Didactic-Mathematical Knowledge Research Articles

Articulation Between a Technological Model and an Educational Model to Deepen the Reflection of Prospective Mathematics Teachers

<p style="text-align:justify"><span style="font-size:9.0pt"><span style="color:black">This article is aimed at integrating the Technological Pedagogical Content Knowledge (TPACK) system with the Didactic Suitability Criteria (DSC) of the Didactic-Mathematical Knowledge and Competences (DMKC) system to improve the reflection of prospective mathematics teachers on online classes. Thirteen prospective teachers, divided into two subgroups, participated in a training cycle that addressed both models. Each participant used and created indicators of reflection of the assigned model to analyze an online class on functions, and subgroups exchanged reflections to examine the class from the other model’s perspective. It was noted that the DMKC model allows for a broad analysis of the class but has limitations in assessing technology and the teacher's technological knowledge, while TPACK’s emphasis is on technology and teacher knowledge but does not explicitly address mathematical interaction or affective aspects. It is concluded that combining the TPACK model and the DSC of the DMKC model can generate more complete tools to reflect on online math classes and consequently allow for a comprehensive evaluation that covers both the mathematical content and the technological and pedagogical skills of the teacher.</span></span></p>

The impact of a training cycle on statistics and probability for future primary schoolteachers with a gender-focused approach within the framework of education for sustainable development

This study explores a training cycle in statistics and probability for future primary school teachers, aimed at fostering gender equality and education in sustainable development. By implementing methodologies such as lesson study and didactic-mathematical knowledge and competencies model, an intervention was conducted with two groups of future teachers: the first group comprised 16 participants, while the second had nine, with the latter receiving an enhanced version of the training cycle. The assessment was carried out using content analysis and tests before and after the cycle, in addition to focus groups with the second group. The findings indicate improvements in pedagogical skills, gender equality, and knowledge in statistics and probability, underscoring the effectiveness of the training cycle. It is concluded that, despite the lack of notable differences between the two versions of the training cycle, it is effective in its educational aim and in raising awareness on sustainability issues.

A model for problem creation: implications for teacher training

AbstractThe invention of problems is a fundamental competence that enhances the didactic-mathematical knowledge of mathematics teachers and therefore should be an objective in teacher training plans. In this paper, we revise different proposals for categorizing problem-creation activities and propose a theoretical model for problem posing that, based on the assumptions of the Onto-Semiotic Approach, considers both the elements that characterize a problem and a categorization of different types of problem-posing tasks. In addition, the model proposes a description of the mathematical processes that occur during the sequence of actions carried out when a new problem is created. The model is illustrated by its application to analyze the practices developed by pre-service teachers in three problem-posing tasks aimed at specific didactic-mathematical purposes (mobilizing certain mathematical knowledge or reasoning, contributing to achieving learning goals, or addressing students’ difficulties). We conclude discussing the potential of our model to analyze the mathematical processes involved in problem creation from the perspective of teacher education.

University teachers’ didactic-mathematical knowledge for teaching the effect of coefficient <i>b</i> on the quadratic function

Literature shows a tendency to relegate the role of coefficient <i>b</i> to second place in the teaching of the quadratic function. We report an experience with Chilean university teachers, who designed a teaching and learning sequence with this function for construction engineering students. Our focus was on the didactic-mathematical knowledge about the effects of varying coefficient <i>b</i> on the graphical representation of this function that the participating teachers made evident. We constituted a focus group with 10 teachers and then qualitatively analyzed their dialogues using the mathematics teacher’s didactic-mathematical knowledge and competencies model. We highlight the following results: (a) the importance of mathematical knowledge and that of the epistemic facet to interpret the effect of coefficient <i>b</i> on the graphical representation of the quadratic function and (b) the proposal of an interpretation for the graphical behavior of coefficient <i>b</i> that contributes to the teaching of the quadratic function.

