ABSTRACT
The results of a training experience with future mathematics teachers, focusing on the didactic analysis of curricular materials on probability, are presented. Although there was progress in identifying mathematical objects, difficulties were encountered in recognizing problemsituations, procedures, propositions, and arguments, as well as the meanings of probability. In the expert assessment of the didactic suitability of the standards, shortcomings were observed; however, the participants did not adequately evaluate these shortcomings, particularly in the epistemic and cognitive facets. These limitations could be attributed to a lack of specific training and insufficient time to become familiar with the suitability indicators.
Keywords
Didactic Analysis; Didactic Suitability; Curricular Material; Probability; Teacher Training
RESUMEN
Se presentan resultados de una experiencia formativa con futuros profesores de matemáticas, centrada en el análisis didáctico de materiales curriculares sobre probabilidad. Aunque hubo progreso en la identificación de objetos matemáticos, se encontraron dificultades para reconocer situacionesproblema, procedimientos, proposiciones y argumentos, así como los significados de probabilidad. En la valoración experta de la idoneidad didáctica de la normativa, se observaron deficiencias, pero los participantes no evaluaron adecuadamente dichas carencias, especialmente en las facetas epistémica y cognitiva. Estas limitaciones podrían deberse a la falta de formación específica y al poco tiempo para familiarizarse con los indicadores de idoneidad.
Palabrasclave
Análisis Didáctico; Idoneidad Didáctica; Material Curricular; Probabilidad; Formación de Profesores
Introduction
The training of mathematics teachers emphasizes the importance of developing skills to professionally describe, explain, and assess the teaching and learning processes (Breda; PinoFan; Font, 2017BREDA, Adriana; PINOFAN, Luís Roberto; FONT, Vicenç. Meta didacticmathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice. EURASIA – Journal of Mathematics, Science and Technology Education, London, v. 13, n. 6, p. 18931918, 2017.; Giacomone et al., 2018GIACOMONE, Belen et al. Desarrollo de la competencia de análisis ontosemiótico de futuros profesores de matemáticas. Revista Complutense de Educación, Madrid, v. 29, n. 4, p. 11091131, 2018.; PinoFan; Assis; Castro, 2015PINOFAN, Luis Roberto; ASSIS, Adriana; CASTRO, Walter. Towards a Methodology for the Characterization of Teachers DidacticMathematical Knowledge. Eurasia – Journal of Mathematics, Science and Technology Education, London. v. 11, n. 6, p. 14291456, 2015.). Since promoting a critical and reflective approach to the effective use of curriculum materials is essential (Braga; Belver, 2016BRAGA, Gloria; BELVER, José Luís. El análisis de libros de texto: una estrategia metodológica en la formación de los profesionales de la educación. Revista Complutense de Educación, Madrid, v. 27, n. 1, p. 199218, 2016.), Shawer (2017)SHAWER, Saad. Teacherdriven curriculum development at the classroom level: Implications for curriculum, pedagogy and teacher training. Teaching and Teacher Education, Philadelphia, v. 63, p. 296313, 2017. suggests fostering training in curriculum development, including management of resources such as curriculum programs and teaching guides, among others. Teachers must interpret information in curriculum materials and make adaptations according to contextual needs (Taylor, 2013TAYLOR, Megan Westwood. Replacing the “teacherproof” curriculum with the “curriculumproof” teacher: Toward more effective interactions with mathematics textbooks. Journal of Curriculum Studies, London, v. 45, n. 3, p. 295321, 2013.; Thompson, 2014THOMPSON, Dennise. Reasoningandproving in the written curriculum: Lessons and implications for teachers, curriculum designers, and researchers. International Journal of Educational Research, Oxford, v. 64, p. 141148, 2014.; Yang; Liu, 2019YANG, KaiLin; LIU, XinYi. Exploratory study on Taiwanese secondary teachers critiques of mathematics textbooks. Eurasia Journal of Mathematics, Science and Technology Education, Belgrade, v. 15, n. 1, p. 116, 2019.).
Despite this importance, both preservice and inserve teachers often encounter difficulties in critically analysing curriculum materials (Shawer, 2017SHAWER, Saad. Teacherdriven curriculum development at the classroom level: Implications for curriculum, pedagogy and teacher training. Teaching and Teacher Education, Philadelphia, v. 63, p. 296313, 2017.; Yang; Liu, 2019YANG, KaiLin; LIU, XinYi. Exploratory study on Taiwanese secondary teachers critiques of mathematics textbooks. Eurasia Journal of Mathematics, Science and Technology Education, Belgrade, v. 15, n. 1, p. 116, 2019.), making it important to develop specific tools that enable the development of their reflective competence regarding such resources (Remillard; Kim, 2017REMILLARD, Janine; KIM, OkKyeong. Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, New York, v. 96, n. 1, p. 6581, 2017.).
This paper describes a formative experience with future teachers of secondarylevel mathematics in Peru, focused on the didactic analysis of curriculum guidelines. We analysed a curriculum program about probability, considering cognitive, social, cultural, and axiological aspects of teaching. Probability is essential in mathematics and has gained importance in recent curricula (CCSSI, 2010CCSSI. Common Core State Standards Initiative. Common Core State Standars for Mathematics. Washington, DC, 2010. Disponible en: https://learning.ccsso.org/wpcontent/uploads/2022/11/ADACompliantMathStandards.pdf. Accedido el: 23 ene. 2023.
https://learning.ccsso.org/wpcontent/up...
; MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016.). In particular, the Peruvian education system includes the study of probability from the earliest educational cycles (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016.). However, there is an observed bias towards the classical approach rather than the frequency or subjective meaning, as well as a lack of representativeness of the proposed situations (Cotrado; Burgos; BeltránPellicer, 2022COTRADO, Bethzabe; BURGOS, María; BELTRÁNPELLICER, Pablo. Idoneidad Didáctica de Materiales Curriculares Oficiales Peruanos de Educación Secundaria en Probabilidad. Bolema, Rio Claro, v. 36, n. 73, p. 888922, 2022.; Vásquez; Alsina, 2015VÁSQUEZ, Claudia; ALSINA, Ángel. El conocimiento del profesorado para enseñar probabilidad: Un análisis global desde el modelo del Conocimiento DidácticoMatemático. Avances de Investigación en Educación Matemática, Madrid, v. 7, p. 2748, 2015.).
For the development of the research, we relied on the Ontosemiotic Approach (OSA) to mathematical knowledge and instruction (Godino; Batanero; Font, 2007GODINO, Juan; BATANERO, Carmen; FONT, Vicenç. The ontosemiotic approach to research in mathematics education. ZDM, Berlin, v. 39, n. 12, p. 127135, 2007.). The OSA provides theoretical and methodological tools for the analysis of mathematical activity in curriculum materials and the assessment of the didactic suitability of instructional processes (Godino, 2013GODINO, Juan. Indicadores de la idoneidad didáctica de procesos de enseñanza y aprendizaje de las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática, San Pedro, v. 8, n. 11, p. 111132, 2013.; Breda; Font; PinoFan, 2018BREDA, Adriana; FONT, Vicenç; PINOFAN, Luis Roberto. Criterios valorativos y normativos en la Didáctica de las matemáticas: el caso del constructo idoneidad didáctica. Bolema, Rio Claro, v. 32, n. 60, p. 255278, 2018.). It also offers a model of knowledge and competencies for mathematics teachers, allowing for the definition of the type of professional knowledge that prospective teachers must acquire in this regard (Godino et al., 2017GODINO, Juan et al. Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, Rio Claro, v. 31, n. 57, p. 90113, 2017.). These tools are briefly described in the following section.
Theoretical Framework
The research is based on the model of DidacticMathematical Knowledge and Competencies (DMKC model) for mathematics teachers, developed within the OSA framework by Godino et al. (2017)GODINO, Juan et al. Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, Rio Claro, v. 31, n. 57, p. 90113, 2017.. This model considers two key competencies for mathematics teachers, mathematical competence and the competence of analysis and didactic intervention, which involves “designing, applying, and evaluating learning sequences developed by oneself and others, using didactic analysis techniques and criteria of quality, in order to establish cycles of planning, implementation, assessment, and propose improvements” (Breda; Pinofan; Font, 2017BREDA, Adriana; PINOFAN, Luís Roberto; FONT, Vicenç. Meta didacticmathematical knowledge of teachers: Criteria for the reflection and assessment on teaching practice. EURASIA – Journal of Mathematics, Science and Technology Education, London, v. 13, n. 6, p. 18931918, 2017., p. 1897). This competence is articulated through five subcompetencies associated with the conceptual and methodological tools of the OSA: analysis of global meanings, ontosemiotic analysis of practices, management of didactic configurations and trajectories, normative analysis, and didactic suitability analysis. In our work, we focus on the subcompetencies of analysis of global meanings, ontosemiotic analysis, and didactic suitability analysis (Godino et al., 2017GODINO, Juan et al. Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, Rio Claro, v. 31, n. 57, p. 90113, 2017.).
The analysis of global meanings competency involves identifying problemsituations and operational, discursive and normative practices implied in their resolution. The teacher must recognize the different meanings of probability: intuitive, laplacian, frequencybased, subjective, and axiomatic (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.; Batanero et al., 2016BATANERO, Carmen et al. Research on Teaching and Learning Probability. Cham: Springer, 2016.), how they relate to each other, and how they are addressed in school curricula at different educational levels. The competency of ontosemiotic analysis of mathematical practices allows the teacher to identify the diversity of objects and processes involved in mathematical practices necessary for solving problemsituations. This recognition enables the
[…] anticipation of potential and effective learning conflicts, assessment of students' mathematical competencies, and identification of objects (concepts, propositions, procedures, arguments) that must be remembered and institutionalized at opportune moments in the study processes
(Godino et al., 2017GODINO, Juan et al. Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, Rio Claro, v. 31, n. 57, p. 90113, 2017., p. 94).
The competency of analysis of the didactic suitability of mathematical study processes allows the teacher to assess the appropriateness of the planned or implemented instructional processes and make wellfounded decisions for improvement (Godino et al., 2017GODINO, Juan et al. Enfoque ontosemiótico de los conocimientos y competencias del profesor de matemáticas. Bolema, Rio Claro, v. 31, n. 57, p. 90113, 2017.).
