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As is mentioned in Chapter VII, part of the analysis of the data involved making Mathematical Activity Traces (MATs) which chart the episodes which occurred in a session. This made it easier to see shifts in reasoning, and to associate reasoning with needs. The analysis of data progressed to different stages, depending on its usefulness to my research at that time. For this reason not every group had a MAT made for them. These are included both to give examples of this way of recording the events in a problem solving session, and to organize the episodes I have referred to in the text.
Mathematical Activity Traces for the Math 13 Pair, Bill & John
| 5.3 | J suggested dividing sums of digits by 3. B responded that they will divide evenly by 3, but no one did the division. | 17-18 |
| 6 | B checked F3n is even conjecture for larger values. | |
| 7.1 | J observed that F4n ∏ F4n-4 ª 6. He used this to calculate: 987 ∏ 144 ª 6; 6 ¥ 21 ª 144 | 20 |
| 7.2 | B found next F4n to test J’s conjecture | 20 |
| 7.3 | J checked conjecture with 4181. It failed as 4181 is not the right number. | 21 |
| 7.4 | B checked sequences to see if they had the right number | 23 |
| 7.5 | They checked J’s conjecture. | 24 |
| 7.6 | J formulated his conjecture. | 24 |
| 7.7 | B commented on its lack of practical utility. | |
| 8.1 | B commented that his digit sum conjecture had not been disproved. J suggested reexamining it. B suggested a counter example would soon occur. | 25 |
| 8.2 | J noticed a pattern in the sums of digits: 3, 3, 9, 24, 24, | |
| 8.3 | B calculated the next numbers. | 26 |
| 8.4 | They checked the next number. J took 10946 to be the next and found it didn’t work. B took 46368 to be the next number and found it worked. They clarified which number is F4n. | 27-28 |
| 9.1 | They discussed J’s ratio conjecture again. J commented on its inexact nature, B commented on its uselessness. | 28 |
| 10.1 | B checked F3n for even numbers again. J watched. The conjecture was confirmed for several more values, but B was still not certain. | |
| 10.2 | B considered his use of digit sums. He was uncertain as to its applicability to mathematics as he had only seen the technique used in astrology/numerology. | 30 |
| 10.3 | B on F3n: “We could go on forever but we can’t know that it’s always even” | 31 |
| 11.1 | J asked DR if they are on the “right track”. DR said yes, and asked for a recap. | |
| 11.2 |
They listed 3 conjectures they found: 1.
the recursive rule for the sequence 2.
B’s digit sum conjecture |
|
| 11.3 | B formulated J’s conjecture | 35 |
| 11.4 | B added 0 to beginning of sequence and gave a deductive justification for its being correct. | 35 |
| 12.1 | B noticed pattern in sums: 3,3,9,6,6,9. | 36 |
| 12.2 | J became confused between Fn and F4n. B corrected J’s confusion | 36-37 |
| 12.3 | B began to extend sequence systematically, recognizing need for some organization in their work. J was working on a dividing pattern using digit sums which he never described. | 38-40 |
| 12.4 | B found a contradiction to his digit sum conjecture; 260497, which was due to an addition error in extending the sequence. | 41 |
| 12.5 | B rejected his conjecture on the basis of his counter example. | |
| 12.6 | Both began looking for patterns in differences of the digit sums. They noted they are all multiples of 3. | 44 |
| 12.7 | B suggested a pattern of alternating divisibility by 3 and 2, in groups of six. | |
| 13.1 | DR pointed out addition error in sequence. | 47 |
| 13.2 | B recalculated sequence. J watched. | |
| 13.3 | B checked next value of F4n: 317811. “It still works” | 50 |
| 13.4 | DR talked about potential for surprise in the sequence | 51 |
| 13.5 | B proposed trying next value of F4n to test digit sum conjecture, and did so. | 52 |
| 14.1 | DR asked about surprise. | |
| 14.2 | B commented that there is no reason to be surprised at a counter example and expressed doubt of F3n even conjecture. | 56 |
| 14.3 | J investigated sums of digits for other numbers. | |
| 15.1 | B suggested that differences between F4ns are divisible by 3 | |
| 15.2 | Checked several cases, but doubted generality. | |
| 16.1 | DR pointed out that 3, 21, 144, ... are all multiples of 3. B didn’t think 144 was. | 62 |
| 16.2 | DR asked B if he can give a reason for every third number to be even. B suggested it is because the sum of two odd numbers is even. | 64 |
| 16.3 | B checked sequences to see if pairs of odds continue, and to see if O+O=E rule holds. | 66 |
| 3.2 | Looked for patterns | 11:06-11:08 |
| 3.3 | J wondered if 11, 18, 27 are all multiples of something. B formulated the task as finding a simpler way (better than trial and error) to find the solution. | 11:08 |
| 4.1 | B returned to his “old theory” (2.1) which worked in two cases. | 11:09 |
| 4.2 | J reported that the sum of the sides is 56. He divided by 3, because the triangle has three sides, to get 18. | |
| 4.3 | B looked for factors in common. | 11:10 |
| 5.1 | J suggested looking at square (they didn’t) | 11:13 |
| 5.2 | Looked for patterns | 11:14-11:16 |
| 6.1 | B found a possible method: 27 ∏ 18 ¥ 11 ª16.5 ª 17, which is one of the numbers needed. He tried this method with other sides, and it failed, but he reasoned that he only needed one corner to find the others. He tried his method on a 3-6-17 triangle . J worked in silence. | 11:16 |
| 6.2 | B tried his method on a 41-86-92 triangle. | 11:19 |
| 6.3 | J asked B what he was doing. B described his conjecture that A ¥ B ∏ C = x, and said he was trying other examples. | |
| 6.4 | B tried a 14-29-56 triangle. | 11:22 |
| 7.1 | J said that the “minus 1 thing” doesn’t work. They had conjectured earlier that one vertex was a side minus 1. B gave a reason “It only worked here [11-18-27] because this is 1”. | 11:22 |
| 7.2 | B: “Maybe this would work with the 56 theory” | 11:23 |
| 8.1 | B wondered if J tried the square. J said he did but it “didn’t go” | 11:25 |
| 8.2 | B considered solving squares, but decided to do triangles first. Conjectured that the solution was related to the number of sides. | |
| 9.2 | B conjectured the pattern of odd-odd-even is important. | 11:28 |
| 9.3 | B conjectured (A - C) + (B ∏ C), rounded off, which in 11-18-27 triangle gives him B, 18, again. He recognized that this is not his goal. | 11:30-11:31 |
| 9.4 | J conjectured 2 + 27 (in 14-29-56 triangle) gives 29, but realized he was reasoning from corners to sides, and he didn’t know corners to start with. | 11:32 |
| 10.1 | J showed B the 11-17-21-15 square he was working on at some previous time. B worked on it, conjecturing that the solution of the square might help solve the triangle. | 11:33 |
| 10.2 | J suggested adding all the numbers. | |
| 10.3 | B commented squares are harder than triangles. Wondered if there is really a link between squares and triangles. Decided there is, based on both occurring in the same situation. | 11:34-11:35 |
| 10.4 | B returned to triangles, suggesting that the only reason he has for thinking there is an easier way, the need to solve large number triangles, could be eliminated by employing SI units, or scientific notation. | 11:37-11:38 |
| 11.1 | DR suggested looking at triangles with smaller numbers. | 11:38 |
| 11.2 | B worked on 2-3-3 triangle. | 11:39 |
| 11.3 | B conjectured, based on 3-2=1, 3 ¥ 1 = 3. He generalized and checked with another triangle | 11:40 |
Session ended due to time constraint.
