# Network psychometrics and cognitive network science open new ways for
detecting, understanding and tackling the complexity of math anxiety: A review

M. Stella

CogNosco Lab, Department of Computer Science, University of Exeter, UK
<EMAIL_ADDRESS>
###### Abstract

Math anxiety is a clinical pathology impairing cognitive processing in math-
related contexts. Originally thought to affect only inexperienced, low-
achieving students, recent investigations show how math anxiety is vastly
diffused even among high-performing learners. This review of data-informed
studies outlines math anxiety as a complex system that: (i) cripples well-
being, self-confidence and information processing on both conscious and
subconscious levels, (ii) can be transmitted by social interactions, like a
pathogen, and worsened by distorted perceptions, (iii) affects roughly 20% of
students in 63 out of 64 worldwide educational systems but correlates weakly
with academic performance, and (iv) poses a concrete threat to students’ well-
being, computational literacy and career prospects in science. These patterns
underline the crucial need to go beyond performance for estimating math
anxiety. Recent advances with network psychometrics and cognitive network
science provide ideal frameworks for detecting, interpreting and intervening
upon such clinical condition. Merging education research, psychology and data
science, the approaches reviewed here reconstruct psychological constructs as
complex systems, represented either as multivariate correlation models (e.g.
graph exploratory analysis) or as cognitive networks of semantic/emotional
associations (e.g. free association networks or forma mentis networks). Not
only can these interconnected networks detect otherwise hidden levels of math
anxiety but - more crucially - they can unveil the specific layout of
interacting factors, e.g. key sources and targets, behind math anxiety in a
given cohort. As discussed here, these network approaches open concrete ways
for unveiling students’ perceptions, emotions and mental well-being, and can
enable future powerful data-informed interventions untangling math anxiety.

## Introduction

Anxiety is a distressing feeling aimed at avoiding a potential threat [71, 4].
By eliciting a sense of danger, anxiety contributes to avoiding negative
experiences [3]. Imagine walking alone in a wild forest. The thought of
meeting a predator would boost anxiety and favour a risk-averse behaviour,
e.g. avoiding shady spots. In clinical populations, anxious feelings of
distress can be triggered also in absence of a real threat [67]. When the risk
of encountering a wild predator is replaced by having to solve a mathematical
equation, some people might still perceive high levels of anxiety [2]. A
growing body of literature identifies math anxiety as a pathological feeling
of tension - different from general anxiety - which deeply inhibits access to
memory and thus critically impairs thinking, learning and remembering
mathematical knowledge [4, 10, 50, 71, 2, 103].

Math anxiety is shown to block a person’s reasoning when confronted with
mathematical situations, even simple ones like performing basic calculations
or understanding a simple plot [10]. This definition appeared back in the
1950s, with the pioneering work of Dreger and Aiken [27], who pointed out math
anxiety as a key factor behind students’ attitudinal difficulty with maths.
Subsequent works found out that math anxiety occurs at all educational levels,
from primary school to Higher Education [100, 50, 4, 24]. Overwhelmed by math
anxiety, students often end up failing/repeating mathematics, statistics or
physics courses relying on mathematical skills [82] and this translates into
lower academic performance and more time required for completing a degree [7,
47, 103], with annexed issues of increased economic burden and debt [10, 29].
Besides affecting students’ well-being and academic careers, math anxiety can
solidify negative attitudes of closure towards mathematics [100] and thus
discourage students from further pursuing careers in STEM [1]. Since it is
estimated that over 7 million new job positions will require math-related
skills by 2025 [73], these trends motivate an urgent need to better detect and
overcome math anxiety [1].

This review adopts a complex systems approach [51] to reviewing and
understanding key sources, impact and detection techniques relative to math
anxiety. Complexity stems from the fact that math anxiety does not affect a
single typology of maths students and it is not limited to the emotional
sphere [99, 24, 50, 58, 72, 1]. Crucially, math anxiety can be transmitted
along social interactions [62] and it can affect cognitive processing at both
conscious and unconscious levels in ways that we know relatively little about
[4, 50, 79]. To better understand such a complex clinical pathology, affecting
the mental well-being and career prospects of even well-performing students in
multiple ways [58, 1], quantitative models of knowledge are crucial [99, 81,
88, 23]. Merging behavioural and cognitive data together with models and
quantitative techniques from multiple fields is key to complexity modelling
[51, 75] and can be a powerful weapon for understanding and reducing math
anxiety, as discussed in this review. To this regard, particularly promising
are novel techniques from cognitive data science [83, 80, 91, 36, 95, 9],
which can map how individuals affected by math anxiety perceive and link
together different aspects of their experiences in potentially distorted and
stress-inducing ways.

### Review outline

This work is divided in two main parts. In the first part, a review of recent
literature from educational psychology identifies sources and targets of math
anxiety, outlining how even well performing students can be affected by such
clinical condition. The review outlines how this pathology can: (i) stem from
and get diffused along social interactions, and (ii) alter cognitive
mechanisms of memory load management. The second part contains a review of
psychometric scales, psychometric networks and cognitive network methods
measuring math anxiety. The review of these quantitative tools has the
potential to highlight key relationships and contexts both promoting and being
promoted by math anxiety. Reviewing these techniques across fields is crucial:
In the hands of professionals in Education, these psychological detection
techniques can enable simple yet powerful data-informed interventions for
detecting and fighting specific aspects of math anxiety.