The didactic suitability of an instructional process is defined as the extent to which this process (or a part of it) possesses certain characteristics that allow it to be classified as optimal or suitable for achieving alignment between students' personal meanings attained (learning) and intended or implemented institutional meanings (teaching), taking into account circumstances and available resources (environment). This involves the coherent and systemic articulation of six facets that affect the instructional process: epistemic (specialized mathematical knowledge), cognitive (students' prior knowledge, difficulties, and reasoning), affective (attitudes, beliefs, and emotions of students), interactional (classroom discourse management), mediational (technological and temporal resources), and ecological (curricular adaptation, interdisciplinary and societal relationships) (Godino, 2013GODINO, Juan. Indicadores de la idoneidad didáctica de procesos de enseñanza y aprendizaje de las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática, San Pedro, v. 8, n. 11, p. 111132, 2013.; Godino et al., 2007GODINO, Juan; BATANERO, Carmen; FONT, Vicenç. The ontosemiotic approach to research in mathematics education. ZDM, Berlin, v. 39, n. 12, p. 127135, 2007.).
For didactic suitability criteria, understood as “[…] corrective norms that establish how a teaching and learning process should be carried out” (Breda; Font, Pinofan, 2018BREDA, Adriana; FONT, Vicenç; PINOFAN, Luis Roberto. Criterios valorativos y normativos en la Didáctica de las matemáticas: el caso del constructo idoneidad didáctica. Bolema, Rio Claro, v. 32, n. 60, p. 255278, 2018., p. 264), to be operational, it is necessary to define a set of observable indicators that allow for the assessment of their degree of achievement (Godino, 2013GODINO, Juan. Indicadores de la idoneidad didáctica de procesos de enseñanza y aprendizaje de las matemáticas. Cuadernos de Investigación y Formación en Educación Matemática, San Pedro, v. 8, n. 11, p. 111132, 2013.). Furthermore, indicators of didactic suitability should be enriched and tailored, considering the specific content (Breda; Font, Pinofan, 2018BREDA, Adriana; FONT, Vicenç; PINOFAN, Luis Roberto. Criterios valorativos y normativos en la Didáctica de las matemáticas: el caso del constructo idoneidad didáctica. Bolema, Rio Claro, v. 32, n. 60, p. 255278, 2018.) and the particularity of the instructional process under analysis. For this reason, in Cotrado, Burgos and BeltránPellicer (2022)COTRADO, Bethzabe; BURGOS, María; BELTRÁNPELLICER, Pablo. Idoneidad Didáctica de Materiales Curriculares Oficiales Peruanos de Educación Secundaria en Probabilidad. Bolema, Rio Claro, v. 36, n. 73, p. 888922, 2022., a systematic review of didactic suitability criteria and indicators is conducted to develop a guide for assessing curriculum materials related to probability.
The aim of our research is to study how the competency of didactic analysis is mobilized and developed in prospective secondary education teachers through formative action, focused on the analysis of the curriculum program related to the topic of probability. Didactic analysis is understood within the OSA as “[…] the systematic study of the factors that condition the teaching and learning processes of a curricular content – or partial aspects of it – with specific theoretical and methodological tools” (Godino et al., 2006GODINO, Juan et al. Análisis y valoración de la idoneidad didáctica de procesos de estudio de las matemáticas. Paradigma, Maracay, v. 27, n. 2, p. 221252, 2006., p. 4). It involves, therefore, the analysis of meanings through the identification of practices, ontosemiotic analysis or recognition of the objects involved in these practices, and the assessment of the didactic suitability of the intended or planned instructional process.
Next, we describe the design of the formative action and the process of evaluating the responses provided by the participants.
Methodology
The research approach of this work is interpretive and exploratory in nature, characteristic of design research (Kelly; Lesh; Baek, 2008KELLY, Anthony; LESH, Richard; BAEK, John (Ed.). Handbook of design research in methods in education: innovations in science, technology, engineering, and mathematics learning and teaching. New York: Routledge, 2008.). It takes place in an authentic classroom context, based on the planning, implementation, and retrospective analysis of an intervention (Godino et al., 2014GODINO, Juan et al. Ingeniería didáctica basada en el enfoque ontológicosemiótico del conocimiento y la instrucción matemáticos. Recherches en Didactique des Mathématiques, Grenoble, v. 34, n. 2/3, p. 167200, 2014.). Additionally, the methodology of content analysis (Cohen; Lawrence; Morrison, 2011COHEN, Louis; MANION, Lawrence; MORRISON, Keith. Research methods in education. London: Routledge, 2011.) is employed to examine transcripts of class recordings, as well as participants' response protocols.
Research Context, Participants and Data Collection
The formative experience was conducted at the Faculty of Education of the Universidad Nacional del Altiplano (Peru), involving 14 prospective teachers (PT) from the Mathematics, Physics, Computer Science, and Informatics Program who were taking Descriptive Statistics during the fourth semester of 2021. The intervention comprised theoreticalpractical activities and group discussions, utilizing Google Meet and Classroom for synchronous and asynchronous sessions.
Four virtual synchronous sessions were conducted, each lasting two hours. The first two sessions addressed the analysis of global meanings and ontosemiotic analysis, while the last two sessions focused on the analysis of didactic suitability based on curriculum norms.
