Search

Positive education takes the study of happiness and wellbeing to inspire and help learners, schools and societies to progress. This study investigated the significant differences in the means of the pretest and posttest of students’ academic achievement and their life satisfaction in Mathematics. The relationship of students’ academic achievement and their life satisfaction in Mathematics was also measured. A total of one hundred twenty students (60 in the experimental group and 60 in the control group) from the two secondary schools in Maguindanao, Philippines living in a hostile environment were chosen and involved in the study. Results showed that there is a significant difference in the means of the pretest and posttest results of academic achievement of students in Mathematics both in the control and experimental group. The means of the pretest and posttest results of students’ life satisfaction in Mathematic were only significant in the experimental group of one school. Furthermore, there is no significant relationship between the academic achievement of students and their life satisfaction in Mathematics. The practical implication of the findings suggests that positive education intervention is also an effective strategy to improve academic achievement. Life satisfaction on the other hand is subjective and perceived differently by the learners.

A simple forward approach for solving linear Diophantine equations is presented, which does not involve using backward substitutions. It is suitable for teaching undergraduate students as an altern...

A simple approach for solving linear Diophantine equation in two variables is presented, which does not involve backward substitutions. Some illustrative examples are provided.

A simple method for computing the inverse of Vandermonde matrix via synthetic divisions is introduced. It can be applied to compute each row of the inverse of Vandermonde matrix systematically and effectively. Some illustrative examples are provided.

The Fagnano’s problem is a famous historical problem in plane geometry, which involves finding an inscribed triangle with minimal perimeter in a given acute triangle. We discuss how to solve this problem via a dynamic geometry approach and derive a simple formula for finding the perimeter of the orthic triangle, which is the solution of the Fagnano’s problem. Some illustrative examples are included.

The two jugs problem is a classic problem in mathematics and computer sciences. In this paper, we introduce a new algorithm to solve the general two jugs problem. This algorithm has the advantage that one can easily apply it to determine all measurable amounts by the given jugs and provides us insight on the hidden relationship between the two different approaches to tackle the problem, namely the attempt to fill the smaller jug or the larger jug first. We also discuss how to implement this algorithm in Excel and visualize the generated outputs by means of staircase diagrams. Some basic properties of the measurable amounts by the jugs will also be described, together with some examples.

In this article, we report a study on how students learn elementary number theory concepts via pre-constructed activities of curve stitching on a unit circle. By using a dynamic learning approach, students were able to observe and guess the patterns of curve- stitching figures and explore how they were related to the total number of points on the circle and the interval between any pair of connected points. The subjects were pre-service and in-service primary mathematics teachers majored in mathematics or information technology in a BEd programme offered by the Hong Kong Institute of Education. We also discuss some teaching and learning issues related to this topic.

Partial fraction decomposition is a useful technique often taught at senior secondary or undergraduate levels to handle integrations, inverse Laplace transforms or linear ordinary differential equations, etc. In recent years, an improved Heaviside's approach to partial fraction decomposition was introduced and developed by the author. An important feature of this approach is that there is no need to solve a system of linear equations or to use differentiations to find the unknown coefficients of the partial fractions. In order to study its potential application in mathematics education at the undergraduate level, a pilot study of tryout at the Hong Kong Institute of Education has been conducted. The data are collected via quizzes, questionnaires and face-to-face interviews. In this article, the results and implications will be discussed. In general, according to the responses and feedbacks from the instructors and students concerned, the improved Heaviside approach is suitable to be introduced at the unde...

In this article, we present an integer sequence approach to solve the classic water jugs problem. The solution steps can be obtained easily by additions and subtractions only, which is suitable for manual calculation or programming by computer. This approach can be introduced to secondary and undergraduate students, and also to teachers and lecturers involved in teaching mathematical problem solving, recreational mathematics, or elementary number theory.

In this article, we discuss how to use a diagrammatic approach to solve the classic sailors and the coconuts problem. It provides us an insight on how to tackle this type of problems in a novel and intuitive way. This problem-solving approach will be found useful to mathematics teachers or lecturers involved in teaching elementary number theory, recreational mathematics or mathematics problem solving.

