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REVIEW ARTICLE
A Computational Model ofSchool Achievement
BrendanA.Schuetze1
Accepted: 16 January 2024 / Published online: 8 February 2024
© The Author(s) 2024
Abstract
The computational model of school achievement represents a novel approach to the-
orizing school achievement, conceptualizing educational interventions as modifica-
tions to students’ learning curves. By modeling the process and products of educa-
tional achievement simultaneously, this tool addresses several unresolved questions
in educational psychology through computational modeling. For example, prior
research has highlighted perplexing inconsistencies in the relationship between time
spent on task and academic achievement. The present simulation reveals that even
under the assumption that time-on-task always positively contributes to achieve-
ment, the correlations between time-on-task and achievement can vary substantially
across different contexts and, in some cases, may even be negative. Analysis of the
correlation between prior knowledge and knowledge gains uncovers similar patterns.
The computational model of school achievement presents a framework, bolstered
through simulation, enabling researchers to formalize their assumptions, address
ongoing debates, and design tailored interventions that consider both the school
environment and individual student contexts.
Keywords Self-regulated learning· Heterogeneity· Formal model· Simulation
“All learning, whether done in school or elsewhere, requires time.”
Benjamin Bloom (1974, p. 682).
In the education sciences, there is a rich tapestry of theories (see Hattie et al.,
2020; Murphy & Alexander, 2000; Urhahne & Wijnia, 2023). Many of these theo-
ries can be characterized as what Davis etal. (2007) describe as “simple theories.”
Davis et al. define simple theories as theories that encompass a “few constructs
and relatedpropositions” (p. 482), each of which has empirical or analytic backing
but often lacks a certain level of necessary detail to make strong predictions from
* Brendan A. Schuetze
brendan.schuetze@gmail.com
1 Department ofEducational Science, University ofPotsdam, Potsdam, Germany
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Educational Psychology Review (2024) 36:18
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case-to-case. It is important to emphasize that “simple theories” should not be mis-
construed as lacking value. Rather, this terminology emphasizes that these theories
are usually described in writing (not mathematical equations) or depicted in flow-
chart style graphics. And though these theories are “in all likelihood correct” (Davis
etal., 2007, p. 482), there is often room for increasing precision linking constructs
more comprehensively, and refining their theoretical logic.
Compared to simple theory, formal modeling and simulation methods aim to elu-
cidate all relationships and variable distributions using mathematical equations, usu-
ally instantiated in a programming language. This approach is also largely in line
with what has been called “model-centric” science. Here, the emphasis is less on
collecting individual results, but rather on conceptualizing the goal of research as
building progressively better models of the phenomena of interest (Devezer & Buz-
bas, 2023). While there are some notable contributions in this direction within edu-
cational sciences (e.g., Doroudi etal., 2019; Frank & Liu, 2018; Hull etal., 1940),
such modeling traditions are more frequently observed in neighboring disciplines
like cognitive science (e.g., Hintzman, 1991; Navarro, 2021).
The potential upsides of the formal modeling approach have long been recognized.
As Bjork (1973) notes, there are four main reasons for a theorist to use mathemati-
cal models: (a) mathematical models are “more readily falsifiable”; (b) require theo-
retical precision; (c) illuminate the consequences of the different model assumptions;
and (d) improve our data analysis by generating rich data (pp. 428–429). Bjork also
argues mathematical models might be more practically useful, noting applications
in intelligent tutoring and medical decision-making. Current proponents of mathe-
matical models generally offer similar reasons for their use (e.g., Devezer & Buzbas,
2023; Navarro, 2021; van Rooij & Blokpoel, 2020).
Practically, simple or verbal theories allow certain details to remain implicit in
the mind of the researcher. In the context of simple theory, indicating that an inter-
vention enhances student outcomes might suffice without specifying the underlying
behavioral or cognitive mechanism (see Yan & Schuetze, 2023). However, when
employing formal computational models, there is a need to enumerate the exact
relationships linking input and output factors. Navarro (2021) points out one of the
strengths of formal modeling, stating that it “tells us which measurement problems
we need to solve” (p. 710). What’s more, the transparency inherent in formal mod-
eling encourages constructive dialogues. If a researcher disagrees with a model’s
assumptions, they can implement, simulate, and compare between alternative mod-
eling assumptions (Devezer & Buzbas, 2023).
Building on these recent developments in formal theory, the present paper intro-
duces the Computational Model of School Achievement (CoMSA), which attempts
to act as a unifying model between cognitive, metacognitive, and motivational the-
ory. Specifically, the CoMSA is a formal (mathematical) model of the relationship
between learning over time and exam achievement. The CoMSA simulates the tra-
jectories of individual students over time, allowing for the analysis of learning as
both a process and as a product. It also allows for the analysis of student achieve-
ment at both the individual-student and group-classroom levels. More broadly, this
model introduces an alternate approach to developing theory that relies on compu-
tational approaches, not as commonly seen in educational psychology.The use of
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Educational Psychology Review (2024) 36:18 Page 3 of 38 18
computational modeling allows the CoMSA to predict complex interactions between
time-on-task, motivation, learning strategies, and achievement that cannot be easily
captured in a simple verbal model, thus improving educational researchers’ abili-
ties to make nuanced predictions concerning where, when, and why a student may
or may not benefit from a particular cognitive, motivational, or metacognitive
intervention.
Theoretical Background
The present formal mathematical theory of goal-oriented exam achievement builds
upon the foundational works of Bloom (1968, 1976), Carroll (1962, 1963, 1977,
1989), Glaser (1982), and Gettinger (1984a, 1984b), on what have been collectively
known as “models of school learning” (Harnischfeger & Wiley, 1978). These theoret-
ical and sometimes computational models attempt to relate teacher and student char-
acteristics (aptitudes, motivations, and instructional quality) to learning outcomes.
Similarities can also be seen with the performance-resource functions analyzed
by Norman and Bobrow (1975), as both approaches focus on the tradeoff between
invested effort and achievement Additionally, the CoMSA shares theoretical affini-
ties with the work of Dumas and colleagues on dynamic measurement (Dumas &
McNeish, 2017; Dumas et al., 2020), as both approaches focus on modeling educa-
tional achievement over time. Modeling approaches put forth by Ackerman and col-
leagues also have focused on the curvilinear relationship between time investment
and achievement, but in the context of skill-learning, automaticity, and reaction times
(see Ackerman, 1987; Kanfer & Ackerman, 1989).
Nevertheless, of all these models, Carroll’s (1962, 1963) model of school learn-
ing is the closest predecessor to the CoMSA. Carroll’s model takes five variables as
input: aptitude, ability to understand instruction, perseverance (motivation), oppor-
tunity (how much time is available), and quality of instruction. Carroll’s model
emphasized the idea that different students need different amounts of time to reach
satisfactory performance on the same task. Some students will need more time, and
some will need less time—even under ideal instructional conditions. However, the
quality of instruction can also vary, with poorer quality instruction leading to slower
uptake. Carroll also notes that students have a certain level of motivation (time will-
ing to spend studying) and a certain amount of time available to them to study.
Carroll (1963) formalizes the relationship between these five variables through
the following equation (n.p.):
The numerator is determined by computing the minimum value of the following
variables: opportunity, perseverance, and time needed to learn (which is modulated
up/down by instructional quality). This means that if motivation or time available
to learn is less than the time needed to learn, the degree of learning will be lower
than the student’s potential. If perseverance and time available are greater than the
Degree of Learning =
f
(
time act ually spent
time needed
)
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Educational Psychology Review (2024) 36:18
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time needed to learn, then potential will be reached. In other words, the degree of
learning is a fraction with the actual time invested on the top and the time necessary
(potential) on the bottom.
Though Carroll’s model provides a strong conceptual foundation for understand-
ing learning, it is notoriously complex to understand (Harnischfeger & Wiley, 1978).
Additionally, it does not incorporate educational psychological theorizing developed
since the 1960s, leaving room for improvement. For example, Carroll’s model of school
learning assumes a linear relationship between time spent and educational achievement
(until the ceiling is reached), but Son and Sethi (2010) argue that curvilinear learning
curves are more likely than not. Nor does his model take into account individual and
contextual differences in floor performance and pre-requisite knowledge.
Another aspect that is missing from Carroll’s model is the metacognitive moni-
toring aspect of self-regulated learning. The outcome measure of Carroll’s (1963)
model is “degree of learning,” but learning is rarely achieved independent of a goal
(see Ariel & Dunlosky, 2013; Deci & Ryan, 2000). School students are almost
always tasked with learning for a specific exam, assignment, or other situation in
which their knowledge will be applied. But metacognitive monitoring of progress
towards these goals is imperfect. Heuristics are often used to determine if complex
material has been adequately learned (Undorf, 2020). This means that learners have
an imperfect understanding of their knowledge state. Accordingly, the CoMSA
attempts to account for the metacognitive (in)accuracy of learners’ metacognitive
judgments.
