Dynamic Algebra Research Articles

The Impact of GeoGebra in Analytic Algebra: A Bibliometric Review

GeoGebra is a software that is often used in mathematics learning. GeoGebra is an electronic program that contains a set of tools that equip students with mathematical skills. This research aims to look at research trends related to algebra learning that integrates GeoGebra. This research is literature review research, there are 95 publications collected from the Scopus database which are then analyzed using the bibliometric analysis method assisted by the Vos viewer application. Data taken from the Scopus database was refined in 4 stages, namely identification, filtering, eligibility, and inclusion, resulting in 95 publications. The research results show that Spain and Austria are the most influential countries and have high cooperation with other countries in this field. The focus of research related to GeoGebra on algebra material is, 1) geometry and students; 2) GeoGebra and algebra; 3) elementary geometry and computational theory; 4) dynamic geometry and computer algebra. The new themes in this research are symbolic computation, quantifier elimination and real quantifier elimination. This research not only enriches the scientific literature in the fields of GeoGebra and algebra but also provides practical guidance for educators and researchers to improve mathematics teaching and learning methods in the future.

Algebraic Structures Formalizing the Logic of Quantum Mechanics Incorporating Time Dimension

AbstractAs Classical Propositional Logic finds its algebraic counterpart in Boolean algebras, the logic of Quantum Mechanics, as outlined within G. Birkhoff and J. von Neumann’s approach to Quantum Theory (Birkhoff and von Neumann in Ann Math 37:823–843, 1936) [see also (Husimi in I Proc Phys-Math Soc Japan 19:766–789, 1937)] finds its algebraic alter ego in orthomodular lattices. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of Quantum Mechanics are depending on time. The aim of the present paper is to show that tense operators can be introduced in every logic based on a complete lattice, in particular in the logic of quantum mechanics based on a complete orthomodular lattice. If the time set is given together with a preference relation, we introduce tense operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections in an orthomodular lattice. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.

Analysing Instrumented Justification: Unveiling Student’s Tool Use and Conceptual Understanding in the Prediction and Justification of Dynamic Behaviours

The study advances the instrumental approach to mathematics education (Drijvers et al., 2013; Trouche, 2003), aiming to elucidate the interplay between students’ reasoning competency, conceptual knowledge and tool utilisation in dynamic digital geometry and algebra environments. The dynamic properties of these environments pose a nuanced predicament, as the outsourcing of translation between visual and algebraic representations raises concerns regarding students’ conceptual development and reasoning competency. To mitigate this issue, a prediction task is proposed, focusing on the dynamic behaviour of variable points in GeoGebra. I introduce a comprehensive framework adapting Toulmin’s argumentation model into the instrumental approach, emphasising processes of justification. This is complemented by the application of components of Vergnaud’s (1998) scheme concerning generative and epistemic ways to approach how students’ conceptual knowledge has played a part in these processes. Through a case study of a student pair solving a prediction task, I explore the links between instrumented justification, students’ mathematical reasoning competency and conceptual understanding, and how students’ use of GeoGebra tools is intertwined with their justification processes. The analysis reveals the intricate interplay between data production and interpretation, and it is grounded in inference drawn regarding students’ implied theorems about concepts, dynamic behaviour and progression in terms of techniques. The results indicate that the progression of technique is driven by the experience of the inefficiency of techniques and artefacts related to the goal of justification. Essentially, the framework links students’ reasoning competency to their use of tools and conceptual knowledge, as well as demonstrates that predicting dynamic behaviour can enhance knowledge-based justification.

Exploring mathematical technological knowledge of teachers with Geogebra application

Geogebra is a mathematical software that can be used in mathematics, especially geometry, calculus, and dynamic algebra, with various features, especially visualization. Geogebra helps students identify and improve their visual skills in understanding mathematical concepts. This study aims to explore the mathematical technology knowledge of teachers based on Geo-Gebra applications in mathematics learning by integrating content and technology. This research is qualitative research that uses a single-case design to explore the teacher's mathematical technology knowledge and involved teachers (N = 8) from the field of mathematics at universities in the Central Region of Indonesia. Three types of data were collected during one semester, weekly observations, progress report description; and the developed learning design. This research shows that there are two categories of teachers' mathematical technology knowledge characteristics. First, moderate teachers produce simple designs and are still influ-enced by numerical and algebraic approaches. Second, expert teachers can use analytical skills to design step-by-step learning, involving animation to explain concepts, and the resulting designs are quite complex and structured. These two findings are useful for building teacher competencies regarding using GeoGebra ap-plication-based technology in learning mathematics.

