PädiUAQ es una revista semestral de la Facultad de Ingeniería, publicada por la Universidad Autónoma de Querétaro, a través del Fondo Editorial Universitario, dirigida a investigadores, docentes, estudiantes y profesionales. Los contenidos están asociados con los procesos de enseñanza-aprendizaje curriculares y escolares de Matemáticas, Ciencias Experimentales e Ingeniería.
Mathematical Modeling in the Training of Mathematics Teachers
Mathematical modeling provides future professors with a practical comprehension of mathematical concepts, allowing them to apply these notions through activities that connect with other areas of knowledge in order to teach mathematics effectively. The teaching of this discipline should go beyond the memorization of formulas and involve students in solving real-world problems through mathematical models. This paper presents the application of an experiment that incorporates chemistry with the aim of developing modeling skills by applying a methodology focused on experimentation and discovery. The proposal, activities, and elements deployed for its implementation with preservice teachers in the Bachelor’s Degree in Mathematics Education imparted at the University of Colima, Mexico, are detailed. It is important to highlight that mathematical modeling promotes the development of critical thinking and analytical skills in preservice teachers, setting them up for the challenges that may occur in the classroom. The study concludes that integrating mathematical modeling into teacher education programs is essential, as it contributes to the quality of mathematics education at both basic and higher levels. And its effects are more beneficial insofar as strong links are established with other disciplines.
Copyright (c) 2025 PädiUAQ https://creativecommons.org/licenses/by-nc-sa/4.0
Subjectivity and experience. Training to teach mathematics
The article examines the initial training experiences of two primary school teachers in teaching mathematics in Mexico, emphasizing how didactic theories and educational practices influence the development of their professional identities and skills. Through narrative interviews, aspects such as their relationship with mathematics, the deployment of pedagogical strategies, and the incorporation of theory and practice into instruction in teacher schools are studied in depth. Both teachers gave an account of their positive experiences with mathematics from an early age, that enabled their learning during teacher training. Problem-solving and challenging the established techniques were instrumental in their development, while strategies based on critical reflection and practical experiences (such as ludic activities and theoretical validation) stood out in their training. It is observed that although educational institutions built a basic theoretical framework —the theory of didactic situations— institutional limitations often hindered the full application of theoretical knowledge. Despite these constraints, both teachers managed to infuse this theory into their professional work, adapting and improving it with their own experiences. The conclusion is that a solid initial training, based on a balanced amalgamation of theory, practice, and critical reflection, is essential to foster an institutional ecology that allows for the effective development of future teachers. Moreover, it emphasizes the importance of further exploring the subjective experiences of trainers and trainees, specifically in regular and daily training contexts.
What do Neuroscience and other disciplines offer us for the teaching of Mathematics?
This text presents some examples of how disciplines such as Neuroscience, Cognitive Psychology and Anthropology can provide clues about the learning and teaching of Mathematics. Examples of learning language and the notion of quantity from childhood are presented as a basis for the later development of written expressions and much more complex and abstract concepts.
Continuous functions and their derivatives
The derivative of a function is a central concept that is approached in the upper secondary and higher education levels in Mexico. This concept is closely linked to the principle of function and instrumental in numerous situations where it is essential to analyze the changes of the variables involved. As an example, derivatives are crucial in the study of motion and velocity, the optimization of functions and obtaining straight lines. This didactic dissertation focuses on the concepts of function, derivative, geometric interpretation and determination of maxima and minima. The main goal is to demonstrate a few properties of this function type through theorems and to lay out a set of problems and exercises associated to these concepts. Hopefully, our readers will find interest and usefulness in the theoretical study of derivatives and their application in practical situations.
interpretations of the derivative of a function
Algebraic language is a means by which situations from many contexts can be represented (school, professional, real life, etc.) since it allows us to represent and manipulate mathematical objects that have their counterpart in very diverse contexts. This is why it constitutes a means of representation necessary for the academic development of students. Furthermore, like any language, the student must internalize it to be able to use it “automatically” so that it does not become an obstacle to learning other more complex concepts in which this language is a tool, a means, to be able to apprehend it. With this intention, in this work some problems and exercises are presented that could be solved in various ways, but it would be worthwhile for the author to try to express them with algebraic language and thus solve them. That is also the reason why they are grouped according to theme.