Purpose andFunction oftheCoMSA
The essential purpose of the CoMSA is to predict exam achievement based on stu-
dent and contextual characteristics, bringing together motivational, cognitive, and
metacognitive research. The CoMSA incorporates much of the same foundational
ideas as Carroll’s (1962) and Bloom’s (1976) models but refines them and refocuses
around the notion of the learning curve, as opposed to the learning ratio that defines
Carroll’s model of school learning. In addition to offering a more intuitive graphical
orientation to the model, this approach better allows for the disentangling of prior
knowledge, learning rate, and floor/ceiling effects. The present work also extends
previous models of school learning to incorporate newer theorizing about the impact
of metacognitive monitoring and agenda-based regulation as they relate to school
achievement. For ease of interpretation, the CoMSA uses test grades as the outcome
of interest; this outcome metric also allows for easier linkage to empirical results
and individual goals operationalized as norms of study. Because performance-
related goals are often driving student behavior (Elliot etal., 2011), it is crucial to
include them into the model as key inputs. Lastly, the computational resources avail-
able now, 60years later, allow us to conduct broader simulations and analyses of the
implications of this family of models.
The CoMSA offers two primary modes of interaction, graphical and mathemati-
cal. The graphical mode visualizes educational interventions as manipulations to the
learning curve (see the progression of figures throughout this paper for example).
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Here, we might ask, “how would the learning curve change if we modify the learn-
er’s strategy use, motivational beliefs, or metacognitive control?” Such visualization
exercises can serve as beneficial tools for planning and conceptualizing modifica-
tions to the learning environment.
In addition to its visual component, the CoMSA can also be instantiated as a
formal mathematical model, manipulated, and evaluated through computer simula-
tion. In this computational modeling approach, assumptions are explicitly encoded
in computer code, enabling modification, theory testing, and novel predictions
(van Rooij & Blokpoel, 2020). Furthermore, data can be simulated from formal
models to better understand dynamics over time that would be difficult to under-
stand using conventional methods. CoMSA can simulate student-level and also
group/classroom-level data. Here, classroom-level data is calculated by aggregat-
ing over data simulated at the individual level.
The following analyses do not directly test the ability of the CoMSA to model
individual-level data. Rather, we assume that the S-shaped curve is generally accept-
able and analyze the implications that follow from this assumption. This assumption
is justified; see the work of the strong modeling tradition of analyzing learning curves
using logistic or S-shaped curves; see Koedinger etal. (2023) and Murre (2014) and
the broader item response theory approach (Cai etal., 2016). These studies all show
success using S-shaped or logistic models to model student achievement. In the fol-
lowing set of analyses, the CoMSA is utilized to simulate and analyze classroom data
from two meta-analytic sources. The first analysis looks at the correlation between
pre- and post-test gains (Simonsmeier etal., 2022). The second analysis investigates
the correlation between time-on-task and academic achievement (Godwin et al.,
2021). The subsequent pages detail the reasoning behind the structure and assump-
tions of the CoMSA.
Model Structure
The central unit of analysis in the CoMSA is the learning curve. Following a con-
vention that dates at least back to Thorndike (1916), the learning curve maps the
relationship between time spent studying and task performance. In this case, the task
of interest is an examination. See Fig.1 for an example. By manipulating this curve,
we can think about different learners and contexts, with varying levels of test diffi-
culty, motivation, and learning rates.
Note that the learning curve depicted in Fig. 1 should be understood as relat-
ing the number of hours spent studying to anticipated achievement for an individual
student in a specific exam context. For example, “If student A with characteristics
X and Y insituation Z were to study 4h, they would be expected to achieve a grade
of 30%, but if the same student were to study 6h, they might be expected to obtain
a grade of 80% on the exam.” It also should be understood that the metric on the
X-axis is measured in hours of engagement. Spending time simply sitting with the
material while procrastinating on one’s phone would not increment the student along
their learning curve (see Carroll, 1963; Karweit & Slavin, 1981). In other words, the
learning curve shown in Fig.1 attempts to answer all things equal—for this student
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in this context—what would the trajectory of their exam grade look like as a func-
tion of time-on-task?
In the present modeling exercise, the learning curve is assumed to take on the shape
of a logistic curve. In line with previous work (e.g., Ackerman, 1987; Nijenkamp
etal., 2016, 2022; Son & Sethi, 2010), the logistic curve is a natural choice for the
present modeling exercise as it fits within the traditional 0–100 percent scale of grad-
ing systems and can flexibly capture a wide array of learning situations with its three
parameters1: intercept, slope, and inflection point. With these three parameters, logis-
tic curves can be made to look S-shaped, like straight lines, or even power curves if
they are properly shifted to the left or right.
Here, readers more familiar with the cognitive psychology literature may ask
whether it would be more accurate to model learning as an exponential or power
function. Indeed, a variety of different mathematical equations have been used to
model learning curves (Stepanov & Abramson, 2008), but for the present purposes,
the logistic curve strikes a useful balance of parsimony and analytic power. As
Murre (2014) shows, the emphasis on power or exponential learning curves may
be an artifact of cognitive psychology’s focus on simple word-pair learning experi-
ments. Rather, complex tasks, such as learning to juggle (Qiao, 2021) or learning a
foreign language (Daller etal., 2013), are most likely to show an S-shaped learning
curve (Son & Sethi, 2010). Even the learning of nonsense syllables can exhibit an
S-shaped curve when the list length is sufficiently large, as shown by Hull etal.’s
(1940) data from the learning of 205 nonsense syllables over multiple presentations
(see Figs.26 and 27 of Hull etal., 1940).
Conceptually, as shown in Fig.1, the flat initial part of the learning curve may be
thought of as any necessary time to learn pre-requisite material that is not being tested
but is necessary to understand the tested material. In other words, there is a possibility
Fig. 1 Phases of the logistic learning curve
1 More complex extensions to the logistic equation exist, but for the purpose of the present investigation,
the curve is limited to the three-parameter logistic (3PL) model commonly used in psychometrics.
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Educational Psychology Review (2024) 36:18 Page 7 of 38 18
of a warm-up period before improvements may be seen. Following the warm-up period,
there is a period of increased learning, where the slope steepens. And finally—similar
to the assumption of Karweit and Slavin (1981)—there is a period of mastery where
learning may still occur but may not be captured by in-class exams—which are not gen-
erally psychometrically validated for large ranges of knowledge (Slepkov etal., 2021).
Research on item response theory (Cai etal., 2016) and additive factors model
(and related models such as performance factors analysis; Koedinger etal., 2023;
Pavlik etal., 2021) have shown that logistic curves are able to accurately model the
acquisition of knowledge by individual students in real-world educational contexts.
In this way, the student-level modeling assumptions have been quite well validated
in many empirical papers (Pelánek, 2017). But less is known about how these logis-
tic longitudinal dynamics over time modulate the relationship between inputs (moti-
vation, learning strategies, and interventions) and outputs (achievement, time spent)
at the aggregate level, which is typically how educational interventions are evaluated
(e.g., through t-tests and correlations across groups of students). We return to these
open questions with the analyses presented later in this paper.
Modeling Individual andContextual Dierences withtheCoMSA
The flexibility of the logistic learning curve allows for the modeling of learning
strategy, metacognitive, and motivational interventions on populations that vary on
their baseline cognitive, contextual, and motivational variables.
Modeling Differences inLearning Rate
Using the logistic learning curve as a base, we can then think about learners as they
vary in terms of learning contexts, strategies, motivation, and metacognition. Differences
in learning strategies and contexts can be thought of as differences in the shape of the
learning curve, in terms of slope, intercept, and inflection point. For example, a medi-
cal student studying for a comprehensive licensing exam might need to study for hun-
dreds of hours to achieve a passing score. This context would be represented by a much
shallower learning curve than, say for example, a fifth-grade student studying for a ten-
word vocabulary quiz, which may only require an hour of study to master. The vocabu-
lary quiz might look more similar to the red curve in Fig.2, while the medical student’s
licensing exam might look like an even more extreme version of the blue curve.
It is critical to remember that the slope and all parameters of the learning curve
are not directly mappable to any one contextual or psychological variable. As
with Lewin’s (1936) equation, the assumption here is that learning over time is a
function of both the student and their environment (and this function is not nec-
essarily a simple or additive combination of a student and their context). Even
within a classroom of medical students all studying for the same exam, some
students may learn more efficiently than others due to individual differences
(McDermott & Zerr, 2019; Zerr et al., 2018) or differences in study strategies
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Educational Psychology Review (2024) 36:18
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(Carpenter et al., 2022; Carroll, 1977). Conversely, the same student may learn
more or less quickly in different learning contexts (see Fig.3).