In this trace each group of episodes is given a heading indicating the general nature of the episodes, or the stage in deductive exploration
| 1 | Stating unknowns and givens | (Time of day) 1:59-2:05 |
| 1.1 | DR pointed to B+C=18; B added A+C=27 | 2:00-2:01 |
| 1.2 | DR pointed to working with a particular corner, as it relates to the others. | 2:03-2:05 |
| 2 | Building on givens | 2:05-2:08 |
| 2.1 | DR: combined 27+11=38, related to A+B+A+C | 2:05-2:07 |
| 2.2 | B: made a false start solving from new relation. | 2:07-2:08 |
| 3 | Stating more givens | 2:09-2:12 |
| 3.1 | DR pointed to B & C | 2:09-2:10 |
| 3.2 | DR wrote known relations | 2:10 |
| 3.3 | B asserted that finding one corner A will be enough to solve all. | 2:10 |
| 3.4 | J summed 11+17+27+38 and was corrected by B | 2:11-2:12 |
| 4 | Building from givens with a new triangle | 2:13-2:16 |
| 4.1 | B stated that knowing the solution interferes with solving in a new way. DR offered 1-4-12 triangle as an alternative puzzle, and established known relations. | 2:12-2:13 |
| 4.2 | DR suggested subtracting (A+B)-(A+C)=3 and simplified to B‑C=3. B claimed to follow, J was uncertain. | 2:14 |
| 4.3 | B pointed out that now that the sum and difference of B & C were both known, a solution could be found. | 2:16 |
| 5 | Review: Digression as DR reviewed simplification of difference for J. | 2:17-2:18 |
| 6 | Building from given continued | 2:19-2:23 |
| 6.1 | B continued: Now that sum and difference of B & C were both known, solution could be found, but he was stumped when no pairs of whole numbers summing to 12 had a difference of 3. | 2:19-2:21 |
| 6.2 | DR suggested looking for numbers in between, and B arrived at 7.5 and 4.5 | 2:21 |
| 6.3 | B verified solution | 2:22-2:23 |
| 7 | B solved a new triangle at DR's suggestion | 2:24-2:30 |
| 7.1 | DR offered 3-18-63 triangle for solution. Suggested B “explain” to J | 2:24 |
| 7.2 | B consulted previous work, and recreated the derivations. | 2:24-2:26 |
| 7.3 | B asked for confirmation that he was proceeding correctly | 2:27 |
| 7.4 | B was worried about C-B=45, as difference should be less than sum. | 2:28 |
| 7.5 | DR suggested adding C-B and C+B. B did so and simplified to C=24 | 2:28-2:30 |
| 7.6 | B verified C=24 in puzzle | 2:31 |
| 8 | Review | 2:31-2:34 |
| 8.1 | J asked how B found the other sides once C was known. B explained | 2:31-2:32 |
| 8.2 | J suggested replacing A+B with a single variable. | 2:33 |
| 8.3 | B recapitulated the derivation. | 2:33-2:34 |
| 9 | Generalization | 2:35-2:37 |
| 9.1 | B asked if the process will always simplify to 2C | 2:35 |
| 9.2 | DR pushed him to try to explain why it would be. | 2:35 |
| 9.3 | B explained his goal: to produce a simple equation. | 2:36 |
| 9.4 | DR asked where the 2 came from. | 2:36 |
| 9.5 | B recapitulated the simplification, and still wondered if the 2C is general | 2:36-2:37 |
| 10 | Formulating | 2:38-2:39 |
| 10.1 | B wrote (A+C)-(A+B)-(B+C) / 2 | 2:38 |
| 10.2 | B explained that this formula simplifies to C. | 2:38 |
| 10.3 | B used formula with numbers. | 2:39 |
| 11 | Checking formula | 2:40-2:41 |
| 11.1 | J suggested solving 11-18-27 triangle with formula | 2:40 |
| 11.2 | B did so. | 2:40-2:41 |
| 12 | Testing B's strength of conviction | 2:41-2:48 |
| 12.1 | B stated that formula works for the second triangle, but that says nothing about general case. | 2:41 |
| 12.2 | DR pointed to the canceling in B’s derivation of the formula and asked if it indicated anything about when the formula would work. B said “I couldn’t say”. | 2:42-2:43 |
| 12.3 | B commented that the canceling eliminated the negatives; the “angles” are positive. | 2:43 |
| 12.4 | DR asked if substituting -21 for B would effects anything. B asserted that the formula would still work with numbers. | |
| 12.5 | DR asked if there are circumstances in which the formula wouldn’t work. | |
| 12.6 | B digressed into a comparison with the area formula for the triangle. | 2:46 |
| 12.7 | B commented: he couldn’t see why it wouldn’t work for all triangles, but he might be wrong. | 2:47 |
| 12.8 | J commented that they don’t need to figure out the other sides once they’ve figured out C with the formula. | 2:47 |
| 12.9 | DR asserted that he can see no reason the formula would not work in general. B concluded that they had found it out. | 2:47 |
| 13 | Review |
| 13.1 | J went through derivation trying to understand where the 48 came from. | 2:49 |
| 13.2 | B explained the origin of the 48 to J. | 2:49 |
| 14 | Formulating | 2:52-2:54 |
| 14.1 | J replaced B’s formula with E-D+F / 2 | 2:52 |
| 14.2 | B asked J if he understood where the 2 came from. | 2:52 |
| 14.4 | B claimed his notation shows why the formula works, while J’s is tidier, easier to learn | 2:53 |
| 14.5 | DR asked if this was related to their school experience. B commented he liked to know why. | 2:54 |
| 15 | Videotape follow-up | 2:55-2:58 |
| 15.1 | DR showed video tape of 13-F; 16.2-16.3 | 2:55-2:57 |
| 15.2 | DR asked why two odds make an even. B gave examples of n+n=even and asserted all numbers would end in digits which determine parity. | |
| 15.3 | J commented that figuring out why is hard, so it’s easier “just to believe” | 2:58 |
| 16 | Starting from a proof | 2:59-3:00 |
| 16.1 | D asked why there’s always two odds in a row in Fn. B questioned odd+even; claimed it wouldn’t always give an odd, then checked a single case and claimed it would. | |
| 17 | Proof analysis, Lemma | 3:00-3:04 |
| 17.1 | DR asked why odd+even=odd? No reply. | 3:00 |
| 17.2 | DR asked what “even” means. B replied “divide by 2” | 3:00 |
| 17.3 | B related odd+even to positive times negative. | 3:01-3:02 |
| 17.4 | J made a comment on evenness. | 3:03 |
| 17.5 | DR gave a weird explanation about odd+even. | 3:04 |
| 18 | Return to an example | 3:04-3:06 |
| 18.1 | DR listed the Fibonacci numbers and labeled each O or E as appropriate. | 3:04 |
| 18.2 | B asserted that F3n is even and F4n is odd. | 3:04 |
| 18.3 | B commented that the pattern is “working out so far ... if I’m correct” | 3:06 |
| 19 | Testing conviction | 3:06-3:08 |
| 19.1 | DR asked if the OOE pattern would continue forever. | 3:06 |
| 19.2 | B said “Yes”, and explained that the sequence started out that way and the pattern repeats, but qualified with “I think” | 3:06-3:07 |
| 19.3 | DR asked how the pattern could change. B said that from what they had seen that would be inconceivable. | 3:08 |
| 19.4 | DR asserted that the pattern did continue. | |
| 20 | Reflections on school math | |
| 20.1 | B & J asked what DR could say about them. | 3:09 |
| 20.2 | DR commented they didn’t work well together, and asked if they had done group work in class. | 3:10 |
| 20.3 | B commented that math is an individual activity. | 3:10 |
| 1 | Conjecture | Time of day 12:59 |
| 1.1 | Given sheet for n3-n | 12:59 |
| 1.2 | B conjectured that they are all factors of 6. | 12:59 |
| 2 | Exploring factors | 1:00?-1:05 |
| 2.1 | DR gave choice: to verify, explain or explore. | 1:00? |
| 2.2 | B and J chose “why” | 1:02 |
| 2.3 | B observed the values of n3-n are multiples of 3 as well. | 1:02 |
