Mathematical Language Research Articles

Multi-factor model to predict surface chloride concentration of concrete based on fuzzy logic system

Surface chloride concentration (Cs) is one of the most critical parameter to determine the chloride concentration in concrete structure by Fick’s second law. The prevalent prediction models don’t consider the coupling effect of multiple factors and their comprehensive influence on Cs, which limits their calculation precision. Therefore, this paper uses the test data from published literatures to formulate fuzzy rules, and employs the membership function to transform literal descriptions of the service environments of concrete structures into mathematical language. The quantitative prediction model takes water-to-binder ratio, mineral admixtures type, temperature, exposed marine environment (or solution concentration of simulated seawater in the experiment) and duration of exposure as input variables, and takes Cs as output variable. Simultaneously, quantitative effects of these input variables on Cs are also discussed. It is shown that the goodness-of-fit of the prediction results obtained by the proposed model was greater than 0.9, which indicates the prediction accuracy of the proposed model is much higher than that of other prevalent models. In addition, the calculated results show that the duration of exposure is the most significant factor affecting Cs. Under different exposed environmental conditions, the time when the chloride concentration on surface of concrete tends to stable is different. This critical time in submerged zone or splash zone is longer than it in tidal zone.

The Formal Presentation Language of Mathematics and Communication Ethics

Mathematics employs a formal language where symbols, verbs, and nouns serve to express terms, concepts, and rules that concatenate to definitions, problem-solving procedures, and proofs. Taken together these constitute the expository language of mathematics found in journals, textbooks, and demonstrations. As a communication given to informing, there are epistemological and ethical considerations that deserve examination. For in keeping with the commitment to an aesthetic of concision promulgated by tradition, the formal presentation language and style of mathematics, while valuable in furthering the body of knowledge, provides only the conclusion of an inquiry, completely excluding the language of investigation that informed the many steps of decision making involved in the process, so that little if any insight into the creative process is made available. Here we explore this problem of communication in the context of mathematics presented to students early in their education as well as at university level. The argument is made that the language of investigation -- the heuristic actions instrumental for their formulation -- ought to accompany the language of formal demonstration so as to provide a communication that is in the best interests of all students and members of the mathematics profession.

Estimation of Transformed Continuous Assessment Scores of Rural and Urban Secondary Schools

This study examined the transformed continuous assessment CA scores of Osun State junior
 secondary schools with a view to providing information on the procedure for transforming CA scores
 for better comparability of scores among rural and urban secondary schools. The study adopted the
 ex-post-facto research design. The population for the study comprised all junior secondary school
 class III students who had CA scores during 2010/2011 session in both rural and urban secondary
 schools in Osun State. The sample consisted of 1200 students selected using multistage sampling
 technique. A proforma titled Continuous Assessment Scores Retrieval was used to collect CA scores
 of the selected students in English Language and Mathematics. Data collected were analyzed using
 percentages and Pearson’s Product moment correlation statistics. The results showed that the
 transformed CA scores of rural and urban secondary schools were above the mean score in
 Mathematics (62.70%) and English Language (66.0%). The result also showed that there was a
 significant positive relationship between the transformed CA scores and raw CA scores obtained by
 students in Mathematics and English Language (r=1.00; p

Three Ways to Enhance Tasks for Multilingual Learners

Enacting these considerations supports integrated thinking about how to attend to both mathematics content and language development.

