Epistemic emotions and pre-service mathematics teachers’ knowledge for teaching

Abstract

Affective and cognitive processes may be jointly researched to better understand mathematics learning, paying special interest to emotions related to knowledge acquisition. However, it remains necessary to explore these processes in studies linked to the education of pre-service mathematics teachers. This study aims to characterize epistemic emotions in different practices linked to the practice of mathematics teaching: problem-solving, anticipating what would happen with the students and reflecting on classroom implementation. It considers the theory of Mathematical Working Spaces to describe the mathematical and cognitive dimensions generated by epistemic emotions, paying special attention to the cognition-affect interaction and the workspace created. The results indicate that the epistemic emotions of the pre-service mathematics teachers associated with the distinct practices were different. Differences are observed in the interaction between emotions and cognitive epistemic actions, depending on whether the pre-service mathematics teachers analyze them within the framework of their own solving or anticipate them in their students. This reveals how personal work relates to what is considered to be suitable for students. Specifically, certain antecedents and consequences have been specified for the emotions of surprise and boredom in relation to the characteristics of the optimization problems and the cognitive activity of the subject when solving them. These results highlight the need to enhance the education of pre-service mathematics teachers through training that helps regulate their epistemic emotions and model effective strategies for regulating their own emotions and those of their students.

Full text

Vol.:(0123456789)
ZDM – Mathematics Education
https://doi.org/10.1007/s11858-024-01624-5
ORIGINAL PAPER
Epistemic emotions andpre‑service mathematics teachers’ knowledge
forteaching
InésM.Gómez‑Chacón1 · JoséM.Marbán2
Accepted: 5 August 2024
© The Author(s) 2024
Abstract
Affective and cognitive processes may be jointly researched to better understand mathematics learning, paying special interest
to emotions related to knowledge acquisition. However, it remains necessary to explore these processes in studies linked to
the education of pre-service mathematics teachers. This study aims to characterize epistemic emotions in different practices
linked to the practice of mathematics teaching: problem-solving, anticipating what would happen with the students and
reflecting on classroom implementation. It considers the theory of Mathematical Working Spaces to describe the mathemati-
cal and cognitive dimensions generated by epistemic emotions, paying special attention to the cognition-affect interaction and
the workspace created. The results indicate that the epistemic emotions of the pre-service mathematics teachers associated
with the distinct practices were different. Differences are observed in the interaction between emotions and cognitive epis-
temic actions, depending on whether the pre-service mathematics teachers analyze them within the framework of their own
solving or anticipate them in their students. This reveals how personal work relates to what is considered to be suitable for
students. Specifically, certain antecedents and consequences have been specified for the emotions of surprise and boredom
in relation to the characteristics of the optimization problems and the cognitive activity of the subject when solving them.
These results highlight the need to enhance the education of pre-service mathematics teachers through training that helps
regulate their epistemic emotions and model effective strategies for regulating their own emotions and those of their students.
Keywords Epistemic emotions· Mathematical Working Spaces· Pre-service mathematics teacher· Teacher education·
Surprise· Boredom
1 Introduction
The articulation between theory and practice continues to
be a key issue in research, given that the knowledge of pre-
service teachers and the use of this knowledge are dependent
constructs. Multiple studies have suggested that the practices
making up initial teacher training are an important scenario
in which theoretical knowledge converges with practical
knowledge (Bastian etal., 2024; Potari, 2021). Furthermore,
the need to jointly consider the affect and cognitive dimen-
sions to study the learning of PMTs has become evident
(Barnes, 2021; Chamberlin & Sriman, 2019; Kuntze & Dre-
her, 2015; Sakiz etal., 2012).
Some authors, such as Pekrun (2014), have suggested the
need for additional research on the emotions of pre-service
mathematics teachers within the educational context. Emo-
tions differ depending on the events and objects triggering
them (Pekrun & Loderer, 2020). In terms of object focus,
general groups of emotions and moods have been typified
based on multiple inputs that may be the most important for
learning: achievement emotions, general and specific moods,
epistemic emotions, topic emotions, and social emotions.
Few studies have examined epistemic emotions. Extensive
research has been conducted on pre-service mathematics
teachers’ achievement emotions [e.g., Bekdemir (2010)
on anxiety], social emotions [e.g., Jenßen etal. (2022) on
shame] and emotions in relation to teaching mathematics
* Inés M. Gómez-Chacón
igomezchacon@mat.ucm.es
José M. Marbán
josemaria.marban@uva.es
1 Instituto de Matemática Interdisciplinar, Facultad de
Ciencias Matemáticas, Universidad Complutense de Madrid
(UCM), Madrid, Spain
2 Dpto. Didáctica de las Ciencias Experimentales, Sociales y
de la Matemática, Facultad de Educación y Trabajo Social,
Universidad de Valladolid, Valladolid, Spain

