Algebraic Symbols Research Articles

How to Support The Algebraic Thinking Skills of Elementary School Students Using The Generative Multi-Representation Learning Model Modification Schema-Based Instruction?

Elementary school teachers still do not fully utilize effective learning models, especially in teaching algebraic concepts. Lack of understanding of algebraic concepts has caused students to have difficulty mastering algebra, including calculation, representation, and mathematical modeling, as well as recognizing algebraic symbols and variables. The purpose of this study is to answer the research question (RQ) How does the Generative Multi Representation Learning Model Modified Schema Based Instruction (MGMRM-SBI) affect the algebraic thinking skills of elementary school students? In this study, the experimental method was used with a posttest-only control group design, where the sample was selected using Cluster Random Sampling with a total sample of 128 students. The research instrument used was an algebraic thinking ability test in the form of description test questions, and data analysis was carried out using a t-test with the help of SPSS statistical software. The results of the t-test showed a significant difference between the algebraic thinking skills of students in the experimental and control classes, with significance values (2 sides) = 0.000 < sig. = 0.05. Based on the results of the study, it can be concluded that the application of the Generative Multi Representation Learning model Modified Schema Based Instruction has a significant effect on the algebraic thinking ability of elementary school students, and is more effective than the expository model.

Errors and Misconceptions in Linear Inequalities Among Senior High Students in Mfantseman Municipality

This study aimed to investigate the different errors and misconceptions made by students when dealing with linear inequalities. The goal was to uncover the nature and causes of these errors and misconceptions among students in Senior High Schools within the Mfantseman Municipality in the Central Region of Ghana. The research employed an explanatory sequential mixed methods design and was conducted in two public Senior High Schools selected from the Municipality. A total of 180 Senior High School students and teachers participated in the study, including 10 teachers. The sample was selected using a random sampling technique, which yielded 170 students from the two chosen public Senior High Schools. Data collection encompassed results from students' tests on linear inequalities, interviews with students, and questionnaires given to mathematics teachers. The collected data was coded and analyzed using descriptive statistics. The study's findings revealed common errors made by students, such as multiplying/dividing by a negative number, substituting inequality symbols with "equal to" symbols, performing operations on only one side or different numbers on the two sides of a compound inequality, as well as errors in algebraic operations, simplification, and arithmetic. Misconceptions observed included confusion between equality and inequality, misconceptions when dividing or multiplying through an inequality by a negative number, and struggles with compound inequalities. Students' difficulties arose from an inadequate understanding of basic inequality concepts, overgeneralization, limited mastery of inequality rules, and insufficient exposure to compound inequalities. Translating word problems into algebraic symbols posed a significant challenge. The study also highlighted that mathematics teachers were aware of the errors made by students. Consequently, teachers made efforts to address these errors during linear inequality classes. The findings suggest that teachers not only need assistance in identifying errors but also in understanding how errors can emerge during the learning process. One of the recommendations is to enhance teacher education by emphasizing diverse teacher-student interactions that thoroughly consider students' mathematical ideas. This approach aims to support teachers in effectively utilizing students' experiences in the learning process.

Algebra discourses in mathematics and physics textbooks: comparing the use of algebraic symbols

It is agreed that algebra has an important role in physics, particularly through handling symbols. A lot of previous research has focused on how mathematics is used in physics from perspectives where mathematics is taken for granted, and not addressing potential differences of mathematics in the physics classroom and in the mathematics classroom. Studies addressing differences between both subjects have been based on researchers’ own experiences of mathematics in both subjects. Thus, more focused empirical research is needed. The purpose of this study is to clarify similarities and differences between mathematics and physics concerning the use of algebraic symbols. Analyses were based on comparisons between upper secondary textbooks in mathematics and in physics from a discourse perspective. Statistical methods were used to decide if there were any significant differences between the subjects. Results showed an overlap in the algebra discourse in both subjects, but also several differences concerning core aspects of algebra. For example, a higher number of different algebraic symbols in equations in physics than in mathematics, and algebraic symbols are more seldom referred to by words in mathematics than in physics. This can make it difficult for students to identify similarities in the algebraic discourses in both subjects.

