This study analyses the didactic-mathematical knowledge for probability of 32 Chilean prospective early childhood education teachers. An exploratory-descriptive study was carried out with mixed methodological approach, which analyses the mathematical practices from the answers to the CDM-Childhood Probability questionnaire. This analysis allowed us to study the different facets involved in this type of knowledge. The results show insufficient didactic-mathematical knowledge, although more complete in the case of common content knowledge. The conclusion is the need to strengthen the opportunities for acquiring and deepening the didactic-mathematical knowledge associated with probability in the training programs for early childhood education teachers.
The objective of this article is to assess the didactic-mathematical competence of a teacher in identifying and resolving semiotic conflicts that arise in chemical engineering students when faced with a problem involving the differential of a function of one variable. The experience described in this study was carried out in an Analysis I class with 70 first-year engineering students. The analysis of task resolutions and the teacher's interventions is based on the Onto-semiotic Approach. The results show the emergence of four cognitive conflicts, which were addressed with the class group under the teacher's coordination. In conclusion, the importance of preliminary task analysis is highlighted as a fundamental tool for making appropriate teaching interventions and competently managing them.
Proof is a fundamental activity in mathematics, and its integration across different educational levels, with appropriate approaches, is relevant. Engaging in proof-related activities enhances students' mathematical understanding, making it crucial for prospective teachers to develop a deep knowledge of this practice. In this study, using a qualitative approach, we aim to understand the knowledge of three prospective secondary mathematics teachers as they engage in proof-related tasks. We identify their knowledge about specific proofs, proof as a mathematical practice, and the didactic aspects of proof. We highlight the recognition of proof as a practice that fosters mathematical learning by these future teachers and, in this regard, emphasize the importance of including proof in initial teacher education.
This work is part of a research project on algebraic thinking in early childhood and primary education students. The research objective of this document is to analyze the relationships between variables, representations, and strategies that preschool students demonstrate when working with a generalization task. We implemented a teaching experiment of 4 to 5 sessions for preschool students. We analyzed the work in one of the sessions, where the number of children invited to a birthday party was related to the number of juices needed, considering that one child does not drink (f(n) = n-1). The session took place in three moments: (a) introduction, (b) individual work, and (c) assembly. In the students' written productions, pictorial representation predominated, and we identified physical and cognitive strategies. There were children who generalized, noting that "we will not give this child juice because he does not like it" to reflect the "-1" implied in the function.
Traditionally, logarithms are introduced in classrooms as the inverse of the exponential function. Weber (2016) argues that this definition leads to errors and hinders understanding, suggesting teaching based on alternative basic models. We present a lesson designed to guide students toward constructing the concept of logarithm as repeated division and to provide empirical evidence of its explanatory potential. Students' discussions were analyzed using the Abstraction in Context theoretical-methodological framework to identify the epistemic actions that occurred during the activities. The results indicate that this model supports concept construction: students recognize its necessity and reach the construct before receiving a formal definition. They were also able to understand and reason about some of its basic properties.
This article presents the assessment of a mathematics textbook by a Colombian rural school teacher. This assessment is analyzed based on the didactic suitability facets of the ontosemiotic approach to didactic-mathematical knowledge, such as epistemic, cognitive, interactional, mediational, affective, and ecological suitability. The study involved a content analysis using Atlas.ti software. The research participant provides information on using a textbook to prepare and implement mathematics lessons. She uses the textbook as a guide for lesson planning but supplements it with additional resources. In addition, this participant makes some criticisms of this resource. Overall, the study highlights the importance of considering didactic appropriateness in analyzing textbooks for teaching mathematics in rural elementary schools.
Neste artigo procuramos responder à questão de investigação: que relações existem entre a realização de conjeturas, generalizações e justificações e o uso da linguagem algébrica com compreensão pelos alunos? A partir de uma experiência de ensino e numa perspetiva qualitativa e interpretativa, os dados são recolhidos através de respostas escritas e orais de alunos no trabalho de sala de aula e em entrevistas. Os resultados mostram que as conjeturas, generalizações e justificações têm um papel fundamental na reflexão sobre os significados dos símbolos algébricos e sobre como estes podem expressar relações matemáticas. A promoção destes processos de raciocínio contribui para que os alunos atribuam significados aos símbolos algébricos com base em múltiplas perspetivas, concorrendo para o uso da linguagem algébrica com compreensão.