Copyright (c) 2024 PädiUAQ https://creativecommons.org/licenses/by-nc-sa/4.0
Visualizing the derivative function in a dynamic geometry environment
A proposal for the teaching of derivative functions, which in many Calculus courses does not receive a reflection of enough depth, is presented. This lack of attention is usually due to the primordial place that procedural skills occupy over conceptual meanings. An alternative is proposed by means of a GeoGebra dynamic construction, where the elements of visualization and covariational reasoning are combined as arguments that consolidate the relationship between the local (derivative at a point) and the global (derivative function). The essence of the proposal is that the derivative of a function at a given point can be seen as the slope of the tangent line at any given value. In that context, the construction of a duo point-slope of the tangent line as a multiplicative object enables the covariation among these relations and the conception of the derivative as a function in itself. This could visually prove certain relations, specifically in polynomial or trigonometric functions. The proposal is expected to become a point of reference of digital technology integration into the classroom to enhance the meanings of the notion of derivative, and a valuable tool for the teaching community to exploit.
Design of a user interface for the modeling and analysis of electrical circuits as a learning process
This research work focuses on characterizing the abstract logical thinking skills (ALTS) developed by college students; to that end, an electrical circuits simulation software compatible with the learning process of university level students was created; moreover, the cognitive and metacognitive actions exhibited by the participants when operating it were characterized. Dynamic mathematical constructive modeling techniques are employed in the field of metacognition from a perspective of linearization (MPL). An advantage of such an infrastructure design is that the interface deployed is considered affordable and versatile, since it is programmed in open source language (C#) to be accesed online or installed on conventional operating systems. The participants' performance along the preliminary tests was satisfactory. They were presented with a novel user interface executable on Windows or accessible via web browsers. The results made it possible to rate the development of thinking skills and to establish teaching-learning methods and strategies. Their evaluation, argumentation and interpretation represent a meaningful contribution to teaching in general and to educational mathematics in particular.
metacognition from a linearization perspective
Implementation of a trigonometry didactic proposal designed in GeoGebra
The results of the implementation of a didactic proposal in the field of trigonometry, designed using GeoGebra software, are presented. It was carried out with high school students from the Conca campus of the Autonomous University of Queretaro (AUQ), and a group of first-semester students of the Degree in Digital Animation (DDA) belonging to the Faculty of Engineering of AUQ. At both levels, information and communication technologies (ICT) were incorporated into the teaching of trigonometry. The proposal consisted of two scenarios with activities based on the Theory of Didactical Situations (TDS). This study aims to demonstrate how, through the didactic intervention supported by GeoGebra and the proposed scenarios, it is possible to establish a learning environment for the construction and analysis of trigonometric functions. According to the data gathered, the high school students accurately related the concepts of trigonometry; similarly, they understood the process of constructing trigonometric function graphs thanks to the GeoGebra software. On the other hand, the undergraduate students reinforced their knowledge in this area.
Mathematics Education and Reflective Thought
What are we? Teachers, professors, trainers, educators, masters, instructors? What is our scope of action? Mathematics education, mathematics didactics, mathematics teaching, educational mathematics, mathematics teaching? What role should pedagogy and didactics play in the educational process of students, understood as their insertion into society as citizens who contribute to their development? Can the study of mathematics at school contribute to this process? In this essay we will seek answers to these questions from the Deweyan perspective on reflective thinking and learning. The reflection ends with the approach of the bases of a didactic intervention model in which reflective thinking is encouraged in an environment of tolerance, respect, and cooperation; an environment in which five equally important dimensions are considered: curricular content; cognitive demand; equitable access to content; identity and belonging; and formative feedback. All to simplify didactics and tasks in the classroom.