How Students Generate Patterns in Learning Algebra? A Focus on Functional Thinking in Secondary School Students

<p style="text-align: justify;">This research aims to describe secondary school students' functional thinking in generating patterns in learning algebra, particularly in solving mathematical word problems. In addressing this aim, a phenomenological approach was conducted to investigate the meaning of functional relationships provided by students. The data were collected from 39 ninth graders (13-14 years old) through a written test about generating patterns in linear functions. The following steps were conducting interviews with ten representative students to get detailed information about their answers to the written test. All students' responses were then analyzed using the thematic analysis software ATLAS.ti. The findings illustrate that students employed two types of approaches in solving the problem: recursive patterns and correspondence. Students favored the recursive patterns approach in identifying the pattern. They provided arithmetic computation by counting term-to-term but could not represent generalities with algebraic symbols. Meanwhile, students evidenced for correspondence managed to observe the relation between two variables and create the symbolic representation to express the generality. The study concludes that these differences exist due to their focus on identifying patterns: the recursive pattern students tend to see the changes in one variable, whereas the correspondence ones relate to the corresponding pair of variables.</p>

Practical algebra and hydrostatics: the legacy of Thomas Harriot

ABSTRACT This article provides a glimpse of the unpublished work of Harriot’s colleague Walter Warner. The main topic is the use of hydrostatic weighing and algebraic symbolism to determine the constitution of a mixture of metals. This was an important matter in the context of contemporary debates about currency and exchange. Recent discoveries have revealed the extent of Warner’s writings in this field, and the involvement of other notable seventeenth-century figures, such as Charles Thynne, Henry Robinson, John Dury, Samuel Hartlib, Justinian Isham, and David Ramsay. Their work illustrates many facets of scientific enquiry at that time, such as the contemporary concepts of weight and density, and the introduction of the new symbolic algebra in Harriot’s style.

Investigating the Mathematics Teachers’ Pedagogical Content Knowledge in Teaching Algebraic Symbols

Secondary level mathematics teachers’ pedagogical content knowledge (PCK) in teaching algebra is identified as a crucial factor of mathematics teachers’ knowledge that influences on the accuracy of the students’ understanding. The aim of this study is to examine the extent of the mathematics teachers’ PCK in teaching basic algebraic concepts, algebraic symbols. Necessity of learning algebra for advanced conceptualization is asserted by the global literature. Researchers found that the mathematics teachers’ PCK in teaching algebra is not satisfactory. Equally, students’ learning difficulties and their struggle in learning algebra can be identified as a global issue. The sample (N=281) mathematics teachers in the Galle education zone were administered with a questionnaire. The mean value was 6.81 and the standard deviation was 2.74. The study found that the mathematics teachers’ PCK in teaching algebraic symbols is very poor in Sri Lanka. Moreover, the mathematics teachers use inadequate algebraic thinking in the teaching practicum that can restrict the students’ creativity. Students’ understanding in algebraic symbols at the acquisition stage of learning algebra should essentially be developed through well designed professional development programs.

Examining the influences of the mathematics teachers’ pedagogical content knowledge on learning algebra in grade six

This mixed method study findings indicate that the pedagogical content knowledge (PCK) of the teachers is an influencing factor of learning algebra in grade six in Sri Lankan schools. Reliable and validated two questionnaires followed by 24 of eight mathematics teachers’ classroom observations and post- lesson interviews were administered for collecting data in the explanatory sequential mixed methods research design. Field notes, lesson video recordings and the interview data were triangulated in terms of the reliability of the qualitative data while the two questionnaires were piloted and designed by Delphi method before collecting quantitative data. The Cronbach’s alpha of the questionnaire-2 was 7.59. SPSS statistical software was used in the first phase while the reflexive thematic analysis method was used in the second phase of the data analysis process. The study results of the first phase indicate that the mathematics teachers’ PCK in teaching algebra is at a medium level and not satisfactory. Teachers’ PCK in algebra was found to be multi-functional, indicating no positive relationship with academic qualification, professional qualification and professional experiences. Further analysis showed that there was no correlation between the PCK and the PCK knowledge dimensions, declarative, procedural and conditional knowledge. Qualitative study results revealed that the mathematics teachers’ PCK is low, in building algebraic concepts in the students’ mind, addressing the students’ algebraic misconceptions and understanding of students’ algebraic thinking. Finally, learning algebra in algebraic symbols, variables and algebraic expressions were observed and found to be less effective. Therefore, content knowledge and pedagogical knowledge based professional teacher development programs are recommended for enhancing the students’ learning algebra in the classrooms.

