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Article |
Las matemáticas en la vida profesional ecuatoriana:
alfabetización cuantitativa y un análisis bayesiano del desempleo por nivel
educativo
María Aurora Parrales Gallo[*]
Byron Oviedo-Bayas*
Abstract
Objective: To reflect on the role of mathematics in
professional practice in Ecuador and to demonstrate, using open data, how
quantitative reasoning (particularly Bayesian inference) supports
decision-making. Methodology: A thematic review was combined with an empirical
study based on open data from the World Bank (World Development Indicators,
based on ILO estimates) on unemployment rates by educational level in Ecuador
between 2005 and 2024 (n = 57 country-year observations). Two Bayesian models
were estimated using PyMC—a comparison of means by educational level and a time
trend—with weakly informative priors and Hamiltonian Monte Carlo (NUTS)
sampling. Results: The average unemployment rate was higher among those with
intermediate education (6.15%; 94% credibility interval: [5.73, 6.59]) and
advanced education (4.95%; [4.57, 5.34]) than among those with basic education
(2.49%; [2.20, 2.80]); the posterior probability that unemployment among those
with advanced education would exceed that of those with basic education was
1.00, and unemployment among those with advanced education showed an upward
trend (0.068 percentage points per year; [0.009, 0.128]; probability of a
positive slope = 0.98). Conclusions: Far from denying the value of education,
this pattern reflects informality and the wait for formal employment, and
demonstrates why careful statistical interpretation is an indispensable
professional skill in contemporary Ecuador.
Keywords: mathematics education; quantitative literacy;
Bayesian inference; labor market; Ecuador.
Resumen
Objetivo: reflexionar sobre el
papel de las matemáticas en el ejercicio profesional en el Ecuador y mostrar,
con datos abiertos, cómo el razonamiento cuantitativo (en particular la
inferencia bayesiana) sostiene la toma de decisiones. Metodología: se combinó
una revisión temática con un estudio empírico basado en datos abiertos del
Banco Mundial (Indicadores del Desarrollo Mundial, con base en estimaciones de
la OIT) sobre las tasas de desempleo por nivel educativo en el Ecuador entre
2005 y 2024 (n = 57 observaciones país-año). Se ajustaron dos modelos
bayesianos con PyMC —una comparación de medias por nivel educativo y una
tendencia temporal— con priores débilmente informativos y muestreo de Monte
Carlo hamiltoniano (NUTS). Resultados: la tasa media de desempleo fue mayor en
la educación intermedia (6.15 %; intervalo de credibilidad del 94 %: [5.73,
6.59]) y avanzada (4.95 %; [4.57, 5.34]) que en la básica (2.49 %; [2.20,
2.80]); la probabilidad posterior de que el desempleo con educación avanzada
supere al de educación básica fue de 1.00, y el desempleo con educación avanzada
mostró una tendencia creciente (0.068 puntos porcentuales por año; [0.009,
0.128]; probabilidad de pendiente positiva = 0.98). Conclusiones: lejos de
negar el valor de la formación, el patrón refleja la informalidad y la espera
por un empleo formal, y evidencia por qué la interpretación estadística
cuidadosa es una competencia profesional insoslayable en el Ecuador
contemporáneo.
Palabras clave: educación matemática; alfabetización cuantitativa; inferencia
bayesiana; mercado laboral; Ecuador.
Introduction
One need only take a close look at a typical workday
in Ecuador to discover that mathematics is everywhere, even though it is almost
never explicitly mentioned. The accountant balancing a ledger, the agricultural
engineer calculating fertilizer doses per hectare on farms in Los Ríos, the
doctor interpreting the sensitivity of a diagnostic test, the vendor estimating
her profit margin from memory at the Quevedo market, or the technician
adjusting the turbine at a hydroelectric plant: all of them reason with quantities,
proportions, rates, and probabilities. Mathematics is, in this sense, a kind of
invisible infrastructure of professional life—so commonplace that we rarely
notice it until it fails.
Research on numeracy in the workplace has shown that
the mathematics used in professional practice is not a simple replica of what
is taught in the classroom. Wake (2015) documents that, to function effectively
in a job, people need context-specific modeling—the ability to translate
reality into useful representations and to return to reality with a decision.