ANALISIS LITERASI MATEMATIS KEISLAMAN MAHASISWA TADRIS MATEMATIKA DITINJAU DARI TIPE KEPRIBADIAN KEIRSEY

This study aims to determine the relationship between Islamic mathematical literacy level and Keirsey personality type (Artisan, Idealist, Guardian, and Rational) in prospective mathematics teacher students, Teacher Training Faculty of IAIN Kediri. In addition, this study also aims to describe the Islamic mathematical literacy ability of prospective mathematics teacher students at the Teacher Training Faculty of IAIN Kediri, in terms of Keirsey's personality type. This research design uses a mixed-methods approach. In the quantitative approach, the type of research is correlational with a total sample of 151 students. While in the qualitative approach, the respondents used were 4 students who represented each Keirsey personality type. Sampling technique in the form of purposive sampling. There is no relationship between the level of mathematical literacy with Keirsey's personality type in prospective mathematics teacher students at the Teacher Training Faculty of IAIN Kediri. The correlation between personality type and the level of Islamic mathematical literacy of prospective mathematics teachers at IAIN Kediri was obtained with a significance of 0.22986. This means that the strength of the relationship between the two variables is included in the weak category. In the process of formulating context mathematically, there is no difference between research subjects in all personality types. All subjects can formulate context, understand, and translate problems into mathematical language or appropriate representations. In the process of using concepts, facts, procedures, and reasoning, the findings show that subjects with rational personality types are able to determine strategically in the use of mathematical concepts and are successful compared to other personality types. In the process of interpreting, applying, and evaluating the results, the findings show that subjects with rational and idealistic personality types can check the reasonableness of the solution. Meanwhile, in the artisan and guardian types, the contrary happens.

Implementation of Predicting Student Performance Using Linear Regression Algorithm

Abstract: This data approach student achievement in secondary education of two Portuguese schools. The data attributes include student grades, demographic, social and school-related features) and it was collected by using school reports and questionnaires. Two datasets are provided regarding the performance in two distinct subjects: Mathematics (mat) and Portuguese language (por). In [Cortez and Silva, 2008], the two data sets were modeled under binary/five-level classification and regression tasks. Important note: the target attribute G3 has a strong correlation with attributes G2 and G1. This occurs because G3 is the final year grade (issued at the 3rd period), while G1 and G2 correspond to the 1st and 2nd period grades. It is more difficult to predict G3 without G2 and G1, but such prediction is much more useful (see paper source for more details). Keywords: Decision Tree Algorithm, Naïve Bayes Algorithm, Support Vector Machine

An Introductory Tutorial on Cohort State-Transition Models in R Using a Cost-Effectiveness Analysis Example.

Decision models can combine information from different sources to simulate the long-term consequences of alternative strategies in the presence of uncertainty. A cohort state-transition model (cSTM) is a decision model commonly used in medical decision making to simulate the transitions of a hypothetical cohort among various health states over time. This tutorial focuses on time-independent cSTM, in which transition probabilities among health states remain constant over time. We implement time-independent cSTM in R, an open-source mathematical and statistical programming language. We illustrate time-independent cSTMs using a previously published decision model, calculate costs and effectiveness outcomes, and conduct a cost-effectiveness analysis of multiple strategies, including a probabilistic sensitivity analysis. We provide open-source code in R to facilitate wider adoption. In a second, more advanced tutorial, we illustrate time-dependent cSTMs.

Analysis of Learning Obstacle to Numeracy Ability Based on Mathematical Resilience of Elementary Students

The difficulty felt by students in the learning process when students face a question and students feel difficulties in using their knowledge, we can say it to be a learning obstacle. Meanwhile, in learning, mathematical resilience can be said to be a positive attitude of students in the learning process. The relationship of mathematical resilience to mathematics learning will make students more interested and motivated in learning mathematics. This type of research is descriptive qualitative. This study studies and explores the characteristics of learning obstacles to numeracy ability based on students' mathematical resilience in geometry materials. The data collection techniques used are numeracy ability tests with geometry materials, student mathematical resilience questionnaires, and interviews. The subjects in this study were 24 students of class V B of SDN Mekarjaya 01. In this study, the data were analyzed by identifying the answers of 24 respondents to the numeracy ability test and mathematical resilience questionnaire that students had conducted. The result is that we can see some of the characteristics of learning barriers found in grade V students of SDN Mekarjaya 01, the characteristics of these learning barriers include difficulties and weaknesses in calculating, difficulties in transferring knowledge, and difficulties in understanding mathematical language. Looking at the characteristics of learning barriers, it is concluded that students have resilience in mathematics can affect students' attitudes in facing problems in learning mathematics.

