Knowledge Of Algebra Research Articles

The next number in the sequence

Readers familiar with numbers and their patterns will most likely know of the game-playing exercise where one is invited to spot the next number in a sequence, only to find that after giving an answer there can be a multiplicity of possibilities, each of which is valid according to its own rule of sequence construction. This is an instance of partial sequence matching and the purpose of this note is to show how a simple method of constructing a partial sequence match can be achieved. It requires only a knowledge of polynomials and matrix algebra, which are topics within the grasp of an undergraduate, and will be instanced against the backdrop of some wellknown sequences.

Combinatorics of Double Grothendieck Polynomials

We give a thorough combinatorial treatment of double Grothendieck polynomials that is accessible to anyone with a basic knowledge of algebra and combinatorics. The text includes many new combinatorial models for these and related polynomials and provides detailed combinatorial proofs. We generalize the Giambelli formula for double Schubert polynomials to a $k$-theoretic version and our proof specializes to a new combinatorial proof of the former.

Algebraic knowledge for teaching test: An adaptation study

In this study, the Mathematical Knowledge for Teaching-Elementary Patterns Functions and Algebra-Content Knowledge (MKT-PFA) test, originally developed in English as part of the "Learning Mathematics for Teaching Project" at Michigan University, was adapted into Turkish. The test comprises two equivalent forms, A and B, each translated into Turkish and culturally adapted through consultations with two mathematics education academics and five secondary school math teachers pursuing doctoral studies. A total of 328 pre-service teachers at a Turkish public university's elementary school mathematics teaching department were administered form A (14 questions, 29 items) and form B (12 questions, 27 items) at a one-week interval. Psychometric analyses revealed high reliability (KR-20: A=0.712, B=0.735; Lord reliability: A=0.733, B=0.756), and strong correlations (rpbi) with the original English forms, indicating suitable adaptation. Item difficulties analyzed using a one-parameter Item Response Theory model showed a normal distribution, affirming the tests' validity for assessing pre-service teachers' algebra teaching knowledge in Türkiye.

Transition from Knowledge of Algebra for Teaching (KAT) to Technological Knowledge of Algebra for Teaching (T-KAT)

Transition from Knowledge of Algebra for Teaching (KAT) to Technological Knowledge of Algebra for Teaching (T-KAT)

How do testing and test-potentiated learning versus worked example method affect medium- and long-term knowledge in abstract algebra for pre-service mathematics teachers?

The retention of foundational knowledge is crucial in learning and teaching mathematics. However, a significant part of university students do not achieve long-term knowledge and problem-solving skills. A possible tool to increase further retention is testing, the strategic use of retrieval to enhance memory. In this study, the effect of a special kind of testing versus worked examples was investigated in an authentic educational setting, in an algebra course for pre-service mathematics teachers. The potential benefits of using tests versus showing students worked examples at the end of each practice session during a semester were examined. According to the results, there was no difference between the effectiveness of the two methods in the medium term—on the midterm that students took on the 6th week and the final that students took on the 13th week of the semester, the testing group performed the same as the worked example group. However, testing was more beneficial regarding long-term retention in studying and solving problems in abstract mathematics. Analyzing the results of the post-test that students took five months after their final test, the authors found that the improvement of those students who learned the material with testing was significantly larger than that of the worked example group. These findings suggest that testing can have a meaningful effect on abstract algebra knowledge and a long-lasting impact on solving complex, abstract mathematical problems.

Mathematical Knowledge of Early Algebra Exhibited by Pre-Service Early Childhood Education Teachers

AbstractThis study analyzes the mathematical knowledge to teach early algebra exhibited by pre-service early childhood education teachers, from the perspective of the Mathematical Knowledge for Teaching (MKT) model. The research adopts a mixed exploratory-descriptive methodological approach, based on the application of the MKT-early algebra questionnaire (3–6), consisting of six open-ended items that place teachers in various teaching situations reflecting the knowledge that characterizes early algebra at this stage of schooling. The analysis of the answers given by the pre-service teachers of early childhood education revealed a level general of insufficient mathematical knowledge, with the common content knowledge exhibiting fewer limitations compared to the other subdomains that comprise the model, and the horizon content knowledge the weakest. We conclude that it is necessary to offer teacher training programs that deepen the didactics of early algebra and provide tools to further the effective teaching of this content block in early childhood education.

