Use Of Mathematical Knowledge Research Articles

Multivariable calculus textbook representation of non-cartesian coordinates: a misalignment between multivariable calculus textbook content and upper-division physics application

Upper-division undergraduate physics coursework necessitates a firm grasp on and fluid use of mathematical knowledge, including an understanding of non-cartesian (specifically polar, spherical and cylindrical) coordinates and how to use them. A limited body of research into physics students’ thinking about coordinate systems suggests that even for upper-division students, understanding of coordinate system concepts is emergent. To more fully grasp upper-division physics students’ incoming understanding of non-cartesian coordinates, the prevalence of non-cartesian content in seven popular Calculus textbooks was studied. Using content analysis techniques, a coding scheme was developed to gain insight into the presentation of coordinate system content both quantitatively and qualitatively. An initial finding was that non-cartesian basis unit vectors were absent in all but one book. A deeper analysis of three of the calculus textbooks showed that cartesian coordinates comprise an overwhelming proportion of the textbooks’ content and that qualitatively the cartesian coordinate system is presented as the default coordinate system. Quantitative and qualitative results are presented with implications for how these results might impact physics teaching and research at the middle and upper-division.

ON THE USE OF THE METHOD OF EXPEDIENT PROBLEMS TO ACTIVATE STUDENTS' COGNITIVE ACTIVITY IN THE PROCESS OF TEACHING MATHEMATICS IN THE NEW UKRAINIAN SCHOOL

Formulation of the problem. The motivation of students to study a subject at school largely depends on the methodology used by the teacher in the teaching process, and his or her ability to convey new knowledge to students in an interesting and accessible form. This problem becomes especially relevant in the context of the New Ukrainian School project, which involves shifting the emphasis in the learning process from the teacher to the student and puts in the forefront the needs of the latter. For a child to be interested in learning and sufficiently motivated, a teacher should have an answer to the question "Why do we need to learn this?" The method of expedient problems gives one of the possible answers to this question, helps the teacher to motivate the student to study mathematics, and indicates directions for further use of mathematical knowledge. The importance of developing students' ability to apply mathematical knowledge in practice is further confirmed by the results of the International Comparative Studies PISA 2018 and PISA 2022, as well as statistical reports of the Ukrainian Center for Educational Quality Assessment in Mathematics. Materials and methods. To achieve our goal, we use a theoretical analysis of the methodological literature related to the chosen research topic. We also implement empirical methods, including conducting our own survey via Google Forms, observing the educational process in secondary schools, and analyzing students’ performance. Results. To study the feasibility and possibility of using the method of expedient problems in teaching mathematics in the NUS, we surveyed 290 mathematics teachers with different pedagogical experiences and qualifications from all regions of Ukraine. In particular, we assumed the effectiveness of using expedient problems in the process of teaching mathematics. The survey was conducted anonymously and online via the Google Forms platform. The results of the survey show that a significant number of respondents are familiar with the method of expedient problems and regularly or constantly use such tasks in the learning process. In addition, they are convinced of the need to include such tasks in existing math textbooks. The survey also shows at what stages of the lesson and when studying which content areas of the mathematics course, in the opinion of the respondents, it is most appropriate to use the method of expedient problems. Conclusions. To ensure the proper quality of students' acquisition of the necessary competencies envisaged by the New Ukrainian School project, the use of appropriate tasks at different stages of the lesson and in the study of different content areas of the school mathematics course is quite appropriate and possible. Although only half of the surveyed teachers currently use such tasks in their pedagogical activities regularly, most of the respondents consider them useful. In addition, teachers express a wish to provide methodological support for the effective implementation of appropriate tasks in the educational process, in particular, in the form of ready-made didactic materials and methodological recommendations for their creation.

Mathematical Knowledge and Skills in E-Business Transactions and in Curbing Electronic Fraud

The digitalization of commerce and finance has brought us to the era of e-business and e-transactions. E-business transactions refer to the use of electronics in the carrying out of commercial activities (sales and purchases). Unfortunately, the rise of e-business transactions has been plagued by the issue of electronic fraud, which has become prevalent now more than ever. Mathematics as a field is fundamental to life and present in virtually every aspect of man’s life and is an essential tool required and used in problem-solving. The business world is not exempt from the influence of mathematics, and considering this, this study explores the relevance of mathematical knowledge and skills in e-business transactions, as well as in solving the problem of electronic fraud. Mathematics was shown to be useful in fraud prevention tools and strategies like cryptography, data management, and optimization, risk assessment, fraud prevention algorithms, etc. We suggest that all the stakeholders in every field should teach the importance and use of mathematical knowledge and skills to all learners in the entire education system and efforts toward encouraging the development of mathematical knowledge and skills for individuals across all fields, especially in science, technology, business world and in form of training among others.

