Knowledge Of Algebra Research Articles

Don’t eliminate the negative: Influences of negative number magnitude knowledge on algebra performance and learning.

Numerical magnitude knowledge is related to current math abilities and predictive of future math learning. However, this relationship has previously been shown only for knowledge of positive magnitudes; the present study is the first, to our knowledge, to examine the connection between magnitude knowledge and math ability with negative integers. In the current study, 8th Grade Algebra I students (n = 154) were given assessments of their algebra and negative sign knowledge and 2 number line tasks, an all negative (−10,000 to 0) and a bidirectional scale (−1,000 to 1,000) as part of a pretest and posttest surrounding their unit on 2-step equation solving. Results show that students’ negative sign and numerical magnitude knowledge are associated with their current, or pretest, algebra performance. However, only negative magnitude knowledge from the bidirectional number line was predictive of improvement in students’ overall algebra knowledge. These findings illustrate differences between knowledge of positive and negative magnitudes and provide support for the integrated theory of numerical development. (PsycINFO Database Record (c) 2020 APA, all rights reserved)

ФОРМУВАННЯ ІНТЕГРАЦІЙНОЇ КОМПЕТЕНТНОСТІ СТУДЕНТІВ ПРИ ВИВЧЕННІ КУРСУ «НАРОДНІ РЕМЕСЛА»

The article considers some aspects of the problem of integration of students' knowledge and the implementation of interdisciplinary links in the modern educational process. It is established that these problems at different times were the subject of consideration of many scientists in whose works interdisciplinary, interdisciplinary connections were studied from different points of view: pedagogical, psychological, philosophical. In particular, the works of teachers, psychologists studied the integration of science, technology and education, a detailed analysis of the structure of the educational process, considered the problems of interdisciplinary links on the cognitive activity of students, classified types and types of interdisciplinary links by various criteria. Also, a significant amount of scientific work is devoted to identifying ways to eliminate duplication of educational material, methods of conducting classes using interdisciplinary links, the problems of integration of knowledge in the training of future teachers. This publication offers elements of the formation of integration competence in students when they study the course "Folk Crafts". The content and structure of the discipline "Folk Crafts", which is a discipline of professional training of future teachers of labor education and technology, describes the interdisciplinary links in the study of this discipline. There are specific examples of problems in solving which students must use previously acquired knowledge in other disciplines. In particular, when studying geometric wood carving, carving of complex ornaments there is a need to build high-quality drawings, which are associated with geometric constructions on the plane: dividing the circle into a number of equal parts, drawing parallel, perpendicular lines, dividing the segment in half, tangent to angle bisectors and so on. In performing such tasks, students use knowledge of algebra, descriptive geometry, computer graphics and other disciplines. It is proved that such an approach contributes to the formation of subject and integration competencies as the ability of students to holistically perceive the acquired knowledge, to use their professional potential in future pedagogical activities.

The Application of Linear Ordinary Differential Equations

In this article, we will explore the applications of linear ordinary differential equations (linear ODEs) in Physics and other branches of mathematics, and dig into the matrix method for solving linear ODEs. Although linear ODEs have a comparatively easy form, they are effective in solving certain physical and geometrical problems. We will begin by introducing fundamental knowledge in Linear Algebra and proving the existence and uniqueness of solution for ODEs. Then, we will concentrate on finding the solutions for ODEs and introducing the matrix method for solving linear ODEs. Eventually, we will apply the conclusions we’ve gathered from the previous parts into solving problems concerning Physics and differential curves. The matrix method is of great importance in doing higher dimensional computations, as it allows multiple variables to be calculated at the same time, thus reducing the complexity.

