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Using DGS in Investigating Synchronised Registers of Representations of Extrema Problems
Abstract
On the one hand, mathematical software is ubiquitous in mathematics education. On the other hand, word problems are an important part of the curriculum, and they often require modelling skills. This is especially true with optimisation and extrema problems proposed to high school and undergraduate students. We propose two activities around extrema problems, modelling with Dynamic Geometry Software (DGS). The exploration relies on the synchronised representations offered by the DGS. We discuss the different registers of representations used, their synchronisation and the limitations of the models versus the concrete occurrence.
... Although students can use any graphing software to create function art, we selected GeoGebra which is a dynamic mathematics software that combines algebra, geometry, calculus, and spreadsheets in one package (Hohenwarter et al., 2008). GeoGebra has multiple and synchronized semiotic registers of representation (Bautista et al., 2023;Duval, 2006). Additionally, the geometric tools within GeoGebra can seamlessly integrate the function graphs within a single window, thereby offering students a diverse array of choices and opportunities to combine algebraic and geometric elements in one artwork. ...
... The first three questions were partially addressed by two studies (Bautista et al., 2023;Bautista et al., 2024). In the first study, Bautista et al. (2023) examined the functions and strategies that students used to create their artwork. ...
... The first three questions were partially addressed by two studies (Bautista et al., 2023;Bautista et al., 2024). In the first study, Bautista et al. (2023) examined the functions and strategies that students used to create their artwork. The study showed that the most popular functions used are quadratic, sine, and cosine functions. ...
This paper explores "function art" as a teaching tool in STEAM education, integrating technology, mathematical function graphs, and artistic expression. Our study seeks to find ?) how function art enhances mathematical understanding, B) the strategies employed by students in creating their artwork, D) suitable assessment methods, and E) the potential for cultivating art and culture appreciation. Our initial findings highlighted the prevalent use of quadratic, sine, and cosine functions, alongside diverse creative approaches. Furthermore, we introduced a rubric for assessing creativity in function art with a focus on its mathematical aspects. We anticipate that our follow-up study will delve into the aesthetic dimensions of function art aspiring to deepen our understanding of its role within the STEAM education framework.
... Dynamic geometry software is good support in creating multiple and synchronous registers of representations (Bautista, et. al, 2023). They can show the intermediate processes in motions and changes (Finzer & Jackiw, 1998). Each mathematical knowledge has usually some different types of representations as symbolic, visual, or lingual. According to Tadao (2007), different representations have different advantages for displaying and manipulating information. The study o ...
PowerPoint is a learning aid, developed for presentations. PowerPoint has “dynamic” features for creating presentations with dynamic actions. The knowledge of the infinite backward multiplier is difficult to understand. One of the difficulties in understanding this knowledge is infinity, “dynamic”. In this study, we use the dynamic features of PowerPoint to design the sum of an infinite geometric sequence teaching. The study was conducted with 32 students in 11th grade. Students experimented with the dynamic model in PowerPoint. Research results show that the flexible use of PowerPoint’s dynamic features allows the creation of dynamic models of the sum of an infinite geometric sequence. Dynamic models have made it easier for students to build knowledge, creating geometric images of infinite sums. Moreover, with the dynamic features of PowerPoint, creating artistic images, attracts students to actively participate in learning.
... Secondly, working with functions in art enables students to perform transformations, viewing functions as objects rather than individual points. Lastly, functions offer multiple representations (Bautista et al., 2023), fostering inventiveness and deepening comprehension of fundamental concepts (Yerushalmy, 1991). ...
This study explores the integration of mathematics and visual arts through "function art" within the STEAM education framework. Utilizing GeoGebra software, 335 students from Grades 8 to 12 in the Philippines participated in an online webinar, resulting in the creation of 235 function art pieces. The analysis identified quadratic, linear, and sine/cosine functions as the predominant choices for symmetry, straight lines, and waves, respectively. These findings align with constructionist principles, emphasizing the significance of foundational functions and practical considerations in art creation. Function art presents a promising avenue for engaging students in meaningful mathematical exploration while nurturing artistic expression. Nonetheless, further research is necessary to evaluate its impact on learning within the STEAM context.
Tejera, M. (2021). Modelos matemáticos mediados por GeoGebra para el desarrollo del pensamiento variacional. Reloj de agua, 24, 39-49.