While the following sets of figures contained in this paper isolate transforma-
tions to the learning curve to ensure proper communication, in real-world con-
texts, these transformations could always happen concurrently with one-another.
In other words, one classroom may have learners with a high slope on average,
but also low motivation, and also high metacognitive monitoring accuracy, while
another classroom may show low learning slopes, low motivation, and low meta-
cognitive monitoring accuracy. None of these variables is necessarily assumed to
correlate with one-another under the present simulation, but this assumption can
be relaxed as needed. Indeed, analyses presented later on in this paper show that
at the very least, it appears that we should assume that prior knowledge and moti-
vation are at least somewhat correlated within classrooms.
Modeling Differences inPrior Knowledge
Learning curves do not always need to start at zero. Differences in interest, teacher
effectiveness, or previous experiences may lead to students starting with different
levels of prior knowledge. In the context of the CoMSA, prior knowledge is opera-
tionalized as the predicted output of the model when time spent equals zero. Two
Fig. 2 Three learning curves showing different learning rates. Note: the dashed line indicates per-
formance when the number of hours spent studying is equal to eight. The vertical separation between
the curves is not stable over the amount of time spent studying, meaning that shifting the curve’s slope
would have heterogeneous effects dependent on the learner’s number of hours spent studying
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Educational Psychology Review (2024) 36:18 Page 9 of 38 18
inputs to the model shift this prior knowledge value: the floor (intercept) or inflec-
tion point of the learning curve.
Shifting theFloor
The floor or intercept of the learning curve (Logis c) can be modulated up and down
due to individual and contextual factors (see Fig. 4).2 Examination policies that
reduce the ability to score below a certain threshold are one simple way of influenc-
ing the floor of the learning curve. For example, it is difficult to score below 25% on
traditional four-choice multiple-choice exams. Similarly, some teachers adopt class-
room policies of assigning 50 percent as the lowest possible score on assignments,
rather than zero. Additionally, the floor may be impacted by teacher or instructional
effectiveness. The more effective in-class learning is, the more knowledge a student
may start out with when they begin self-regulated studying.
Fig. 3 Illustration of individual student learning over time. Note: here, ten learners have been simulated
from the CoMSA with mean slope = 1 (SD = 0.2), mean inflection point = 4 (SD = 2), mean intercept = 0.47
(alpha = 3.5, beta = 4), norm of study = 100 (constant, no metacognitive error or bias), and mean maxi-
mum willingness to study = 5 h (SD = 1.5). Each student exhibits their own trajectory, with some students
requiring more or less practice to reach the same level of proficiency
2 Throughout this paper, the terminology Logis a, c, and d is used to refer to the inputs of the logistic
equation as defined in the psych package for R (Revelle, 2023). Thus, in psychometric terms, a refers to
the slope, c to the guessing parameter, and d to the discrimination parameter.
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An extreme example of shifting the floor of the learning curve would be to con-
sider giving a college student an examination designed for a middle school student.
The college student will likely start at a relatively high proficiency for simple alge-
bra quizzes, meaning that the floor of their learning curve should be higher than
that of a middle school student facing novel mathematics problems. Altogether, it
is critical to remember that the floor of the learning curve varies from student to
student and from context to context. Regardless of the upstream influences, the floor
of the learning curve most directly reflects the grade a student would achieve if they
were not to study at all—with one caveat, the relationship between floor and prior
knowledge also depends on the inflection point. Shifting the inflection point shifts
the entire learning curve to the left or right. Thus, changes to the inflection point can
lead to modifications of the relationship between the floor of the curve and the stu-
dent’s predicted achievement when time spent equals zero.
Shifting theInflection Point
The inflection point of the learning curve determines the left versus right shift of
the learning curve. In terms of educational theory, this component of the CoMSA
relates to the minimum amount of time needed for a student to study before they
see improvement in their capabilities as measured by the classroom examination of
interest. For example, classroom exams may require students to know some level of
prerequisite knowledge before they can begin to learn the content being tested in the
class. Bloom (1976) calls these necessary prerequisite skills “cognitive entry behav-
iors,” because they are necessary to enter the learning phase of subsequent skills. If
all the cognitive entry behaviors have been mastered, a student’s inflection point will
be closer to zero. The more prerequisite behaviors needed to master before learning
the focal material, the further the curve gets shifted to the right (away from zero).
Fig. 4 Two approaches to modeling prior knowledge. Note: on the left are two curves with the same
inflection point, but different starting floor performance. On the right are two curves with the same floor
performance, but different inflection points
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As a more concrete example, for a native speaker of Ukrainian, a Ukrainian vocabu-
lary test may have an early inflection point, requiring very little additional background
knowledge. On the other hand, for someone learning Ukrainian as a second language,
the same vocabulary test may have a much later inflection point. If the new learner
does not know how Cyrillic letters match to their respective phonemes, they will have
to gain a basic understanding of this knowledge before they will be able to easily
acquire new vocabulary. In other words, the more prerequisite knowledge required to
do well on an examination, the further the curve will be shifted to the right.
Modeling Differences inMotivation
In the model, motivational variables determine how far each student progresses
along their learning curve, but this is not to say that all motivational interventions
only affect time-on-task. They may also have other behavioral effects on students’
slope or metacognitive monitoring biases. Motivational variables are instantiated in
two ways in the CoMSA. Students can be limited either by their goals (in terms of
achievement) or by the amount of time they are willing to spend studying. As with
Carroll’s (1962, 1963) model, there is a circuit breaker mechanism, meaning that
once either one of the two criteria is met, a student will disengage from studying.
The first criterion is called the norm of study. Under this assumption, students are
thought to be working towards individualized performance goals (i.e., some students
may be seeking an 80% on this exam, while another student will only be happy with
a 95%). The norm of study criterion derives from models of study time allocation,
going back to at least the work of Le Ny etal. (1972) and adopted in subsequent the-
orizing by Carver and Scheier (1982, 1990), Thiede and Dunlosky’s (1999) discrep-
ancy reduction model, and Ackerman’s (2014) diminishing criterion model. In these
models, differences in motivation are almost entirely represented by differences in
norms of study.
However, each student also has a second criterion related to the maximum time
they are willing to spend studying (see Fig.5). In this way, students are self-regu-
lated towards their goals—but only to an extent. They direct their efforts towards
achieving a certain grade until it becomes too costly to obtain time-wise (or the time
has run out). Intuitively, the performance goal circuit breaker is necessary because it
seems unlikely students would continue studying even after they know themselves to
have mastered the material. The maximum time spent criterion—what Carroll called
“perseverance”—is necessary because it is commonplace to observe students who
not only spend much time studying, but also underperform the grades they purport
to seek (Carroll, 1962, 1963).
An interesting implication of the CoMSA is that not every hour of study time
is equal. As shown in Fig.1, if we assume a logistic curve relating time spent to
achievement, some hours of study time will be more productive than others (Son
& Sethi, 2010). Depending on the warm-up period (see earlier section on the
inflection point), where the curve is relatively flat, sometimes the most efficient
hours of study will come early in the study session. In other cases, the most
beneficial hours of study will require many hours of studying of pre-requisites
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before they can be reached. This has important ramifications for educational
interventions that seek to increase the amount of time dedicated to education
(i.e., motivational interventions), as some students will benefit greatly from
spending an additional hour of studying, while others are not anticipated to get
much benefit at all from an additional hour of study (e.g., those at the ceiling or
floor of the test).
Modeling Differences inMetacognition
Under the current model, motivation and cognition do not exist in isolation.
Rather, in line with existing theorizing, metacognition acts as a “bridge between
decision making and memory” and also “learning and motivation” (Nelson &
Narens, 1994, p. 1). Metacognitive factors most likely affect the CoMSA in a
multi-faceted way through both affecting students’ perceptions of their achieve-
ment and also their study behavior. Here, the impacts of metacognitive monitor-
ing, beliefs, and control are discussed as they relate to the CoMSA.
Metacognitive Monitoring
As shown in Fig. 6, the model assumes that metacognitive monitoring faculties
determine how accurately students are able to anticipate their exam performance. In
the CoMSA, metacognitive monitoring is modulated by two different factors: bias,
which is also known as “calibration” in the metacognition literature, and precision
(“resolution” or “sensitivity”; Fleming & Lau, 2014). Here, these terms refer to the
Fig. 5 Modeling differences in motivation on the learning curve. Note: motivation determines how long
students will progress upon their learning curves. Highly motivated students will be willing to spend
more time studying and thus are expected to achieve higher grades, all else equal. Here, the norm of
study is not depicted (i.e., assumed to equal 100%)
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Educational Psychology Review (2024) 36:18 Page 13 of 38 18
metacognitive monitoring of the entire test, that is, the students’ judgments about
the mastery of the entire corpus of tested content. However, students also make
metacognitive judgments of learning concerning individual items, which will have
more multifaceted effects on the model through several different inputs. For exam-
ple, better monitoring at the item-level could lead to faster acquisition via targeted
study of materials that are more likely to be tested (see Schuetze & Yan, 2022). The
bias and precision parameters defined in the following paragraphs represent just two
potential knobs which monitoring abilities could impact.