## Part 1: A socio-cognitive profiling of math anxiety

Learning endeavours often include moderate levels of anxiety, due to a variety
of elements, including the educational challenge of passing a test or the
effort required for producing an essay [86, 53]. This first part of the
current work reviews the key elements that make math anxiety different from
general anxiety. Attention is devoted to recent studies outlining math anxiety
as a clinical pathology, affecting the cognition of even well-performing and
adult students [58].

### Math anxiety is an excessive response to math-related tasks

When performing a learning task or an exam, moderate levels of anxiety are
normal and can correlate positively with better academic performance [3, 12,
71]. Moderate anxiety boosts the production of adrenaline in the brain and it
activates a sequence of threat-reaction mechanisms that increase reactiveness
and quick thinking [71]. These aspects help interpreting the original finding
by Cassady and Johnson [12] that moderate anxiety had a positive effect over
test performance in a population of 168 undergraduate students. These results
were replicated by Wang and colleagues [103]. However, the interplay between
anxiety and performance was found to be non-linear. In fact, both works found
that even stronger levels of anxiety created reactions of panic and ended up
critically impairing, rather than enhancing, students’ performance [12, 103],
in agreement with other independent investigations [3, 2]. This negative
effect has been replicated multiple times in students performing mathematical
tasks [4, 79, 72].

Academic performance is not the only element influenced by math anxiety. A
survey of 40 high-school students by Espino and colleagues [29] highlighted
how math anxiety can drastically decrease students’ confidence in their
mathematical abilities. This lowered confidence translates into students
actively avoiding math-related courses, with detrimental effects over the
development of their mathematical literacy. Similar effects were found in
University [63] and other high school settings [40, 28]. This indicates that
widening the number of courses offering math-related skills might have only
limited positive effects over promoting mathematical literacy [32]. To boost
computational thinking [92, 104] and mathematical reasoning [100] a first
crucial step would be detecting math anxiety in a given learning environment,
including University settings [7].

### Math anxiety affects also well-performing students

As a debilitating negative emotional reaction towards maths, a preliminary
hypothesis might be that math anxiety stems from poor academic performance
[27]. Recent studies show that this claim is not supported by the data. An
investigation involving almost 1800 students in elementary and middle school
settings [24] showed that 77% of the students who experienced math anxiety
turned out to be normal-to-high achievers on standard curriculum tests. Hill
and colleagues [40] found that correlations between levels of math anxiety and
school performance surfaced at later educational stages. In a cohort of 1104
students attending middle and high schools in Italy, the authors found weak
negative correlations between arithmetic test scores and math anxiety. In a
population of 90 adults, Douglas and LeFevre showed how basic number and
spatial skills failed to predict the occurrence of math anxiety [26],
suggesting for the presence of more complex factors in addition to numeracy
skills behind math anxiety. Foley and colleagues reviewed the link between
math anxiety and academic performance through results from over 3,300 students
in the 2012 edition of PISA tests (Program for International Student
Assessment) [32]. Although a weak negative correlation was found between math
anxiety and academic performance, several countries in East Asia identified
patterns where high-achieving students exhibited also higher levels of math
anxiety (between 0.44 and 1.41 standard deviations above the mean [32]).
Analogous results were replicated by Yi and colleagues [106]. Foley and
colleagues also underlined that countries like Japan or Switzerland exhibited
close levels of average maths performance (536 and 531, respectively), yet
Swiss students exhibited considerably lower levels of math anxiety compared to
Japanese learners [32]. These differences indicate how the prevalence of math
anxiety in high performing students is a complex phenomenon, only weakly
correlated and thus partially explained by academic performance. Additional
evidence to this comes from the cognitive study of Beilock and colleagues [6],
who found that, in a population of 93 undergraduates, individuals with higher
cognitive skills (i.e. working memory capacity) displayed stronger levels of
math anxiety.

The above studies are relevant also for grasping a view about the prevalence
of math anxiety across different educational systems. This clinical condition
was detected in 63 out of 64 educational systems that participated in PISA
2012 and it affected roughly 17% of the over 3,300 students involved in the
studies [32, 106]. In a country with advanced STEM programmes like the US,
almost 93% of adults reported having experienced math anxiety [8]. These
quantitative results suggest that math anxiety is a global phenomenon that
might be vastly undetected in student populations when only academic
performance is considered [58].

To sum up, the above findings indicate that maths performance is only a
partial, limited estimator for the prevalence of math anxiety in student
populations. Whereas math anxiety can impact academic performance, the above
results indicate that even well performing learners can experience high levels
of distress [32, 6, 24, 26, 40]. This underlines the need for detecting math
anxiety via alternative approaches.

### Math anxiety, social interactions and cognitive mechanisms of working
memory

As an emotional condition, math anxiety affects the mental well-being of an
individual and influences their perception of themselves, the others and their
environments [3, 4, 50]. The process of understanding math anxiety should thus
occur on multiple levels, adopting in parallel: (1) an ecological psychology
approach [65, 62], e.g. math anxiety can stem from social interactions, and
(2) a cognitive framing [30], e.g. math anxiety can stem from specific
mechanisms of cognitive information processing. We will follow these two
directions in the remainder of this section.