Session 1: Development of the competency in the analysis of global meanings and ontosemiotic analysis. Pragmatic meanings of probability and the network of characteristic mathematical objects were presented. Subsequently, the PTs analysed the curriculum program (CP) (MINEDU, 2023MINEDU. Currículo Nacional de la Educación Básica. LimaPerú: Ministerio de Educación, 2023. Disponible en: http://www.minedu.gob.pe/curriculo/. Acceso el: 2 ene. 2023.
http://www.minedu.gob.pe/curriculo/...
) for the data management and uncertainty block. The CP is divided into 11 units of analysis: NC6 (expected competency level by the end of cycle VI), DG1.1DG1.5 (firstgrade performances), and DG2.1DG2.5 (secondgrade performances). The task for this session involved identifying the mathematical objects and relating them to the meanings of probability emerging in NC6. The session included a group discussion and proposed asynchronous work, which entailed the analysis of DG1.1.
Session 2: Putting into practice. The PTs shared and compared their analysis of DG1.1. They continued by individually analysing DG1.2, DG1.3, and DG1.4, and then discussed the results. As an individual asynchronous evaluation task, they were assigned to analyse the units of analysis related to secondgrade probability.
Session 3: Introduction of the didactic suitability analysis tool. Didactic suitability and its system of components and general empirical indicators were presented as a means of reflection and a rubric for analysing study processes in teaching practice.
Session 4: Application of the didactic suitability analysis tool. The PTs applied the suitability indicators to the CP using the guide from Cotrado, Burgos and BeltránPellicer (2022)COTRADO, Bethzabe; BURGOS, María; BELTRÁNPELLICER, Pablo. Idoneidad Didáctica de Materiales Curriculares Oficiales Peruanos de Educación Secundaria en Probabilidad. Bolema, Rio Claro, v. 36, n. 73, p. 888922, 2022., examining the units of analysis and evaluating whether the suitability indicators were always satisfied, sometimes satisfied, or never satisfied.
Session recordings, trainer's annotations, and written responses to specific tasks were used as data collection instruments.
A priori analysis of the Curricular Programme
The authors conducted didactic analysis of the curriculum program (CP) as a basis for examining the participants' productions. Initially, they analysed the meanings and mathematical objects involved in the units of analysis NC6, DG1.1 to DG1.5, and DG2.1 to DG2.5 of the CP.
In the analysis of NC6, it was determined that the meanings of probability are not clearly identified through the objects associated with each of them (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.), such that situations can be related to both the classical and frequencybased approaches, or even to intuitive aspects (qualitative assessment, predictions). Students are expected to use verbal and symbolicnumeric language registers (fractional, decimal, and integer representations). Specifically, the concepts of random event or situation, probability, sample space, event, certain event, likely event, and impossible event are involved. Procedures were identified such as “enumeration of elementary events,” “relating the value of probability to certain, likely, or impossible events,” and “predicting the occurrence of events.” Propositions such as “the certain event always occurs” and “the probability of an event is associated with a number between 0 and 1” were also found. To justify the propositions in NC6, students are expected to use arguments that support the occurrence of events and the assignment of values between 0 and 1 to certain, likely, and impossible events.
Next, Table 1 exemplifies the expert analysis of the firstgrade units of analysis in relation to the mathematical objects observed in the CP.
Regarding the meanings of probability, it was observed that in the DG1.1, DG1.4, DG2.1, and DG2.4 analysis units, terms and expressions related to both the classical and frequencybased approaches are mentioned, such as the use of the Laplace’s rule and the calculation of frequencies or relative frequency. On the other hand, DG1.2, DG1.3, DG1.5, DG2.2, DG2.3, and DG2.5 do not clearly establish a specific approach, which could suggest an orientation towards the classical, frequencybased, or even intuitive approaches. Regarding the frequencybased approach, statistical procedures are included, but not experimentation and simulation. The intuitive approach is identified through qualitative assessments of probability.
After analysing the meanings and mathematical objects, the researchers independently evaluated the suitability of the CP, using criteria and indicators of didactic suitability. This expert analysis will serve as a reference for interpreting the assessments of the PTs. For the degree of fulfillment of an indicator, never, sometimes, and always are assigned 0, 1, and 2 points, respectively. Table 2 illustrates how the different indicators are reflected and to what degree, in relation to the problemsituation components and languages of the epistemic facet, exemplifying the expected analysis by the PTs.
Assessment and identification of indicators according to the components of epistemic suitability
Epistemic Suitability:The CP presents various types of problemsituations without specifying their connection to the meanings of probability, leading to a “sometimes” rating. There is no mention of problematizing situations where students can experiment with or simulate random experiences, resulting in a “never” rating. Concerning languages, the CP promotes the use of suitable registers for the educational level, with an “always” rating. Key concepts, propositions, and procedures are missing in first and second grades, which could introduce biases in learning (Vásquez; Alsina, 2017VÁSQUEZ, Claudia; ALSINA, Ángel. Proposiciones, procedimientos y argumentos sobre probabilidad en libros de texto chilenos de Educación Primaria.Profesorado, Revista de currículum y formación del profesorado, Granada, v. 21, n.1, p. 433457, 2017.). Clarifications about relationships between meanings of probability through assimilable mathematical objects are not provided (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.).