Bias refers to the average distance between the students’ metacognitive judg-
ment of their anticipated test performance and their true state of knowledge. In
other words, it represents their average over- or underconfidence concerning their
upcoming test. If bias is high, it may mean that a student will study too long or stop
studying too early relative to their goals. An overconfident student will overestimate
their current state of knowledge, assess that their goal has been achieved, and thus
discontinue studying. Conversely, an underconfident or perfectionist student may
continue studying far past the time necessary to master the material. Though such
perfectionistic striving can be adaptive and actually lead to increased achievement
outcomes (Rice etal., 2013; Stoeber & Rambow, 2007), this increased performance
in absolute terms may come at the cost of using study time relatively inefficiently by
studying beyond the time necessary to achieve the student’s norm of study.
As opposed to bias, precision refers to the relative consistency of these judg-
ments—how noisy they are, for example, whether a student is always making judg-
ments relative to the ground truth in a similarly underconfident, accurate, or over-
confident manner. Consistency of metacognition is instantiated as the error variance
Fig. 6 Modeling differences in metacognitive monitoring on the learning curve. Note: gray dots repre-
sent metacognitive judgments, while the black curve represents the ground truth. Bias refers to the aver-
age error of the judgments, while the precision refers to the deviation of the judgments around the curve
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around the judgment of learning relative to the ground truth (see Fig. 6). Practi-
cally, low precision (high error variance) might mean that a student has perfectly
calibrated metacognitive monitoring on average, but each individual judgment does
not afford much information about the state of their knowledge.
Metacognitive Beliefs andControl
In addition to metacognitive monitoring, researchers often are interested in the
effects of metacognitive knowledge, beliefs, and control strategies on self-regulated
learning. These facets of metacognition may be best instantiated on the CoMSA as
changes to the slope of the model. For example, students need to use metacognitive
knowledge to adapt their choice of study strategies to the learning context. If a stu-
dent does not know about retrieval practice, they may not use it for this reason, but
even if they know about the benefits of retrieval practice, other metacognitive and
motivational beliefs may lead them to failing to adopt this beneficial strategy (Rea
etal., 2022). Although there may be a metacognitive knowledge component to the
choice of study strategy, the downstream effect would be to change the slope of the
learning curve, as retrieval practice is more beneficial per unit of time spent study-
ing than more passive controls such as rereading. Similar analysis can be applied to
metacognitive control at the item level (which is distinct from the test-level bias and
precision parameters discussed above). Knowing which items to prioritize is one
way of increasing academic achievement (Schuetze & Yan, 2022; Son & Kornell,
2008), but ultimately, this will lead to increased slope also.
How Do theComponents oftheModel Come Together?
As shown in Fig.7, there are three general steps to go from inputs to the CoMSA to
predicted achievement outcomes. First, each student’s learning curve is established.
Establishing the learning curve means determining the slope, intercept, and inflec-
tion point of the curve. Steeper slopes indicate faster learning than shallower slopes.
Higher intercepts indicate either that the exam has a functional floor on how low a
student can achieve (e.g., due to guessing on a multiple-choice examination), or that
the student is starting out with some existing level of prior knowledge. The inflec-
tion point determines how much prerequisite knowledge must be learned before
improvement on the tested material will be reflected in the student’s grade.
In the second stage, after establishing the functional form of the learning curve, the
curve is modulated with regard to the student’s metacognitive bias and precision. This
modulation of the curve is most important when it comes to the third stage, which
determines how long the student is willing to study. Under the present model, students
discontinue studying due to one of two reasons. Either (1) they have assessed that they
have reached their goal in terms of anticipated achievement (norm of study). These
assessments are where metacognitive monitoring comes into play. Students who are
over- or underconfident will assess their competence inaccurately relative to their true
performance. Or (2) they have run out of willingness to spend any more time (or they
simply do not have any more time). Together, the norm and time constraints act as
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Fig. 7 CoMSA and simulation overview
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circuit breakers. Students stop studying whenever the first constraint of the two con-
straints is tripped. Once the shape of the student’s learning curve and their time spent
studying has been established, a grade can be predicted from the CoMSA.
Note that these three steps are laid out in a strict sequence for the purpose of mak-
ing the present modeling and simulation study tractable. In real-world educational
settings, when students are actively engaging in self-regulated learning, there are
undoubtedly contingencies between each of the steps. For example, metacognitive
information is known to alter learners’ study choices in real time (Thiede & Dunlosky,
1999). In this case, step 2 would lead into step 1. Theoretically, the CoMSA can be
extended to deal with such complexities, but for the sake of parsimony in the present
study, all inputs to the model are assumed to be set once at the beginning of the study
session(s) and do not change over the course learning period.
How Does This Model Integrate Contemporary Work onPsychological
Constructs?
It would be understandable for the approach outlined in this paper to be somewhat
counterintuitive to those with deep backgrounds in the contemporary theory and
practices of educational psychology. As mentioned in the introduction, contempo-
rary educational psychology theory often centers on verbal theories, describing how
psychological constructs relate to educational achievement. For example, dominant
theories of motivation tend to focus on the links between a belief or construct (e.g.,
expectancies, values, and self-efficacy) and an outcome of interest (e.g., achievement,
persistence in school, or well-being). Though this theoretical approach is common
(see Hattie et al., 2020), it is not infallible and potentially crowds out alternative
modes of theorizing in educational psychology (see Kaplan, 2023; Murayama, 2023).
For this reason, there have been few references to these sorts of constructs in the
previous pages. This is not an oversight. Rather, the CoMSA acts as an interface
between traditional construct-focused educational psychology and the achievement
outcomes of interest (see Fig.8). In other words, the CoMSA models the downstream
behavioral effects of cognition, motivation, and metacognition. Most contemporary
research focuses on the beliefs/motivations/cognitions → outcome link without speci-
fying how exactly intermediate or learner behavior changes (see Yan & Schuetze,
2023). The CoMSA focuses on the behavior → outcome link but does not make theo-
retical commitments regarding how these changes to the shape of the learning curve
or motivation of the learner occur. Rather, the model asks “assuming these changes do
occur, what should researchers expect to happen?”.
A common area of research known as utility value interventions can act as a case
study for understanding where the CoMSA fits with contemporary motivation schol-
arship. Utility-value interventions encourage students to find the information they
are learning in school to be more valuable to their long-term success. For example,
Harackiewicz etal. (2016) engaged students in a task that involved writing an essay
that emphasized the personal relevance of their introductory biology coursework.
The theory here is that students in the utility value condition will see more value in
their schoolwork and thus achieve better academic outcomes.
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Educational Psychology Review (2024) 36:18 Page 17 of 38 18
Accordingly, Harackiewicz etal. found that the intervention increased students’
self-reported value for their coursework and grades, with particular benefits to first-
gen underrepresented minority students. The intuitive link to the CoMSA would be
to hypothesize that the increase in the value of biology to these students leads them
to increase their norm of study. If a student finds a course more valuable to their
long-term goals, they might be less likely to accept lower performance (i.e., adopt a
higher norm of study or more stringent achievement-focused goals). Where perhaps
a B was an acceptable course grade before, a student may now only disengage study-
ing when they assess their anticipated performance as reflecting an A- or A.
It could also be that a student would set aside more time for a higher-valued
course. This increase in time would be represented on the CoMSA by shifting stu-
dents’ willingness to study further along the curve to the right (see Fig.5). If some-
one values a course more, they may be more likely to prioritize this course over
other courses, extracurricular activities, or outside hobbies. But then, we can also
use the CoMSA to ask “how effective would adding an additional increment of time
to this student’s study time be?” The effectiveness depends on a host of factors, but
primarily on where the student currently stands in relation to their learning curve. If
the student has already mastered the material (Fig.1), they may not benefit from an
additional hour of study at all. If the student is too early in their learning journey—
for example, they do not have the basics of biology or chemistry nailed down but are
being tested on advanced biochemical concepts—an additional hour of study may
not be enough to show any progress on the classroom exam of interest (despite the
fact that the student is improving their biology knowledge more generally).