| 2.4 | DR asked if all multiples of 6 are even. | 1:03? |
| 2.5 | J gave explanation: 6 is even and adding evens makes evens. B extended this with the example 6¥4=6+6+6+6. | 1:03? |
| 2.6 | DR asked if all multiples of 6 are multiples of 3. | 1:04 |
| 2.7 | J replied that 3 goes into 6 so 3 goes into “these”. B observed that 3 can be a factor of both even and odd numbers. | 1:04-1:05 |
| 3 | Failed formalization | |
| 3.1 | DR asked what they knew about n3-n. | 1:06 |
| 3.2 | B responded that it was a general expression for expressions like 33-3 | 1:06 |
| 3.3 | DR asked if they knew about factoring. They said maybe; probably not. | 1:06-1:07 |
| 4 | Exploring n3-n | 1:07-1:11 |
| 4.1 | DR asked if 53-5 is a multiple of 5 | 1:07 |
| 4.2 | B responded “yes” and noted that 60 (43-4) is a multiple of 4 | 1:08 |
| 4.3 | J suggested finding 63-6. DR said it is 210. | 1:08 |
| 4.4 | B observed that 210 is a multiple of 3, and asked DR what he was trying to get them to see. | 1:08-1:11 |
| 5 | Exploring factors of 120 | 1:11-1:15 |
| 5.1 | DR asked what goes into 120 | 1:11 |
| 5.2 | B and J listed all the factors and thought in silence. | 1:12-1:14 |
| 5.3 | DR observed 120=5¥24, and asked if there is anything special about the factors of 24 | 1:14 |
| 5.4 | B listed the factors. DR asked if 5 was among them, and B replied “No”. | 1:15? |
| 6 | Exploring factors of other numbers | 1:16-1:20 |
| 6.1 | DR suggested looking at the factors of other numbers. B and J wrote out factors | 1:16-1:17 |
| 6.2 | B suggested a false pattern. | 1:18 |
| 6.3 | DR pointed out the sequence 123, 1234, 12345, 123456 | 1:18 |
| 6.4 | B predicted, 1234567, tried it and rejected the conjecture. | 1:19-1:20 |
| 7 | Exploring groups of factors | 1:20-1:23 |
| 7.1 | DR asked what the factors of 210 are. | 1:20 |
| 7.2 | B wrote them out. | 1:21 |
| 7.3 | DR pointed out the groups 123 and 567. | 1:22 |
| 7.4 | B Saw no pattern, and suggested investigating differences. | 1:22-1:23 |
| 8 | Formalizing with a generic example | 1:24-1:34? |
| 8.1 | DR factors 6 out of 63-6, and then redistributes 62-1 | 1:24-1:28 |
| 8.2 | DR asked if breakdown of 63-6 into (6-1)(6)(6+1) would work for 5 | 1:28 |
| 8.3 | B said “yes” and checked. | 1:28-1:29 |
| 8.4 | DR asked if it would work in general. B said yes | 1:30 |
| 8.5 | DR attempted to explain that n3-n is always a multiple of 6, based on the factoring. | 1:30-1:32 |
| 8.6 | B wondered if all the work was really needed. He pointed out that it only gave them a few factors, which DR claimed was enough to show the conjecture. | 1:32-1:33 |
| 9 | Explaining O+O=E | 1:35-1:38? |
| 9.1 | DR asked why adding rules for odd and evens work. | 1:35 |
| 9.2 | B gave his analogy with integer multiplication, including implicit use of the parity adding rules to justify the integer multiplication rules. | 1:35-1:36 |
| 9.3 | DR rejected his argument because it was based on the notation which is of recent introduction. | 1:36 |
| 9.4 | DR asked J for his ideas. He had nothing to add. | 1:38? |
| 10 | Exploring evenness | 1:39 |
| 10.1 | DR asked what they knew about even numbers. | 1:39 |
| 10.2 | B said they were multiples of 2, except for 0, and they can be divided by 2. | 1:39 |
| 10.3 | J argued that E+E=E because apple+apple=apple. DR gave O+O, orange+orange, as a counter example. | 1:39 |
| 11 | Explaining E+E formally | 1:40-1:43 |
| 11.1 | DR went through argument that E+E=E based on denoting E as “2xsomething” and redistributing. | 1:40-1:42 |
| 11.2 | J asked for clarification: if you multiply 2 by an odd you get an even. | 1:43 |
| 12 | Explaining E+O formally | 1:44-1:48 |
| 12.1 | DR asked B to do E+O | 1:44 |
| 12.2 | B wondered if the factor of 2 for an even would be 1 or 3 for an odd. | 1:44 |
| 12.3 | DR asked what they knew about odds. J offered that they had two factors, and then realized he was thinking of primes and composites. | 1:44 |
| 12.4 | DR went through argument with generic example of 2+3=5 | 1:45 |
| 12.5 | DR generalized to 2n+2m+1 | 1:45 |
| 13 | Explaining formally O+O | 1:48?-1:51 |
| 13.1 | DR asked B to do O+O | 1:48? |
| 13.2 | B began with 2(3)+2(3) | 1:48? |
| 13.3 | DR prompted “odds are...?” B Replied “1 more than even” | 1:49 |
| 13.4 | DR went through argument | 1:51 |
| 14 | Explaining graphically E+E | 1:53-1:54 |
| 14.1 | DR drew arrays of pairs of dots, and asked whether the total number was even or odd. | 1:53 |
| 14.2 | J explained that there are 2 dots in every group, so “it’ll always be even” | 1:53 |
| 14.3 | DR formalized the number of pairs to be n and m and asked how many dots in each column (2n: J) and how many in the total (2n+2m: B) | 1:53-1:54? |
| 15 | Explaining graphically E+O | 1:55-1:57 |
| 15.1 | DR drew arrays for E+O | 1:55 |
| 15.2 | J said the total would be odd. B began assigning symbols to the number of dots. | 1:55 |
| 15.3 | B worked through the formal argument for odd | 1:56 |
| 15.4 | DR asked for clarification of why the answer was odd, and B gave it, based on the total being one more than a multiple of 2. | 1:57 |
| 16 | Explaining graphically and formally O+O=E | 1:58-1:59 |
| 16.1 | DR asked what the pictures for O+O would look like. | 1:58 |
| 16.2 | B drew picture, adding formal labels as he went, and argued formally that the sum was even. | 1:58 |
| 16.3 | DR asked why 2 more than an even number would be even. | 1:59 |
| 16.4 | J explained it based on the alternation of evens and odds. | 1:59 |
| 16.5 | B explained it based on it being the sum of two even numbers. | 1:59 |
| 17 | Debriefing | 2:00-2:01 |
| 17.1 | DR explained that his plan had been to take them from unformulated proving to formulated proving. | 2:00 |
| 17.2 | DR commented on the value of pictures to make things less formal. | 2:01 |
| 18 | Does proof explain? | 2:02 |
| 18.1 | DR asked if proof explained that O+O=E B said yes, with a good summary of argument | 2:02 |
| 19 | Testing conviction | 2:03 |
| 19.1 | DR proposed that for very large odds their sums are odd. | 2:03 |
| 19.2 | B accepted this idea. | 2:03 |
| 19.3 | DR said it’s not actually true. | 2:03 |
| 20 | Winding down | 2:04-2:16 |
| 20.1 | DR discussed transfinite numbers. | 2:04 |
| 20.2 | B wondered why he and J were picked. DR explained. | 2:05-2:07 |
| 20.3 | B said he didn’t like proofs, he doesn’t care why. | 2:13 |
| 20.4 | B wondered if new math is always being discovered. | 2:16 |
| 1 | Finding Answers | (time of day)10:57-11:01 |
| 1.1 | Given problem sheet | 10:57 |
| 1.2 | Found recursive rule for sequence and formalized it. | 10:58 |
| 1.3 | Claimed F3n is always even. | 10:59 |
| 1.4 | Claimed F4n is always a multiple of 3 | 11:00-11:01 |
| 1.5 | C wrote out “answers” neatly on problem sheet. A asked if there was more to it. DR said yes. | 11:02 |
| 2 | Data gathering. | 11:02-11:06 |
| 2.1 | They extended the sequence to 24 terms, making an error at F10 | 11:02-11:06 |
| 3 | Finding a rule for F3n | 11:07-11:12 |
| 3.1 | C noticed 8x4+2=32. Conjectured F3n = 4F3n-3 + F3n-6. | 11:07 |
| 3.2 | Encountered counter examples due to error of F10=600. | 11:08-11:12 |
| 3.3 | Found error, formalized rule for F3n | 11:12 |
