The Aha! experience

255 Pages·2005·2.49 MB·English

by  Liljedahl, Peter Gunnar.

THE AHA! EXPERIENCE: MATHEMATICAL CONTEXTS,
PEDAGOGICAL IMPLICATIONS
Peter Gunnar Liljedahl
B.Sc. Simon Fraser University 1993
THESIS SUBMllTED IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
In the Faculty
of
Education
O Peter Gunnar Liljedahl2004
SIMON FRASER UNIVERSITY
February 2004
All rights reserved. This work may not be
reproduced in whole or in part, by photocopy
or other means, without the permission of the author.
APPROVAL
NAME Peter Gunnar Liljedahl
DEGREE Doctor of Philosophy
TITLE The Aha! Experience: Mathematical Contexts, Pedagogical
Implications
EXAMINING COMMITTEE:
Chair Kieran Egan, Professor
Rina Zazkis, Professor
Senior Supervisor
Thomas O'Shea, Associate Professor
Member
Peter Borwein, Professor, Mathematics and Statistics
Member
Charles Bingham, Assistant Professor
External Examiner
Dr. Anna Sfard, Professor, Department of Mathematics,
Michigan State University
Wells Hall D-325, East Lansing, MI 48824-1029
Examiner
Date February 20,2004
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Bennett Library
Simon Fraser University
Burnaby, BC, Canada
ABSTRACT
The AHA! experience is a term that captures the essence of the experience of
illumination. In the context of 'doing' mathematics it is the EXPERIENCE of
having an idea come to mind with brevity, suddenness, and a sense certainty. In
the studies presented here I examine this extra-logical process in pursuit of the
answers to three questions: What is the essence of the AHA! experience? What
is the effect of an AHA! experience on a learner? Can the AHA! experience be
controlled, and if so can it be invoked?
The data for this pursuit comes from three distinct sources; the anecdotal
reflections of 76 undergraduate students, the anecdotal reflections of 25
prominent mathematicians, and the mathematics journals of 72 preservice
teachers. The results indicate that, although the AHA! experience is precipitated
by the sudden coming to mind of an idea, what actually sets the AHA! experience
apart from other mathematical experiences is the affective components of the
experience. and only the affective component. That is, what serves to make the
experiences extraordinary is the affective response invoked by the experience of
an untimely and unanticipated presentation of an idea or solution, not the
mystery of the process, and not the idea itself. Hence, the AHA! experience has
a positive and, sometimes profound, transformative effect on a learner's beliefs
and attitudes about mathematics as well as their beliefs and attitudes about their
iii
ability to do mathematics. The results also indicate that a measure of control can
be exercised over the AHA! experience through the manipulation of a problem
solving environment. As such, the results provide a pedagogical approach to
problem solving that can be used in the classroom. Furthermore, a
methodological contribution in the form of a new form of journaling for the
tracking of students' problem solving processes is presented.
For my wife, Theresa.
Once again you have found it in you to give me the time and the
space to pursue my dreams, my passions, and my musings.
For my children, Anders, Connor, and Lena.
Thank you for your patience, your love, and your unfailing smiles.
Disgusted at my want of success, I went away to spend a few days
at the seaside, and thought of entirely different things. One day, as
I was walking on the cliff, the idea came to me, again with the same
characteristics of brevity, suddenness, and immediate certainty .. .
- Henri Poincare (1952, pp. 53-54)
My attitude towards mathematics is that most of it is lying out there,
sometimes in hidden places, like gems encased in a rock. You
don't see them on the surface, but you sense that they must be
there and you try to imagine where they are hidden. Suddenly,
they gleam brightly in your face and you don7 know how you
stumbled upon them. Maybe they always were in plain view, and
we all are blind from time to time.
- Enrico Bombieri, mathematician'
1 must admit that math is challenging for me ... after the AHA!
experience you feel like learning more, because the joy of obtaining
the answer is so exhilarating. It almost refreshes one's mind and
makes them want to persist and discover more answers. It gave me
the inspiration and the determination to do the best that I can do in
the subject.
- Kristie, mathematics student2
1 Excerpt from data presented in chapter six.
2 Excerpt from data presented in chapter five.
vi
I would like to acknowledge the collegiality of Rina Zazkis, without
whom none of this would have been possible. From our many
collaborations I learned much.
I would also like to recognize Peter Borwein for his collaborative
involvement in the solicitation of responses from prominent
mathematicians (the results of which are presented in chapter six)
and Nathalie Sinclair for her timely suggestions of relevant
literature.
I extend my thanks to the Social Sciences and Humanities
Research Council of Canada (SSHR C) and the private scholarships
provided by the Rotary Club of Burnaby and the Lis Welch
Endowment Fund, that funded part of my research.
vii
TABLE OF CONTENTS
..
....................................................................................
Approval Page 11
...
...........................................................................................
Abstract 111
.........................................................................................
Dedication v
.........................................................................................
Quotations vi
..
.........................................................................