Of course, it is also possible that a value intervention affects both willingness to
study (maximum study time) and norm of study concurrently. Or perhaps the interven-
tion works on an entirely different aspect of the CoMSA. For example, higher value
may lead students to think more deeply and make connections between the biology
content and prior knowledge. Or perhaps, as Xu etal. (2021) found, a growth mindset
intervention reduces cognitive load, thus enabling students to better learn from a text
Fig. 8 Linking the CoMSA to existing psychological theory
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on the Doppler effect. In these cases, increased cognitive engagement might be better
instantiated as a shift to the slope of the learning curve, because the students benefit-
ing from the motivational interventions are using their time more efficiently. Similarly,
one might think that a retrieval practice intervention might shift the slope upwards
(increase learning rate) and reduce metacognitive bias, because retrieval practice has
been shown to improve learning rate and metacognitive accuracy over restudy controls
(Schuetze etal., 2019). These interventions represent just a selection of many con-
struct- or strategy-focused theories in educational psychology, but a similar analysis
of the interleaving effect (Brunmair & Richter, 2019), expectancy-value (Wigfield &
Eccles, 2000), or self-determination theories (Deci & Ryan, 2000) could be facilitated
through the use of the CoMSA as a theoretical model for anticipating and understand-
ing where and when students would be benefited by various interventions.
Simulation Methods
The mathematical model defined in the previous sections formed the basis of the pre-
sent simulation. Overall, 22 total model input parameters were manipulated in the sim-
ulation. Fourteen of these parameters define the baseline characteristics of the simula-
tion. A further seven parameters have intervention (Δ, read “delta”) counterparts. For
the present paper, these intervention parameters are not of interest, as we are focused
on understanding the patterns of outcomes in the control data. However, these interven-
tion parameters were used for analyses described further in Schuetze (2023). See the
Appendix for a full listing and description of the 22 inputs to the simulation.
Simulation Inputs
In a period of iteration and consultation with subject matter experts, the simulation
ranges for the 22 parameters in the simulation were defined. Possible values were
selected with the overall imperative to err on the side of expansiveness (i.e., test
wider ranges of possible values rather than more limited ranges). The values for
each parameter were sampled from a uniform distribution of equally spaced values.
For example, the slope variable could have one of 26 possible values: 0.25, 0.30,
0.35, 0.4 … 1.45, 1.50. Uniform distributions were used as a sort of uninformative
prior, to explore a wide range of equally probable possible values.
Data Generation
Generating Reasonable Classroom‑Level Data
Naively, one might use grid expansion to enumerate all possible parameter combi-
nations and evaluate the effects of every possible intervention in the stimuli space
(see, theAppendix Table4). If we multiply out each possible combination of levels
of input parameters (e.g., 26 possible levels for the slope × 32 levels for the max-
imum willingness to study, and so forth…), there are 6.73 × 1028 possible unique
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Educational Psychology Review (2024) 36:18 Page 19 of 38 18
combinations. This number was not computationally tractable (in terms of both
generating and analyzing the data). Given the computationally expensive number
of possible combinations, a final sample size of 100,000 combinations of inputs was
chosen. Because not all combinations of inputs will be valid (e.g., standard devia-
tions were negative), in the first step, 200,000 potential combinations were sampled
from the overall possibility space.
To arrive at N = 100,000, these 200,000 combinations were filtered through three
steps to assure reasonable and computable combinations to arrive at the final data-
base of 100,000 simulation runs. In the first step, runs were filteredfor being uncal-
culable. For example, in some runs the metacognitive SD was negative after the
intervention was applied; since standard deviations cannot be negative, these runs
were removed before the simulation was run (this criterion filtered 8% of the 200k).
Second, during the simulation, the code checks whether the control and treatment
curves cross over one another. These runs were also filtered as to only focus on posi-
tive interventions to the system (this criterion removed 37% of the runs). This sec-
ond filtering step did not appreciably change the results of the simulation, save for
analyses concerning the slope (Logis a) parameter, where the filtering mechanism
resulted in more parsimonious results. For ease of interpretation, the final database
was limited to a random sample of the remaining 100,000 simulation runs (this third
criterion removed 4.5% of runs). Repeated simulation indicated that the results of
the following analyses stabilized after as little as 10,000 simulation runs, indicating
that the final sample size of 100,000 was more than sufficient to avoid overfitting.
Generating Student‑Level Data
After generating 100,000 different simulated classroom contexts, 5000 data points
were generated within each classroom. The N of 5000 students was chosen to bal-
ance sampling error with computational burden. These groups of 5000 students can
be thought of as counterfactual classrooms. Within these counterfactual classrooms,
each student’s learning curve input parameters and motivational characteristics were
sampled from the relevant distribution for each parameter (see, Table1).
For example, for each of the 100,000 simulation runs, a mean and standard devia-
tion was set for the value of willingness to study. Then, for each of the 5000 students
within each simulation run, a value for willingness to study was sampled dependent
on the mean and standard deviation set for that particular simulation run (this struc-
ture resembles that of a multilevel model). This means that within each classroom,
there will be variation in the maximum amount of time each student will spend stud-
ying, but generally, students from the same classroom will be more similar to one
another than students in a separate classroom with a different underlying set of mean
and standard deviation parameters for willingness to study.
All other inputs to the simulation, including the logistic parameters slope (Logis
a), intercept (Logis c), and inflection point (Logis d) were also drawn from distribu-
tions, where the parameters (means, SDs, etc.…) of these distributions were manip-
ulated between simulated classrooms (see the Appendix for more detail). In sum-
mary, students within each simulation run (classroom) were drawn from the same
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Table 1 Inputs to the CoMSA
Input name Approximate terms in the Carroll model Technical definition Possible psychological interpretation
Slope (Logis a) Learning rate (aptitude & quality of
instruction)
Slope of the learning curve This changes how quickly students acquire
information per unit of time invested. For
example, retrieval practice changes the rate
of learning per trial
Intercept (Logis c) No match Y-intercept of the learning curve This changes the floor performance when
time spent studying is zero. For example,
it is difficult to achieve less than 25%
accuracy onmultiple-choice tests, meaning
there is a floor of at least Logis c = 0.25.
It is also associated with prior knowledge.
Students with higher floors will do better
on tests without studying than those with
lower floors
Inflection point (Logis d) Ability to understand instruction Determines mid-point of learning
curve where probability of suc-
cess = 0.5
This changes the amount of prerequisite
learning needed to achieve before the stu-
dent sees performance gains on the present
test (e.g., a student needs an algebra review
before a calculus exam)
Metacognitive bias (calibration) No match Mean error around norm of study Over- or under-confidence when it comes to
evaluating one’s performance relative to
their norm of study goals
Metacognitive precision (SD) No match Error variance around norm of study General variation of the metacognitive judg-
ments, around the bias
Norm of study (performance criterion) Opportunity to learn (partial match) Self-regulated stopping point This is the grade the student hopes to achieve
and will stop studying when thought to be
achieved
Time-on-task (willingness to study) Perseverance Input to the logistic learning curve A student will only spend up to this amount
of time studying and no more
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Educational Psychology Review (2024) 36:18 Page 21 of 38 18
underlying distributions but had differing motivational and learning curve character-
istics depending on their draw.
Simulation Outputs
Finally, the performance of students within each simulation run in terms of achieve-
ment and time spent studying was calculated. The database can be queried, filtered,
plotted, and analyzed to better understand the effects of different individual and con-
textual factors on the CoMSA. The full database is available at the following OSF
project: https:// osf. io/ z8n3a/. This database includes N = 184,032 simulation runs,
representing over 1.8 billion individual simulated students. This number includes
runs rejected due to the second and third criterion (see simulation input section
above). The results of the present project were derived through an iterative model
building and refining process and thus were not pre-registered.
Results
For the following proof-of-concept analyses, data is used from two different meta-
analyses to compare the results from the simulation of the CoMSA to empirical data
from a wide variety of educational contexts. One of the advantages of the current
modeling approach is that it allows for both simulation of individual students and
groups of students (i.e., classrooms). In both examples, we simulate classroom-level
descriptive statistics by simulating individual students and then aggregating over
this student-level data.
The Relationship Between Time‑on‑Task andAchievement
An open question in educational psychology relates to the importance of time spent on
task for predicting achievement. The relationship between time-on-task and achieve-
ment is a classic question of educational psychology and one that has produced seem-
ingly inconsistent findings over years of study (Godwin etal., 2021; Vu etal., 2022).
In 1981, Karweit and Slavin hypothesized “that the choice of model linking time and
learning may be implicated in the inconsistent findings for time-on-task” (p. 171). The
following results suggest that the CoMSA could be one such model. Here, data sum-
marized by Godwin etal. (2021) is used as an empirical benchmark against which data
from the CoMSA simulation can be compared against.