| 4 | Finding a rule for F4n | 11:13-11:20 |
| 4.1 | Conjectured F4n = 7F4n-4 - F4n-8. | 11:14 |
| 4.2 | Encountered problems verifying due to errors in sequence. Re-calculated the sequence correctly. | 11:15-11:17 |
| 4.3 | Encountered counter example due to miscalculation. | 11:18 |
| 4.4 | Tried more cases. | 11:19 |
| 4.5 | C formalized rule for F4n | 11:20 |
| 5 | Looking for another F3n rule | 11:21-11:24 |
| 5.1 | A considered looking at F5n but didn’t as it was not on the sheet. | 11:21 |
| 5.2 | They both looked for other patterns in F3n, without finding any. | 11:21-11:24 |
| 6 | Finding a rule for F5n | 11:25-11:31 |
| 6.1 | C made a variety of calculations attempting to find a new rule for F5n. | 11:25-11:27 |
| 6.2 | A joined him in considering F5n | 11:28-11:31 |
| 6.3 | C described F5n rule. A commented it was the same rule as they had found for F3n and F4n. | 11:31 |
| 7 | Finding a rule for F7n | 11:32-11:33 |
| 7.1 | C made a variety of calculations attempting to find a rule for F7n. | 11:32-11:33 |
| 8 | Formalizing | 11:33-11:35 |
| 8.1 | C wrote out the rules for F3n, F4n and F5n | 11:33-11:34 |
| 8.2 | C saw a pattern in 4, 7, 11 as 4+3=7, 7+4=11, 11+5=16. A saw the alternating addition and subtraction in the rules. | 11:35 |
| 9 | Predicting from a formalism | 11:36-11:38 |
| 9.1 | C predicted F6n = 16F6n-6 - F6n-12. | 11:36 |
| 9.2 | When they’re calculations failed they rechecked them several times for errors. | 11:36-11:38 |
| 10 | Summary | 11:39-11:41 |
| 10.1 | DR asked them to summarize their results and how they had arrived at them. | 11:39 |
| 10.2 | C Attempted to locate the formalisms he had written down, and read them. A indicated that they had tried their conjectures out and they had worked. | 11:40-11:41 |
| 1 | Solving by mechanical deduction | (time of day) 12:50-12:53 |
| 1.1 | Given sheet. | 12:50 |
| 1.2 | A labeled triangle and set up system of equations. | 12:51 |
| 1.3 | A found an incorrect solution. | 12:52 |
| 1.4 | A asked DR of their solution was correct. DR asked them to check it, and they did. | 12:53 |
| 2 | A second approach | 12:54-12:55 |
| 2.1 | C attempted to solve by isolating x and equating the expressions 18‑y and 27-z. | 12:54-12:55 |
| 3 | Redoing the system of equations | 12:55-12:57 |
| 3.1 | C asked A to explain how he had obtained x-z=7 in his derivations | 12:55 |
| 3.2 | A solved the system of equations again, obtaining the correct solution and checked it. | 12:56-12:57 |
| 4 | Considering the general case | 12:57-12:58 |
| 4.1 | C set up a general triangle with equations | 12:57 |
| 4.2 | A suggested solving C’s equations, but decided they couldn’t. | 12:58 |
| 5 | Searching for patterns | 12:59-1:04 |
| 5.1 | C set up a 23-30-33 triangle, by beginning with known corners and adding to find the sides. Both looked for patterns in this triangle. | 12:59-1:00 |
| 5.2 | A noticed that the sum of the sides is equal to twice the sum of the corners. C wrote out this relation formally. | 1:00-1:01 |
| 5.3 | They attempted to use the sum relation to solve a 23-60-100 triangle. They determined the sum of the corners, but concluded that the triangle had no solution. | 1:01-1:03 |
| 5.4 | C solved a 4-4-4 triangle, then proposed a 4-4-6 triangle. A systematically checked all positive pairs of numbers adding to 6, and then rejected the triangle as unsolvable. | 1:03-1:04 |
| 6 | Searching for a pattern in the original triangle | 1:05-1:09 |
| 6.1 | C Suggested that solvable triangles might require the ‘prime plus two multiples of three’ pattern. He attempted to solve a 13-9-12 triangle. C found solution by guessing, but A wanted a formula. | 1:05 |
| 6.2 | They re-examined the solution to the original triangle, and observed that 17=18-11+10. They conjectured a general formula of x=a-b+10. | 1:06-1:08 |
| 6.3 | A suggested trying another triangle. C drew 3-15-24 triangle and they tried it with their formula. | 1:08-1:09 |
| 7 | Searching for more patterns | 1:09-1:16 |
| 7.1 | C suggested investigating 13-16-17 triangle, which he had set up from known corners. | 1:09 |
| 7.2 | They calculated the sum of the sides, and A divided the triangle into sum triangles and calculated sums of corners for them. | 1:10-1:11 |
| 7.3 | Solved 10-16-26 triangle | 1:13-1:14 |
| 7.4 | A solved 20-30-22 triangle using a system of equations. | 1:14-1:15 |
| 7.5 | C noticed relationship between differences of sides and differences of corners. He related this to the sharing of corners. | 1:15-1:16 |
| 8 | Investigating squares | 1:17-1:18 |
| 8.1 | C assumed opposite sides must be equal. | 1:17 |
| 8.2 | A divided a 3-4-3-4 square to make a triangle, and found a solution. | 1:18 |
| 9 | Discovering a method | 1:19-1:22 |
| 9.1 | Found solution to 5-12-13 triangle based on differences. | 1:19 |
| 9.2 | Found solutions to 20-30-36 triangle, and 24-14-16 triangle. | 1:20-1:22 |
| 10 | Formalizing | 1:23-1:27 |
| 10.1 | C labeled 11-18-27 triangle. | 1:23 |
| 10.2 | A suggested using x, X to represent opposite sides and corners. | 1:24 |
| 10.3 | C wrote out Z-X = x-z. | 1:25 |
| 10.4 | C commented that their method was not a general solution, and began deriving formally. | 1:26 |
| 10.5 | C derived Z-X+Y=2x, but thought that they had already found this. | 1:27 |
| 11 | Back to squares | 1:28-1:31 |
| 11.1 | Investigated squares with all sides equal. Found that there were many solutions. Formalized this conclusion. | 1:28-1:29 |
| 11.2 | Introduced diagonals. They found a conflict in a 3-3-3-3 square with 1-2-1-2 as the solution. C concluded that squares could be solved in various ways. | 1:30-1:31 |
| 12 | Formalizing their method for triangles | 1:32-1:38 |
| 12.1 | C drew a general triangle. | 1:32-1:33 |
| 12.2 | A attempted to investigate triangles by adding sides to make a square. | 1:34 |
| 12.3 | C wrote difference relationship as a ratio relationship, and began trying to solve 11-18-27. | 1:35-1:36 |
| 12.4 | A asked if they were close to a formula, and then set up another general triangle. | 1:37 |
| 12.5 | C concluded that his ratios wouldn’t work. | 1:38 |
| 13 | Refining the method | 1:39-1:41 |
| 13.1 | A declared that they were “stuck”. They looked for more patterns. | 1:39 |
| 13.2 | They solved 10-16-44 triangle. C solved by taking half the difference ( =3) and adding or subtracting it from half the other side (22), to find the two adjacent corners (he found 20 and 24, by mistake, and did not check). | 1:40 |
| 13.3 | C concluded some triangles were impossible because the differences would have to be divisible by 2. | 1:41 |
| 14 | Formalizing the method | 1:42-1:43 |
| 14.1 | A wrote out difference relation as B-C = b<c | 1:42 |
| 14.2 | C changed it to |B-C | = |b-c |. | 1:43 |
| 15 | Refining the method again | 1:43-1:46 |
| 15.1 | C gave A a 21-24-33 triangle to solve. A did a systematic search based on 33-24=9 and found solution. | 1:23-1:46 |
| 15.2 | A described the method as taking the difference of the largest side and another side, and finding two numbers with that difference, whose sum is the third side. |
Session ended due to time constraint.