#### Math anxiety can be aggravated by and transmitted through social
interactions

Ecological psychology is a core branch of psychological studies indicating the
relevance of social interactions in determining human personal, cognitive and
emotional identities [65]. Several recent studies indicated how math anxiety
is tightly connected with social interactions. Vast evidence from the
literature report that social learning environments and educational curricula
might be key sources of math anxiety in students [65, 72, 71, 50, 63]. Mutodi
and Ngirande [72] found that, in a random sample of 120 students, negative
experiences of failure and classroom settings devoid of inclusiveness both
boosted the prominence of math anxiety. The important role of teachers and
teaching practice in favouring the growth of math anxiety was confirmed also
by other recent studies [71, 105]. Moore and colleagues [71] pointed out how
the creation of positive learning environments - providing students with
feasible goals and rewards - could make maths more attractive and less prone
to induce anxiety. The authors underlined how instructors should focus on
creating a learning experience that stimulates students to have a positive
expectation about their learning, re-framing failure in maths as an enriching
experience, useful for personal growth and improvement, rather than as a
source of shame [105]. In fact, Wilson showed how eliciting shame over poor
academic performance bolstered math anxiety in classroom settings [105].
Several works [72, 65, 50, 86, 63] underlined that a more positive perspective
on maths failure should be built also in family settings, since parents were
found to play an important part in mitigating or worsening children’s and
teenagers’ fears about mathematics achievements [106, 65]. Maloney and
colleagues [62] found evidence for a contagion of math anxiety, which can be
transmitted within a family from parents to children. Once infected with
negative attitudes towards mathematics, children would grow up internalising
stereotypical perceptions (e.g. ”maths is hard” [82]) that could ultimately
favour the prevalence of maths anxiety and maths disengagement, making it
harder for teachers to promote math literacy [100, 63]. Maloney and colleagues
also found that this negative inter-generational feedback loop was weakened
whenever parents with high levels of math anxiety helped children less
frequently [62]. Analogous negative effects of parents’ attitudes over
children’s math anxiety were found also in other independent investigations
(cf. [65, 58]). Even simple pressure from parents and teachers was found to be
a significant predictor of math anxiety in over 3355 students from 65
countries [106]. In particular, pressure stemmed from students perceiving that
their educational role models, e.g. teachers and parents, valued performance
more than other skills relative to self-confidence and conceptual
understanding. In other words, performance anxiety [63, 69] was found to be
promoted by distorted perceptions of role models, with exacerbating effects
over math anxiety itself [106, 69]. Distorted perceptions of role models could
also amplify negative attitudes towards a student’s self-perceived confidence
in math skills, which co-occurred together with enhanced levels of maths
anxiety [58].

The above findings underline the complexity of math anxiety as a global
phenomenon, going well beyond educational places like classroom settings.
Nonetheless, classroom interactions are found to contribute to math anxiety.
Interactions between peers were found to transmit or worsen math anxiety by
facilitating emotional states where students might feel inferior to their
peers [72, 69]. Social interactions between peers were also found to transmit
and boost “stereotype threats” in maths [79, 45], in particular the myth of
“male superiority” in achieving success in mathematics [56]. In social
psychology, stereotype threats are subconscious biases representing incomplete
yet quick-to-grasp information about specific groups [84, 68, 85]. These
stereotypes can inform people’s behaviour on subliminal levels [85]. People
might be aware that some piece of knowledge about a group or category (e.g.
girls) might be completely unfounded and thus be a myth/stereotype [56] (e.g.
girls are worse at maths than boys). And yet, by being aware of the existence
of such stereotype, that group might concretely under-perform in tasks related
to that sphere of knowledge [84, 45, 28, 68]. For instance, girls aware of
their stereotypical portrayal as low-performing students in maths - compared
to “mythical” better performances of white male students in maths [56] \- were
found to perform worse than boys with analogous mathematical literacy in
computational tasks [79, 45]. This effect was attributed to anxiety stemming
from subconscious processing of stereotypical expectations [79] and was
replicated in different contexts [85, 28]. Crucially, a study over a sample of
144 students (75 women and 42 men) by Johns and colleagues [44] found that
raising awareness about stereotype threat could significantly improve
students’ computational performance, opening the way to future pedagogic
interventions reducing such effect.

The above complex interplay between stereotype threat, self-perceived
competence and differences in academic performance calls for the crucial need
to highlight key cognitive mechanisms at work in learning environments, as
potentially related to math anxiety and students’ performance.