I.M.Gómez-Chacón, J.M.Marbán
[e.g. Marbán etal. (2021) on enjoyment]. Many of these
studies have been carried out during pre-service in primary
school, but very few works on epistemic emotions have been
conducted on pre-service teachers in secondary school.
Future research is necessary to better understand certain
factors: epistemic emotions when performing mathematics,
effects on pre-service teachers' knowledge of mathematics,
or effects or predictions on subsequent teaching.
Recent studies on teacher assessment of students' emo-
tions (Kanefke & Schukajlow, 2023) indicate that pre-ser-
vice teachers establish differences based on the situational
interest and type of mathematics tasks. They also highlight
that, when faced with the same problem, the judgments
made by pre-service teachers regarding students' emotions
can be different from the emotions that they actually feel
when solving it. These studies conclude that it is neces-
sary to consider perspectives regarding the implications for
theoretical models of pre-service teacher judgments and the
consequences that they have on teacher education. Other
studies (Cai & Leikin, 2020; Gómez-Chacón, 2018; Schin-
dler & Bakker, 2020) have highlighted the need to describe
the combined nature of cognition, motivation and emotion
in the learning and teaching of mathematics. Therefore, it
may be interesting to consider perspectives for the train-
ing of PMTs that consider the cognition-affect interaction
in their professional knowledge for teaching. To recognize
epistemic action when performing mathematics is to recog-
nize cognitive epistemic actions (actions belonging to the
observable mathematical knowledge of participants) and
affect epistemic actions (epistemic emotions,1 epistemic
beliefs, values and epistemic goals).
Epistemic emotions are defined as emotions arising
when their object is knowledge and the process of know-
ing (Gómez-Chacón, 2017; Muis etal., 2015; Pekrun etal.,
2017). Epistemic emotions (surprise, curiosity, enjoyment,
confusion, anxiety, frustration or boredom) are triggered by
knowledge, the characteristics of the tasks and the cognitive
activity of the subject when solving them. Distinct studies
(Chevrier et. al., 2019; D'Mello & Graesser, 2012, Gómez-
Chacón, 2017; Schukajlow etal., 2017) have explored how
some of these epistemic emotions are beneficial or counter-
productive to learning, identifying the origin that generates
them. Our perspective assumes the influence of emotions in
the configuration of personal mathematical work at two lev-
els (Person and Task-Person). Emotions may differ from one
person to another (trait emotions) but a specific task can also
bring out state emotions, dynamics that unfold during the
phases of performing a mathematical task at the Task-Per-
son level. Theories such as control-value theory (Pekrun &
Perry, 2014) suggest that the development of emotions can
be explained by evaluations (appraisals) of situations. Study
results (Chevrier etal., 2019; Muis etal., 2015; Pekrun etal.,
2014; Schubert etal., 2023; Vogl etal., 2019) highlight con-
trol, value, novelty, complexity and achievement (or epis-
temic goals) as antecedents of epistemic emotions as ele-
ments for a theoretical model. They also have considered the
effects on planning, motivation, cognitive and metacognitive
strategies and learning outcomes, which raises the question
whether, and how, they relate to each other. In the context of
Mathematics Education, an attempt has been made to detail
various antecedents in relation to mathematical knowledge.
For example, epistemic emotions such as surprise, confu-
sion, frustration and curiosity are considered important in
students’ mathematical thinking (Goldin, 2004; Gómez-
Chacón, 2000; Hannula, 2012; Schukajlow etal., 2017), in
the processes of demonstration (Schubert etal., 2023), in
visualization processes (Gómez-Chacón, 2018) and in the
use of heuristics in problem-solving (Gómez-Chacón, 2017).
Studies from the perspective of pre-service teachers are
necessary to understand how emotions act on mathematical
reasoning, their antecedents and consequences. Numerous
studies have indicated that the epistemic emotion of surprise
occurs as an initial reaction to information that is inconsist-
ent with prior knowledge. Likewise, they highlight bore-
dom as a crucial point at which students disengage from
activities, often a consequence of the frustration emotion
(D'Mello & Graesser, 2012). These results may contribute to
the knowledge of the PMT for teaching. Therefore, it would
be interesting to examine questions regarding the emotional
experience of the PMT (self-awareness), but also with regard
to the perception of the emotional experience of others in
mathematics (awareness of the task and of accompanying
another).
Achievement and epistemic emotions are the dominant
emotions that are triggered in the solving of a specific task,
dynamically influencing the regulatory phases of learning.
Therefore, regarding the professional knowledge to be devel-
oped by a PMT, it is important to pay attention to the char-
acteristics and mathematical processes of solving tasks that
may be catalysts of emotions. In this study, we focus on the
solving of optimization tasks. Mathematical optimization
is an essential mathematical concept to address everyday
problems. In the Spanish curriculum, they are studied at a
high school level (16–18years of age) (Balcaza-Bautista
etal., 2017) and a PMT must be aware of this knowledge
and manage it in the classroom).
The theoretical foundations of the study are presented
below, describing the methodological design used as well as
the results obtained in order to, finally, present conclusions
derived from this research and suggestions for future studies.
1 In this study, we focus on epistemic emotions.

