS research cycle typically involves the following phases:
1. Formalization: Converting a real-world problem into a mathematical object (e.g., a
graph, a logic formula, or a state machine).15 This step is critical because any error in
formalization invalidates the subsequent proofs.
2. Deduction and Proof: Using axioms and proof techniques to derive properties of the
model. Common techniques include inductive proofs, reduction (mapping a problem
to a known solved/unsolved problem), and diagonalization.1
3. Verification: Proving that an algorithm or system design meets its specification for all
possible inputs. This offers a guarantee of correctness that empirical testing can never
provide, as testing can only show the presence of errors, not their absence.16
36
This methodology is most frequently used to prove facts about algorithms (e.g.,
time/space complexity) and systems (e.g., safety, liveness).14 It underpins fields such as
computational complexity theory, cryptography, and programming language theory.1
2.3 Formal Methods vs. Empirical Testing: A Fundamental Tension
A fundamental tension exists in computer science between formal methods and
empirical testing. Formal methods offer the promise of absolute correctness—proving a
system is error-free—whereas testing is inherently limited by the finite number of test cases
that can be run.16
However, formal methods face significant challenges:
● The "Gap" Problem: A mathematical proof applies to the model, not the implementation.
A recent trend in mathematical modeling is to publish computer code together with
research findings, but this raises the question of whether the code faithfully implements
the model.18 Implicit assumptions in the code (e.g., regarding memory limits or
floating-point precision) may not be reflected in the formal model, severing the causal
link between the proof and the actual system behavior.18
● Scalability: Formal verification is often computationally expensive and difficult to
scale to large, complex software systems. This has led to the development of
"lightweight" formal methods and their integration with automated tools like model
checkers.12
● Learning Curve: The mathematical disciplines used to formally describe
computational systems (e.g., discrete mathematics, logic, automata theory) are often
outside the domain of traditional engineering education, creating a barrier to
adoption.19
Despite these challenges, the "Analytical Paradigm" remains essential. In critical
systems—such as avionics, nuclear power, and medical devices—no serious engineer would
build without mathematical modeling because of the risks involved. The cost of correcting
37
design errors after implementation is prohibitive, making the predictive power of
mathematical analysis invaluable for both safety and efficiency.16
2.4 Deterministic vs. Stochastic Modeling
Mathematical modeling in CS is not monolithic; it varies based on the nature of the
system being studied (see Table 2).21
Table 2: Deterministic vs. Stochastic Modeling
Modeling Type
Application Domain
Methodology
Deterministic Modeling
Algorithm Analysis, Formal
Verification, Logic
Uses deductive logic to prove
properties that hold true under
all conditions (e.g., sorting
algorithm correctness). 21
Stochastic Modeling
Networks, Performance
Evaluation, Finance
Uses probability theory and
statistics to model systems with
inherent uncertainty or
randomness (e.g., packet
arrival rates, stock prices). 21
Stochastic modeling is particularly important in systems research, where researchers
must account for variable workloads and environmental noise. Here, the mathematical
model (e.g., a queuing theory model) provides a theoretical basis for extrapolation, allowing
predictions about system behavior under loads that cannot be easily tested empirically.4
3. Prototyping and Constructive Research: The
Engineering Pillar
While mathematical modeling dominates the theoretical side of CS, Prototyping and
Constructive Research dominate the engineering and applied sides. This methodology is
concerned with "building" as a primary research activity. It involves the construction of
38
artifacts—software, hardware, methods, conceptual frameworks, or tools—to demonstrate
feasibility, solve practical problems, or generate new knowledge through the act of
creation.14
3.1 Constructive Research Methodology
Constructive research is a research procedure for producing innovative
constructions, intended to solve problems faced in the real world and, by that means, to
make a contribution to the theory of the discipline in which it is applied.22 It differs from
"scientific method" research, which aims to explain why something happens, and from pure
"consulting," which aims to solve a problem without necessarily generating new theoretical
knowledge.22
The constructive approach implies building an artifact (practical, theoretical, or
both) that solves a domain-specific problem in order to create knowledge about how the
problem can be solved in principle.23 This methodology gives results which can have both
practical and theoretical relevance, solving knowledge problems concerning feasibility,
improvement, and novelty.23
3.1.1 The Constructive Research Cycle
The process of constructive research is often described as a multi-phase cycle.
According to 24 and 22, the key phases are:
1. Problem Identification: Find a practically relevant problem that also has research
potential. The problem must be significant enough to warrant a research effort.22
2. Pre-understanding: Obtain a general and comprehensive understanding of the topic.
This involves literature reviews and studying existing solutions to ensure the new
artifact will be novel.24
3. Innovation (Construction): Construct the solution idea. This is the creative phase
where the researcher designs the artifact (algorithm, system, model).24
39
4. Demonstration: Demonstrate that the solution works. This is often achieved through
a "Proof of Concept" or a prototype implementation.24
5. Theoretical Contribution: Show the theoretical relevance of the construct. The
researcher must articulate why the solution works and linking it back to the body of
knowledge.22
6. Validation/Applicability: Examine the scope of applicability and generalize the
findings. This often involves testing the artifact in a real-world or simulated
environment.22
3.2 Design Science Research (DSR)
Closely related to constructive research is Design Science Research (DSR), a
methodology prevalent in Information Systems and Software Engineering. DSR focuses on
the creation and evaluation of IT artifacts intended to solve identified organizational
problems.25 DSR distinguishes itself by explicitly requiring a contribution to the "knowledge
base" while solving an "environment" problem. DSR operates through three distinct cycles
that ensure the research remains grounded and rigorous 22:
● The Relevance Cycle: Connects the research to the environment (people, systems,
organizations). It defines the problem requirements and provides the context for field
testing.
● The Design Cycle: The core iterative process of building, testing, and refining the
artifact. This is where prototyping takes place.
● The Rigor Cycle: Connects the research to the knowledge base (foundations,
methodologies, past theories). It ensures the design is not just a "hack" but is grounded
in existing scientific knowledge and contributes new knowledge back to the field.
3.2.1 Validity in DSR
A major challenge in DSR is ensuring the validity of the resulting artifacts. Artifacts
are often criticized for lacking rigorous evaluation. To address this, DSR frameworks
40
propose five essential validity types 26:
1. Instrument Validity: Does the artifact function correctly? (Technical correctness).
2. Technical Validity: Does the artifact solve the technical problem it claims to?
3. Design Validity: Is the design structure sound and justifiable?
4. Purpose Validity: Does the artifact actually help the user or solve the real-world
problem? (Utility).
5. Generalization: Can the artifact or its underlying principles be applied to other
contexts?
Crucially, "instrument validity" and "design validity" are often the least developed
in research papers, posing a risk of overlooked flaws that threaten research credibility.26
3.3 The Role of Prototyping
Prototyping is the central activity within the constructive and DSR methodologies.
A prototype is an early sample, model, or release of a product built to test a concept or
process.28 It serves as a tangible representation of an idea, allowing for iterative testing and
refinement.5
3.3.1 Research vs. Industrial Prototypes
It is vital to distinguish between Research Prototypes and Industrial Prototypes, as
they serve different goals and are evaluated by different criteria (see Table 3).
Table 3: Research vs. Industrial Prototypes
Feature
Research Prototype
Industrial (Commercial)
Prototype
Goal
Knowledge production; Proof
of concept; Feasibility 29
Product design; Usability;
Manufacturability; Market
viability 28
41
Nature
"Good enough" to serve as a
vehicle for knowledge; Often
uses "shortcut technology" or
Wizard-of-Oz techniques 29
Robust; Reflects final product;
Focus on user experience and
scale 28
Lifespan
Short-lived; Often discarded
after the concept is proven 29
Iterative; Evolves into the final
product or beta release 28
Evaluation
Novelty; Theoretical
contribution; Feasibility 22
Reliability; Return on
investment; Market demand;
User satisfaction 30
User Role
Participants in an experiment
or demonstration 7
Beta testers or potential
customers 28
Research prototypes are not "deficient" versions of industrial prototypes; they are
purpose-built instruments for inquiry.29 Their quality is secondary to their ability to
generate insight. For example, a research prototype might use a clumsy interface if the
research question is about the underlying algorithm, whereas an industrial prototype would
fail if the interface were poor.29
3.3.2 Proof of Concept (PoC)
A Proof of Concept (PoC) is an advanced form of prototyping often used to "prove"
that a new technology, service, or idea is viable.30 In computing literature, "experiment" is
sometimes used synonymously with "demonstration" or "PoC".3 A PoC differs from a
standard prototype in that it may not be a working model of the entire system but rather a
focused experiment to validate a specific critical function or claim.30 It is the "existence
proof" of the engineering world—demonstrating that a system with certain properties can
be built.14
4. The Scientific Method: The Empirical Pillar
The third pillar, the Scientific Method, represents the empirical branch of computer
science. While CS involves artificial objects, the behavior of complex software systems,
42
networks, and human-computer interactions is often too complex to model analytically or
predict solely through construction. Thus, researchers must observe the world, propose
models/theories, measure, analyze, and validate hypotheses.4
4.1 Adapting the Scientific Method to the Artificial
The scientific method in CS follows the classical hypothetico-deductive pattern but
with domain-specific adaptations.32 The general cycle includes:
1. Observation: Noticing a phenomenon (e.g., "users struggle with this interface" or "this
network protocol slows down under load").33
2. Hypothesis Formulation: Proposing a testable explanation or prediction. A hypothesis
must be falsifiable (e.g., "Algorithm A is faster than Algorithm B for datasets of type
X").34
3. Experimentation: Designing a controlled test (benchmarking, randomized control
trial) to gather data. This involves identifying independent variables (what is changed),
dependent variables (what is measured), and nuisance variables (what must be
controlled).35
4. Analysis: Using statistical methods to accept or reject the hypothesis. This often
involves Null Hypothesis Significance Testing (NHST).34
5. Conclusion/Reporting: Publishing results to contribute to the body of knowledge.33
In computer science, this method is applied across several sub-disciplines, each with its own
norms and "experimental cultures".3
4.2 Empirical Software Engineering (ESE)
Empirical Software Engineering (ESE) applies the scientific method to the software
development process itself. It aims to move the field from "advocacy research" (where
researchers propose a method and claim it is good based on persuasion) to "evaluation
research" (where claims are tested).4
43
4.2.1 ESE Strategies
ESE employs a variety of empirical strategies, broadly categorized into fixed and
flexible designs 10:
● Controlled Experiments: Performed in laboratory settings (e.g., using students as
subjects) to test specific hypotheses with high internal validity. However, they often
suffer from low external validity because the setting is artificial and the tasks are
small.10
● Case Studies: In-depth inquiries into real-world projects. These have high external
validity because they study professionals in their natural environment, but they are
hard to control and generalize.10
● Surveys: Collecting qualitative or quantitative data from a broad population to
understand trends, opinions, or adoption rates.10
● Action Research: A collaborative approach where the researcher actively participates
in the development process to solve a problem while studying it. This blends
constructive and empirical methods.10
4.2.2 The "ABC" Framework
To navigate the trade-offs in ESE strategies, researchers use the ABC Framework,
which identifies three desirable aspects of research that cannot be maximized
simultaneously 36:
1. A (Actors): Generalizability over the population (e.g., all software engineers).
2. B (Behavior): Precision of control over the behavior (e.g., controlling the exact task
duration).
