Development of Didactic Analysis Competence in Prospective Mathematics Teachers

Abstract

In this article, we describe the implementation and results of a formative experience with prospective mathematics teachers, focused on developing the competence for didactic-mathematical analysis of curriculum materials, specifically student workbooks related to probability. The research design follows a methodological approach typical of design-based research, utilizing content analysis to examine participants' responses. The study was conducted with 16 Peruvian students preparing to become mathematics teachers at the National University of the Altiplano. The responses from the prospective teachers revealed deficiencies in their common content knowledge. They also encountered difficulties in distinguishing mathematical practices and recognizing the mathematical objects involved in the study process, especially propositions and their respective arguments. Furthermore, they struggled to differentiate between intuitive, classical, and frequentist meanings of probability. To improve these outcomes, it is necessary to reinforce didactic-mathematical knowledge regarding probability.

Full text

Journal of Research in Mathematics Education
Volume 14, Issue 1, 24th February, 2025, Pages 1–29
The Author(s) 2025
http://dx.doi.org/10.17583/redimat.15471
Development of Didactic Analysis Competence in
Prospective Mathematics Teachers
Bethzabe Cotrado1, María Burgos2, Pablo Beltrán-Pellicer3, & Alfredo Carlos Castro1
1) Universidad Nacional del Altiplano, Peru
2) Universidad de Granada, Spain
3) Universidad de Zaragoza, Spain
Abstract
In this article, we describe the implementation and results of a formative experience with
prospective mathematics teachers, focused on developing the competence for didactic-
mathematical analysis of curriculum materials, specifically student workbooks related to
probability. The research design follows a methodological approach typical of design-based
research, utilizing content analysis to examine participants' responses. The study was
conducted with 16 Peruvian students preparing to become mathematics teachers at the National
University of the Altiplano. The responses from the prospective teachers revealed deficiencies
in their common content knowledge. They also encountered difficulties in distinguishing
mathematical practices and recognizing the mathematical objects involved in the study process,
especially propositions and their respective arguments. Furthermore, they struggled to
differentiate between intuitive, classical, and frequentist meanings of probability. To improve
these outcomes, it is necessary to reinforce didactic-mathematical knowledge regarding
probability.
Keywords
Teacher education, curriculum materials, probability, ontosemiotic analysis
To cite this article: Cotrado, B., Burgos, M, Beltrán-Pellicer, P., & Castro, A. C. (2025).
Development of didactic analysis competence in prospective mathematics teachers. Journal of
Research in Mathematics Education, 14(1), pp. 1-29.
http://dx.doi.org/10.17583/redimat.15471
Corresponding author(s): Bethzabe Cotrado
Contact address: bcotrado@unap.edu.pe

Journal of Research in Mathematics Education
Volumen 14, Número 1, 24 de febrero de 2025, Páginas 1–29
Autor(s) 2025
http://dx.doi.org/10.17583/redimat.15471
Desarrollo de la Competencia de Análisis Didáctico
en Profesorado de Matemáticas en Formación
Inicial
Bethzabe Cotrado1, María Burgos2, Pablo Beltrán-Pellicer3, y Alfredo Carlos Castro1
1) Universidad Nacional del Altiplano, Perú
2) Universidad de Granada, España
3) Universidad de Zaragoza, España
Resumen
En este artículo describimos la implementación y los resultados de una experiencia formativa
con futuros profesores de matemáticas, centrada en el desarrollo de la competencia de análisis
didáctico-matemático de materiales curriculares, específicamente cuadernos de trabajo del
estudiante sobre probabilidad. El diseño de la investigación sigue un enfoque metodológico
típico de la investigación basada en el diseño, utilizando el análisis de contenido para examinar
las respuestas de los participantes. El estudio se llevó a cabo con 16 estudiantes peruanos que
se preparan para ser profesores de Matemáticas en la Universidad Nacional del Altiplano. Las
respuestas de los futuros profesores revelaron deficiencias en su conocimiento común del
contenido. También encontraron dificultades para distinguir las prácticas matemáticas y
reconocer los objetos matemáticos involucrados en el proceso de estudio, especialmente las
proposiciones y sus respectivos argumentos. Además, tuvieron problemas para diferenciar
entre los significados intuitivo, clásico y frecuencial de la probabilidad. Para mejorar estos
resultados, se concluye que es necesario reforzar el conocimiento didáctico-matemático con
respecto a la probabilidad.
Palabras clave
Formación de profesores, materiales curriculares, probabilidad, enfoque ontosemiótico
Cómo citar este artículo: Cotrado, B., Burgos, M., Beltrán-Pellicer, P. y Castro, A. C. (2025).
Desarrollo de la Competencia de Análisis Didáctico en Profesorado de Matemáticas en
Formación Inicial. Journal of Research in Mathematics Education, 14(1), pp. 1-29.
http://dx.doi.org/10.17583/redimat.15471
Correspondencia Autores(s): Bethzabe Cotrado
Dirección de contacto: bcotrado@unap.edu.pe