RESUMEN
La siguiente investigación basada en diseño se enmarcó dentro de la linea de investigación del Pensamiento y Lenguaje Variacional. La preocupación inicial vino dada por las dificultades ampliamente reportadas en la literatura que afrontan los estudiantes al enfrentarse a sus primeros cursos de cálculo en la universisdad. Se advirtió en la revisión bibliográfica que estas están asociadas a una excesiva mecanización y poco desarrollo de las ideas fundantes del calculo. Es por eso que se trabajó en el diseño y validación de una serie de tareas, mediadas por GeoGebra, centradas en lo conceptual, la conexión entre registros de representación y con la idea de variación como eje. La secuencia permitió a los estudiantes desarrollar sus ideas variacionales a través de diferentes registros asistidos por los recursos tecnológicos, y conectarlas de forma significativa para elaborar sus propios argumentos variacionales dando respuesta a las tareas. Por lo que se considera que el trabajo desde una mirada variacional del cálculo escolar aporta significativamente a la construcción de significados ricos sobre los conceptos centrales del mismo.
PALABRAS CLAVES: pensamiento variacional, razonamiento covariacional, cálculo, investigación basada en diseño.
ABSTRACT
The following design-based research was framed within the research line of Variational Thinking. The initial concern was given by the difficulties widely reported on the literature about students' difficulties when facing their first calculus courses at university. It was noted in the literature review that these are associated with excessive mechanization and little development of the grounding ideas of calculus. That is why we worked on the design and validation of a series of tasks, mediated by GeoGebra, focused on conceptual ideas, the connection between representation registers, and with the idea of variation as an axis. The sequence allowed students to develop their variational ideas through different records assisted by technological resources, and connect them in a meaningful way to elaborate their own variational arguments responding to the tasks.
1 Magíster en Matemática Educativa (IPN, México), Profesor de Matemática (IPA, Uruguay).
Combining new educational approaches and educational technologies can
make mathematics education more adaptable to pupils‘ needs in the 21st
century. Our explorative educational study aimed to identify how learning
settings and learning environments should be designed to facilitate
synthesising flipped approaches to education and using GeoGebra. To
discover how to combine flipped approaches and GeoGebra in mathematics
education, we conducted a nine-month educational study at a Viennese
secondary school. In our study, we focused on pupils‘ needs, as pupils are key
to combining successfully new educational approaches and using
technologies. Analysing our qualitative research data following design-based
and grounded theory approaches indicates that the categories (a) clear task
definition and task design, (b) feedback, (c) context and benefits, and (d)
single-source learning environments are important for pupils when utilising
GeoGebra for enhancing flipped education.
The article is dedicated to solving extrema problems in teaching mathematics, without using calculus. We present and discuss a wide variety of mathematical extrema tasks where the extrema are obtained and find their solutions without resorting to differential. Particular attention is paid to the role of arithmetic and geometric means inequality in solving these problems.
In this article, we develop a theoretical model for restructuring mathematical tasks, usually considered advanced, with a network of spatial visual representations designed to support geometric reasoning for learners of disparate ages, stages, strengths, and preparation. Through our geometric reworking of the well-known “open box problem”, we sought to enrich learners’ conceptual networks for optimisation and rate of change, and to explore these concepts vertically across curricula for a variety of grades. We analyse a network of physical, geometric spatial visual representations that can support new inferences and key understandings, and that scaffold these advanced concepts so that they could be meaningfully addressed by learners of various ages, from elementary to university, and with diverse mathematical backgrounds.
This book offers a new conceptual framework for reflecting on the role of information and communication technology in mathematics education. Borba and Villarreal provide examples from research conducted at the level of basic and university-level education, developed by their research group based in Brazil, and discuss their findings in the light of the relevant literature. Arguing that different media reorganize mathematical thinking in different ways, they discuss how computers, writing and speech transform education at an epistemological as well as a political level. Modeling and experimentation are seen as pedagogical approaches which are in harmony with changes brought about by the presence of information and communication technology in educational settings. Examples of research about on-line mathematics education courses, and Internet used in regular mathematics courses, are presented and discussed at a theoretical level. In this book, mathematical knowledge is seen as developed by collectives of humans-with-media. The authors propose that knowledge is never constructed solely by humans, but by collectives of humans and technologies of intelligence. Theoretical discussion developed in the book, together with new examples, shed new light on discussions regarding visualization, experimentation and multiple representations in mathematics education. Insightful examples from educational practice open up new paths for the reader.
En este artículo mostramos cómo a través de la interacción entre un colectivo de investigadores con el software GeoGebra surgieron algunas ideas para el diseño de estrategias que potencian el desarrollo del pensamiento variacional. Usamos el constructo teórico de seres-humanos-con-medios propuesto por Borba y Villarreal (2005) para analizar dos episodios de nuestra experiencia como investigadores. Desde dicho análisis pudimos observar cómo a través de la necesidades de la investigación se pudieron generar algunas “herramientas” del software, que a su vez permitieron estudiar, establecer y demostrar nuevas conjeturas acerca de algunos conceptos matemáticos.