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
2 Theoretical framework
Regarding the characterization of the professional devel-
opment of the teacher, the theory of Mathematical Work-
ing Spaces (MWS) has been considered in distinct stud-
ies (Gómez-Chacón, 2022; Gómez-Chacón etal., 2016a,
2016b). The dynamism of the theory has been demon-
strated, as well as how it can provide methodological tools
to advance the description, understanding and creation of
the mathematical work, establishing the connection between
the practices of the teachers and the development of the stu-
dents’ learning in the real-world classrooms. This theory
focuses on the detailed analysis of the mathematical work
in which both students and teachers actively participate.
It emphasizes mathematical content, and the term work
involves three aspects related to its execution and develop-
ment: the goal of the work, distinguishing between a simple
activity, assigning an objective to a long-term action; the
processes that are related to the procedures and the limita-
tions of the execution of certain tasks and the results of the
work that should be validated and coherent within the math-
ematics domain under consideration (Kuzniak etal., 2022).
Within the framework of the theory of MWS, mathemati-
cal situations/problems or tasks are considered as objects
involved in mathematics practice, from whose solution the
concepts and procedures emerge. In this study, the theory of
MWS is used to describe the mathematical workflow in the
cognitive-affect interaction. Through the lens of the MWS,
the individual’s development of appropriate mathemat-
ics work is a gradual and progressive process that builds
bridges between the epistemological and cognitive planes.
The epistemological plane consists of three poles (Fig.1):
referential (in close relation to the mathematical content,
based on definitions, theorems, properties and axioms), rep-
resentamen (semiotic signs used, which may be geometric
images, algebraic symbols and graphics) and artifacts (mate-
rial or symbolic). This plane refers to the theoretical part
of the mathematical work that will support the deductive
discourse of proof and justification. The cognitive plane
refers to processes and procedures used by individuals in
the task-solving activity. It is structured around three cogni-
tive processes: visualization, construction and proving. The
functioning should not be understood as the union of indi-
vidual components located on the epistemological and cog-
nitive levels, but as links activated by three geneses, semi-
otic, instrumental and discursive genesis, which articulate
both planes. Therefore, in the configuration of objects and
processes associated with a mathematical practice, different
associated dimensions are established: semiotic (referring
to natural language, mathematical symbols, visualization),
instrumental (establishes the link between artifacts and con-
struction processes) and discursive (mathematical reason-
ing in the proof, using theoretical concepts and discursive
validations).
The productions of the PMTs related to the optimization
problems are analyzed while considering the relationship
between the research model of the Mathematical Working
Spaces (MWS) (Fig.1) and the modeling cycle (Blum &
Borromeo-Ferri, 2009) (Fig.2), which may be considered a
first cycle for solving the modeling of a task. When a student
begins at a given situation, it is assumed that a horizontal
mathematization process starts, which is the basis for bring-
ing the problem situation to a mathematical domain. Next,
a vertical mathematization process takes place in which the
MWS framework and the modeling cycle can interact with
each other.
Developing modeling skills (Blum & Borromeo Ferri,
2009) mobilizes mathematical notions and objects from
distinct mathematical domains, where the knowledge that
can be learned by students is based on arguments belong-
ing to different domains (analysis, algebra, geometry, etc.).
This gives rise to different forms of mathematical work
within an arithmetic/geometric, calculation or real analysis
paradigm (see optimization task, Sect.4.2). The modeling
cycle (Fig.2) is not taken in its entirety: the focus is on
phases 3 to 5 of the cycle and the objective is to analyze the
Fig. 1 Mathematical Working
Spaces (MWS), geneses and
vertical planes

I.M.Gómez-Chacón, J.M.Marbán
mathematization process through the MWS model when stu-
dents solve a modeling task (e. g., Montoya Delgadillo etal.,
2017,Nechache &Gómez-Chacón, 2022). The theoretical
elements provided by the MWS model allow us to identify
the mathematical domain of resolution, representations and
knowledge in the work of PMTs, paying special attention to
the cognition-emotion interaction.
In the theory of Mathematical Working Spaces, there
are three types of mathematical work associated: Reference
MWS, defined by theoretical criteria aimed to organize cer-
tain knowledge according to the epistemological principles
of the mathematical domain studied; personal MWS, related
to the individual's own work, without necessarily having a
teaching intention, defined by the way in which the individ-
ual constructs their own mathematical work and, finally, suit-
able MWS, related to an individual (teacher or researcher)
who makes sense of mathematical content designed for
teaching in a given context (Kuzniak etal., 2022).
Teachers aim to configure the suitable MWS so that their
students can perform mathematical work according to the
reference MWS. It is at this level of the suitable MWS where
mathematical learning occurs. In effect, when solving prob-
lems, both high school students and PMTs face the math-
ematical task with their own knowledge and cognitive and
affect processes, which are shaped by their personal MWS.
However, the work carried out may not conform to what is
expected and defined by the educational organization. There-
fore, the teacher's goal in implementing the suitable MWS
is to more closely align students' personal MWS with the
expectations of the teaching organization (Menares-Espinosa
& Vivier, 2022).
In this context and considering the indicated concepts,
issues related to the description, characterization or (trans)
formation of the mathematical work into distinct practices
linked to the practice of teaching mathematics are key to
characterizing epistemic emotions. Analyzing the mathemat-
ical work of PMTs through the lens of the MWS allows us to
reflect on the interaction between cognition and affect in a
context of optimization problems. It reveals how to consider
epistemic emotions in teaching practice in order to configure
suitable the MWS. It also helps understand the connections
between theory and practice in the affect dimension and
between the personal MWS of the PMT and the configura-
tion of the suitable MWS.
3 Research questions
This study examines how PMTs characterize the epistemic
emotions associated with the different practices linked to
the practice of mathematics teaching: in problem-solving
(personal MWS), anticipating what was happening with the
students, and reflecting on classroom implementation (con-
figuration of the suitable MWS). Based on this objective, the
following question was presented:
How do the epistemic emotions reported by the PMTs
when solving optimization problems differ from those when
they anticipate the emotions and strategies of high school
students when solving those same problems?
This research question seeks to contribute to the under-
standing of PMTs regarding epistemic emotions (Kanefke
& Schukajlow, 2023), as well as the knowledge that PMTs
should develop regarding epistemic emotions for the practice
of mathematics teaching. Therefore, it is necessary to pay
attention to the cognitive and epistemic aspects of math-
ematics knowledge (Gómez-Chacón, 2017; Schubert etal.,
2023) and to the importance of personal MWS on the estab-
lishment of suitable MWS.
Fig. 2 The modeling cycle
(Blum & Borromeo-Ferri,
2009)