REDIMAT – Journal of Research in Mathematics Education, 14(1)
3
he work of a mathematics teacher is a complex activity that requires mastery of various
types of knowledge and competencies (Chapman, 2014). They must possess
mathematical knowledge that allows them to solve the problems outlined in the
curriculum at the educational level where they teach. Additionally, they need specialized
knowledge about the subject matter, the transformations required for the teaching and learning
processes, as well as psychological and pedagogical factors, among others, which influence
these processes.
When implementing a specific instructional process, a teacher should be able to interpret
the information in the curriculum materials based on their ability to assist students in achieving
the learning objectives set forth in the study programs and curriculum guidelines (Thompson,
2014). Textbooks, teacher manuals, student workbooks, and other resources, serve as tools that
support teachers in making educational decisions, acting as sources of learning and a means of
interaction with students (Pepin & Gueudet, 2018; Remillard & Kim, 2020). Therefore,
research in mathematics education emphasizes the need for the analysis of such resources to
be one of the competencies included in teacher training (Burgos et al., 2020; Lloyd & Behm,
2005; Yang & Liu, 2019). These actions in teacher training are significant for the teachers' own
learning because the analysis of materials requires a deep dive into the characteristics related
to mathematical objects, their representations, and relationships, as well as knowledge about
the teaching and learning of mathematics, prompting teachers to question some of their own
beliefs (Lloyd & Behm, 2005).
In response to this demand, prior research (Breda et al., 2017; Font et al., 2010; Giacomone
et al., 2018; Pino-Fan et al., 2023; Pino-Fan et al., 2018; Pochulu et al., 2016) suggests the use
of tools from the Ontosemiotic Approach (OSA) to mathematical knowledge and instruction
(Godino et al., 2007) to develop in teachers the specific competence of didactic analysis of
instructional processes. This competence involves, among other aspects, the teacher's ability to
describe and explain the mathematical practices at play when solving problems and studying
the intended mathematical content (Burgos et al., 2019; Burgos & Godino, 2021; Giacomone
et al., 2018; Godino et al., 2017). Recognizing the mathematical practices carried out when
solving tasks in curriculum materials, as well as the mathematical objects and processes
involved, allows the teacher to become aware of potential learning conflicts, assess them, and
make the necessary adjustments to address their limitations (Yang & Liu, 2019).
Much of the literature on curriculum materials has focused on textbooks, neglecting other
components such as student workbooks. The importance of student workbooks lies in that they
constitute tools: a) for practice, providing students with structured and sequenced tasks or
activities that challenge and expand their conceptual understanding of the content and skills
developed in class or in textbooks; b) for assessment, as they include tasks covering a variety
of content with different levels of complexity using answers, solutions, or rubrics, and c) for
monitoring, allowing the determination of how much curriculum content is covered and at what
levels of cognitive demand (Hoadley & Galant, 2016). In addition to these characteristics, in
contexts such as Peruvian secondary education, which is the context of this research,
administrations provide students with workbooks updated according to current curriculum
regulations, while textbooks have not yet been revised, so workbooks allow for the
implementation of curriculum regulations in teaching practice. However, we have not found in
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Cotrado et al.– Development of Didactic Analysis Competence
4
previous literature any research aimed at developing prospective teachers' competence in the
didactic analysis of student workbooks, which motivates the study presented in this paper.
Workbooks differ from textbooks in that the latter include "explanatory" configurations (that
is, they introduce definitions, procedures, etc.,) that do not appear in workbooks. Tasks in
textbooks are, to some extent, determined by these prior configurations, as they aim to reinforce
the knowledge introduced earlier. In this study, the distributed workbooks are not accompanied
by a textbook or a teacher's guide, so it is expected that the teacher understands and interprets
the mathematical elements required in the tasks included. In this sense, although the proposed
tasks may be similar to those found in other textbooks, the material analysed here involves
greater complexity. Since the mathematical content included in these workbooks is very
extensive, so it is relevant to focus on a specific content, in our case, probability.
The limited existing studies on the treatment of probability content in the curriculum and
textbooks highlight deficiencies. For instance, they reveal a bias towards scenarios involving
games of chance, insufficiently representative and balanced situations, a lack of experimental
and simulation-based scenarios utilizing manipulatives or software, among other issues. These
deficiencies hinder the development of adequate probabilistic literacy (Cotrado et al., 2022;
Vásquez & Alsina, 2015).
The objective of this research is to describe and analyse the results of implementing a
formative experience with prospective Peruvian mathematics teachers aimed at developing
their competence in didactic-mathematical analysis of curricular materials in probability.
Specifically, we focus on identifying practices, objects, and meanings of probability considered
in student workbooks. We consider this type of analysis essential for trainee teachers to
recognize deficiencies in the material that may require action decisions on their part to ensure
adequate teaching of probability.
Theoretical Framework
The research is based on the model of Didactic-Mathematical Knowledge and Competencies
(the DMKC model) for mathematics teachers developed by Godino et al. (2017). This model
is supported by the system of theoretical tools established in the OSA framework (Godino,
2024). Below, we describe the key tools that will be fundamental in our research.
Pragmatic Meaning and Ontosemiotic Configuration
From the perspective of the OSA (Godino, 2024), the systematic analysis of instructional
processes, whether planned, anticipated, or implemented, requires an understanding of the
mathematical practices involved in the content and the identification of the network of objects
and processes that these practices mobilize. These analyses are built upon the notions of
pragmatic meaning and ontosemiotic configuration. These tools have been employed in teacher
education, utilizing various strategies and within different mathematical contexts (Burgos et
al., 2020; Burgos & Godino, 2021; Burgos et al., 2018; Giacomone et al., 2018).
In the OSA, the notion of mathematical practice serves as the starting point for the analysis
of mathematical activity (Font et al., 2013). Mathematical practices, or systems of

REDIMAT – Journal of Research in Mathematics Education, 14(1)
5
mathematical practices, involve various types of mathematical objects as actions carried out
by a subject to solve problems, communicate and/or generalize their solutions. Mathematical
objects that are involved in and emerge from systems of mathematical practices are interrelated,
forming ontosemiotic configurations of practices, objects, and processes. In the OSA, a
mathematical process is any sequence of actions carried out over a certain period to achieve an
objective, typically the solution of problem situations or the communication of their solutions.
Mathematical objects such as languages, problems, concept-definitions, propositions,
procedures, and arguments emerge from mathematical practices through their respective
mathematical processes of communication, problematization, definition, enunciation,
algorithmization, and argumentation. Other processes like modeling or problem-solving can be
understood as mega-processes, as they involve one or more of the aforementioned processes.
Practices and processes have many aspects in common (concatenation, time, etc.), which is
why they are sometimes confused. However, they have enough differences to not be identified
as the same. In the OSA, a distinction is made between practice, procedure, and process. Being
a mathematical object means participating in some way in mathematical practice. Thus, at first
glance, anything that can be 'individualized' in mathematics will be considered an object. The
analysis of mathematical activity reveals a first type of objects that intervene in mathematical
practices problems, concept-definitions, propositions, etc.—which we will refer to here as
primary objects. Primary objects are related to each other and form configurations, which can
be defined as networks of objects that participate in systems of practices and emerge from
them.
Font et al. (2013) develop an ontology of mathematical objects, their different types, the
configurations they form, their ways of being in mathematical practices, their forms of
existence, etc. Based on these considerations, Font et al. (2013) explain how mathematical
practices can produce a referent that, implicitly or explicitly, is considered a mathematical
object, and that appears to be independent of the language used to describe it (called a
secondary object, which in this article would be probability). In other words, this object would
be the content to which, explicitly or not, the pair—mathematical practices and the
configuration of primary objects that activate them—globally refers. Put another way, a
configuration of primary objects is considered as definitions, representations, properties of a
secondary object that is independent of these primary objects. In turn, a secondary object can
only be made operational through the use of a configuration of primary objects. Given this
symbiosis between primary and secondary objects, we will use the term 'object' and will only
distinguish between primary and secondary when necessary.
Since a mathematical object is conceived as emerging from the practices performed by an
institution (person) associated with a field of problems in which the object is involved, the
institution (person) meaning of the object is determined by the institution (person) practices
associated with the field of problems from which the object arises in a given moment and
context (Godino et al., 2007). In particular, we will mean the reference meaning as the system
of practices used as a reference to elaborate the intended meaning. In a specific educational
institution, this reference meaning will be part of the holistic meaning of the mathematical
object. Determining this overall meaning requires conducting a historical-epistemological