3. C (Context): Realism of the setting (e.g., a real company under deadline pressure).
A lab experiment maximizes B but sacrifices C and often A. A field study maximizes
C but sacrifices B. A survey maximizes A but sacrifices B and C (since it relies on self-
reporting). This framework helps researchers choose the right strategy for their specific
44
research question.
4.3 Human-Computer Interaction (HCI)
HCI research relies heavily on the scientific method, borrowing techniques from
psychology and social sciences to study how humans interact with technology.34
HCI studies are often categorized as:
● Formative Evaluation: Exploratory tests done during the design process to identify
usability problems and refine the design. These are often qualitative and iterative.38
● Summative Evaluation: Controlled experiments done after the design is complete to
measure performance (e.g., time on task, error rate) against a benchmark or a
competitor. These are quantitative and hypothesis-driven.38
In HCI, the "scientific method" is often synonymous with Null Hypothesis
Significance Testing (NHST). Researchers set up a null hypothesis ($H_0$: "there is no
difference between Interface A and B") and try to reject it using statistical tests (t-tests,
ANOVA) to show that a new design is significantly better.34 However, the field is
increasingly embracing qualitative methods (ethnography, grounded theory) to capture the
richness of user experience that numbers alone cannot describe.37
4.4 Systems Research: The "Experimental" Debate
In the sub-fields of Computer Systems (Operating Systems, Networks, Databases),
"experimentation" often refers to performance evaluation. However, the methodological
rigor of this work has been a subject of intense debate.
4.4.1 The Rejuvenation of Experimental CS
In the 1980s and 90s, reports by Feldman and Sutherland critiqued the field for
undervaluing experimental work and for a lack of methodological rigor.3 This led to a
movement to "rejuvenate" experimental CS, distinguishing between "building components"
45
(prototyping) and "doing science" (experimenting).3
4.4.2 Benchmarking vs. Science
A common criticism in systems research is that "evaluation" often consists of ad-hoc
benchmarks rather than hypothesis-driven inquiry. Evaluators should look for work that
forms a clear hypothesis, constructs reproducible experiments to shed light on that
hypothesis while controlling other variables, and analyzes the data to prove or disprove the
claim.40
Methodological rigor in systems requires:
● Workload Generation: Creating realistic inputs that mimic real-world usage (e.g.,
using traces of actual web traffic).
● Isolation of Variables: Ensuring that observed differences are due to the system design
and not background noise (e.g., OS jitter, network interference).
● Reproducibility: Ensuring that other researchers can run the same code and obtain the
same results. This is a current crisis in the field, as many systems papers rely on
proprietary code or hardware that is not available to others.40
5. Comparative Analysis: Tensions and Trade-offs
The coexistence of these three methodologies—Mathematical, Constructive, and
Empirical—creates a vibrant but sometimes fractured discipline. Each has different values,
validation criteria, and definitions of "success."
5.1 Comparing the Three Methodologies
The table 4 summarizes the key differences between the three pillars.
Table 4: Comparing the Three Methodologies
46
Feature
Scientific Method
(Empirical)
Primary Goal
Explanation,
Prediction, &
Falsifiability
Core Activity
Observing,
Measuring, Testing
Key Output
Data, Plots,
Accepted/Rejected
Hypotheses
Validation
Statistical
Significance,
Reproducibility
Primary Risk
Internal validity
(uncontrolled
variables)
Typical Discipline
HCI, Empirical SE,
Performance Analysis
Source of Truth
Empirical Data
(Observation)
5.2 The Tension Between Theory and Practice
A significant tension exists between the "theoretical" (math-based) and "practical"
(prototype-based) camps.41
● Theory's Critique of Practice: Theoreticians may view prototypes without formal
proofs as "hacking"—unreliable, specific to one implementation, and lacking
47
generalizable insight.42 They argue that empirical testing can only show the presence
of bugs, never their absence.16
● Practice's Critique of Theory: Practitioners often view formal models as "vacuous"
because they rely on restrictive or false assumptions that ignore real-world constraints
(e.g., memory caches, human behavior, unpredictable network latency).43 A model that
proves an algorithm is $O(n \log n)$ is useless if the constant factors make it too slow
for real-time application.
This tension is often described as the "Gap" between formal specifications and
running code.18 A system might be verified correct against a model, but if the model
simplifies the hardware too much, the system may still fail. Conversely, empirical results
without a theoretical model may be non-generalizable—knowing that a system is faster is
less useful than knowing why.4
5.3 Rigor vs. Relevance
In Constructive and Empirical research, there is a constant trade-off between rigor
(control, precision) and relevance (realism, usefulness).36
● High Rigor, Low Relevance: A lab experiment where students use a simplified tool in
a classroom allows for perfect control of variables but tells us little about how that tool
would perform in a complex, messy industrial software development project.
● Low Rigor, High Relevance: A case study observing a team at a major tech company
is highly relevant to industry but nearly impossible to control or replicate. The findings
may be specific to that one company's culture.
The ABC Framework discussed in Section 4.2.2 explicitly acknowledges this,
suggesting that researchers must choose their strategy based on which trade-off is
acceptable for their specific research question.36
5.4 The "Valley of Death"
48
In Constructive Research, there is a phenomenon known as the "Valley of Death" for
prototypes. This refers to the difficulty of transitioning a Research Prototype (which is often
a "duct-taped" proof of concept) into an Industrial Prototype or product.44 Academia
incentivizes novelty and publication, not the robust engineering required to make a
prototype production-ready. This gap often prevents promising research ideas from having
a practical impact.
6. Synthesis and Integration: Toward a Unified
Methodology
The most impactful computer science research often integrates these methodologies,
creating a virtuous cycle of modeling, building, and evaluating. The interplay among these
methodologies enriches the research landscape, offering diverse perspectives and tools for
tackling challenges.5
6.1 Synergy: The Virtuous Cycle of Research
A robust research project often traverses all three pillars in a "virtuous cycle":
1. From Model to Prototype: A researcher uses mathematical modeling to design a new
algorithm (e.g., a new consensus protocol) and verifies its safety properties formally.45
2. From Prototype to Experiment: The researcher implements this algorithm in a research
prototype. This implementation reveals practical issues (e.g., unexpected network
latency, memory overhead) that were ignored by the abstract model.46
3. From Experiment to Theory: Empirical benchmarking (scientific method) reveals
unexpected behaviors or performance bottlenecks. The researcher analyzes the data to
refine the mathematical model, making it more realistic and adding new constraints.47
This synergy is particularly evident in fields like Formal Verification of Systems,
where researchers use formal methods to verify core components (like the seL4 microkernel)
but rely on empirical testing for the un-modeled parts (such as hardware behavior or user-
49
level applications).48
6.2 Mixed Methods Research
Increasingly, CS researchers employ Mixed Methods, combining qualitative and
quantitative data to provide a more complete picture of a phenomenon.50 This is distinct
from simply using multiple algorithms; it involves mixing methodological paradigms.
● Sequential Explanatory: Running a quantitative experiment (e.g., a benchmark) and
then conducting qualitative interviews with users or developers to understand why the
results occurred.
● Triangulation: Using both logs (quantitative) and surveys (qualitative) to validate a
finding from two different angles. If both the system logs and the user reports indicate
a problem, the finding is much more robust.52
Mixed methods are particularly vital in HCI and Software Engineering, where
"human factors" are as important as "system performance." For example, a new
programming tool might be theoretically efficient (Model) and functionally complete
(Prototype), but if developers find it confusing to use (Empirical/Qualitative), it will fail.50
6.3 Artifact Evaluation and Reproducibility
A major methodological shift in the last decade is the institutionalization of the
"Constructive" methodology through Artifact Evaluation (AE) tracks at major conferences
(e.g., POPL, OOPSLA, SIGMOD).54 Historically, CS papers were judged solely on the text
(the theory or the reported results). This encouraged "paper-ware"—systems that sounded
good on paper but didn't actually exist or run.
Now, independent committees evaluate the artifact (the code, data, and scripts) to
ensure it exists, runs, and supports the claims in the paper. This process:
● Validates the "Constructive" contribution: The artifact is recognized as a scholarly
output in itself.
50
● Ensures "Scientific" reproducibility: It guarantees that the experiments described in the
paper can be replicated.
● Bridges the "Theory-Practice Gap": It forces authors to make their implementation
robust enough for others to use, moving it slightly out of the "research prototype" stage
toward usability.
Papers that pass this process receive badges like "Artifact Functional" and "Artifact
Reusable," signaling the quality of the engineering work underpinning the scientific
claims.54
6.4 Future Directions: AI and Methodological Evolution
The rise of Artificial Intelligence (AI) and Machine Learning (ML) is introducing new
methodological challenges. AI models are often "black boxes"—they are empirical objects
that work (high predictive accuracy) but lack a theoretical model explaining why they work
(low interpretability).18
This is shifting the field towards a more empirical direction, where "science" looks
more like biology (observing the emergent behavior of a neural network) than mathematics
(proving the correctness of an algorithm). Future research methodologies will likely focus
on:
● Explainable AI (XAI): Trying to derive mathematical models from empirical black
boxes.