| 1 | Tape Viewing | Time of day 12:04-12:16 |
| 1.1 | Watched tape of Arithmagon session from 1:20-1:22. | 12:04-12:06 |
| 1.2 | C described how they solved triangles using their method. | 12:06 |
| 1.3 | Watched tape of Arithmagon session from 1:22-1:24. | 12:07-12:09 |
| 1.4 | Discussed which triangle they had been solving on tape. It was a 11-18-27 triangle. | 12:09 |
| 1.5 | DR asked how their differences method worked. C explained. | 12:10 |
| 1.6 | DR asked how they knew the differences were equal. A replied that they had tested many cases. | 12:10 |
| 1.7 | DR asked how knowing the difference helped in solving the triangles. C replied that they plugged in numbers with that difference until a pair worked. | 12:11 |
| 1.8 | DR asked what they would do if the difference were a million. A replied that they would know where to start checking based on the sum they were trying to get. | 12:12 |
| 1.9 | Watched tape of Arithmagon session from 1:24-1:28. | 12:12-12:16 |
| 1.10 | A commented “We were stuck so we were just trying to do anything”, referring to their derivation of Z-X+Y=2x. | 12:15 |
| 1.11 | DR asked for an interpretation of the derivation. C attempted to explain it, but discovered an error. | 12:16 |
| 2 | Mechanical deduction of the formula | 12:17-12:22 |
| 2.1 | C corrected second line to y = X+Z. | 12:17 |
| 2.2 | C derived new formula. DR asked what it meant. C noted that the formula gave one of the corners. | 12:18 |
| 2.3 | A suggested trying the formula on an example to see if it worked. C tried it with the original 11-18-27 triangle and concluded the formula worked. | 12:19-12:20 |
| 2.4 | DR asked which side would be subtracted in the formula. C answered that the side opposite the corner the formula gave would be subtracted. | 12:20 |
| 2.5 | A suggested using the formula to find a different corner. After they had done so DR asked if they had expected the formula to fail. C claimed that the formula should work in the same way for all the corners, and wrote a verbal formulation of it in terms of adjacent and opposite sides. | 12:21 |
| 2.6 | A asked if the formula was the correct one. DR said it was one correct formula. | 12:22 |
| 3 | “Proving” the formula | 12:23-12:26 |
| 3.1 | DR asked why the formula worked. A said they had tested it with examples. C suggested “proving” it. | 12:23 |
| 3.2 | C identified the difference relation as important to the formula, and asked why it worked. He then made the connection with the common corner, which requires that any difference in the sides be due to the difference between the other two corners. | 12:24-12:25 |
| 3.3 | DR commented on the informality and meaningfulness of C’s argument, compared with the mechanics of the algebraic derivation. A connected this with the contrast between explaining and finding a formula. | 12:26 |
| 4 | Extending new ideas to squares | 12:27-12:29 |
| 4.1 | C gave an example of a square with negative corners to show negative numbers could occur. | 12:27 |
| 4.2 | DR suggested they drop their requirement that all sides of a square should be equal. | 12:28 |
| 4.3 | A asserted that they could draw in a diagonal and use their triangle formula to solve squares. He tried to do this with a 3-4-3-4 square, arriving at a false solution. C concluded that the formula didn’t apply to squares. | 12:28-12:29 |
| 5 | Searching for a pattern for squares. | 12:30-12:33 |
| 5.1 | They tried to solve a 3-4-5-6 square by trial and error. | 12:30-12:31 |
| 5.2 | C set up a 7-13-16-10 square by starting with known corners. They looked for patterns in this square, and labeled the corners and sides. | 12:31-12:33 |
| 5.3 | DR asked what relations they had found, and asked them to write equations for them. | 12:33-12:34 |
| 6 | Mechanical deductions on squares | 12:34-12:40 |
| 6.1 | A derived expressions from the relations he wrote. | 12:34-12:36 |
| 6.2 | DR pointed out that they also knew that corners added up to sides, and A added equations for these relations to his. | 12:36 |
| 6.3 | A produced the equation 27 = 2C-2A+w+z-y. DR asked if that equation meant anything. A continued with his derivations | 12:36-12:37 |
| 6.4 | DR asked how A had arrived at 27, and pointed out that it was based on the sides of the particular square they were investigating. A replaced the numerical values of the sides with the variables which stood for them. | 12:38-12:39 |
| 6.5 | A continued his derivations, mentioning that he was trying to isolate the variables for the corners. | 12:39-12:40 |
| 7 | Formalizing the proof for the triangle formula. | 12:40-12:45 |
| 7.1 | DR asked them to return to the derivation of the triangle formula, and to try to formalized the informal explanation of the difference relation, or to try to explain informally the formal derivation of the formula. | 12:40 |
| 7.2 | A commented that some things were more easily done one way than the other. | 12:41 |
| 7.3 | C explained the difference relation informally. | 12:42 |
| 7.4 | A set up equations for the relations along the sides in question. | 12:43 |
| 7.5 | C commented that one could see from the equations that the differences must be the same. | 12:44 |
| 7.6 | DR asked him to show it algebraically, and C subtracted the two equations to arrive at the difference relation. | 12:45 |
| 8 | Giving meaning to the algebra | 12:46-12:49 |
| 8.1 | DR asked them to say what the algebra of their formula derivation meant. | 12:46 |
| 8.2 | C attempted to do so, but ended up reciting the algebra, rather than giving meaning to it. | 12:47 |
| 8.3 | C focused on the meaning of the subtraction of the two equations. A commented that they had learned to isolate variables. C pointed out that they still didn’t know what it meant. They continued to consider it. | 12:48-12:49 |
Session ended due to time constraint.
| 1 | Gathering data | Time of day 11:03-11:07 |
| 1.1 | A and C both thought they had seen something similar before. Tried to remember. | 11:03-11:04 |
| 1.2 | A conjectured they are all even, and calculated the next term to confirm. | 11:04 |
| 1.3 | C calculated several terms and differences between them. | 11:05 |
| 1.4 | A calculated all the terms up to n=10. | 11:05-11:06 |
| 1.5 | DR asked why the needed more terms. C answered that it kept them from identifying “fluke patterns”. | 11:06 |
| 1.6 | A calculated terms for -4 < n < 0 | 11:07 |
| 2 | Noticing patterns | 11:08-11:10 |
| 2.1 | C conjectured they are all even. | 11:08 |
| 2.2 | A noticed 0 0 0 6 4 pattern in final digits and pointed it out to C. | 11:08-11:09 |
| 2.3 | A calculated terms for 10 < n < 15 | 11:10 |
| 2.4 | C conjectured that they are all multiples of 6 and DR said that was the pattern he was thinking of. | 11:10 |
| 3 | What do we know about n3-n? | 11:11-11:13 |
| 3.1 | DR asked if they would like to explain or verify that n3-n is always a multiple of 6, or explore for more. They chose to explain. | 11:11 |
| 3.2 | DR asked what they knew about expressions like n3-n. They replied they could graph it or factor it. | 11:12 |
| 3.3 | C factored n3-n. | 11:13 |
| 4 | What do we know about those three numbers? | 11:13-11:15 |
| 4.1 | DR asked what they knew about n(n-1)(n+1). C replied that they were three consecutive numbers. | 11:13 |
| 4.2 | DR asked why the product of three consecutive numbers would be a multiple of 6. | 11:13 |
| 4.3 | A observed that either they would have two odd numbers or two even numbers in the three. | 11:13 |
| 4.4 | C pointed out that they needed to show that the product was divisible by both 3 and 2. | 11:14 |
| 4.5 | DR asked if they knew one of the numbers is even. C answered that the numbers were either even-odd-even, or odd-even-odd. A concluded this meant the product must be even. | 11:14 |
| 4.6 | C commented that the middle number of the three is the original n from n3-n. | 11:15 |
| 5 | Is there a factor of 3? | 11:15-11:17 |
| 5.1 | DR asked if they could find a factor of 3 in the three numbers. C examined several examples. He then claimed that they would always have a number divisible by 3 because every third number is divisible by 3. | 11:15-11:16 |
| 5.2 | DR asked if that explained why n3-n is always a multiple of 6. A said yes, and C explained that n3-n is always the product of three consecutive numbers, at least one of which is even and one of which is a multiple of 3. | 11:17 |