Cotrado et al.– Development of Didactic Analysis Competence
6
study on the origin and evolution of the object in question, as well as considering the diversity
of contexts in which this object is used (Godino et al., 2007, p.3).
Reference Meanings of Probability
The OSA considers that mathematical objects emerge from practices, which entails their
complexity (Font et al., 2013; Rondero & Font, 2015), understood as a multiplicity of
meanings. From this principle, it follows that this complexity must be considered, as much as
possible, in the design and redesign of didactic sequences. For this reason, various studies have
been conducted using OSA as theoretical framework to deepen the understanding of the
multiplicity of meanings of different mathematical objects and to explore students' and
teachers' comprehension of this complexity (Burgos et al., 2018, 2020; Calle et al., 2021, 2023).
The reference meanings of probability considered in current secondary education
curriculum programs are intuitive, subjective, frequentist, classical, and axiomatic (Batanero
et al., 2016; Beltrán-Pellicer et al., 2018). Each of these meanings has specific differences, not
only in the definition of probability itself but also in the objects and processes involved in the
practices used to solve or model various specific real-world problems or phenomena (Batanero
et al., 2016). Intuitive meaning corresponds to the intuitive ideas that children may have about
uncertainty and the everyday use of terms stemming from experiences and contexts related to
random phenomena. Subjective meaning develops the concept of probability as a degree of
belief based on personal judgment that can be revised based on an individual's knowledge and
experience. The first mathematical definition of probability is associated with the classical
meaning. It considers probability as the ratio of the number of favourable outcomes to the total
number of possible outcomes, provided that all elementary events are equally likely. This
definition is valid only for sample spaces with a finite number of equiprobable elementary
events and gave rise to Laplace's rule and the calculation of probability in situations involving
games of chance, where combinatorial reasoning is often applied (Batanero et al., 2016). In the
frequentist meaning, probability is defined as the hypothetical value toward which the relative
frequency of an event stabilizes when the experiment is repeated a large number of times.
Compared to the classical approach, it has the disadvantage that the true value of probability is
never actually calculated, i.e., it is only estimated through relative frequency, which can lead
to confusion with probability (Batanero et al., 2016). However, it has the advantage of being
applicable to experiments with non-equally likely events. Finally, the axiomatic theory
addresses the problem of organizing and structuring the other partial meanings of probability
and allows for the development of all known results at the time regarding probability
calculations. While some textbooks, typically at the end of secondary education, incorporate
Kolmogorov's axioms, the axiomatic meaning is too formal and is generally recommended for
university-level study (Batanero et al., 2016).
Each meaning of probability entails different systems of practices and mathematical objects
that must be considered together and integrated into the processes of teaching and learning
probability. In works by Batanero (2005) and Gómez (2014), for every meaning, the field of
problems from which it arises is described, as well as the concepts, languages, properties,
procedures, and arguments involved in the mathematical practices that address them.
Specifically, the analysis of the Peruvian curriculum program conducted by Cotrado et al.

REDIMAT – Journal of Research in Mathematics Education, 14(1)
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(2022) reveals a greater representation of mathematical objects (concepts, procedures,
propositions) related to the classical and frequentist meanings of probability, to the detriment
of the intuitive meaning. The problem-situations considered lead to recognizing the conditions
defining a random situation, expressing the value (decimal or percentages) of probability as
more or less likely, determining the sample space, calculating the probability of events using
Laplace's rule or by calculating their relative frequency, interpreting information from texts
involving probabilistic situations, or drawing conclusions about the probability of event
occurrences. Each of these meanings allows for the resolution of different types of tasks;
therefore, if the goal is for students to become competent in solving a variety of problems,
particularly extra-mathematical ones where the mathematical object of probability plays a
determining role, it is necessary for students to have a well-connected network of partial
meanings.
Teacher Knowledge and Competencies Model in the OSA
The DMKC model (Godino et al., 2017) developed within the OSA framework can serve as a
basis for guiding the training of mathematics teachers. This model acknowledges that a teacher
should have a common mathematical knowledge related to the educational level where they
teach and expanded knowledge that enables them to connect it to higher levels. Furthermore,
as mathematical content is brought into play, the teacher must possess didactic-mathematical
knowledge of the various facets involved in the educational process. In particular, teachers
should have the necessary knowledge to recognize, on the one hand, the different meanings
(understood as systems of practices) of the corresponding content and their interconnection,
and on the other hand, the diversity of objects and processes involved (ontosemiotic
configuration) for the different meanings. Additionally, teachers should be able to use this
knowledge competently in the processes of didactic design.
In the DMKC model, it is considered that the two key competencies of a mathematics
teacher are mathematical competence and the competence of didactic analysis and intervention,
which essentially involves "designing, applying, and evaluating learning sequences for oneself
and others, using techniques of didactic analysis and quality criteria, to establish cycles of
planning, implementation, evaluation, and propose improvements" (Breda et al., 2017, p.
1897). This overall competence of didactic analysis and intervention of the mathematics
teacher is articulated through five sub-competencies, associated with the OSA conceptual and
methodological tools: competence in the analysis of global meanings, competence in
ontosemiotic analysis of practices, competence in the management of didactic configurations
and trajectories, competence in normative analysis, and competence in the analysis of didactic
suitability (Godino et al., 2017). In your work, the focus is on the sub-competencies of the
analysis of global meanings and ontosemiotic analysis.
In the competence of global meaning analysis, the focus is on "identifying problem
situations that provide partial meanings or senses to the mathematical objects or topics under
study, and the operational and discursive practices that must be employed in their resolution"
(Godino et al., 2017, p. 99). This competence allows teachers to answer questions such as:
What are the meanings of the mathematical objects involved in the study of the intended

Cotrado et al.– Development of Didactic Analysis Competence
8
content? How are they interconnected? On the other hand, the competence of ontosemiotic
analysis of mathematical practices enables teachers to identify the objects and processes
involved in the mathematical practices necessary for solving problem situations. This
recognition facilitates "anticipating potential and effective learning conflicts, evaluating
students' mathematical competencies, and identifying objects (concepts, propositions,
procedures, arguments) that need to be remembered and institutionalized at the appropriate
moments in the study processes" (Godino et al., 2017, p. 94).
In line with the work of Pino-Fan et al. (2023), we consider the following levels of
development for the competence in didactic-mathematical analysis (onto-semiotic analysis of
practices and global meanings):
L0 The teacher identifies some evident mathematical elements: languages, procedures, or definitions
of certain concepts used in the analysed practices, without recognizing the partial meanings
involved.
L1 The teacher analyses mathematical practices, recognizing most types of mathematical objects
involved, as well as partial meanings, based on their experience but without considering any
theoretical-methodological analysis tool.
L2 The teacher uses theoretical tools, particularly the onto-semiotic configuration, to analyse
mathematical practices and recognize the network of emerging mathematical objects but does not
use them to differentiate meanings.
L3 At this level, the teacher is familiar with the onto-semiotic configuration and uses it as a tool to
analyse mathematical practices. Additionally, based on this analysis and in relation to the context,
the teacher identifies the different meanings involved.
Existing Research
Studies that specifically address teachers' mathematical and didactic-mathematical knowledge
about probability reveal deficiencies that may hinder its effective teaching (Batanero &
Álvarez-Arroyo, 2024; Vásquez & Alsina, 2015). In this section, we summarize the results of
studies that have explicitly examined the competence for didactic analysis as part of their
research with teachers.
Contreras (2011) examined 183 prospective primary teachers in relation to a task involving
the calculation of simple, compound, and conditional probabilities, finding that although they
managed to identify some mathematical objects, they did not recognize all those necessary to
solve the problem. The participants had no difficulty recognizing language types, with
definition-concepts being the most frequently identified mathematical objects, followed by
procedures, while properties and arguments went unnoticed. Nonetheless, overall results were
poor and highly variable among students.
Mohamed (2012) assessed the specialized knowledge of 31 groups of prospective teachers
on fair play and sample space, revealing that only a small group was able to correctly identify
key concepts such as fraction comparison, proportionality, and randomness. Concepts such as
expected value and combinatorial reasoning were not identified, highlighting deficiencies in
their understanding of these topics.