In this article, we report a study of two-term unit fraction expansions using a geometric approach. It provides us insight on how to find all two-term unit fraction expansions and identify the property of the expansions with smallest maximal denominators, for a given unit fraction. These findings will be useful to lecturers or teachers involved in teaching history of mathematics or elementary number theory.

In this article, we give a brief history of the Feynman's Triangle problem and describe a simple method to solve a general version of this problem, which is called the Routh Theorem. This method could be found useful to school teachers, instructors or lecturers who are involved in teaching geometry.

In this note, a simple proof of the Generalized Ceva Theorem in plane geometry is presented. The approach is based on the principle of equilibrium in mechanics.

A simple algorithm for computing the partial fraction expansions of proper rational functions with multiple poles is presented. The main idea is to use the Heaviside's cover-up technique to determine the numerators of the partial fractions and polynomial divisions to reduce the multiplicities of the poles involved successively, without the use of differentiation.

Since the last decade, the use of computer algebra systems at the Hong Kong school level is still very limited. Among various reasons behind, the lack of exposure of this kind of software to local school teachers should be taken into account. In this article, we describe how to introduce MAPLE in a BEd module of a local teacher-training programme. Reflection on the effectiveness of the module and barriers faced by the school teachers at present are discussed.

In this note, a modified Golomb algorithm for computing unit fraction expansions is presented. This algorithm has the advantage that the maximal denominators involved in the expansions will not exceed those computed by the original algorithm. In fact, the differences between the maximal denominators or the number of terms obtained by these two Golomb algorithms could be very significant, as illustrated by the examples chosen. Some examples on how to use this new algorithm in teaching are also included.

The Prelle-Singer procedure for determining elementary first integrals of two-dimensional autonomous systems of ordinary differential equations is introduced and how it can be generalized to higher dimensions is discussed. The application of this procedure in several dynamical systems is reported.

We present a synthetic division approach to compute partial fraction decompositions of rational functions. This method can determine the unknown partial fraction coefficients successively, without the need to use differentiation or to solve a system of linear equations. Examples of its applications in indefinite integration, Laurent series, inverse Laplace transform, linear ordinary differential equations, and linear recursive relations are provided.

An algorithm for computing the dispersion of one or two polynomials is described, based on irreducible factorization. It is demonstrated that in practice it is faster than the “conventional” resultant-based algorithm, at least for small problems. It can be applied to algorithms for indefinite summation and closed-form solution of linear difference equations. A brief survey of existing mostly resultant-based dispersion algorithms is given and the complexity of the resultant involved is analysed. The effectiveness of the proposed algorithm applied to indefinite summation is demonstrated by some examples that are not easily summed by the standard facilities in several computer algebra systems.

In this communication, we present a method for computing the Liouvillian solution of second-order linear differential equations via algebraic invariant curves. The main idea is to integrate Kovacic's results on second-order linear differential equations with the Prelle–Singer method for computing first integrals of differential equations. Some examples on using this approach are provided.

We present an approach to computing the Darboux polynomials required in the Prelle–Singer algorithm which avoids algebraic extensions of the constant field, and describe a partial implementation in REDUCE in which the leading terms of the polynomials are obtained by a modified version of the method described by Christopher and Collins.

The Prelle-Singer procedure is an important method for formal solution of first order ODEs. Two different REDUCE implementations (PSODE versions 1 & 2) of this procedure are presented in this paper. The aim is to investigate which implementation is more efficient in solving different types of ODEs (such as exact, linear, separable, linear in coefficients, homogeneous or Bernoulli equations). The test pool is based on Kamke's collection of first order and first degree ODEs. Experimental results, timings and comparison of efficiency and solvability with the present REDUCE differential equation solver (ODESOLVE) and a MACSYMA implementation (ODEFI) of the Prelle-Singer procedure are provided. Discussion of technical difficulties and some illustrative examples are also included.

A decision procedure for finding closed forms for indefinite summation of polynomials, rational functions, quasipolynomials and quasirational functions is presented. It is also extended to deal with some non-hypergeometric sums with rational inputs, which are not summable by means of Gosper's algorithm. Discussion of its implementation, analysis of degree bounds and some illustrative examples are included.