Time‑on‑Task Analysis
The correlation between time spent and achievement was computed for each of the
100,000 simulation runs. This data shows a mean correlation between time spent
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and achievement of r = 0.42. However, this correlation is context-dependent and is
not stable across all of the simulation runs. The 5th percentile r equaled − 0.06, and
the 95th percentile r equaled 0.83. The minimum r was − 0.67, while the maximum
was r = 0.97 (see Fig. 9). Given that the present simulation assumes a strong causal
link between time spent and achievement at the level of the individual student, one
might expect it to overestimate the relationship between these two variables. This
was not the case.
In fact, the r of 0.42 in the present simulation is slightly lower than the figures
reported by Cury et al. (2008), who conducted a laboratory investigation, finding
a raw correlation between time-on-task during a practice period and achievement
as measured by performance on a decoding task on an IQ subtest of r = 0.49. Per-
haps more educationally relevant, Godwin etal. (2021) report a summary table of 17
studies examining the relationship between time-on-task and performance in K-12
classroom environments. The estimates of the individual studies ranged from − 0.23
to 0.78, with a mean of approximately r = 0.31. As shown by Fig.9, the overall dis-
tributions of the empirical and simulated data showed clear resemblance, with the
Godwin etal. (2021) data showing slightly smaller correlations on average.
Interim Discussion
The present model suggests that the correlation between time spent and achieve-
ment will be highly dependent not only on the noise in the data (the more noise, the
Fig. 9 Density plots of simulated and empirical correlations between time spent and achievement. Note:
dashed lines indicate means of respective distributions. The simulation captures the same general trend
of the empirical correlation coefficients (mostly positive, left skewed, with a thin negative tail below
zero) without any fitting procedure. Density plots can be interpreted as histograms scaled such that the
area of the distribution is equal to one
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Educational Psychology Review (2024) 36:18 Page 23 of 38 18
less likely a correlation is to be high), but also on the features of the learning curve,
such as slope, prior knowledge, and intercept. If the slope is high, all else equal,
we might expect to see a stronger relationship between time spent and achievement.
As for the instances where there is a negative correlation (approximately 8.5% of
the simulation data), this could be explained by situations where the norm of study
constraint causes already high-performing students not to study much at all, while
those with lower prior knowledge starting points may need to invest more time to
get to a lower end result. The existence of a negative correlation does not necessarily
mean that adding an additional hour of study time is actually leading to worse per-
formance. Rather, the need for study time is confounded with prior knowledge, cre-
ating an illusory negative relationship between time-on-task and achievement. These
findings help unify data from the cognitive psychology literature which clearly sup-
ports the time-on-task hypothesis (e.g., Zerr etal., 2018) and real-world educational
data (Godwin etal., 2021) that previously seemed to contradict the notion that more
time-on-task creates higher achievement.
Relationship Between Prior Knowledge andKnowledge Gains
Another open question in educational psychology concerns the seemingly incon-
sistent relationship between prior knowledge and knowledge gains in academic set-
tings. In the following analyses, the distribution of effect sizes generated through
the simulation of the CoMSA is compared to the data collected in a meta-analysis
conducted by Simonsmeier etal. (2022). In this meta-analysis, Simonsmeier and
colleagues were interested in assessing the “knowledge is power hypothesis.” In
other words, the hypothesis is that higher domain-specific prior knowledge is one of
the best predictors of future knowledge acquisition. To better understand the rela-
tionship between prior knowledge and knowledge acquisition, Simonsmeier et al.
collected meta-analytic data from a wide variety of studies from all domains of edu-
cation (though the majority of studies came from K-12 populations). Essentially,
their analyses included any study where at least a pre- and post-test correlation
could be calculated using a semi-objective measure of domain-specific knowledge.
They did not include studies where knowledge is derived via self-report Likert scale
metrics or broad indicators such as IQ scores or metacognitive knowledge.
In terms of primary outcomes of interest, Simonsmeier etal. calculated correlations
between prior knowledge with final test performance (rp), absolute knowledge gains
[rag = cor(pre-test, post-test − pretest)], and normalized knowledge gains [rng = cor(pre-
test, (post-test − pretest) / (scale max − pretest))]. In essence, Simonsmeier etal. sought
to understand whether prior knowledge was correlated with increased rate of knowledge
acquisition (i.e., a positive feedback loop or “rich gets richer” effect). Critically, they
found that there was a high pre-post test correlation (rP = 0.53), but a small negative
correlation between prior knowledge and normalized knowledge gains (rNG = 0.06).3
3 Note that these numbers differ somewhat from the correlations presented in the following table,
because the data supplied by Simonsmeier etal. are a subset of the overall data. The numbers in the pre-
sent manuscript also differ somewhat from those presented in the author’s dissertation due to the iterative
improvements made to the model in the time since the submission of the first document.
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The CoMSA allows for the calculation of these quantities across the dataset and thus
the comparison of the predicted correlations and true correlations found by Simons-
meier etal. This meta-analysis is particularly suited to this task because the three out-
come measures give relatively independent information about the classroom environ-
ment. Furthermore, Simonsmeier et al. analyzed studies where the primary research
question did not relate to the issues of prior knowledge and post-test performance. Thus,
there are potentially fewer worries about publication bias in this sample of studies.
After personal communication with the first author of this study, a subset
of the data reported by Simonsmeier et al. (2022) was received, composed
of the data from all the studies where all three correlations were able to be
computed (rP, rAG, and rNG). This data consisted of 270 correlation triplets.
The average sample size was 139, ranging between N = 8 and N = 1508. These
three correlations all give unique insight into the dynamics of knowledge
acquisition between pre- and post-test. In the words of Simonsmeier etal., “rP
cannot be inferred from rNG or rAG (without additional information) and vice
versa” (p. 46).
Thus, the present data provide at least three relatively independent data
points for each of these studies that either can or cannot be reproduced by the
CoMSA. Although no guarantees about the representativeness of these stud-
ies can be made, these data points represent real learning situations and the
type of data that can result from student learning between pre and post-tests.
One other interesting, yet counterintuitive aspect of gain scores is that they are
not easily understood in terms of conventional statistical modeling (Ackerman,
1987). As Clifton and Clifton (2019) show, a gain score’s correlation with a
pretest is inevitable due to regression to the mean and the fact that a gain score
is calculated from the pretest score. In other words, because the pre-test factors
into the calculation of both sides of the correlation rAG, the metric is partially
a pre-test score’s correlation with itself. Clifton and Clifton write that “… the
change score is always correlated with the baseline score [] and with the post
score []. This is purely a statistical artefact, and it exists regardless of whether
the treatment is effective or not” (p. 3). In other words, even when there is
no causal relationship between pre-test and post-test, the correlation between
pre-test and gainsmay appear negative (and this does not even account for any
further complexity introduced by ceiling effects). For this reason, it may be
best to simulate a data-generating mechanism at the individual student level
and then compare the gain scores that result to empirical data, rather than to
interpret positive gain scores in empirical samples as evidence for the rich gets
richer effect.
Rich‑Gets‑Richer Analysis
As shown in Table2 and Fig.10, the data before matching had varying levels of
similarity to the Simonsmeier data, between 7 and 15% error. While the rP data
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Educational Psychology Review (2024) 36:18 Page 25 of 38 18
looked relatively similar to the Simonsmeier data before matching, there were
some discrepancies between the simulation and Simonsmeier’s data when it came
to rAG and rNG. In particular, the unmatched simulated data from the CoMSA was
more likely to produce negative values of rNG and rAG than found in the empiri-
cal data. To better understand why the source of these discrepancies, a match-
ing process was engaged such that better-fitting inputs to the simulation could be
found. In other words, the following analysis attempts to answer two questions:
(1) Can the CoMSA produce realistic patterns of data? And (2) of the inputs to
the CoMSA in the current simulation, which of these inputs is most likely to pro-
duce realistic data patterns?
In this matching process, each row from the Simonsmeier data was matched one-
by-one to the closest simulation run in terms of rP, rAG, and rNG, such that the root
mean squared error between the three pairs of correlations derived from each empir-
ical study and the matched simulation run was minimized. This approach was rela-
tively successful in finding 270 matching pairs of empirical data and simulated data,
as evidenced by the descriptive statistics presented in Table2.
Table 2 Mean correlations for
simulated and empirical data
and chi-squared tests of the
equality of correlation matrices
chi-square represents the test of correlation matrices of the three
variables against the Simonsmeier correlation matrix. For these two
tests, df = 3.