Available resources and mathematical practices
In this text, a brief reflection is made on the importance of the resources available in the development of Mathematics for the realization of mathematical practices that have led to the formulation of notions that are studied in secondary school. Emphasis is placed on the importance of the use of languages and means of representation appropriate for mathematical developments and their understanding.
En este número se presentan dos artículos. En uno se expone un cuestionario orientado a la valoración del conocimiento matemático de los profesores en cuanto al concepto de función, el cual puede servir como referencia para proponer actividades de enseñanza para la mejora de la práctica educativa. En el segundo artículo se presentan los resultados de un proceso de indagación sobre el pensamiento algebraico de estudiantes cuando generalizan patrones lineales, el cual puede ser tomado como base para futuras investigaciones o para el diseño de actividades de enseñanza relacionadas.
Characterization and typification of errors to solve linear equations of one variable
The purpose of this paper is to point out some of the most common errors in characterization and typing made by CECYTEM students when they are solving linear equations in one variable in physical-mathematical field. To carry out this research it was applied a diagnostic test to 30 students which was applied on February 16th, 2023, at the school facilities. Some qualities found by characterization were randomly attempting to determine the value of the unknown quantity. unknown, associate the number that goes with the unknown quantity as its value, stand the addition of a variable and a number as a product, separate a product, and express it as a sum and the last one found was the failure to employ proper algebraic language. In the other hand according to typification terminology it was found in the sample that 59.52% of the errors were to poor learning, 19.04% accuracy of thought, 9.54% illogically deductions, 7.14% math language problems and 2.38% unintelligible concepts, as well a lack of results verification.
Didactic proposal to promote algebraic reasoning through figural patterns in GeoGebra
This study's objective is to present a didactic proposal to promote algebraic reasoning in high school students working with GeoGebra. A qualitative methodology, design-based research, was used because of its foundation on three axes: didactic, mathematical, and technological. The design considers three stages: sequence selection, adaptation to a figural pattern and finally generation of the figural pattern in GeoGebra. As a result, design of activities related to figural patterns and their adaptation to GeoGebra's environment is presented. It is concluded that patterns turn out to be a tool to introduce school algebra since algebraic notation and notion of succession are related. In addition, using proposed software contributes teacher and student's approach to apply technology as a support tool for learning and teaching algebra.
Algebraic thinking through pattern generalization
The present study aims to expose the characteristics of high school students' algebraic thinking when solving a task on generalization of linear patterns. The notions of pattern generalization strategies and the levels of cognitive demand for this type of tasks are used, as theoretical framework. These notions were used for task proposal and data analysis. A qualitative methodology based on case studies and developed with two high school students, is used. Results show, that the students mainly used the recursive. In the case of student A, the difficulty in the declaration and denotation of the variables and the algebraic treatment is identified. As for student B, there are fewer difficulties to algebraically solve the task. The information generated is expected to be useful for future research and for active teachers who decide to implement this type of task in the classroom.
The disciplinary training of the teacher, a necessary or sufficient condition?
This paper presents a reflection on the complexity of teacher training in terms of the knowledge and skills that must be learned and developed to carry out this professional work. The case of what is related to disciplinary aspects is mainly addressed, but it is shown that although they are necessary, they are not enough to participate in the teaching process properly.
Copyright (c) 2023 PädiUAQ https://creativecommons.org/licenses/by-nc-sa/4.0
Questionnaire on the mathematical knowledge for teaching of mathematics teacher
The present study is about a questionnaire to assess mathematical knowledge for teaching of mathematics teachers about the concept of polynomial function. A qualitative methodology based on design is used, based on three elements: the mathematical knowledge for teaching, the semiotic representation registers and function meanings. This study is situated in the context of high school mathematics teachers because, according to research, there isn’t a homogeneous professional training, as well as insufficient specific knowledge for the teaching profession at this educational level. In this way, the design of a questionnaire is presented, which is composed of six task that unify teaching knowledge, representation registers and function meanings. Finally, this type of proposal becomes a tool that characterizes the necessary knowledge for the function concept teaching. In addition, this proposal serves as a reference to strengthen teacher training activities to improve their educational practices.
