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### Modelling recursion

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Ammari-Allahyari, Mojtaba
(2008)
*Modelling recursion.*
PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b2337419~S15

## Abstract

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.

Item Type: | Thesis or Dissertation (PhD) |
---|---|

Subjects: | L Education > LB Theory and practice of education Q Science > QA Mathematics |

Library of Congress Subject Headings (LCSH): | Recursion theory -- Study and teaching, Recursion theory -- Computer programs, Computer-assisted instruction, College teaching -- Aids and devices |

Official Date: | March 2008 |

Institution: | University of Warwick |

Theses Department: | Institute of Education |

Thesis Type: | PhD |

Publication Status: | Unpublished |

Supervisor(s)/Advisor: | Pratt, David |

Sponsors: | International Mathematical Union (IMU) ; Unesco ; Iran. Ministry of Science, Research & Technology (MSRT) |

Extent: | 383 leaves : ill. |

Language: | eng |

URI: | http://wrap.warwick.ac.uk/id/eprint/3602 |

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