Linear Equation and Linear Function: An Alternative Material Based on Visual Representation in Mathematics

Linear equations and linear functions are important materials for students in developing their in-depth knowledge of arithmetic and algebra. However, there is a gap between potential development and actual development, which causes obstacles for students in understanding the concepts of these two materials. This gap results in other obstacles, including a fixed mindset in mathematics. The results of interviews with secondary school teachers in Indonesia also show that students have problems in number operations and motivation to learn mathematics. The design used in this study uses a descriptive qualitative study based on the results of literature studies, interviews with subject teachers, and the author's experience as an educator. This study also uses the theory of zone proximal development and mathematical mindset as a frame for preparing alternative material for equations and linear functions based on visual representations in mathematics. The results of this study contribute to providing alternative materials to reduce these barriers using a visual representation approach in mathematics. The author also provides alternative teaching arts to anticipate obstacles that arise when implementing mathematics material based on visual representation.

Calculating toxic pressure for mixtures of endocrine disruptors

Incidence of autoimmune disorders, birth defects, and neurological diseases rose over the past 50 years due to increasing variety and quantity of pollutants. To date, there appear few methods capable to evaluate and predict mixture effects by endocrine disruptors (EDs). For the first time, we have developed calculus to determine mixture effects by all kinds of EDs.Our method uses the golden ratio ϕ and draws from bifurcation and chaos theory. Using also the concept of molecular mimicry, we developed the equation: effect=100%1+e5·∑KiCi−niϕ3. We successfully tested the equation using a range of cohort studies and biomarkers, and for different pollutants like heavy metals, thyroid hormone mimickants, chromate/chlorate, etc.The equation is simple enough to use with only minor prior knowledge and understanding of basic algebra. The method is universal and calculation is data 'light', requiring only pollutant concentrations [C], potencies K and an integer n for endocrinal involvement. This study offers a comprehensive framework to assess the health effects of pollutant exposure across diverse populations, envisioning far-reaching impact, and presenting practical examples and insights.

Introduction to Haar Measure Tools in Quantum Information: A Beginner's Tutorial

The Haar measure plays a vital role in quantum information, but its study often requires a deep understanding of representation theory, posing a challenge for beginners. This tutorial aims to provide a basic introduction to Haar measure tools in quantum information, utilizing only basic knowledge of linear algebra and thus aiming to make this topic more accessible. The tutorial begins by introducing the Haar measure with a specific emphasis on characterizing the moment operator, an essential element for computing integrals over the Haar measure. It also covers properties of the symmetric subspace and introduces helpful tools like tensor network diagrammatic notation, which aid in visualizing and simplifying calculations. Next, the tutorial explores the concept of unitary designs, providing equivalent definitions, and subsequently explores approximate notions of unitary designs, shedding light on the relationships between these different notions. Practical examples of Haar measure calculations are illustrated, including the derivation of well-known formulas such as the twirling of a quantum channel. Lastly, the tutorial showcases the applications of Haar measure calculations in quantum machine learning and classical shadow tomography.

Student profiles based on in‐game performance and help‐seeking behaviours in an online mathematics game