***p < 0.001
Dataset rPrAG rNG χ2
Simonsmeier meta 0.46 − 0.14 0.02 –
Simulation (full) 0.59 − 0.45 − 0.23 158.49***
Simulation (matched) 0.48 − 0.18 0.04 5.02
Fig. 10 Comparison of correlation coefficients pre- and post-matching. Note: rP = correlation between
pre- and post-test. rAG = correlation between pretest and absolute gains. rNG = correlation between pretest
and net gains. The goal of the matching procedure was for the red distribution (matched data) to resem-
ble the black line indicating Simonsmeier’s data
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The last column of Table2 presents a series of Jennrich (1970) tests of the equal-
ity of correlation matrices. Perhaps somewhat confusingly, these tests are testing the
correlation matrix computed from outcome variables, which themselves are correla-
tions. The resulting chi-squared values from these tests can be understood as a sort
of model fit index. The results shown in the right-most column of Table2 indicate
the following: Before the matching process, the correlation matrices between the
three variables of interest (rp, rAG, and rNG) differed significantly between the sim-
ulated and empirical data, indicating relatively poor model fit. After the matching
process, the correlation matrix from the matched data was significantly better fitting
than that of the full simulation data. Furthermore, the matched data was no longer
significantly different from the Simonsmeier data. Thus, the matching process
resulted in a model that both mimicked the means and correlations of the Simons-
meier data.
This was further confirmed by visual inspection of the density plots compar-
ing the true versus matched data. After matching, the distributions looked rela-
tively similar for all three correlational metrics, except for a few regions in the
extremes, with only 1–2% error. Specifically, the empirical data tends to be more
extreme than would be anticipated by the simulated data. This seems at least par-
tially attributable due to sampling error in the empirical data that does not exist
in the simulated data (since the simulated data is estimated with N = 5000 stu-
dents, it is much more precise than the average study contained in the Simons-
meier meta-analysis).
After building the matched dataset, it was of interest to determine if the
matching procedure favored some of the input values of the CoMSA over others.
If some values of the inputs to the CoMSA were more common in the matched
data than in the non-matched data, this would suggest that these values for the
CoMSA might be more reasonable than the relatively flat input distributions
used to run the simulation. In other words, can we use empirical meta-analytic
data to better understand what the inputs to the CoMSA might look like in real
education settings?
To test whether the matched simulation rows differed from the overall inputs
to the simulation, t-tests of means and the Kolmogorov–Smirnov tests of equal
distributions between the matched and unmatched data were conducted. As sum-
marized in Table3, the largest difference between the matched and unmatched
data was found in the distribution of the input of the learning curve intercept
(Logis c). All but four variables showed significant differences between the
matched and unmatched data at the p < 0.001 level. These four variables were:
performance criterion, metacognitive precision, floor-time correlation, and crite-
rion-time correlation.
What does it mean when there is no significant difference between the matched
and unmatched rows? The lack of significant difference indicates that the subject-
matter-expert-derived inputs to the simulation were “hitting the mark,” rather than
needing to be shifted during the matching process. For example, the current simula-
tion tests a range of correlations between learning curve intercept and motivation
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Educational Psychology Review (2024) 36:18 Page 27 of 38 18
Table 3 Comparison of matched and unmatched rows from the simulation for simulation inputs
Only baseline variables are included in this analysis because the correlations were calculated only for the untreated groups, which were not affected by treatment variables
t-test KS test
Variable Mean unmatched SD unmatched Mean matched SD matched t p D p
Floor 0.43 0.17 0.18 0.17 − 24.86 < 0.001 0.57 < 0.001
Slope 0.88 0.37 1.08 0.36 9.28 < 0.001 0.26 < 0.001
Inflection Pt 2.25 2.08 3.39 2.03 9.25 < 0.001 0.27 < 0.001
Time limit 4.12 2.3 3.57 1.97 − 4.59 < 0.001 0.16 < 0.001
Metacog bias 0 0.12 − 0.03 0.12 − 4.07 < 0.001 0.14 < 0.001
Criterion-time corr 0.44 0.25 0.42 0.24 − 1.38 0.168 0.06 0.328
Metacog SD 0.07 0.03 0.07 0.03 0.82 0.415 0.05 0.505
Performance criterion 0.73 0.17 0.73 0.18 0.5 0.616 0.08 0.05
Floor-time corr 0.41 0.24 0.41 0.25 0.45 0.653 0.04 0.739
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Educational Psychology Review (2024) 36:18
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(willingness to study) from r = 0 to r = 0.85. This nonsignificant result simply means
that the mean of these intercept-time correlations in the matched data did not signifi-
cantly differ from 0.43, the mean of the correlation in the full set of simulation data.
Given that the mean may not be the only parameter of interest in understand-
ing the differences between the matched and unmatched distributions, and sig-
nificance testing may be a somewhat blunt instrument for the present purposes, it
may be best to visually inspect each of the relevant distributions visually, which
are shown in Fig.12. Overall, we see that the matched simulations were more
likely to come from simulations with low starting floor performance (Logis c),
relatively high criterion-time correlations, and relatively low baseline metacogni-
tive bias (underconfidence). We also see that the baseline time limit tends towards
the lower end of the input distribution suggesting that this data comes from stud-
ies where students are putting at least a moderate amount of learning effort into
between the pre- and post-tests.
Interim Discussion
The above analyses show that the CoMSA could reproduce the data collected in
the Simonsmeier etal. (2022) meta-analysis. However, caution should be taken,
as in-depth analysis would have to be undertaken to determine if there are com-
peting theories that would better explain this data. A prerequisite for such an
analysis would be for competing theories to also be computationally formalized
(which would be a positive step for the literature). What can be inferred from the
present analyses is as follows: After the matching process, the data resulting from
the CoMSA is statistically indistinguishable from the empirical data collected by
Simonsmeier, while also being generated by a relatively plausible model of learn-
ing over time.
The matching process indicated that the empirical data was best matched by
simulated learning situations where relatively high amounts of learning were
occurring. For example, the matched data almost uniformly showed low start-
ing prior knowledge (Logis c), relatively high slopes (Logis a), and relatively
high norms of study (most between 75 and 95 percent). This makes conceptual
sense, as one might expect teachers to scaffold content, such that they are engag-
ing students at the edge of their zone of proximal development (Vygotsky, 1978).
Theoretically, this is where students are expected to feel the most motivated and
for the most learning to occur—this is as opposed to potential scenarios outside
the zone of proximal development where learning is too hard or too easy and
very little payoff can be made (Puntambekar, 2022). Furthermore, the simula-
tion lends support to the notion that there is a positive correlation between time-
on-task and one’s starting point. Indeed, the mean matched correlation between
the intercept of the learning curve and motivation is r = 0.41 and is significantly
greater than zero. This constraint is in line with work on self-efficacy theory that
finds reciprocal relationships between achievement and self-efficacy (Schunk &
DiBenedetto, 2020, 2021). In other words, the present study adds evidence for the
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Educational Psychology Review (2024) 36:18 Page 29 of 38 18
“big-fish-little-pond” relationship between within-class achievement ranking and
motivation, which has been recently shown strong support by a national census
analysis of entire grades of Luxembourgish elementary students by van der West-
huizen etal. (2023).
General Discussion
The purpose of the present paper is twofold. The first purpose was to introduce
the CoMSA, which represents a novel computational model of school achieve-
ment that provides a framework through which researchers can understand stu-
dent achievement and where it may be best to intervene in order to improve out-
comes. The second purpose was to show how formal modeling approaches, as
embodied by CoMSA, can be used to test and expand upon psychological theory
and potentially explain two open questions in the educational psychology litera-
ture as a proof of concept. In this section, these analyses are discussed first, and
then the paper concludes with the broader aims and limitations of the present
modeling approach.
Proof‑of‑Concept Analyses
The proof-of-concept analyses showed that two open research questions, with het-
erogeneous findings in the literature, could both be modeled as resulting from pop-
ulations of students defined by varying distributions of learning curve shapes and
motivation. In the first proof-of-concept analysis, the simulation data was compared
to the highly variable empirical results concerning the relationship between time-
on-task and academic achievement (the so-called time-on-task hypothesis). Godwin
etal. (2021) summarize this variation in correlations across studies, from r = − 0.23
to 0.78, indicating that “the relationship between time and learning remains elusive
as prior research has obtained mixed findings” (p. 503). Although potentially a sin-
gle value is elusive, this variation reported by Godwin etal. is in line with the funda-
mental predictions of the CoMSA and may not be much of a conundrum as has been
suggested by earlier reviews in the literature (e.g., Godwin et al., 2021; Karweit,
1984; Vu etal., 2022). Put simply, the data summarized by Godwin etal. (2021) can
be created even when assuming that time-on-task is always a positive influence on
learning.
In the second question of interest, we looked at how well the data generated from
the CoMSA matched the results of Simonsmeier etal.’s (2022) data concerning the
relationship between prior knowledge and post-test performance. Here, the CoMSA
was used to calculate potential correlations between raw performance, absolute
gains, and net gains. Using a matching process, we determined that the CoMSA
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Educational Psychology Review (2024) 36:18
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could recreate the patterns of correlations seen in the empirical data summarized by
Simonsmeier etal. (2022) with less than 2–4% error if the correct ranges of inputs
were used.