Characterization of university students’ understanding when relating a function with its derivative function through a graphic approach
This paper aims to show how university students relate a function with its derivative function in a task with a purely graphical approach. This task is part of a diagnostic test used in a broader investigation whose objective is to build the concept of derivative in university students with the help of the APOE theory, which provides mental structures required to understand a mathematical concept: action, process, object, and scheme. Initially the paper shows the details of the test and the analyzed task, then it shows the qualitative and quantitative analysis of the observed responses. In general, the results obtained show deficiencies in students when faced with graphic focus exercises. Likewise, the justifications observed in students show an inclination towards solving exercises from an algebraic aspect, thus leaving aside the graphics of the activity. We assume that future research may use these results as background for their works, and active teachers may encourage their students to resolve this type of conceptual items in the classrooms.
The problem of assessing the didactic suitability of a massive study process
The objectives of this work are: 1) to present a quantitative model of assessment of the didactic suitability of a massive study process of mathematical content in the period of admission to the university, as a resource for the person who coordinates said process to reflect on its own practice and make decisions aimed at improvement; 2) discuss the possibilities that the model offers and its limitations. The model was developed and tested within the framework of a mixed approach research with quantitative preponderance, and of a descriptive and exploratory natur, framed in the Onto-semiotic Approach of Mathematical Knowledge and Instruction. It is based on a device consisting of two questionnaires (for teachers, for students) that operationalize the didactic suitability construct and allow it to be measured on an interval scale. The main results of the research are: the identification of the facets with greater (respectively, lesser) suitability, and the detection of relative achievements and deficits in each facet from the teachers´ perspective; the exploration of the structure of the student's questionnaire through factor analysis, and the assessment of the suitability of the study process in terms of each factor from the perspective of the students; the identification of specific problems of the study process from the dialogue between both questionnaires. As for the limitations, they are those associated with the comparability of the answers of those who responded to the questionnaires, to their objectivity and to the global nature of the measurement.
Configuration of a methodology of historical-didactic study of textbook
Diverse research works from Mathematics Education have given evidence about how methodological strategies for the significative learning of mathematics can be grounded from the analysis of text books. This study is part of a broader project with based on the Socioepistemological Theory, that surveys about the didactical transformation of the significance on trigonometric notions in ancient books (from the 19th and early 20th centuries) and contemporary books (21st century) that are used on engineering educational programs. Given this, for delimitation in this article we aim to explain the task of the configuration of a method for the historical-didactical análisis of text books for the study of inverse trigonometric ratio and function. In the results we show the implementation of the method in four phases: book selection, task selection, individual analysis and transversal analysis. The fourth phase is omitted, due to the project being in phase three at the moment. We also give deep insight on the instrument for the individual analysis of text books and exemplify its use with two books. This instrument for the historical-didactical analysis of text books consists in three categorires: characterization of the text, characterization of the mathematics and teaching product; and can be used as a theorical-methodological tool in other research projects with the pertinente adjustments in line with the theory and mathematics of interest.
Mathematics Education; Socioepistemological Theory; Book analysis; Engineering Education, Trigonometry.
Número temático sobre el Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos
El Enfoque Ontosemiótico (EOS) del conocimiento y la instrucción matemáticos es, en la actualidad, una teoría propia de la Didáctica de las Matemáticas que surge a inicios de los 90’ con el propósito de articular diferentes puntos de vista y nociones teóricas sobre el conocimiento matemático, su enseñanza y aprendizaje. Así, a nivel internacional, por ejemplo en la Encyclopedia of Mathematics Education, el EOS se presenta como “un sistema teórico inclusivo que trata de articular diversas aproximaciones y modelos teóricos utilizados en la investigación en Educación Matemática” (Presmeg, 2014).
Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos
Copyright (c) 2022 PädiUAQ https://creativecommons.org/licenses/by-nc-sa/4.0
Epistemic suitability indicators of the probability contents in the Costa Rican Mathematics curriculum
In this paper we analyse the epistemic suitability of the probability contents in the Mathematics curricular guidelines for Costa Rican General Basic Education. We rely on the theory of didactic suitability from the Ontosemiotic Approach to mathematical knowledge and instruction, developed to support the design and evaluation of curricula and teacher training. Through content analysis, norms are identified from which indicators of epistemic suitability are inferred, which are compared with those proposed by this theory and others generated in previous works, in order to identify their concordances and complementarities. In general, the inferred indicators show a high epistemic suitability of the probability teaching guidelines for these educational levels; that is, a good match between the intended institutional meaning and the institutional reference meaning of probability. Furthermore, the value of problem situations as an integrator and catalyst of language, rules and argumentation is highlighted. The indicators identified can guide and support teaching action in the design and implementation of probability instruction processes and constitute a useful resource for the initial and continuous training of educators.
Epistemic suitability; probability; curriculum; General Basic Education; Costa Rica
Study of the chilean curriculum with respect to division as an isomorphism of measurement in the case of fifth grade elementary school
Abstract The current work to analyze the perspective around the division promoted by the Chilean curriculum and justify the importance of carrying out an analysis on the topics of this concept in basic education. Due to the relevance that this material acquires for teachers at the time of carrying out their classes. This time we will focus our work on 5th grade. To carry out this analysis we have relied on two theoretical frameworks, on the one hand, we have the multiplicative structure problems (measurement isomorphism) proposed by Vergnaud (1997) and verify what are the intended and holistic meanings of references proposed in the Ontosemiotic Approach. On this occasion, we will give an account of the analysis presented by the Ontosemiotic Approach. Keywords: Chilean Curriculum, Division, Measurement Isomorphism, Ontosemiotic Approach.
Thematic issue on The Theory of Mathematical Work Spaces. Presentation
Looking at the socio-epistemology from the onto-semiotic approach in mathematics education
The articulation of theoretical frameworks is a topic of growing interest in mathematics education, since, although the diversity of theories serves to enrich the foundations of research, at the same time it can constitute a hindrance to its consolidation as a scientific and technological field. This paper analyses the theoretical framework of socio-epistemology in mathematics education from the point of view of the onto-semiotic approach to mathematical knowledge and instruction. The comparison and possible articulation of both theories is approached through the analysis of two examples of research carried out within the socio-epistemology framework: dealing with contradiction in mathematics by university students when they are confronted with the existence of the logarithm of negative numbers and the study of cultural, historical, institutional, and cognitive aspects related to periodicity. As a result of the analysis, similarities, differences, and complementarities of these theoretical models are identified, as well as their relations with the theory of didactic situations and the anthropological theory of didactics. It is also shown in what sense the mathematical and didactic ontology proposed within the onto-semiotic approach can contribute to the progress and coherent articulation of these theories.
Socio-epistemology; onto-semiotic approach; networking theories
Principios para el diseño de tareas en un sistema de evaluación en línea
En esta contribución se presentarán algunos resultados de tres casos sobre diseño de tareas en sistemas de evaluación en línea y utilizados en contextos virtuales. Con estas tareas se ha estudiado el trabajo matemático personal de los estudiantes. Se analiza cuál es el trabajo matemático a partir de la variación de ciertas variables didácticas de las tareas. Se muestra cómo los registros de representación semiótica y los números que los definen los objetos matemáticos, la apertura de las tareas y el feedback afectan el trabajo matemático.
