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This article presents the partial results obtained in the first stage of the research, which sought to answer the following
questions: (a) What is the role of intuition in university students' solutions to optimization problems? (b) What is the role
of rigor in university students' solutions to optimization problems? (c) How is the combination of i...
Contexts in source publication
Context 1
... this paper optimizing intuition refers to an a priori characterization of "optimizing intuition", and confirming the existence (or not) of this type of intuition is the aim of the experiment designed herein (see sections 4 and 5). To achieve this goal let us consider the following question as a context for reflection: Why are there people who consider it evident that a graphic which looks like a parabola that is shown to them (Figure 2(A)) has a maximum? To answer this question we use three of the processes considered in Figure 1 (idealization, generalization and argumentation), image schemas and metaphorical mappings in the way they were applied in Acevedo (2008) and Malaspina and Font (2009). ...
Context 2
... all, intuition has to do with the process of idealization . Let us suppose that the teacher draws Figure 2(A) on the blackboard and that he talks about it as if he were displaying the graphic of a parabola, while simultaneously expecting that the students interpret the figure in a similar way. The teacher and students talk about Figure 2(A) as if it were a parabola. ...
Context 3
... us suppose that the teacher draws Figure 2(A) on the blackboard and that he talks about it as if he were displaying the graphic of a parabola, while simultaneously expecting that the students interpret the figure in a similar way. The teacher and students talk about Figure 2(A) as if it were a parabola. However, if we look carefully at Figure 2(A) we can see that it is not actually a parabola. ...
Context 4
... teacher and students talk about Figure 2(A) as if it were a parabola. However, if we look carefully at Figure 2(A) we can see that it is not actually a parabola. Clearly, the teacher hopes that the students will go through the same idealization process with respect to Figure 2(A) and draw it on the sheet of paper as he has done. ...
Context 5
... if we look carefully at Figure 2(A) we can see that it is not actually a parabola. Clearly, the teacher hopes that the students will go through the same idealization process with respect to Figure 2(A) and draw it on the sheet of paper as he has done. In other words, Figure 2(A) is an ideal figure (explicitly or implicitly) for the type of discourse the teacher and students produce about it. ...
Context 6
... the teacher hopes that the students will go through the same idealization process with respect to Figure 2(A) and draw it on the sheet of paper as he has done. In other words, Figure 2(A) is an ideal figure (explicitly or implicitly) for the type of discourse the teacher and students produce about it. Figure 2(A), drawn on the sheet of paper, is concrete and ostensive (in the sense that it is drawn with ink and is observable by anyone who is in the classroom) and, as a result of the process of idealization, one has a non-ostensive object (the parabola) in the sense that one supposes it to be a mathematical object that cannot be presented directly. ...
Context 7
... other words, Figure 2(A) is an ideal figure (explicitly or implicitly) for the type of discourse the teacher and students produce about it. Figure 2(A), drawn on the sheet of paper, is concrete and ostensive (in the sense that it is drawn with ink and is observable by anyone who is in the classroom) and, as a result of the process of idealization, one has a non-ostensive object (the parabola) in the sense that one supposes it to be a mathematical object that cannot be presented directly. On the other hand, this non-ostensive object is particular. ...
Context 8
... that intuition is usually considered as a clear and swift intellectual sensation of knowledge, of direct and immediate understanding, without using conscious and explicit logical reasoning, we can assume that in intuition there is no explicit argumentation even though there is an implicit inference. In the case shown in Figure 2 the inference could be, for instance, "as in the curve there is first a part that goes up and then a part that goes down, so there must be a point of maximum height". ...
Context 9
... that case, the student's answer must contain some indicator of the three components of the intuition "vector". It is plausible to assume that processes of idealization and materialization have occurred in the students' solutions (Figure 12), because the figures of the square and the rectangle on the left-hand side of the answer are the materialization of the mathematical objects "square" and "rectangle". The student writes about Figure 12(A) as if it were one square and one rectangle. ...