FitzSimons and Wedege (2024) emphasize that this workplace numeracy is both
technical and social: it is interwoven with the tools, norms, and relationships
specific to each trade. The gap between school mathematics and workplace
mathematics does not mean that the former is useless, but rather that it
requires a process of appropriation that vocational training should explicitly
support.
This phenomenon has become more pressing with
digitalization. Karaali (2023) warns that, in the age of artificial
intelligence, quantitative literacy—far from becoming dispensable—is becoming
the skill that allows one to discern when to trust an automated calculation and
when to question it. Van Laar et al. (2019) and Rakowska and de Juana-Espinosa
(2021), drawing on research into 21st-century competencies, agree that data
skills and analytical thinking currently top the lists of skills most in demand
by employers. And economic evidence suggests that these skills pay off: Lee and
Wie (2017) estimate significant wage returns associated with cognitive and
quantitative competencies in Asian labor markets—a finding consistent with a
long tradition in the economics of education.
This article pursues two intertwined objectives. The
first, of a reflective nature, is to argue why mathematics—and statistical
reasoning in particular—constitute a core competency for professional practice
in Ecuador. The second, of an empirical nature, is to demonstrate this with a
concrete and reproducible example: to analyze, using Bayesian inference and
open data, a feature of the Ecuadorian labor market that directly challenges
those who train professionals. The choice of the Bayesian approach is not coincidental;
as will be seen, it allows uncertainty to be expressed in the plain language of
probability—something a professional needs when making decisions without
certainty.
The text is organized as follows: Section 2 reviews
the literature on labor numeracy, statistical literacy, educational performance
in the region, and the link between higher education and the labor market in
Ecuador, as well as Bayesian reasoning as a professional practice. Section 3
describes the open data and models. Section 4 presents and discusses the
results. Section 5 concludes with findings and implications for teaching and
public policy.
From School Mathematics to Numeracy
in the Workplace
The distinction between school-based mathematical
knowledge and workplace numeracy underpins much of the literature. Wake (2015)
proposes a modeling-based perspective: the competent professional does not
mechanically apply formulas but rather constructs provisional models of their
situation, uses them to make decisions, and revises them in light of the
results. FitzSimons and Wedege (2024) complement this idea by showing that
numeracy in the workplace is a social practice, shaped by the tools, formats, and
conventions of each sector. A pedagogical implication emerges from both works:
training numerically competent professionals requires authentic situations, not
just decontextualized exercises.
Statistical and Quantitative Literacy
as a Competency
If anything characterizes contemporary work, it is
the ubiquity of data. Statistical literacy—the ability to read, interpret, and
question quantitative information—has become an essential component of
citizenship and professional performance. Sabbag et al. (2018) distinguish
between and empirically measure statistical literacy and statistical reasoning,
showing that they are related but not identical constructs. Gómez-Blancarte et
al. (2021) document, at the Mexican secondary level, the gap that still separates
the statistics curriculum from its effective teaching—a gap that resonates
throughout the region. In the field of adult competencies, Tunstall (2020)
analyzes how numeracy is measured within the framework of the Program for the
International Assessment of Adult Competencies (PIAAC), and Curry (2019)
translates that framework into concrete instructional guidelines. Karaali
(2023) links these discussions to the current technological landscape:
quantitative literacy is what enables us to engage critically with automation.
Performance, Attitudes, and Equity in
Mathematics Education
Performance in mathematics and its social
distribution have been extensively studied using international assessments.
Gamboa and Waltenberg (2012) show, using data from PISA 2006–2009, that in
Latin America a considerable portion of the inequality in outcomes is explained
by circumstances beyond individual effort (socioeconomic background, family
environment), which constitutes an inequality of opportunity. Gamboa and Krüger
(2016) delve deeper into the role of early childhood education in later
achievement. Using more recent data, Guerra et al. (2026) analyze PISA 2022 and
highlight socioeconomic and digital gradients in mathematics performance across
Latin America and the Caribbean. Martins and Veiga (2010) confirm the
significance of parents’ education in score gaps. Regarding school-related
factors, Liu et al. (2024) demonstrate, across a decade of PISA data, the
association between perceived instructional quality and mathematics
achievement. These findings are important for Ecuador because they describe the
background from which future professionals emerge.