Mathematics teacher argumentation in a didactic perspective

The teacher’s argumentation is considered essential to promote the student’s mathematical education; the teacher’s argumentation is linked to the teacher’s didactic-mathematical knowledge and the achievement of instructional objectives. In this sense, a pragmatic proposal is required that links the teacher’s didactic-mathematical knowledge with argumentation. Toulmin’s (2007) model makes it possible to study the structure of teacher or student argumentation; in contrast, pragma-dialectic considers that in the face of a difference of opinion, the parties involved try to persuade their counterpart while observing argumentative discourse standards. In the context of an argumentative class, both argumentation models do not consider the particularities of the argumentation of a didactic nature that the teacher uses. This document presents a proposal for argumentation suited to mathematics education, based on a mapping of literature and a model of the teacher’s knowledge, contrasted in an actual class. The analysis and discussion of data let us conclude that a teacher’s argumentation is characterized by epistemic, cognitive, interactional, emotional, and communicative features that reaffirm not only the complexity of educational practices but also the teacher’s argumentation for educational purposes and the adequacy of the proposed definition.

Didactic-mathematical knowledge of future mathematics teachers in Ecuador when developing tasks based on ethnomathematical practices

Este trabajo pretende inferir características del conocimiento didáctico-matemático de futuros profesores de matemáticas al desarrollar tareas basadas en prácticas etnomatemáticas locales. Se identificaron las prácticas etnomatemáticas y los conocimientos matemáticos de estos futuros profesores, estudiantes de una universidad ecuatoriana y, posteriormente, se analizaron sus reflexiones con relación al llevar estas prácticas al aula de clases. Los resultados indican una contemplación positiva por parte de los futuros profesores en dos aspectos: primero, en conectar las prácticas etnomatemáticas con las actividades profesionales locales y segundo, en presentar fragilidades en las características del conocimiento tanto matemático como didáctico-matemático. Como camino a seguir, el artículo sugiere que estas percepciones deben incorporarse en la formación de los futuros profesores, además de los conocimientos didáctico-matemáticos y las perspectivas de las Etnomatemáticas.

Strategies of Pre-Service Mathematics Teachers when Articulating Representations of a Function

Background : Several investigations have shown the difficulties that pre-service mathematics teachers present in articulating the elements of a function, in particular, its different representations. Objective : To analyse pre-service mathematics teachers’ articulation of semiotic representations and partial meanings of the notion of function. Design: This research uses qualitative design. Setting and participants : The respondents’ sample comprised 37 pre-service mathematics teachers from a Chilean university. Data collection and analysis : To collect the information, interviews were applied based on the answers given by the pre-service teachers when solving problem situations that involve functional relationships, of which they had to produce and connect their representations, both with conceptual elements and with the sociocultural context. The information was processed using the content analysis technique. Results: The results show a good state of development of the mathematical dimension of the participants’ didactic-mathematical knowledge. Regarding didactic knowledge, while some managed to articulate the partial meanings of function and the representations produced adequately, others show limitations, by not putting at the service of mathematical knowledge the knowledge with which they can potentially carry out teaching situations, establishing few connections between the elements of the representations produced, with corresponding conceptual and sociocultural elements. Conclusions: There is a need for implementing strategies that allow pre-service teachers to articulate partial meanings and representations of a function with conceptual and sociocultural elements that leads them to envision potential teaching activities of this concept.

Didactic–Mathematical–Computational Knowledge of Future Teachers When Solving and Designing Robotics Problems

The social, scientific and technological development of recent years has encouraged the incorporation of computational thinking in the school curriculum of various countries progressively, starting from early childhood education. This research aims to characterize future kindergarten teachers’ traits of didactic-mathematical and computational knowledge presented when solving and posing robotics problems. Firstly, aspects of the mathematical and computational knowledge of the participants (97 students of the subject of Didactics of Mathematics of the Degree in Early Childhood Education at a Spanish university) were identified when they solved problems as users of the Blue-Bot didactic robot. Secondly, we analyzed their justifications for reflecting on the design of robotics problems. The results indicate that future teachers present characteristics of didactic-mathematical knowledge when solving and designing robotics problems, although errors and ambiguities are evident, especially in the procedures and representations of the programming. These shortcomings significantly influence the didactic suitability of the robotics problems they design. From a future perspective, in the training of future teachers, it is considered relevant to incorporate didactic-mathematical and computational knowledge that allows them to develop logical, spatial and computational thinking.

