APPENDIX D

TABLES OF RESULTS

The following tables summarize some of the information presented in Chapters II, III and IV. Table 6 shows the distribution of needs and reasoning according to problem situations. Table 7 shows the distribution of needs and reasoning according to the participants involved. The remaining tables show distributions for individual participants, indicating problem situations.

1. Needs and proving in different problem situations

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self	F	A
Explaining to others	A, F	A, N, O		A, O
Exploring to a goal	A	A	A	A	A, F, G, N, O
Exploring without goal	A, F	A	A, F		A, F, G
Verifying	A		A		A, F, G, N, O	A, O
Teacher-game		A, N

Table 6: Distribution of needs and reasoning according to problem situations.

Key to problems:

A: Arithmagon

F: Fibonacci

G: GEOworld

N: Patterns in n³-n

O: Sum rule, Odd+Odd=Even

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2. Needs and proving by different participants

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self	K	E, K, R
Explaining to others	Be, Bi	A, Be, Bi, Co, E, Jo, R		Be, Bi, St, W
Exploring to a goal	Sa	E, Sa	Co, E, K, St, R	W	A, Be, Bi, Co, E, Jo, K, R, St, W
Exploring without goal	St	R	Ja, Ch		A, Be, Bi, Co, E, Jo, K, R, St, W
Verifying	Bi		Co		A, Be, Bi, Co, E, K, R, St, W	A, Bi
Teacher-game		Co, K, R

Table 7: Distribution of needs and proving according to participants involved.

Key to participants:

A: Anton (Math 30 student, worked with Colin)

Be: Ben (undergraduate student, worked with Wayne)

Bi: Bill (Math 13 student, worked with John)

Ch: Chris (undergraduate student, worked with Jane)

Co: Colin (Math 30 student, worked with Anton)

E: Eleanor (Undergraduate student, worked with Rachel)

Ja: Jane (undergraduate student, worked with Chris)

Jo: John (Math 13 student, worked with Bill)

R: Rachel (undergraduate student, worked with Eleanor)

Sa: Sandy (grade 6 student, informal interview)

St: Stacey (undergraduate student, worked with Kerry)

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3. Needs and proving for individual participants

In the following tables letter codes for the problem situations in which the reasoning took place are used. The key for Table 6 should be consulted. An “X” indicates that such reasoning occurred several times, but that no specific example is present in Chapter II. In all other cases a description of the episode in question can be found in the appropriate section of Chapter II or, in the case of Stacey and Kerry, in Chapter III.

First clinical study

Ben

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others	A	A		A
Exploring to a goal					X
Exploring without goal					X
Verifying					X
Teacher-game

Table 8: Needs and reasoning — Ben.

Wayne

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others				A
Exploring to a goal				A	X
Exploring without goal					X
Verifying					X
Teacher-game

Table 9: Needs and reasoning — Wayne.

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Stacey

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others				A
Exploring to a goal			A		X
Exploring without goal	A				X
Verifying					X
Teacher-game

Table 10: Needs and reasoning — Stacey.

Kerry

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self	F	A
Explaining to others
Exploring to a goal			A		X
Exploring without goal					X
Verifying					X
Teacher-game		A

Table 11: Needs and reasoning — Kerry.

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Eleanor

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self		A
Explaining to others		A
Exploring to a goal		A	A		X
Exploring without goal					X
Verifying					X
Teacher-game

Table 12: Needs and reasoning — Eleanor.

Rachel

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self		A
Explaining to others		A
Exploring to a goal			A		X
Exploring without goal		A			X
Verifying					X
Teacher-game		A

Table 13: Needs and reasoning — Rachel.

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Jane

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others
Exploring to a goal			A		X
Exploring without goal			X		X
Verifying					X
Teacher-game

Table 14: Needs and reasoning — Jane.

Chris

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others
Exploring to a goal			A
Exploring without goal			X
Verifying
Teacher-game

Table 15: Needs and reasoning — Chris.

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North School study

Bill

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others	F	O		O
Exploring to a goal					X
Exploring without goal	F				X
Verifying	A					O
Teacher-game

Table 16: Needs and reasoning — Bill.

John

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others		O
Exploring to a goal					X
Exploring without goal					X
Verifying					X
Teacher-game

Table 17: Needs and reasoning — John.

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Colin

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others		A, N
Exploring to a goal			A	X
Exploring without goal				X
Verifying			A	X
Teacher-game		N

Table 18: Needs and reasoning — Colin.

Anton

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others		N
Exploring to a goal					X
Exploring without goal					X
Verifying					A	A
Teacher-game

Table 19: Needs and reasoning — Anton.

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Sandy

	Proving Unformulated	Formulated	Mech. Deduction	Reasoning by analogy	Inductive reasoning	Referring to an authority
Explaining to self
Explaining to others
Exploring to a goal	A	A
Exploring without goal
Verifying
Teacher-game

Table 20: Needs and reasoning — Sandy.

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