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The Need to Prove


FIGURES

Figure 1: Intersecting perpendiculars. 8
Figure 2: The Arithmagon prompt. 19
Figure 3: The Fibonacci prompt. 20
Figure 4: First rank of the proving network. 23
Figure 5: Paths related to explaining. 24
Figure 6: Diagrams corresponding to Rachel’s (left) and Eleanor’s (right) equations. 26
Figure 7: Eleanor’s proving to explain her method. 26
Figure 8: Rachel’s proving to explain her method. 26
Figure 9: Labeling of Arithmagon for description of Ben’s explanation. 27
Figure 10: Labeling of triangles for description of Colin and Anton’s explanation. 28
Figure 11: Colin’s written proof from the second interview. 30
Figure 12: Labeling used by Bill and John for their formulae. 31
Figure 13: Representation of addition of odd numbers used in Bill’s proving. 32
Figure 14: Paths related to exploring. 39
Figure 15: Sandy’s formula for the Arithmagon. 41
Figure 16: Rachel’s proving to explore. 42
Figure 17: Work related to Eleanor’s formulated proving to explore. 44
Figure 18: Triangles drawn by Wayne while exploring, relating to geometric properties. 46
Figure 19: Output of GEO 100 100 3. 47
Figure 20: Paths related to verification. 48
Figure 21: Paths related to teacher-games. 53
Figure 22: Laura’s "proof". 55
Figure 23: The complete network. 56
Figure 24: The Arithmagon prompt. 59
Figure 25: Stacey’s triangle with "extended lines". 63
Figure 26: Stacey’s third triangle. 65
Figure 27: Values on the outer triangles. 68
Figure 28: Stacey's four triangles. 69
Figure 29: The proof shown to Stacey and Kerry in the interview session. 70
Figure 30: The Arithmagon prompt. 77
Figure 31: The Fibonacci prompt. 78
Figure 32: Fawcett’s diagram. 90
Figure 33: Perpendiculars to tangents meet at the center. 91
Figure 34: Output of GEO 100 100 3. 145
Figure 35: The Arithmagon prompt. 184
Figure 36: Labeling the Arithmagon for a system of equations. 185
Figure 37: Stacey and Kerry’ general solution. 186
Figure 38: Colin and Anton’s difference relation. 187
Figure 39: Labeling of triangle for the basic relations in Eleanor’s method. 187
Figure 40: Eleanor’s "middle" number. 188
Figure 41: The Fibonacci prompt. 188

 


TABLES
Table 1: Alternate readings. 4
Table 2: Use of systems of equations in solving the Arithmagon. 74
Table 3: Summary of participants’ activities in the Arithmagon situation. 78
Table 4: Schedule of the sessions for the first clinical study. 140
Table 5: Schedule of sessions for the second clinical study. 140
Table 6: Distribution of needs and reasoning according to problem situations. 175
Table 7: Distribution of needs and proving according to participants involved. 176
Table 8: Needs and reasoning – Ben. 177
Table 9: Needs and reasoning – Wayne. 177
Table 10: Needs and reasoning – Stacey. 178
Table 11: Needs and reasoning – Kerry. 178
Table 12: Needs and reasoning – Eleanor. 179
Table 13: Needs and reasoning – Rachel. 179
Table 14: Needs and reasoning – Jane. 180
Table 15: Needs and reasoning – Chris. 180
Table 16: Needs and reasoning – Bill. 181
Table 17: Needs and reasoning – John. 181
Table 18: Needs and reasoning – Colin. 182
Table 19 Needs and reasoning – Anton. 182
Table 20: Needs and reasoning – Sandy. 183

  

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