A course in mathematics appreciation

The purpose of this study is to create a mini-course in mathematics appreciation at the senior high school level. The mathematics appreciation course would be offered as an elective to students in the 11th or 12th grade, who are concurrently enrolled in trigonometry or calculus. The topics covered in the mathematics appreciation course include: systems of numeration, congruences, Diophantine equations, Fibonacci sequences, the golden section, imaginary numbers, the exponential function, pi, perfect numbers, numbers with shape, ciphers, magic squares, and root extraction techniques. In this study, the student is exposed to mathematical proofs, where appropriate, and is encouraged to create practice problems for other members of the class to solve. Also, areas for research are suggested so that the student may explore, even more deeply, areas which hold a particular interest for that student. These topics are treated with a three-pronged approach – historical, recreational, and practical. It is the author\u27s contention, supported by research, that this approach, along with the choice of topics, will assist in developing and enhancing the mathematics potential of the student to the highest possible extent

The Four Phases of Philosophy

Introduction and translation of “The Four Phases of Philosophy” by Franz Brentano

Accurate Parametric Inference for Small Samples

We outline how modern likelihood theory, which provides essentially exact inferences in a variety of parametric statistical problems, may routinely be applied in practice. Although the likelihood procedures are based on analytical asymptotic approximations, the focus of this paper is not on theory but on implementation and applications. Numerical illustrations are given for logistic regression, nonlinear models, and linear non-normal models, and we describe a sampling approach for the third of these classes. In the case of logistic regression, we argue that approximations are often more appropriate than `exact' procedures, even when these exist.Comment: Published in at http://dx.doi.org/10.1214/08-STS273 the Statistical Science (http://www.imstat.org/sts/) by the Institute of Mathematical Statistics (http://www.imstat.org