Cognitive Suitability:The CP offers general expressions about the progressive treatment of content but does not fully address the diverse meanings of probability (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.). The curriculum focuses on the classical meaning, with little consideration for frequencybased and intuitive meanings, and does not mention common reasoning biases (Lecoutre, 1992LECOUTRE, Marie Paule. Cognitive models and problem spaces in “purely random” situations. Educational Studies in Mathematics, Dordrecht, v. 23, n. 1, p. 557568, 1992.).
Affective Suitability: The assessment of alignment with student needs and interests is “always,” as the CP clearly states: “[…] that the student analyses data on a topic of interest or study or random situations, allowing them to make decisions, formulate reasonable predictions, and draw conclusions supported by the produced information” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 273). Although the curriculum does not provide specific evidence about emotions, attitudes, and beliefs of students toward random situations, a general record exists for the entire mathematics area that “Emotions, attitudes, and beliefs act as driving forces of learning” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 148); therefore, its alignment is rated as “never.”
Interactional Suitability: All indicators in this dimension are only partially observed because the CP does not provide specific interaction guidelines between teacherstudent or promotes very generic orientations that encourage communicative interaction among students, as seen in the following expression: “Provide spaces for students for dialogue, debate, discussion, and decisionmaking, related to their actions or others' actions in various situations” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 25). Autonomy is also promoted through the crosscutting competency “Manages their learning autonomously” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 29), which applies to all areas.
Mediational Suitability: The use of manipulative and computational resources is not explicitly stated for probability in the curriculum. However, through the expression “[...] use strategies and procedures to collect and process data” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 170), it can be inferred that resources facilitating the calculation of probabilistic measures are implied. Regarding the proper use of classroom space and resources, the CP generally mentions that “The organization of educational spaces, the appropriate and relevant use of educational materials and resources, as well as the teaching role, provide environments and interactions that create a favourable climate for learning” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 54). There are no indications in the CP suggesting the management of suitable schedules or timeframes for addressing probability.
Ecological Suitability: International research and guidelines indicate that meanings of probability should be treated progressively in school curricula (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.; Giacomone et al., 2018GIACOMONE, Belen et al. Desarrollo de la competencia de análisis ontosemiótico de futuros profesores de matemáticas. Revista Complutense de Educación, Madrid, v. 29, n. 4, p. 11091131, 2018.; Vásquez; Alsina, 2017VÁSQUEZ, Claudia; ALSINA, Ángel. Proposiciones, procedimientos y argumentos sobre probabilidad en libros de texto chilenos de Educación Primaria.Profesorado, Revista de currículum y formación del profesorado, Granada, v. 21, n.1, p. 433457, 2017.). However, this is only partially fulfilled in the CP. No expressions are observed that promote innovation. Socioprofessional dimensions, values education, and interdisciplinary connections are presented in a very generic manner. For example, concerning values education, the CP generalizes that
[…] crosscutting approaches are the observable concretization of the values and attitudes that teachers, students, [...] are expected to demonstrate in the daily dynamics of the educational institution, and which extends to the various personal and social spaces in which they operate
(MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 20).
Analysis of Results and Discussion
In this section, we examine the identification of meanings and mathematical objects by the FPs and their success in evaluating the didactic suitability in the curriculum using the specific indicator guide by Cotrado, Burgos and BeltránPellicer (2022)COTRADO, Bethzabe; BURGOS, María; BELTRÁNPELLICER, Pablo. Idoneidad Didáctica de Materiales Curriculares Oficiales Peruanos de Educación Secundaria en Probabilidad. Bolema, Rio Claro, v. 36, n. 73, p. 888922, 2022.. We also analyse the relationship between the relevance of ontosemiotic analysis and the assessment of didactic suitability compliance in the CP.
Development of the competence in meaning analysis and ontosemiotic analysis
To assess the degree of development achieved in the competence of meaning analysis and ontosemiotic analysis, we begin by examining the difficulties encountered in the initial task (pretraining).
Initial Exploration of Meanings and Mathematical Objects
The initial task aimed to determine the initial conceptions of the PTs regarding mathematical practice and object, using NC6 as the unit of analysis. The PTs correctly identified the presence of verbal and symbolicnumerical language and concepts like probability and sample space, but encountered difficulties with the problemsituations, procedures (only one FP indicated that assigning 0 or 1 as probability could be a procedure), propositions, arguments, as well as recognizing the involved meanings of probability in NC6. Concerning the meanings of probability, four indicated the frequency meaning based on the appearance of statistical tables or predictions. Two PTs related it to the intuitive meaning, while another two considered the classical meaning without justification. Moreover, PT12 suggested that the meaning could be either frequency or classical, depending on the sample space. These limitations might stem from regulatory frameworks prescribing student actions, necessitating the interpretation of descriptions in terms of mathematical objects involved in practices. This finding guided subsequent reflection.