ANALYSIS OF THE DIFFICULTIES OF CLASS X STUDENTS OF SMKN 4 PALANGKA RAYA IN SOLVING TRIGONOMETRY PROBLEMS

This research was motivated by the low scores obtained by students in the trigonometry material for the area of ​​triangles in class X GP of SMK Negeri 4 Palangka Raya. In the 2019/2020 school year, only 48% of the 34 students achieved the school's Minimum Completeness Criteria, which is 65. The purpose of this study is to describe the difficulties experienced by students in solving math problems on trigonometry material and the factors that students experience difficulties. This research is a qualitative descriptive study. The results showed that the difficulties experienced by students in solving trigonometry problems were: (a) Difficulties in counting were difficulties in performing number operations. The factor that causes students to be less careful. (b) Difficulty in transferring knowledge, namely determining the formula and sine value. The reason is that students do not understand the material being taught so they cannot distinguish the formula used in determining the area of ​​a triangle according to what is known in the problem. (c) Difficulty in understanding mathematical language, namely understanding mathematical symbols correctly. The reason is that students are less careful in writing down the numbers that match the existing mathematical symbols. (d) Difficulty in visual perception, namely making and understanding sketches or drawings based on existing problems. The reason is that students do not understand how to change what is asked in the question of the shape of a triangular image and vice versa.

Research on Mathematical Dyslexia of Middle School Students in China

Mathematics reading ability is an important factor affecting students' mathematics learning. Current studies have found that middle school students have mathematical dyslexia. Mathematical dyslexia often leads to students' difficulties in learning and even frustrates students' enthusiasm and initiative in learning mathematics. Therefore, many researchers have begun to pay attention to mathematical dyslexia problems in middle school students. This study used the literature analysis method to summarize and sort out the previous research on mathematical dyslexia of middle school students and found that: (1) Current domestic studies mainly focus on the following six aspects: Types of mathematical dyslexia; manifestations of mathematical dyslexia; analysis of causes of mathematical dyslexia; Analysis of the content of the material causing mathematical dyslexia; suggestions and strategies to overcome mathematical dyslexia. (2) The main types of mathematical dyslexia are comprehension and translation obstacles; Previous studies have discussed the manifestations of mathematical dyslexia from learning attitude, learning ability, and knowledge mastery. The formation of mathematical dyslexia is mainly influenced by the characteristics of mathematical language itself, students' factors, teachers' factors, and teaching modes. There are few studies on the influencing factors of mathematical dyslexia and the content of the materials causing mathematical dyslexia. Most studies put forward improvement suggestions for teachers and students. (3) Researchers mainly used self-written test papers and self-written interview outline to conduct investigations, and some researchers drew conclusions based on theoretical speculation or teaching experience. Therefore, based on existing research, future research needs to adopt scientific research tools, develop a localized research framework, expand the scope of research, increase the verification research of suggestions and strategies, and put forward practical and effective overcoming strategies, to make this research more comprehensive, systematic and in-depth.

Characteristics of de Bruijn’s early proof checker Automath

The ‘mathematical language’ Automath, conceived by N.G. de Bruijn in 1968, was the first theorem prover actually working and was used for checking many specimina of mathematical content. Its goals and syntactic ideas inspired Th. Coquand and G. Huet to develop the calculus of constructions, CC ([1]), which was one of the first widely used interactive theorem provers and forms the basis for the widely used Coq system ([2]). The original syntax of Automath ([3, 4]) is not easy to grasp. Yet, it is essentially based on a derivation system that is similar to the Calculus of Constructions (‘CC’). The relation between the Automath syntax and CC has not yet been sufficiently described, although there are many references in the type theory community to Automath. In this paper we focus on the backgrounds and on some uncommon aspects of the syntax of Automath. We expose the fundamental aspects of a ‘generic’ Automath system, encapsulating the most common versions of Automath. We present this generic Automath system in a modern syntactic frame. The obtained system makes use of λD, a direct extension of CC with definitions described in [5].