Quantum Technology for Economists

Research on quantum technology spans multiple disciplines: physics, computer science, engineering, and mathematics. The objective of this manuscript is to provide an accessible introduction to this emerging field for economists that is centered around quantum computing and quantum money. We proceed in three steps. First, we discuss basic concepts in quantum computing and quantum communication, assuming knowledge of linear algebra and statistics, but not of computer science or physics. This covers fundamental topics, such as qubits, superposition, entanglement, quantum circuits, oracles, and the no-cloning theorem. Second, we provide an overview of quantum money, an early invention of the quantum communication literature that has recently been partially implemented in an experimental setting. One form of quantum money offers the privacy and anonymity of physical cash, the option to transact without the involvement of a third party, and the efficiency and convenience of a debit card payment. Such features cannot be achieved in combination with any other form of money. Finally, we review all existing quantum speedups that have been identified for algorithms used to solve and estimate economic models. This includes function approximation, linear systems analysis, Monte Carlo simulation, matrix inversion, principal component analysis, linear regression, interpolation, numerical differentiation, and true random number generation. We also discuss the difficulty of achieving quantum speedups and comment on common misconceptions about what is achievable with quantum computing.

ASSESSING SENIOR SECONDARY STUDENTS’ KNOWLEDGE OF BASIC MATHEMATICAL SKILLS AND ALGEBRA

This study assessed the extent of the senior secondary students’ knowledge of basic mathematical skills and algebrainObio/Akpor Local Government Area (LGA) of Rivers State, Nigeria. Four research questions and four hypotheses guided the study. A minimum sample of 400 students obtained from a population of 18,087 using Taro Yamane formula was selected by simple random sampling. Basic Mathematical Skills Test (BMST) and Algebra Performance Test (APT) were used for data collection. The instruments were validated by experts in Mathematics Education while their reliabilities were established using test-retest method. The reliability coefficients of 0.74 and 0.78 were obtained for BMST and APT respectively by correlating the test and re-test scores of BMST and APT using the Pearson product moment correlation. Mean and standard deviation were used to answer the research questions while sample t-test and analysis of variance (ANOVA) were used to test the hypotheses at 0.05 significant level. Major result of the study showed that students’ knowledge of basic mathematical skills and algebra is significantly high. The result also revealed that the differences in the male and the female students’ knowledge of basic mathematical skills and algebra were not significant respectively. It was recommended among others that the current instructional process and practice in Mathematics at the basic and senior secondary education levels should be sustained and instead improved upon by the Mathematics teachers. Article visualizations:

중학생들의 분수 지식과 대수식 표현 사이의 관계 분석

The aim of this study is to explore various ways in which students’ algebraic representations relate to their fraction knowledge. From clinical interviews with seven middle school students, three students each of whom represented distinct ways of reasoning were chosen as the focus of the present paper. The in-depth analysis of the data suggests that students’ quantitative reasoning related to their iterative fraction schemes was involved in their ways of representing situations into algebraic equations. More specifically, students who did not construct iterative fraction schemes, only used whole numbers as multipliers for unknowns. Students who weakly constructed iterative fraction schemes, conceived whole numbers, unit fractions, and proper fractions as multipliers for unknowns. Lastly, students who constructed iterative fraction schemes, conceived whole numbers, unit fractions, proper fractions, and improper fractions as multipliers for unknowns, and could represent given situations in various forms of algebraic representation. We finalize our paper by discussing how the study complements past research on students’ knowledge of fractions and its relationship to algebraic knowledge and by providing implications for further research.

Assessing the didactic and mathematical knowledge of prospective mathematics teachers in Namibe, Angola

In this work the results of applying a questionnaire to assess didactic-mathematical knowledge of prospective secondary school teachers are analyzed. The sample consists of 31 students, prospective teachers, from the School of Teacher Training of Namibe in Angola. The questionnaire is made of 20 items that assess algebraic content and didactic content knowledge in secondary education. The quantitative analysis of the results show a low level of algebraic and didactic content knowledge and we conclude that it is necessary to improve the teacher training program to enable these teachers to develop the mathematics and didactic competences they need.