#### A cognitive outlook on math anxiety

In addition to the distorted perceptions of social agents and interactions
that might aggravate math anxiety, one should also consider the key cognitive
mechanisms at work in such clinical pathology [2, 58]. Math anxiety alters
cognitive performance in accessing, processing and producing information [3,
4]. All these aspects are strongly related to human memory, the cognitive
system apt at storing and processing knowledge, stemming from neural activity
in the brain circuitry [102, 41]. From a cognitive perspective, human memory
can be classified in different sub-components (cf. [71, 3, 54, 70]):

  * •

episodic memory deals with storing recollections of personal experiences;

  * •

semantic memory generates conceptual knowledge about the world and deals with
the linguistic description of facts, ideas and events;

  * •

long-term memory deals with events that occurred in the distant past and are
crucial for determining future behavioural patterns;

  * •

working memory stores and processes simple facts and bits of information that
need to be combined while performing a computation.

Emotional processing [70], personality traits [5, 60] and psycho-pathologies
[6, 13, 14] can heavily alter access and retrieval from these interconnected
cognitive systems. Overwhelming evidence [71, 10, 3, 4, 12] has shown that
math anxiety in young adults (undergraduate students enrolled in Higher
Education) can severely slow down recall of knowledge from semantic memory
(e.g. remembering definitions or theorems) and also impair the maximum
processing load from working memory (e.g. combining many entries in an
equation in order to solve it).

Let us explore these two cognitive mechanisms through a computer science
analogy. Like a computer with limited random access memory (RAM) for running
applications, working memory in the human mind has a limited, finite capacity
for processing information [3, 12]. Such capacity is strongly affected by the
attention level devoted to each task and by stimuli present in the environment
[6, 30]. Attentional Control Theory [30] posits that anxiety can disrupt this
balance between attention and environmental checks/stimulation. In general,
working memory deals with processing not only information but also background
tasks, e.g. sitting in a lecture room and paying attention to other colleagues
or to the lecturer requires spatial coordination and reacting to others with
emotional/memory responses [2]. Multiple tasks, including emotion regulation
[70, 74] or information processing [12, 6], have to be performed in parallel
according to different capacities, as dictated by attention, up until maximum
load is reached [71, 4, 30]. Through a review of several cognitive
experiments, Moore et al. [71], Buckley et al. [10] and Luttenberger et al.
[58] indicated that math anxiety can lower down the maximum capacity of
working memory. This decrease corresponds to boosting resources allocated
mainly to two processes: (i) emotion regulation [37, 74, 70], i.e. a process
apt at containing or managing switches between positive and negative emotional
states, and (ii) negative rumination, i.e. an abnormal persistence of negative
thoughts affecting working, semantic and episodic memories [37, 10]. In other
words, current empirical studies and psychological coding theories of human
memory converge towards a model in which students affected by math anxiety end
up being overwhelmed with negative thoughts, which drain learners’ mental
capacity to perform computations [3, 4, 12, 6, 71, 76]. Notice that this
emotion-related draining could be only weakly correlated with levels of domain
knowledge of maths [4, 76, 46], addressing why math anxiety has been reported
to affect also well performing students (cf. [24] and [76]).

The above interplay underlines that math anxiety does not spawn from poor
domain knowledge or competence only but it is rather a complex phenomenon that
strongly depends on different emotional and information processing aspects.
Hence, methods powered by cognitive psychology might considerably improve the
detection, understanding and action against math anxiety.

## Part 2: Innovative approaches in the literature for detecting math anxiety

The previous Part outlined how math anxiety is a global, complex clinical
condition affecting almost all educational systems across the globe and poorly
predicted by academic performance [50, 65, 29, 24, 58]. These aspects
underline how crucial it is to measure math anxiety via alternative mechanisms
beyond performance itself [106, 32]. Promoting measurements of math anxiety
particularly in Higher Education would have mainly three beneficial effects:
(i) it would improve students’ learning experience [7], (ii) freer from
anxiety, students could be facilitated and encouraged to pursue successful
careers in a growing field like STEM [73], (iii) instructors could devise
class-tailored techniques specifically targeting key aspects of anxiety in
their classes. These points would disrupt a vicious bidirectional positive
feedback loop, where increased math anxiety lowers mental capacity and
academic performance, which in turn promote math anxiety itself [32]. This
disruption would contribute to the erasure of distorted, conflicting ideas
about maths [100] and foster more creative connections between mathematics and
other fields of knowledge discovery [91, 92, 104]. Let us proceed with a
review of novel psychological approaches to detecting and understanding maths
anxiety mainly in terms of psychometrics [36, 9, 39] and cognitive network
science [83, 41, 89].

### Psychometric approaches to detecting math anxiety

Psychometrics is a key branch of psychology dealing with measuring
psychological constructs like anxiety and, more in general, symptoms of
psycho-pathologies through quantitative scales [15, 64, 35]. As a pathological
state, math anxiety can be measured on a psychometric scale indicating the
severity of the condition in a given individual. These scales are often self-
assessed surveys [42, 43], where students need to self-evaluate the overall
severity of their anxiety and match their own experience against a given
description, i.e. an item. For instance, an item might be “taking an
examination in a math course” and students should have to rate how anxious
they feel when in the situation described by such item, on a Likert scale from
MIN (not anxious at all) to MAX (extremely anxious). Usually 5-point Likert
scales are used, so that MIN equals 1 and MAX equals 5. The Abbreviated Math
Anxiety Scale (AMAS) [42] outlined 20 different items relative to students’
learning of maths. The scale was validated through measures of internal
consistency across samples. In a population of over 1239 undergraduate
students enrolled in math-related programmes in Higher Education [42], the
items with stronger discriminative power for detecting high levels of math
anxiety in self-reports were found to be “listening to a lecture in math
class” and “taking an examination in a math course”. These items relate with
the above connection between academic performance and math anxiety [32].