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
4 Method
4.1 Participants andcontext
Given the characteristics of the research, two different par-
ticipant groups were necessary. On the one hand, there was
a group of 24 PMTs, all of whom were graduates in math-
ematics and were studying a specialized master's degree in
mathematics education. And, on the other hand, there was
a group of 31 high school students (16–17years old), all of
whom were part of a classroom in which one of the PMTs
from the first group was teaching as part of their teaching
internship.
The study was carried out in accordance with the Design
Experiment principles (Cobb et al., 2003), following a
clearly qualitative approach. The observation processes were
associated with the university training of the PMTs and the
internship teaching.
PMT training was carried out over four 90-min sessions.
Three of these sessions focused on the solving of five opti-
mization problems and the creation of a report anticipating
what would happen in the classroom with a group of high
school students if these same problems were applied. The
fourth session was carried out after the participating PMT
implemented the mentioned problems with a group of high
school students. In this session, two researchers acted as
non-participant observers. This fourth session was devoted
to reflection on the personal MWS of the PMTs upon solving
two of the optimization problems. The reflection and discus-
sion focused on the domains and resolution paradigms used,
as well as the difficulties and emotions experienced. Their
solutions were compared with what was expected and with
the actual teaching practice and with that reported by the
two researchers during the teaching practice. This permitted
reflection on the interaction between cognition and affect
in a context of optimization problems and how to consider
epistemic emotions in the practice of mathematics teaching
in order to configure the suitable MWS. This fourth session
was recorded on video, and it highlighted the connections
between theory and practice in the affect dimensions and
between the personal MWS of the PMT and the configura-
tion of the suitable MWS.
4.2 Instruments
As a data collection instrument, a Questionnaire was devel-
oped (Fig.3). It consisted of two parts. In the first part, the
PMTs had to solve an optimization problem detailing the
mathematical processes followed, the thoughts that arose,
and the feelings and emotions experienced. They were
explicitly asked to solve it using different methods and to
describe the mathematical actions and epistemic emotions
experienced.
In the second part, the PMTs were to anticipate the
behavior of the high school students upon solving the prob-
lem: problem-solving strategies, difficulties and epistemic
emotions. For the epistemic emotions, a Mood Map of the
problems was followed (Gómez-Chacón, 2000). The Mood
Map is an iconic instrument that imitates weather maps,
establishing a code to express different emotional reactions
experienced by the student in the course of mathematical
activity. In this study, it was adapted using emoticons for
the epistemic emotions. The marks registered by the student
using these emoticons allow the identification of their emo-
tions during the process of solving a mathematical problem
and its origins, antecedents, and consequences.
They also had to indicate which methods they considered
most appropriate in teaching and how they could help them
overcome difficulties.
This problem was used in the classroom by one of the
PMTs having 31 high school students (aged 16–17) in the
teaching internships. This PMT proposed that this group of
students solve the problem and explain the processes used in
detail, as well as the emotions experienced according to the
Mood Map emoticons or if they felt no emotion at all when
working on the mathematical task.
Prior analysis of the problem This optimization problem
requires a geometric interpretation of what the problem asks
of us. The statement does not give the direct function to
optimize. The situation is realistic and easy to understand,
but the translation of the geometric situation into a symbolic
one requires the semiotic dimension and visual reasoning, as
well as mathematical knowledge (function, formula, deriva-
tion). It may be solved using distinct domains and differ-
ent strategies or methods: arithmetic/geometric, calculation
techniques or real analysis techniques or using distinct arti-
facts such as GeoGebra to represent the optimal point of the
function. A more common resolution is:
The volume of the box is (36- x) (36-x) x= (36-x)2 x
To actually have a box, x must be positive and less than
36. The function to be modeled is f(x)= (36-x)2 x; the
critical points of the function are calculated by setting
the first derivative equal to zero and using the second
derivative to check if they are maximums or minimums,
obtaining the y the second derivative is used to check if
they are maximums or minimums, obtaining the solu-
tion x=12 cm. (Analytical method resolution).
4.3 Data analysis
The data analyzed for this research are the PMT’s responses
to the questionnaire, the recording of the reflection ses-
sion (session 4 of the training process) and the high school

I.M.Gómez-Chacón, J.M.Marbán
students’ responses to the problem and questions indicated
in the previous section.
Given that two types of data were used (the discur-
sive responses to the questionnaire and the Mood Map
responses), two analytical approaches were employed. For
the discursive data, a qualitative approach was used (Strauss
& Corbin, 1990) based on cross-checking of solutions by
three researchers. The PMTs' responses to the questionnaire
were analyzed by differentiating between two practices:
resolution, referring to the process of problem-solving by
the PMTs, and anticipation, referring to what the PMTs
expected to find with regard to the students (Fig.4).
Emotional episodes were classified according to the epis-
temic emotions expressed by the subjects, which occurred
during the process of searching for the answer to the math-
ematical task by both the PMTs and the high school students,
highlighting the alleged reasons and the mathematical work
of mathematization and modeling (Figs.4 and 5). The data
Fig. 3 Questionnaire proposed to the PMTs

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
from the mood maps of both the PMTs and the high school
students were considered for the interrelation cognition and
affect. The evidence from the Mood Map in the protocols
(Fig.5) reveals patterns of sequences of emotions that inter-
act with heuristics during the resolution of mathematical
problems or mathematical task processes. This permits the
qualification of the interaction between cognition and affect
in the characterization of the epistemic emotions.
The inductive analysis process generated the follow-
ing system of categories, organized in three dimensions:
epistemic emotions, paradigms and resolution methods, and
epistemic cognitive actions.
1. Epistemic emotions surprise, curiosity, confusion,
boredom, frustration, enjoyment, anxiety, interest. In addi-
tion, the category none of them as considered.
2. Paradigms and resolution methods The methods and
procedures were categorized into five sections:
• Analytical: Search for the “Volume of the box as a func-
tion of the side of the square to be cut” function and
Fig. 4 Analysis of the solution as reported by PMT24 (PMTs were named from PMT1 to PMT24 and the high school students from A1 to A31.)
in the resolution practice and in the anticipation and categories of analysis by the research