Interpreting literal symbols in algebra under the effects of the natural number bias

ABSTRACT In this study, we investigated how secondary students interpret algebraic expressions that contain literal symbols to stand for variables. We hypothesized that the natural number bias (i.e., the tendency to over-rely on knowledge and experiences based on natural numbers) would affect students to think that the literal symbols stand for natural numbers only rather than for any rational or real number (integrity effect); and that the arithmetical values of the algebraic expressions are of the same sign as the expressions’ phenomenal sign (phenomenal sign effect). The participants (138 8th and 9th graders) were asked to evaluate 48 statements about numbers that can or cannot be assigned to six algebraic expressions that contained literal symbols (e.g., a, -d-4). The results supported the main hypothesis of the study with respect to the integrity as well as the phenomenal sign effect and also indicated that the former was stronger than the latter. Additionally, the most salient characteristics of the form of each expression, such as its sign, appeared to affect students’ responses regarding the arithmetical values they may represent. Theoretical and educational implications are discussed.

Mendel G.J.: life and scientific activity (on the 200th birth anniversary)

The main life and activity moments of the founder of modern genetics, the world-famous Austrian biologist and monk Gregor Mendel are discussed in the article. He developed the scientific principles of hybrids and their offspring description and research, introduced the concept of alternative traits – dominant and recessive, first applied the algebraic symbol system and trait designations, and formulated the basic inheritance trait laws in a number of generations too. His scientific report Experiments on plant hybrids (1865, Brun), as well as the article based on his results of research published the following year, remained underestimated and not fully understood by biologists at that time. The recognition Mendel’s outstanding works took place 35 years later, after the almost simultaneous article publications by three European scientists – H. de Vries, C. Correns and E. Tschermak, who independently confirmed his data with their own experiments. G. Mendel’s success in discovering the first laws of genetics is due to the fact that he chose research objects with alternative morphological features, involved a sufficient number of individuals and used mathematical methods to generalize the experiment results.

From process to object in teachers’ introductory algebra discourse

Generalization is one of the core aspects of algebraic thinking, and research literature points to the importance of objectification when generalizing mathematical discourses. To increase the field’s understanding of the objectification process in various algebra discourses, our study analyses Grade 6 teachers’ use of words, symbols, routines and narratives when they introduce various algebraic concepts. Our results show that there exist instances of objectification in the teachers’ discourses, but these are all implicit, which can make them difficult for students to notice. Thus, teachers need to be made more aware of the importance of making the objectification process explicit. Furthermore, our results show that instances of objectification are tightly connected to the use of formal algebraic symbols. This highlights the importance of the transition from concrete/iconic to formal/arbitrary representations.

Students Mistakes Analysis of the Tenth- Grade in Interpretation Verbal Expressions into Algebraic Equations and Quantities: تحليل أخطاء طلبة الصّف العاشر الأساسي في ترجمة التعابير اللفظية إلى معادلات ومقادير جبرية

The study aims to detecting the mistakes of the tenth- grade student's interpretation verbal expressions into algebraic equations and quantities and vice versa, identify and analyze common students’ mistakes. The researcher used the descriptive approach and applied the study to a sample that formed of (100) female students of the tenth grade from Fatima Bint Al Khattab Secondary School, that affiliated to the Directorate of Education for First Zarqa District. A test was prepared - in two models, each model included 3 paragraphs - in phrasing of algebraic equations and quantities and vice versa into verbal expressions and vice versa, and it was applied to two divisions of the tenth- grade. The results have indicated to a classification of students' most common mistakes at the level of all test items, which were arranged in descending order: Common mistakes in interpretation the verbal phrases of addition, subtraction or multiplication into algebraic symbols: repeated by (39%) within the six test paragraphs, followed by mistakes in mathematical concepts, repeated by (29%), followed by the mistake: Confusion between an equation and an algebraic expression, repeated by (25%). The study also found, through checking the percentages of common mistakes, that the students were unimproved in an interpretation the verbal formulation within a story or phrase into algebraic symbols, or formulating equations or algebraic expressions in a verbal phrase, in a story or a life’s situation. In light of the results, several recommendations were suggested, the most important were as following: Application and execution of the teachers for the teaching strategy by using common students’ mistakes, and the committees based on curriculum development and design should take into account the selection of appropriate words for each age group, when phrasing verbal mistakes, and to ensure that they are linguistically and orthographically correct. Based on the development and design of curricula, choosing appropriate words for each age group, when formulating verbal mistakes, and ensuring that they are linguistically and orthographically correct. It was also proposed to hold training courses for teachers, in order to train them on the use of the developed programs to address common students' mistakes in mathematics, such as Drive program and Roll Space program.

