Factors Affecting Algebraic Problem-solving Skills of Secondary School Students

ABSTRACT The study is done to investigate the proficiency level of students’ basic skills (BS) and attitude towards algebra (ATA) and their relation with algebraic problem-solving skills (PSS). Basic Skills Test (BST) and Problem Solving Test (PST) are done to know proficiency level of BS and PSS, and the Algebraic Attitude Scale (AAS) is designed to know the ATA. The reliability analysis of the questionnaire is 0.842 measured by Cronbach alpha. The proficiency level of students’ BS and level of attitude is determined by using frequency, percentage distribution, mean and weighted mean. To determine the relationship between PSS and the factors (ATA and BS), Chi-Square test is used. The result shows that students’ PSS are significantly related to their level of ATA and BS.

Metaphorical Thinking of Students in Solving Algebraic Problems based on Their Cognitive Styles

One of thinking concepts which connects real life to mathematics is called metaphorical thinking. Metaphor and modelling are two closely related concepts. Besides, each individual performs different cognitive styles, such as field independent (FI) and field dependent (FD) cognitive styles. This factor possibly leads to different metaphorical thinking in solving algebraic problems. The participants of this qualitative research consist of two students at grade 7 of one of junior high school in Banda Aceh, Indonesia, with FI and FD as their cognitive styles. Based on the findings, it is found that: 1) Metaphorical thinking of the student with FI cognitive style in solving the algebraic problem in the stage of understanding the problem, devising a plan, carrying out the plan, and looking back is considered to achieve the target for each criteria of CREATE; 2) Metaphorical thinking of the student with FD cognitive style in solving the problem in the all four stages but could not reveal all criteria mentioned in CREATE. This happens as the student is unable to find the appropriate metaphor to the algebraic problem. Therefore, the student does not need to explain the suitability of the metaphor to the algebraic problem.

THE POSITION OF STUDENTS' ERRORS IN ALGEBRAIC PROBLEM-SOLVING BASED ON FIELD DEPENDENT AND INDEPENDENT

There is a strong relationship between field-dependent (FD), field-independent (FI) cognitive styles, and problem-solving performance. FD students are more oriented towards the outside world, while FI students rely more on their knowledge and experience. The present study aimed to reveal the position of the FI and FD student's errors in algebraic problem-solving. The subjects of this study were 27 students of class VII in one of the Junior High Schools in Kefamenanu, Indonesia, Academic Year 2018/2019. Data collection involved tests of algebraic problem-solving ability, interviews, and Group Embedded Figure Test. The case study results showed that the algebraic problem-solving abilities of FI students were better than FD students. The scores of algebraic problem-solving abilities of FI students were dominant in the medium and high categories. In contrast, the FD students were dominant in the medium and low categories. Also, FI students predominantly committed procedural errors. Whereas, most FD students made errors on all types of errors, specifically factual, conceptual, and procedural errors. Thus, it is recommended that FI and FD students' algebraic problem-solving ability become the focus of attention and importance to characterize them as a basis for further research.

A note on letters of Yangian invariants

Motivated by reformulating Yangian invariants in planar $$ \mathcal{N} $$ 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 $$ \mathcal{A} $$ 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.

Overview

This chapter provides an overview of equivariant cohomology. Cohomology in any of its various forms is one of the most important inventions of the twentieth century. A functor from topological spaces to rings, cohomology turns a geometric problem into an easier algebraic problem. Equivariant cohomology is a cohomology theory that takes into account the symmetries of a space. Many topological and geometrical quantities can be expressed as integrals on a manifold. Integrals are vitally important in mathematics. However, they are also rather difficult to compute. When a manifold has symmetries, as expressed by a group action, in many cases the localization formula in equivariant cohomology computes the integral as a finite sum over the fixed points of the action, providing a powerful computational tool.

Simulation of an incomplete algebraic problem of eigenvalues and vectors by the method of frequency-dynamic condensation based on FEM in the form of the classical mixed method

Relevance . Dynamic analysis of complex structures using numerical methods leads to the solution of the algebraic problem of eigenvalues and the corresponding eigenvectors of high orders. The solution of this problem for high order matrices is performed using reduction methods. One of the most effective methods is the method of sequential frequency-dynamic condensation, which allows partial consideration of the dynamic properties of the structure in the minor degrees of freedom. This allows for more accurate results compared to static condensation. Frequency-dynamic condensation is traditionally used to reduce frequency equations derived from the finite element method in the form of the displacement method or the force method. Methods. The authors have developed an algorithm for the frequency-dynamic condensation method for the frequency equation obtained on the basis of the FEM in the form of the classical mixed method. That allows to obtain not only the spectrum of the lower vibration frequencies, but also the corresponding vibration modes and the stress-strain state of the structure. Results . This article describes the algorithm and its practical implementation in the problem of dynamic analysis of a rectangular plate. The results of the numerical analysis of the problem are presented. An assessment of the accuracy of the method and recommendations for its use are given.

The Solution of a Spectral Task on Variable Section Compressed Beams Vibrations by Numerical Methods

Free flexural free vibrations of variable section are considered. The vibrations mathematical model represents the boundary value problem consisting of the hyperbolic type and boundary conditions main equation. By means of separation method of variables the task at the beginning comes to homogeneous differential equation of the fourth order for fundamental function with the corresponding boundary conditions. The grid area of an argument change and fundamental function in it are applied. That leads to an algebraic problem of eigenvalues. Multimodal non-negative function which null values match its eigenvalues is designed. The finite differences methods and coordinate descent in combination with the specified function sections graphic visualization at a small amount of descents with an adequate accuracy for eigenvalues practice are given. The known ways to define fundamental functions are applied.

Almost Exact Computation of Eigenvalues in Approximate Differential Problems

Differential eigenvalue problems arise in many fields of Mathematics and Physics, often arriving, as auxiliary problems, when solving partial differential equations. In this work, we present a method for eigenvalues computation following the Tau method philosophy and using Tau Toolbox tools. This Matlab toolbox was recently presented and here we explore its potential use and suitability for this problem. The first step is to translate the eigenvalue differential problem into an algebraic approximated eigenvalues problem. In a second step, making use of symbolic computations, we arrive at the exact polynomial expression of the determinant of the algebraic problem matrix, allowing us to get high accuracy approximations of differential eigenvalues.