Another scale commonly used for detecting math anxiety is the Mathematics
Anxiety Scale-UK (MAS-UK) [43]. This scale included 23 items and was validated
within a population of 283 British undergraduate students. Key groups of items
(i.e. factors) explaining the most variance in the observed measures of math
anxiety were: (i) self-evaluation of maths skills, (ii) calculations occurring
in everyday life, and (iii) observation of math-related contexts and actors.
These factors reflect the above discussed bidirectional influence that math
anxiety has over reducing self-confidence in mathematical skills [58, 106].

### Psychometric network science approaches to detecting math anxiety

Items should not be considered as separate elements of students’ experience
but rather as deeply interconnected elements of potentially distorted
perceptions [9, 78]. This is the key innovation of network psychometrics [9,
78, 64], which models how individuals respond to specific items as a network
combination of one or more latent variables, which cannot be measured directly
by the experimenters. Hence, network psychometrics postulates the existence of
a network structure driving the responses to items as a probability
distribution associated with latent variables [64]. The selection of a model
regulating responses to items is encoded by Item Response Theory [77], which
has fascinating analogues with the Ising model and the theory of interacting
spins in statistical mechanics [64]. In physics, spins interacting on a given
network topology have the tendency to align because of an underlying energy
minimisation principle or in presence of an external magnetic field [39]. When
aligned, individual spins can create collective, global phenomena like non-
zero magnetisation [77, 39]. Analogously to spins aligned in a certain way,
items might showcase specific responses (e.g. symptoms of anxiety). In
presence of multiple items showcasing network-related responses, collective
effects on the whole psychology of individuals might become evident (e.g.
anxiety as a disorder) [39].

Identifying a specific network model relating item responses to collective
clusters of symptoms is the key innovation behind graph exploratory analysis,
a technique introduced by Golino and Epskamp [36]. Graph exploratory analysis
deals with estimating a limited number of dimensions that retain the highest
explainable power of the variability observed in a large set of possibly
correlated features, as expressed by items. Network estimation is performed
through Markov Random Fields [78], where nodes represent random variables
connected by links indicating correlations or more general interaction metrics
between node-level variables. Golino and Epskamp [36] adopt Gaussian graphical
models [78] for capturing node interactions through an inverse co-variance
matrix between normally distributed variables. In their model, network
connections indicate the presence of a dependence between variables present
after conditioning for all the other network variables. Once built, this
network structure frames the problem of estimating psychological dimensions in
terms of network community detection, i.e. finding clusters of tightly
connected nodes [98, 20, 16].

In more mathematical terms, Item Response Theory [77] assumes that the item
responses $\textbf{y}_{P}$ produced by a person $P$ are the outcome of a
linear transformation of $M$ latent variables, encoded in $\eta_{P}$, through
a factor loading matrix $\Lambda$ \- to be estimated from the data - plus a
random error term $\epsilon_{P}$:

$\centering\mathbf{y}_{P}=\Lambda\mathbf{\eta}_{P}+\mathbf{\epsilon}_{P}.\@add@centering$
(1)

Considering the variance-covariance matrix $\Psi=Var(\eta)$ and the diagonal
matrix $\Theta=Var(\epsilon)$ (which assumes local independence), graph
exploratory analysis [36] reconstructs network topology of correlations
between items through the matrix K:

$\centering\textbf{K}=(\Lambda\Psi\Lambda^{T}+\Theta^{-1})^{-1},\@add@centering$
(2)

whose entries $k_{ij}$ indicate the local correlation between any two node-
level variables $i$ and $j$. In fact, it can be shown analytically [36] that
the off-diagonal elements of K can be interpreted as partial correlation
coefficients, whose clustering within communities can be identified through
network science techniques like walktrap community detection [34] or other
recent approaches in community detection for feature-rich networks [16].

The advantage of performing dimensionality/factor analysis through a complex
network is in being able to immediately visualise and interpret key patterns
of inter-dependencies through network science metrics [9]. This network
approach to understanding psychological multivariate data is quickly growing
within psychology [9, 78], with pioneering applications in intelligence
analytics [23], estimation of key traits in attitudes from political discourse
[20] and modelling of general anxiety symptoms in student [82] and non-student
populations [67]. The suitability of graph exploratory analysis in
investigating data from students is supported by a recent study performed by
Golino and colleagues [35]. Through the social networks of 7 teachers, the
authors investigated a population of 128 students between 2 and 11 years of
age, reconstructing their concentration skills as an interconnected four-
factor structure, i.e. emotion regulation, task engagement, empathy and
imagination [35]. An analogous adoption of graph exploratory analysis for
investigating the key dimensions of math anxiety in specific student
populations remains an unexplored gap, suitable for relevant future research.
Interested readers might explore this methodology with an R package available
online [34] and capable of extracting factors from both multivariate
psychometric data and text.