Compilation of dynamic sparse tensor algebra

Many applications, from social network graph analytics to control flow analysis, compute on sparse data that evolves over the course of program execution. Such data can be represented as dynamic sparse tensors and efficiently stored in formats (data layouts) that utilize pointer-based data structures like block linked lists, binary search trees, B-trees, and C-trees among others. These specialized formats support fast in-place modification and are thus better suited than traditional, array-based data structures like CSR for storing dynamic sparse tensors. However, different dynamic sparse tensor formats have distinct benefits and drawbacks, and performing different computations on tensors that are stored in different formats can require vastly dissimilar code that are not straightforward to correctly implement and optimize. This paper shows how a compiler can generate efficient code to compute tensor algebra operations on dynamic sparse tensors that may be stored in a wide range of disparate formats. We propose a language for precisely specifying recursive, pointer-based data structures, and we show how this language can express many different dynamic data structures, including all the ones named above as well as many more. We then describe how, given high-level specifications of such dynamic data structures, a compiler can emit code to efficiently access and compute on dynamic sparse tensors that are stored in the aforementioned data structures. We evaluate our technique and find it generates efficient dynamic sparse tensor algebra kernels that have performance comparable to, if not better than, state-of-the-art libraries and frameworks such as PAM, Aspen, STINGER, and Terrace. At the same time, our technique supports a wider range of tensor algebra operations---such as those that simultaneously compute with static and dynamic sparse tensors---than Aspen, STINGER, and Terrace, while also achieving significantly better performance than PAM for those same operations.

Visualizing a Cubic Linkage through the Use of CAS and DGS

Our goal is to discuss the different issues that arise when attempting to visualize a joints-and-bars cube through GeoGebra, a widespread program that combines dynamic geometry (DGS) and computer algebra systems (CAS). As is standard in the DGS framework, the performance of the graphic model (i.e., the positions of the other vertices when dragging a given one) must correspond to a mathematically rigorous, symbolic computation-driven output. This requirement poses both computational algebraic geometry and dynamic geometry programming challenges that will be described, together with the corresponding proposed solutions. Among these, we include a complete determination of the dimension of the cubic linkage from an algebraic perspective, and introduce advanced 3D visualizations of this structure by using the GeoGebra software.

The challenges of visualizing DMLA models

Abstract Multi-layer metamodeling is a metamodeling method that provides great flexibility for domain prototyping, while it also ensures rigid validation of domain rules. Visualization of multi-layer models would be essential in order to promote this new modeling technique in industrial setups; however, the traditional way of representing model entities is not sufficient here. Dynamic Multi-Layer Algebra is a multi-layer metamodeling approach that addressed the challenges of multi-layer modeling, but uses a text-based language to create models. This paper presents Dynamo, a visual notation for Dynamic Multi-Layer Algebra, which is also applicable to other multi-layer metamodeling approaches. The paper elaborates the graphical syntax and the design decisions.