REDIMAT – Journal of Research in Mathematics Education, 14(1)
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Vásquez and Alsina (2015) applied a questionnaire to assess the didactic-mathematical
knowledge of 93 in-service Chilean teachers for teaching probability. Results showed a limited
performance level across different knowledge types involved, particularly regarding the notion
of a sure event, the calculation and comparison of probabilities of elementary events, and
understanding event independence. Significant difficulties were found in identifying concepts
or properties involved in situations requiring probability calculation and comparison or sample
space analysis.
On the other hand, studies by Batanero et al. (2021), Gómez (2014), and Parraguez et al.
(2017) focus on understanding different meanings of probability. Gómez (2014) analysed the
probability knowledge of 81 prospective primary education teachers, noting that while they
have an adequate understanding of the classical meaning of probability and can recognize
fundamental concepts such as probability, sample space, and fair play, they face limitations
with the frequentist meaning, struggle with understanding variability in small samples, and
exhibit equiprobability bias. Results from Parraguez et al. (2017), with a group of 60
prospective primary education teachers, and Batanero et al. (2021), with 139 secondary
education prospective teachers, highlight their difficulties in linking the classical probability
concept to the relative frequencies of an experiment due to challenges in understanding the
notion of convergence.
In our research, we follow the line of studies by Burgos et al. (2018) and Giacomone et al.
(2018), which use the onto-semiotic configuration tool to develop the competence for didactic
analysis in prospective teachers for tasks involving proportionality or diagrammatic
visualization and reasoning, respectively, guiding them in the identification of practices,
objects, and meanings.
Methodology
The research problem is the design, implementation, and evaluation of a formative action with
prospective secondary education teachers to develop their competence in didactic-
mathematical analysis about probability, through the identification of practices, objects, and
meanings of probability considered in student workbooks. Therefore, the methodological
approach follows the typical phases of design-based research, as proposed by Godino et al.
(2014) within the OSA framework: 1) preliminary study; 2) design of the didactic trajectory
(selection of problems, sequencing, and a priori analysis); 3) implementation of the didactic
trajectory (observation of interactions between individuals and assessment of achieved
learning); 4) evaluation or retrospective analysis derived from the contrast between what was
planned in the design and what was observed in the implementation. The study is essentially
qualitative, as it collects and analyses information based on the actions of prospective teachers
in a real classroom context. Content analysis methodology (Cohen et al., 2011) is used to
examine class recording transcripts and participant response protocols1.

Cotrado et al.– Development of Didactic Analysis Competence
10
Research Context, Participants, and Data Collection
The formative experience was conducted with a group of 16 students from the Secondary
Education Program specializing in Mathematics, Physics, Computer Science, and Informatics
at the National University of the Altiplano in Peru. This program spans 10 academic semesters
(5 years). The participating prospective teachers (referred to as PT) were in their fourth
semester, taking the Descriptive Statistics course through the virtual platform LAURASIA.
The PTs had only studied probability in secondary education and had not covered probability
topics during their university training. The course planning included synchronous sessions via
Google Meet and asynchronous activities to provide study materials and assignments
(Classroom). During the workshop's implementation, which consisted of three virtual
synchronous sessions, each lasting two hours, all 16 participants were present in the first
session, while 13 participants attended the second and third sessions (three students left the
course). The instructor responsible for managing the workshop also serves as a researcher.
Data collection instruments include recordings of the class sessions through Google Meet,
the instructor's notes, and written response protocols from the participants during the
synchronous and asynchronous sessions of the workshop.
Implemented Sessions and Teaching Resources
The formative workshop was organized into three virtual synchronous sessions, each lasting
two hours, which included theoretical-practical training and group discussions. In these
sessions, asynchronous activities were also provided to be completed by the participants as a
complement to the two hours of synchronous work. These activities involved reading
documents and solving tasks that are part of the participants' final assessment.
Session 1: Initial Exploration of Mathematical Practices and Objects in Probability
Tasks
The session began with the resolution of three selected problems from the assessment section
of worksheets 9 (initial diagnostic problems 1 and 2) and 13 (initial diagnostic problem 3) that
deal with probability in the Mathematics Problem-Solving Workbook for Secondary
Education. Problem 1 (Figure 1) presents a simple random experiment described through a pie
chart, where the events are not equiprobable.

REDIMAT – Journal of Research in Mathematics Education, 14(1)
11
Figure 1
Initial Diagnostic Problem 1
Source. MINEDU, 2019a, p. 129. Authors’ translation.
The graph (Figure 1) represents the population of 100 students of a sports academy and the
disciplines they practice. The PTs must calculate the probability that a student randomly chosen
on any given day does not practice basketball.
The context of problem 2 involves a compound random experiment consisting of three
simple experiments (flipping a coin three times): “Problem 2. A coin is tossed three times.
What is the probability of getting "heads" exactly twice?” (MINEDU, 2019a, p. 129). In
problem 3, a compound random experiment is presented:
Problem 3. There are 200 workers in a company, 100 of whom are men, and the rest are women.
The workers who read the magazine "La Estación" are 30 men and 35 women. If an employee is
randomly selected, calculate the probability that: (a) He is male and does not read "The Station"
magazine. (b) That he reads the magazine "La Estación" (MINEDU, 2019b, p. 182).
Once the participants solved these problems, the session continued with the initial
exploration of the personal meanings of the PTs regarding the nature of mathematical objects
and their ability to identify these objects in mathematical practices. To do this, the participants
individually described and listed the practices they used to solve problem 1 and identified the
concepts, symbols, graphs, or tables used, mentioning any difficulties encountered if
applicable. Subsequently, they shared and presented their answers in class, comparing them
with the analysis of the solution to problem 1 proposed by the instructor. Before concluding
the session, the PTs were instructed to read the article by Batanero (2005) concerning pragmatic
meanings of probability, and based on this, create a summary table proposing examples of
problem situations associated with the different meanings of probability considered. This
reading was part of the asynchronous activity, providing flexibility to participants to respond
according to their availability.