AbstractWell‐designed online educational games can improve students' math knowledge, skills and engagement; however, more research is needed to understand how to formatively assess components of students' mathematical understanding and learning as students solve problems in online educational games. In this study, we examined how 7th‐grade students' (N = 760) in‐game performance (ie, strategy efficiency, mathematical validity of the first action, mathematical errors) and help‐seeking behaviours (ie, hint requests) correlated with their algebraic knowledge in an online mathematics game. The k‐means cluster analysis identified four groups of students based on their in‐game metrics, and some variabilities in their in‐game performance were found. Although hint requests were available, only a few students showed a high percentage of hint requests during their gameplay. The regression analysis revealed that students' in‐game performance and help‐seeking behavioural patterns explained a significant amount of variance in students' algebraic knowledge above and beyond their prior knowledge. Our results suggest that how students interact with the online learning game is related to their later math performance, and qualitative aspects of student behaviours captured using log data provide meaningful information to students' algebraic understanding. Practitioner notesWhat is already known about this topic Well‐designed online educational games can improve students' math knowledge, skills, and engagement. Several studies found that some in‐game metrics, such as in‐game progress or correctness of tasks, are positively associated with learning outcomes. However, many studies used simple metrics like the correctness of each task or the number of problems completed to measure students' performance in the game. Few studies measured more in the moment or qualitative aspects of student in‐game performance, for example, how efficiently students solved problems in the game. What this paper adds In‐game metrics measuring more in the moment and qualitative aspects of students' problem‐solving explained a significant amount of variance in students' algebraic knowledge above and beyond students' prior knowledge. Students' strategy efficiency and strategic planning were positively associated with students' math problem‐solving performance. Although hints were available for almost all problems, only a few students exhibited a high level of help‐seeking behaviours, and the students with the lowest in‐game performance who needed help the most showed low help‐seeking behaviours. Implications for practice and/or policy Classifying students into profiles could provide ways to identify students with different levels of problem‐solving skills and help teachers provide more personalized instruction. Teachers could use this information as a formative assessment to identify the core areas of intervention needed, particularly for students struggling with math concepts taught in the game. Instructional strategies that make hints more appealing or engaging to students might be needed.

A Golden Exploration of Almost Congruent Triangles

Explore and expand your knowledge in geometry and algebra with our exciting sequence of exploratory tasks to create two 5-con triangles with integer side lengths.

Unpacking the Connections Between Fractions and Algebra: The Importance of Fraction Schemes and Units Coordination

ABSTRACT Researchers from multiple disciplines have found that fractions and algebra knowledge are linked. One major strand of research has identified children’s units coordination structures as crucial for success with fractions and algebra via multiplicative reasoning, whereas a second strand of research points to magnitude knowledge as a central conceptual structure supporting fractions and algebra knowledge. In an online study with 59 eighth graders, we compared the associations of these two foundational factors with fractions and algebra knowledge by assessing students’ units coordination skills, magnitude knowledge, and several general cognitive skills alongside multiple aspects of fractions and algebra knowledge. In regression models controlling for math anxiety and general cognitive skills, units coordination and fraction schemes were better predictors of students’ algebra knowledge than were fraction magnitude or fraction arithmetic skills. We replicated findings from previous studies that showed units coordination and fraction schemes were closely related and showed for the first time that units coordination was significantly associated with fraction number line estimation and fraction arithmetic scores, even after accounting for general cognitive skills. We emphasize the need for interdisciplinary teams to continue investigating relations among these constructs to further our understanding and ultimately support student learning of fractions and algebra.

Remote monitoring of implementation fidelity using log-file data from multiple online learning platforms

The impact of educational programs on student learning is contingent upon the quality and fidelity of their implementations. Yet, the most reliable method of implementation monitoring, direct observation, may be infeasible for large-scale efficacy studies. Here, we use log-file data from an efficacy study assessing the impact of gamified learning technologies on algebraic knowledge to measure the fidelity of implementations remotely within and across multiple programs. We employ exploratory-data-analysis to identify measures of fidelity, multilevel models to evaluate which levels of implementation contexts (i.e. classrooms, teachers, students) were associated with variance in fidelity metrics, and latent profile analysis to identify high and low-fidelity contexts. These analyses illustrate how log-file data may be used to remotely monitor and evaluate the implementations of online learning platforms.