| 6 | Formalizing | 11:17-11:22 |
| 6.1 | DR asked if they could write out the argument. | 11:17 |
| 6.2 | C wrote out the argument. | 11:17-11:21 |
| 6.3 | C gave examples of the two cases of n not a multiple of 3. | 11:22 |
| 7 | Testing confidence | 11:23 |
| 7.1 | DR asked them if they now knew that 4173-417 is a multiple of 6. | 11:23 |
| 7.2 | A checked on a calculator. | 11:23 |
| 7.3 | C argued that it would be 416x417x418, which includes an even number and a multiple of 3. | 11:23 |
| 8 | Did we explain/explore? | 11:24-11:25 |
| 8.1 | DR asked if they would use their argument to explain why n3-n is always a multiple of 6. C said yes. | 11:24 |
| 8.2 | DR asked if they had discovered anything new about n3-n by working out the argument. They didn’t think so. DR commented on the discovery that n3-n is always the product of three consecutive numbers. | 11:25 |
| 9 | What about the converse? | 11:25-11:26 |
| 9.1 | DR asked if the product of three consecutive numbers would always be n3-n for some n. C answered yes, and that n would be the middle number of the three. He also worked an example. | 11:25-11:26 |
| 10 | Another statement to explain | 11:27-11:30 |
| 10.1 | DR asked why the sum of two odd numbers is even. | 11:27 |
| 10.2 | C and A independently determined the sum of (2n-1) and (2n‑1), concluding that 4n-2 must be even. | 11:27-11:28 |
| 10.3 | DR pointed out that they had only shown that the sum of two identical odd numbers is even. C calculated (2n-1)+(2n-7). A calculated (2n-1)+(2n+1). | 11:28 |
| 10.4 | DR pointed out that these were still special cases, and suggested using a different variable for the second number. C and A independently added (2n-1)+(2x-1) arriving at 2(n+x-1). | 11:29-11:30 |
| 11 | Winding down
Various discussions occurred after the second interview. Only key episodes are listed here. |
11:30+ |
| 11.1 | DR showed how Arithmagon squares work. | 11:34 |
| 11.2 | A commented on doing math in school. “I can’t remember a formula unless I understand it.” | 11:37 |
| 11.3 | A commented that Mr. B was happy to explain anything they asked about, at great length, and they were disappointed when they found an elaborate explanation would not be on the examination. A described this as a “waste of time.” | 11:38 |
Ben
| 0 | Given Problem Sheet | Time elapsed from start of tape 4:08 |
| 1 | Solved Problem “intuitively” | 4:38 |
| 2.1 | Explored relations between the numbers. | ~6:00 |
| 2.2 | Found pattern in differences: A-B=a-b, etc. | 8:11 |
| 3 |
W: “Can you use negative numbers?” B: “Sure.” W: “You can’t have negative length of a side though.” B: The triangle is irrelevant |
9:40 |
| 4 | Compared Solutions with R & E | |
| 5.1 | B asked E & R how they used algebra to solve problem; | |
| 5.2 | B Explained his constraints method to E & W | |
| 6 |
W said he was “playing” with properties of triangles. E & K told W the triangle was irrelevant. E: “I guess you could [treat sides as lengths]” B: “I don’t think you could.” B rejected taking triangle as important as angles didn’t work |
|
| 7 | Discussion of B’s method. B reconstructed his thinking and solved another triangle. | |
| 8.1 | DR gave 1-4-12 triangle. | ~22:30 |
| 8.2 | B declared it impossible. He explained that only 0+1 gives 1, and neither order works. E & W suggested negative numbers | ~23:00 |
| 9 |
W: “Do the three numbers represent angles or something?” E, B, & DR: “No.” |
~23:30 |
| 10.1 | B Decided 1-4-12 triangle could be solved with negative numbers. | ~24:30 |
| 10.2 | B Proposed that E or R solve 1-4-12 algebraically, E started | 25:30 |
| 10.3 |
B: “I’ve determined that it is impossible.” E: “You think it’s impossible?” B: (to DR) Is it impossible? |
|
| 10.4 |
B Asked for E’s algebraic solution. E got it wrong. |
25:40 |
| 10.5 | B Suggested fractions involved. | 26:11 |
| 10.6 | All worked on 1-4-12 independently ~ | 26:30-29:00 |
| 10.7 | E gave B solution to 1-4-12 triangle. | ~29:00 |
| 10.8 | B checked her solution. | ~30:00 |
| 11 | Everyone listed to R describe progress. | |
| 12.1 | W described what he was doing to E and B. | ~32:00 |
| 12.2 | E noticed 6-6-6 [A+a=B+b=C+c] in W’s work. | ~32:30 |
| 12.3 | W checked [A+a=B+b=C+c] on other triangles. They continued to explore. | ~33:00 |
| 12.4 | E noticed that a+b+c=12=A+a in a particular triangle (Announced that a+b+c=12) | 37:14 |
| 12.5 | W enunciated rule: a+b+c=2(A+B+C) | ~37:30 |
| 12.6 | Several examples were checked. | 38:50 |
| 13 | K & W worked on solving 11-8-15 using new found relations. Found that sum/2 =12, but then became stuck. | ~40:00 |
| 14.1 | E explained her method. W interrupted with a new problem. | |
| 14.2 |
E gave solution. W: “No.” |
|
| 15 | B, W & E worked on relation of division by 2 to area formula for a triangle. | |
| 16 |
R announced her formula
to the group |
|
| 17 |
Tried with W to Confirm R’s formula for 0-1-2 triangle |
|
| 18 | Everyone discusses connection of R’s formula with cosine law | |
| 19.1 | Discussed R’s formula. W: “I understand everything except why you divide by 2” | |
| 19.2 | W repeated operational version of R’s formula. | |
| 19.3 | Exchange of explanations for division by 2. B’s link to a+b+c=2(A+B+C) accepted as explanation. | |
| 20 | Watched W work examples | |
| 21 | All explained E’s method to TK | |
| 22 | Discussed relation with angles, with E & W | |
| 23 | Worked out 3-7-9 triangle by E’s method. |
Wayne
| 0 | Given Problem Sheet | time elapsed from start of tape 4:08 |
| 1.1 | Explored properties of triangles. | ~5:30-9:30 |
| 1.2 |
W: “Can you use
negative numbers?”
B: “Sure.”
W: “You can’t have negative length of a side though.” |
9:40 |
| 2 | Compared Solutions with R & E | ~10:30 |
| 3.1 | Explored properties of triangles. | ~11:00-12:00 |
| 3.2 |
W said he was “playing” with properties of triangles. E & K told W the triangle was irrelevant E: “I guess you could [treat sides as lengths]” B: “I don’t think you could.” B rejected taking triangle as important as angles didn’t work. |
~13:00 |
| 3.3 | Working on diagrams of triangles. | ~14:30 |
| 3.4 | Announced that he was: “Frustrated...no idea what to do.” | ~15:00 |
| 4.1 | DR gave 1-4-12 triangle. | ~22:30 |
| 4.2 | B declared it impossible. He explained that only 0+1 gives 1, and neither order works. E & W suggested negative numbers | ~23:00 |
| 5 |
W: “Do the three numbers represent angles or something?” E, B, & DR: “No.” |
~23:30 |
| 6 | Explained his “page 1” to DR. Began with 3-4-5 because it is a right triangle. Expanded out. Shrunk in. Proposed 0-1-2 triangle to B. | ~26:30-29:00 |
| 7 | Everyone listed to R describe progress. | |
| 8.1 | W described what he was doing to E and B. | ~32:00 |
| 8.2 | E noticed 6-6-6 [A+a=B+b=C+c] in W’s work. | ~32:30 |
| 8.3 | W checked [A+a=B+b=C+c] on other triangles. They continued to explore. | ~33:00 |
| 8.4 | E noticed that a+b+c=12=A+a in a particular triangle (Announced that a+b+c=12) | 37:14 |
| 8.5 | W enunciated rule: a+b+c=2(A+B+C) | ~37:30 |
| 8.6 | Several examples were checked | 38:50 |
| 9 | K & W worked on solving 11-8-15 using new found relations. Found that sum/2 =12, but then became stuck. | ~40:00 |
| 10.1 | E explained her method. W interrupted with a new problem. | ~40:30 |
| 10.2 | E gave solution. W: “No.” | |
| 11 | Confirmed that A+a rule holds for original Arithmagon. Recorded rule. | |
| 12 | B, W & E worked on relation of division by 2 to area formula for a triangle. |
| 13 | Suggested link to Golden Ratio. DR discouraged this idea. | |
| 14 | R announced her formula to the group. | |
| 15 | Tried with B to confirm R’s formula for 0-1-2 triangle | 49:30 |
| 16 | Everyone discussed connection of R’s formula with cosine law | ~50:00-51:00 |
| 17.1 | Discussed R’s formula. W: “I understand everything except why you divide by 2" | ~52:20 |
| 17.2 | W repeated operational version of R’s formula. | ~53:30 |
| 17.3 | Exchange of explanations for division by 2. B’s link to a+b+c=2(A+B+C) accepted as explanation. | ~54:00 |
| 18 | Worked examples, others watched. | ~55:00 |
| 19 | All explained E’s method to TK | ~57:00 |
| 20 | Discussed relation with angles, with E & B | ~58:30-59:30 |