Química analítica básica

A maioria dos estudantes de Química sempre foi induzida a pensar e a trabalhar os equilíbrios iônicos em solução aquosa usando o tratamento algébrico tradicional, como nos exemplos mostrados recentemente para equilíbrios ácido-base [1-3]. Entretanto, a solução algébrica exata deste tipo de problema frequentemente requer a resolução de polinômios de mais altos graus. Por exemplo, o cálculo exato do pH de uma simples solução aquosa de um ácido monoprótico fraco requer a solução de uma equação cúbica e no caso de um ácido triprótico (ex.: ácido fosfórico), a solução de um polinômio de quinto grau [1,2]. Isto induz o uso de aproximações para simplificar o trabalho, o que sempre leva a resultados aproximados. Um outro procedimento proposto por Birk [4], que emprega uma formulação matricial baseada na extensão de reação, também pode ser utilizada para este propósito, mas é um método que exige conhecimentos mais aprofundados de química e de álgebra linear, razão pela qual não é apresentado em livros textos e também não será discutido neste trabalho. Uma abordagem alternativa, mais fácil, é a utilização de cálculos numéricos e a apresentação gráfica dos resultados, procedimento bastante facilitado pelo uso de planilhas eletrônicas. Isto permite uma visão geral do sistema ácido-base, reforça princípios importantes e permite o tratamento de uma ampla gama de sistemas químicos.

Pre-service mathematics teachers’ professional learning in a pedagogy course: Examining changes in beliefs and confidence in teaching algebra

This study examined changes in beliefs and confidence of 24 pre-service mathematics teachers after they participated in professional learning in an algebra pedagogy course, which addressed an under-researched area about their domain-specific beliefs in teaching algebra. A single-group pretest–posttest design was adopted. Quantitative data were collected to measure beliefs and confidence in teaching algebra, and qualitative data were used to help interpret and understand the results obtained from the quantitative research. Quantitative data analysis demonstrated significantly less change in beliefs than in confidence. Qualitative data analysis identified two themes accounting for changes in beliefs (learning various pedagogies and conceptually oriented teaching) and four themes accounting for changes in confidence (understanding students’ learning needs, acquisition of pedagogical knowledge and approaches, designing engaging mathematical activities and tasks, and discussion with teacher and students). The course helped shift teachers’ beliefs from an emphasis on performance to one on understanding and an increase in pedagogical content knowledge in algebra resulted in increased teachers’ confidence. This study highlights the key to changing beliefs and confidence and the necessity of progressing from a dualistic to a relativistic orientation towards mathematics teaching.

The role of rational number density knowledge in mathematical development

Many students still have not developed a robust understanding of rational number concepts at the end of primary school, despite several years of instruction on the topic. The present study aims to examine the patterns, predictors, and outcomes of the development of rational number knowledge in lower secondary school. Latent transition analysis revealed that rational number development from primary to lower secondary school (N = 362) appears to follow similar patterns as in younger students. In particular, a majority of students had poor knowledge of the density of the rational number set. Whole number magnitude knowledge appeared to be an important predictor of the development of rational number size knowledge, but not density knowledge. Finally, fraction density knowledge appeared to be related to concurrent algebra knowledge. Together these results point to an important role for density knowledge in mathematical development.

Enhancing the Conceptual, Procedural and Flexible Procedural Knowledge of Pre-Service Mathematics Teachers in Algebra

The study investigated the effects of Solve-Reflect-Pose Strategy (SRP) on pre-service mathematics teachers’ algebraic knowledge for teaching in Nigeria. A pre-test-post-test quasi experimental design was employed. Intact classes were used and in all, 182 pre-service mathematics teachers’ participated in the study (92 in the experimental group taught with the SRP and 90 in the control group taught using the Modified Conventional Method (MCM). One research instrument manipulated at three levels namely: Conceptual Knowledge Test (CKT), Procedural Knowledge Test (PKT) and Flexible Procedural Knowledge Test (FPKT), was used for the quantitative data and interview protocol for qualitative data. The two research questions formulated were analysed using descriptive statistics while independent sample t-test was used to analyse the two hypotheses. Results showed that there were statistically significant differences in the mean post-test achievement scores on conceptual knowledge test, procedural knowledge test and flexible procedural knowledge test between pre-service teachers exposed to the SRP and those exposed to the MCM, all in favour of the SRP group. Based on the results, SRP should be adopted as an instructional strategy and efforts should be made to integrate the philosophy of SRP into the pre-service teachers’ curriculum at the teacher-preparation institutions.