I.M.Gómez-Chacón, J.M.Marbán
calculate its maximum through the derivative function
(first and second derivative).
• Algebraic: Divide the side of the piece by three and pro-
ceed with algebraic resolution using a similar problem
as a model.
• Geometric: Search for the “Volume of the box as a func-
tion of the side of the square to be cut” function to rep-
resent it graphically and find its maximum point, using
GeoGebra in some cases.
• Numerical-arithmetic trial: Compare the volumes of the
boxes of dimensions n·(36-n)·(36-n), with n between 0
and 18.
• Theoretical: This way of proceeding consists of reason-
ing that in order for the product to be maximum, the
factors must be as large as possible.
3. Epistemic cognitive actions related to MWS and the
modeling of optimization problems. Four epistemic actions
are identified, some of which have specific characteristics:
Fig. 5 Analysis of the resolution protocol of high school student A25

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
• Understanding of the statement and the problem (CEA1)
• Representamen, semiotics and mathematical objects: cre-
ation of a mathematical model of the situation (CEA2):
• Selection of variables and specification of what each of
them represents (CEA21)
• Obtain the expression of the function to be analyzed to
calculate its extreme value (CEA22)
• If the previous function is expressed with more than one
variable, look for the relationship that may exist between
these variables to express the function with a single vari-
able (CEA23)
• Mathematical content: application of mathematical tech-
niques to the model (CEA3):
• Calculate the derivative of the function (CEA31) and cal-
culations to obtain the values of x that cancel it (CEA32)
• Instrumental-discursive dimension: translation to the real
situation to analyze its validity (CEA4):
• The domain of the function (CEA41) is considered, often
conditioned by the problem statement, and the values
obtained in the previous step, determining the absolute
extremes using the sign of the second derivative as neces-
sary (CEA42).
5 Results
The results are organized in two sections. First, the rela-
tionship between epistemic emotions and problem-solving
approaches, identifying paradigms and methods in the per-
sonal MWS of the PMTs and their relation to the suitable
MWS. And, secondly, the differences between epistemic
emotions in different teaching practices: resolution, anticipa-
tion and students’ epistemic emotions in the implementation.
5.1 Paradigms andmethods inthepersonal MWS
ofthePMTs andtheir relation toasuitable MWS
The actions inherent to the modeling and resolution of opti-
mization problems can be associated with different work
domains: arithmetic, geometric, calculation and real analy-
sis. We identify five PMTs’ approaches to solving modeling
problems (Table1). Table1 shows the percentage of use
of each approach of the 24 PMTs. The personal MWS dif-
fers between the PMTs, although there is a trend to use one
specific method, the analytical approach which is the most
institutionalized approach used in teaching. The optimiza-
tion problem resolution may remain in a numerical-arithme-
tic trial MWS (see the solution in Fig.4). However, the intro-
duction of a function changes the MWS, and this involves
specific analysis techniques (for example, derivation) and
representations (graphs, table of values). The choice of each
MWS requires the use of diverse knowledge based on the
mathematical domains. It is found that various PMTs have
difficulty investigating other domains of mathematical work
that combine the geometric and algebraic registers. This lack
of knowledge triggers more negative epistemic emotions or,
as shown in the following section, the emotion of boredom.
Regarding what they consider to be suitable MWS for teach-
ing practice, 62% of PMTs believe that the analytical approach
is the best working paradigm and method since it allows for
work on the official curriculum content and is considered
more accurate. It refers to the application of a more standard
technique, which simplifies the student’s learning process and
provides them with more confidence. However, almost 40%
of the PMTs responded that, when solving this problem with
students, other ways that encourage reasoning would also be
useful. For example, PMT21 indicated: “Different types of reso-
lution expand our concept of the same knowledge. They help us
see all its sides, relate them and become aware of the abstract
structures underlying mathematics, thus providing a broader
view of mathematics. As a personal opinion, I also believe that
it develops the students’ creativity. Therefore, it is important to
consider this when teaching” (PMT21).
In addition, the PMTs indicate that the use of a paradigm
and resolution method will also depend on the students’
level of knowledge. For example, two PMTs indicated the
following:
“Once the problem has been modeled, we proceed to
solve it in two different ways. The first way could be done
by 9th or 10th graders and is based on use of the GeoGe-
bra mathematical tool to solve the problem. The students
would do what is shown in the following image in GeoGebra
[includes resolution with GeoGebra]. And a second way,
which could be done by high school students, is based on
the use of notions of derivatives. The high school students,
upon hearing the word “maximum” will think of functions
and their derivatives, in this case, the use of the derivative of
V(x)
to calculate its maximum as taught in class” (PMT23).
“Regarding the learning obtained, this is an optimization
exercise which, depending on the method used to solve it,
can be studied at various levels. If functions and the deriva-
tive are used to calculate the maximum, we will be talking
about a high school level, since they will also need some
knowledge of basic geometry of calculating volumes and
areas. On the other hand, solving it by giving different values
to x is much costlier and lacks interest beyond providing the
students with an idea of what the result will be” (PMT24).
Table 1 Identified paradigms and resolution methods
Analyti-
cal
Alge-
braic
Geomet-
ric
Numeri-
cal-arith-
metic
trial
Theoretical
Percent-
age
100% 8.3% 29.1% 16.6% 8.3%