The purpose of the study is to analyze the process of mathematical knowledge co-construction from a sociocultural perspective. Using a qualitative methodology, we present the results of implementing five research situations and how these results relate to the covariation between variables, this as a prelude to the concept of function. Using a multiple case study design, we present the results for four high school students aged 14 to 15. The results focus on the production process of functional-spontaneous representations and their evolution towards Socially constructed representations. This evolution process is determined by the discussion of ideas across the five stages of the ACODESA method. We also discuss the concept of habitus in the mathematics classroom. The study concludes that Socially constructed representations in mathematics classrooms are both constructed and transformed through a process of communication and objectification of signs or concepts.
Problem posing is central to mathematics education and, like other practices, is challenged by teachers' motivational beliefs and the use of artificial intelligence (AI). In this study, we applied a proprietary instrument, the ForPro-IA questionnaire, to assess the perceptions as problem formulators and AI users of 175 prospective primary school teachers. As a result, we found that participants found problem formulation more useful and cost-effective compared to the use of AI. The design, validation and application of the instrument are also presented as an original contribution to the field.
Problem posing promotes problem comprehension and solving, but its application in statistics is limited in the literature. This study analysed how 164 Pre-service Teachers (PSTs) posed statistical problems based on graph reading levels, following the Slow Reveal Graphs approach. The results showed that the PSTs were able to contextualise their proposals, align them with the curriculum, and use appropriate language, but they made errors in identifying the graph reading levels. The study represents a novel line of inquiry in statistical education, highlighting ways to strengthen the didactic-mathematical knowledge of PSTs in problem posing and graph reading levels.
In this article, we aim to address the following research questions: What algebraic understandings are evidenced when a group of 9-10-year-old students invent numerical sentences and mathematical stories? And what contributions do these tasks make to the development of algebraic thinking from both structural and analytical perspectives? We analyse students' responses by focusing on the operations and properties involved, the meanings attributed to the equal sign, and the ways in which students refer to and reason about unknown quantities. The results show that students invent sentences involving numbers and unknown quantities, drawing on different properties of operations and demonstrating relational understandings of the equal sign. When inventing stories, they relate unknown quantities to everyday situations and construct narratives that are coherent with the given equation. We discuss the role of problem invention in the construction of algebraic thinking in the early years of primary education.
infinity Problem posing in mathematics has been regarded for several decades as both a research tool and a teaching and learning strategy, not only with students at different educational levels but also with pre-service and in-service teachers. The articles included in this special issue highlight that research on problem posing is gaining increasing importance in Mathematics Education. The diversity of perspectives and theoretical frameworks supporting studies on problem formulation, as well as the variety of objectives addressed, provides a broad view that may be of interest to the international research community, with particular emphasis on studies conducted in Spain.
Using the didactic-mathematical knowledge model of the ontosemiotic approach, we assessed the knowledge of 70 prospective secondary school teachers during their training when creating probabilistic problems from news reports. Participants were asked to choose issue news, create and solve probabilistic questions about the issue, indicate the educational level at which the problem was aimed, and identify possible difficulties. The identified problems were appropriate, addressed topics of interest to students, used news from various sources and covered PISA contexts. Most questions were solved correctly, and, in addition to probabilistic calculations, some included reasoning and decision-making. However, the participants demonstrated lower competence in predicting potential student difficulties. The results provide new information about the knowledge of teachers by creating probabilistic problems in the context and identifying areas for teacher training improvement.
