A Mathematical Analysis of Student-Generated Sorting Algorithms

Sorting is a process we encounter very often in everyday life. Additionally it is a fundamental operation in computer science. Having been one of the first intensely studied problems in computer science, many different sorting algorithms have been developed and analyzed. Although algorithms are often taught as part of the computer science curriculum in the context of a programming language, the study of algorithms and algorithmic thinking, including the design, construction and analysis of algorithms, has pedagogical value in mathematics education. This paper will provide an introduction to computational complexity and efficiency, without the use of a programming language. It will also describe how these concepts can be incorporated into the existing high school or undergraduate mathematics curriculum through a mathematical analysis of student-generated sorting algorithms

Kompetensi Guru dalam Penerapan Kemahiran Berfikir Aras Tinggi dalam Pengajaran Pendidikan Sejarah

Kajian yang dijalankan ini adalah untuk melihat tahap kompetensi guru dalam pelaksanaan Kemahiran Berfikir Aras Tinggi (KBAT) dalam pengajaran pendidikan sejarah dengan melihat perbezaan antara jantina guru dan opsyen mengajar. Kajian tinjauan dengan pendekatan kuantitatif ini melibatkan seramai 100 orang guru Pendidikan Sejarah dalam Daerah Hulu Langat yang dipilih sebagai responden kajian. Borang soal selidik telah digunakan untuk mendapatkan data atau maklumat yang berkaitan dengan kajian. Maklumat data yang diperoleh telah dianalisis menggunakan SPSS Version 22.0 (Statistical for Social Science). Hasil daripada dapatan kajian ini dihuraikan dalam bentuk deskriptif dan inferensi. Ujian T tidak bersandar telah digunakan untuk melihat tahap kompetensi guru dalam melaksanakan Kemahiran Berfikir Aras Tinggi (KBAT) yang dilihat dari tiga konstruk utama iaitu pengetahuan, kemahiran serta sikap guru. Ujian Manova satu hala pula dijalankan bagi mengenal pasti perbezaan penerapan KBAT dari tiga konstruk utama pedagogi, kurikulum dan pentaksiran dalam pengajaran dan pembelajaran sejarah berdasarkan opsyen mengajar. Dapatan kajian melaporkan bahawa tahap kompetensi guru dalam melaksanakan Kemahiran Berfikir Aras Tinggi (KBAT) dalam pengajaran pendidikan sejarah berada pada aras yang tinggi. Kajian ini juga membuktikan tiada perbezaan yang signifikan terhadap pengajaran guru opsyen Pendidikan Sejarah dan bukan opsyen Pendidikan Sejarah terhadap pelaksanaan KBAT. Ini bermakna guru-guru yang mengajar Sejarah dapat mengamalkan KBAT semasa sesi pengajaran dan pembelajaran Sejarah dengan baik

A Pre-Programming Approach to Algorithmic Thinking in High School Mathematics

Given the impact of computers and computing on almost every aspect of society, the ability to develop, analyze, and implement algorithms is gaining more focus. Algorithms are increasingly important in theoretical mathematics, in applications of mathematics, in computer science, as well as in many areas outside of mathematics. In high school, however, algorithms are usually restricted to computer science courses and as a result, the important relationship between mathematics and computer science is often overlooked (Henderson, 1997). The mathematical ideas behind the design, construction and analysis of algorithms, are important for students' mathematical education. In addition, exploring algorithms can help students see mathematics as a meaningful and creative subject. This study provides a review of the history of algorithms and algorithmic complexity, as well as a technical monograph that illustrates the mathematical aspects of algorithmic complexity in a form that is accessible to mathematics instructors at the high school level. The historical component of this study is broken down into two parts. The first part covers the history of algorithms with an emphasis on how the concept has evolved from 3000 BC through the Middle Ages to the present day. The second part focuses on the history of algorithmic complexity, dating back to the text of Ibn al-majdi, a fourteenth century Egyptian astronomer, through the 20th century. In particular, it highlights the contributions of a group of mathematicians including Alan Turing, Michael Rabin, Juris Hartmanis, Richard Stearns and Alan Cobham, whose work in computability theory and complexity measures was critical to the development of the field of algorithmic complexity. The technical monograph which follows describes how the complexity of an algorithm can be measured and analyzes different types of algorithms. It includes divide-and-conquer algorithms, search and sort algorithms, greedy algorithms, algorithms for matching, and geometric algorithms. The methods used to analyze the complexity of these algorithms is done without the use of a programming language in order to focus on the mathematical aspects of the algorithms, and to provide knowledge and skills of value that are independent of specific computers or programming languages. In addition, the study assesses the appropriateness of these topics for use by high school teachers by submitting it for independent review to a panel of experts. The panel, which consists of mathematics and computer science faculty in high school and colleges around the United States, found the material to be interesting and felt that using a pre-programming approach to teaching algorithmic complexity has a great deal of merit. There was some concern, however, that portions of the material may be too advanced for high school mathematics instructors. Additionally, they thought that the material would only appeal to the strongest students. As per the reviewers' suggestions, the monograph was revised to its current form