Mathematical Notation Research Articles

WRITTEN MATHEMATICAL COMMUNICATION SKILLS ON OPEN-ENDED PROBLEMS: IS IT DIFFERENT BASED ON THE LEVEL OF MATHEMATICS ABILITY?

This study aims to describe students' written mathematical communication skills in open-ended problems based on their mathematical abilities. This research was conducted at grade VII-A of SMPN 1 Pamekasan. This research uses descriptive qualitative research. The instruments used were the Mathematical Ability Test (MAT), Written Mathematical Communication Skills Test (WMCST) on open-ended problems, and Interview Guidelines. The results showed that subjects with high and moderate mathematical ability were able to meet three indicators of written mathematical communication skills, namely being able to express mathematical ideas through writing, demonstrate and convey visually; able to understand, interpret and evaluate mathematical ideas in writing, as well as in other visual forms; and able to use mathematical notations in the structure to write down ideas and relationships with situation models in writing. Meanwhile, subjects with low mathematics ability can fulfill 2 out of 3 indicators of written mathematical communication skills, namely being able to express mathematical ideas through writing, demonstrate and convey visually; unable to understand, interpret and evaluate mathematical ideas in writing or other visual forms; and able to use mathematical notations in the structure to write ideas and relationships with situation models.

Mathematical Communication Profiles on Learning Styles in Solving High School Level Mathematics Problems

This study aims to see the mathematics communication of students in terms of learning styles in solving high school mathematics problems. The focus of this research is: students' mathematics communication in terms of visual learning styles, mathematics communication in terms of auditory learning styles, mathematics communication in terms of kinesthetic learning styles. The subjects of this study were students at SMA Negeri 1 Gowa who were grouped based on their learning styles. The research instrument is the researcher himself, because the researcher himself plans what to ask the research subjects. Data collection techniques are carried out by providing learning style questionnaires, carrying out written tests to see mathematical communication skills, and interview guidelines. This study used technical triangulation with data analysis techniques, namely data presentation, data reduction, and conclusions. The results of this study indicate that the level of mathematics communication can be seen from the learning styles of students. Students with visual and auditory learning styles are able to solve mathematical problems completely and in detail, are able to use mathematical ideas, are able to use mathematical notations and symbols to determine answer conclusions. Students who have a kinesthetic learning style are also able to solve mathematical problems in their own style and way, namely a shorter way, being able to use mathematical ideas, symbols and mathematical notations in solving problems, but they do not provide conclusions from the answers that are done

Iconicity in mathematical notation: Commutativity and symmetry

Mathematical notation includes a vast array of signs. Most mathematical signs appear to be symbolic, in the sense that their meaning is arbitrarily related to their visual appearance. We explored the hypothesis that mathematical signs with iconic aspects – those which visually resemble in some way the concepts they represent – offer a cognitive advantage over those which are purely symbolic. An early formulation of this hypothesis was made by Christine Ladd in 1883 who suggested that symmetrical signs should be used to convey commutative relations, because they visually resemble the mathematical concept they represent. Two controlled experiments provide the first empirical test of, and evidence for, Ladd’s hypothesis. In Experiment 1 we find that participants are more likely to attribute commutativity to operations denoted by symmetric signs. In Experiment 2 we further show that using symmetric signs as notation for commutative operations can increase mathematical performance.

Analysis and Design of Modules to Generate Students' Mathematical Communication Skills through Problem Based Learning Models

The ability of mathematical communication is the ability of students to express mathematical ideas both oral and written in the form of language, symbols, and mathematical notations that are easier for students to understand mathematical problems. Problem based learning is a learning model that can be used to improve students' mathematical communication. In implementing the Problem Based Learning (PBL) model, teachers need appropriate modules. This study analyzes and designs teaching materials in the form of modules, based on the Problem Based Learning approach, allowing improving mathematical communication skills. The subjects of this study were students of class XI SMK Muhammadiyah 1 Sentolo, Kulonprogo. This type of research is research and development with the ADDIE development model. The collection instruments used were interview guides, classroom observations, and tests. Data analysis uses data reduction, data presentation, and drawing conclusions. This research produces an module needs analysis and module design according to the needs analysis. Modules are designed according to Problem Based Learning (PBL). Modules are designed to improve students' mathematical communication skills. The results of the study can be used as a guide when the teacher develops a learning module for a problem-based learning model. The novelty of this research is the integration of mathematical communication skills into module design.