Analyses of algebra knowledge and difficulties among Ghanaian Junior High School learners: Insights from standard-based mathematics curriculum implementation

Learners' algebra knowledge significantly determines their performance in mathematics. Based on the critical role of learners’ algebra knowledge in their mathematics achievements, the current study sought to investigate the algebra knowledge and difficulties of Junior High School learners in public and private schools in Ghana. Utilising the learning behaviour dimensions of the recently implemented standard-based mathematics curriculum in the Ghanaian context, the study employed the descriptive survey design and involved the participation of 441 learners using an achievement test instrument. The findings revealed a continuous decline in average scores and the number of learners operating at higher levels of the dimensions. Public school learners demonstrated proficiency up to the Evaluating level, whilst private school learners exhibited proficiency up to the Creating level, with private school learners exhibiting higher knowledge across all six levels and the overall algebra knowledge as compared to their counterparts in public schools. Furthermore, the study identified ten major algebra difficulties among Junior High School learners. The study provides valuable insights into the depth of algebra knowledge and specific difficulties among JHS learners in Ghana. The findings underscore the significance of teacher preparation, professional development and teaching practices in addressing the identified challenges in algebra.

Ensuring Validity and Reliability in Algebra Midterm Assessment: A Comprehensive Approach to Test Development and Analysis

First-year mathematics education students often face challenges with College and Advanced Algebra assessments. This study aimed to develop and validate a 100-item multiple-choice midterm test for College and Advanced Algebra, ensuring the test's validity and reliability. The test was designed following a structured process that included creating a Table of Specifications (TOS) based on the approved syllabus. To ensure content accuracy and relevance, the test was reviewed by three subject matter experts and evaluated for clarity by 15 students. Pilot testing was conducted with 82 fourth-year Bachelor of Secondary Education Mathematics (BSEd Mathematics) students. The pilot test results underwent detailed item analysis, focusing on metrics such as the difficulty index, discrimination index, and overall reliability using the Kuder-Richardson Formula 20 (KR-20). Of the 100 items, 22 were retained, 48 were revised, and 30 were discarded. The test achieved a reliability coefficient of 0.876, indicating strong internal consistency. The findings suggest that the validated test questionnaire is a dependable tool for accurately assessing students’ knowledge in College and Advanced Algebra, providing valuable feedback for educators and students.

Application of Hierarchical Analysis of the Efficacy of Digital Resource Integration in Mathematics Teaching and Learning

Abstract At present, the construction and application of digital teaching resources is no longer an educational development trend but a normalized teaching tool. The purpose of this paper is to analyze the principle of digital integration of resources for teaching mathematics and develop the FAHP-Entropy model. Determine the weights of the first-level indicators in the hierarchical model constructed in this paper, then determine the weights of the second-level indicators by the entropy method, and use the fuzzy comprehensive evaluation method to carry out a comprehensive evaluation of the established indicators and then get the results. The study found that the total score of the fuzzy comprehensive evaluation of algebraic knowledge points is 83.825 points, and the secondary indicators are all above 75 points, with the highest being that D1 can make students’ analytical problem-solving ability improve significantly, up to 91.21 points. Most of the secondary indicators are in the first and second quadrants, of which A1 and B1 continue to maintain the zone in the first quadrant with high importance and good evaluation results. The fourth quadrant focuses on the improvement area with two secondary indicators, A4 and C1, indicating that A4 media presentation is diversified and C1 media is well-produced and attractive, both of which need to be further improved. The scores of the four first-level indicators in the comprehensive evaluation results of the geometry category are 81.111, 86.061, 82.525, and 82.957, respectively, and the highest among the second-level indicators is D3, with a score of 87.76. A1, A4, and C1 are in the fourth quadrant of the key improvement area, which is an indicator that does not match the importance of the results of the evaluation and is in urgent need of improvement. The Probability Statistics category had a slightly lower overall evaluation score than the Algebra and Geometry category of knowledge points. The total score of the comprehensive evaluation results was 81.898. The four primary indicators have scores of 80.438, 84.01, 79.901, and 83.281. The evaluation results of the A4 and C1 indicators are not aligned with their importance, and the other indicators perform better or have lower importance. This study provides new ideas for the analysis of the effects of digital resource integration.