An Investigation of Mathematics Teachers' Conceptions of Mathematical Literacy Related to Participation in a Web-Based PISA Course

In today's mathematics education, teachers are expected to have an adequate understanding of mathematical literacy and to know how to incorporate mathematical literacy understanding into their instructional practices. Therefore, the purpose of this study was to investigate mathematics teachers' conceptions of mathematical literacy related to participation in a web-based learning environment about mathematical literacy and Programme for International Student Assessment (PISA). The case study method was used and the participants consisted of 20 mathematics teachers determined by the criterion sampling method. The data were collected by semi-structured interviews. In the data analysis process content analysis method was used. The results revealed that, the participants who participated the web-based PISA education conceptualized mathematical literacy in relation to the use of mathematical knowledge and skills to deal with real-life problems and associated supporting mathematical literacy with supporting development of problem solving skill. However, the others conceptualized mathematical literacy as having mathematical knowledge and skills to solve math problems and considered problem solving as a goal in supporting mathematical literacy.

Does further-mathematics performance predict student's achievement in Physics? A correlational diagnosis in senior secondary schools

The awareness and application of the knowledge of Further Mathematics transcends the boundary of Physics as a discipline. The need to explore means of improving students’ achievement in physics remains valid in the parlance of research. The use of mathematical knowledge in resolving physics is established in the literature. Over the years, students offer general mathematics and physics at the same time, yet, perform unsatisfactorily in physics. The application of further mathematics in physics and vice versa is evident in the curriculum. This study explored predictively, the correlation in the achievement of students in physics and further-mathematics. Ex post facto research of the co-relation type was employed in the comparison of pro forma of respondents in West African Senior School Certificate examination (WASSCE). Purposive sampling technique was employed in the selection of 103 respondents for the study. PPMC and MANOVA was employed in the analysis of data. The study observed a significant prediction in the achievement of students offering further mathematics and physics. However, gender and school type were not significant influencer the correlation between these two subjects. This study recommended that equal opportunities exist for male and female students, as well as, in public and private schools in these subject areas.

Characterizing prospective secondary teachers’ foundation and contingency knowledge for definitions of transformations

One promising approach for connecting undergraduate content coursework to secondary teaching is using teacher-created representations of practice. Using these representations effectively requires seeing teachers' use of mathematical knowledge in the work of teaching. We argue that the dimensions of Rowland's (2013) Knowledge Quartet, especially Foundation and Contingency, form a fruitful framework for this purpose. We contribute an analytic framework to characterize the quality of mathematical knowledge observed in the Foundation and Contingency dimensions, developed using a purposive sampling from over 300 representations. These representations all featured geometry teaching. We showcase the framework with examples of "high" and "developing" Foundation and Contingency.Then, we compare our coding along these dimensions with performance on a measure of mathematical knowledge for teaching geometry. Finally, we describe the potential for generalizing this framework to other domains, such as algebra and mathematical modeling.

Developing a Number Sense-Based Instructional Design to Eliminate Student Errors Based on Mathematical Misconceptions

The purpose of this research was to develop and evaluate a number sense-based curriculum aiming to eliminate student errors due to mathematical misconceptions. A mixed method research model as one of the qualitative research approaches was adopted in this study aiming to develop a number sense-based instructional design to eliminate student errors arisen from mathematical misconceptions in secondary school students. In preparation stage of the instructional design, 27 seventh grade students studying at a state school in Mersin province were determined as the study group. At the stage of developing and evaluating the instructional design, 24 fifth grade students studying at a state school in Mersin province were determined as the study group. In the process of developing the instructional design, "Needs Analysis Student Interview Form," "Instructional Design Teachers’ Board Interview Form," and "Expert Interview Form" were used. When all these results were evaluated together, it would be correct to say that number sense was an important prerequisite for the interpretation and use of mathematical knowledge, and it would be efficient on eliminating errors arisen from the mathematical misconceptions when it was integrated into an effective instructional design.It was possible to suggest that further researches should be conducted for determining the misconceptions of the students related to different mathematics sub-topics.

