##### Extended Rayleigh Distribution: Properties and Application to Failure Data

Asian Research Journal of Mathematics, Page 1-8
DOI: 10.9734/arjom/2021/v17i730313

This paper introduces a new three parameter Rayleigh distribution which generalizes the Rayleigh distribution. The new model is referred to as Extended Rayleigh (ER) distribution. Various mathematical properties of the new model including ordinary and incomplete moment, quantile function, generating function are derived. We propose the method of maximum likelihood for estimating the model parameters. A real life data set is used to compare the flexibility of the new model with other models.

##### Examining the Influence of University Teachers’ Qualification and Experience on Students’ Academic Achievement in Mathematics

Sonam Gyeltshen

Asian Research Journal of Mathematics, Page 9-18
DOI: 10.9734/arjom/2021/v17i730314

Aims: This study aims to examine the effect of qualification and experience level on students’ academic achievement in mathematic in five colleges under the Royal University of Bhutan.

Study Design: It is a quantitative study where the researcher tests the significance of the difference                             in the mean academic achievement of the students concerning teachers’ qualifications and experience                  level.

Place and Duration of Study: Five colleges under the Royal University of Bhutan spread across five districts in Bhutan were considered for the study between July 2020 and July 2021.

Methodology: A sample of 25 university mathematics teachers (21 males, 4 females) who taught mathematics from 2018 to 2020 were selected through a simple random sampling. The data was collected in two parts: one the mathematics marks of the modules taught by the 25 teachers from the year 2018 to 2020 was obtained from the respective college's exam records through an official correspondence; second, the information on the perception of these 25 teachers was obtained online through self-administered survey questionnaires.

Results: The result from the two way ANOVA revealed no statistically significant difference between qualifications (Masters and PhD) on students’ academic achievement (F (1, 19) = 3.653, P = 0.07) and between experience levels on students’ academic achievement (F (3, 19) = 0.493, P = 0.69). In contrast, the perception of the teachers is that the higher level of qualifications and experience levels would have a more positive effect on students’ academic achievement.

Conclusion: The qualification and experience have a positive effect on students academic achievement but the effects are not significant.

##### Wigner Distribution Analysis Applied to Lehmer’s Conjecture on the Ramanujan tau Function

Takaaki Musha

Asian Research Journal of Mathematics, Page 19-28
DOI: 10.9734/arjom/2021/v17i730315

Wigner distribution is a tool for signal processing to obtain instantaneous spectrum of a signal. By using Wigner distribution analysis, another representation of the Euler product can be obtained for Dirichlet series of the Ramanujan tau function. From which, it can be proved that the Ramanujan tau function never become zero for all numbers.

##### A new Model for Solving Three Mixed Integral Equations with Continuous and Discontinuous Kernels

M. A. Abdou, M. I. Youssef

Asian Research Journal of Mathematics, Page 29-38
DOI: 10.9734/arjom/2021/v17i730316

In this paper, we discuss a new model to obtain the answer to the following question: how can we establish the different types of mixed integral equations from the Fredholm integral equation? For this, we consider three types of mixed integral equations (MIEs), under certain conditions.  The existence of a unique solution of such equations is guaranteed. Using analytic and numerical methods, the three MIEs formulas yield the same Fredholm integral equation (FIE) formula of the second kind. For continuous kernel, the solution of these three MIEs, via the FIEs, is discussed analytically. In addition, for a discontinuous kernel, the Toeplitz matrix method (TMM) and Product Nyström method (PNM) are used to obtain, in each method, a linear algebraic system (LAS). Then, the numerical results are obtained, the error is computed in each case, and compared as well.

##### Perceptions of Pre-service Teachers toward Teaching and Learning Mathematics in Bhutan

Karma Dorji, Karma Wangchuk

Asian Research Journal of Mathematics, Page 39-48
DOI: 10.9734/arjom/2021/v17i730317

This study was carried out to investigate the perceptions of teaching and learning Mathematics by the pre-service Mathematics teachers in Bhutan. The research has employed quantitative methods. A five-point Likert scale survey questionnaire was administered to collect the data. The participants included both male and female pre-service Mathematics teachers from 2 colleges of education in Bhutan aged 18-29 (N = 261). In order to examine the perceptions of Mathematics, the mean interpretation developed by Roble and Bacabac (2016) was used and found that perceptions of pre-service teachers teaching and learning Mathematics were high indicating they enjoy teaching and learning Mathematics. The analysis also showed that there was no significant difference between male and female pre-service teachers perceptions toward teaching and learning Mathematics in Bhutan. Further, the findings also indicated there was no significant difference in perceptions of teaching and learning Mathematics between pre-service teachers at College A and College B. The College of Education might consider devoting additional attention to improving teaching practices in order to experience the real classroom scenario.

##### Preservice Teachers’ Content Knowledge and Learning Obstacles in Shape and Space

Peter Akayuure

Asian Research Journal of Mathematics, Page 49-60
DOI: 10.9734/arjom/2021/v17i730318

The notion of shape and space is the building block for geometry reasoning. Accordingly, it is expected that those trained to teach would possess substantial content knowledge for teaching planar and spatial notions. This study investigated the level of content knowledge and sources of learning obstacles encountered by preservice teachers prior to their first practice teaching of shape and space.  The study adopted a cross-sectional survey design involving 757 second-year preservice teachers from 12 colleges of education in Ghana. Data were collected through written responses to basic shape and space tasks. Quantitative data were analyzed descriptively while qualitative data were analyzed thematically into matrix index. Participants’ content knowledge were categorized into declarative, conditional and procedural knowledge types. The result shows that participants were largely operating at moderate levels of declarative, procedural and conditional content knowledge. However, participants demonstrated higher procedural knowledge than declarative and conditional knowledge. Tasks on prisms and pyramids were more difficult for preservice teachers than those on angles, triangles and quadrilaterals. The learning obstacles encountered were mostly didactical followed by epistemological with few being ontogenetic. The study recommends that tutors should employ investigative didactic strategies to promote the three content knowledge types identified and to address the epistemological and ontogenetic obstacles in preservice teachers’ learning of shape and space.