Cotrado et al.– Development of Didactic Analysis Competence
12
Session 2: Pragmatic Meanings and Ontosemiotic Configurations in Probability
The session began with the following questions:
• What types of probability meanings does the reading propose?
• What characteristics should a problem have to relate to a specific partial meaning of
probability?
• What meanings of probability do the problems you solved in the previous session
correspond to?
The intention was to engage the participants in a reflection on the different meanings of
probability and how they could be characterized based on the network of mathematical objects
emerging in associated practices. After a detailed explanation by the instructor about the
problem situations and elements that characterize the various meanings of probability, as well
as their progressive inclusion in the teaching of this content in secondary education, the
participants were asked to work individually to respond to the following instructions:
1. Analyse the problem-situation 2, describing its solution process.
2. Identify the mathematical objects involved and relate them to a specific meaning of
probability.
3. Mention any difficulties encountered.
The written responses were shared in class. Then, the instructor presented the a priori
analysis of problem 2 (MINEDU, 2019a, p. 129), using the ontosemiotic configuration tool.
After breaking down the solution into elementary practices (units of analysis), the use and
intentionality of each of them were identified, along with the mathematical objects involved.
At the end of the session, participants were assigned an individual asynchronous task to analyse
Situation A, which appears alongside its solution in worksheet 9 of the problem-solving
workbook for first-degree problems (Figure 2). In this case, the ontosemiotic analysis was not
conducted on their practices but on the solution proposed by the author of the curricular
material, using the same type of table employed by the instructor when presenting the
ontosemiotic configuration.
In addition, they were asked to link situation A to one of the meanings of probability and to
specify the difficulties they encountered in relating the situation to the meanings of probability
and in identifying the mathematical objects.

REDIMAT – Journal of Research in Mathematics Education, 14(1)
13
Figure 2
Situation A Proposed in the Testing Section of the First Grade Worksheet 9
Source. MINEDU, 2019a, p. 120-121. Authors’ translation.
Session 3. Sharing and Proposal of Assessment Tasks
In this session, the participants presented the results of applying the ontosemiotic configuration
tool to conduct a detailed analysis of the mathematical practices described in Situation A
(Figure 2). Subsequently, similar to the previous sessions, the instructor shared and explained
the a priori analysis of Situation A, allowing the participants to compare their responses and
discussing the challenges encountered. At the end of the third session, they were asked to
individually complete two assessment tasks.

Cotrado et al.– Development of Didactic Analysis Competence
14
Figure 3
Situation B Included in the Testing Section of Worksheet 9 of First Grade
Source. MINEDU, 2019a, p. 122-123. Authors’ translation.
In the first assessment task, participants were required to perform an analysis using the
ontosemiotic configuration tool of the solution provided by the instructor for problem 3. In the
second task, they should identify the meaning, develop the ontosemiotic configuration, and

REDIMAT – Journal of Research in Mathematics Education, 14(1)
15
recognize possible errors or conflicts in the solution provided by the author of the workbooks
for Situation B (Figure 3) concerning the simultaneous roll of two dice. To promote
independent work and responsibility for submission, participants had one week to submit their
analysis report through the virtual platform.
Results and Analysis
Although the formative experience is aimed at developing competence in analysing meanings
and ontosemiotic analysis, it's important to consider the shortcomings in the common
knowledge of probability among the participants as a potential obstacle to progress in this
competence. It's also of interest in this regard to understand their initial ideas about
mathematical practices and objects.
Initial Assessment of Common Content Knowledge
It is observed that of the 16 participants, two did not solve the diagnostic problem 1, and 14
provided partially correct answers since they correctly obtained the value 0.9 but did not justify
how they calculated it or only made use of Laplace's rule. Furthermore, an incorrect use of
Laplace's rule as a problem-solving strategy is noted. Participants did not consider the
distribution of absolute frequencies and assume that the four possible outcomes of the random
experiment forming the sample space are equiprobable, thus displaying an equiprobability bias
(Lecoutre, 1992). This bias appeared in studies like those of Mohamed (2012), Gómez (2014),
Parraguez et al. (2017), and Batanero et al. (2021).
Regarding problem 2, errors in probability calculations resulted from an incorrect
enumeration of the sample space and the inappropriate use of counting schemes to calculate
the number of favourable and possible cases (Gómez, 2014; Mohamed, 2012). In other cases,
participants correctly identified all possible outcomes but make mistakes in counting
favourable cases because they considered two different outcomes as identical, or vice versa,
erroneously thinking that two distinct results are the same. This type of combinatorial error was
identified by Batanero et al. (1997) as a confusion of the type of object.
None of the participants solved problem 3 correctly. The common error observed in this
statement is the confusion of conditional probability with compound probability (Contreras,
2011; Estrada & Díaz, 2007). That is, the PT incorrectly restricted the possible outcomes of the
random experiment, considering only all the men instead of the entire sample.
All these limitations, as also noted in previous research, suggest the need to enhance the
training of PTs in probability (Batanero et al., 2021; Parraguez et al., 2017; Vásquez & Alsina,
2015). As we show below, the participants themselves highlight these shortcomings in common
knowledge when asked about the difficulties encountered while solving the task.

Cotrado et al.– Development of Didactic Analysis Competence
16
Development of the Ontosemiotic Analysis Competence
The study of the responses provided by the PTs the analysis of meanings and ontosemiotic
configuration allows us to observe the difficulties in understanding the requirement of the tasks,
the achievements reached, and the possibilities offered by the situations presented in the
training workshop sessions.
Initial Exploration of Mathematical Practices and Objects
During the first session, the PTs were not clear about the nature of mathematical practices and
primary mathematical objects, as observed in Giacomone et al. (2018). When asked to describe
or distinguish mathematical practices and identify mathematical objects (primarily focusing on
concepts and languages, which we expected they would be more familiar with) in the solutions
or responses given to problem 1, eight participants did not respond, three did not distinguish
the sequences of practices but mentioned some mathematical objects, and five attempted to
sequence mathematical practices based on their use and intentionality. In this case, they
endeavoured to describe mathematical practices through their intentionality based on Polya's
generic problem-solving strategies. For example, they used expressions such as "identifying
the problem data," "verifying the problem," "reading the problem," "gathering data,"
"representing data, and calculation." Overall, in this initial session, none of the 16 PTs
distinguished elementary practices in describing the mathematical activity carried out to solve
the task. This demonstrates the initial difficulties of the participants in distinguishing and
sequencing the units of elementary practices.
After describing the mathematical practices, the participants were required to identify some
mathematical objects, such as concepts or languages. In this case, it is observed that only four
of them partially and somewhat hesitantly ("The concepts will be, the ones I am using in
addition or percentages, that part I don't understand," PT3) recognized the following concepts:
fractions, total cases, favourable cases, percentages, probability, and population.
The identification of concepts by the PTs during the sharing phase also allowed us to
recognize deficiencies in their knowledge of probability, which they often associated solely
with the classical approach and Laplace's rule. For example, PT6 wrote, "uses the concept of
probability 'favourable cases/possible cases'," and similarly, PT3 commented:
Regarding percentages and probabilities, I always use it as a formula, you could say, on the top of
the fraction is what I want or what the problem asks us for, and at the bottom is the total, in this case,
on top, it asks for those who do not play basketball.
Additionally, participants were asked to identify types of linguistic representations used in
mathematical practices. Most of them indicated "none" or left the response blank. Only one PT
recognized the pie chart ("none except for the pie chart that was given, PT4"), and another
identified procedural elements as languages ("sum of data, division of total data," PT7). The
initial difficulty of the participants in recognizing concepts and languages used in probability
mathematical practices would be associated with their lack of knowledge of these mathematical
objects.