Playground for multi-level modeling constructs

In recent years, multi-level modeling has become more and more popular. It is mainly due to the fact that multi-level modeling aims to reduce or even totally eliminate any accidental complexity inadvertently created as by-product in traditional model design. Moreover, besides reducing model complexity, multi-level modeling also improves on general comprehension of models. The key enablers of multi-level modeling are the concepts of clabjects and deep instantiation. The latter is often governed by the potency notion, of which many different interpretations and variations emerged over the years. However, there exist also some approaches that disregard the potency notion. Thus, multi-level modeling approaches tend to take advantage of different theoretical and practical backgrounds. In this paper, we propose a unifying framework, the Multi-Level Modeling Playground (MLMP), which is a validating modeling environment for multi-level modeling research. The MLMP environment is based on our multi-layer modeling framework (the Dynamic Multi-Layer Algebra), which provides useful mechanisms to validate different multi-level modeling constructs. Since beyond the structure also the well-formedness rules of the modeling constructs can be specified, our proposed MLMP environment delivers several practical benefits: i) well-formedness is always verified, ii) multi-level constructs can be experimented with independently of any concrete tool chains, and iii) relationships (i.e., correlations or exclusions) between different multi-level constructs can be easily investigated in practice. Also, the capability of the environment is demonstrated via complete examples inspired by state-of-the-art research literature.

A spatially dynamic network algebra

AbstractMap algebras are used for the manipulation of spatial data and form the basis for many types of spatial analyses and modeling efforts. The most basic form of a map algebra applies the same function (e.g., addition, subtraction) across the study area. To account for local variation of modeling parameters, we present a spatially dynamic map algebra to demonstrate the need for and utility of algebraic functions where the function is determined based on a specific and relative location. The need for such an algebra comes from the growth of complex models where the values of variables or parameters are not fixed across space. Locally derived parameters are not new, as shown by geographically weighted regression. This type of algebra is needed in cases of complex models, such as those found in flow networks, where network dynamics vary across space. This includes hydrologic processes, movement of people and goods, and the transmission of ideas. We test the approach on a case study of nitrogen flow in the Niantic River watershed, Connecticut. Findings show that the results from a spatially dynamic map algebra can differ from a fixed function. Fixed functions can result in model outputs that under‐ or overestimate by up to 50%.

Obstacle avoidance dynamic control of manipulator based on space operator algebra

The traditional manipulator obstacle avoidance dynamics control method ignores the acceleration constraints and speed constraints of the joint obstacle avoidance, and judges the equivalent moment of inertia inaccurately, which leads to large fluctuations in the driving torque of the manipulator motor, and high maximum driving torque and acceleration peaks. Therefore, a dynamic control method of manipulator avoiding obstacles based on space operator algebra is proposed. Using the space recursion method, according to the dynamic space operator algebra system, the manipulator dynamics model is obtained, PID parameters are adjusted, and the model is controlled by three-loop closed-loop PID control to obtain the joint torque under different obstacle avoidance states, and calculate the torque feedforward. The channel transfer function enables the moment to act on the joints of the manipulator to achieve the equivalent moment of inertia control target, to identify and finely control the moment parameters, and to plan the obstacle avoidance trajectory of the manipulator. A parallel manipulator was selected for comparison experiments. The results show that the design method reduces the motor torque ripple, the maximum driving torque, and the acceleration peak in the motion trajectory plane, so that the manipulator has better motion smoothness, reduces motor energy loss, and improves dynamic performance.

Tense and dynamic algebras related to GFA

The present paper is an attempt to suggest and scrutinize tense operators in the dynamic logic $textbf{B}$ which is regarded as a set of propositions about the general fuzzy automaton $ tilde{F} $, in which its underlying structure has been a bounded poset. Here, the operators $ T_{delta}, P_{delta}, H_{delta}$ and $ F_{delta} $ are proposed regardless of what propositional connectives the logic comprises. For this purpose, the axiomatization of universal quantifiers is applied as a starting point and these axioms are modified. In this study, firstly, we demonstrate that the operators can be identified as modal operators and the pairs $ (T_{delta},P_{delta}) $ are examined as the so-called dynamic pairs. In addition, constructions of these operators are attained in the corresponding algebra and in the following a transition frame is suggested. Besides, the problem of finding a transition frame is solved in the case when the tense operators are given. Specifically, this study shows that the tense algebra $ textbf{B} $ is representable in its Dedekind-MacNeille completion. Representation theorems for dynamic and tense algebra are explicated in details in the related given theorems.