The Foundation of the Romany Students’ Algebra Knowledge through Text-Based Problems

The Foundation of the Romany Students’ Algebra Knowledge through Text-Based Problems

As ações epistémicas na construção do novo conhecimento matemático e no desenvolvimento do pensamento algébrico

This article discusses how 5th grade students construct new mathematical knowledge, from knowledge acquired in previous learning experiences. We adopted the AiC (Abstraction in Context) theoretical model and the RBC+C (Recognizing, Building-with, Constructing, Consolidation) theoretical and methodological model in order to understand how the abstraction process occurred during students’ completion of algebraic tasks. We focused on how students mobilized and reorganized mathematical knowledge and procedures to solve the tasks. The aim was to understand how the epistemic actions of the RBC+C model and the subcategories of analysis defined in the scope of this research emerged, how they related to each other, and what influence those relationships had in the construction of new mathematical knowledge. In this study, new mathematical construction is associated with the development of algebraic thinking. Following a qualitative approach, we concluded that the relationship between the epistemic actions Recognizing and Building-with was essential for the development of new mathematical knowledge. The manifestation of the Constructing and Consolidation epistemic actions revealed that students of this age level can successfully establish some algebraic relations, and they can understand and use symbolic language. These students, who still had no significant knowledge in algebra, evidenced mathematical gains with the resolution of tasks, which intended to promote algebraic thinking, thus reinforcing the approach of early algebra.

What algebraic knowledge may not be learned with CAS-a praxeological analysis of Faroese exam exercises

We are interested in the potentials and pitfalls of introducing computer algebra systems in lower secondary school, investigating the case of the Faroese Islands. In order to identify what algebraic knowledge is tested in the final written exam in mathematics after the ninth grade, and how this would change if computer algebra systems were allowed at that exam, we analyse all exam exercises from the past 10 years in terms of the techniques required to solve the exercises both with and without symbolic tools. The comparison suggests that fundamental algebraic structures may not be learned if students consistently use computer algebra systems for the tasks given in the exam

Scaling up innovative learning in mathematics: exploring the effect of different professional development approaches on teacher knowledge, beliefs, and instructional practice

Professional learning experiences (PLEs) provide teachers with opportunities to improve their understanding of mathematics content and teaching practices. However, PLEs are often conducted in person and in small groups—hence costly and localized. The purpose of the current study was to explore different ways for teachers to engage in PLEs and how these approaches might enable the field to scale up these efforts in a sustainable manner. We compared the impact of three PLE formats on the early algebra knowledge and teaching practices of elementary mathematics teachers: (1) a facilitated summer workshop, (2) a multimedia course completed on teachers’ own time, and (3) learning resources provided in the algebra curriculum unit that teachers used individually. Our findings suggest that all three formats can be mapped against a set of principles for quality professional learning. Analysis of pre- and post-treatment measures indicate that participating teachers’ knowledge of algebra content and best practices significantly increased, regardless of the PLE format with which they engaged. Interviews with a subset of the teachers from the three groups point to the key features of each of the formats that can be capitalized on by designers of PLEs.

Mathematical Flexibility: Aspects of a Continuum and the Role of Prior Knowledge

Psychologists and mathematics educators have long viewed flexibility as critical to students’ mathematical development. In this paper, we focused on the multidimensional nature of flexibility to better understand how preference, knowledge, and use of effective methods for solving algebra problems are related. In Study 1, we identified research-based aspects of flexibility with algebra and assessed students on them following a two-step equations unit. Results indicated that certain aspects of flexibility develop prior to others and that prior knowledge of algebra plays a significant role in that development. Study 2 confirmed and elaborated on these results using a larger sample size. Implications for theory and for supporting flexibility in classrooms are discussed.