REDIMAT – Journal of Research in Mathematics Education, 14(1)
17
Finally, participants were required to mention the complications they have encountered
during problem-solving. Four PTs pointed out little familiarity with the type of problem,
difficulties in understanding the problem statement, or were unsure about which procedure was
suitable to solve it.
Meanings of Probability and Ontosemiotic Analysis: Initial Progress
Following a reflection on the meanings of probability and their associated mathematical
objects, PTs were required to discuss their identification of the sequence of mathematical
practices and mathematical objects. This identification should encompass not only concept-
definitions and languages but also procedures, propositions, and arguments involved in solving
problem 2, connecting it with one of the meanings of probability. The results showed that five
participants did not complete the task, and of the eight who did, four did not manage to
sequence the practices properly, while the other four only partially described the elementary
units. This observation is consistent with findings from Burgos et al. (2018), where only half
of the participants in their study failed to distinguish practice units within the resolution
sequence or provided limited configurations.
In this session, we saw a slight improvement compared to the results from the first session
regarding the identification of concept-definitions and languages (the only types of
mathematical objects considered in the initial task as an introduction). While analysing problem
2, five PTs appropriately recognized graphical, symbolic, and numeric languages. Another
three mentioned them, referring to how they were used in mathematical practices. For example,
PT7 stated, "I use the [tree diagram] method to solve probabilities."
Recognition of the mathematical object "concept-definition" continued to be problematic in
this session. None of the PTs mentioned the concept-definitions of compound random
experiment, sample space, possible outcomes, and favourable outcomes, which, however, are
recurrent concepts in this task. Only five participants recognized the concept-definitions of
probability, event, percentages, and fractions, which appear explicitly in their solutions.
However, when they did, they often described these concepts as the mathematical practices
that involve them or the intention behind these practices. For example, PT1 mentioned,
"concepts: multiplying fractions," and PT7 stated, "the concepts I used were first to identify
the data in the problem and the probability," where it seems that they understood "data from
the problem" as a mathematical concept-definition.
PTs had difficulty recognizing procedures in their practices. In fact, 11 of them did not
respond, and two assimilated procedures with the practices in the sequencing (see Figure 4).

Cotrado et al.– Development of Didactic Analysis Competence
18
Figure 4
Object Identification by PT1
Note. Authors' translation.
Only PT12 identified "the probability of getting only heads is 3/8" as a proposition.
Similarly, regarding the mathematical object argument, only one PT mentioned it in his report,
and he did it incorrectly. This is the case of PT1 who, as observed in Figure 4, identified the
(object) argument as the process of generalization. Therefore, despite the training received on
mathematical practices and objects involved in probability tasks, the participants still faced
challenges in sequencing elementary mathematical practices and identifying the objects of
procedures, propositions, and arguments. However, there were some small achievements in
recognizing concepts and languages.
Regarding the meaning of probability, six PTs managed to relate this task to the classical
meaning, but they did not justify it based on equiprobability conditions or the finiteness of the
sample space, although one of them attributed it to the context of gambling. In some cases,
they exhibit a confused idea about the nature of the meaning of probability, as seen in PT3's
response:
In the types of probability meanings, I put that there is a lower probability of this process occurring
since it is less than 50%... I put it as classical.
Finally, the participants were asked to mention any difficulties they encountered while
working on the task. In this case, only three PTs indicated troubles in understanding the
problem they had to solve. None of them referred to specific difficulties regarding the
identification of mathematical objects.
Analysis of situation A
At the beginning of the final session, the PTs were asked to discuss the analysis of Situation A
(Figure 2), using the ontosemiotic configuration tool. It was expected that the training received
and providing them with a means to sequence, identify intentionality, and recognize objects in
elementary practices would improve the analysis of a situation, moreover, resolved by the
author of the textbook (expert solution).

REDIMAT – Journal of Research in Mathematics Education, 14(1)
19
In general, there was some improvement in the sequencing of elementary practices and the
identification of mathematical objects (especially in languages, concept-definitions, and
procedures) when analysing the solution provided in the workbook. Specifically, 12 PT
managed to divide the solution into three or four units of analysis. Additionally, the 13 PTs
who analysed this task correctly recognized the types of languages used in mathematical
practices (symbolic and numeric most frequently, with natural language being mentioned less
frequently).
All the participants successfully recognized some concept-definitions. In this case, as in the
studies by Gómez (2014) and Vásquez y Alsina (2015), the most frequently mentioned
concepts were probability, event, sample space, favourable cases, and possible cases, while the
less recognized ones included random situation, likely event, certain event, unlikely event,
impossible event, and possible outcomes. However, three of them incorrectly identified
Laplace's rule as a concept rather than a property. This confusion may be due to the usual
overrepresentation of the classical meaning in curricular materials, which can lead to an
assimilation of concept, property, and procedure (Cotrado et al., 2022; Gómez, 2014; Gómez
et al., 2015). Recognizing Laplace's rule as a concept rather than a property leads PTs to not
reflect on the conditions that justify the use of this property.
They also successfully identified some of the procedures considered in the a priori analysis.
For example, 11PTs identified the procedure "construct or list the sample space" and "calculate
the probability using Laplace's rule," the latter in line with what was observed in Mohamed
(2012). However, arithmetic procedures or conversion between fractions and decimal numbers
went unnoticed. For example, none of the PTs mentioned the procedure for reducing fractions
and expressing the probability value as a decimal, although one mentioned "expressing
probability as percentages."
Although the participants recognized most of the concept-definitions, languages, and some
procedures involved in Situation A, they still struggled to recognize propositions and
arguments, as seen in the studies by Giacomone et al. (2018) and Burgos et al. (2018). Only
two participants (PT1, PT13) partially identified a proposition and its associated argument,
while two others (PT5, PT7) mentioned the proposition as the intention or requirement of the
practice ("rolling a die once, determining the events"). For example, PT1 indicated the
proposition "The number of favourable cases in event A is 3" and as an argument "because in
the 6 events of the sample space, only 3 meet what event A requires-." Also, PT13 identified
the proposition "the probability value ranges from 0 to 1" and the argument "Laplace's rule."
In other cases, generic arguments are mentioned before the propositions and are not relevant.
In this sense, it can be concluded that, although there are achievements in identifying the
objects of concepts-definition, languages, and procedures, PTs continue to have difficulties in
recognizing propositions and arguments.
Situation A was incorrectly related to the intuitive meaning of probability by six PTs, four
of them associated it with the classical meaning, one with the classical and intuitive meaning,
and two did not respond to this question. Except for PT8, who justified that the associated
meaning was classical because it was a game of chance, the others did not provide reasoning
for their responses. They also did not do so during the discussion, despite questions from the
instructor, although in some cases, they mentioned their difficulties regarding this. In this