IMPLEMENTATION OF A SYSTEM OF PROFESSIONALLY ORIENTED TEACHING OF MATHEMATICS AND EXPERIMENTAL VERIFICATION OF ITS EFFECTIVENESS

The relevance of this article is due to the fact that the changed socio-economic conditions in the country, competition in the labor market in a new way raise the question of the need to significantly improve the quality of training of modern specialists in the field of engineering and technology. Increasing automation of modern production, its intensification, the need to improve the quality of manufactured products require a graduate of a technical university to be able to solve the problems of optimizing technological processes and modes on a solid scientific basis, calculate the parameters of their stability, the probability of rejection, and also implement a number of complex tasks of tool design, machines and mechanisms. Such production questions and creative tasks can be successfully solved only on the basis of the wide practical use of mathematical knowledge. As specially conducted studies and existing practice show, many graduates of technical specialties of universities, unfortunately, do not know how to creatively apply mathematical knowledge to solve new engineering and applied problems.

LEARNING SITUATIONS AND ICT SKILLS IN MATHEMATICS IN THE FIRST YEAR OF THE SECONDARY SCHOOL FOR ECONOMICS

This research presents one of many possibilities of how to check and repeat knowledge at the end of the school year and at the same time provide an answer to questions often asked by students who are disinterested in learning mathematics: "Where will we need it?" or "Why do we learn this?" To provide them with an answer and motivate them at the same time, this research focused on actively encouraging students to find the answers themselves and thus find the importance of learning mathematics. With a changed way of repeating and consolidating the material at the end of the school year, the aim of this research is to reduce the fear of mathematics and increase the motivation of students in the following year. Students, divided into groups, choose the topic or examples of the use of mathematical knowledge of the first year of secondary technical and professional education in everyday life, and thus shape the learning situation (LS). The goal of preparing the LS is for students to make sense of the subject matter with examples from everyday life. For the selected LS, they prepare a short story with tasks that they solve by calculation, prepare presentations and also, present the LS to the classmates. During the formation of the LS and the preparation of presentations, students are active (active learning methods: task search, knowledge of the subject matter, interviews, problem solving, use of mathematical applications, teamwork, problem solving…) and cooperate with each other. They are constantly developing more 21st century competencies (self-regulation, collaboration, problem solving) and digital competencies. With the formation of the LS, the world of mathematical knowledge gets a little closer to students, they lose their fear of mathematics and become more motivated. Keywords: learning situations, collaborative work, active learning methods, mathematics in everyday life, deviation from the established

Research On The Correlation Between Mathematics And Physics Of The Senior High School Students

Materialist dialectics holds that the material world is generally connected, and the realization of the calculation, verification and other goals involved in physics requires the use of mathematical knowledge as the research tool and language, and the use of mathematical methods and mathematical ideas for reasoning and analysis, which shows that there is a close relationship between mathematics and Physics. Analyzing the influence of mathematics on physics from the perspective of data is helpful for teachers to improve the teaching process of physics, promote students' ability of mathematics application and physics learning, and improve the quality of high school teaching. In this paper, through statistical analysis and questionnaire research. The scores of mathematics and physics in a school were collected and analyzed by SPSS22.0 software. It is found that mathematics achievement has a significant influence on physics achievement. Then through the questionnaire survey of whether mathematics has an impact on physics, we get a positive answer from the perspective of students. The final conclusions are as follows: (1) there is a positive correlation between mathematics achievement and physics achievement; (2) more than half of the students can use mathematical methods to solve problems when learning physics; (3) most students hope that teachers can teach them how to use mathematical methods in physics learning. Through these conclusions, we can know that mathematics achievement has a very important impact on physics achievement. It reminds us that we should pay attention to students' mathematics education.

Использование математических знаний в практической деятельности будущих геодезистов

Использование математических знаний в практической деятельности будущих геодезистов

A percepção dos alunos do Ensino Médio de uma escola pública a respeito da aplicabilidade dos conhecimentos matemáticos nas profissões almejadas

É comum os professores de Matemática ouvirem dos alunos questionamentos sobre o motivo de estarem aprendendo e desenvolvendo determinado conteúdo, além de perguntarem sobre quando na vida irão utilizá-lo. As respostas a estes questionamentos, típicos da adolescência, os quais devem ser recebidos e respondidos com a devida atenção pelo educador, são definitivos para tomadas de decisão, pelo aluno, a respeito da escolha de sua futura profissão. Esta é uma abordagem considerada importante no conjunto de documentos que norteiam a educação a nível nacional. No entanto, em alguns momentos, o educador não dispõe do tempo necessário e aprofundamento suficiente do conteúdo para adentrar na grande diversidade de aplicações dos conhecimentos matemáticos em diferentes profissões. Nesse sentido, o presente trabalho visou: 1) realizar um levantamento a respeito dos possíveis cursos de graduação que um grupo de alunos do Ensino Médio de uma escola pública pretendia realizar, e reconhecer sua percepção sobre a utilização dos conhecimentos matemáticos no cotidiano das profissões dos cursos citados; 2) intervir com estes alunos, apresentando ao grupo as aplicações matemáticas específicas para os cursos pretendidos e 3) permitir que a turma avaliasse a intervenção no sentido de a mesma ter acrescentado conhecimentos novos sobre as profissões. Foi possível observar que, no grupo em questão, mais da metade da turma pretendia cursar uma graduação e, para a grande maioria, a intervenção foi positiva, trazendo novas informações a respeito da utilização dos conhecimentos matemáticos em diferentes profissões.