● AI-assisted Formal Methods: Using AI to help generate proofs, bridging the gap
between intuition and formalism.47
● Simulated Environments: Using "Digital Twins" and high-fidelity simulations to
prototype and test systems in silico before building physical artifacts.56
Methodological diversity is not a weakness of Computer Science but its defining
strength. The field has evolved from "hacking" and pure mathematics into a sophisticated
discipline that blends mathematical modeling to ensure correctness and abstraction,
51
prototyping to demonstrate feasibility and innovation, and the scientific method to validate
hypotheses and measure performance in the real world.
Successful computer science research often requires a traversal of all three pillars. A
rigorous theoretical foundation provides the "why," a robust prototype provides the "how,"
and a careful empirical evaluation provides the "so what." As systems become more
complex—incorporating AI, human behavior, and physical environments—the ability to
triangulate findings using this tripartite methodological framework will become
increasingly essential. The future of the field lies not in choosing between math, engineering,
or science, but in mastering the synthesis of all three to solve the complex problems of the
artificial world.
52
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58
Chapter III.
Mathematical Modeling and the
Educational Research Process: Theoretical
Foundations, Methodological Architectures,
and Empirical Trajectories
1. Epistemological Foundations of the Models and
Modeling Perspective
The trajectory of mathematics education research over the past three decades has
been characterized by a decisive shift away from the "acquisition metaphor" of learning—
which views knowledge as a commodity to be transmitted and stored—toward a
perspective that emphasizes the development of conceptual systems. This paradigm shift is
most rigorously articulated in the Models and Modeling Perspective (MMP). The MMP
fundamentally reconfigures the ontology of mathematical learning, positing that the
primary goal of instruction is not merely the mastery of isolated procedural skills or the
memorization of algorithms, but the cultivation of powerful, shareable, and reusable
conceptual models that enable learners to describe, explain, and predict the behavior of
complex real-world systems.1
This perspective emerged as a response to the limitations of traditional
constructivism. While constructivism successfully argued that learners actively build
knowledge, it often lacked a specific description of what is being constructed and how those
constructions evolve in the context of complex problem-solving. The MMP fills this void by
identifying "models" as the fundamental unit of cognition.3 In this view, models are not
static mental pictures but dynamic conceptual tools consisting of elements, relationships,
operations, and rules that learners project onto their experiences to make sense of them.1
59
1.1 The Nature of Models in Educational Contexts
Within the MMP framework, a model is understood as a system of thinking that is
fundamentally social and iterative. Unlike the traditional definition of a mathematical
model—which might be limited to a set of differential equations or a statistical regression—
the educational definition is broader. It encompasses the internal conceptual systems
(mental models) and the external representations (graphs, diagrams, equations, metaphors)
that learners use to externalize their thinking.1
The distinction between internal and external representation is critical for the
research process. Internal conceptual systems are inaccessible to direct observation.
However, when students are engaged in Model-Eliciting Activities (MEAs), they are
compelled to externalize these internal systems into visible artifacts. This process of
externalization serves a dual purpose: it stabilizes the student's own thinking, allowing for
self-reflection and revision, and it provides researchers with a tangible "audit trail" of
cognition.4 Thus, the model becomes the interface between the private world of the learner's
mind and the public world of social negotiation and educational assessment.
The iterative nature of these models is another defining characteristic. Models are
rarely formed in a finished state. Instead, they evolve through a series of "modeling cycles."
Learners begin with primitive, unstable, and often barren interpretations of a problem
situation. Through interaction with the problem's constraints and with peer feedback, these
initial models are tested, revised, differentiated, and integrated into more robust and stable
systems.5 This evolution mimics the scientific process itself, where theories are proposed,
tested against empirical data, and refined.
1.2 Holistic vs. Atomistic Approaches to Learning
The MMP challenges the "atomistic" curriculum design prevalent in many
educational systems, which breaks mathematics down into discrete, decontextualized skills
(atoms) to be mastered sequentially. The assumption behind atomistic instruction is that
60
once students have collected enough "atoms" of knowledge, they will spontaneously
assemble them into complex understanding. Empirical evidence suggests this rarely
happens; students often fail to transfer these isolated skills to novel contexts.2
In contrast, the MMP advocates for a holistic approach where the "whole" (the
model) gives meaning to the "parts" (the specific mathematical procedures). A student does
not learn about ratios in isolation and then apply them; rather, the need to quantify the
steepness of a ramp or the density of a population creates the need for a ratio model. The
model provides the structural context in which specific mathematical tools (arithmetic
operations, algebraic notation) become meaningful and necessary.2 This aligns with findings
that students demonstrate significantly higher engagement and retention when
mathematics is presented as a tool for solving authentic, client-driven problems rather than
as a set of abstract rules.8
1.3 Mathematical Modeling as a Research Methodology
Perhaps the most significant contribution of the MMP is its integration of
instructional design and research methodology. In traditional educational research, the
"treatment" (instruction) and the "measurement" (testing) are separate events. The
researcher teaches a concept, then administers a post-test to see if it stuck. This approach
often fails to capture the process of learning, offering only a snapshot of the product.
The MMP dissolves this dichotomy. The Model-Eliciting Activity (MEA) acts
simultaneously as the instructional vehicle and the research instrument. Because MEAs
require students to document their decision-making processes and justify their solutions to
a client, the "data" is generated in real-time as the learning occurs.4 This "thought-revealing"
quality allows researchers to trace the genesis of mathematical ideas—from their messy,
informal origins to their formal, mathematical expressions—without the interference of
artificial testing protocols.10
The following table summarizes the fundamental distinctions between traditional
61
problem-solving research and the Models and Modeling Perspective (see Table 5).
Table 5: Traditional Problem Solving Research Vs. Models and
Modeling Perspective (MMP)
Dimension
Traditional Problem Solving
Research
Models and Modeling
Perspective (MMP)
Unit of Analysis
Isolated skills or procedural
fluency.
Conceptual systems (models)
and their evolution.
View of Learning
Acquisition of facts and
procedures (Linear).
Iterative development of
models (Recursive/Zig-Zag).
Problem Context
Well-defined, "puzzle-like"
problems with single answers.
Ill-structured, complex, real-
world situations with multiple
solutions.
Role of Social Interaction
Often viewed as "cheating" or
noise; focus on individual
cognition.
Essential; models are co-
constructed and negotiated in
groups.
Research Data
Pre-test and Post-test scores
(Outcome-based).
Transcripts, student drafts, and
final model documentation
(Process-based).
Goal of Instruction
Mastery of curriculum
standards.
Development of shareable,
reusable, and transferable
conceptual tools.
Validation Authority
The teacher or the answer key.
The problem constraints and
the "client's" needs (Self-
assessment).
2. The Architecture of Model-Eliciting Activities
(MEAs)
62
The operational heart of the MMP is the Model-Eliciting Activity (MEA). These tasks
are meticulously engineered to simulate real-world professional contexts, such as
engineering, urban planning, or business consulting. Unlike standard "word problems,"
which typically require the application of a previously taught procedure, MEAs are
designed to be "model-eliciting"—that is, they require the learner to invent, extend, or refine
a mathematical system to solve the problem.9
The design of these activities is governed by six fundamental principles, first
articulated by Lesh, Hoover, Kelly, and Post (2000). These principles serve as a quality
control mechanism for instructional design and as a validity check for research data. If a
task violates these principles, it is unlikely to reveal significant insights into student
thinking.12
2.1 The Reality Principle
The Reality Principle dictates that the problem context must be meaningful and
accessible to the students based on their existing life experiences. This does not mean the
problem must be "real" in the sense of happening right now, but it must be realistic—
students must be able to imagine the situation and care about the outcome.12
Crucially, the Reality Principle ensures that students can use their "real-world"
knowledge to validate their mathematical results. In traditional textbook problems (often
critiqued as "pseudo-contexts"), the reality is stripped away; students learn to ignore
common sense (e.g., assuming a bus can travel at constant velocity forever). in an MEA,
reality is the first line of defense against error. If a model predicts that a "Giant" is 40 feet tall
based on a footprint, the Reality Principle compels the student to pause and reject that
conclusion based on their knowledge of human biology.12 This grounding in reality lowers
the barrier to entry (the "low floor") while allowing for sophisticated mathematical analysis
(the "high ceiling").2
2.2 The Model Construction Principle
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This principle mandates that the task must require the creation of a system, not just
the calculation of a value. The question asked of the student should not be "What is the
answer?" but "How should this be decided?" or "What rule works best?".12 For example, in
the "Summer Jobs" MEA, students are not asked to calculate the sum of wages. They are
asked to develop a method for hiring the "best" employees based on conflicting data sets
involving hours worked, money earned, and consistency.15 This shift forces students to
construct a model that defines "best"—is it total accumulation? Is it efficiency (earnings per
hour)? Is it reliability? The resulting product is a decision-making algorithm, which is a
mathematical model in its purest form.2
2.3 The Self-Assessment Principle
Perhaps the most critical principle for fostering autonomy, the Self-Assessment
Principle requires that the problem statement contains the criteria for success. Students
should be able to judge for themselves whether their solution is good enough without
asking the teacher, "Is this right?".4
In practice, this is often achieved through "test data" provided by a fictional client. If
the students' model works for Employee A but fails for Employee B, the data itself provides
the feedback. This feedback loop drives the iterative nature of the modeling cycle. Students
test their model, see it fail against the data, and revise it. This creates a "need for revision"
that comes from the task, not the instructor, shifting the locus of authority from the teacher
to the mathematics itself.12
2.4 The Model Documentation Principle
The Model Documentation Principle ensures that the students' thinking is made
visible. The task must explicitly require students to describe their process, usually in the
form of a memo, letter, or "user guide" to the client.4
This principle is the bridge between instruction and research. Without
64
documentation, a researcher might see a correct answer but have no idea how it was
derived. By requiring a written explanation of the "toolkit" or "procedure," the MEA forces
students to externalize their internal cognitive state. This documentation serves as the
primary data source for researchers, allowing for the analysis of the process of model
development rather than just the final product.4
2.5 The Generalizability (Shareability) Principle
Models are powerful because they are reusable. The Generalizability Principle states
that the solution created by the students should not just work for the specific instance in the
problem (e.g., this specific footprint), but should be shareable with others and usable in
similar situations (e.g., any footprint found at a crime scene).4
This pushes students toward mathematical abstraction. A solution that says, "It's 10
feet tall because I measured it with my finger" is not shareable. A solution that says,
"Measure the foot length, multiply by 6.5, and add 2 inches" is a generalizable algorithm.