A related approach is the one by Siew and colleagues [82], who adopted network
psychometrics for investigating stats anxiety in undegraduate students. Stats
anxiety is an abnormal response to anticipating or performing statistical
computations, analogous but not fully overlapping with math anxiety [61]. Siew
and colleagues [82] built networks of items from the Statistical Anxiety
Rating Scale and correlated them according to the responses of two subgroups
of Higher Education students, suffering from high ($N_{1}$ = 115) and from low
($N_{2}$ = 113) stats anxiety. These two networks of item correlations
displayed a difference of crucial relevance for math anxiety. Students
suffering from higher stats anxiety organised their own responses to make the
idea “maths is hard and necessary to excel in stats” considerably more central
(in terms of network metrics like strength or closeness [83]) than what was
found in students with lower stats anxiety. This finding was validated also in
another study adopting a cognitive network approach [80]. The evidence
presented by Siew and colleagues [82] indicates that math anxiety might thus
be a crucial aspect of stats anxiety, underlining the need to address and
alleviate negative perceptions of mathematics across disciplines.

The above works underline how network psychometrics [78, 36] has recently
produced a range of powerful techniques that could highlight and connect
different complex aspects of math anxiety. This represents a fascinating,
quantitative research direction that has been explored for other types of STEM
anxiety [82] but not yet for math anxiety.

### Cognitive network science approaches to detecting math anxiety

Scales are powerful tools for investigating psychological constructs [55].
However they suffer from some limitations like potential biases and
uncertainty in rating decisions [11]. Another key limitation of scales is that
they rely on pre-determined items, that have to be identified a priori by the
experimenters [42]. However, the experience of specific groups of individuals
might substantially differ and present aspects that might escape from a pre-
determined/fixed list of items [78]. Differently put, a single list of items
pre-determined by experimenters might not capture the full complexity of the
ways specific elements are perceived, both cognitively and emotionally, by a
given specific audience.

This limitation can be overcome by accounting quantitatively for the different
ways in which groups or individuals structure their own knowledge and emotions
around concepts [95, 17, 94]. Giving structure to knowledge is a paramount
task of cognitive network science [83, 13, 60, 41], a research area dealing
with modelling knowledge representations in the human mind as networks of
interconnected conceptual elements, e.g. concepts linked by semantic [54],
syntactic [38] or phonological associations [102]. This field originated
within psycholinguistics, where models of concept representations in the human
mind employed network structures way before the advent of network science
[25]. Overwhelming theoretical and empirical research over the years has shown
that mental representations of knowledge are highly structured [25, 102] and
such structure influences a variety of phenomena related to knowledge
acquisition [38, 90] and processing [60, 59, 55, 14]. For instance, the length
of the shortest path between any two concepts in a network, i.e. network
distance [90], was shown to be predictive of normative language learning in
young children [90, 89, 102], creativity levels and curiosity in healthy
populations [93, 59, 5, 49], word production, picture naming and lexical
access in semantic and working memory in healthy [54, 25] or clinical
populations [14, 48, 107].

From the perspective of Education Science [81], models of cognitive networks
were used for mapping both the mindsets of educational agents (e.g. students
[80, 91, 46]) and of educational supports (e.g. textbooks [52]). Koponen and
Nousiainen [52] showed that two textbooks containing the same knowledge about
physics, i.e. the same set of concepts, portrayed it with widely different
connections, giving rise to different clusters of concepts and connections
between them. Similarly, Kapuza and colleagues [46] showed that the cognitive
networks of expert and novice learners portrayed the same sets of concepts as
connected in different ways, with experts identifying key concepts as being
network nodes with more connections. Siew [80] showed that networks of cue-
target word recalls from semantic memory (i.e. free associations [21]) were
found to be correlated with academic performance in a population of 104
undergraduate students [80]. After controlling for network size, cognitive
networks displaying larger network distances between concepts were associated
with better academic performance [80]. This indicated a positive trend between
academic scores and being able to build long chains/cycles of conceptual
associations in a mental representation of knowledge. Longer network distances
in cognitive networks of memory recalls were found to characterise more
semantically remote associations [54, 101] and thus identified people with
higher levels of creativity [5, 93, 49]. These approaches underline how much
variety and insights can be encapsulated within conceptual associations as
coming from different groups or sources. Cognitive network science provides a
powerful framework for extracting information out of such associations [83,
41, 88].

The work of Stella and colleagues [91] implemented a quantitative detection of
math anxiety via cognitive network science. The authors reconstructed
associative knowledge and emotional perceptions around STEM concepts in two
populations: (i) 159 high school students enrolled in math-heavy curricula and
(ii) 59 STEM researchers. Memory recalls from semantic memory were extracted
by participants through a cue-target continued free association game [21].
Participants were provided with a list of 50 words that were used as cues for
stimulating semantic memory and they could respond to each cue by writing up
to three free associates, e.g. reading “maths” made a participant immediately
think of and react with “passion”, “creativity” and “hard”. These free
associations, representative of semantic memory patterns [54, 21, 49, 48],
were enriched with emotional labels obtained from individual self-assessments
(e.g. how positively would you rate the concept of “maths”?). Both these tasks
required a simple spreadsheet and less than 1 hour for completion [91]. The
resulting network linking cues and targets was coined as forma mentis (Latin
for ”mindset”) network (FMN). Two FMNs were built, one representing students’
and one representing STEM professionals’ knowledge and emotions.