The Wittgensteinian Perspective and Ethnomathematics: An Analysis of Language Games and the Rules Governing their Uses in Certain Work Activities

This work, adopting a Wittgensteinian perspective, aims to analyze the language games that involve mathematical concepts present in certain work activities, as well as the rules of use of such concepts, comparing them with the existing rules in School Mathematics. The studies analyzed used Ethnomathematics as a research method to understand the generation, organization and dissemination of mathematical knowledge in certain professions, in particular carpenters, fishermen, farmers and artisans. In considering the language games present in the mathematical practices existing in these professions, it is possible to show that in some games rules are presented that have strong family similarities to the games that make up the School Mathematics when they need a written mathematics, however, the expression of language games orally assume different meanings for terms present in both grammars. In addition, it presents examples of the use of mathematical knowledge without the formalism and rigor present in the language games of School Mathematics. It is a way of doing mathematics generated by another grammar that uses other rules, in this case estimation and rounding, a type of rationality distinct from that which constitutes School Mathematics, but which is effective in that form of use.

Функциональная математическая грамотность: что под этим понимать и как формировать

The article reveals the characteristic of the concept of “functional mathematical literacy” and its essential component - the student’s readiness to apply the acquired knowledge and methods of action in practical situations. Based on the results of international studies, the problem of forming skills and skills in the use of mathematical knowledge in practice, the representation of universal learning activities in them and the features of the methodology for their development are discussed. Questions of maintenance of continuity of the maintenance of functional mathematical literacy between initial and basic school are considered.

Middle School Teachers’ Use of Mathematics to Make Sense of Student Solutions to Proportional Reasoning Problems

Mathematics teacher education aims to improve teachers’ use of mathematical knowledge to support teaching and learning, an aspect of pedagogical content knowledge (PCK). In this study, we interviewed teachers to understand how they used mathematics to make sense of student solutions to proportional reasoning problems. The larger purpose was to find accurate ways of categorizing teachers’ ability to do this vital aspect of teaching and thereby to inform assessment, teacher education, and professional development. We conjectured that teachers’ PCK for proportional reasoning could be reliably described in terms of attention to quantitative meanings in story problem contexts and in terms of understanding naive forms of proportional reasoning. Instead, our findings reveal that individual teachers used a variety of means to make sense of (1) cognitively similar student solutions to different tasks and (2) mathematically related steps of a student solution within a single task. These findings illustrate the complexity of PCK for the topic of proportional reasoning and suggest the limits of what can be inferred about teacher knowledge from teachers’ evaluations of student solutions. We discuss implications for teacher education and assessment.

What Does it Take to Develop Assessments of Mathematical Knowledge for Teaching?: Unpacking the Mathematical Work of Teaching