Context 10
... is plausible to assume that processes of idealization and materialization have occurred in the students' solutions (Figure 12), because the figures of the square and the rectangle on the left-hand side of the answer are the materialization of the mathematical objects "square" and "rectangle". The student writes about Figure 12(A) as if it were one square and one rectangle. If we look carefully at Figure 12(A) we observe that: (1) the figure on the left looks more like a trapezium than a square; (2) in the two figures the lines are curved rather than straight like the segments of a line; (3) there are sides that are not connected, etc. Figure 12(A), which is drawn on the sheet of paper, is concrete and ostensive, and as a result of the process of idealization there are two non-ostensive objects (square and rectangle), in the sense that they are assumed to be mathematical objects that cannot be presented directly. ...
Context 11
... student writes about Figure 12(A) as if it were one square and one rectangle. If we look carefully at Figure 12(A) we observe that: (1) the figure on the left looks more like a trapezium than a square; (2) in the two figures the lines are curved rather than straight like the segments of a line; (3) there are sides that are not connected, etc. Figure 12(A), which is drawn on the sheet of paper, is concrete and ostensive, and as a result of the process of idealization there are two non-ostensive objects (square and rectangle), in the sense that they are assumed to be mathematical objects that cannot be presented directly. On the other hand, these non-ostensive objects are particular. ...
Context 12
... student writes about Figure 12(A) as if it were one square and one rectangle. If we look carefully at Figure 12(A) we observe that: (1) the figure on the left looks more like a trapezium than a square; (2) in the two figures the lines are curved rather than straight like the segments of a line; (3) there are sides that are not connected, etc. Figure 12(A), which is drawn on the sheet of paper, is concrete and ostensive, and as a result of the process of idealization there are two non-ostensive objects (square and rectangle), in the sense that they are assumed to be mathematical objects that cannot be presented directly. On the other hand, these non-ostensive objects are particular. ...
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Citations
... (Schupp, 1992), but many others miss the connection, e.g., OECD (2016). Some research has investigated certain cognitive aspects of solving optimization problems (e.g., Malaspina & Font, 2010). Moreover, there is some research investigating optimization with technology, e.g., Bushmeleva et al. (2018). ...
Optimization is a fundamental idea of mathematics, yet it is not that much in the focus of widespread digital teaching tools. The paper first explores the relevance of optimality and then shows how the idea is supported in the FeliX dynamic geometry online system. The idea is to include optimality statements together with equality statements as means to describe geometric scenes. Some reports from teaching experiences are discussed as well.
... proposicions, procediments i arguments/justificacions que es produeixen a través dels respectius processos matemàtics de comunicació, problematització, definició, enunciació, elaboració de procediments i argumentació que són activats en les pràctiques matemàtiques que desenvolupa el professorat en la resolució d'un problema (Malaspina & Font, 2010), o com a part de la planificació d'un problema per a l'aula (Malaspina, Torres & Rubio, 2019). A més a més, en el coneixement matemàtic es contempla la descripció de les possibles errades i ambigüitats comeses pel professorat des d'un punt de vista matemàtic (Sánchez, Breda, Font & Sala Sebastià, 2021). ...
... Following this line of helping to improve the didactic design, the analysis carried out with the tools provided by the OSA allows us to see the configuration of primary objects that intervene in the solving process. The analysis of these configurations provides information concerning the anatomy of the solution to the problem posed (see Badillo et al., 2015;Malaspina & Font, 2010;Malaspina et al., 2019), in addition to making it possible to create new problems, among other aspects, through variation of those initially proposed or directly, in such a way that a certain (or several) primary object(s) must be used in their solution. ...