| 21 | Wrote out rule. Gave verbal version of rule to DR. | ~61:00 |
| 0 | Given problem sheet. | time elapsed from start of tape 2:10 |
| 1.1 | Kerry chose to “deduce”; to use algebra rather than trial and error. | 2:45 |
| 1.2 | Solved by using simultaneous equations. | 3:15-4:30 |
| 1.3 | Checked their solution: Stacey—”Is that right?” | 4:30 |
| 1.4 | Kerry observed that if method gives solution, it must work in original puzzle. | |
| 2.1 | Stacey—”What happens if you add the middle numbers together?” | 6:50 |
| 2.2 | She added up the “middle” numbers (those on the sides), and then considered how that is related to the secret numbers on the corners. | 7:10 |
| 2.3 | As each secret number is added into two of the numbers on the sides (“So you add each of those twice”) she deduced that the sum of the numbers on the sides is twice the sum of the numbers on the corners. | 7:30 |
| 2.4 | Kerry checked her assertion, but did not see her argument. | 8:10 |
| 2.5 | For Stacey the relationship must hold “for all of them” because she has deduced it. For Kerry it is only the unlikeliness of such a relationship occurring by chance that convinces him that the relationship is a general one. | 8:20 |
| 3.1 | Kerry solved again, by using matrices. | 8:30-15:00 |
| 3.2 | Stacey observed that 0 0 1 12 is wrong, as she attached the meaning C=12 to it. Kerry was proceeding formally. | c. 11:00 |
| 3.3 | Due to an arithmetic error, they obtained a different “solution”. After checking over their work, they briefly considered the possibility of two solutions, but rejected the idea when they checked their answer in the original problem. | 14:15 |
| 3.4 | Kerry claimed matrix should give solution as the number of variables equals the number of equations. | 16:15 |
| 4 | “Generalized” their solution by describing their actions in general terms. | 18:00-21:00 |
| 5.1 |
Stacey extended sides: “just trying something”. Around the original triangle she drew a sequence of triangles, using the corner numbers from each triangle as the side numbers for the next larger triangle. |
21:45 |
| 5.2 | Explored relationships between nested triangles. | c. 23:00 |
| 6.1 | Stacey observed that the differences 27-13, 17-3, and 11-(-3), are all 14. She predicted that the numbers in the next triangle would also be 14 less than those in the 1-10-17 triangle. For example, she predicted that the number at the lower right hand corner of the largest triangle would be 3. | 25:15 |
| 6.2 |
When this prediction was disproved both Stacey and Kerry advanced new predictions. Kerry predicted that the next difference would be 3.5, an induction based on 7 being half of 14. Stacey suggested that the differences might alternate: 14,7,14,7,... |
26:40 |
| 6.3 | Tested Stacey’s prediction | 33:25 |
| 6.4 | Kerry suggested trial and error to determine next triangle’s solution | 33:40 |
| 6.5 |
Stacey observed that her prediction was only based on one trial, so it was not surprising it failed. She suggested they work out the next triangle’s solution to give them another trial to base predictions on. |
33:50 |
| 6.6 | Kerry’s prediction of 3.5 was confirmed. | 36:00 |
| 6.7 | Kerry predicted 1.75, tested and confirmed his prediction. | 36:30-37:45 |
| 7.1 | They extended halving principle to a doubling principle (going in). | 41:00 |
| 7.2 | They discussed the limit of the values for the triangles. | 43:00 |
| 7.3 | Stacey suggested deriving a formula. This idea was rejected due to the large number of variables. | 45:15 |
| 8.1 | Kerry expressed interest in finding a reason for the initial difference being 14 and for failure of the matrix. | 47:00 |
| 8.2 | Stacey observed that (11+18+27)÷14=4. | 48:00 |
| 8.3 | Stacey compared 4 with the number of sides of the triangle. | 48:45 |
| 8.4 | Kerry compared dividing by 4 with averaging. | 50:00 |
| 9.1 | TK intervened—”Is 14 special, or is one fourth of the sum special?” | 50:30 |
| 9.2 | Investigated an exterior triangle, and another triangle based on new numbers. | c. 51:00 |
| 9.3 | Described a general method for solving triangles. | 59:00 |
| 10.1 | Both expressed continuing concern over the number 4. | 60:30 |
| 10.2 | Stacey observed that the act of nesting the original triangle in a larger triangle created four triangles approximately the same size as the original. | 61:00 |
| 10.3 | Kerry was unhappy with this as an explanation—”You can’t just say that, you have to explain that. Why are those 4 triangles important?” | 61:30 |
| 11 | Continued exploration (of Arithmagons of more than 3 sides) after research session ended. | |
| 12 | In a follow up interview three weeks later, Kerry and Stacey were shown a formal proof of the correctness of their method. For Kerry this proof explained the occurrence of the 4. He commented “That’s where we get the 1/4 from”. It is not clear whether Stacey understood the proof as an explanation. |
| 1.1 | They tried to remember the rule, arriving at Fn = Fn-1 + Fn-2. | time elapsed 1:35 |
| 1.2 | Kerry rephrased as Fn+2 = Fn+1 + Fn |
|
| 1.3 | Kerry added Fn= Fn+2-Fn+1 to allow determination of F1 and F2 | |
| 1.4 | Verified by cases. | until 6:00 |
| 1.5 | Looked for other rules. | 7:00-7:45 |
| 2.1 | Examined F3n for pattern. Tried to used differences and ratios. | 8:00 |
| 2.2 | Determined they are even, by induction. | finished 12:40 |
| 3.1 |
Examined Fp for pattern. Stacey determined they are all odd, by induction. |
began12:45 12:55 |
| 3.2 | Revised conjecture to: Fp is always prime. | 13:45 14:15 |
| 3.3 | Made a list of Fibonacci numbers to examine. | 14:15-15:15 |
| 3.2 | Observed that converse does not hold. | 15:50 16:15 |
| 3.4 | Tested more cases. | 16:15-20:30 |
| 4.1 |
TK asked which are even Examined F3n in table. |
20:30 |
| 4.2 | Kerry provided O+O=E proof when prompted. | 23:25-25:30 |
| 5.1 | Examined groups of four consecutive Fibonacci numbers at TK’s prompting. | 27:00-28:00 |
| 5.2 | Investigated sequence starting with 7,7 at Stacey’s suggestion. | 28:00 |
| 5.3 | Found sums of groups of four make a Lucas sequence. | 28:40-32:00 |
| 5.4 | Investigated negative sequence | 32:00-35:00 |
| 6 | Explored groups of three consecutive Fibonacci numbers at Kerry’s suggestion. Found sums make a Lucas sequence. | 36:00-43:00 |
| 7 | DR suggested investigating products of three consecutive Fibonacci numbers. | 43:00 |
| 7.2 | Found a false pattern. | 47:00 |
| 7.3 | Found product of end numbers equal to middle number squared ± 1 | 47:30 |
| 7.4 | Checked for 7,7 sequence at Stacey’s suggestion. | 48:15-50:00 |
| 7.5 | Find product of end numbers equal to middle number squared ± F1 . | 50:00 |
| 7.6 | Investigated other sequences. | 53:00 |
| 7.7 | TK and Kerry debate whether sequences start n,n or 0,n . | 56:00 |
| 7.8 | TK suggests making a list of sequences considered so far. | |
| 8. | Gave F3n even and Fp prime as their discoveries when asked to summarize. | 100:00+ |
Eleanor
| 0 | Given Problem Sheet | time elapsed from start of tape 3:55 |
| 1.1 | Worked with R on solution by system of equations. | -6:00 |
| 1.2 | Worked independently on solution by system of equations. | 6:00-7:55 |
| 1.3 | Wondered if the solution is unique. | 8:00 |
| 1.4 | Decided that the algebraic solution showed only one solution is possible | 9:38 |
| 2.1 | Compared Solutions with R, B & W | 10:20 |
| 2.2 | B asked E & R how they used algebra to solve problem; | ~11:00 |
| 3 |
W said he was “playing” with properties of triangles. E & K told W the triangle was irrelevant. E: “I guess you could [treat sides as lengths]” B: “I don’t think you could.” B rejected taking triangle as important as angles didn’t work. |
|
| 4.1 | Discussion of B’s method. | |
| 4.2 |
E & R watched B B reconstructed his thinking and solved another triangle. |
1400-1500 |
| 5 | Worked independently | 1500-1600 |
| 6.1 | R & E analyzed B’s method. | 16:30-~17:00 |
| 6.2 | Tried a triangle by B’s method, with R | ~17:00-21:00 ~20:00 gave solution |
| 7.1 | DR gave 1-4-12 triangle. | 22:15 |
| 7.2 | B declared it impossible. He explained that only 0+1 gives 1, and neither order works. E & W suggested negative numbers. | |
| 7.3 | W: “Do the three numbers represent angles or something?” E, B, & DR: “No.” | ~23:30 |