Resumo
Neste artigo apresentamos como, por meio da interação entre um coletivo de investigadores com o software GeoGebra, surgiram algumas ideias para o desenho de estratégias que potencializam o desenvolvimento do pensamento variacional. Usamos a construção teórica de seres-humanos-com-mídias proposta por Borba e Villarreal (2005) para analisar dois episódios de nossa experiência como investigadores. Desta análise pudemos observar como por meio das necessidades da investigação foi possível gerar “ferramentas” do software, que por sua vez permitiram estudar, estabelecer e demonstrar novas conjecturas de alguns conceitos matemáticos.
Palavras chave:Pensamento variacional; seres-humanos-com-mídias; GeoGebra; vizualização.
The last decade has seen the development in France of a significant body of research into the teaching and learning of mathematics
in CAS environments. As part of this, French researchers have reflected on issues of ‘instrumentation’, and the dialectics
between conceptual and technical work in mathematics. The reflection presented here is more than a personal one – it is based
on the collaboration and dialogues that I have been involved in during the nineties. After a short introduction, I briefly
present the main theoretical frameworks which we have used and developed in the French research: the anthropological approach
in didactics initiated by Chevallard, and the theory of instrumentation developed in cognitive ergonomics. Turning to the
CAS research, I show how these frameworks have allowed us to approach important issues as regards the educational use of CAS
technology, focusing on the following points: the unexpected complexity of instrumental genesis, the mathematical needs of
instrumentation, the status of instrumented techniques, the problems arising from their connection with paper & pencil techniques,
and their institutional management.
The pandemic has demonstrated more than ever that citizens around the world need to understand how mathematics contributes to understanding global challenges and ways of overcoming them. People need to understand that predictions are based on models that make use of assumptions and the best inputs available. They also need to learn to critically evaluate reports based on the outcomes of models to make effective decisions and deal with the inherent uncertainty in an appropriate way. These capabilities make it clear that mathematical modelling is a key element of citizenship education. Given this fundamental role of modelling, we take a closer look at its definition, its history, its connection to other teaching approaches, as well as the competences students need to carry through modelling processes and the competences teachers need for teaching modelling.KeywordsMathematical modellingCitizenship educationModelling competencesTeaching modellingLarge scale implementation in day-to-day teaching
This paper details an exploratory study of ten elementary preservice teachers (PSTs) involvement in the problem-solving process in a mathematics methods course. The dynamic, messy and nonlinear nature of this process was demonstrated by discussing PSTs’ solution process of an open box construction problem. Effective approaches to solve this problem were discussed. Polya’s (1957) four-step framework was presented to illustrate the problem-solving process as a more realistic and authentic endeavor, including the importance of using this framework with discretion. The eight Mathematical Practices (Common Core State Standard for Mathematics, CCSSM, 2010) were used to explore the PSTs’ problem solution process.
Full article available at http://www.k-12prep.math.ttu.edu/journal/2.pedagogy/volume.shtml or http://www.k-12prep.math.ttu.edu/journal/2.pedagogy/ortiz01/article.pdf.
Mathematical models are conceptual processes that use mathematics to describe, explain and/or predict the behaviour of complex systems. This article is written for teachers of mathematics in the junior secondary years (including out-of-field teachers of mathematics) who may be unfamiliar with mathematical modelling, to explain the steps involved in developing a model and the differences between modelling and traditional problem solving.
In this book, Raymond Duval shows how his theory of registers of semiotic representation can be used as a tool to analyze the cognitive processes through which students develop mathematical thinking. To Duval, the analysis of mathematical knowledge is in its essence the analysis of the cognitive synergy between different kinds of semiotic representation registers, because the mathematical way of thinking and working is based on transformations of semiotic representations into others. Based on this assumption, he proposes the use of semiotics to identify and develop the specific cognitive processes required to the acquisition of mathematical knowledge. In this volume he presents a method to do so, addressing the following questions:
• How to situate the registers of representation regarding the other semiotic “theories”
• Why use a semio-cognitive analysis of the mathematical activity to teach mathematics
• How to distinguish the different types of registers
• How to organize learning tasks and activities which take into account the registers of representation
• How to make an analysis of the students’ production in terms of registers
Building upon the contributions he first presented in his classic book Sémiosis et pensée humaine, in this volume Duval focuses less on theoretical issues and more on how his theory can be used both as a tool for analysis and a working method to help mathematics teachers apply semiotics to their everyday work. He also dedicates a complete chapter to show how his theory can be applied as a new strategy to teach geometry.
“Understanding the Mathematical Way of Thinking – The Registers of Semiotic Representations is an essential work for mathematics educators and mathematics teachers who look for an introduction to Raymond Duval’s cognitive theory of semiotic registers of representation, making it possible for them to see and teach mathematics with fresh eyes.”
Professor Tânia M. M. Campos, PHD.