The aim of this article is to carry out a work of networking theories which combines two perspectives on the mathematical activity involved in a modelling process, in order to answer the following question: To what extent does the application of the onto-semiotic tools complement the analysis from a cognitive perspective of a mathematical modelling process? To this end, we considered two theoretical frameworks: on the one hand, the onto-semiotic approach, which provides tools for the analysis of any mathematical activity and which here we applied to the activity of modelling; on the other hand, the modelling cycle from a cognitive perspective, which is a reflection on the specific mathematical activity of modelling. Then, we took a modelling problem that we applied to prospective mathematics teachers (at undergraduate and postgraduate level), and we analysed it from the perspective of both frameworks, in order to identify concordances and complementarities between these two ways of analysing the mathematical activity involved in the modelling process. The main conclusion is that both frameworks complement each other for a more detailed analysis of the mathematical activity that underlies the modelling process. Specifically, the analysis with the tools provided by the onto-semiotic approach reveals the phases and transitions of the modelling cycle as a conglomerate of mathematical practices, processes, and the primary objects activated in these practices.
... En primer lugar, el análisis presentado en la Tabla 1 permitió ver la configuración de objetos primarios intervinientes en el proceso de resolución. Este tipo de análisis aporta información sobre la anatomía de la solución del problema propuesto (véase Badillo et al., 2015;Malaspina y Font, 2010), además de posibilitar la creación de nuevos problemas, entre otros aspectos, mediante la variación de aquéllos inicialmente propuestos. ...
En este capítulo se sintetiza un modelo de análisis de la actividad matemática subyacente al proceso de modelización desde una perspectiva semiótico-cognitiva, el cual permite revelar las fases y transiciones del ciclo de modelización como un conglomerado de prácticas, procesos, y objetos matemáticos (activados en dichas prácticas). Este modelo emerge de la articulación teórica entre dos marcos: el ciclo de modelización desde una perspectiva cognitiva (marco teórico específico) y las herramientas del enfoque onto-semiótico (marco teórico general). Para comenzar, se explican brevemente los fundamentos teóricos del modelo; luego, se describe y ejemplifica su metodología de análisis; y, finalmente, se discuten algunas implicancias para la enseñanza de la modelización matemática.
... Ejemplos de estudios cualitativos que utilizaron métodos propios del marco teórico que declararon fueron Malaspina y Font (2010), en su análisis del rol de la intuición en matemáticas introductorias con el Enfoque Ontosemiótico de la Cognición e Instrucción Matemática (EOS) en 38 estudiantes de ingeniería; Hernandez-Martinez y Vos (2017), por su parte, exploraron la experiencia que tuvieron estudiantes noruegos en tareas de modelación matemática a través de la teoría de la actividad histórico-cultural, y diseñaron tareas de modelación a partir de una clase dictada por un profesionista; y González-Martín y Hernandes-Gomes (2019) identificaron con la TAD cómo las diferencias en la formación académica y profesional de profesores brasileños de ingeniería influye en el uso que dan a la matemática. ...
El objetivo de este estudio fue conocer el estado del arte de la Matemática Educativa en la formación matemática de ingenieros. Se configuró un método empírico que permitió delimitar la revisión a 132 artículos que cubren el período de 1968 a 2020. Los resultados mostraron once temáticas, destaca que el 70.5% de los artículos atiende la enseñanza y aprendizaje de la matemática en Ciencias Básicas. Los resultados evidencian lo siguiente: la formación de ingenieros articula tanto varias matemáticas como el conocimiento disciplinar en cuestión; la profesionalización del profesor de matemáticas que forma ingenieros es un aspecto pendiente; y hacer investigación en y fuera de la escuela es indispensable para identificar otros significados de la matemática que el ingeniero requiere para su desempeño.
... For example, the "ontosemiotic configuration" tool is available for the development of instruments to systematically analyze teachers' knowledge of the mathematical dimension and the epistemic facet of the DMK. This tool allows the description and characterization of the primary mathematical objects-representations/language in its different registers; problem situations; concepts and definitions; propositions; procedures; and arguments/justifications-that are produced through mathematical processes or as part of the planning of a task (or sequence of tasks) for the classroom (Malaspina & Font, 2010). In addition to this, mathematical knowledge contemplates the description of errors and ambiguities from a mathematical point of view. ...