Using Graph Software in Teaching the Subject of Functions: The Case of Azerbaijan

Bu çalışmada matematik derslerinde kullanılan Graph 4.4 yazılımının özellikleri tanıtılmakta ve bu yazılımın fonksiyonların grafiklerinin öğretiminde kullanılması sonucunda öğrencilerin fonksiyon grafiklerini oluşturma ve yorumlama becerilerindeki gelişim incelenmektedir. Çalışma deneysel bir çalışma olarak tasarlanarak yürütülmüştür. Örneklemi 46. (n=24 deney grubu, n=22 kontrol) 8. sınıf ortaokul öğrencileri oluşturmaktadır. Deney grubunda fonksiyonlar konusu Graph 4.4 yazılımı kullanılarak işlendi. Uygulamanın sonunda her iki gruba cevaplamaları için grafik oluşturmayı ve yorumlamayı gerektiren 2 soru soruldu. Öğrencilerin yazılı cevapları doğru, kısmen doğru ve yanlış şeklinde üç kategoride puanlanarak analiz edilmiştir. Yapılan karşılaştırmada deney grubunun ortalamalarının daha yüksek olduğu belirlenmiştir. Bulgular matematik derslerinde dinamik cebir yazılımlarının kullanılmasının öğrencilerin grafik becerilerini geliştirdiğini ortaya çıkarmıştır. Küçük bir örneklemden elde edilen bu sonuç genellenemese bile, bu çalışma matematik derslerinde Graph 4.4 yazılımı kullanımının temel öğretim araçlarından biri olabileceğini göstermektedir.

ЗАСТОСУВАННЯ ПРОГРАМИ GEOGEBRA У ПРОЦЕСІ ВИВЧЕННЯ РОЗДІЛІВ «МНОГОГРАННИКИ» ТА «ТІЛА ОБЕРТАННЯ»

The school has always been and remains a kind of model and a mirror that reflects the life and problems of the society. The idea of implementing innovative technologies in education provides for achieving the goal of high quality education, in other words, education that is competitive, able to provide each person with conditions for independent achievement of a goal, creative self-affirmation in various social spheres. Interactive technologies in learning help to understand better the learning material, to develop communication skills and activity, to create opportunities for independent research. One such environment for dynamic mathematics is GeoGebra which is based on the principles of dynamic geometry and computer algebra that allows to combine design, modeling, dynamic variation and experiment, build geometrically accurate images, print documents and publications on the Internet. This article considers the various functional capabilities of the environment which should be used in the construction of geometric bodies. In particular such as scans, sections and combinations of geometric bodies. The article considers the advantages of the interactive environment GeoGebra in comparison with other environments of dynamic mathematics, the directions of using the package of dynamic mathematics GeoGebra in geometry lessons. There are also various functionalities of this environment, which should be used in the construction of geometric bodies. In particular, such as polyhedra, their scans and sections, combinations of geometric bodies. GeoGebra program allows you to show the sequence of images of geometric shapes. The construction of cross sections of geometric figures is shown in specific examples.

Exploration of Dual Curves Using a Dynamic Geometry and Computer Algebra System

This submission presents a particular example of the use of the free dynamic mathematics software GeoGebra to determine the dual curve to the given curve and to inspect its properties. The example is aimed at students of mathematics teaching. The combination of dynamic geometry tools with computer algebra functions, namely the functions for the computation of Groebner bases for polynomial ideals and a function for eliminating variables from the system of polynomial equations, based also on the method of Groebner bases, allows them to apply naturally the knowledge they acquire during their study of mathematics.

Categorical Equivalence Between Orthomodular Dynamic Algebras and Complete Orthomodular Lattices

This paper provides a categorical equivalence between two types of quantum structures. One is a complete orthomodular lattice, which is used for reasoning about testable properties of a quantum system. The other is an orthomodular dynamic algebra, which is a quantale used for reasoning about quantum actions. The result extends to more restrictive lattices than orthomodular lattices, and includes Hilbert lattices of closed subspaces of a Hilbert space. These other lattice structures have connections to a wide range of different quantum structures; hence our equivalence establishes a categorical connection between quantales and a great variety of quantum structures.