Developing Covariational Reasoning Among Students Using Contexts of Formulas

Multiple studies have been conducted to assess students’ ability to apply covariational reasoning to sketching graphs in physics. This study is supported by research on developing students’ skills in sketching functions in mathematics. It attempts to evaluate physics students’ ability to apply these skills to identify critical algebraic attributes of physics formulas for their potential to be sketched. Rather than seeking formulas’ physical interpretation, this study is posited to challenge students’ skills to merge their mathematical knowledge within physics structures. A group of thirty ([Formula: see text]) first-year college-level physics students were provided with two physically identical equations that described the object’s position. However, one equation was expressed in functional mathematical notation, whereas the other in a standard formula notation. The students were asked to classify the symbols in each formula as variables or parameters and determine these formulas’ potential to be graphed in respective coordinates. The analysis revealed that 93% of these students considered function notation as possessing sketchable potential against 13% who envisioned such potential in the standard formula notation. Further investigations demystified students’ confusion about the classification of the symbols used in the formula notation. These results opened up a gate for discussing the effects of algebraic notations in physics on activating students’ covariational skills gained in mathematics courses. Suggestions for improving physics instructions stemming from this study are discussed.

An Algorithmic Approach to Solve Continuum Hypothesis

The continuum hypothesis has been unsolved for hundreds of years. In other words, can I answer it completely? By refuting the culturally responsible continuum [1], one can link the problem to the mathematical continuum, and it is possible to disproof the continuum hypothesis [2] . To go ahead a step, one may extend our mathematical system (by employing a more powerful set theory) and solve the continuum problem by three conditional cases. This event is sim-ilar to the status cases in the discriminant of solving a quadratic equation. Hence, my proposed al-gorithmic flowchart can best settle and depict the problem. From the above, one can further con-clude that when people extend mathematics (like set theory — ZFC) into new systems (such as Force Axioms), experts can solve important mathematical problems (CH). Indeed, there are differ-ent types of such mathematical systems, similar to ancient mathematical notation. Hence, different cultures have different ways of representation, which is similar to a Chinese saying: “different vil-lages have different laws.” However, the primary purpose of mathematical notation was initially to remember and communicate. This event indicates that the basic purpose of developing any new mathematical system is to help solve a natural phenomenon in our universe.

Analysis of series RL and RC circuits with time-invariant source using truncated M, Atangana beta and conformable derivatives

Complex processes of the physical world require novel and sophisticated mathematical notions to get deep insights. In this research analysis, two standard mathematical models for the series RL and RC circuits having time-invariant sources taken from the discipline of electrical engineering have been investigated with the help of differential operators known with the name of truncated M- derivative, Atangana beta-derivative, and the conformable derivative operators. The exact solutions for these two models have been found in terms of the transcendental exponential function of time under the truncated M-derivative, Atangana beta-derivative, and the conformable derivative operators. The numerical simulations carried out via MATLAB ”9.4.0.813654 (R2018a)” have been interpreted to explore new behavior for solutions of the models not possible to obtain through standard classical calculus wherein one is restricted to have integer-order derivatives unlike the differential operators used in the present research study. The models under consideration for the three differential operators have been investigated with varying parameters’ values including the differential orders α and β whereupon the classical case is resumed for α=β=1 and τ=0.

On the diversity and frequency of code related to mathematical formulas in real-world Java projects

In this paper, the term formula code refers to fragments of source code that implement a mathematical formula. We present empirical studies that analyze the diversity and frequency of formula code in open-source-software projects. In an exploratory study, we investigated what kinds of formulas are implemented in real-world Java projects and derived syntactical patterns and constraints. We refined these patterns for sum and product formulas to automatically detect formula code in software archives and to reconstruct the implemented formula in mathematical notation. In a quantitative study of a large sample of engineered Java projects on GitHub we analyzed the frequency of formula code and estimated that one of 700 lines of code in this sample implements a sum or product formula. For a sample of scientific-computing projects, we found that one of 100 lines of code implements a sum or product formula. To assess the need for tool support, we investigated the helpfulness of comments for program understanding in a sample of formula-code fragments and performed an online survey. Our findings provide first insights into the characteristics of formula code, that can motivate further studies on the role of formula code in software projects and the design of formula-related tools.