I.M.Gómez-Chacón, J.M.Marbán
5.2 Differences between
resolution‑anticipation‑implementation
The epistemic emotions of the PMTs associated with the
different practices related to the practice of mathemat-
ics teaching (resolution, anticipation and implementation)
were different. According to the established categorization
of cognitive actions and epistemic emotions (Sec 4.3), we
proceeded to analyze the frequency of appearance and the
cognitive epistemic action with which it was associated.
The results are shown in Table2 and Fig.6, distinguishing
between the practices linked to the teaching of the optimiza-
tion problem. Thus, Table2 shows the percentage of PMTs
that indicate experimenting each of the emotions making up
the Mood Map when solving the problem; the percentage of
PMTs that anticipate those emotions in the high school stu-
dents and, finally, the percentage of the high school students
that indicated each of the emotions during the PMT’s teach-
ing internship. As previously mentioned, these students
solved the problem and indicated the emotions experienced
(see resolution protocol in Fig.5).
Figure6 shows the relative weight of the different emo-
tions according to the practices related to the PMTs during
the resolution as well as their anticipation of what could be
expected from the students, and the high school students’
epistemic emotions when solving the problem.
Figure6 can be interpreted as a flowchart of epistemic
emotions through the three analyzed practices. This suggests
possible transfers from personal MWS to suitable MWS,
as well as their characteristics. This transfer is exemplified,
considering the specific case of the emotions of surprise
and boredom, two emotions typified by different valence
and activation levels. Table 3 shows the percentages of
both emotions in relation to cognitive epistemic actions. In
anticipation, it is the percentage of PMTs that indicate it
and in implementation, it is the percentage corresponding
to the high school students who indicated it when solving
the problem.
Table3 shows the differences between the emotion of
surprise experienced by PMTs and what they anticipate that
the high school students would experience. In the resolution
of the PMTs, only 12.5% say that they experience this emo-
tion (Table2). They do, however anticipate its appearance
in their students in 37.5% of the cases, and it ultimately
appeared in the real implementation undertaken in the class-
room in 42% of the high school students (Table3).
Table3 shows the differences between the emotion of sur-
prise experienced by PMTs and what they anticipate that the
high school students would experience. In the resolution of
Table 2 Percentage of epistemic emotions indicated during resolu-
tion, anticipation and implementation
During the
PMTs resolu-
tion
PMTs anticipat-
ing the students’
emotions
Students’ emotions
in the implementa-
tion
Surprise 12.5 37.5 42
Curiosity 37.5 37.5 22.6
Interest 45.8 41.7 9.8
Enjoyment 37.5 45.8 38.7
Confusion 25 66.7 77.41
Frustration 45.8 70.8 42
Anxiety 8.3 66.7 22.6
Boredom 20.8 45.8 6.45
Fig. 6 Relative weights of the
epistemic emotions according to
the practices
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
Resolution Anticipation Implementation
EpistemicEmotions
Surprise Curiosity Interest Enjoyment
Confusion FrustrationAnxietyBoredom

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
the PMTs, only 12.5% say that they experience this emotion
(Table2); however, they anticipate its appearance in their
students in 37.5% of the cases, and it ultimately appeared in
the real implementation undertaken in the classroom in 42%
of the high school students (Table3).
Although the epistemic emotion of surprise is considered
to be an initial reaction to information that is incongruent
with prior knowledge (CEA1), in the anticipation made by
PMTs they also relate it more broadly to cognitive processes.
They also refer to it in the application of techniques to the
model (CEA3 and CEA32)—ignorance of the concept of
derivative and calculation techniques—and in the analysis of
the solution’s validity (CEA4). Furthermore, of the indicated
epistemic actions that generate surprise, one PMT points out
the discrepancy that the high school student may experience
with their beliefs with regard to this type of problem.
In the implementation, surprise is indicated with a greater
frequency by high school students in the creation of the
model to be optimized (CEA2) and in the execution of cal-
culation techniques (CEA3.2).
If we are to further examine what was expressed by the
PMTs during the problem resolution, the emotion of sur-
prise is linked to the onset and to the understanding of the
problem (semiotic and visualization dimension) and also
to the determination of model variables and the validation
and confirmation of the technique. For example, accord-
ing to PMT23: "I was surprised to discover that there were
equivalent resolution methods. I saw that I could make this
problem equivalent to one that I could solve numerically
and by arithmetic trial. Its statement would be: “Decom-
pose number 36 into two positive addends to maximize the
product of the first and the square of the second”, this would
be easier for the students to solve. This has been interesting
for my future work with the students since at first glance,
they appear to be totally different problems and I think this
is what the students would feel ". In this case (PMT23), it
is interesting to observe where the surprising or significant
event takes place, the confirmation of an intuition and the
establishment of a technique. In the anticipation reported
by PMT23, this will be significant in terms of performing
epistemic actions with the high school students. She seeks
to find and re-experience surprise as a reaction to something
that has become routine and obvious, and to use it to decide
what to emphasize, which can make a big difference in the
development of a topic.
In the anticipation, some of the PMTs describe surprise as
an emotion that can be generated in validation and in the link
to self-confidence: “The most important thing at the begin-
ning is to have a spatial view of how the box is assembled.
Students who get overwhelmed and do not know how to plan
the exercise well will feel anxiety and frustration over not
knowing how to advance. Either nothing occurs to them or
the approach that they have taken is incorrect, and therefore,
they will get confused and even bored since they don't know
what to do or how to move beyond this mental block. On the
other hand, those who were able to solve everything more or
less the first time will feel enjoyment and satisfaction, and
even surprise if they did not have much faith in themselves.”
(PMT4).
Regarding the epistemic emotion of boredom, considered
a crucial point where students disengage from activities and
often a consequence of the emotion of frustration, we find
that, in the implementation, only 6.45% of the high school
students noted this emotion. Some of them associated it with
performing the operations to find the maximum of the objec-
tive function, while the others mentioned it when conclud-
ing the problem and validating the results. The latter, even
though they knew that their solution was incorrect, spent
time looking for the error in their resolution process. How-
ever, in the anticipation made by the PMTs, they consider
that with this problem, there will be an increased appearance
of the epistemic emotion of boredom, which is linked to the
following epistemic actions: the understanding of the state-
ment and the problem (CEA1), the execution of a sequence
of steps in the calculation technique (CEA3, CEA32) -igno-
rance of these-, the execution of calculations, the recognition
Table 3 Percentage of the surprise and boredom emotions and cog-
nitive epistemic actions in the anticipation by the PMTs and that
manifested by the high school students [CEA1: understanding of the
statement and the problem; CEA2: Representamen, semiotics and
mathematical objects: creation of a mathematical model of the situa-
tion; CEA3: Mathematical content: application of mathematical tech-
niques to the model, CEA32: calculations to obtain the values of x
that cancel out the derivative; CEA4: Instrumental-discursive dimen-
sion: translation to the real situation to analyze its validity]
Anticipation
emotions
Anticipation Cognitive Epistemic Action Implementation. High
school students’ Emotions
Implementation. High school
students’ Cognitive Epistemic
Action
Surprise 37.5% CEA1 (8.3%), CEA3 (12.5%), CEA 4(20.8%) 42% CEA1 (25.8)
CEA2 (6.5%)
CEA32 (3.2%)
Not specified (6.5%)
Boredom 45.8% CEA1 (8.3%), CEA2 (4.2%), CEA3 (8.3%),
CEA4 (12.5%), Other reasons (12.5%)
6.45% CEA32 (3.2%), CEA4 (3.2%)