This principle forces students to move from ad-hoc reasoning to formal mathematical
syntax, facilitating the transfer of learning to new contexts.18
2.6 The Effective Prototype Principle
The final principle ensures that the resulting model is simple yet powerful—a
"prototype" or metaphor that students can carry forward to interpret other situations.12 If
the model is overly complex or messy, it is less likely to be retained as a cognitive tool. An
effective prototype serves as a mental hook. For instance, once students develop a "rate of
change" model to solve a problem about filling a water tank, that model becomes a
prototype they can apply to problems about population growth or velocity. The MEA acts
as the genesis event for this conceptual prototype.2
3. The Modeling Cycle: Cognitive Dynamics and
65
Student Trajectories
Learning in the MMP is conceptualized as a "Modeling Cycle." This cycle is rarely
linear; it is a "zig-zag" path of development where students move through phases of
expressing, testing, and revising their thinking.6 Researchers have mapped these cycles
extensively, identifying how groups negotiate the conflict between their initial intuitive
ideas and the formal demands of the problem.
3.1 Phases of the Modeling Cycle
1. Interpretation and Expressing: Students first attempt to understand the problem
situation. They filter the messy data, identifying which variables matter. This often
results in a "barren" or overly simple initial model.
2. Mathematization and Testing: Students attempt to map their initial ideas onto
mathematical structures (e.g., deciding to add numbers or find an average). They then
"run" this model against the data provided in the problem.
3. Revision and Refinement: The testing phase usually reveals flaws. The model might
give a ridiculous answer (violating the Reality Principle) or fail to account for a specific
data point. This failure triggers a revision phase, where students differentiate their
concepts (e.g., realizing that "big" can mean "tall" or "heavy") or integrate new
variables.2
3.2 Case Study Analysis: The Bigfoot MEA
The "Bigfoot" MEA is a classic example used in MMP research to illustrate the
evolution from qualitative to multiplicative reasoning. In this activity, students are
provided with a photograph of a large footprint and asked to develop a "toolkit" for police
to estimate the height of the person who made it.14
Phase 1: Qualitative and Intuitive Reasoning
Research transcripts reveal that student groups often begin with purely descriptive,
66
non-mathematical observations.
● Student Quote: "Wow! This guy is huge... You know any girls that big?".14
● Student Quote: "Those're Nike's... The tread's just like mine."
At this stage, the students are engaging with the context but have not yet formulated a
mathematical model. They are using their personal experiences (Reality Principle) to
orient themselves.
Phase 2: Additive Reasoning (The First Primitive Model)
As they move to quantification, students frequently default to additive strategies,
which are cognitively less demanding than proportional ones.
● Observed Strategy: A student places his own foot next to the footprint. He uses his
fingers to mark the gap—let's say it's 3 inches. He then reasons, "His foot is 3 inches
longer than mine, so he must be 3 inches taller than me".14
● Testing and Failure: When the group tests this model (Self-Assessment), they realize that
a person only 3 inches taller than a middle schooler is not a "giant." The model produces
a result that violates the "Bigfoot" premise. This cognitive dissonance—the failure of
the additive model—is the catalyst for the next leap in learning.
Phase 3: Proportional Reasoning (The Paradigm Shift)
Forced to abandon the additive model, students search for a relationship that
preserves the "bigness."
● Emerging Insight: "It's not about how much longer, it's about how many times longer."
● Mathematization: Students begin to calculate ratios. "My foot is size 8 and I am 5 feet
tall. This foot is size 16. That's double. So he must be double my height."
● Refinement: This proportional model (Height = k \times FootLength) is robust. Students
then refine it by gathering data from the whole class to find an average ratio (k \approx
6.6), satisfying the Generalizability Principle.18
This trajectory demonstrates that proportional reasoning is not just "applied" but
"invented" in response to the limitations of additive thinking.
67
3.3 Case Study Analysis: The Summer Jobs MEA
The "Summer Jobs" MEA exposes the conflict between different conceptual
definitions of value and the difficulty of quantifying them. The problem asks students to
select the best employees based on data tables of earnings and hours worked.15
The Conceptual Conflict: Accumulation vs. Efficiency
Research on this MEA highlights a divergence in student hypotheses:
● Veli's Hypothesis (Accumulation Model): "The time is not important. I think the ones
earning the highest amount of money should be the best." This model posits that Total
Value = Total Earnings. It ignores the cost of time.
● Sıla's Hypothesis (Efficiency Model): "We will also find the one who brings the
highest amount of money in the shortest time." This model introduces a rate: Value =
Earnings / Time.
The Modeling Paradox: Conceptual Agreement vs. Mathematical Execution
The group conceptually agreed that Sıla's efficiency model was superior. However,
when they moved to the "mathematization" phase to create a scoring system, a critical error
occurred.
● The Scoring System: The students decided to award points: 3 points for the highest
earnings, and also 3 points for the longest working hours.
● The Contradiction: By awarding points for long hours, they mathematically penalized
efficiency—contradicting their own conceptual agreement.
● The Outcome: A worker named Ahmet received the maximum score (3 points for
earnings + 2 points for other factors) because he worked long hours, even though he
was inefficient.
● Student Reflection: Veli remarked, "This scoring system has good mathematics. I am glad
I found a scoring system".15
This finding is profound for educational researchers. It illustrates "the seduction of
quantification." The students were so relieved to have a system that produced numbers
68
("good mathematics") that they failed to notice the numbers contradicted their logic. This
disconnect between the conceptual model and the computational model is a fertile ground for
teacher intervention and learning.15
4. Methodology: The Multi-Tiered Teaching
Experiment
To capture the complexity of these modeling cycles, MMP researchers utilize a
specialized methodological design known as the "Multi-Tiered Teaching Experiment." This
design acknowledges that educational settings are complex systems where learning occurs
simultaneously at multiple levels: the student level, the teacher level, and the researcher
level.4
4.1 Tier 1: The Student Tier
At the first tier, the focus is on the students interacting with the real-world context
provided by the MEA.
● Agent: Students.
● Task: Model-Eliciting Activity (e.g., Bigfoot).
● Goal: Construct a model of the physical reality.
● Data Generated: Audio/video transcripts of group discussions, draft notes, final client
reports.
● Research Focus: Tracking the evolution of mathematical concepts (e.g., from additive
to proportional).23
4.2 Tier 2: The Teacher Tier
The second tier focuses on the teacher. Just as students must model the Bigfoot
problem, teachers must model the students' thinking.
● Agent: Teachers.
69
● Task: Model-Eliciting Activities for Teachers (MEA-Ts). For example, a teacher might
be given a set of student responses to the Summer Jobs problem and asked to design
an assessment rubric.
● Goal: Construct a model of student cognition and pedagogy.
● Data Generated: Teacher interview transcripts, lesson plans, teacher reflections on
student work.
● Research Focus: How do teachers interpret student errors? Do they see them as
mistakes to be corrected or as windows into developing logic?.4
4.3 Tier 3: The Researcher Tier
The third tier involves the researchers themselves.
● Agent: Researchers.
● Task: Analyzing the interactions between Tier 1 and Tier 2.
● Goal: Construct a model of the entire educational ecosystem (theory building).
● Data Generated: Academic papers, theoretical frameworks, coding schemes.
● Research Focus: Developing generalizable theories about learning and instruction.22
This recursive design ensures that the research is not extractive but collaborative.
Researchers are not just observing "subjects"; they are participants in a system where their
own theories are constantly being tested and revised against the reality of the classroom.7
4.4 Data Analysis: Protocol Analysis and Triangulation
Analyzing the massive amount of qualitative data generated by these experiments
requires rigorous protocols.
● Protocol Analysis: Researchers parse transcripts of student dialogue into "episodes" of
reasoning. They look for specific "markers" of cognitive shifts, such as changes in
language (e.g., from "it looks like" to "the data shows").24
● Coding Schemes: Detailed coding schemes are developed to categorize student
70
statements. For example, in the Summer Jobs problem, codes might distinguish
between "absolute comparisons" (Employee A made 500) and "relative comparisons"
(Employee A made 10/hour).25
● Triangulation: To ensure validity, findings are triangulated across data sources. A
researcher's interpretation of a student's confusion is cross-referenced with the
teacher's field notes and the student's own written reflection. If all three align, the
finding is considered robust.12
● Inter-Rater Reliability: When multiple researchers code transcripts, they must achieve
a high level of agreement (typically >70% or >0.8 Cohen's Kappa). Disagreements are
resolved through discussion, which further refines the coding definitions.12
5. Quantitative Assessment and Engineering
Education Outcomes
While the MMP is rooted in qualitative inquiry, it bridges the gap to quantitative
assessment, particularly in the context of engineering education and ABET (Accreditation
Board for Engineering and Technology) accreditation. MEAs are increasingly used in
undergraduate engineering courses to measure "professional skills" that are difficult to
capture with traditional exams.8
5.1 The Quality Assurance Guide (QAG)
To quantify performance on MEAs, researchers developed the Quality Assurance
Guide (QAG). This rubric assesses student products not on "right/wrong" binary scales, but
on the quality and utility of the model (see Table 6).30
Table 6: Dimensions of the Quality Assurance Guide (QAG)
Dimension
Description
Score Range
71
Usefulness
Does the solution meet the
client's needs? Is it practical?
1 (Not useful) - 5 (Very useful)
Justification
Are the decisions supported by
data and mathematical
reasoning?
1 (No justification) - 5 (Fully
justified)
Shareability
Can the model be understood
and used by others without the
authors present?
1 (Not shareable) - 5 (User-
friendly)
Reusability
Can the model be applied to
similar problems with different
data?
