Frontispiece	ii
Title page	iii
Dedication	v
Abstract	vii
Acknowledgements	ix
Figures & Tables	xiv
INTRODUCTION	1
A note on transcripts and diagrams	4
CHAPTER I: PROOF AND PROVING	6
1. What is proving?	6
What is proving in school mathematics?	7
What is proving for professional mathematicians?	8
2. Why do people prove?	10
Why do mathematicians prove?	10
Proving does not always verify	11
Other ways mathematicians verify	13
Why do students prove?	14
Do students prove?	14
Uses of proving	15
Researching the need to prove	17
CHAPTER II: NEEDS AND PROVING	21
1. Explaining	24
Explaining – Formulated proving – Preformal proof	25
Rachel and Eleanor explain the formulae found inthe Arithmagon situation	25
Ben’s formulated proving to explain to others	27
Colin and Anton explain the origin of their Arithmagon formula	28
Colin and Anton’s formulated explanation that n3 - n is a multiple of 6	29
Bill and John’s explanations by formulated proving	30
Explaining–Unformulated Proving	32
Kerry’s short explanation	32
Ben’s unformulated explaining to others	33
Bill’s unformulated explanation that F3nis even	34
Explaining–Analogy	35
Bill’s explanation by analogy	35
Rachel, Ben and Wayne attempt to explain "Why 2?"	36
Summary	38
2. Exploring	39
Exploring – Unformulated proving	39
Bill’s unformulated exploration	40
Sandy’s formulation of his unformulated proving to explore	40
Exploring – Formulated proving – Preformal proof	41
Rachel’s explorations by formulated proving	42
Eleanor’s explorations by formulated proving	43
Exploring – Mechanical Deduction – Preformal proof	44
Exploring – Analogy	45
Exploring – Inductive reasoning	46
Summary	47
3. Verifying	49
Verifying – Unformulated proving	48
Verifying – Mechanical deduction – Preformal proof	49
Verifying – Inductive reasoning	50
Verifying – Authority	50
Anton’s reference to authority	51
Bill’s rejection of analogy, induction, and deduction in the face of authority	51
Verifying in school	52
Summary	52
4. Teacher-games	52
Teacher-game/Verifying – Formulated proving	53
Colin verifies a particular case by reference to formulated proving	53
Kerry proves to verify that F3nis even	54
Teacher-game/Verifying – Formulaic proof making	54
Summary	55
5. Synthesis	56
Verification	56
Explanation	57
Exploration	57
Teacher games and other social contexts	58
CHAPTER III: KERRY, STACEY AND THE ARITHMAGON: REASONING IN ACTION	59
1. Mechanical deduction to explore	59
2. Unformulated proving to explore	60
3. Mechanical deduction to explore	61
4. "Generalizing"	62
5. Inductive and deductive exploring	62
6. Inductive reasoning to explore	66
7. Unformulated proving to explore	67
8. Reasoning by analogy to explain	67
9. Inductive reasoning to explore	68
10. Reasoning by analogy to explain	69
11. Further exploring	70
12. Formulated proving to explain, interpreting a semi-formal proof	70
CHAPTER IV: CONSTRAINTS ON PROVING	72
1. Individual Differences	72
Summary	74
2. Social constraints	75
Eleanor and Rachel	75
Rachel	75
Eleanor	76
Bill and John	77
3. The prompts	77
Arithmagon	77
Fibonacci	78
GEOworld	79
Summary	79
CHAPTER V: TEACHING PROVING	80
1. Proving in the Curriculum	80
Alberta	80
The Standards	82
2. Current practices in teaching proving	83
Schoenfeld on teaching for examinations	83
Teaching at North and South Schools	84
3. Experiments in teaching proving	85
Fawcett’s research	85
Research by Balacheff et al.	87
Research by Lampert in the United States	88
Weaknesses of teaching based on the courtroom metaphor	88
4. Speculations on improving teaching	89
5. Why teach proving?	92
CHAPTER VI: PROVING IN SOCIETY	95
1. What is Rationalism?	95
2. Rationalism’s Weaknesses	97
Deduction and Absolute truth	97
Significance of Gödel’s Theorem outside ofmathematics	100
Rationalism is not universally applicable	103
Dangers of misapplied Rationalism	104
3. Rationalism and other modes of thought	106
CHAPTER VII: REASONING AND RESEARCH FROM AN ENACTIVIST PERSPECTIVE	110
1. Enactivism	110
2. Enactivism and reasoning	114
3. Enactivism and research	119
Basic principles	120
Some specifics	121
4. A few parting words on Enactivism, proving, and teaching	122
REFERENCES	124
APPENDICES
A: RELATED LITERATURE	132
1. Other publications by researchers referenced in the main text	132
2. Discussions of the nature of proof in mathematics	134
3. Research on students’ understanding of the conceptof proof	134
4. Useful literature reviews	136
B: DESIGN AND DÉROULEMENT OF THE RESEARCHSTUDIES	137
1. The North School study	137
Context	137
Outline	137
Participants	138
2. The South School study	138
Context	138
Outline	139
Participants	139
3. The first clinical study	139
Context and outline	139
Participants	140
4. The second clinical study	140
Participants	141
5. Other Studies	141
Sandy	141
Central High School	141
6. Methods	141
7. Prompts	142
Arithmagon	143
Fibonacci	143
GEOworld	144
C: MATHEMATICAL ACTIVITY TRACES	146
1. MATs from the study at North School	146
Mathematical Activity Traces for the Math 13 Pair, Bill & John	146
Fibonacci	146
Arithmagon	148
First Interview	150
Second Interview	153
Mathematical Activity Traces for the Math 30 Pair, Colin & Anton	155
Fibonacci	155
Arithmagon	157
First Interview	159
Second Interview	161
2. MATs from the first clinical study	163
Group I: Ben and Wayne	163
Arithmagon	163
Group II: Stacey and Kerry	166
Arithmagon	166
Fibonacci	168
Group III: Eleanor and Rachel	169
Arithmagon	169
Fibonacci	172
Group IV: Jane & Chris	173
Arithmagon	173
D: TABLES OF RESULTS	175
1. Needs and proving in different problem situations	175
2. Needs and proving by different participants	176
3. Needs and proving for individual participants	177
First clinical study	177
Ben	177
Wayne	177
Stacey	178
Kerry	178
Eleanor	179
Rachel	179
Jane	180
Chris	180
North School study	181
Bill	181
John	181
Colin	182
Anton	182
Sandy	183
E: METHODS OF SOLUTION	184
1. Arithmagon	184
Specific solutions to the puzzle	184
The constraints method	184
Systems of equations	185
The method of false position	185
General solutions	186
The usual formula	186
Stacey and Kerry’s method	186
Colin and Anton’s method	187
Eleanor’s method	187
General problems	188
2. Fibonacci	188
Properties of every nthFibonacci number	189
Generalizations of the Fibonacci sequence	189

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