Early Mathematics Learners’ Numerical Errors: Consequence of Poor Learners’ Comprehension and Teachers’ Instructions

While the coronavirus disease 2019 (COVID-19) is still considered as a pandemic in recent human history, evidence from World Health Organization (2021) so far has recorded a total of 116,521,281 confirmed cases of COVID-19 with 2,589,548 as a total of deaths from over 215 countries or territories worldwide. Recognizing that COVID-19 is not only pandemic since March 11, 2020, but spreading worldwide at unprecedented rate, number of sectors including schools and universities as a measure to minimize person-to-person transmission closed their services. Such an uncertain closure warranted restructuring of services provided by schools and universities. The challenges therefore have necessitated the current research to investigate and alleviate challenges brought about by the COVID-19. In essence, the present research’s aim was to report on early mathematics learners (foundation phase) numerical errors, which is as a consequence of poor learners’ comprehension and teachers’ instructions. Based on the aim, the study was positioned within a cognitive theory in order to examine processing of numerical competence among early mathematics learners. A case study via 80 grade 3 learners with ages 8 and 9 was sampled. A textual analysis was used in unpacking and de-contextualizing processing of numerical competence by early mathematics learners. The evidence revealed learners’ mathematical mistakes were caused from limited reading skills and ill-presented problems via teachers. Due to the need to teach children at home (home school) due to the COVID-19, it is hoped that the findings thus assist audience, including non-academic and parents, who grapple with poor instructions coupled with poor learners’ comprehension.

Structure-preserving analysis on Gaussian solitary wave solution of logarithmic-KdV equation

The existence of the Gaussian solitary wave solution in the logarithmic-KdV equation has aroused considerable interests recently. Focusing on the defects of the reported multi-symplectic analysis on the Gaussian solitary wave solution of the logarithmic-KdV equation and considering the dynamic symmetry breaking of the logarithmic-KdV equation, new approximate multi-symplectic formulations for the logarithmic-KdV equation are deduced and the associated structure-preserving scheme is constructed to simulate the evolution of the Gaussian solitary wave solution. In the structure-preserving simulation process of the Gaussian solitary wave solution, the residuals of three conservation laws are recorded in each step. Comparing with the reported numerical results, it can be found that the Gaussian solitary wave solution can be simulated with tiny numerical errors and three conservation laws are preserved perfectly in the simulation process by the structure-preserving scheme presented in this paper, which implies that the proposed structure-preserving scheme improved the precision as well as the structure-preserving properties of the reported multi-symplectic approach.

Interference Errors in Numerical Expression by Vietnamese EFL Speakers

This paper aims to investigate numerical expression by Vietnamese speakers of English as a foreign language (EFL). The study identifies and explains the causes of interference errors in expressing number of nouns. A descriptive-cognitive research design was conducted error-oriented investigation of 62 high-school students and 30 employees working in English-speaking companies participating in writing a 45-minute essay for numerical errors from the essays collected. The findings revealed that Vietnamese EFL speakers had difficulty in expressing the number of the entities represented by the nouns due to differences in means and manner of numerical expression in English whose sentences are numerically compulsory and grammatically relevant as opposed to those in Vietnamese whose numerical category is grammatically unimportant, but lexically relevant, and seen with number-neutral nouns or general numbers. Errors also occurred as Vietnamese EFL speakers failed to acquire the count-uncount distinction due in part to differences in perceptualizing the numerical meaning of the entities represented by nouns, ascribing the countability wrong and keeping the same property of countable/uncountable nouns despite having referred to different referents. The paper ended with some pedagogical implications to help Vietnamese EFL speakers improve numerical errors when using English.

Computational Analysis of Lung and Isolated Airway Bifurcations under Mechanical Ventilation and Normal Breathing

Mechanical ventilation is required for many patients who cannot breathe normally as a result of lung disease and other factors that result in reduced lung function. In this study, we investigated the effects of mechanical ventilation and normal breathing on whole lung geometry as well as isolated bifurcations through computational fluid dynamic (CFD) simulations. Results of flow characteristics (airflow velocity, wall pressure, and wall shear stress) obtained from the CFD simulations are presented. Similar flow patterns and pressure drops were obtained between the whole lung geometry and isolated bifurcations under both normal breathing and mechanical ventilation, respectively. Results obtained from simulations suggest that analyzing specific local bifurcations may be a more feasible alternative as it may reduce the computational time and numerical errors resulting from computations as compared to simulating a complex whole lung geometry. The approach presented in this study also demonstrated that analyses of isolated bifurcations gave similar flow characteristics to that of whole lung geometry. Therefore, this approach may be useful for quickly obtaining results that will assist in making clinical predictions and other applications.

An adaptive memory method for accurate and efficient computation of the Caputo fractional derivative

A fractional derivative is a temporally nonlocal operation which is computationally intensive due to inclusion of the accumulated contribution of function values at past times. In order to lessen the computational load while maintaining the accuracy of the fractional derivative, a novel numerical method for the Caputo fractional derivative is proposed. The present adaptive memory method significantly reduces the requirement for computational memory for storing function values at past time points and also significantly improves the accuracy by calculating convolution weights to function values at past time points which can be non-uniformly distributed in time. The superior accuracy of the present method to the accuracy of the previously reported methods is identified by deriving numerical errors analytically. The sub-diffusion process of a time-fractional diffusion equation is simulated to demonstrate the accuracy as well as the computational efficiency of the present method.

Subtractive modeling using the reverse condensed transfer function method: influence of the numerical errors

Decoupling procedures based on substructuring methods allow to predict the vibroacoustic behaviour of a given system by removing a part of an original system that can be easily modelled. The reverse Condensed Transfer Function (rCTF) method has been developed to decouple acoustical or mechanical subsystems that are coupled along lines or surfaces. From the so-called condensed transfer functions (CTFs) of the original system and of the removing part, the behaviour of the system of interest can be predicted. The theoretical framework as well as a numerical validation have been recently published. In the present paper, we focus on the influence of numerical errors on the results of the rCTF method, when the CTFs are calculated using numerical models for the original system and/or the removed part. The rCTF method is applied to a test case consisting in the scattering problem of a rigid sphere in an infinite water domain and impacted by an acoustic plane wave. Discrete green formulation and finite element method are used to estimate the CTFs. Numerical results will be presented in order to evaluate the sensitivity of the method to model errors and the potential promises and limitations of the method will be highlighted.

Numerical Methods I

Textbook, established at several universities. Second edition. *** Written primarily for students at technical studies. Valuable handbook for engineers, PhD students and scientists. *** Published in several variants. *** Seven chapters. In the first chapter, a historical background and basic properties of various numeral systems, as well as conversion of numbers from one system to another are briefly explained. In the second chapter, numbers in digital computers, namely integers and floating point numbers are described. This helps the reader to choose precision and range limits of stored numbers. The third chapter explains constant variables and related numerical errors, including error propagation and algorithm instability. The fourth and fifth chapters explain random variables and related random errors, uncertainty, confidence level, as well as propagation of random errors. Various types of regression analyses of experimental data are described in the sixth chapter. Direct methods for finding roots of the third and fourth degree polynomials are described in the seventh chapter, followed by general iterative methods for polynomials of any degree. *** Why the presented topics are so important? Simply, they are common to all numerical methods. *** Practical application is supported by 84 examples and 17 algorithms. For reasons of simplicity, algorithms are written in pseudo-code, so they can easily be implemented in any computer program. Finally, the given text with 98 figures and 52 tables represents a valuable background for understanding, applying and developing various numerical analyses.