Researchers in psychology emphasize the importance of games during children's learning and development processes; on the other hand, researchers in mathematics education emphasize the importance of problem posing in mathematics learning. This leads us to integrate both points of view and explore game invention as a means to stimulate probabilistic thinking in children, using the framework of problem posing. In this sense, we qualitatively analyze the reactions of children in primary education when playing games invented by them, by modifying the rules of a game in one case and based on the material presented to them in another. Through the implemented processes, we found that inventing games using the problem posing framework helps stimulate probabilistic thinking in children.
ao In this study, developed within the framework of the Anthropological Theory of Didactics, we present and discuss the potential of Study and Research Paths as didactic means of a modality of study that, governed by the paradigm of mathematical modelling, promotes the integration of problem posing in the mathematical activity of students considered globally. Working with students in the third year of compulsory secondary education, we study the spatial problem of packaging design within the field of determination and construction of geometric solids. We analyse the crucial role played by the questions and answers maps elaborated by the students as evidence of the posing of new problems, and we show some institutional constraints that hinder the implementation of this new modality of study.
infinity This research analyzes how the use of GeoGebra strengthens future teachers' ability to reformulate problems. A qualitative analysis is applied, based on four phases of the problem formulation process (orientation, connection, generation and reflection), along with the categories established by Baumanns: reformulating to solve the problem and to inquire or investigate; formulating to generate new problems or to design them for didactic purposes. The results show that GeoGebra enabled participants to evaluate prior ideas and assess the feasibility of the problems they proposed. Additionally, visualization and dynamic approaches influenced both the reformulation and the formulation of new problems.
ao In this paper, we present the results of research that aims to signify the criteria of the first and second derivatives through a variational analysis of a motion modeling situation that incorporates technological elements. To construct the evidence, a movement-modeling situation was applied to a group of undergraduate students. The results show that the discussion on the criteria is based on how the motion (position, velocity, and acceleration) of a mobile must reproduce a certain proposed position graph and to reach a dialogue on how to generate the different behaviors (constant, increasing, and decreasing) observed in it and, therefore, of the variational characteristics (represented in the monotony and concavity of the curve). The importance of the variational strategies of comparison and seriation on the significance of the criteria is evident.
oe Vygotsky's sociocultural theory states that interaction with peers will expand students' Zone of Proximal Development (ZPD) to think critically. Collaborative Problem Solving (CPS) is an activity that requires interaction between team members, and several studies state that CPS has an impact on increasing students' critical thinking abilities. This study describes students' critical thinking skills when solving collaborative math problems. Students' critical thinking skills when solving problems collaboratively appear in two conditions: when working independently (individual space) and when interacting with other team members (collaborative space). Students' critical thinking skills are triggered by the problems given in the individual space. In collaborative space, students' critical thinking skills emerge more because they are triggered by two things, namely the problem given and responses from other group members. A description of how students' critical thinking skills work when solving problems collaboratively is explained in more detail in the research results section.
A framework of reference on levels of comprehension was refined based on the establishment of mathematical connections. To this end, the networking between the Ontosemiotic Approach to Knowledge and Mathematical Instruction and the Extended Theory of Mathematical Connections is used as a theoretical reference. This qualitative research is a case study in which the productions and responses to a questionnaire of three high school students were analyzed. The data were analyzed using ontosemiotic analysis. The results showed that the refined frame of reference allowed to assess the level of understanding of the students, who, through the establishment of connections, evidenced a different level of understanding concerning the exponential and logarithmic functions. The case studies did not show a high level of understanding; some reasons were due to the lack of time to socialize and deepen some characteristics of the functions during the sessions.
In this paper, we present a bibliometric analysis whose objective is to quantify and describe the scientific production of algebraic thinking in early childhood and primary education. We developed this analysis within the Scopus database, as it is one of the databases with the greatest coverage at the level of journals and volume of citations at the international level. We considered the initial open and final search ranges until December 2022. The results show a growing interest in the research community in Didactics of Mathematics in algebraic thinking at these educational levels. We identified authors who have produced a large amount of research on this topic, as well as groups of authors who have collaborated on different papers. In addition, key terms such as early algebra, functional thinking and generalization stand out as prominent topics in this area of study.