Advancements in the Competence of Meaning Analysis and Ontosemiotic Analysis
In the second session, the PTs analysed the firstgrade units (DG1.1 to DG1.4), demonstrating progress in identifying mathematical objects. The quality of the analysis was scored as follows: 0 (no response or all incorrect), 1 (at least one correct object, but less than half), 2 (at least half of the objects correct, but not all), and 3 (all objects correct). Table 3 summarizes the frequency of quality levels exhibited by the PTs in identifying mathematical objects in each firstgrade unit of analysis. A slight improvement was observed compared to the first session.
Frequency of the quality of identification of mathematical objects in the analysis conducted by the PTs in the different firstgrade units of analysis
In the initial analysis of DG1.1, only five out of the 14 PTs actively participated, and only two correctly identified at least one problemsituation (recognizing the conditions defining a random situation) or half of the correct problemsituations. The main difficulties were in identifying procedures, propositions, and arguments. For instance, only two FPs recognized the application of the Laplace's rule as a procedure. The PTs did not recognize that probability is a calculable value and confused the Laplace's rule as a concept rather than a proposition or linked to a procedure.
In the analysis of DG1.2, eleven out of the 14 PTs actively participated, but difficulties in accurately capturing the problemssituations persisted (sometimes descriptions or intent of mathematical practices, e.g., “calculate the probability of situations”), and they confused procedures with propositions (e.g., considering “determine the value of probability” as a proposition). They also had trouble recognizing graphical and tabular language. During the analysis of DG1.3 and DG1.4, there were slight improvements in identifying situations, procedures, and propositions, and justifications started to be indicated as arguments.
After identifying mathematical objects, the preservice teachers were required to link each unit of analysis with underlying probability meanings. Three PTs associated DG1.1 and DG1.4 with the classical meaning, based on the presence of Laplace's rule, and with the frequency meaning due to the inclusion of the relative frequency term. Others attributed an intuitive meaning by using expressions like “more or less likely,” “unlikely,” or “very likely.” Units DG1.2 and DG1.3 were related to classical, frequency, and intuitive meanings, but none of the PTs managed to justify their responses.
In general, the PTs expressed their insecurities and limitations in identifying objects (primarily with propositions and arguments) and meanings. This difficulty may be due, on the one hand, to the lack of training and, on the other hand, to the fact that the curriculum does not explicitly define mathematical entities; rather, it prescribes the actions that the student should perform, requiring the interpretation of emerging practices and objects from these activities. Similarly, in some units of analysis, the standard does not explicitly state expressions that refer to the probability meanings that should be addressed.
Assessment Task: Results of Secondgrade Analysis Units by Prospective Teachers
Throughout the workshop implementation, progress was observed in the ontosemiotic analysis capacity of the preservice teachers. The secondgrade analysis units (DG2.1 to DG2.5) were used as final assessment instruments following the training sessions. Table 4 presents the frequency of quality levels exhibited by the PTs when identifying mathematical objects in each secondgrade analysis unit.
Frequency of the quality of mathematical object identification in the analysis conducted by the PTs in different secondgrade analysis units
In this activity, the majority of the PTs identified at least one correct problemsituation within the analysis units. However, only a few PTs recognized propositions. In fact, despite reflecting on Laplace's rule as a property, the PTs continued to identify it as a concept. Most procedures were categorized as problemsituations (e.g., calculating probability using Laplace's rule, calculating relative frequencies, comparing event frequencies, reading tables or histogram graphs). Regarding the argument object, only two PTs partially mentioned “justifying results” in DG 2.4 (for example, PT7 considered the presence of arguments in “justifying using the information obtained, and their statistical and probabilistic knowledge”), but none could recognize it adequately.
In all analysis units, the meanings of probability were related to intuitive, classical, and frequency meanings. However, in no case was it justified why they corresponded to these meanings of probability, except for PT6 and PT11, who mentioned that DG2.5 was related to the frequency meaning due to the inclusion of the term relative frequency.
Analysis of didactic suitability carried out by the PTs on the curricular programme
In this section, we present the results of the analysis of the didactic suitability of the curricular regulations carried out by ten PTs, using the analysis tool by Cotrado, Burgos and BeltránPellicer (2022)COTRADO, Bethzabe; BURGOS, María; BELTRÁNPELLICER, Pablo. Idoneidad Didáctica de Materiales Curriculares Oficiales Peruanos de Educación Secundaria en Probabilidad. Bolema, Rio Claro, v. 36, n. 73, p. 888922, 2022.. Tables 5 to 10 summarise the frequencies of the evaluations given by the PTs regarding the fulfilment of suitability indicators in each of the facets. The ones highlighted in italics coincide with the researchers' assessment to verify their correctness.
Regarding epistemic suitability, Table 5 shows a significant disparity with the expert evaluation in aspects related to problemsituations, concepts, procedures, and arguments.
Most of the PTs had difficulties identifying correct problemsituations in the ontosemiotic analysis; however, nine of them considered through the fulfilment of I1 the presence of multiple situations in the CP associated with the different meanings of probability. It is possible that the PTs may have misinterpreted this indicator or focused on general expressions. For example, PT3 indicated that the CP promotes the development of competencies to solve data management and uncertainty problems. Although the CP emphasises problemsituations, it lacks precision concerning the meanings of probability that should be integrated into teaching and learning (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.; BeltránPellicer; Godino; Giacomone, 2018BELTRÁNPELLICER, Pablo; GODINO, Juan; GIACOMONE, Belén. Elaboración de indicadores específicos de idoneidad didáctica en probabilidad: aplicación para la reflexión sobre la práctica docente. Bolema, Rio Claro, v. 32, n. 61, p. 526548, 2018.).