Attitudes and emotions also matter. Lim and Chapman
(2015) demonstrate that teaching interventions can simultaneously improve
attitudes, anxiety, motivation, and performance, suggesting that the affective
dimension is not merely an afterthought but a factor with measurable
consequences. Finally, gender equity remains an unresolved issue: Beekman and
Ober (2015) document how trends in math test scores may or may not position
young women for careers in science, technology, engineering, and mathematics
(STEM), and Martínez et al. (2023) analyze, from an interregional perspective,
the persistently low representation of women in these fields. Without equity in
access to mathematical proficiency, professional life reproduces existing
inequalities.
Higher Education and the Labor Market
in Ecuador
The link between education and employment in Ecuador
has well-documented characteristics. In a recent econometric study, Rivera
Ávalos (2026) finds that higher education is a determinant of formal
employment; that is, a higher level of education increases the probability of
obtaining a formal job. This finding coexists, however, with a structural
history: MacIsaac and Rama (1997) had already shown that labor market
regulations play a significant role in determining hourly wages in the country.
The quality and governance of the university system have been the subject of
successive reforms; Jameson (1997) described early on the tensions within a
higher education system subject to multiple pressures, and Jiménez Cabrera
(2021) analyzes how current regulations seek to ensure quality. Acosta and
Stefos (2021) compare the quality assurance systems of Colombia and Ecuador,
while Reinoso Avecillas (2023) examines dual programs in Ecuador’s public
technical higher education system—a model that directly link the classroom and
the workplace. At the regional level, Ontaneda Jiménez and Mendieta Muñoz
(2022) show that institutional factors influence subnational economic growth,
reinforcing the idea that human capital formation operates within structures
that either enhance or limit it. These institutional conditions partly explain
why the returns on education—well established internationally by Lee and Wie
(2017)—do not automatically translate into better employment outcomes, and why
the digital and analytical skills highlighted by Van Laar et al. (2019) and
Rakowska and de Juana-Espinosa (2021) take on strategic value.
Bayesian Reasoning as a Professional
Practice
Bayesian statistics offers a framework particularly
attuned to the way professionals reason: one starts with prior knowledge,
updates it with evidence, and arrives at a conclusion expressed as a
probability. Zellner (1995) contrasts the Bayesian and frequentist approaches
to inference and decision-making, emphasizing the former’s consistency for
decision-making under uncertainty. Reilly (1976) represents the calculus
tradition that made Bayesian inference operational long before modern
computing. In professional practice, examples abound: Berry (2006) and Etzioni
and Kadane (1995) demonstrate the role of Bayesian methods in medicine and
public health; Kostoulas and Doi (2024) explain how likelihood ratios,
interpreted Bayesianly, guide clinical diagnosis; and Takramah et al. (2022)
apply hierarchical Bayesian models to estimate neonatal mortality with
spatiotemporal variation. In engineering and management, Akhavan Niaki and
Fallah Nezhad (2007) integrate Bayesian inference with stochastic dynamic
programming to make decisions in production processes, and Ferrara et al.
(2017) use robust optimization for stock market investment decisions. Abdallah
(2025) illustrates how Bayesian inference and probabilistic graphical models
support decision-making in artificial intelligence systems. However, Bayesian
reasoning is not intuitive: Talboy and Schneider (2016) demonstrate that even
professionals make systematic errors when updating probabilities, and that
brief training significantly improves their performance. This last point
connects the argument to education: if probabilistic reasoning is a
professional skill, then it must be taught deliberately.
Materials and methods
The study adopts a quantitative approach, with an
aim that is both illustrative and demonstrative: rather than providing an
exhaustive explanation of the labor market, it seeks to show how Bayesian
reasoning allows us to formulate professional questions and answer them using
open data. Transparency and reproducibility were therefore prioritized.
Open data from the World Bank (World Development
Indicators, WDI) were used, which in turn are based on modeled estimates from
the International Labor Organization (ILO). Three unemployment rate indicators
by educational level were used for Ecuador: basic education (SL.UEM.BASC.ZS),
intermediate education (SL.UEM.INTM.ZS), and advanced education
(SL.UEM.ADVN.ZS), each expressed as a percentage of the economically active
population with the corresponding educational level. The time series covers the
period from 2005 to 2024; data for 2020 are not available in the source, so the
final dataset comprises 57 observations (19 per level). The data are publicly
available and can be downloaded from the World Bank’s open data portal
(https://data.worldbank.org), allowing any reader to replicate the analysis.