Acknowledgements VII
...
............................................................................
Table of Contents vii~
..
...................................................................................
List of Tables xi1
...
................................................................................
List of Figures xi11
..........................................................................................
Foreword xiv
.......................................................................................
Introduction I
. ...................
Chapter One Invention. Creativity. and the AHA! Experience 8
'Doing' Mathematics ......................................................................9
Mathematical Invention and Discovery ............................................I 0
Creativity ..................................................................................1 8
Intuition ....................................................................................2 1
Imagination ...............................................................................2 2
How Does the AHA! Experience Fit (or not fit)? ..................................2 3
. ............................................................
Chapter Two Problem Solving 27
Problem Solving by Design ..........................................................2 8
George Polya: How to Solve It ......................................................2. 9
Alan Schoenfeld: Mathematical Problem Solving ..............................3 1
David Perkins: Breakthrough Thinking ............................................3 3
John Mason: Thinking Mathematically ............................................3. 8
Gestalt: The Psychology of Problem Solving ....................................3 9
Mathematical Problem Solving: Perceptions Versus Reality ................4 0
viii
. ...................................................
Chapter Three Research Questions 42
Research Question One ...............................................................4 3
...............................................................
Research Question Two 44
.............................................................
Research Question Three 45
Summary of Research Questions ...................................................4 6
. .......
Chapter Four The Journey: An Overview of Research Methodology 47
A Look Towards Anecdotal Data ............................................................- 50
Reflective Anecdotal Accounting: Undergraduate Students ................. 51
Reflective Anecdotal Accounting: Prominent Mathematicians ...............5 2
Journaling: Preservice Teachers ....................................................5 2
.
Chapter Five AHA! and Affect: The Case of Undergraduate
........................................................................
Mathematics Students 55
......................................................................................
Methodology -56
The Course. the Participants. and the Teaching Assistant ...................5 6
Creating the Survey .................................................................... 59
..................................................................................
Response Rates 62
An Overview of the Responses ...............................................................6 4
The Affective Domain ........................................................................... 66
Participant's Responses ....................................................................... -70
Anxiety ....................................................................................- 71
Pleasure ...................................................................................7 3
Change in Beliefs ...................................................................... -74
Change in Attitudes .....................................................................7 6
Analysis ............................................................................................ -77
Conclusions .......................................................................................- 80
Chapter Six: AHA! Experiences: The Case of
..................................................................
Prominent Mathematicians 82
Hadamard's Survey Resurrected ............................................................ 84
Constructing the First Survey ................................................................. 86
Selecting the Recipients .......................................................................- 90
Response Rates ..................................................................................9 2
The Second Survey ............................................................................- 93
Responses ....................................................................................... -94
Question One .........................................................................-.9 5
Question Two ..........................................................................I 00
Question Three ........................................................................1 04
Question Four .........................................................................1. 07
Question Five ...........................................................................1 08
Question Six ............................................................................ I1 0
Question Seven ........................................................................1 13
Emergent Themes ............................................................................. I1 6
Sleep .....................................................................................1. 17