The Difference Between a Probability and a Probability Density

Learning introductory quantum physics is challenging, in part due to the different paradigms in classical mechanics and quantum physics. Classical mechanics is deterministic in that the equations of motion and the initial conditions fully determine a particle’s trajectory. Quantum physics is an inherently probabilistic theory in that only probabilities for measurement outcomes can be determined. Prior to studying quantum physics, students will typically have little experience with probabilistic analyses of physical systems, and thus probability may be a conceptual hurdle for introductory quantum physics students. This article describes two interactive simulations developed as part of the QuVis Quantum Mechanics Visualization Project that aim to bridge the gap between classical mechanics and quantum physics using probabilistic analyses of classical systems. The simulations illustrate how a probability density can be obtained for two classical systems well known to students. The key learning goals of the simulations are to introduce students to probability densities and to help students distinguish between a probability and a probability density. The simulations build on previous work by Bao and Redish, who developed an activity that used pseudo-random video frames of a glider in harmonic motion to derive a classical probability density for this system, and a University of Washington quantum mechanics tutorial focusing on probability and probability density for a classical system. Interactive simulations allow students to easily carry out experiments and change variables that would be difficult to do with real equipment, and help students connect multiple representations by showing explicitly how they are linked. The simulations described here only require basic knowledge of algebra and classical mechanics. They run on touchscreen devices as well as desktop computers, and can be run in a standard web browser from the QuVis website or downloaded for offline use.

Type of mathematical proof: personal preference or adaptive teaching behavior?

In our study, 32 German and Swiss 8th/9th-grade classes of lower-secondary school worked with their teacher on the same proving problem. The sample belongs to the Swiss-German study “Quality of Instruction, Learning Behavior and Mathematical Understanding”. Our data analyses relate to the teachers’ approaches to generating a specific form of evidence with their classes when dealing with a particular elementary number-theory problem. We address the question of how the different strategies can be characterized as manifestations of a certain approach to proving and try to clarify in which way the observed approach can be interpreted as adaptive teaching behavior. For this purpose, we searched for possible correlations between three main strategies or types of generating a specific form of evidence (experimental, operative, formal-deductive approach) on the one hand and (a) the teachers’ beliefs and personal characteristics and (b) the students’ prior knowledge of algebra and mathematics in general on the other hand. As our analyses show, three main approaches to proving occurred but not in equal proportions: there is a predominance of the approach that entails the highest extent of formalization and abstraction. Nevertheless, an operative way of proving is widespread too. On the whole, the findings indicate that one particular approach to proving can be interpreted as a personal preference of a specific group of teachers and, at the same time, with respect to the students’ mathematical skills as a manifestation of adaptive teaching behavior.

Differenciálgeometria a mérnökképzésben

The goal of classical differential geometry is to study curves and surfaces in R3 uses the techniques of differential and integral calculus. This mathematical discipline is strongly based on knowledge of linear algebra and calculus, hence the elements of classical differential geometry appeared in advanced analysis courses in former university level education in engineering. The implementation of the Bologna process, the admission of the two-cycle system have led almost completely to the exclusion of differential geometry from bachelor's degree programs. This paper is devoted to a short overview of the above-mentioned topics and to give some applications.

A Case Study Examining Links between Fractional Knowledge and Linear Equation Writing of Seventh-Grade Students and Whether to Introduce Linear Equations in an Earlier Grade

As a part of the larger study, a case study of two seventh grade students, Peter and Willa, was conducted. To examine links between their fractional knowledge and algebraic reasoning, the students were interviewed twice, once for their fractional knowledge and once for their algebraic knowledge in writing linear equations that required explicit use of unknowns. Peter and Willa’s fractional knowledge influenced the linear equations they wrote to represent quantitative situations. In particular, the findings showed that a reversible iterative fraction scheme is important to understand reciprocal relationships between two quantities in writing a basic linear equation of the form ax=b. Also, considering fractions as multipliers on unknowns is important to write linear equations. Implications for the possibility of introducing linear equations in an earlier grade and how to support the introduction are suggested.