Since these are modeled estimates, the results should be interpreted as a
description of trends, not as a labor force census.
Two models were specified. The first compares the
average unemployment rate across educational levels. Let the observed rate for
level g in year t be; we assume the model in Equation (1), with a mean and
standard deviation specific to each level, and weakly informative priors
defined in Equation (2), consistent with the percentage scale of the phenomenon
and deliberately uncommitted to any particular outcome.
The differences in means across educational levels
were calculated as derived quantities, along with the posterior probability
that each difference would be positive.
The second model estimates the temporal trend in
unemployment among those with advanced education using a Bayesian linear
regression (Equation 3), where b represents the annual change in percentage
points and the covariate is centered on the temporal mean. The priors are
defined in Equation (4); the prior for b is centered on the absence of a trend.
The posterior probability that the slope is positive was reported.
The estimation was performed using Hamiltonian Monte
Carlo sampling, via the NUTS algorithm implemented in the PyMC library (4
chains, 2,000 warm-up iterations, and 4,000 sampling iterations per chain;
target_accept = 0.95). Convergence was assessed using the R̂ statistic and the
effective sample size (ESS). Credibility intervals are reported as 94% highest
posterior density (HDI) intervals. The analysis, random seeds, and derived
datasets are preserved to ensure reproducibility; the choice of the Bayesian
framework follows the reasoning presented by Zellner (1995) and Reilly (1976).
Results
Figure 1 shows the evolution of the three
unemployment rates. At first glance, a persistent pattern is evident:
unemployment is systematically higher among those with intermediate education,
followed by those with advanced education, and significantly lower among those
with only basic education. Table 1 summarizes these characteristics.
Figure 1. Unemployment rates by educational level in Ecuador (2005–2024).
Note: Prepared by the authors using data from the
World Bank (WDI). Data for 2020 are not available in the source.
Table 1
Descriptive statistics on the unemployment rate by
educational level (2005–2024)
|
Educational level |
n |
Mean (%) |
SD |
Min. |
Max. |
|
Basic |
19 |
2.49 |
0.64 |
1.51 |
3.73 |
|
Intermediate |
19 |
6.15 |
0.92 |
5.00 |
8.32 |
|
Advanced |
19 |
4.95 |
0.80 |
3.83 |
6.49 |
Note: SD = standard deviation. Author’s own
calculations based on World Bank (WDI) data.
Bayesian comparison of levels
The comparison model converges without issues (R̂ =
1.00 and ESS greater than 10,000 for all parameters). Figure 2 presents the
posterior distributions of the average unemployment rate by level, and Table 2
summarizes the main results. The posterior mean was 2.49% for basic education
(HDI 94%: [2.20, 2.80]), 6.15% for intermediate education ([5.73, 6.59]), and
4.95% for advanced education ([4.57, 5.34]).
Figure 2. Posterior distributions of the average unemployment
rate by educational level.
Note: Each curve summarizes the uncertainty
surrounding the mean for the respective level.
The differences are clear. The gap between advanced
and basic education is estimated at 2.46 percentage points (94% HDI: [1.98,
2.95]), with a posterior probability of 1.00 that it is positive (Figure 3).
The difference between intermediate and basic education is even greater: 3.66
points ([3.10, 4.17]), also with a probability of 1.00. Between advanced and
intermediate education, the difference is −1.20 points ([−1.79, −0.63]): the
probability that unemployment among those with intermediate education exceeds
that among those with advanced education is 0.9998. In other words, the data
are practically conclusive in a way that, at first glance, seems
counterintuitive: in Ecuador, having more education is associated with a
higher—not lower—open unemployment rate.
Figure 3. Post-estimation difference between the mean for advanced education and
that for basic education.
Note. The lower band indicates the 94th percentile
of the HDI; the vertical line marks the zero value.