Exploring the Meaning-Making Process in Conceptual Mathematics: The Case of Linguistic Markers in Algebraic Equations

As research established, linguistic features are essential to understanding mathematical word problems, and these features are seen as a significant factor of students’ difficulty in word problems. Students are unable to translate the words into numbers or formula to work on the needed mathematical operations. This present study’s aim is two-fold. First, the analysis of the syntactic features of mathematical language is hoped to contribute to the understanding of mathematical sentence by making its lexis, syntax and semantics explicit. Second, by focusing on the linguistic markers in algebraic equation, the meaning￾making process in the mathematical language was described by determining how these linguistic markers prompted the mathematical conditions in the expressions and statements. Results show that algebraic sentences of equalities were characterized by inflectional phrase (InflP) projection, and ‘if…then’ syntax, which are considered in lexico-grammatical precision. Predication in algebraic sentence of equalities was signified essentially by the linking verb ‘is’ in definition, description and illustration frames.

Developing Pre-service Mathematics Teachers' Mental Rotation Skills through Dynamic Geometry Software-Supported Instruction

Spatial skills, including mental rotation, are necessary to understand, comprehend, explore and learn mathematics, but research in mathematics education points out insufficient spatial skills of pre-service mathematics teachers. In this study, the effectiveness of the instruction, designed by taking Cognitive Load Theory (CLT) as the reference frame and executed in the Dynamic Geometry Learning Environment (DGLE), was investigated. The study sample was four pre-service mathematics teachers determined by the purposeful sampling technique. Individual teaching experiment, executed in three steps (pre-evaluation— teaching sessions/instruction— post-evaluation), was used as the method. The collected quantitative and qualitative data were analyzed through content analysis. Findings showed that participants' mental rotation skills were basic or below, before the instruction. They had low mental imagery skills, inadequacies in mathematics language use, and misconceptions about rotation and its components. After the instruction, their mental rotation skills, compassing the mental rotation performance, mental imagery skills, mathematical language use, and concept schemes about rotation and its components, developed. It is recommended that cognitive loads be considered while designing instruction executed in a DGLE.

MODERN ASPECTS OF STUDYING RUSSIAN TERMINOLOGY BY INTERNATIONAL STUDENTS AT THE PREPARATORY DEPARTMENT IN A TECHNICAL UNIVERSITY

In this study, one of the aspects of teaching Russian as an international language, namely the teaching of mathematical terms in the preparatory department of a technical university in Russia, is considered. The relevance of this subject has arisen due to the growing number of international students in Russian universities and the need for them to master the Russian language as soon as possible to continue their studies at Russian universities. This work aims to describe the learning process of Russian scientific vocabulary and terminology of technical disciplines, especially the language of mathematics, by international students from the countries of the former Soviet Union studying at the Preparatory Department of Tomsk State University of Architecture and Civil Engineering. The complexity of mastering scientific vocabulary in Russian arises from the characteristic differences between Russian and Turkic languages. A linguistic experiment was conducted in which students heard and wrote down scientific terms, words, and phrases in Russian. Students were given a dictation test rich in scientific vocabulary. The results have been gathered, and numerous errors in spelling scientific terms have been pointed out. Studies in Russian universities take place mainly in Russian, with active use of scientific vocabulary, which is extremely difficult for people who are just beginning to learn Russian. After correcting the mistakes and explaining the correct spelling of the words, the dictation is repeated. The results obtained are compared. The features of deviations from the norm are identified, described, and structured, and options for working on the implementation of a program for the development of normative writing skills in international students are suggested. For example, it is quite challenging to determine the boundaries of a word, to identify differences between prefixes and prepositions, the normative spelling of vowels in the endings of nouns, adjectives, and verbs, and the spelling of fricative consonants.