Didactic-Mathematical Knowledge and Teacher Education: An Investigation with Pre-Service and In-Service Mathematics Teachers

Background: The mathematical and didactic knowledge in mathematics teachers’ domain should constitute a permanent focus of investigations and reflections. Objective: To investigate the development of knowledge and procedures concerning geometry teaching among pre-service and in-service mathematics teachers engaged in the final years of elementary school. Design: The investigation is inserted in a qualitative perspective and follows the assumptions of action research. Setting and Participants: Fifty-four pre-service and in-service mathematics teachers from the Brazilian city of Paulo Afonso, Bahia, participated in a teacher education process. Data Collection and Analysis: Participant observation, data collection and analysis following an analysis protocol guided by the constructs of the onto-semiotic approach; analysis of productions. Results: Development and application of 24 sequences of activities focusing on the study of geometry. High epistemic adequacy was observed based on the application of the analysis protocol concerning structuring sequences of activities. Regarding the other dimensions, concerning the analysis of the application of the activities (cognitive-affective, interactional-mediational, ecological), high adequacy was observed. Conclusions: It is relevant to invest in teacher education courses proposing the expansion of didactic-mathematical knowledge to enhance the use of different ways to teach and learn mathematics.

Future math teachers ‘knowledge about fair games

This work reports the results of a study whose objective was to analyze the didactic-mathematical knowledge of future mathematics teachers when solving an unequal game. The sample was made up of 24 future professors from a Chilean university. Task-based interviews were used to collect the information. The results show that future teachers have adequate intuitive ideas about expected value, prioritize the classical approach to solve situations and determine if the game is fair or not, taking into account both the probability of winning or losing the game, and the value that is gained or lost, using few solution strategies, which makes it easier for them to combine their statistical thinking with its teachableness. It concludes on the need to improve the teachability of probability, in both disciplinary and didactic aspects, suggesting support for teachers, both in training and in practice.

Knowledge and Competencies of Prospective Teachers for the Creation of Proportionality Problems

Background: Problem posing is a fundamental competence that enhances the didactic-mathematical knowledge of the mathematics teacher, so it should be an objective in teacher education plans. Objectives : This paper describes and analyses a training intervention with prospective teachers to develop such competence using proportionality tasks. Design: This qualitative and interpretative study adopts a methodology characteristic of didactic design or engineering research. The design of the intervention and the content analysis of the participants’ answers use theoretical and methodological tools from the onto-semiotic approach to mathematical knowledge and instruction. Context and participants : The training action was carried out with 127 undergraduates attending a primary education teaching degree in the framework of the Design and Development of the Mathematics Curriculum in Primary Education subject in a Spanish university. Data collection and analysis: The prospective teachers, organised in 33 working teams, were asked to create two problems based on a given situation and to identify objects and difficulties, the solution of which was analysed by the research team. Results: The results show that the participants encounter difficulties in elaborating relevant proportionality problems from the given situation, identifying the associated level of complexity, recognising the mathematical objects interacting in the solution to their problems and the difficulties that these could cause to primary school pupils. Conclusions: It is mandatory to reinforce problem creation competence and proportional reasoning in teacher education.

Proposal to Systematize the Reflection and Assessment of the Teacher’s Practice on the Teaching of Functions

This research proposes theoretical-methodological strategies and tools to guide teaching, reflection, and assessment on the practice of the prospective teacher of mathematics when performing teaching and learning processes on the notion of function. It exemplifies our proposal’s use, through the reflection and characterization of the didactic-mathematical knowledge, of a prospective Chilean teacher, underlying his practice of teaching the power function. For the development of our proposal, we used the theoretical-methodological notions of the Didactic-Mathematical Knowledge (DMK) model and micro-teaching contexts, understood as rich and essential spaces for teacher education. As a conclusion, we could observe that the theoretical-methodological proposal here presented serves to systematize the reflection and assessment processes of teaching practice (one’s own or that of others) for the teaching of functions, and, thus, determine (or anticipate) actions that improve the teaching processes on this important mathematical notion.