The kinetics of the processes of extracting the Cu(II) and Fe(III) ions from aqueous solutions by the biosorbents based on pea processing waste

Effective purification of natural and wastewater from heavy metals is a relevant environmental and national-economic problem. It can be solved by using plant-waste-derived biosorbents in water treatment technologies. They are formed in large quantities by agricultural and food enterprises. Taking into consideration data on the peculiarities of mechanical and thermal effects on the components of plant biomass, the techniques have been substantiated to obtain biosorbents from pea processing waste. It has been shown that the dehydration of the waste, its carbonization, and the crushing of char can produce biosorbents with different sorption properties. The nature of influence exerted by the process parameters of the Cu(II) and Fe(III) ions biosorption from model aqueous solutions on a change in the concentration of the solution, the value, and adsorption uptake has been established. In particular, the effect of the process duration, the type and initial content of metal in the solution, dosage, and a biosorbent production technique was studied. It has been shown that 38 to 98 % of heavy metals can be removed from solutions at their initial concentration between 2 and 20 mg/dm3 and a biosorbent dosage between 1 and 30 g/dm3. It was found that char is more efficient at removing heavy metals. It was also determined that the biosorbents made from pea processing waste are better at removing the Cu(II) ions from aqueous solutions than the Fe(III) ions. The generalization of the results of kinetic research is represented in the form of a multifactor regression equation. The equation makes it possible to calculate a change in the concentration of heavy metal in the solution depending on its initial concentration, the duration of the biosorption process, and the dosage of a biosorbent. For the mathematical notation of the experimental adsorption isotherms, values of the coefficients in a Langmuir equation have been determined. The derived equations could make it possible to optimize the technological parameters of the process

Devising a procedure for calculating the designed strength of a kingpin­type load­carrying system for an articulated tractor container carrier

The development of a methodology for calculating the designed strength of an unconventional kingpin-type load-carrying system for the articulated tractor container carrier on pneumatic wheels is an important and relevant task due to the unique layout of a given specialized vehicle. The purpose of this work is to devise a procedure for calculating the designed strength of a clip load-carrying system for the articulated tractor container carrier on pneumatic wheels, aimed at building an improved structure with rational metal consumption. The results of the theoretical and experimental studies have determined the most common estimation cases. They include the movement over the technological roads' irregularities along the horizontal section of the road ‒ an estimation case of the semi-trailer frame girders. climbing a high curb at an angle was also considered as an estimation case for the semi-trailer frame girders and cross member. Starting a tractor container carrier forward with an insurmountable obstacle in front of the semi-trailer's wheels was analyzed as an estimation case for the cross member of the frame and the suspension attachment units in the semi-trailer of a tractor container carrier. The result of the reported theoretical and experimental research is the determined load that acts on one cradle structure of the load-carrying system; it forms the torsional rigidity of the frame structure. Underlying the derived mathematical model for each case are the torsional moments and inertia moments that act on the structural elements of the girder in different planes. The mathematical models have been built on the basis of strength conditions for each estimation case and thus form a mathematical formula for determining the wall thickness of the frame girder. The mathematical notation demonstrates the degree of advantage of climbing a high curb at an angle over other estimation cases for the semitrailer frame member and girders. The estimation parameters underlying the mathematical model are meant to prevent the breaking and torsion of the semi-trailer frame in a tractor container carrier

Особливості лексико-семантичної характеристики товаронімії відапелятивного походження

Tovaronym is mainly an artificial onym related to material sphere of human activity, which characterizes proper name of a product, i.e. object designed for trade or exchange. When analyzing proper names of industrial products that function in modern Ukrainian, we registered 825 names based on appellative. They include: names of apparatuses related to sound and image transmission (594; 72%); proper names of household appliances that are used to facilitate household chores (131; 15,88%); names of means of transportation (100; 12,12%). A significant amount of proper names of industrial products is interrelated with approximation to the appellative, which can be explained by the name’s functioning peculiarity, namely such onym does not need an additional explanation; it is generally known, easy to remember, which is considered to be a positive feature in terms of advertisement. We assume that appellative-based machinonyms motivate such groups of lexeme: anthropocentric names (225; 27,27 %), zoo-lexemes (183; 22,18 %), floro-lexemes (142; 17,21 %), names of natural phenomena and processes (88, 10,66 %), art notions (45; 5,45 %), astronautic lexemes (31; 3,76 %), space notions (27; 3,27 %), everyday-character notions (24; 2,91 %), technical notions (23; 2,79 %), temporal lexemes (15; 1,82 %), mathematical notions (11; 1,33 %), sports terminology (11; 1,33 %). Appellative-based tovaronyms are mostly motivated by lexemes that have positive semantics, render pleasant associations for consumers, and have symbolic meaning for ethnos, indicate on the place of production of a certain denotative. Regarding separate pejorative names, they are mostly widespread in slang forms, their sounding or spelling being the reason of their appearance.