Figure 1 portrays the semantic/emotional frames of students (left) and STEM
experts (right) around ”mathematics”. Students displayed an overwhelmingly
negative perception of “maths” and associated it predominantly with other
methodological ideas (e.g. “equation”, “formula”, “integral”, etc.). These
concepts were rated by students as being mostly negative. Through cross-
validation with an independent valence dataset by Fairfield and colleagues
[31], Stella and colleagues found that concentrations of negative associations
corresponded to increased levels of anxiety. Not only maths, but also other
STEM concepts like physics [91], statistics [88] and computational thinking
[92] featured a concentration of negative associations and thus reported
increased levels of anxiety. These trends were absent in the representations
of knowledge coming from STEM experts. Hence, the quantitative reconstruction
of students’ mindsets via cognitive network science provided further evidence
that math anxiety persists even in populations of students well trained in
mathematics, in agreement with previous findings [50, 7, 32, 106]. Anxiety did
not affect students’ perceptions of science [91], teachers [95] or reasoning,
in general [92]. Negative relationships were almost completely replaced by
positive ones in the representations of knowledge from STEM researchers, who
rather associated maths with positive links to “creativity” and “art”. These
cognitive associations were missing in the ways of thinking of students [91,
95]. Future pedagogic research could use such quantitative data for enhancing
students’ learning experience by favouring those specific creative conceptual
associations present in the minds of STEM professionals [92] but absent in
student one’s. Differently put, by reading the cognitive networks coming from
STEM experts, education professionals might have an additional data compass
for enhancing students’ learning experiences and reducing math anxiety through
specific associations.

Figure 1: Cognitive networks showing how students (left) and STEM experts
(right) frame and perceive the concept of ”mathematics”. Colours indicate
negative (red), positive (cyan) and neutral (black) valence perceptions as
expressed by students and STEM experts. Links indicate memory recalls, showing
which concepts are associated to ”mathematics” in the minds of students and
STEM experts.

### Recommendations from the literature for fighting math anxiety

To sum up, the above review indicates that math anxiety is a complex
phenomenon, spreading along social interactions [62] and affecting memory,
information processing and self-confidence of both low-achieving and well-
performing students [3, 4, 40, 106]. Such complex and globally diffused
phenomenon [32] cannot be detected by considering academic performance only
but it should rather be identified through psychological inquiries of
students’ minds [81]. Recent achievements from psychometrics [36, 82] and
cognitive network science [91, 95, 80] can be powerful tools up to the task.
Psychometric scales can measure the gravity of math anxiety in students
through a self-assessment of distressing experiences [42], going well beyond
simplistic detection of math anxiety based on performance [106]. Novel methods
from psychometric network science like graph exploratory analysis [36, 34]
could shed light on the key dimensions of students’ perceptions and
experiences of math anxiety [82]. An analogous objective can be fulfilled by
cognitive network science [83, 13, 41, 95], which pointed out that math
anxiety persists even in the mindsets of students enrolled in math-focused
teaching curricula [91] and it is related to a dry perception of the
discipline, lacking creative links present in the mindsets of STEM
professionals [88]. Combined together, cognitive networks might identify
evident and latent concepts in students’ experience, which would then be
identified as suitable items for building psychological constructs through
graph exploratory analysis. The resulting cognitive multi-layer network
structure [22] could shed light on the organisation of key sources and targets
of math anxiety in students, enabling a new generation of effective
interventions. Notice that this synergy might include not only conceptual
relationships, like in concept maps [46], but also feature affective/emotional
perceptions and thus enable a complete conceptual/emotional profiling [94, 33]
of students’ mindsets.

Both these techniques are simple enough that instructors might quickly use
them for their own pedagogic practice. Forma mentis networks require simple
spreadsheets and less than 1 hour for data gathering [91]. Cues might be
selected as to replicate Stella and colleagues’ studies [91, 88, 95, 92] or to
address specific needs of a given classroom. Graph exploratory analysis can be
automated via a convenient R package [34] and follow items provided by math
anxiety scales [42, 43].

Using these models as a compass, teachers might then access not only what type
of knowledge is available to their students but also what kind of positive or
negative emotional states, like math anxiety or test anxiety [28, 88], might
be affecting their class in a given context. These cognitive/psychological
networks might provide also quantitative landscapes of students’ creativity
and personalities [101, 49, 5, 41], going well beyond student assessment based
on quizzes only. Why would these analyses be useful? Letting students face
their own emotions, as highlighted by network methodologies, and training them
in emotional regulation would be a powerful way for reducing the negative
effects of math anxiety, as suggested by recent studies [74].

In this way, a key question is relative to understanding more in detail which
educational interventions might be guided by the above quantitative
techniques. Let us discuss this aspect by concentrating over math anxiety and
by following four distinct directions of relevance for students’ learning
[51]: (i) teaching environments, (ii) teaching styles, (iii) assessment
methods and (iv) pedagogic innovation of maths curricula.

