The purpose of my research is to examine and explore the ways that undergraduate students understand the concept of recursion. In order to do this, I have designed computer-based software, which provides students with a virtual and interactive environment where they can explore the concept of recursion, and demonstrate and develop their knowledge of recursion through active engagement. I have designed this computer-based software environment with the aim of investigating how students think about recursion. My approach is to design digital tools to facilitate students' understanding of recursion and to expose that thinking. My research investigates students' understanding of the hidden layers and inherent complexity of recursion, including how they apply it within relevant contexts. The software design embedded the idea of functional abstraction around two basic principles of: 'functioning' and 'functionality'. The functionality principle focuses on what recursion achieve, and the functioning dimension concerns how recursion is operationalised. I wanted to answer the following crucial question: How does the recursive thinking of university students evolve through using carefully designed digital tools? In the process of exploring this main question, other questions emerged: 1. Do students understand the difference between recursion and iteration? 2. How is tail and embedded recursion understood by the students? 3. To what extent does prior knowledge of the concept of iteration influence students' understanding of tail and embedded recursion? 4. Why is it important to have a clear understanding of the control passing mechanisms in order to understand recursion? 5. What is the role of functional abstraction in both, the design of computer-based tools and the students' understanding of recursion? 6. How are students' mental models of recursion shaped by their engagement with computer-based tools? From a functional abstraction point of view almost all previous research into the concept of recursion has focused on the functionality dimension. Typically, it has focused on procedures for the calculation of the factorial of a natural number, and students were tested to see if they are able to work out the values of the a function recursively (Wiedenbeck, 1988; Anazi and Uesato, 1982) or if they are able to recognize a recursive structure (Sooriamurthi, 2001; Kurland and Pea, 1985). Also, I invented the Animative Visualisation in the Domain of Abstraction (AVDA) which combines the functioning and functionality principles regarding the concept of recursion. In the AVDA environment, students are given the opportunity to explore the hidden layers and the complicated behaviour of the control passing mechanisms of the concept of recursion. In addition, most of the textbooks in mathematics and computer sciences usually fail to explain how to use recursion to solve a problem. Although it is also true that text books do not typically explain how to use iteration to solve problems, students are able to draw on to facilitate solving iterative problems (Pirolli et al, 1988). My approach is inspired by how recursion can be found in everyday life and in real world phenomena, such as fractal-shaped objects like trees and spirals. This research strictly adheres to a Design Based Research methodology (DBR), which is founded on the principle of the cycle of designing, testing (observing the students' experiments with the design), analysing, and modifying (Barab and Squire, 2004; Cobb and diSessa, 2003). My study was implemented throughout three iterations. The results showed that in the AVDA (Animative Visualisation in the Domain of Abstraction) environment students' thinking about the concept of recursion changed significantly. In the AVDA environment they were able to see and experience the complicated control passing mechanism of the tail and embedded recursion, referred to a delegatory control passing. This complicated control passing mechanism is a kind of generalization of flow in the iterative procedures, which is discussed later in the thesis. My results show that, to model a spiral, students prefer to use iterative techniques, rather than tail recursion. The AVDA environment helped students to appreciate the delegatory control passing for tail recursive procedures. However, they still demonstrated difficulties in understanding embedded recursive procedures in modelling binary and ternary trees, particularly regarding the transition of flow between recursive calls. Based on the results of my research, I have devised a model of the evolution of students' mental model of recursion which I have called – the quasi-pyramid model. This model was derived from applying functional abstraction including both functionality and functioning principles. Pedagogic implications are discussed. For example, the teaching of recursion might adopt 'animative' visualization, which is of vitally important for students' understanding of latent layers of recursion.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:526157 |
| Date | January 2008 |
| Creators | Ammari-Allahyari, Mojtaba |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/3602/ |
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