1 (One-time use) - 5
(Generalizable)
Studies utilizing the QAG have demonstrated that students who engage in MEAs
show significant improvements in problem-solving scores over the course of a semester
compared to control groups.30
5.2 Measuring Incremental Validity and ABET Outcomes
Engineering programs often struggle to assess "soft skills" like teamwork, ethical
reasoning, and communication (ABET outcomes). Research indicates that MEAs provide a
valid instrument for these measures.
● Incremental Validity: A study by Kim & Moore (2019) found that MEA scores added
significant predictive capacity to models of student success, explaining variance that
GPA and standardized test scores could not. This suggests that MEAs measure a
distinct construct—"modeling competency" or "engineering design thinking"—that is
independent of rote academic ability.33
● Pre/Post Concept Inventories: When MEAs are paired with Concept Inventories (tests
of fundamental conceptual understanding), researchers observe that MEAs help
students repair misconceptions. The iterative testing phase of the MEA forces students
72
to confront their misunderstandings in a way that lectures do not.8
6. The Teacher's Role and Professional Development
The implementation of MMP places significant demands on teachers, requiring a
shift from "implementer" to "investigator." The "Multi-Tiered" design explicitly includes
teachers as learners, acknowledging that they too must develop new models of pedagogy.28
6.1 The Challenge of "Letting Go"
Traditional teaching often involves "smoothing the path" for students—removing
obstacles to ensure they reach the correct answer efficiently. In the MMP, obstacles are the
engine of learning. Teachers must learn to "let go" and allow students to struggle with the
ambiguity of the MEA.2
● Didactic Tension: Teachers often feel a strong urge to intervene when they see students
heading down a "wrong" path (like Veli's accumulation model). However, MMP
research shows that premature intervention robs students of the opportunity to self-
correct via the Self-Assessment Principle.
● Pedagogical Content Knowledge: Facilitating an MEA requires a deep understanding
of the multiple ways a problem can be solved. A teacher must be prepared to evaluate
a graphical solution, an algebraic solution, and a statistical solution simultaneously.28
6.2 Model-Eliciting Activities for Teachers (MEA-Ts)
To support this shift, professional development programs utilize MEA-Ts. These are
simulations where teachers are the "students."
● Example Task: Teachers are given a packet of "Student Work" (responses to the Bigfoot
problem ranging from poor to excellent) and a "Client Request" from the school
principal: "Design a grading rubric to assess these students' mathematical thinking."
● Outcome: By working together to build the rubric (a model), teachers must externalize
and negotiate their beliefs about what constitutes "good" mathematics. Do they value
73
the correct number? Or the reasoning process? This activity mirrors the student's
modeling cycle, allowing teachers to experience the same intellectual struggle.1
6.3 Shifting Beliefs and Dispositions
Research confirms that engagement with MEAs changes teacher dispositions.
Teachers who use MEAs tend to move away from "deficit models" (focusing on what
students lack) toward "competency models" (recognizing the diverse resources students
bring). They begin to see student errors not as failures of memory, but as rational attempts
to model complexity based on limited data.5
7. Technology, STEM, and Future Directions
The future of MMP research is increasingly intertwined with technology and
interdisciplinary STEM education.
7.1 Technology as a Modeling Tool
Digital tools like GeoGebra, dynamic spreadsheets, and simulation software act as
"amplifiers" for student modeling.
● Dynamic Modeling: In calculus and algebra, tools like GeoGebra allow students to
manipulate parameters and instantly see the effect on the model (e.g., changing the
slope of a line). This tightens the feedback loop of the modeling cycle, allowing for more
rapid iterations of testing and revision.19
● Data Analysis: For MEAs involving large datasets (like the Summer Jobs problem),
spreadsheets enable students to handle complexity that would be impossible by hand.
The ability to sort, filter, and graph data allows students to move beyond arithmetic
and engage in statistical reasoning.32
7.2 Interdisciplinary STEM Connections
MEAs are naturally interdisciplinary. The "Bigfoot" problem connects biology
74
(proportions) with math; the "Summer Jobs" problem connects economics (efficiency vs.
cost) with data analysis.
● STEM Integration: Research shows that MEAs are effective vehicles for STEM
integration because they require the simultaneous application of knowledge from
multiple domains. A study by Armutcu and Bal (2017) demonstrated that MEAs
improved students' ability to synthesize science and math concepts to solve
engineering challenges.11
● Science Literacy: MEAs that tackle controversial topics (e.g., environmental data,
pseudoscience like Bigfoot) provide a platform for teaching scientific literacy and
critical thinking. Students learn to question the source of data, the validity of
assumptions, and the limits of models—skills essential for navigating the modern
world.37
7.3 Policy and Assessment Implications
The greatest barrier to the widespread adoption of MMP is the misalignment with
standardized testing. Most high-stakes tests value speed and procedural accuracy, whereas
MEAs value depth, iteration, and revision.
● The "Assessment Gap": Research indicates that while MEAs improve deep conceptual
understanding, these gains do not always translate to higher scores on multiple-choice
standardized tests, which measure different constructs. This creates a policy dilemma:
schools are incentivized to teach to the test, potentially at the expense of the modeling
competencies required for the 21st-century workforce.38
● Future Research: A critical area for future MMP research is the development of
scalable, standardized assessments that can measure modeling competency reliably,
providing a viable alternative to traditional testing.34
The Models and Modeling Perspective represents a maturing field of educational
inquiry that offers a robust alternative to traditional instructional and research paradigms.
By centering the "model" as the fundamental unit of analysis, the MMP provides a
75
theoretical framework that accounts for the complex, non-linear, and social nature of human
learning.
The methodological innovations of the MMP—specifically the Model-Eliciting
Activity (MEA) and the Multi-Tiered Teaching Experiment—have dissolved the artificial
boundary between instruction and assessment. MEAs serve as powerful lenses, revealing
the microscopic evolution of student thinking from intuitive guesses to sophisticated
mathematical systems. The case studies of "Bigfoot" and "Summer Jobs" serve as empirical
validation of this framework, demonstrating that even young students are capable of
profound mathematical invention when the task structure supports their autonomy.
However, the path forward is not without challenges. The rigorous demands of
qualitative analysis, the need for intense teacher professional development, and the friction
with current assessment policies present significant hurdles. Yet, as the demand for a
workforce capable of navigating complex systems grows, the relevance of the Models and
Modeling Perspective only increases. It offers not just a way to teach mathematics, but a
way to research the very nature of how human beings organize their experience of the
world.
76
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19. Kovtoniuk, M. M., Soia, O. M., Kosovets, O. P., & Tiutiun, L. A. (2025). Virtual
educational mathematics park as a platform for educational and research activities
at the university. Journal of Physics: Conference Series, 3105(1).
https://doi.org/10.1088/1742-6596/3105/1/012024
20. Laja, Y. P. W., Delvion, E. B. S., & Siahaan, M. M. L. (2025). Chatbot Artificial
Intelligence (AI) to Supporting Differentiated Mathematics Learning. Jurnal
Pendidikan MIPA, 26(1). https://doi.org/10.23960/jpmipa.v26i1.pp118-130
21. Martins Bessa, S., Soares Oliveira, F. W., & Floriano Paulino, O. (2025). Mathematical
modeling: a study on its incorporation into basic education from the proceedings of
SIPEM. Educação Matemática Pesquisa, 27(1).
22. Matorin, D. D., & Cherepkov, A. Yu. (2025). Numerical analysis of multicomponent
dynamic models of stage-by-stage knowledge assimilation. Nonlinear World.
https://doi.org/10.18127/j20700970-202501-04
23. Mu’arif, W., Mustaqimah, Prasetyawati, A., Suryani, Maryati, T., & Rokhimah, S.
(2025). Penerapan Model Realistic Mathematic Education (RME) untuk
Meningkatkan Hasil Belajar pada Pembelajaran Matematika. Jurnal Educazione :
Jurnal Pendidikan, Pembelajaran Dan Bimbingan Dan Konseling, 2(2).
24. Nyirenda, J. L. Z., Mbakaya, B. C., Wagner, D., Stete, K., Jaeger, V. K., Bartz, A.,
Karch, A., Nanyinza, E., Soko, M., Chimbatata, N., Chirambo, G. B., & Lange, B.
(2025). NOZGEKA—a collaborative curriculum development process for a Master
of Science in Public health with a focus on infectious diseases epidemiology in
Malawi: a perspective. Humanities and Social Sciences Communications, 12(1).
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https://doi.org/10.1057/s41599-025-05654-6
25. Orazali, G., Dauletkulova, A., & Mekebayev, N. (2025). METHODOLOGY OF
TEACHING GEOMETRY IN THE CONTEXT OF DIGITALIZATION OF
EDUCATION. Журнал Серии «Педагогические Науки», 76(1).
https://doi.org/10.48371/peds.2025.76.1.029
26. Pérez-Mojica, R., & Osorio-Gutiérrez, P. A. (2025). Diseño de actividades de
modelación matemática para solucionar problemas reales de la función cuadrática.
Praxis, 21(2). https://doi.org/10.21676/23897856.6079
27. Podkhodova, N., Sheremetyeva, O., & Soldaeva, M. (2025). How to teach
mathematics so that students truly understand it. South Florida Journal of
Development, 6(9). https://doi.org/10.46932/sfjdv6n9-001
28. Saeki, A., Kawakami, T., & Lamb, J. (2025). The role of children’s analogies in the
transfer of their knowledge in mathematical modelling processes. ZDM -
Mathematics Education, 57(2). https://doi.org/10.1007/s11858-025-01675-2
29. Sahani, S. K., & Karna, S. K. (2025). Program for Cultural Education Optimization by
the Application of Numerical Methods. Journal of Posthumanism, 5(1).
https://doi.org/10.63332/joph.v5i1.2687
30. Sakibayeva, B., & Sakibayev, S. (2025). Enhancing professional competence in
physics and mathematics education through information technology. Educational
Research and Evaluation, 30(7–8). https://doi.org/10.1080/13803611.2025.2502813
31. Sardor, S., & Uldona, S. (2025). THE IMPORTANCE OF STUDYING
MATHEMATICS AND MODERN APPROACHES. JournalNX- A Multidisciplinary
Peer Reviewed Journal, 11.