Improving Mathematics Achievement and Attitude of the Grade 10 Students Using Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS)

It has become a fact that fluency and competency in utilizing the advancement of technology, specifically the computer and the internet is one way that could help in facilitating learning in mathematics. This study investigated the effects of Dynamic Geometry Software (DGS) and Computer Algebra Systems (CAS) in teaching Mathematics. This was conducted in Zamboanga del Sur National High School (ZSNHS) during the third grading period of the school year 2015-2016. The study compared the achievement and attitude towards Mathematics between those students taught with DGS and CAS and those students who were taught without DGS and CAS. A quasi-experimental pretest-posttest control group design was used with two groups which was matched according to their mathematical level. An achievement test was developed and administered to assess the students’ achievement in Mathematics. Aiken’s attitude scale was also used to assess the students’ attitude towards Mathematics. The data were treated with One-Way Analysis of Covariance (ANCOVA) and t-test at 0.05 level of significance. The study revealed a significant difference between the mathematics achievement between students taught with DGS and CAS and those who were taught without it. The integration of DGS and CAS in Mathematics lessons leads to better academic achievement result compared with that of the conventional method of teaching. Moreover, there was no significant difference on the attitude level of the students between the two groups.

Formalism for Static Aspects of Dynamic Metamodeling

The viability of any multi-level meta-modeling technology clings on the correct and exact definition of the underlying instantiation concept. Although multi-level instantiation has been well researched and conceptualized, current methodologies mostly limit their scopes to design-time modeling. Hence, state-of-the-art meta-modeling seriously neglects important practical needs of run-time modeling. In this paper, we present two incarnations of our Dynamic Multi-Level Algebra (DMLA), a modular, semantically correct multi-level meta-modeling formalism consisting of (i) an algebraic ASM foundation, (ii) a flexibly replaceable bootstrap mechanism for defining modelled entities, (iii) a corresponding set of logical validation formulae for establishing instantiation semantics. The expected features of DMLA clearly derive from practical modeling needs so that the resulting mechanism be applicable for both design-time and run-time modeling. The core components of the theory are explained both individually and by comparing their two versions of DMLA. Hence, the aim of this style of presentation is to showcase the balance struck between parsimoniousness and wanted modularity within our semantically correct, practical multi-level meta-modeling approach. A simplified networking model is also included to demonstrate the approach.

Concurrent Dynamic Algebra

We reconstruct Peleg’s concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an axiomatization of concurrent dynamic algebras from that basis. In this context, sequential composition is not associative. It interacts with parallel composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg’s axioms are shown to be derivable in these algebras, and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg’s axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.

Investigating Why Teachers Reported Continued Use and Sharing of an Educational Innovation After the Research Has Ended

We investigated prospects for reported sustainable adoption and sharing of an educational innovation through survey research including online questionnaires and telephone interviews. This investigation is part of the Scaling-Up SimCalc experimental program, which combines dynamic representational algebra software (SimCalc MathWorlds) with integrated curriculum and small professional development workshops focused on how to use the software and how to integrate this intervention into the larger year-long curriculum. Teachers who (1) perceived the usefulness of the SimCalc professional development as being consistent with personal aims and (2) perceived specific affordances of the software/curriculum to be valuable were more likely to report continued use of the innovation after the research had ended and that they had shared it with colleagues over time.

Dynamic Mereotopology. III. Whiteheadean Type of Integrated Point-Free Theories of Space and Time. I

Some Whiteheadean type point-free theories of space and time are presented. Here the term “point-free” means that neither space points nor time moments are assumed as primitives. The theory has an algebraic formulation, called dynamic contact algebra (DCA), which is a Boolean algebra whose elements symbolize dynamic regions changing in time, with several spatio-temporal relations between the regions: space contact, time contact, preceding, and some others. In the second part of the paper, a class of intended standard models of DCAs of topological kind will be introduced, which is a reason for calling DCAs dynamic mereotopologies. The main result of the paper is a kind of representation theorem saying that each DCA in a given class of DCAs is isomorphic to some DCA of standard type in the same class (see the third part of the paper). The first part contains a historical introduction and some facts on static mereotopology needed in succeeding parts—namely, the definitions of contact and precontact algebras, their models and representation theory.