Conocimiento de futuros profesores chilenos de matemática sobre los números decimales

Mathematics teachers' knowledge of the different numerical sets is essential for their appropriate teaching. We analysed and evaluated the knowledge of a group of 25 prospective mathematics teachers at a Chilean university about decimal numbers. Participants were given two tasks, adapted from previous research (Konic, 2011), to describe the strategies they use in relation to the concept and properties of decimal numbers. The study used some elements of the Didactic-Mathematical Knowledge Model proposed by Godino and cols. As results of the content analysis of the answers, the participants show a limited knowledge of common and specialised contents, and different semiotic conflicts are identified, such as the application of the idea of successor or the identification of the decimal number as a number expressed by writing with commas.

Prospective Primary Teachers’ Didactic-Mathematical Knowledge in a Service-Learning Project for Inclusion

An analysis of the didactic-mathematical knowledge put into play by a sample of 30 prospective primary teachers during their participation in a Service-Learning program focused on mathematical stimulation as a measure of attention to adolescents at risk of social exclusion is presented. The program aimed to respond to current training demands by promoting the development of mathematics-specific professional competencies, as well as social skills to interact with students, and to positively influence the prospective teachers’ affection towards mathematics and its teaching. It was developed during three academic years in two phases: the learning module was oriented to the presentation of the mathematical stimulus program and the didactical analysis of the STEAM activities knocking it into shape; in the service module, the participants implemented the activities in an educational center with students at risk of social exclusion. The analysis was carried out from the video recordings of the sessions, the future teachers’ written reports with the analysis of the activities, questionnaires on the didactic-mathematical knowledge and a satisfaction test. The results show difficulties on the part of the future teachers to analyze some of the activities of the program, which seems to have its origin in their poor command of the common knowledge of the mathematical content, which also limits them when it comes to managing the activities in the way expected for the stimulus program. All in all, participation in the program was positively assessed by them in relation to its usefulness for their training. The potential benefit of Service-Learning programs such as the one addressed here in the face of a positive impact on the affections towards mathematics is concluded, as a necessary step for a more effective acquisition of didactic-mathematical knowledge and skills in the initial training.

Planning of training spaces and the mobilization of Didactic-Mathematical Knowledge: a look at the Pedagogical Residency Program

This article aims to present a discussion about the planning of formative spaces that focus on the initiation to teaching, since these can provide the mobilization of knowledge and, in particular, of Mathematical-Didactic Knowledge (CDM). To this end, we chose the Pedagogical Residency Program (PRP), as a Public Policy managed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/BRASIL) as a training context to base our reflections. The reflections are anchored in the structure proposed by CAPES for the implementation of PRP and in the composition of modules (Environment, semi-structured observation and regency) supporting, theoretically, in the discussions about the teaching initiation process and in the Ontosemiotic Approach - EOS, in particular, the discussions related to the Didactic Adequacy Criteria.

Prospective high school mathematics teachers’ assessment of the epistemic suitability of a proportionality textbook lesson

Background: Every teacher should be able to use curriculum materials to guide instructional design and make reasoned pedagogical decisions about the limitations these resources may have. Objectives : In this paper, we describe and analyse a formative intervention with prospective high school mathematics teachers, aimed at developing their competence of didactic suitability analysis of mathematics textbook lessons. Design: The methodology followed is didactic engineering, furthermore, the content analysis methodology is applied to examine the response protocols of the participants. Setting and Participants : The experience was carried out within the framework of a University Master's Degree in Compulsory Secondary Education and High School; the sample was made of 30 students. Data collection and analysis : We proposed these prospective teachers systematically and critically analyse a lesson on proportionality. The written reports of the lesson suitability produced by 14 work-teams are compared with the a priori analysis carried out by the researchers. Results : The results suggest that the prospective teachers usually make more descriptive and less analytical analyses even while using a guide. The participants did not clearly identify the epistemic deficiencies of the lesson, thus revealing their limited didactic-mathematical knowledge on proportionality and their lack of critical evaluation of the textbook. However, based on the analysis previously conducted, prospective teachers managed to be quite accurate in preparing their proposals for the use of the textbook lesson. Conclusions : In this article we show the interest and usefulness of providing future teachers with a tool to systematically analyse a specific textbook lesson. However, in order for future teachers to acquire the necessary skills in critical analysis of the lesson, it is necessary to reinforce their didactic-mathematical knowledge related to proportionality.