Symbol alphabets from tensor diagrams

We propose to use tensor diagrams and the Fomin-Pylyavskyy conjectures to explore the connection between symbol alphabets of n-particle amplitudes in planar mathcal{N} = 4 Yang-Mills theory and certain polytopes associated to the Grassmannian Gr(4, n). We show how to assign a web (a planar tensor diagram) to each facet of these polytopes. Webs with no inner loops are associated to cluster variables (rational symbol letters). For webs with a single inner loop we propose and explicitly evaluate an associated web series that contains information about algebraic symbol letters. In this manner we reproduce the results of previous analyses of n ≤ 8, and find that the polytope {mathcal{C}}^{dagger}left(4,9right) encodes all rational letters, and all square roots of the algebraic letters, of known nine-particle amplitudes.

Mathematics in the archives: deconstructive historiography and the shaping of modern geometry (1837-1852).

This essay explores the research practice of French geometer Michel Chasles (1793-1880), from his 1837 Aperçu historique up to the preparation of his courses on 'higher geometry' between 1846 and 1852. It argues that this scientific pursuit was jointly carried out on a historiographical and a mathematical terrain. Epistemic techniques such as the archival search for and comparison of manuscripts, the deconstructive historiography of past geometrical methods, and the epistemologically motivated periodization of the history of mathematics are shown to have played a crucial role in the shaping of Chasles's own theories. In particular, we present Chasles's approach to the 'material history' of algebraic symbolism and argue that it motivated and informed his subsequent invention of a novel notational technology for the writing of geometrical proofs and propositions. In return, this technology allowed Chasles to carry out a programme for the modernization of geometry in keeping with epistemic requirements he had also delineated via a form of historical writing.

Recognizing and Verifying Mathematical Equations using Multiplicative Differential Neural Units

Automated mathematical reasoning is a challenging problem that requires an agent to learn algebraic patterns that contain long-range dependencies. Two particular tasks that test this type of reasoning are (1)mathematical equation verification,which requires determining whether trigonometric and linear algebraic statements are valid identities or not, and (2)equation completion, which entails filling in a blank within an expression to make it true. Solving these tasks with deep learning requires that the neural model learn how to manipulate and compose various algebraic symbols, carrying this ability over to previously unseen expressions. Artificial neural net-works, including recurrent networks and transformers, struggle to generalize on these kinds of difficult compositional problems, often exhibiting poor extrapolation performance.In contrast, recursive neural networks (recursive-NNs) are,theoretically, capable of achieving better extrapolation due to their tree-like design but are very difficult to optimize as the depth of their underlying tree structure increases. To over-come this, we extend recursive-NNs to utilize multiplicative,higher-order synaptic connections and, furthermore, to learn to dynamically control and manipulate an external memory.We argue that this key modification gives the neural system the ability to capture powerful transition functions for each possible input. We demonstrate the effectiveness of our pro-posed higher-order, memory-augmented recursive-NN models on two challenging mathematical equation tasks, showing improved extrapolation, stable performance, and faster convergence. We show that our models achieve 1.53% average improvement over current state-of-the-art methods in equation verification and achieve 2.22% top-1 average accuracy and 2.96% top-5 average accuracy for equation completion.

Quantifying Solid Findings in Mathematics Education: Loss of Meaning for Algebraic Symbols

We report an example of a research approach aimed at gathering quantitative pieces of evidence of solid findings in mathematics education. The main goal of this project is to provide an additional perspective on solid findings in education, to be used by teachers and by researchers in their work. As a case study, we present a situation of “loss of meaning” in algebra, exploring it with data coming from a large-scale assessment interpreted by means of theoretical lenses. We are able to give information about the extent of the phenomenon and to highlight how the phenomenon is relevant also for high-level students. This approach can provide a link between large-scale assessment results, educational research, and teachers’ practices, and suggests further research issues.