Configural reasoning of highschool students when solving geometric problems in the distance modality
En este artículo se muestra el diseño de una serie de problemas geométricos que involucran alguna de las aprehensiones propuestas por Duval: perceptiva, operativa, discursiva o secuencial. También se muestran las respuestas dadas a esos problemas por estudiantes de segundo semestre de un Bachillerato en el que actualmente se trabaja a distancia; y se analizan los tipos de respuestas dadas por los estudiantes, para determinar el razonamiento configural que llevan a cabo, a partir de conocer los diferentes tipos de aprehensión presentes en sus procesos de visualización, así como los conocimientos Geométricos con los que cuentan.
Evaluación y resultados del aprendizaje con enfoque en competencias profesionales de Cálculo Diferencial para ingenierías del TecNM
The topic of competencies, due to its recent implementation, has generated changes in the conception of training, instruction, teaching, and evaluation of learning, which leads to new approaches in the design, development and evaluation of such information. The evaluation of a professional competence is evidenced by exploration, knowledge, understanding, application, analysis, synthesis and evaluation of the project management or academic activity that the student´s works on, individually or collectively, in each subject. Consequently, working by competencies means that the student must understand learning as a multidirectional circuit where he/she they has to take the initiative and stimulate critical, ethical, creative and sensitive capacity in the management of his/her learning at all levels to favor his/her integral formation. That is why it is important to implement adequate instruments that allow a reliable evaluation of the competencies acquired by students. In this paper we present the results obtained by implementing an evaluation with a “competency-based approach” in several undergraduate groups in the subject of differential calculus. The results obtained at the end of the course reflect the way in which the evaluation by competencies achieves in the students a better appropriation of the acquired knowledge thanks to what the whole process entails, in addition to an increase in the approval rate.
Didactic Sequence Design Based on Problems
The overall objective of this research is to design a didactic sequence based on the Problems Based Learning (PBL), which includes contextualized situations to the administrative field; For which we took as reference the planning of the Mathematics Applied to Administration subject of the Business Administration and Management career of the Universidad Politécnica of Victoria, specifically analyzing unit II: Application of differential calculus, and topic II of the same: Derivatives of functions. The methodology that guides this research is qualitative with a descriptive approach. The design of the didactic sequence will follow the process of Problem Based Learning theory (PBL).
The Physical-Mathematical Working Space as a theoretical proposal to analyze the semiotic processes that are carried out during the modeling of a dampened harmonic motion
The modeling process is very important for teaching and learning both physics and mathematics and establishes their interaction. Here we propose a Physical-Mathematical Working Space (PMWS) as an extension to the theory of Mathematical Working Spaces (MWS) because its structure allows to explain the processes that arise when such tasks are carried out. We consider that modeling the movement of a damped harmonic oscillator allows students to reflect on their own representations. We present the results of a didactic implementation based on the construction of representation, a pedagogical approach to targeted research that requires students to interpret and build representations of scientific concepts, affirmations, and processes. Since these are students in work teams to motivate scientific debate and self-reflection, the ACODESA methodology was used. On the other hand, we show elements that allow us to account for how they are activated in semiotic, experimental, and discursive genesis within the PMWS. This is the relationship between PMWS and MWS. The analysis of the data in this qualitative study on how a student transitions from understanding a physical phenomenon to its mathematical description through modelling has made it possible to propose a contribution to the theoretical framework of the MWS. He worked with a group of 14 second semester engineering students on a comprehensive calculus course at a University of Mexico City.
How do students perceive their own learning by online- based education during the Covid-19 Pandemic?
Due to the pandemic generated by Covid-19 in March 2020, students from Mexico were forced to stop face-to face learning. As an emerging response, some schools opted for online education to continue teaching activities. This involved the use of information technologies and alternative pedagogical methodologies, which generated new experiences for students. After more than a year of online education, this paper seeks learning more about these experiences. Through the analysis of new surveys of students who are currently experiencing online learning, the general perception of students about this new learning process is being investigated.
Copyright (c) 2022 PädiUAQ https://creativecommons.org/licenses/by-nc/4.0