Cotrado et al.– Development of Didactic Analysis Competence
20
regard, seven participants mentioned having conflicts in recognizing mathematical objects ("To
be honest, I didn't have many difficulties, just a little trouble recognizing mathematical objects
because studying mathematical objects is something new to me," PT4), three in relating the
situation to a meaning of probability; two referred to difficulties in understanding the procedure
of the solved situation, and four did not record responses.
Difficulties in identifying the meanings of probability implied in the proposed situations,
especially in distinguishing between the intuitive and classical meanings, were not only
observed in the analysis of PTs written answers, but also during the sharing. For instance, in
the group discussion, PT5 mentioned that although they initially thought the meaning was
intuitive because "the intuitive had events, sure, possible, impossible," they later began to have
doubts if it was the classical meaning "because there were favourable and possible cases."
Final Assessment Tasks
While systematic assessment of participants' progress was carried out throughout the
workshop, Problem 3 and Situation B (both including an expert solution, in the first case
provided by the instructor and, in the second case included in the textbook, Figure 3) are used
as final evaluation instruments to determine the degree of ontosemiotic analysis competence
achieved with the training action.
Results of the Ontosemiotic Analysis of the Expert Solution to Problem 3
The ontosemiotic analysis of the solution to item 3, facilitated by the instructor, was carried
out by 10 PTs As a result, it was observed that seven of them distinguished sequences of
mathematical practices into two or three units of analysis, while the other three did not
differentiated them. Regarding the identification of objects, eight participants correctly
identified verbal and symbolic-numerical languages in different elementary units, but they
encountered difficulties in recognizing tabular language (identified this way by only one PT),
which half of them referred to as graphical language (see Figure 5).
Nine participants recognized explicit concepts in this task, such as probability, randomness,
favourable cases, and possible cases. However, concepts like population (Figure 5), event,
sample space, double, marginal and conditional frequency were mentioned very rarely. Similar
to Contreras (2011), the participants did not identify the concept of compound probability in
the solution (PT1 did not indicate simple probability nor compound probability in the last
elementary practice) They also mentioned as concept-definitions essential elements in the
information provided by the problem (e.g., "total workers" instead of possible cases).

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Figure 5
Ontosemiotic Analysis of the Expert Solution to Problem 3 (PT1)
Note. Authors’ translation.
The mathematical object "procedure" was recognized in various practices by seven
participants. In line with Contreras' study (2011) and like previous tasks, most referred to the
method of calculating probability using the "application of Laplace's rule." In the practice unit
"we build a contingency table, place the variables, and distribute the compound, marginal and
total absolute frequencies in each cell," only two participants partially mentioned procedures
like "building a two-way table" and "transferring data." In this case, participants indicated
arithmetic calculations as a procedure, which had not been mentioned in previous analyses.
Participants continued to face difficulty in recognizing arguments and propositions (Burgos
et al., 2018; Contreras, 2011; Giacomone et al., 2018; Gómez, 2014). Only three correctly
pointed out some propositions in the given solution, for example, "the probability of being a
man and not reading the magazine is 0.35" (PT1 in Figure 5, PT4). In this case, they associated
it with an explicit argument, for example, "Because out of 200, only 70 are men and do not
read" (PT4), although not always appropriately, for instance, "Argument: graphical support"
(PT1), referring to the information included in the two-way table (Figure 5). This was the only

Cotrado et al.– Development of Didactic Analysis Competence
22
argument indicated by the participants ("table support to verify the data," PT5), in addition to
the one based on Laplace's rule (PT3). The rest of the participants confused the proposition
with what the problem asks, indicated propositions related to the two-way table, or did not
provide responses.
Lastly, it's worth mentioning that three participants related this task to the frequentist
meaning, and two to the classical meaning, but, as in previous tasks, they did not provide
justifications. In particular, as shown in Figure 5, despite associating mathematical practices
with frequentist meaning, PT1 indicated that Laplace's Rule was applied to determine the
probabilities required in questions (a) and (b). Regarding difficulties, only PT13 quoted having
difficulties identifying mathematical objects, while the rest did not indicate any or maintained
their initial responses (Session 1) regarding the comprehension of the problem itself.
Results of the Analysis of Situation B Prepared by the Prospective Teachers
We observed that out of the 12 participants who analysed Situation B, nine broke down the
sequences of mathematical practices into two or three units of analysis, while three did not
distinguish elemental practices. Although 11 PT adequately recognized the presence of
language of different types: verbal, symbolic and numerical, only two participants identified
the graphic or tabular language involved in the two-way table. At this point, participants are
starting to more successfully recognize emerging concept-definitions in mathematical
practices, with the most frequently mentioned one being "sample space" and "event". This
progress in identifying concept-definitions following the training aligns with previous
observations by Giacomone et al. (2018) and Burgos et al. (2018). However, they still do not
consider "possible outcome" and "probable outcome," or, as shown in Figure 6, do not
recognize "compound probability," "random variable," "distribution," and "mode" as concepts
involved in Situation B. However, similar to previous tasks, four PTs persisted in recognizing
the Laplace’s Rule as a concept-definition, rather than a property. In this task, as in Contreras'
study (2011), the PTs s recognized concepts as elements that describe the problem: colour, die.
The mathematical object "procedure" was recognized by seven participants, primarily in the
practice unit "We determine the probability of A applying the Laplace's rule." The most
frequently mentioned procedure was "calculation of probability using Laplace's rule" (in all
cases), followed by "building/interpreting a two-way table" (four PTs) and "listing the sample
space" (three PTs) (see Figure 6). Two PTs indicated "finding the sum" or arithmetic
calculations as procedures involved in determining the number of possible cases or in applying
Laplace's rule as procedures. Other participants did not mention any type of procedure or
continue to incorrectly identify a procedure with the intention or use of mathematical practices.
For example, they pointed: "identifying the given data regarding how to write the data obtained
in the two-way table and number line" (FP5); "identifying how much the sample space will be"
(FP1); "using the table method to solve" (FP7).
Although no participant to correctly identified all the propositions and arguments involved
in situation B, and they continued to confuse propositions with questions posed in the problem
("How many elements does the respective sample space have?"; "Which sum is more likely to
occur?"), there was some progress in the identification of these objects. As shown in Figure 6,