... For example, for the development of instruments that allow the evaluation and systematic analysis of the knowledge of teachers regarding the mathematical dimension (common and expanded knowledge) and the epistemic facet of DMK. The "ontosemiotic configuration" tool allows the description and characterization of the primary mathematical objects-representations/language (terms, expressions, notations, graphs) in their diverse registers; problem situations (intra-or extra-mathematical applications, exercises); concepts and definitions (introduced through definitions or descriptions); propositions (statements about concepts); procedures (algorithms, operations, calculation techniques); and arguments/justifications (statements used to validate or explain propositions or procedures)-that are produced through the respective mathematical processes of communication, problematization, definition, enunciation, elaboration of procedures (creation of algorithms and routines) and argumentation that are activated in the mathematical practices that teachers develop in solving a problem [32,33], or as part of a problem planning (or problem sequence) for the classroom [34,35]. In addition, mathematical knowledge includes the description of errors and ambiguities committed by teachers from a mathematical point of view. ...
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.
... The analysis of data considers the degree of task's correctness (correct, partially correct, and incorrect answers) and type of cognitive configuration (resolution by the pre-service and in-service teachers, specifying objects and process brought into play). The analysis of data concerning the cognitive configuration is carried out by the technique known as semiotic analysis (e.g., Malaspina & Font, 2010;Godino, Font, Wilhelmi, & Lurduy, 2011), which allows describing systematically both the mathematical activity performed by teachers while the primary mathematical objects (linguistic elements, concepts/definitions, propositions/properties, procedures, and arguments) are put in place to solve the problems. ...
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.
... We found it useful because it emphasizes the role of processes and aspects related to problem solving. Malaspina and Font (2010) suggest that complex processes can be decomposed into simpler processes. In their study, intuition is considered a complex process. ...
This research investigates the significance that the development of creativity in mathematics classes has for the preservice teachers from a master's program in secondary school teaching, who are not trained on how to develop creativity. In the master's final projects, preservice teachers reflect on their teaching practice at school and propose changes to improve it. In their reflection, some preservice teachers mention creativity and its enhancement. We did a document analysis and classified the comments about creativity from 198 master's final projects focusing on the question: Which aspects of the teaching and learning process are related to creativity? Categories are based on didactic suitability criteria, the tool that participants use to analyze their teaching practice. Preservice teachers relate creativity and its development to several elements of the teaching and learning process. Most of the comments are associated with tasks where students can develop mathematical processes (argumentation, problem posing, modeling). They also identify the use of technology or manipulatives and the cooperative work as factors that could foster students' creativity. We conclude that a significant number of preservice teachers assume that students' creativity can be developed at school, particularly in mathematics classes. For them, creativity indirectly stems from the work on other aspects that are actually implemented in the classes.
... Then, a question emerges: how is creativity generated in a teaching and learning process? Regarding this question, the first step is assuming that creative thinking is a complex process that can be studied as a process that emerges from others, as Malaspina and Font (2010) explained with the intuitive process. These authors propose that a method to investigate complex processes is decomposing them into more simple processes and they use the metaphor of the vector space: the more complex process, creative thinking in this case, can be understood as a linear combination of the basis vectors (other processes). ...
... Then, when we concluded that the way to conceptualize the creative process in mathematics is decomposing it into other processes, we considered, together with the MC2 team, that the best way to go further in its study was adopting the method proposed by the Ontosemiotic Approach [Malaspina & Font (2010)] …but introducing an important variant. Instead of using a "basis" of processes defined a priori by the theoretical framework, we would use the "basis" that would be the result of a brainstorm performed by the participants in the project …MC2 when each of them explained what they understood by "creative process"…. ...
What do preservice teachers think about creativity and the ways to promote it in their classes? In this study, we analyze the references to creativity that appear in the master’s degree final projects of a group of secondary school preservice teachers of mathematics. The projects hold the preservice teachers’ reflections on their own practice, but it is important to note that the master’s degree does not include any specific training in creativity. In our analysis, we first searched for explicit references to creativity in the projects and registered them. Then, we classified the comments depending on the elements of the teaching and learning process that are related to creativity in each case. We observe a good variety of comments and more than half of the preservice teachers mention creativity, although some of them just do it superficially.