The Accessibility of Mathematical Notation on the Web and Beyond

This paper serves two purposes. First, it offers an overview of the role of the Mathematical Markup Language (MathML) in representing mathematical notation on the Web, and its significance for accessibility. To orient the discussion, hypotheses are advanced regarding users’ needs in connection with the accessibility of mathematical notation. Second, current developments in the evolution of MathML are reviewed, noting their consequences for accessibility, and commenting on prospects for future improvement in the concrete experiences of users of assistive technologies. Recommendations are advanced for further research and development activities, emphasizing the cognitive aspects of user interface design.

Mathematical Analysis in Static Equilibrium of Economics: as Support to Microeconomics Course

Microeconomics studies the economic behavior of individual decision-makers. To be able to make sound decisions in economics, the economist has designed mathematical models based on the differentiation of simple functions. Economics and mathematics are directly related as changes in quantities and variables affect the relationship and the direction of the consumer behavior to become better –off or worse-off. The main objectives of this study are microeconomic theories that may simplify or easily understandable through the mathematical calculation to illustrate the examples and calculate the equilibrium positions numerically. The study also aims to provide mathematical tools that manage to determine all the information necessary for decision making from a broad vision. The relationship between quantity and price may be better explained through mathematical notations. The supply and demand curve is designed to find the equilibrium point and it is better explained through mathematical equations. To explain, the methodology is used for a simple mathematical function of price with the help of a simple linear and nonlinear model of static analysis of the market equilibrium of Qd=Qs. This study concludes that it will greatly help college students, professionals, entrepreneurs, and in general anyone presents solved exercise and proposes problems, for the student to solve cases create graphs, table, and interpret and analyze results in the market and thus fixing the knowledge.

Robust design from systems physics

A crucial challenge in engineering modern, integrated systems is to produce robust designs. However, quantifying the robustness of a design is less straightforward than quantifying the robustness of products. For products, in particular engineering materials, intuitive, plain language terms of strong versus weak and brittle versus ductile take on precise, quantitative meaning in terms of stress–strain relationships. Here, we show that a “systems physics” framing of integrated system design produces stress–strain relationships in design space. From these stress–strain relationships, we find that both the mathematical and intuitive notions of strong versus weak and brittle versus directly characterize the robustness of designs. We use this to show that the relative robustness of designs against changes in problem objectives has a simple graphical representation. This graphical representation, and its underlying stress–strain foundation, provide new metrics that can be applied to classes of designs to assess robustness from feature- to system-level.

Identifying individuals associated with organized criminal networks: A social network analysis

The past couple of decades has witnessed an unprecedented rise in organized crime. This rise, coupled with increasing intricacies of organized crime, poses significant and evolving challenges for international law enforcement authorities. With the passage of time, authorities such as Interpol, have discovered that modern criminal organizations have adopted a networked structure, a shift away from the traditional hierarchical structure. Fluid networked structures make it difficult for the authorities to apprehend individuals associated with each network, and consequently, to disrupt the operations of the network. Various research groups have analyzed prison/courtroom transcripts, to create an organizational structure of known individuals, or a social network of individuals, suspected to be a part of a major drug/terrorist organization. These social networks have been studied fairly extensively from network centrality perspectives, to understand the role of suspect individuals in the network. Additionally, with drug and terror offenses increasing globally, the list of suspect individuals has also been growing over the past decade. As it takes significant amount of technical and human resources to monitor a suspect, an increasing list entails higher resource requirements on the part of the authorities, and monitoring all the suspects soon becomes an impossible task. In this paper, we primarily focus on two types of networks – (i) Drug Trafficking Organizations, and (ii) Terrorist Organizations, and present a methodology for the surveillance of individuals associated with these networks. Our methodology is based on the mathematical notion of Identifying Codes, which ensures reduction in resources on the part of law enforcement authorities, without compromising the ability to uniquely identify a suspect, when they become “active” in drug/terror related activities. Furthermore, we show that our approach requires far lesser resources when compared to strategies adopting standard network centrality measures for the unique identification of individuals. In other words, we show that the strategy of monitoring individuals in such networks, by utilizing centrality measures is wasteful, on the part of the authorities. Finally, we evaluate the efficacy of our approach on real world datasets.