32. Shcherbatykh, S. v., Lykova, K. G., Igonina, E. v., & Simonovskaya, G. A. (2025).
Development of cognitive mobility when teaching probability theory and
mathematical statistics to students using digital solutions. Perspektivy Nauki i
Obrazovania, 74(2). https://doi.org/10.32744/pse.2025.2.12
33. Smirnov, E. I., Skornyakova, A. Y. U., & Tikhomirov, S. A. (2025). The ontology and
engineering of professional self-determination of school students in synergy
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34. Solod, L. V., Tkachova, V. V., Prokofieva, H. Ya., & Bereziuk, H. H. (2025).
MODELING THE STUDY OF SOLID MATERIALS’ THERMAL CONDUCTIVITY
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Chapter IV.
Epistemological Convergence: The
Integrated Role of Prototyping,
Mathematical Modeling, and the
Scientific Method in Complex Systems
Engineering
1. The Teleological Nature of the Artifact
The contemporary engineering landscape is defined by a fundamental convergence
between the abstract rigor of the scientific method and the pragmatic utility of artifact
creation. Historically, science was perceived as the pursuit of knowledge—understanding
"what is"—while engineering was viewed as the pursuit of utility—creating "what is not
yet." However, the increasing complexity of modern systems, from the gravitational wave
detectors of LIGO to the bio-fidelic digital twins of the human heart, has necessitated a
unification of these domains. The act of engineering has become an experimental science,
where the prototype serves as a physical hypothesis and the mathematical model acts as the
theoretical framework governing that hypothesis.
This report explores the intricate relationship between prototyping, mathematical
modeling, and the scientific method. It posits that the design process is, at its core, a rigorous
exercise in hypothesis testing. Whether the "experiment" is a Monte Carlo simulation
running on a supercomputer or a physical crash test of an automotive chassis, the
epistemological goal remains the same: the reduction of uncertainty and the validation of
predicted behavior against empirical reality. We must begin, however, with a philosophical
caution rooted in the limitations of representation. As noted by S. I. Hayakawa and
reiterated in the foundational texts of mathematical modeling, "The map is NOT the
82
territory it stands for".1 A model is an abstraction, a deliberate simplification of reality
designed to answer specific questions.2 The danger in modern R&D lies not in the use of
models, but in the confusion of the symbol with the thing symbolized. Thus, the validation
of models against the "territory" of physical reality remains the central challenge of systems
engineering.
In examining this convergence, we must recognize the teleological nature of
modeling and simulation. Unlike a work of art, which may exist for its own sake, a model
is defined by its purpose. As defined in the literature, a model helps us to answer questions
and solve problems.2 Beginners in the field often fall into the trap of believing that a "good"
model is one that mimics reality as closely as possible in every dimension. This is a fallacy.
Modeling aims at simplification, not the useless production of complex copies of a complex
reality.2 The utility of a model is derived from its ability to strip away the noise of the real
world to reveal the signal of the underlying mechanism. This report will traverse the
theoretical foundations of this abstraction, the rigorous structures of the V-Model in systems
engineering, the economic imperatives driving the shift from physical to virtual testing, and
the detailed case studies that illustrate the triumph—and occasional failure—of this
integrated methodology.
2. Epistemological Foundations: Engineering as
Experimental Science
2.1 The Hypothesis of the Artifact
The scientific method, in its traditional formulation, involves a cycle of observation,
hypothesis formulation, experimentation, and conclusion.3 A scientist observes a
phenomenon, formulates a hypothesis to explain it, and designs an experiment to falsify
that hypothesis. If the experiment fails to refute the hypothesis, confidence in the theory
grows.5 In the context of engineering, the "hypothesis" takes a tangible form. The engineer
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hypothesizes that a specific configuration of materials, geometry, and logic will satisfy a set
of functional requirements within a given set of constraints.4 The prototype—whether it is
a physical assembly of steel and silicon or a virtual assembly of code and equations—is the
experiment designed to test this hypothesis.
There is a distinct parallel between the "Strong Inference" method in biology and the
iterative design process in engineering. Strong Inference involves devising alternative
hypotheses, devising a crucial experiment to exclude one or more of them, and carrying out
the experiment to get a clean result.6 Similarly, in engineering, prototyping is the mechanism
for excluding design candidates that fail to meet performance criteria. When a prototype
fails, it is an instance of epistemic refutation. It demonstrates that the engineer's
understanding of the system dynamics, the material properties, or the environmental
interactions was incomplete or incorrect.7 Therefore, failure in prototyping is not merely an
operational setback; it is a successful generation of new knowledge regarding the
boundaries of the design space.8
The distinction between the scientific method and the engineering design process is often
drawn in terms of their outputs: science produces knowledge, while engineering produces
solutions. However, the process by which these outputs are achieved is increasingly
identical.
● Scientific Hypothesis: "If X conditions are met, nature will behave in Y manner."
● Engineering Hypothesis: "If I construct X artifact, it will perform function Y under
constraints Z."
In both cases, the rigorous testing of the hypothesis determines the validity of the
work. The "experiment" in engineering is often a test of the prototype against the
requirements defined in the design phase.10 If the prototype functions as predicted, the
hypothesis is corroborated. If it malfunctions, the hypothesis is falsified, and the engineer
must return to the theoretical drawing board—the mathematical model—to adjust the
parameters of the hypothesis.1
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2.2 Deductive vs. Inductive Modeling Paradigms
Mathematical models serve as the formal language for these engineering
hypotheses. They allow engineers to predict behavior before physical resources are
committed, acting as a filter for ideas that are theoretically unsound. We can categorize these
modeling approaches into two broad epistemological camps: deductive and inductive.
Deductive Modeling relies on a priori information, typically the fundamental laws
of nature.11 These are often referred to as "white-box" or "physics-based" models. They
derive the behavior of a system from first principles—Newton’s laws of motion, Maxwell’s
equations of electromagnetism, or the laws of thermodynamics.13 The strength of deductive
modeling lies in its universality and explainability. If a bridge collapses in a physics-based
simulation, we can trace the failure to a specific stress concentration that exceeded the yield
strength of the material. This allows for "Strong Inference" where the mechanism of failure
is understood and can be corrected.
Inductive Modeling, by contrast, derives relationships from observed data. These
are "black-box" or "data-driven" models, increasingly prevalent with the rise of machine
learning and artificial intelligence.13 In this paradigm, the model "learns" the behavior of the
system by analyzing large datasets of inputs and outputs. While these models can be
incredibly accurate, particularly for complex, chaotic, or poorly understood phenomena
where the governing equations are intractable, they lack the explanatory power of deductive
models. An inductive model might predict that a component will fail, but it may not be able
to explain why in terms of fundamental physics.
The modern scientific method in engineering increasingly relies on a hybrid
approach. Engineers use data-driven methods to calibrate the parameters of physics-based
models, thereby refining the "map" to better fit the "territory".13 This synthesis allows for the
rigor of physical laws to be combined with the empirical precision of real-world data,
creating a "Gray Box" model that is both explainable and accurate.
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2.3 The Role of Failure in Epistemology
In the philosophy of science, particularly the Popperian view, the only relevant
evidence is negative evidence. One can never prove a hypothesis is true; one can only fail to
refute it.5 This logical asymmetry is central to the engineering epistemology of prototyping.
A successful prototype test does not prove that the design is perfect; it merely proves that it
works under the specific conditions tested. A failed prototype test, however, provides
absolute proof that the design (or the model used to create it) is flawed.
This perspective reframes "failure" from a negative outcome to a critical epistemic
event. As noted in the literature on engineering epistemology, failure is an inevitable and
essential part of physical prototype execution because it provides the feedback necessary to
correct the mental and mathematical models of the engineer.7 The "glitch" in the software or
the fracture in the strut is the moment where reality asserts itself against the abstraction of
the design. It is the "counter-example" that refutes the hypothesis.
However, epistemic failure can also occur when engineers believe something "that
just ain't so".14 This occurs when a model is relied upon outside of its domain of validity, or
when "tribal knowledge" in an engineering organization goes unchallenged by empirical
testing. The rigorous application of the scientific method—specifically the requirement for
falsifiability—protects against this form of intellectual complacency. By treating every
design decision as a hypothesis that must be tested, engineers can systematically reduce the
"epistemic uncertainty" (lack of knowledge) in the system, leaving only the "aleatory
uncertainty" (natural randomness) to be managed.15
3. The Mathematical Model: Abstraction, Formalism,
and Classification
3.1 Taxonomy of Mathematical Models
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T o understand how models function as scientific tools within the engineering
workflow, we must categorize them based on their mathematical structure and their
relationship to time, certainty, and state. A rigorous taxonomy allows engineers to select the
appropriate "tool" for the specific "experiment" they wish to conduct.
3.2 The Role of Simulation as Second-Order Experimentation
Simulation is the execution of a mathematical model over time. It has been described
in the philosophy of science as a "second-order experiment".16 If a thought experiment is a
simulation run in the mind, and a physical experiment is a simulation run in matter, a
computer simulation is an experiment run in logic. Simulation allows for the exploration of
scenarios that are dangerous, expensive, or impossible to replicate physically—such as a
nuclear core meltdown, the collision of galaxies, or the spread of a pandemic.17
A crucial technique in modern simulation is the Monte Carlo method, which relies
on repeated random sampling to solve deterministic problems or model physical systems
with significant uncertainty.18 This method is particularly vital in fields like high-energy
physics. In the context of the Large Hadron Collider (LHC), Monte Carlo simulations are
used to generate billions of simulated proton-proton collisions. These simulations provide
the "control group" for the experiment; they predict what the detectors should see if the
Standard Model of particle physics is correct. When the real data collected by the detectors
deviates from these Monte Carlo simulations, it indicates the potential discovery of new
physics, such as the Higgs boson.19
3.3 Fidelity vs. Cost: The Trade-off Landscape
The pursuit of high fidelity in modeling comes at a significant cost—both in terms
of computational resources and the time required to configure and run the simulation. This
creates a fundamental trade-off that engineers must navigate.
● High-Fidelity Models: These models aim to capture the physics of the system with
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minimal abstraction. Examples include Large Eddy Simulation (LES) in computational
fluid dynamics (CFD) or detailed finite element analysis (FEA) of a crash structure.