Exploration of Innovative Teaching Methods in the Teaching Reform of Linear Algebra Course in the Internet Era

Abstract For the task of ternary extraction in unstructured text, an entity recognition model based on BiLSTM-CRF is proposed. By introducing graph convolutional neural network and attention mechanism respectively, a relation extraction model based on attention-guided graph convolutional neural network and a text classification model based on GCN are proposed to realize the automatic extraction of linear algebra knowledge triples. To solve the problem of ignoring semantic information of knowledge points and sparse data, the linear algebra knowledge level matrix of learners is obtained by using the DKVMN model. The knowledge graph’s relationships are assigned weights by experts through evaluation. The semantic information in the knowledge graph and the weights of relationships between nodes are utilized to complement the knowledge level matrix as a way to recommend appropriate linear algebra exercises for learners. Compared with the traditional teaching class, the experimental class of this paper’s method scores higher in algorithmic thinking, collaboration, critical thinking, and problem-solving by 0.67, 0.27, 0.41, and 0.37, respectively. Based on the learning data of each chapter’s knowledge points, it is understood that the standard deviation of the eigenvalues and eigenvectors in the eigenvalue and eigenvector and quadratic form, quadratic form, similarity matrices, positive qualitative and geometric questions, and proofs are respectively 4.076, 3.226, 6.185, 3.292, and 4.105, and the standard deviations are generally large compared to other chapters. This paper demonstrates that the method can be taught accurately depending on the students’ linear algebra level.

Factoring primes and sums of two squares

Mathematicians began to study a series of properties about numbers a long time ago, and a new field of mathematics, the number theory, was born from this. Some special properties of numbers in the number theory make mathematicians use the knowledge of group theory to make some ingenious answers when considering some problems. In the analytic number theory, equations related to numbers have always been a concern of mathematicians. The most famous Fermat's last theorem also brought long-term troubles to countless mathematicians and was finally proved by the British mathematician Wiles. Many famous theorems also prove that some problems in the number theory can be solved by thinking in relation to other algebraic knowledge. This paper focuses on the factoring primes and constructs prime ideals of lying above a prim from irreducible factors of . The paper also shows that these are all prime ideals lying above . Based on these theorems and definitions, as a simple application of the theory, this paper first considers which primes can be written as sums of two squares, then the second part of this paper gives the answer: is a sum of two squares if and only if .

The Impact of Algebra Background on Upper Secondary Students’ Performance in Mathematics: A Case of Study of Ruhango District

This study investigated the impact of algebra background on upper secondary students' mathematics performance. The researcher used nonprobability sampling methods, specifically convenience sampling techniques. The convenience sampling technique was used to select participants who were easily accessible and willing to participate. The sample size consisted of 197 students from three schools of general education in Ruhango District and five teachers teaching mathematics. The pre-test and post-test were carefully designed and tested to ensure that they were reliable and valid. The pre-test and post-test were administered to measure the participants' algebra background and performance in mathematics. The pre-test included questions on algebraic concepts, while the post-test included questions on a range of mathematical topics. The questionnaires were used to collect data from teachers assessing students’ algebra background. Document analysis looked at documents such as textbooks, curriculum materials, and assessments to gain a better understanding of the algebra and mathematics content covered in the upper secondary schools in Ruhango District. The IMB SPSS software was used to analyse the data that was collected. The study found that there is a strong correlation between students' proficiency in algebra and their overall mathematical ability. As a result, the study recommends that educators, students, and researchers should place a greater emphasis on assessing children's mathematical backgrounds, with a particular focus on their familiarity with algebra. The study also provides suggestions for improving both arithmetic proficiency and algebra knowledge.

LATTICE OF A FUZZY PRIME IDEALS IN AN ORDERED SEMIGROUP

This paper introduces the concept of lattices of prime fuzzy ideals, which extends the classical theory of lattices of prime ideals in ring theory to the fuzzy setting. In this paper, we propose some examples and properties related to lattices of prime fuzzy ideals, contributing to the knowledge in fuzzy algebra. Think about an ordered semigroup ℛ. Plotting the function f from ℛ to the closed interval [0,1] defines a fuzzy subset of ℛ, where [0,1] ϵ ℛ. In order to explore prime ideals in rings, semigroups, and ordered semigroups in terms of fuzzy subsets, we build an ordered semigroup ℛ with ordered fuzzy points. To study, we presented the thoughts of ℒ-fuzzy prime ideals of an ordered semigroups ℛ, which forms a completelattice obeying the law relating the operations of multiplication and addition.