Euclid’s Common Notions and the Theory of Equivalence

The “common notions” prefacing the Elements of Euclid are a very peculiar set of axioms, and their authenticity, as well as their actual role in the demonstrations, have been object of debate. In the first part of this essay, I offer a survey of the evidence for the authenticity of the common notions, and conclude that only three of them are likely to have been in place at the times of Euclid, whereas others were added in Late Antiquity. In the second part of the essay, I consider the meaning and uses of the common notions in Greek mathematics, and argue that they were originally conceived in order to axiomatize a theory of equivalence in geometry. I also claim that two interpolated common notions responded to different epistemic needs and regulated diagrammatic inferences.

Students’ cognitive style in mathematical thinking process

The process of mathematical thinking in learning mathematics is very important. One internal factor that affects it is cognitive style. The purpose of this study is to describe how each student’s cognitive style performs mathematical thinking processes. The types of cognitive styles studied are field dependent (FD) and field independent (FI). This research is a literature study by examining journals related to cognitive style and students’ mathematical thinking abilities. The results of journals review show that students with FI cognitive style analyze and orient analytically to process information and are able to process information using mathematical notation and their own language, and are able to model problems using image visualization. Students with FD cognitive styles adopt a global orientation to understand and process information. In solving mathematical problems, students with FI cognitive style were better than students with FD cognitive style.

Views of an Octogenarian on a Path to a Nonagenarian: The Field of General Symmetries and its Applications

I am on the path from my eightieth to my ninetieth birthday, or from an octogenarian to a nonagenarian. It is certainly encouraging to have opportunities on this path to talk about personal recollections and my views on my work and activities towards the development of the field—our field—of understanding and applying models in physical and related sciences based on the infinite set of mathematical notions of symmetries in their most general sense and with all their facets. I plan to talk today about the first part of my path in my native city Berlin, followed by the University of Marburg and jointly at the International Centre for Theoretical Physics in Trieste.

Characterising Brouwer’s continuity by bar recursion on moduli of continuity

We identify bar recursion on moduli of continuity as a fundamental notion of constructive mathematics. We show that continuous functions from the Baire space $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to the natural numbers $${\mathbb {N}}$$ which have moduli of continuity with bar recursors are exactly those functions induced by Brouwer operations. The connection between Brouwer operations and bar induction allows us to formulate several continuity principles on the Baire space stated in terms of bar recursion on continuous moduli which naturally characterise some variants of bar induction. These principles state that a certain kind of continuous function from $${{\mathbb {N}}}^{{\mathbb {N}}}$$ to $${\mathbb {N}}$$ admits a modulus of continuity with a bar recursor. The results for the Baire space are recast in the setting of the Cantor space $${{\{ 0,1 \}}}^{{\mathbb {N}}}$$ using the notion of fan recursor, a bar recursor for binary trees. This yields characterisations of uniformly continuous functions on the Cantor space and fan theorem in terms of fan recursors. The results for the Cantor space hold over the extensional version of intuitionistic arithmetic in all finite types ( $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ ), and those for the Baire space hold over $${\mathsf {E}}\text {-}\mathsf {HA}^\omega $$ extended with the type for Brouwer operations. Our work places Spector’s bar recursion in a proper context of Brouwer’s intuitionistic mathematics and clarifies the connection between bar recursion and bar induction.

Uma conexão geométrica: imagens mentais, visualização e registros matemáticos

No presente artigo, apresenta-se o resultado de uma pesquisa qualitativa que teve o objetivo de investigar se existe conexão entre imagens mentais, visualização e notações matemáticas convencionais. O trabalho foi realizado com um grupo de estudos e pesquisas em geometria, constituído por futuros professores e professores em ação continuada, no primeiro semestre do ano de 2019, no sul do país. Na investigação, foram realizadas duas atividades que partiram da imaginação individual de cada participante do grupo para criar visualização como construto mental. Posteriormente, foram produzidas representações semióticas, tanto em linguagem natural quanto em linguagem simbólica geométrica. A coleta de dados foi realizada por meio de registros escritos dos participantes, e a respectiva análise foi feita à luz da Teoria dos Registros de Representação Semiótica, Criatividade, Visualização e Imaginação, levando em consideração aportes de diferentes estudiosos sobre esses temas. Isso possibilitou aos investigadores concluírem que é possível uma conexão que permita levantar hipóteses e, após, comprová-las por meio do uso de recursos materiais adequados.