They require massive computing power and are slow to solve, making them unsuitable
for rapid iteration or real-time testing.21
● Low-Fidelity Models: These models use abstractions—such as "lumped parameters"
or linearized equations—to reduce computational cost. They are fast enough to run in
real-time, making them ideal for Hardware-in-the-Loop (HIL) testing where a physical
controller must interact with the model at millisecond intervals.21
The choice of fidelity is an economic and epistemological decision. Engineers must
"right-size" the model to the question at hand.23 Using a high-fidelity model to answer a
basic architectural question is a waste of resources (a violation of the "Occam's Razor"
principle in engineering). Conversely, using a low-fidelity model to validate a safety-critical
margin can lead to catastrophic failure if the abstractions mask critical non-linear
behaviors.24
The energy consumption of modeling is also becoming a relevant factor. Higher
fidelity models generally require larger datasets and more complex computations, leading
to a larger environmental footprint.22 This sustainability constraint adds another dimension
to the trade-off matrix, pushing the industry toward more efficient algorithms and "reduced
order modeling" techniques that seek to retain accuracy while slashing computational cost.
4. The Architecture of the Prototype: Physical vs.
Virtual
4.1 From CAD to Virtual Prototyping (VP)
Virtual Prototyping (VP) represents the evolution of Computer-Aided Design (CAD)
from a static drafting tool to a dynamic testing environment. VP allows engineers to
simulate the kinematics, dynamics, and control systems of a product in a purely digital
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environment before any physical material is cut.25 This shift is driven by the desire to "shift
left"—to move the validation and testing phases earlier in the design cycle where the cost of
making changes is orders of magnitude lower.
In the automotive industry, VP has become sophisticated enough to replace physical
prototypes for certain validation steps, such as component interference checking, basic
aerodynamic profiling, and ergonomic assessments.25 The "Zero Prototype" ambition—the
goal of going directly from simulation to production—is now a stated objective for some
manufacturers, particularly in the context of "Zero Prototypes Summits" hosted by
simulation companies like VI-grade.25 However, this ambition is tempered by the reality
that virtual models, no matter how advanced, are still approximations of reality.
4.2 The Digital Twin Paradigm
The concept of the Digital Twin (DT) goes beyond a mere simulation or a virtual
prototype. It implies a persistent, data-driven connection between the virtual model and a
specific physical asset. The IEEE and industrial literature provide a rigorous taxonomy for
this concept, distinguishing between three key states of the Digital Twin 27:
1. Digital Twin Prototype (DTP): This is the DT that exists before the physical product is
built. It contains the essential information to create the physical asset—the Bill of
Materials (BOM), the 3D CAD models, the control code, and the manufacturing
instructions. The DTP serves as the "hypothesis" of the product. It allows for virtual
testing and optimization of the design before capital is committed to production
tooling.
2. Digital Twin Instance (DTI): This is the specific virtual counterpart of an individual
physical asset, created once the product is manufactured. The DTI remains linked to its
physical twin throughout its lifecycle, receiving real-time data from sensors. It captures
the unique history of that specific unit—its operational hours, the stress loads it has
endured, and its maintenance history. The DTI is used for predictive maintenance and
condition monitoring.27
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3. Digital Twin Environment (DTE): This is the integrated platform where multiple DTIs
and DTPs operate, allowing for system-of-systems analysis.28
Leading technology providers like AWS (with IoT TwinMaker) and IBM have
developed platforms to operationalize these concepts. For instance, Carrier uses AWS to
create digital twins of its cold chain solutions, allowing for the rapid development and
testing of new refrigeration systems in a virtual environment.29 Similarly, INVISTA (a Koch
Industries subsidiary) uses digital twins of its manufacturing operations to give staff a
complete digital view of assets and data, enabling optimization of the production process.29
These implementations typically follow a workflow of data collection, virtual modeling, live
data integration, and finally, analysis and simulation.30
4.3 Hardware-in-the-Loop (HIL)
Bridging the gap between pure simulation (Virtual Prototyping) and physical testing
is Hardware-in-the-Loop (HIL) simulation. In an HIL setup, a physical controller (e.g., the
Engine Control Unit of a car or the flight computer of a drone) is connected to a real-time
simulator that models the behavior of the plant (the engine or the aircraft).31
HIL is a critical "integration" step in the V-Model. It allows engineers to validate the
software and control logic without risking the expensive physical plant. For example, testing
a battery management system (BMS) for an electric vehicle using HIL allows engineers to
simulate extreme conditions—such as a thermal runaway or a short circuit—that would be
dangerous and destructive to test with a real battery pack.33 The simulator tricks the physical
controller into believing it is connected to a real battery, allowing for rigorous testing of the
safety logic.
The utility of HIL extends to the power grid as well. Researchers at the DLR Grid
Lab use HIL to emulate grid networks, allowing them to test how renewable energy
inverters interact with the power system without risking stability of the actual grid.32 This
capability is essential for validating the "smart" components of the modern energy
90
infrastructure, where the behavior of software-defined inverters can be radically different
from traditional rotating generators.33
5. Systems Engineering Frameworks: The V-Model
and Validation
5.1 Structure of the V-Model
The management of complexity in modern engineering requires a structured
approach to ensure that the "scientific method" is applied consistently across thousands of
requirements and components. The V-Model is the standard framework for this process. It
bends the linear "Waterfall" model into a V-shape to emphasize the direct relationship
between the design phases (on the left) and the testing phases (on the right).34The V-Model
ensures that every "hypothesis" generated on the left side (e.g., "The system shall stop within
50 meters") has a corresponding "experiment" on the right side (e.g., "Brake Test A"). This
structure prevents the "Big Bang" testing approach, where all testing is left until the end,
often resulting in catastrophic discovery of deep-seated architectural flaws.36
5.2 Verification vs. Validation: The Epistemic Distinction
A critical distinction in systems engineering, often confused by laypeople, is the
difference between Verification and Validation (V&V). This distinction is vital for
understanding the scientific application of engineering.15
● Verification: "Are we building the product right?" This is a check against the
specifications. It asks whether the system meets the requirements defined in the
previous step. Verification is often an internal mathematical or logical check—for
example, verifying that the code compiles without errors or that the stress analysis
shows a safety factor of 2.0 as required.15
● Validation: "Are we building the right product?" This is a check against the user's needs
and the real world. It asks whether the system, even if it meets all specifications,
91
actually solves the problem it was intended to solve. Validation requires external
confirmation, often through physical prototyping or field trials. A product can be fully
verified (it meets every spec) but fail validation (users hate it or it doesn't work in the
actual environment).17
In the context of mathematical modeling, Code Verification ensures the equations
are solved correctly by the computer (the math is right), while Model Validation ensures the
equations accurately represent the physical reality (the physics is right).15 The ASME V&V
40 standard provides a risk-informed framework for assessing the credibility of
computational models in medical device development, illustrating how these concepts are
formalized in regulated industries.40
6. Economic Imperatives: The Cost of Change and ROI
6.1 The 1-10-100 Rule and the Cost of Change Curve
The drive toward mathematical modeling, virtual prototyping, and the rigorous
application of the V-Model is not merely an academic exercise; it is fundamentally driven
by economic imperatives. The Cost of Change curve, often formalized as the 1-10-100 Rule,
dictates that the cost of fixing a defect increases exponentially as the product moves through
its lifecycle.41
● Concept/Simulation Phase (1): Identifying an error here involves changing a line of
code or a parameter in a model. The cost is negligible—time and electricity.
● Design/Drafting Phase (10): Fixing an error here requires updating CAD drawings,
reviewing BOMs, and potentially re-doing some analysis. The cost is administrative
and labor-intensive.
● Production Phase (100): Fixing an error here means scrapping physical tooling,
discarding manufactured parts, and halting the production line. The cost is material
and operational.
● Field/Operation Phase (1,000+): Fixing an error after the product is released leads to
92
recalls, warranty claims, lawsuits, and brand damage. The cost can be existential for
the company.42
This economic reality forces engineers to "shift left"—to move the discovery of
defects as early in the process as possible. Virtual prototyping allows engineers to iterate
hundreds of times at the 1 level (Simulation), whereas physical prototyping often forces
discovery at the 100 level or higher.
6.2 The ROI of Virtualization: Boeing 777 and Beyond
The aerospace industry provides a potent example of the Return on Investment
(ROI) for virtual prototyping. The Boeing 777 was the first commercial aircraft designed
entirely using 3D CAD (CATIA), eliminating the need for a massive, full-scale physical "iron
bird" mock-up that was traditionally used to check for cable and piping interference.44
● The Savings: By trusting the geometric model (the Digital Twin Prototype), Boeing
reduced rework and "fit-up" errors during assembly. The model was the master
reference. This "digital pre-assembly" saved Boeing an estimated 90% of their pre-
production time for certain processes.45
● Data-Driven Assembly: Modern iterations of this process use data from past
assemblies (e.g., 10,076 laser gap measurements) to train PCA (Principal Component
Analysis) models. These models predict "shim gaps" for new aircraft, allowing parts to
be machined to the correct tolerance before they are brought to the assembly line.44 This
is a move from deterministic modeling (nominal CAD) to statistical modeling
(predictive assembly).
6.3 The Economics of Crash Testing
In the automotive sector, the cost differential between physical and virtual testing is
stark. A physical crash test destroys a prototype vehicle that can cost hundreds of thousands
of dollars to hand-build. A virtual crash test using Finite Element Analysis (FEA) costs only
the computing time and the software license.
93
● Scale: Virtual testing allows for the evaluation of thousands of impact angles and
speeds, whereas physical testing is limited to a handful of regulatory scenarios.46
● Accuracy: While virtual testing is cheaper, it must be accurate to be valuable. Studies
comparing physical vs. virtual prototypes have shown that while virtual models are
improving, physical models still hold the edge in certain domains. For example, a study
on ship bridge workstations found that physical models yielded 96% accuracy in
material estimation, while virtual models only achieved 76%.48
● Sensory Validation: The same study noted that physical prototypes elicited more
positive emotional responses (valence) from users. This highlights a limitation of the
virtual ROI calculation: it often fails to account for the "subjective" quality of the
design—how it feels to the user—which is difficult to quantify in a digital model.48
7. Deep Dive Case Studies: The Scientific Method in
Action
To fully understand the interplay of prototyping, modeling, and the scientific
method, we must examine their application in specific, high-stakes domains where the
failure of the "hypothesis" has profound consequences.
