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

MULTI REPRESENTATION SYSTEMS IN THE DESIGN OF A MICROWORLD FOR THE LEARNING OF SORTING ALGORITHMS

This paper presents the design and the features of a Web-based micro-world-the SORTING micro-world- constructed for the learning of sorting algorithms by secondary level education students. The design of this microworld has been based on modern constructivist and social theories of learning. SORTING is an interactive environment that can support learners: a) to express their own sorting procedures in multiple representation systems (MRS), b) to express their own approaches to typical sorting algorithms, c) to correct their sorting actions while actually sorting entities using typical sorting algorithms by receiving immediate feedback by the system, d) to explore typical sorting algorithms in MRS when these algorithms are automatically performed by the system, and e) to give different meanings to a sorting algorithm when it is represented in different RS so as to acquire a broader view of this algorithm. The typical sorting algorithms in focus are: Bubble-sort, Quick-sort and Selection-sort. The Representation Systems (RS) provided are: i) real object-based simulations: the learner can sort simulations of real objects by using hands-on experience, ii) free-text RS: the learner can reflect on their hands-on experience and then express their sorting approach using natural language in the form of freetext, iii) pseudo-code RS: the learner can express their sorting approach using pseudo-code, and c) flow-chart based RS. Animations are also provided to visualize the sorting of elements using the sorting algorithms mentioned above. These elements are simulations of objects taken from the real lives of learners. Moreover, interactive animations demonstratin