Problemsituations should include experimentation and simulation (BeltránPellicer; Godino; Giacomone, 2018BELTRÁNPELLICER, Pablo; GODINO, Juan; GIACOMONE, Belén. Elaboración de indicadores específicos de idoneidad didáctica en probabilidad: aplicación para la reflexión sobre la práctica docente. Bolema, Rio Claro, v. 32, n. 61, p. 526548, 2018.). The standard provides general clues, which could have confused the PTs. Regarding linguistic records, more than half of the PTs identified the different types of language in the CP but did not know whether they were appropriate for the corresponding level (I4).
Most of the PTs identified at least two or more correct concepts in the CP but assigned inappropriate evaluations to I5, overlooking essential concepts for teaching probability (Batanero, 2005BATANERO, Carmen. Significados de la probabilidad en la educación secundaria. Revista Latinoamericana de Investigación en Matemática Educativa RELIME, Ciudad de México, v. 8, n. 3, p. 247263, 2005.). Similarly, they overlooked basic procedures related to the frequentist meaning (experimentation, estimation, and simulation). Half of the PTs adequately evaluated I6 but did not justify their evaluation, although some identified a correct proposition in the CP. In the regulations, there are implicit arguments, but the use of types of arguments (inductive or deductive) is not proposed. This might have led seven PTs to assign a less relevant evaluation to I8. Regarding the relations component, half of the PTs believed that I10 is always met, despite not having clearly identified the meanings of probability in the CP.
Table 6 summarises the PTs’ assessment in the CP concerning the indicators of the cognitive facet. A greater mismatch with the expert evaluation is observed in indicators I11, I14, I15, and I16.
Regarding prior knowledge, most of the PTs believed that I11 is always met. They likely did not take into account expressions associated with the progressive treatment of content according to the meanings of probability. Although they justified that the CP includes the different levels of competency development, they did not notice that these expressions only set out linguistic elements and concepts expected at the end of each school cycle.
The CP does not reflect expressions related to cognitive conflicts, but most of the PTs, without justification, assigned partial or total fulfilment evaluations to the associated I13 indicator. Similarly, I14 received inappropriate and unjustified evaluations, possibly because the PTs could not distinguish metacognitive processes according to individual differences. However, PT2 justified the partial fulfilment of the indicator, arguing that according to the CP, “students learn by themselves when they are able to selfregulate their learning process and reflect on their successes, mistakes, progress, and difficulties that arose during the problemsolving process.”
Regarding assessment, six PTs believed that I15 is always met, although it is observed partially. For the PTs, it was possibly enough that the CP proposes some generic evaluation instruments. Furthermore, half of the PTs indicated that the CP refers to learning outcomes either partially or always, although this indicator does not appear in the curriculum.
On the affective level (see Table 7), the PTs’ assessments in components such as emotions, attitudes, beliefs, and values differ from those of the researchers. The partial or total evaluation of the degree of fulfilment of these indicators was probably due to the identification of imprecise expressions in the CP on the matter, such as features of the same. For instance, PT2 evaluated the indicators I19, I20, I21 with “sometimes”, justifying that “Emotions, attitudes, and beliefs act as driving forces of learning” (MINEDU, 2016MINEDU. Programa Curricular de Educación Secundaria. LimaPerú: Ministerio de Educación, 2016., p. 148).
In Table 8, the discrepancy in the assessments could be due to the fact that the CP does not provide specific guidelines for teacherstudent interaction or between students and the development of autonomy (I26, I27, and I28), leading to different assessments than those of the researchers.
Table 9 presents the assessments of the mediational suitability indicators that the PTs assigned when assessing the CP. The general expressions related to material resources, classroom conditions, and time management in the CP may have made it difficult for most of the PTs to assign appropriate assessments of the degree of fulfilment.
In the ecological facet, Table 10 shows that four PTs believe that I33 is “always” met, suggesting that they possibly did not conduct a prior analysis of the curriculum in relation to its correspondence with international research and regulations. In this facet, the indicators where the greatest disparity with the expert assessment is observed are those related to openness to innovation, values education, and intra and interdisciplinary connections. For instance, in the CP, no expressions related to innovation and reflective practice are observed, but most of the PTs assigned “always” or “sometimes” fulfilment assessments to this indicator. Although the CP does not explicitly state the intra and interdisciplinary relationship, references to statistics are observed, leading most of the PTs to assess I37 as “always” met, when actually it is only partially.
Relationship between Ontosemiotic Analysis and Didactic Suitability
In this section, we analyse the relationship between the degree of accuracy in identifying mathematical objects in the CP and the success achieved in the analysis of didactic suitability. To observe this relationship, Table 11 collects the frequency of the different didactic suitability assessments made by the PTs. Additionally, it displays the relevance of object identification, based on the quantitative scores 0 (no answer or all incorrect), 1 (at least one correct object, but less than half), 2 (at least half of the objects are correct, but not all) and 3 (all objects are correct).