7.1 Case Study: High Energy Physics & Astrophysics (LHC & LIGO)
Fields dealing with the fundamental laws of nature often operate at scales—
subatomic or cosmic—where direct physical prototyping is impossible or prohibitively
expensive.
The Large Hadron Collider (LHC):
The ATLAS and CMS experiments at CERN rely on a complete digital replication of the
detectors to function.
● The Simulation: The Geant4 toolkit is used to simulate the interaction of particles with
the matter of the detector. This is a "Monte Carlo" simulation because particle decay is
94
probabilistic.19
● The Workflow: Before the detector was built, simulations predicted its response to
ensure the design was valid. Now, during operation, "real" data is compared against
"simulated" data. If the real data deviates from the simulation (which represents the
"Standard Model" background), it indicates the potential existence of new particles. The
simulation is the "control," and the physical detector is the "variable".20
● Resource Constraints: The simulation is so computationally expensive that it
consumes the majority of the Worldwide LHC Computing Grid's resources.
Researchers are now exploring "fast simulation" techniques (using Generative
Adversarial Networks - GANs) to approximate the Geant4 results, trading fidelity for
speed—a classic modeling trade-off.19
LIGO (Laser Interferometer Gravitational-Wave Observatory):
LIGO detects ripples in spacetime using mirrors suspended by multi-stage
pendulums. The isolation requirements are so extreme that physical prototyping is difficult
due to environmental noise.
● Mathematica vs. MATLAB: Researchers utilized Mathematica for the exact analytical
derivation of the equations of motion (symbolic modeling) to understand the physics
deeply. They then used MATLAB for numerical state-space analysis to design the
control systems.51 This shows the complementary use of different modeling tools for
different epistemological needs (understanding vs. control).
● The Prototype Refutation: A physical prototype of the "double pendulum" suspension
was built at MIT to verify the models. The mathematical model initially failed to predict
specific "cross-coupling" effects (e.g., longitudinal motion causing pitch rotation). The
physical experiment revealed "unmodeled dynamics"—specifically regarding the wire
attachment points and the flexure stiffness of the blades.52
● The Loop: The experimental data from the physical prototype was used to update the
mathematical model (adding "h" parameters for attachment point offsets). The refined
model was then sufficiently accurate to design the final "Quadruple Pendulum" for
95
Advanced LIGO without building a full-scale intermediate prototype for every
iteration.52
● Insight: This perfectly illustrates the scientific method in engineering. The Model was
the Hypothesis. The Physical Prototype was the Experiment. The discrepancy
(refutation) led to a Theory Update (better model), which enabled the Final Design.
7.2 Case Study: Biomedical Engineering (In Silico Trials & The
Ventilator Splitter)
Highly variable biological systems where "standardization" is impossible and
human life is at risk.
In Silico Clinical Trials (ISCT):
Testing medical devices (e.g., stents, heart valves) typically requires animal and
human trials, which are slow, expensive, and ethically complex. ISCT uses computational
cohorts—"virtual patients"—to test devices before they touch a human.54
● Regulatory Credibility: The FDA and the ASME V&V 40 standard have established a
framework for determining the "credibility" of a model. This is a legal and scientific
acceptance that a simulation can, in specific contexts, stand in for a human life.40
● The Digital Heart: Researchers are creating Digital Twins of the human heart,
modeling electrophysiology (calcium ion flow) and hemodynamics.56 A study showed
that a digital twin of the pulmonary artery could predict pressure with sufficient
accuracy to replace invasive catheterization in heart failure management.56 However,
these models must be calibrated carefully; assuming "well-mixed" compartments for
drugs can lead to significant errors if mass-transfer limitations in tissue are ignored.24
The Failure of the Ventilator Splitter:
During the COVID-19 pandemic, a task force attempted to design a ventilator
splitter to allow one machine to serve multiple patients.
● The Modeling Success (Apparent): Using Computational Fluid Dynamics (CFD),
96
engineers optimized the valve geometry to regulate airflow to 30%, 50%, 70%, etc. The
model predicted success.58
● The Prototyping Failure: When 3D-printed prototypes were tested at Memorial
Hospital with actual ventilators, the splitter failed to modify airflow enough to match
the CFD predictions.58
● The Cause: The boundary conditions of the CFD model likely did not match the
complex, dynamic pressure response of the actual ventilator hardware and the patient's
lung compliance.
● The Outcome: The project was closed. This is a stark reminder that a verified model
(mathematically correct) may not be a validated model (physically correct). The
physical prototype was the ultimate arbiter of truth.
8. Physical Prototyping in the Digital Age: Why We
Still Build
Despite the power of the Digital Twin and the success of "Zero Prototype" initiatives
in specific domains, physical prototyping remains an essential component of the scientific
method in engineering. The reasons for this are epistemological, sensory, and practical.
1. Unmodeled Dynamics (The "Unknown Unknowns"): Mathematical models only
contain the physics we know to include. Reality contains all physics, including the ones
we forgot, ignored, or don't strictly understand (e.g., the wire flexure in LIGO).52 The
physical prototype is the only "perfect" simulation of reality because it is reality. It
integrates all physical laws—thermal, electromagnetic, mechanical, chemical—
simultaneously and without abstraction.
2. Epistemic Opacity: As simulations become more complex (e.g., Deep Learning models
or massive multi-physics simulations), they become "black boxes." We know that they
predict the output, but we may not understand why. Physical testing provides a
"ground truth" that cuts through this algorithmic opacity.13
97
3. Human-Centric Validation (Sensory & Emotional): Haptic feedback, aesthetics, and
ergonomic comfort are difficult to simulate. As shown in the comparison study,
physical prototypes significantly outperform virtual ones in evaluating "emotional"
and "sensory" requirements.48 You cannot touch a digital steering wheel to feel the
grain of the leather or the damping of the switch.
4. Integration Chaos: Components may work individually in simulation, but when
physically connected, manufacturing tolerances, thermal expansion, and
electromagnetic interference can cause system-level failure. The physical prototype
exposes these interface failures.59
9. The Hybrid Epistemology
The integration of prototyping, mathematical modeling, and the scientific method
has created a new epistemology for engineering—one that is hybrid, recursive, and risk-
aware. We no longer view the "scientific method" as the exclusive preserve of natural
philosophers, nor "prototyping" as the exclusive domain of the workshop artisan. Instead,
we see a unified workflow where the digital and the physical are in constant dialogue.
1. Observation: Data is collected from the physical world (via sensors or historical data).
2. Hypothesis (Model/DTP): A mathematical representation is constructed to explain
this data and predict future behavior.
3. Experiment (Simulation): The model is stressed in the virtual domain (Monte Carlo,
FEA) to filter out poor designs (the 1 cost of change).
4. Refinement: The design is optimized based on virtual feedback.
5. Validation (Physical Prototype): The optimized design is instantiated in matter. This
is the critical test.
6. Refutation or Corroboration: If the physical prototype fails (like the ventilator splitter),
the model is refuted and must be updated (as in LIGO). If it succeeds (like the Boeing
777 fit-up), the design is validated.
7. Operation (Digital Twin Instance): The physical and virtual assets continue to exist in
98
a feedback loop, where the physical provides data and the virtual provides insight for
maintenance and optimization.
In this framework, the "failure" of a prototype is not an error; it is a necessary step in
the calibration of our understanding. It is the moment where the map is forced to align with
the territory. As we move toward autonomous experimentation and AI-driven design, this
grounding in the scientific method—the commitment to validating the abstract against the
concrete—will remain the essential safeguard against the hubris of the simulation. The
engineer of the future is neither just a builder nor just a mathematician, but an
experimentalist operating at the interface of the virtual and the real, using every tool
available to rigorously test the hypothesis of the artifact.
99
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106
Conclusion
At the end of this review of the general methodological proposals for
research in Computer Science, it is evident that the historical dichotomy between
theory and practice, or between engineering and science, is an increasingly
blurred and less useful distinction. We have found that rigorous computational
research does not lie in the exclusive choice of a single tool, but in the intelligent
orchestration of prototyping, mathematical modeling, and the scientic method.
The main lesson that emerges from the previous chapters is that these
three methodologies form a continuous, virtuous and necessary feedback loop:
- From Model to Prototype: We saw that mathematical modeling provides us
with the formal language to dene the problem and the theoretical
guarantees of the solution. However, a model without implementation
runs the risk of remaining a sterile abstraction.
- From Prototype to Evidence: Prototyping acts as the bridge to reality,
exposing our ideas to physical, hardware, and usability limitations. But a
prototype without evaluation is simply a technical demonstration, not
science.
- From Evidence to Knowledge: This is where the scientic method closes the
circle. Through controlled experimentation and statistical analysis, we
transform the observed behavior of the prototype into data, and that data
into validated knowledge that renes, in turn, our initial mathematical
models.
This methodological integration is more urgent today than ever. We are
facing an era where software systems (such as deep neural networks or planet-
107
scale distributed systems) exhibit emergent behaviors that cannot always be fully
deduced from rst mathematical principles.
In this context, the researcher must be able to operate as a mathematician
to formalize uncertainty, as an engineer to build the testing tools, and as a natural
scientist to observe and explain digital phenomena that often operate as "black
boxes." The renunciation of any of these three pillars weakens the ability of the
discipline to explain the why and how of technological advances.
Research in Computer Science has ceased to be a purely artisanal activity
to become a mature science with its own epistemological standards. This book
has sought to equip the researcher with the condence to navigate these
standards.
We hope that the reader will close this volume not only with new
techniques in his arsenal, but with a new perspective: the conviction that the best
research is the one that dares to build to understand, that calculates to predict
and that experiments to verify. The future of computing will depend on
researchers who are not afraid to get their hands dirty with code or tire their
minds with abstraction, understanding that true scientic progress lies at that
intersection.
108
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This edition of "General methodological proposals for Computer Science
research: Prototyping, mathematical modeling, and the scientific method", was
completed in the city of Colonia del Sacramento in the Eastern Republic
of Uruguay on October 2, 2025
ISBN 978-9915-698-39-7