IntroductionBroad consensus exists about the importance of teachers' mathematical knowledge (Adler & Venkat, 2014; Ball, Lubienski, & Mewborn, 2001; Baumert et al. 2010; Dohrmann, Kaiser, & Blomeke, 2014). Studies have linked mathematical knowledge for teaching to the quality of teachers' mathematics instruction (Eisenhart, Borko, Underhill, Brown, Jones, & Agard, 1993; Hill et al., 2008). Mathematical knowledge for teaching has also been linked to student achievement gains in the elementary grades (Hill, Rowan, and Ball, 2005). However, many U.S. teachers lack the deep, nuanced, and specialized mathematical knowledge needed for responsible teaching. This finding is persistent over time, grade levels, and both national and international contexts (e.g., Hill & Ball, 2004; Ma, 1999; Tatto et al., 2008). Simultaneously, the Common Core State Standards for Mathematics (National Governors Association Center for Best Practices, 2010), which have been adopted by 47 states and territories, have set out rigorous standards for K-12 mathematics learning that consequently increase the mathematical demands of teaching. To ensure that teachers are well-positioned to help students meet these more challenging learning goals, it is now -- more than ever -- critically important to focus on developing their mathematical knowledge for teaching.To investigate and grow teachers' mathematical knowledge for teaching (MKT) it is crucial to be able to measure and track the development and uses of MKT. Most work to develop measures of MKT has typically been done by groups of experts in relevant fields, such as mathematicians, mathematics educators, and teachers who have worked together to draft and revise assessment items (Hill, Schilling, & Ball, 2004). The early work in this area was focused on developing and refining the construct of mathematical knowledge for teaching while simultaneously and iteratively developing measures of the construct. The process of item development was therefore often time consuming and challenging. Because of the promising results of these earlier efforts, there is now a broad need for assessments of MKT. Building tests at scale means, however, that people who are not deeply immersed in research on MKT will have to be able to write valid MKT items. This will require detailed supports to help test developers understand the nuances of the construct of MKT and ways to assess it. In this paper, we present a framework that identifies the different ways that teachers make use of mathematical knowledge as they go about the work of teaching and provides support to assessment developers. We begin by articulating and specifying what we mean by mathematical knowledge for teaching and its relationship with the mathematical work of teaching that arises in everyday practice.Theoretical FramingConceptualizing Mathematical Knowledge for TeachingBuilding supports for assessment development of mathematical knowledge for teaching (MKT) rests on a clear conceptualization of what we mean by MKT, how MKT is drawn upon in practice, and the specific areas of the work of teaching that we seek to assess. Scholars of mathematical knowledge have examined such knowledge in action as it is used in the practice of teaching (Ball & Bass, 2002; Rowland, Huckstep, & Thwaites, 2005).Our work builds on a particular practice-based perspective on mathematical knowledge for teaching that begins with the premise that, to understand the specific knowledge of mathematics needed in teaching, one must first examine the mathematical work that arises in the context of teachers' instruction in classrooms, a form of job analysis (Ball & Bass, 2002). Through detailed analysis of instruction in a 3rd grade classroom over an entire year, Ball and her colleagues identified mathematical problems that teachers regularly encounter and must solve while teaching, such as interpreting and evaluating students' non-standard mathematical ideas (Ball & Bass, 2002, p. …

Erratum to: A validation study of the use of mathematical knowledge for teaching measures in Ireland

Erratum to: A validation study of the use of mathematical knowledge for teaching measures in Ireland

A validation study of the use of mathematical knowledge for teaching measures in Ireland

Researchers who study mathematical knowledge for teaching (MKT) are interested in how teachers deploy their mathematical knowledge in the classroom to enhance instruction and student learning. However, little data exists on how teachers’ scores on the US-developed measures relate to classroom instruction in other countries. This article documents a validation study of Irish teachers’ scores on measures of MKT that were adapted for use in Ireland. A validity argument is made identifying elemental, structural and ecological assumptions. The argument is evaluated using qualitative and quantitative data to analyse inferences related to the three assumptions. The data confirmed the elemental assumption but confirming the structural and ecological assumptions was more difficult. Only a weak association was found between teachers’ MKT scores and the mathematical quality of instruction. Possible reasons for this are outlined and challenges in validating the use of measures are identified.

Assessing elemental validity: the transfer and use of mathematical knowledge for teaching measures in Ghana

This paper reports on a validation study that investigates the utility of US-developed mathematical knowledge for teaching measures in Ghana. Using three teachers as cases this study examines the relationship between teachers’ mathematical knowledge for teaching responses and their reasoning about their responses. Preliminary findings indicate that although the measures could be used in Ghana with adaptation to determine teachers with high mathematical knowledge, the validity of the findings are influenced by other issues such as the cultural incongruence of the item contexts.

Resolução de problemas matemáticos de alunos da Educação de Jovens e Adultos

Solution of mathematical problems in the education of young and adult pupils. Results of INAF (2004) indicate that only 23% of the Brazilian young and adult population is capable of adopting and controlling a strategy of resolution of a problem to involve the execution of a series of operations. This work aims to investigate the causes of the difficulties faced by adults - enrolled in the Education Program for Youngsters and Adults (EJA), at a public school in the State of Parana - the difficulties analyzed here concern the use of mathematical knowledge for the solution of problems. We are verifying the possible relationship of difficulties with the lack of knowledge on concepts and mathematical algorithms or whether, they originate in the incomprehension of the mother language used in the situation problems. We could verify that the possible causes are: teaching modality never received appropriate attention from competent authorities. The evaluation system and promotion as well as the low ages allowed by LDB 9394/96 for access to the program, have been signaling towards a growing identification between supplemental teaching and mechanisms of acceleration of the regular teaching, of educational proof.