A note on letters of Yangian invariants

Motivated by reformulating Yangian invariants in planar mathcal{N} = 4 SYM directly as d log forms on momentum-twistor space, we propose a purely algebraic problem of determining the arguments of the d log’s, which we call “letters”, for any Yangian invariant. These are functions of momentum twistors Z ’s, given by the positive coordinates α’s of parametrizations of the matrix C(α), evaluated on the support of polynomial equations C(α) · Z = 0. We provide evidence that the letters of Yangian invariants are related to the cluster algebra of Grassmannian G(4, n), which is relevant for the symbol alphabet of n-point scattering amplitudes. For n = 6, 7, the collection of letters for all Yangian invariants contains the cluster mathcal{A} coordinates of G(4, n). We determine algebraic letters of Yangian invariant associated with any “four-mass” box, which for n = 8 reproduce the 18 multiplicative-independent, algebraic symbol letters discovered recently for two-loop amplitudes.

Learning actions indicating algebraic thinking in multilingual classrooms

This article discusses algebraic thinking regarding positive integers and rational numbers when students, 6 to 9 years old in multilingual classrooms, are engaged in an algebraic learning activity proposed by the El’konin and Davydov curriculum. The main results of this study indicate that young, newly arrived students, through tool-mediated joint reflective actions as suggested in the ED curriculum, succeeded in analysing arithmetical structures of positive integers and rational numbers. When the students participated in this type of learning activity, they were able to reflect on the general structures of numbers established as additive relationships (addition and subtraction) as well as multiplicative relationships (multiplication and division) and mixtures thereof, thus a core foundation of algebraic thinking. The students then used algebraic symbols, line segments, verbal, written, and gesture language to elaborate and construct models related to these relationships. This is in spite of the fact that most of the students were second language learners. Elaborated in common experiences staged in the lessons, the learning models appeared to bridge the lack of common verbal language as the models visualized aspects of the relationships among numbers in a public manner on the whiteboard. These learning actions created rich opportunities for bridging tensions in relation to language demands in the multilingual classroom.

Symbol alphabets from plabic graphs

Symbol alphabets of n-particle amplitudes in mathcal{N} = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C ∙ Z = 0 to associate functions on Gr(m, n) to parameterizations of certain cells of Gr(k, n) indexed by plabic graphs. For m = 4 and n = 8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.

Exploring the intersection of algebraic and computational thinking

ABSTRACT This article investigates how the recent implementation of programming in school mathematics interacts with algebraic thinking and learning. Based on Duval’s theory of semiotic representations, we analyze in what ways syntax and semantics of programming languages are aligned with or divert from corresponding algebraic symbolism. Three examples of programming activities suggested for school mathematics are discussed in detail. We argue that although the semiotic representations of programming languages are similar to algebraic notation the meanings of several concepts in these two domains differ. In a learning perspective these differences must be taken into account, especially considering that students have to convert between registers with both overlapping and specific meanings.

Multi-representation raised by prospective teachers in expressing algebra

This study investigates multi-representations raised by prospective teachers in expressing algebra. Through a case study approach, as many as 112 prospective mathematics education teachers from Universitas Negeri Malang of Indonesia participated in this research. researchers use test, interview and research sheet instruments as data collection tools. Then an analysis is carried out so that the following categories are obtained: subjects that symbolically express algebra, pictorial, and geometric. The results obtained that some subjects express symbolically algebra (83 of 112 subjects), while some subjects expressed pictorially (17 of 112 subjects), and geometrically (12 of 112 subjects). The research findings show that there are three forms of representation raised by the subject in expressing algebra, namely the representation of algebraic symbols, image representations, and geometric representations. Most of the participants produced algebraic symbolic representations and some of them experienced obstacles in producing pictorial representations and geometric representations, also researcher found similar patterns in producing geometric representation namely, perception, appearance, strategy, and re-examination. Researchers recommend geometric representations for further research because they tend to be done by subjects with high mathematical abilities and rarely found research that produces geometrical representations when solving algebraic problems.