Table 2
Posterior summary of Bayesian models
|
Parameter |
Mean |
94% HDI |
P(> 0) |
|
Basic μ (%) |
2.49 |
[2.20, 2.80] |
— |
|
Intermediate μ (%) |
6.15 |
[5.73, 6.59] |
— |
|
μ Advanced (%) |
4.95 |
[4.57, 5.34] |
— |
|
Advanced − Basic (pp) |
2.46 |
[1.98, 2.95] |
1.00 |
|
Intermediate − Basic (pp) |
3.66 |
[3.10, 4.17] |
1.00 |
|
Advanced − Intermediate (pp) |
−1.20 |
[−1.79, −0.63] |
0.0002 |
|
Trend b (pp/year) |
0.068 |
[0.009, 0.128] |
0.98 |
Note: pp = percentage points; HDI = upper posterior
density interval; P(> 0) = posterior probability that the parameter or
difference is positive. R̂ = 1.00 in all cases.
Unemployment Trend Among Those with Advanced
Education
The second model estimates that unemployment among
those with advanced education is rising at a rate of 0.068 percentage points
per year (94% HDI: [0.009, 0.128]), with a posterior probability of 0.98 that
the trend is indeed upward (Figure 4). Although the pace is moderate, the
signal is consistent: over two decades, labor market participation among the
most highly educated has not improved in terms of open unemployment—quite the
opposite.
Figure 4. Bayesian trend in unemployment among those with advanced education in
Ecuador.
Note. The band represents the 94% credibility
interval for the mean; the points are the observed values.
How should we interpret the fact that, in Ecuador,
more education is associated with higher open unemployment? This is where the
careful quantitative reasoning that this article advocates as a professional
skill becomes indispensable. A naive interpretation would conclude that
studying “is pointless”—an inference as tempting as it is mistaken. The most
plausible explanation is that offered by the literature on dual labor markets
in economies with high levels of informality: those with less education cannot afford
to be unemployed and accept almost any job—often in the informal sector—so
their measured unemployment rate is low even though their precariousness is
high; in contrast, those with more credentials tend to wait and search longer
for formal employment commensurate with their training, which raises their open
unemployment rate without implying lower well-being. This “queueing” mechanism
for formal employment is consistent with the evidence from Ecuador: Rivera
Ávalos (2026) shows that higher education increases the probability of formal
employment, and MacIsaac and Rama (1997) document the role of regulations in
determining income. The open unemployment rate, on its own, does not capture
the quality of employment; interpreting it without this nuance leads to error.
The professional lesson is twofold. First, the
numbers do not speak for themselves: they require a model and an
interpretation. Bayesian inference helps precisely because it forces us to make
assumptions explicit (the priors) and provides conclusions in the form of
manageable probabilities—“there is a 98% probability that the trend is
upward”—rather than binary verdicts. Talboy and Schneider (2016) point out that
this type of reasoning does not come naturally and must be learned ; Berry
(2006), Etzioni and Kadane (1995), and Kostoulas and Doi (2024) demonstrate its
usefulness in medicine, while Takramah et al. (2022), Akhavan Niaki and Fallah
Nezhad (2007), Ferrara et al. (2017), and Abdallah (2025) do so in public
health, manufacturing, finance, and artificial intelligence. Ecuadorian
professionals who master these tools will be better positioned to make
decisions.
Second, this finding raises questions for education
and economic policy. The fact that returns on education exist—as confirmed by
Lee and Wie (2017)—but do not translate seamlessly into better labor market
outcomes suggests that the problem lies not only in the supply of skills but
also in the structure that absorbs them. Ontaneda Jiménez and Mendieta Muñoz
(2022) emphasize the role of institutional factors in subnational growth;
Reinoso Avecillas (2023) and Acosta and Stefos (2021) point to relevance and quality
assurance; Jiménez Cabrera (2021) and Jameson (1997) remind us that university
governance is a long-term factor. Training good professionals is necessary, but
it is not enough if the productive sector does not generate sufficient formal
employment. And here, the gaps in origin documented by Gamboa and Waltenberg
(2012), Gamboa and Krüger (2016), Guerra et al. (2026), and Martins and Veiga
(2010), along with the gender inequalities identified by Beekman and Ober
(2015) and Martínez et al. (2023), and the influence of attitudes described by
Lim and Chapman (2015), determine who ends up competing for those jobs.
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