Preservice Teachers’ Argumentation and Some Relationships to Didactic-Mathematical Knowledge Features

This paper presents research on the argumentation that preservice teachers perform when designing and teaching geometry. Argumentation is associated with speech acts carrying didactical intentions. This research study features of preservice teachers’ argumentation when explaining geometry tasks both to peers, during preparation and discussion of designed activities, and to students in the classroom. This is qualitative research and the results support establishing relationships between the didactical dimension of the didactic-mathematical knowledge model and some characteristics of the argumentation that preservice mathematics teachers exhibit during their planning and teaching.

Comparing the Didactic-Mathematical Knowledge of Derivative of In-Service and Pre-service Teachers

Background : The knowledge that a mathematics teacher should master has taken an increasing interest in recent years. Very few studies focused on comparing didactic-mathematic knowledge of in-service and pre-service teachers aimed at identifying features of the teachers’ didactic-mathematical knowledge on specific topics that can establish a line between pre-service and in-service teachers’ knowledge for teaching. Objective : The research aims to compare derivative knowledge of pre-service and in-service teachers to identify similarities and differences between teachers’ knowledge. Design : This research is a mixed and interpretative study. Settings and Participants : The participants were 22 pre-service teachers, and 11 in-service teachers enrolled in a pre-service teacher education programme and a master’s programme, respectively. Data collection and participants : Data were collected based on a questionnaire designed purposefully for the study. Results : The results show that pre-service teachers lack both epistemic derivative knowledge, while in-service teachers not only have this knowledge but relates it to its use in teaching. Pre-service teachers may not be making sense of the concept of derivative means, much less related to teaching. Conclusions : The insufficiencies found in pre-service teachers’ knowledge justify the pertinence to design specific formative cycles to develop prospective teachers’ epistemic facet of didactic-mathematical knowledge. It is recommended that both in-service and pre-service teachers discuss activities in which they can identify and reflect on possible mistakes and errors made by students. The development of these formative cycles should consider the complexity of the global meaning of the derivative.

Textbook: an Analysis in the Light of the Ontosemiotic Focus on Rational Numbers

Background: Many teachers consider the textbook the primary guide for the curriculum materialisation. This work analyses the content of the rational numbers in a textbook used in the 7th-grade of elementary school. For this analysis, the theoretical and methodological tools of the Ontosemiotic Approach to Knowledge and Mathematical Instruction (OSA) are used. Objective: Our goal was to understand the level of didactic suitability of the instruction process in the textbook. Design and setting: Thus, we analysed a section of the textbook about rational numbers using the categories described by OSA. Concepts, procedures, problem situations, definitions and arguments used by the authors were also analysed. Results: From the results obtained, we could infer that there are examples of the concepts worked in class, but without the proper definitions and arguments, hindering generalisation. The analysis also allowed us to highlight didactic-mathematical knowledge that can guide the teacher concerning the textbooks' possibilities and limitations, to achieve more didactic suitability in the process of teaching and learning rational numbers. Conclusions: the book under study should not be considered as planning for an instructional process to meet the current curriculum guidelines.

Task (re)design to enhance the didactic-mathematical knowledge of teachers

Background: Currently, studies in the field of design of tasks, especially when linked to teacher training actions, have made significant contributions to the area of Mathematics Education. Objectives: This research aimed to point out the contributions that task (re)design can bring to enhance the didactic-mathematic knowledge of the teachers, as well as to discuss the difficulties and advances of educators when dealing with tasks characterised by challenging situations. Design: Using a qualitative approach, based on the Intervention Research format, we held training meetings with teachers in two main stages: the first consisted of meetings to work on the concept of the design and redesign of tasks; the second focused on redesigning a school project, putting into practice the concepts acquired in the first stage. Setting and participants: The intervention took place in an elementary school of the city of Jaguaquara, Bahia, Brazil, with three Mathematics teachers and the teacher-researcher. Data collection and analysis: The data were produced using a field diary, the materials produced by the teachers, and the audio recording of the training meetings. Results: The results show that designing tasks increases the didactic-mathematical knowledge of teachers. Also, it showed that the participants presented difficulties, some of which they could overcome during the process, while others remained after the training. Conclusions: The remaining difficulties leave room for new training actions. On the other hand, the training contributed to diminish the difficulties of the participating teachers and instigated innovative actions in Mathematics teaching.