Broader Implications oftheCoMSA andtheFormal Modeling Approach Generally
The CoMSA represents one of many potential paths forward through com-
putational modeling within the domain of educational psychology. The pri-
mary strength of this model lies in its ability to simulate the interactions
between motivation, cognitive processes, and achievement over time. With
further refinement and software infrastructure, such counterfactual simula-
tion tools could be invaluable for both researchers and educators. Further-
more, CoMSA distills and builds upon the fundamental notion—embodied
by Carroll’s (1962, 1963) model of school learning—that there are two
general approaches to improving learning: either a student can study bet-
ter or they can study longer (Bloom, 1976). Here, we may slightly modify
this maxim to say that either you can shift the learning curve (i.e., engage
in learning strategy or instructional interventions), or you can shift your
position along the learning curve (i.e., engage in motivational interven-
tions). In this way, hopefully, the CoMSA can facilitate the synthesis of dif-
ferent research threads within educational psychology, providing an over-
arching framework through which new data can be analyzed and findings
interpreted.
More broadly, formal computational models offer several advantages over
traditional verbal models for studying educational phenomena. Perhaps coun-
terintuitively, one of the main benefits of using formal models is that they make
the assumptions and shortcomings of our models become more clear. Though
it is painful to see where our models fail, this increased clarity can help us
better pinpoint areas of improvement. Ultimately, embracing a model-centric
framework in educational psychology can optimize study design and hypoth-
esis testing, in addition to supporting the development of integrative theory and
fostering practical application of education research (Bjork, 1973; Devezer &
Buzbas, 2023).
Limitations, Future Directions, andConcluding Thoughts
Though the CoMSA may be a useful tool, there are undoubtedly limitations to the
present modeling exercise. Learning is a dynamic, chaotic, and unruly process,
which the CoMSA can only attempt to approximate through differences in learning
curves, time spent on task, and metacognitive biases. That said, the economy is a
dynamic, chaotic, and unruly process, too. And economists have gotten substantial
mileage out of shifting supply, demand, and labor curves. Humphrey (1992) calls
supply and demand curves: “Undoubtedly the simplest and most frequently used
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Educational Psychology Review (2024) 36:18 Page 31 of 38 18
tool of microeconomic analysis” (p. 3). Educational psychologists might find that
learning curves could function as one of the simplest means of analysis of academic
achievement.
As van Rooij and Blokpoel (2020) note, computational modeling is an iterative
art, much like sculpting. Formal models represent the creators’ own biases and
understandings of a complex subject matter. For this reason, it would be inap-
propriate to consider the present model to be set in stone. Rather—to paraphrase
Gerald Bullett (1950)—this model is offered up not as a final contribution to the
scholarship of the subject, but as a means of the first (or maybe second; Carroll,
1962) approach to it. The mark of a strong model, then, will be to see its assump-
tions validated and flaws ameliorated by future scientists (see Devezer & Buzbas,
2023).
While the current analysis primarily concentrates on classroom-level descriptive
statistics, future research can extend this model’s capabilities by exploring its capac-
ity to track individual learning trajectories and students’ responses to interventions.
Furthermore, for the sake of parsimony, the current simulation only concerns itself
with self-regulated learning in the pursuit of a single goal. And this goal is assumed
not to change over time. Future studies could also expand upon this assumption
by considering how learners react in real-time to progress or setbacks. Perhaps, a
learner finds that material is much harder than they anticipate (i.e., the slope of the
learning curve is very shallow). They may potentially decide to lower their norm of
study as a response (see Ackerman, 2014).
Additional insights could also be garnered through the simultaneous modeling
of multiple learning curves with shared motivational resources (i.e., to model a
student having to decide between studying for multiple classes). One might also
consider extending the CoMSA model from the test level to the item level. In
this case, researchers could examine how students respond when they realize they
understand item A better than item B. Historically, these types of inquiries have
been central to many self-regulated learning models (e.g., Thiede & Dunlosky,
1999). There is room for progress to be achieved by computationally linking the
CoMSA to these more fine-grained models of self-regulated learning and study-
time allocation.
Altogether, the present simulation results represent just a subset of potential uses
of the CoMSA. Beyond these proof-of-concept analyses, the CoMSA also provides
a mathematical and graphical interface through which researchers, teachers, and
policy makers can consider how achievement is manifested through motivational
and cognitive factors. Furthermore, the CoMSA unites often disparate research on
motivation, cognition, and metacognition into a single computational model. The
CoMSA grounds existing theories from educational psychology in the concrete
behavior changes that can alter academic achievement, making nuanced predictions
about the interplay between prior knowledge, motivation, learning strategies, and
academic outcomes.
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Educational Psychology Review (2024) 36:18
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Appendix
Table 4 Full table of simulation parameters
Type of intervention Parameter name Description Simulation range
Cognitive/study strategies Slope mean Mean of slope (Logis a) distribution 0.25 … 1.5
Slope SD SD of slope dist 0.05 … 0.5
∆ Slope Intervention effect on slope 0 … 1
Intercept shape 1 Beta distribution shape 1 of distribution of y-intercept 0.25 … 5
Intercept shape 2 Beta distribution shape 2 of distribution of y-intercept 2 … 4
∆ Intercept Effect on Logis c (y-intercept) 0 … 0.5
Inflection point mean Mean of inflection point dist − 1 … 6
Inflection point SD SD of inflection point dist 0.01 … 0.25
∆ Inflection point Effect on inflection point − 1 … 0
Metacognitive monitoring Calibration (bias) mean Average over- or under-confidence − 0.20 … 0.20
∆ Calibration Effect on calibration − 0.10 … 0.10
Resolution (precision) Metacognitive monitoring ability 0.02 … 0.12
∆ Resolution Intervention effect on resolution − 0.05 … 0.05
Motivation Maximum time Mean of max time distribution 0.25 … 8
Time SD SD of max time distribution 0.05 … 4
∆ Time limit Effect on t 0 … 4
Perf. criterion shape 1 Shape 1 for PC beta dist 3 … 14
Perf. criterion shape 2 Shape 2 for PC beta dist 3 … 14
∆ Performance criterion Effect on PC 0 … 0.25
Time, criterion correlation cor(max time, PC) r = 0 … 0.85
Time, intercept correlation cor(max time, intercept)and cor(PC, intercept) r = 0 … 0.85
Other Baseline SD Measurement error of test 0.005 … 0.10
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Educational Psychology Review (2024) 36:18 Page 33 of 38 18
Table4.
Figure11.
Figure12.
Fig. 11 CoMSA formaliza-
tion. Note: in step 1 (red text),
student-level variables are
sampled from classroom-level
distributions. In step 2 (blue),
prior knowledge and time-lim-
ited achievement are calculated.
In step 3 (purple), the students’
metacognitively-modulated
norm of study is compared to
the level of achievement they
would obtain if they studied for
their entire max willingness to
study. Their latent performance
is the minimum of these two
quantities, with the condition
that latent performance is never
below their prior knowledge.
Step 4 (black) shows the ceil-
ing and floor effects. In the
above equation: t (max time),
a (slope), c (intercept), and d
(inflection point) are sampled
from the relevant classroom-
level distribution to create each
student’s individual learning
curve. k = prior knowledge,
r = performance at maximum
study time, and l = latent perfor-
mance achieved prior to meas-
urement error and floor/ceiling
effects. N refers to sampling
from a normal distribution. NTR
is a truncated normal distribu-
tion with lower bound of zero,
and beta refers to sampling from
a beta distribution. Note that
the two correlation inputs are
not accounted for in the above
formalization
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Educational Psychology Review (2024) 36:18
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18 Page 34 of 38
Acknowledgements The author would like to extend his thanks to his dissertation committee, particu-
larly chair Veronica Yan, for their intellectual mentorship throughout the course of this project. He also
thanks Bianca Simonsmeier for providing the meta-analytic data used in the present paper.
Author Contribution B.A.S. conceptualized and conducted the simulation study and wrote the
manuscript.
Funding Open Access funding enabled and organized by Projekt DEAL.
Data Availability Simulation code, data, and analysis scripts are available on this project’s OSF page
(https:// osf. io/ z8n3a/).
Declarations
Conflict of Interest The author declares no competing interests.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License,
which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long
as you give appropriate credit to the original author(s) and the source, provide a link to the Creative
Commons licence, and indicate if changes were made. The images or other third party material in this
article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line
to the material. If material is not included in the article’s Creative Commons licence and your intended
use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permis-
sion directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/
licenses/by/4.0/.
Fig. 12 Comparison of simulation inputs pre- and post-matching
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Educational Psychology Review (2024) 36:18 Page 35 of 38 18
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