Frequency of PTs based on object identification scoring and relevance of didactic suitability assessment
In general, all the PTs recognised at least one correct mathematical object in all the analysis units of the CP. However, most were not successful in assessing the suitability of the CP in the epistemic, affective (correctly assessed only by two PTs in each case), interactional (only correctly assessed by one PT), and mediational facets (correctly assessed by three PTs), performing better in the cognitive and ecological facets, correctly assessed by six and five PTs, respectively.
Languages and concepts were the bestreferred objects, but only the indicators associated with the language component were appropriately assessed in the epistemic facet. In contrast, the assessment of the indicators related to the concept component was not very relevant both in the epistemic facet (where the PTs did not recognise essential concepts lacking in the CP that are crucial for adequate probability teaching) and in the cognitive one (in the component of prior knowledge, they failed to adequately observe that the CP proposed a partial progressive treatment and manageable difficulty of the content).
Problem situations, procedures, and propositions presented greater identification challenges, and the assessment of related indicators was also not very relevant. Thus, the limited identification of problemsituations influenced the appropriate assessment of indicators related to the affective facet, such as the proposal of problemsituations based on the needs and interests of students and the contextualisation of these with motivating elements.
Conclusions
In this study, we have described the design, implementation, and outcomes of a formative experience with prospective Peruvian mathematics teachers, aimed at promoting the competence of didactic analysis of curricular materials (normative frameworks) on the topic of probability. While these curricular resources are used for planning instructional processes on a specific content, their analysis is a complex task that involves didacticmathematical knowledge about the content, its teaching, and its learning (Remillard; Kim, 2017REMILLARD, Janine; KIM, OkKyeong. Knowledge of curriculum embedded mathematics: Exploring a critical domain of teaching. Educational Studies in Mathematics, New York, v. 96, n. 1, p. 6581, 2017.).
The distinction of meanings and identification of the mathematical objects involved in these normative frameworks is a challenge for prospective teachers and an essential competence that will allow them to understand and reflect systematically and in detail on the relevance of the teaching and learning processes of probability, considering the educational context. Each meaning of probability involves different systems of practices and, therefore, different challenges and difficulties in its instruction. Properly identifying the types of objects ensures understanding their functionality in these practices. However, as we have seen in our research, participants showed limitations in identifying these objects (especially propositions).
In general, the inadequate identification of mathematical objects in the CP influenced the low relevance of the didactic suitability assessment of the CP. This fact is mainly reflected in the epistemic facet, where only two PTs were successful in their assessment, and it was observed that the PTs overlooked fundamental concepts for adequate probability teaching or decisive procedures related to the frequentist meaning. The difficulty in appropriately assessing indicators from other facets on the affective, interactional, or mediational plane may be motivated both by the challenge in properly interpreting the indicators and by the lack of precision in the programme. Therefore, there is an inferred need for curricular programmes to provide more specific guidelines to teachers and incorporate the results of research in the area of mathematics education so that the standard helps improve the teaching and learning of probability in our case. But also, the need to train prospective teachers to be curriculum developers who reflectively, groundedly, and critically assess the guidelines.
Despite the disparity in assessments between researchers and PTs, the latter were able to identify the main shortcomings and deficiencies in the CP, as had been observed in previous experiences in the context of proportionality (Castillo; Burgos, 2022CASTILLO, María José; BURGOS, María. Competencia de futuros profesores de matemáticas para el análisis de la idoneidad didáctica de una lección sobre proporcionalidad en un libro de texto. Educación Matemática, Guadalajara, v. 34, n. 2, p. 3971, 2022.). Therefore, analysing and evaluating curricular regulations before their use is a good strategy to generate spaces for critical analysis, reflection, and professional development.
The implementation of curricular regulation analysis is a significant contribution to mathematics didactics, as it is not limited to a descriptive aspect and allows analysing the process of contextualising curricula on a specific topic, without neglecting essential aspects such as the affective, interactional, mediational, and ecological. Moreover, it allows diagnosing didacticmathematical knowledge in relation to the content addressed, making action decisions to correct its deficiencies.
The ontosemiotic analysis and the didactic suitability of the curricular regulation on probability have been challenging for many prospective teachers, possibly due to the lack of specific training and the short time to familiarise themselves with the suitability indicators. It is recommended to specify more those difficulttoassess indicators and include prior training on facets, components, and didactic suitability indicators in future formative interventions^{1} 1 Acknowledgments:Research developed within the framework of the projects PID2019105601GBI00 and PID2022139748NB100 funded by MICIU/AEI/10.13039/501100011033 and FEDER, EU; Group S60_23R  Research in Mathematics Education (Government of Aragon, Spain). .

1
Acknowledgments:Research developed within the framework of the projects PID2019105601GBI00 and PID2022139748NB100 funded by MICIU/AEI/10.13039/501100011033 and FEDER, EU; Group S60_23R  Research in Mathematics Education (Government of Aragon, Spain).
Availability of research data:
The dataset supporting the results of this study is published in this article.
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Publication Dates

Publication in this collection
02 Aug 2024 
Date of issue
2024
History

Received
10 Jan 2023 
Accepted
28 Aug 2023