I.M.Gómez-Chacón, J.M.Marbán
that it is similar to other optimization problems (validation
phase (CEA4)). In addition, three PMTs mention other rea-
sons that may cause boredom, such as the belief regarding
its usefulness (believing that these problems are useless) and
the situating of the problem to the student’s level.
In the PMT’s resolution practice, this emotion was linked
to the execution of calculations, to being unable to solve it
using various methods and to the acceptance that repetition
can be of interest to strengthen the acquisition of mathemat-
ics procedures and techniques. In their own words: “When
I solved it I felt interested, then a little bored and frustrated
with the calculations. Also, I'm not able to find a second
method of solving it that isn't conceptually equivalent, look-
ing for function endpoints or something like that. This makes
me anxious and frustrated.” (PMT5). The PMTs mention
how for them, this repetition leads to boredom: “I don't see
the point in having to do it again, this problem is formulated
geometrically, but it is the same as another one that we per-
formed in an arithmetic formulation” (PMT14). However,
it can be seen that they did not go into detail regarding the
understanding of the mathematical work paradigms and
some of these techniques and their didactic qualification for
teaching practice.
6 Discussion
This study explores how PMTs characterize epistemic emo-
tions in different practices linked to the practice of math-
ematics teaching: in the resolution of optimization problems
and in the anticipation of the epistemic emotions that may
arise in high school students when solving these same prob-
lems. In addition, we compare the PMTs’ expectations with
the epistemic emotions indicated by the high school students
upon solving the same problem.
The epistemic emotions of the PMTs that were associated
with the three practices were different. The study reveals
some significant differences in the interaction between emo-
tions and epistemic actions according to practices of solving
and anticipating and provides evidence of the influence of
personal MWS on a suitable MWS.
6.1 Personal MWS ofthePMTs andsuitable MWS
The results regarding the epistemic emotions experienced
during the resolution and the anticipation of how they would
be generated in high school students when solving the same
problem provide us with information on the personal MWS
of the PMTs and their relations to a suitable MWS. The
PMTs refer to the analytical method as the one that would
offer them greater confidence since it has more institutional
recognition and is considered to be an application of tech-
niques, making it the prescribed procedure for use with
students. With this method, the introduction of a function
as a mathematical modeling object marks the MWS with
the derivation techniques and the graph and table of values.
The personal MWS involved also differs from one PMT to
another. Several of the PMTs have difficulty investigating
other domains of mathematical work that combine geometric
and algebraic registers, opting to use the analytical method,
which is considered to generate the most confidence. The
lack of knowledge to solve the problem using other meth-
ods triggers negative emotions in the PMTs, and when they
anticipate it in the students, they expect that it will generate
similar emotions. However, they suggest a positive value for
learning when solving the problem through distinct methods.
This leads us to question whether the modeling process for
some PMTs is a routine that has been normalized by the
institution, especially for certain types of problems and if it
is associated with a positive epistemic emotion.
6.2 Epistemic emotions andepistemic cognitive
actions
The results obtained are in line with those from previous
studies regarding the distance in the emotions that PMTs
analyze in themselves and those that they expect from their
students (Kanefke & Schukajlow, 2023). The PMTs valued
enjoyment, interest, boredom with the situation and consid-
ered it important to avoid frustration in the development of
high school students. The study has provided empirical evi-
dence on the emotions of surprise and boredom as applied
to themselves or in the anticipation made regarding what the
high school students will possibly experience. The analysis
from the theory of MWS allows us to consider the math-
ematical concepts and objects from mathematization identi-
fied in the modeling process in detail. This offers additional
knowledge and specificity to the results of other studies on
epistemic emotions (D'Mello etal., 2012; Munzar etal.,
2021) regarding the task characteristics and the subject’s
cognitive activity when solving them. For example, when
personal knowledge conflicts with external knowledge, that
is, when a cognitive incongruence arises, emotions may be
triggered, not only due to the nature of the knowledge but
also due to gaps in the same. In the study, they are associated
with the determination of the model, the concept of deriva-
tive, the execution of a sequence of steps in the calculation
technique and with validation actions.
As for the epistemic emotions and cognitive interactions
in the personal MWS, the results show how certain influ-
ences of the reference MWS become evident in the personal
MWS of the PMTs, who, in turn, project this onto the suit-
able MWS for teaching. Regarding the cognitive epistemic
actions, interactions with emotions were identified in the
creation of a model of the situation, the application of