DISCUSSION OF DEFINITIONS AND TERMINOLOGY PROBLEMS OF TECHNICAL CONTROL

The directions of development of control as a scientific branch are considered. It is shown that the variety of definitions of control goes beyond terminological inconveniences and hinders communication of specialists and their creative search, and this necessitates the development of a single approach and a single terminology in the theory of control. A detailed analysis of the existing views of experts on control, the results of their discussions and changes in approaches to terminology that existed in previous years and have developed at this time. The analysis of literature sources on the problems of definitions and terminology of technical control is given. Most of the existing dictionary definitions of control and technical control, as well as their modern interpretations are given and analyzed. There are two main forms of technical control: passive and active control. The terminology related to passive and active control is considered and analyzed, taking into account the methods after operational and technical control. It is shown that the universality and capacity of the term control contributed to its penetration into various spheres of human activity, including technology. Control is defined as a generic concept, and this interpretation for the needs of quantitative research has been translated into mathematical language in terms of set theory. An isomorphic mathematical model of the physical model of the object of control is given, in which the nature of the object of control is not specified, ie it can be technical, economic, biological, etc. The parameters of technical objects of control of the corresponding offered model are also given. It is shown that in terms of control capabilities the main interest is its methodological component, which expresses the essence of control, determines its specific varieties. The results of control of technical and other objects as the necessary information base of this or that production procedure are considered: rejection, sorting, protection. It is proved that when the control results are used to control the technological object, in this case it is not the control of the source object, but the controlled process and the process itself is now the object of control, but not the object of control. Thus, the more developed form of participation of control in management of technological object is its active control.

A study on pre-service mathematics teachers’ criteria of proof evaluation

Proof is foundational to mathematics, and constructing proofs and establishing their validity are both important mathematical studies. Determining the validity of a proof is a part of the process of proof evaluation. Proof evaluation contributes to students’ ability to construct and revise their own proofs. The purpose of this study was to determine the criteria pre-service mathematics teachers take into account in evaluating a proof. The study was carried out with 50 first-year university students enrolled in an elementary mathematics teacher education program. The data were collected through activities relating to proving. The results of this study revealed that, when evaluating a proof, the participants regarded use of appropriate definitions, axioms, or theorems in the steps of the proof as the most important criterion with reference to justification, while in terms of mathematical language; they regarded appropriate use of symbolic language as the most important criterion. However, they tended to ignore situations where non-symbolic language was used. To address this issue, it is recommended that mathematics-learning environments include the use of non-symbolic language, as well as symbolic representations, in the definition of mathematical concepts.

Developing Competences in Future Primary School Teachers under the Conditions of Teacher Education Standardization: A Theoretical Review

Mathematical competence plays an essential role among subject-specific competences of primary school teachers. One should be able to think mathematically, understand mathematical proofs, communicate in “the language of mathematics” and use appropriate codification resources, which makes the concept of mathematical competence rather vital. The concept of primary school teachers’ mathematical competence can be specified through its structure. In this regard, this article analyzes scientific approaches to determining the components of this phenomenon. The structure of future primary school teachers’ mathematical competence is viewed as a system of the following components: motivational-axiological, cognitive, activity-related, communicative, reflexive-creative. It must be noted that the process of developing the structure of this competence and its components, especially content- and activity-related ones, required the authors of the article to analyze regulatory documents and syllabi of the mathematics course of six higher education institutions (HEIs) in Ukraine. All the analyzed syllabi indicate which knowledge, skills and abilities future graduates should possess. The chosen HEIs distribute academic workload differently; at the same time, the content of educational material to be studied is almost identical. The elements of future primary school teachers’ mathematical competence are as follows: arithmetic, algebraic, geometric, that of identical transformations of mathematical expressions, logical