Teaching environments can affect the occurrence and gravity of math anxiety
[3, 4, 97, 7]. Considering a blend of face-to-face and online teaching can
have beneficial effects. By cutting the pressure of peer interactions and
embedding students in environments as familiar as their very own houses,
distance learning has been shown to drastically reduce math anxiety while
favouring concentration over mathematical tools and reasoning [97]. Analogous
performance trends were found even during the current health crisis with
COVID-19 [87]. Using psychometric/cognitive techniques to understand how
students exactly perceive and frame distance learning, in relation to
mathematics, could lead to innovative hybrid teaching environments, tailored
around the needs of specific cohorts [7, 75]. Assessing the impact of such
blended techniques, mixing face-to-face and remote learning, could open the
way to substantially reducing math anxiety across different educational
systems.

The psychometric results reviewed above converge towards the idea that math
anxiety corresponds with a lowered self-confidence in maths skills [82, 42,
43]. Cognitive network science, instead, pointed out a dry perception of
maths, where students think of mathematics as a purely dry technical
discipline, devoid of creative or real-world applications [91, 92]. Hence,
teaching styles boosting self-confidence and creative aspects of maths
literacy could have beneficial effects over reducing math anxiety. These
styles could be designed and enhanced by education professionals sharing their
”wisdom of practice” along networks of teaching practice [69, 51].

Assessments and tests can enhance anxiety, a phenomenon known as test anxiety
[61, 3] and linked also to math anxiety [40]. Guided by the psychometric and
cognitive analyses reviewed above [80, 42, 95], a way to reduce math anxiety
could be devising more formative assessments, reducing the fear of failure and
performance anxiety coming from peer pressure [4, 50, 47]. More opportunities
for students to measure their skills without performance anxiety should be
considered. Flipped classroom or peer learning techniques from Education
research [63, 66], where students become active agents of their own learning,
could be beneficial in implementing anxiety-free formative learning
experiences [7].

Fighting math anxiety heavily relies on changing distorted perceptions of
mathematics [80, 82]. This could be tackled by promoting creative associations
between specific concepts, as suggested by previous works [91, 88, 2].
Boosting students’ learning progression through specific semantic frames or
conceptual associations is proving successful in physics learning [53] and it
might prove a powerful technique also for enhancing the perception of maths.
Hard mathematics is not always essential to master other subjects (see the
above discussion of [82, 91, 92]). The fact that specific groups might be
better or worse at it is currently not supported by scientific evidence once
subconscious biases are accounted for (cf. [79]). Fighting these distorted
perceptions can have great repercussions over eliminating the barrier posed by
unconscious biases [85, 79, 50], ultimately favouring access to maths by a
diverse set of students [65]. Channelling mathematics as a tool enabling a
deeper understanding of our world can be a concrete way of disrupting maths
stereotypes, as shown by the NetSciEd’s initiative in the United States [18,
19]. Academics and education professionals brought the mathematics of complex
networks in high schools, involving both teachers and high schoolers into
outreach programmes about understanding the real world with network science.
Through a series of workshop events, scientific output and students’ feedback,
analogous initiatives found that students translating mathematical concepts
like matrices or equations into more concrete network representations ended up
showing an enhanced appreciation of mathematics and decided to pursue a career
in STEM (cf. [19]). The problem with underlining applications of a discipline
like maths is that applied maths does not represent the complete landscape of
mathematical knowledge. As cleverly summarised by Strogatz in [96],
mathematics contains an intrinsic beauty in its abstractness and universality,
beyond its applications. This is why attention should be given towards
presenting students with a more diverse landscape of maths achievements and
relevance. Students should be presented with a complete profile of
mathematical discoveries, both theoretical and applied, encompassing also
historical background and relevance. By highlighting the struggles and
achievements of mathematical physicists, Lommi and Koponen [57] found lectures
about the history of science solidified students’ semantic memory and domain
knowledge around curriculum subjects. Analogously, McIntyre and colleagues
[68] found that computational performance of women in academic tests could be
enhanced by providing successful examples of women scientists, thus countering
stereotype threat [79] and providing contextual knowledge in classroom
settings. These approaches provide evidence that merging mathematical
knowledge with the history or context behind it might boost students’
perceptions, reducing anxious and dry distorted mindsets [95, 81].

### Conclusions

This multidisciplinary review outlined: (i) the socio-cognitive components of
math anxiety, (ii) crucial key ways for measuring such pathological condition,
and (iii) the most relevant investigations that provided evidence useful for
fighting it. Reducing math anxiety in student populations is key to improving
their mental well-being and preparing them better to the opportunities of the
prospective job market. More research is required for understanding the
unconscious/hidden sources of math anxiety, which is widely present even in
well-performing learners [24, 106, 32]. This makes academic performance a
rather poor way for estimating the prevalence of math anxiety among students,
a detection task that should and could be tackled by recent advancements from
network psychometrics [82, 36] and cognitive network science [91, 80]. These
methods are simple enough to be used in classroom settings and can crucially
outline where math anxiety stems from and what kind of processes are drained
by it. Access to such knowledge, tailored around specific cohorts and
settings, can provide powerful guidance for effective strategies enhancing the
appreciation of mathematics in the near generations of learners.

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