Epistemic emotions andpre-service mathematics teachers’ knowledge forteaching
techniques and the translation to the real situation to ana-
lyze its validity.
However, the possibility that different MWS are gener-
ated in optimization problems from the mathematization
process makes it an attractive but complex task to control
for the PMT, especially for a second modeling cycle. In fact,
this study focuses on a first cycle in which PMTs are asked
to solve a modeling task. This reveals the difficulty for a
practical teaching internship to make the combined nature
of cognition, motivation and emotion on mathematical work
explicit, although it is a challenge that has been considered
in distinct studies (Cai & Leikin, 2020; Gómez-Chacón,
2018; Schindler & Bakker, 2020).
Finally, the results of this study may help PMTs connect
specific knowledge on epistemic emotions with their pro-
fessional teaching knowledge. The results highlight three
dimensions: knowledge and awareness of oneself, aware-
ness of the mathematical task and knowledge and awareness
of how to accompany others. The study highlights the con-
trast between what they experience as PMTs and what they
anticipate from their students. Two implications for PMTs
are identified from the proposed PMT training: an increased
mathematical knowledge (awareness of the discipline) and
the reorganization of knowledge in the modeling cycle to
consider the cognition-affect interaction for suitable MWS.
7 Conclusions
The research described in this article has addressed the char-
acterization of epistemic emotions by PMTs in the teaching
internship. We have identified epistemic emotions in optimi-
zation problems, analyzing them with regard to the associ-
ated personal and suitable Mathematical Working Spaces
(MWS). Thus, the identification of epistemic emotions has
been established in distinct practices linked to mathemat-
ics teaching: in solving problems and anticipating what
would happen with the students and reflecting on classroom
implementation.
The results indicate that PMTs display a variety of para-
digms and methods for solving the optimization problem,
although the analytical method predominates. Furthermore,
the influence of the institutionalized reference MWS on the
personal MWS of the PMTs has been confirmed. They, in
turn, project it on the suitable MWS anticipated for the stu-
dents, considered the best for fostering the confidence emo-
tion. The lack of knowledge of several PMTs when faced
with solving the problem using a variety of methods triggers
more negative epistemic emotions.
PMTs rely on distinct perspectives when analyzing emo-
tions in themselves and when anticipating possible emotions
in high school students. They positively value enjoyment and
interest in high school students and consider it important
to avoid frustration and boredom during classes. The study
also shows that the perception and interpretation of emotions
such as surprise and boredom vary in how PMTs personally
experience them and how they expect for them to arise in
students. These findings expand upon past works (D'Mello
etal., 2012; Munzar etal., 2021) on the cognition-affect
interaction. In the specific case of the epistemic emotion of
surprise, in addition to being an initial reaction to informa-
tion that is incongruent with prior knowledge, as typified
in past studies, it is also identified with respect to cognitive
processes and linked to gaps in knowledge regarding the
application of techniques to the model and the analysis of
the validity of the solution.
Both conclusions suggest the need to reinforce teacher
training with educational processes that will help them
face their own epistemic emotions and model appropriate
strategies to regulate their own emotions and those of their
students.
This study has certain limitations that should be consid-
ered. First, the specific typology of problems, in our case
optimization problems, so further studies were required to
address a wide variety of tasks and to contrast how the suit-
able MWS is important for the understanding of dominant
epistemic emotions by PMTs.
In relation to the extension of the work at the methodo-
logical level, it can be implemented with several groups of
high school students. The implementation has been carried
out with a High School group by only one of the PMTs, so
as to better understand the process of anticipating the pos-
sible emotions of the students. At a methodological level,
this would permit a multiple analysis of cases so as to obtain
information on the demands associated with classroom
teaching. Also, another limitation to note is in relation to the
PMT self-reported classroom observed emotion data. The
external observation made of their performance could be
analyzed in more depth and contrasted with their statements
to identify biases related to selective or attribution memory,
among others. Finally, a limitation exists with regard to the
data coding process. This coding was carried out without
using triangulation processes supported by intercoder reli-
ability indicators, based on consensual decisions made by
the participating researchers. The study may be considered
exploratory and is an initial phase of a sequential mixed
strategy. The next stage would be quantitative in nature, in
order to generalize or expand upon the findings of the quali-
tative phase.
Specifically, if confirmed, these findings may have certain
implications on educational interventions at the PMT uni-
versity training level. They also may indicate relevant future
research lines. Given the relationship between emotions and
mathematical knowledge, these results may be of interest
to generalize aspects related to the role of epistemic emo-
tions and their impact on the generation of knowledge for

I.M.Gómez-Chacón, J.M.Marbán
teaching. Furthermore, when reflecting on a suitable MWS,
further study may be needed on the origin of the strong link
made by teachers in initial training programs between cer-
tain epistemic emotions such as boredom and confusion as
limitations of mathematical learning. We believe that both
issues may help to design practical education interventions
that can develop awareness on accompanying others.
Funding Open Access funding provided thanks to the CRUE-CSIC
agreement with Springer Nature. The results and publication are part
of the action Research Project PID2022-138325OB-I00, financed by
MICIU/AEI/10.13039/501100011033 and by FEDER, EU. Also, it was
supported by the INVEDUMAT_uni_2020-2023 Programme, Institute
of Interdisciplinary Mathematics (IMI).
Declarations
Conflict of interest The authors declare that there is no conflict of in-
terest.
Open Access This article is licensed under a Creative Commons Attri-
bution 4.0 International License, which permits use, sharing, adapta-
tion, 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 permission directly from the copyright holder. To view a
copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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