| 7.4 | B Proposed that E or R solve 1-4-12 algebraically, E began to do so. | ~25:00 |
| 7.5 | B: “I’ve determined that it is impossible.” |
|
E: “You think it’s impossible?” B: (to DR) Is it impossible? B Asked for E’s algebraic solution. |
25:40 | |
| 7.6 | worked on 1-4-12 independently | ~26:30-29:00 |
| 7.7 | E gave B solution to 1-4-12 triangle. | |
| 8 | Everyone listed to R describe progress. | |
| 9 | W described what he was doing to E and B. | |
| 10.1 | E noticed 6-6-6 [A+a=B+b=C+c] in W’s work. | |
| 10.2 | W checked [A+a=B+b=C+c] on other triangles. They continued to explore. | |
| 10.3 |
E noticed that a+b+c=12=A+a in a particular triangle (Announced that a+b+c=12) W enunciated rule: a+b+c=2(A+B+C) |
37:14 |
| 10.4 | Several examples were checked. | 38:50 |
| 11.1 | Worked on inventing a method of solution. | |
| 11.2 | E explained her method. W interrupted with a new problem. | 40:00 |
| 12.1 | E gave solution. W: “No.” | |
| 13.1 | Tried to clarify the relations she was working with. | |
| 14 | B, E, & W worked on relation of division by 2 to area formula for a triangle | ~44:30-48:00 |
| 15 | R announced her formula to the group. | |
| 16 | R explained the derivation of her formula to E by recapitulating her calculations. | ~49:30-51:00 |
| 17 | Tried to relate R’s formula to her own relations. | |
| 18 | General discussion of R’s formula. W: “I understand everything except why you divide by 2” | |
| 19.1 | Announced they have found two different methods. | |
| 19.2 | Worked on clarifying her method. | |
| 20 | All explained E’s method to TK | ~56:00-58:00 |
| 21 | Discussed relation to angles with B&W | ~58:00-59:30 |
| 22.1 | Derived R’s formula from her equations. | ~59:30-60:30 |
| 22.2 | Compared results w/ R | ~60:30 |
| 22.3 | Tried to explain equations by algebraic derivation. | ~62:00 |
Rachel
| 0 | Given problem sheet | time elapsed from start of tape 3:55 |
| 1.1 | Worked with E on solution by system of equations. | -6:00 |
| 1.2 | Worked independently on solution by system of equations. | 6:00-7:55 |
| 1.3 | Solved puzzle | 7:55 |
| 2.1 | Compared Solutions with E, B & W | 10:20 |
| 2.2 | B asked E & R how they used algebra to solve problem; | ~11:00 |
| 3.1 | Made a new puzzle | ~13:00-14:30 |
| 3.2 | Looked for patterns. | |
| 4.1 | Discussion of B’s method. | |
| 4.2 |
E & R watched B B reconstructed his thinking and solved another triangle. |
1400-1500 |
| 5 | Worked independently | 1500-1600 |
| 6.1 | R & E analyzed B’s method. | 16:30-~17:00 |
| 6.2 | Tried a triangle by B’s method, with E | ~17:00-21:00 ~20:00 B gave solution |
| 6.3 | Tried a triangle by B’s method, alone | ~21:00-22:15 |
| 7.1 | DR gave 1-4-12 triangle. | 22:15 |
| 7.2 | Worked on solving 1-4-12 triangle | ~23:00-25:30 |
| 7.3 | Watched E & B ~ | 25:30-26:00 |
| 7.4 | worked on 1-4-12 independently | ~26:30-29:00 |
| 8.1 | Began working on algebraic derivations. | ~29:00 |
| 8.2 | Determined that if two sides are equal then two corners are equal. | 29:37 |
| 8.3 | Everyone listed to R describe progress | ~31:00 |
| 9 | Continued to explore deductively. | ~32:00-48:00 |
| 9.2 | Worked with TK’s help | ~34:00-35:00 |
| 9.3 | deduced that if all sides are equal all corners are too. | ~37:30 |
| 9.4 | TK suggested deducing with no constraints. | 42:20 |
| 9.5 | TK Suggested focus on ½ | 4:30 |
| 9.6 | TK helped | 46:00 |
| 9.7 | Found formula, and tested it. | ~46:30 |
| 9.8 | R announced her formula to the group. | ~48:00-49:00 |
| 10 | R explained the derivation of her formula to E by recapitulating her calculations. | ~49:30-49:00 |
| 11 | Watched as: | |
| 11.1 | All discussed R’s formula. W: “I understand everything except why you divide by 2” | |
| 11.2 | W repeated operational version of R’s formula. | ~53:30 |
| 12.1 | Exchange of explanations for division by 2. B’s link to a+b+c=2(A+B+C) accepted by W, and R, as explanation | ~54:00 |
| 12.2 | Watched W work an example. | ~55:00 |
| 13 | All explained E’s method to TK | ~56:00-59:00 |
| 14.1 | Derived a+b+c = 2(A+B+C) algebraically | 59:00 |
| 14.2 | Compared results w/ E | ~60:30 |
| 14.3 | Continued derivations | ~62:00 |
| 1.1 | Making conjectures | time of day, or tape counter Clk 3:10-3:15 |
| 1.1 | Conjecture (R): F3n is even | Clk 3:12 |
| 1.2 | Conjecture (E): F4n is odd, also a multiple of 3 | Clk 3:12 |
| 1.3 | Noted pattern OOEOOE | Clk 3:15 |
| 2 | Discovery of 3s rule |
|
| 3 | Making conjecture (E?): Fp is prime, and Fc is not prime | Clk 3:20 |
| 4.1 | Exploring: R looking at 4s, E looking at 3s | |
| 4.2 |
4s rules discovered (100) verified, (130) formulated (160-730 by R) |
|
| 5 | Making conjecture: F-n = -Fn | Ctr: 660-700 |
| 5.2 | Summarized results | 780-880 |
| 5.3 | TK talked about Fibonacci Quarterly | 880-1000 |
| 6 |
3s rule formulated (1050 by E) (Why 4?, need to explain) (Ctr: 970-1130) Worked independently: E tried to relate the 4s rule and the 3s rule (1150-1350) TK asked for clarification, R answered (1250-1320) E gave report to TK (1350-1400) ? (1400-1625) |
|
| 7 |
E wondering about 4 in 3s rule. R looking at factors (1625) Searching for explanations (1625, E) R looked at 2s rule briefly (1660) |
|
| 8 |
5s rule discovered, (1760) verified, (1760) formulated (1860-1940 by E; 2070 by R) |
|
| 9 | Making conjecture (R): An n-rule exist for all n | Ctr: 1920 |
| 10 | Cycles of discovery, verification, and formulation. | 2040 |
| 10.1 |
6s rule discovered, verified,(2040) formulated (2120 by E; 2190 by R)
|
|
| 10.2 |
2s rule discovered and verified (2150) R formulated her method for discovering rules (2200) |
| 11 | Making conjecture (R): 7s rule will not work as 7 is prime. E: 2&3 are prime. | 2300-2310 |
| 12 | Cycles of discovery, verification, and formulation. | 2360 |
| 12.1 | 7s rule verified | 2420 |
| 12.2 |
8s rule predicted & verified, alternative verification suggested. |
2390-2480 |
They were about to check for an 11 rule when they ran out of time.
| Solving | tape counter | |
| 1.1 | Using system of equations | 140-450 |
| 1.2 | Check solution, it doesn’t work in original problem (it does, C mis-added in his head. ) | 450 |
| 2.1 |
Second attempt, with matrix C knows matrix is the same as equations, but doesn’t have any better ideas |
580-940
580 |
| 2.2 | C Predicts matrix will not reduce, as otherwise it should produce a solution which should work. | 640 |
| 3 | Temporary halt in matrix work, based on knowledge that it’s the same as the equations. Search for error in the equations. | 770-880 |
| 4 | Discovery of solution with matrix | 880-940 |
| Generalizing | ||
| 5 | Conjecture that all numbers which work are of the form 2n, 3n, n+2. Rejected. | 1200-1550 |
| 6 | trying another example: 3-5-2. C predicts any triangle solvable based on 3 equations with 3 unknowns. | 1580-1590 |
| 7 | J Considers 1-1-1 triangle, concludes need for fractions. | 1720-1760 |
| Is the Arithmagon always solvable? | ||
| 8 |
C on general solvability: 3x3=>solution Squares work too. Equations must be linearly independent |
1840
1870 1890 |
| 9Is the triangle always solvable? | ||
| 9.1 | Worked to solve general square. J using equations, C using matrix. Matrix does not reduce. This casts doubt on general solvability of triangles. | 1920-2200
2200 |
| 9.2 |
J Solving triangle in general. C continuing to investigate square’s matrix J concludes square never works |
2270
2375 |
| 9.3 | C solves general triangle with matrix | 2600-2680 |
| 10 | Is the square never solvable? | |
| 10.1 | Solution of triangle now casts doubt on calculations related to square | 2680 |
| 10.2 | C reviewed calculations for square | 2680-2750 |
| 10.3 | Conjecture made: Square is different as corners are unconnected. Conjecture Squares never work | 2750-2800 |
| 10.4 |
Counter example 1-1-1-1 Revised conjecture: It works for some values. |
2825 |
| 11 | What is going on with our matrix? | |
| 11.1 |
Search for error in square’s matrix: Confusion in writing v and y Relation: v=y+z-x discovered, confirmed, and considered. |
2930
300, 3060 &3170 |
| 11.2 |
Conclusion that C and D are arbitrary rejected on grounds it “doesn’t make sense” |
3110, 3160&3300
3130&3300 |
| 12 | Interacting with TK | 3350-3400 |
| 13 | Continuing to explore square’s matrix | 3415-3460 |