REDIMAT – Journal of Research in Mathematics Education, 14(1)
23
some PTs started to correctly recognize propositions and their associated arguments
specifically for elementary practice units. As for the object "argument," other participants, like
FP5, stated that the argument "is supported by the table."
Figure 6
Excerpt from the Ontosemiotic configuration for Situation B (PT4)
Authors’ translation.
Five participants related this task to the classical meaning, two of them said it corresponds
to both the frequentist and classical meanings, and one related it to the intuitive meaning. As
had already happened in the analysis of Situation A, none of them managed to justify why they
relate it to these meanings. Regarding difficulties, two mentioned that they still found the
elaboration of the onto-semiotic configuration challenging, but in this case, none of them
mentioned having had difficulties in relating the task to the meanings of probability (even
though four of them did not respond to that instruction).
Conclusion
Throughout this work, we have described the design, implementation, and results of a
formative experience with prospective Peruvian mathematics teachers aimed at fostering the
competence of didactic-mathematical analysis of student workbooks in probability. Since
teachers often rely on curriculum materials as the institutional knowledge that must be taught
and learned, it seems necessary to ensure that PTs are competent in critically analysing the

Cotrado et al.– Development of Didactic Analysis Competence
24
content that underpins the management and use of such materials (Godino et al., 2017; Shawer,
2017). This competence can only be achieved with a deep mathematical and didactic
knowledge of the intended teaching object, allowing for a microscopic analysis of the sequence
of operational, discursive, and normative practices proposed by the author to ensure learning
by potential students.
Distinguishing between meanings and identifying the objects involved in mathematical
practices is a challenge for PTs. However, it is a competence that will allow them to understand
the progression of learning, manage the necessary institutionalization processes, and assess
students' mathematical competencies.
The results of the initial exploration, as seen in previous research (Contreras, 2011; Gómez,
2014; Mohamed, 2012; Vásquez & Alsina, 2015), showed a deficient common knowledge of
probability among PTs. Their responses exhibited equiprobability bias (Lecoutre, 1992),
incorrect enumeration of the sample space, inappropriate use of counting schemes (tree
diagram) to calculate the number of favourable and possible outcomes, confusion between
conditional and compound probability (Contreras, 2011; Estrada & Díaz, 2007), and
calculation errors.
In addition to the limitations posed by a deficient mathematical knowledge for the progress
of the didactic analysis competence, we found that PTs participating in the training experience
did not have a clear understanding of the nature and function of mathematical objects.
Concerning procedures, they often confused them with the mathematical practices themselves
or their intentions, and they couldn't distinguish arguments from propositions, which
sometimes appeared in interrogative form, referring to the problem statements (Burgos &
Godino, 2021; Burgos et al., 2018). In this regard, as noted by Font et al. (2013), although the
onto-semiotic configuration tool is clear a priori, applying it to analyse mathematical activity
can be complex because primary mathematical objects (conventional rules) appear as
descriptions of secondary mathematical objects. Thus, a procedure or a property may present
itself as the definition of a concept in the mathematical practices included in curricular
materials. Nonetheless, we believe that the training provided, the sharing of experiences, and
discussions of the answers given in each class helped PTs improve their analytical competence.
They managed to distinguish and sequence the units of elementary mathematical practices,
began to recognize concepts and languages, and, to some extent, the various pragmatic
meanings of probability put into play in different proposed tasks.
A significant progression within the levels of development for the competence in didactic-
mathematical analysis was observed among the PTs throughout the sessions. In the first
session, out of the 16 PTs who participated, only four were able to partially recognize some
mathematical objects, placing them at level L0. For the second session, in which situations 2
and A were addressed asynchronously and individually, the group was reduced to 13
participants. Among them, five PTs demonstrated progress by partially recognizing elemental
units and some meanings based on their experience. Thus, they reached level L1, with an
improved understanding of sequencing elementary practices and identifying mathematical
objects, such as concepts, languages, and procedures. In the third and final session, the training
received, experience-sharing, discussions on responses, and the use of the ontosemiotic
configuration tool promoted a notable advancement in didactic analysis competency. Of the 13
participants, more than half managed to distinguish and sequence units of elementary

REDIMAT – Journal of Research in Mathematics Education, 14(1)
25
mathematical practices, appropriately identified concepts, languages, procedures, and partially
recognized the various pragmatic meanings of probability within the proposed tasks. However,
none were able to justify why they related these meanings. This process allowed them to reach
level L2.
The results of the experience highlight the challenges presented by these types of activities
for both PTs and educators. Nevertheless, the recognition of meanings, the description of
practices, and the identification of objects are key elements in training teachers to implement
mathematics study processes that promote students' mathematical competence. Our results are
consistent with those obtained in Rubio (2012), which highlight that the teacher's competence
in analysing mathematical practices, objects, and processes is a "deep knowledge" that enables
the evaluation and development of their students' mathematical competence, However, its
implementation in professional training is not without ambiguities As Godino (2024) indicate,
teaching a mathematical content may be compromised if teachers do not recognize the nature
and role of the objects involved in the mathematical practices associated with the problem-
solving domain: problem situations are the origin of activity; arguments justify procedures and
propositions connecting concepts; languages are the ostensive part of concepts, propositions,
and procedures while also contributing to the elaboration of arguments.
Since a clear limitation of our study is that the sample size may restrict the generalizability
of the results obtained, further research cycles will be necessary. In future experiences, it is
necessary, first and foremost, to reinforce mathematical knowledge related to the content
involved, in our case, probability. More space should be dedicated to reflection on a wider
variety of problem situations that allow achieving an adequate level of competence in the
analysis of meanings and onto-semiotic analysis of mathematical practices.
Acknowledgments
This work is part of the research project PID2022-139748NB-100 founded by
MICIU/AEI/10.13039/501100011033/ and FEDER, UE, with support from the research group
FQM-126 (Junta de Andalucía, Spain) and from Group S60_23R - Investigación en Educación
Matemática (Government of Aragón).
Notes
1 The research complies with the ethical principles established in the guidelines of the Institutional Research Ethics
Committee of the Universidad Nacional del Altiplano. The consent report of the participants is available. Our
research guarantees the strict confidentiality of the information provided by each participant and preserves their
anonymity.

Cotrado et al.– Development of Didactic Analysis Competence
26
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