High schools commonly use a differential approach to teach minima and maxima geometric problems. Although calculus serves as a systematic and powerful technique, this rigorous instrument might hinder students’ ability to understand the behavior and constraints of the objective function. The proliferation of digital environments allowed us to adopt a different approach involving geometry analysis combined with the use of the inequality of arithmetic and geometric means. The advantages of this approach are enhanced when it is integrated with dynamic e-resources tailored by the instructor. The current study adopts the abstraction in context framework to trace students’ knowledge construction processes while solving extremum problems in an e-resource GeoGebra-based environment using a non-differential approach. We closely monitored the learning of 5 pairs of high-track yet low achieving 17-year-old students for several lessons. We further assessed the students’ understanding at the end of the learning unit based on their explanations of extrema problems. Our findings allowed us to pinpoint the contributions (and pitfalls) of the e-resources for student learning at the micro level. In addition, the students demonstrated the ability to solve extrema problems and were able to explain their reasoning in ways that reflect the e-resources with which they worked.
Teachers’ perception of changes to their teaching practice, with respect to digital technology use in secondary school mathematics, during their participation in a research project are reported. Two case studies are presented of teacher perspectives illustrative of the range of perceived changes teachers made to their practice and positions along the ‘path of change’ during their participation in the project. Participating in a project supportive of teacher change and resulting in perception of substantial change was necessary, but not sufficient, to meet the goal of transformative use of digital technologies to increase the level of cognitive demand experienced by students.
Using the AM-GM inequality, we solve the classic open-top box problem from first-semester calculus.
In this paper, we shall report on some of the work that has been, and is being, done in the DISUM project. In §, we shall describe the starting point of DISUM, the SINUS project aimed at developing high-quality teaching. In §, we shall briefly describe the DISUM project itself, and in §3 we shall present and analyse a modelling task from DISUM, the “Sugarloaf” problem. How students dealt with this task will be the topic of §4, the core part of this paper. How experienced SINUS teachers dealt with this task in the classroom will be reported in §5. Finally, in §6, we shall briefly describe future plans for the DISUM project.
Almost fifty years before Leibniz and Newton developed the tools of calculus, Pierre de Fermat was able to solve standard calculus problems. We examine several applications of his methods (providing additional details), offer some additional exercises, and briefly consider Fermat's place in the development of the derivative.
Many physical systems are described by quadratic equations. In such
cases the requirement for a real solution is that the discriminant be
non-negative. This requirement can often lead, directly or indirectly,
to maximizing or minimizing some system variable of interest. Three
examples suitable for a noncalculus physics course are given : (1)
projectile motion, (2) hydrostatic equilibrium (Archimedes' principle),
and (3) resistance matching.
To understand the difficulties that many students have with comprehension of mathematics, we must determine the cognitive
functioning underlying the diversity of mathematical processes. What are the cognitive systems that are required to give access
to mathematical objects? Are these systems common to all processes of knowledge or, on the contrary, some of them are specific
to mathematical activity? Starting from the paramount importance of semiotic representation for any mathematical activity,
we put forward a classification of the various registers of semiotic representations that are mobilized in mathematical processes.
Thus, we can reveal two types of transformation of semiotic representations: treatment and conversion. These two types correspond
to quite different cognitive processes. They are two separate sources of incomprehension in the learning of mathematics. If
treatment is the more important from a mathematical point of view, conversion is basically the deciding factor for learning.
Supporting empirical data, at any level of curriculum and for any area of mathematics, can be widely and methodologically
gathered: some empirical evidence is presented in this paper.
Representation is viewed as central to mathematical problem solving. Yet, it is becoming obvious that students are having
difficulty negotiating the various forms and functions of representations. This article examines the functions that representation
has in students’ mathematical problem solving and how that compares to its function in the problem solving of experts and
broadly in mathematics. Overall, this work highlights the close connections between the work of experts and students, showing
how students use representations in ways that are inherently similar to those of experts. Both experts and students use representations
as tools towards the understanding, exploration, recording, and monitoring of problem solving. In social contexts, experts
and students use representations for the presentation of their work but also the negotiation and co-construction of shared
understandings. However, this research also highlights where students’ work departs from experts’ representational practices,
hence, providing some directions for pedagogy and further work.
KeywordsRepresentation–Mathematical processes
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He is currently a Ph.D. student in STEAM Education at Johannes Kepler University in Linz, Austria. His research interests are the integration of arts and mathematics, the integration of technology in the classroom, the use of technology in professional development for teachers and lesson study
- Guillermo Bautista
Guillermo Bautista, Jr. is a Mathematics Education
Specialist at the University of the Philippines National
Institute for Science and Mathematics Education
Development (UP NISMED). He is currently a Ph.D.
student in STEAM Education at Johannes Kepler
University in Linz, Austria. His research interests are the
integration of arts and mathematics, the integration of
technology in the classroom, the use of technology in
professional development for teachers and lesson study.