Measurement, Evaluation, and Model Construction of Mathematical Literacy Based on IoT and PISA

“Mathematics Curriculum Standard for Ordinary Senior High School” points out that mathematical modelling literacy is the literacy of abstracting real problems mathematically, expressing problems with mathematical language, and building models with mathematical methods to solve problems. It assesses the amount to which students who are about to finish compulsory schooling have the knowledge and aptitude to deal with future life issues, based on the notion of lifelong learning. The usage of a local integrated development environment requires a number of complex processes, including installation and setup, as well as the procurement of necessary hardware. In reality, in order to nurture students, the core literacy of mathematics is to completely execute quality education, further identify the training and development direction of talents, and infiltrate the core literacy material into junior high school mathematics classroom instruction. For such a large-scale international education measurement project, the research and development of test questions is very important. Therefore, detailed technical consideration has been carried out and test volumes have been designed that are relatively suitable for different countries. The test questions are closely related to life, while there are few questions related to real situational problems in the test, which are mathematically processed first. Due to the international background of evaluation, it is a great challenge for researchers to realize the localization of evaluation on the basis of adapting to the domestic situation. At the same time, there is no scientific and perfect mathematical modelling literacy evaluation system in China’s basic education research. How can middle school front-line teachers better evaluate students’ mathematical modelling literacy? As a result, it is critical to develop a scientific and quantitative mathematical literacy measurement model that will aid in the teaching and research of mathematical modelling by middle school front-line teachers, as well as contributing theoretically to the evaluation and research of mathematical modelling literacy in China. Therefore, based on mathematics literacy evaluation, this paper studies the hierarchy of the IoT and the measurement and evaluation model of mathematics literacy based on the IoT, as well as its enlightenment to the compilation of mathematics academic test questions in China.

Knowledgebra: An Algebraic Learning Framework for Knowledge Graph

Knowledge graph (KG) representation learning aims to encode entities and relations into dense continuous vector spaces such that knowledge contained in a dataset could be consistently represented. Dense embeddings trained from KG datasets benefit a variety of downstream tasks such as KG completion and link prediction. However, existing KG embedding methods fell short to provide a systematic solution for the global consistency of knowledge representation. We developed a mathematical language for KG based on an observation of their inherent algebraic structure, which we termed as Knowledgebra. By analyzing five distinct algebraic properties, we proved that the semigroup is the most reasonable algebraic structure for the relation embedding of a general knowledge graph. We implemented an instantiation model, SemE, using simple matrix semigroups, which exhibits state-of-the-art performance on standard datasets. Moreover, we proposed a regularization-based method to integrate chain-like logic rules derived from human knowledge into embedding training, which further demonstrates the power of the developed language. As far as we know, by applying abstract algebra in statistical learning, this work develops the first formal language for general knowledge graphs, and also sheds light on the problem of neural-symbolic integration from an algebraic perspective.

Behavioral SIR models with incidence-based social-distancing

Most available behavioral epidemiology models have linked the behavioral responses of individuals to infection prevalence. However, this is a crude approximation of reality because prevalence is typically an unobserved quantity. This work considers a general endemic SIR epidemiological model where behavioral responses are incidence-based i.e., the agents perceptions of risks are based on available information on infection incidence.The differences of this modeling approach with respect to the standard ‘prevalence-based’ formulations are discussed and its dynamical implications are investigated. Both current and delayed behavioral responses are considered. We show that depending on the form of the ‘memory’ (i.e., in mathematical language, of the information delaying kernel), the endemic equilibrium can either be globally stable or destabilized via Hopf bifurcations yielding to stable recurrent oscillations. These oscillations can have a very long inter-epidemic periods and a very wide amplitude. Finally, a numerical investigation of the interplay between these behavior-related oscillations and seasonality of the contact rate reveals a strong synergic effect yielding to a dramatic amplification of oscillations.