previous table of contents appendix B table of contents next
5.14 Cynthia hesitates about 29. She puts her finger in the spot to "try out" what the marker would look like. She points to the diagonal running up to B5 from E2.
5.15 David, "I got the win! Oh! If you put that there (B5) then I put one there (F4) then you put that there (E5)". The boys start taunting Cynthia to drop her marker.
EHN Multi-Step Deduction
5.16 Teacher, "The boys seem pretty excited about something." 29 is placed.
5.17 Jerome starts smiling about something and places 30b.
5.18 David, "Oh I got a plan. Oh, no, no, no, no, no!" (as 30 is played).
5.19 Cynthia plays 31. "We won!"
5.20 Jerome goes to put a marker above 30b to create four in a row but realizes that Cynthia and the teacher has won.
5.21 David, "No, it's right there by, I put 1 right there (F4) then another one right there (G5), right there (F5) and right there (G6) and that's four!" EHN Simple Deduction
Game 6
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29 r |
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15 r |
30 b |
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23 r |
25 r |
35
r |
14 b |
19 r |
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22 b |
24 b |
17 r |
13
r |
18 b |
31 r |
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21 r |
20 b |
16 b |
12 b |
11
r |
28 b |
34 b |
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8 b |
7 r |
3 r |
9 r |
10 b |
27
r |
33 r |
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1 r |
6 b |
2 b |
4 b |
5 r |
26 b |
32 b |
Note: Seth often placed his marker directly on top of Jared's.
6.0 A new game is started. Seth (red) and Jared (black) play against each other. (3:44)
6.1 Jared, "I know where you're going" after Seth places 5.
6.2 Jared, "Oh no you're not!"
6.3 The teacher asked Seth why he placed 15r where he did. Seth starts to remove the marker.
6.4 Seth shook his head, indicating he didn't know why.
6.5 Jared: "I know good tricks. You think you're gonna win."
6.6 Jared, "Hey, you could have had 4 in a row, but you can only have 3" (after 25 is played).
6.7 After 27 is played, Jared examines the board and points to three black in a row (26, 10, 12). The game slows down a little bit.
6.8 Jared, "How could you [Seth] win?" He puts his head on his hand as he looks at the board. "Oh, he could have won if I hadn't put one there." (29, 25, 17, 12)
6.9 Teacher, "Is there anywhere else he could win now?"
6.10 Jared says, "I know" as he places 30b.
6.11 Jared, "I hope he puts it there, too. So then I can win.
6.12 Seth, (placing 31) "This way?" (Holding the marker above 28)
6.13 Jared, "No! " (Giggles)
6.14 Seth, "This way?" (Holding the marker above 17)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-39
To Beginning of Section | To End of Section
6.15 Jared, "Yes!"
6.16 Seth, "No!" (He puts it back over 28)
6.17 Jared, "Now, how can he win? If I put one right here (C5) then he could put one in up there (C6) and then I'll lose. If this one weren't right there then I'd have Connect 4." (8, 20, 17, 14) "How can I win? … You think you're gonna win" He places #34.
EHN Simple Deduction
6.18 Seth wins by placing 35r.
6.19 Seth, "I win, 1, 2, 3, 4!"
6.20 Jared, "Oh, I didn't see that one! I thought if he put one in there then I could win." (C6, 28, 18, 14) EHN Simple Deduction
6.21 Seth: "Guess what. I'm really winning on these games. I'm really good."
Game 7
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9 b |
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14
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8
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6
r |
10
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13 b |
2 r |
5 b |
7 b |
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12 r |
1 b |
3 b |
4 r |
11 b |
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7.0 Seth (red) and Jared (black) start a new game. (3:48)
7.1 Seth says before he places any markers: "I'm really good at this!"
7.2 Jared says on turn 5: " You think you can win."
7.3 Jared says on turn 7: "You think you can still win."
7.4 Jared says on turn 9: "I figured out how to put 'em."
7.5 Jared says with a grin on his face on turn 13: "I'm gonna win."
7.6 Seth: " 1,2,3,4! I win!"
7.7 Jared: "Oh! I didn't see that one! I thought it was going to be 1,2,3,4,5 (he says while pointing to 14r, 8r, 6r, 10r and the next block in the row." EHN Simple Deduction
7.8 Seth: "I win again! I'm really good at these games. - - I'm really good!"
Game 8
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20 r |
21 b |
17 r |
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19 b |
18 b |
6 b |
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15 r |
13
r |
12
b |
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14 b |
6 b |
11
r |
10
b |
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26 r |
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4 b |
5 r |
7 r |
9
r |
27
b |
25 b |
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1 r |
3 r |
2 b |
8 b |
22
r |
23
b |
24 r |
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-40
To Beginning of Section | To End of Section
Note: Jared ended up placing two markers in a row (18 and 19) because they lost track of who was supposed to lay next while Seth was trying to take out 17r.
8.0 Seth (red) and Jared (black) set up for a new game. (3:50)
8.1 Seth: "I'm really good at this."
8.2 Jared: "You might cheat you know."
8.3 Seth: "No I won't."
8.4 On turn 6, Jared states, "You're not gonna win. Cause I said you're not."
8.5 On turn 10, Jared states, "This time, you're not gonna fool me."
8.6 On turn 14, Jared states, "If I put one in there (where 22 r was later placed) , he'll put one in right there (the place directly above where 22r was placed)" before he decides where to place his checker. EHN Simple Deduction
8.7 On turn 15, Jared states, "Don't put it in there (18). No!"
8.8 As soon as Seth places #17 Jared says, "Wrong spot!"
8.9 Teacher, "Why do you say wrong spot?"
8.10 Jared, "Cause I was gonna win." Seth tries to take it out. EWT
8.11 On turn 20 Jared says, "I can win. No matter what if he puts one there (D4)"
8.12 Teacher, "You better stay away from that one Seth."
8.13 On turn 28, Seth notices that he has already made a connection: "Hey. I think I got a row. 1,2,3,4." (3:52)
8.14 Jared says, "Hey. 1,2, - - I win! Look! 1,2,3,4."
8.15 Teacher: "When did that happen?"
8.16 Jared: "I don't know. Just then. Yeah, when I put this one in, it happened and we didn't even notice.
8.17 Seth: "Actually, I had one before him. 1,2,3,4. We tied!"
8.18 Jared: "How did you win?"
8.19 Seth: "1,2,3,4." Seth points to the row of black.
8.20 Jared is still unsure of the row Seth is referring to so Seth turns the board around so both of them can see and points it out to him.
8.21 Jared: "Oh - - We're out of time."
Game 9
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30
b |
27 r |
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22 b |
28 b |
26
b |
25 r |
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21 r |
19 r |
23 r |
24
b |
29 r |
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20 b |
10 b |
17 r |
13 b |
16
b |
18 b |
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11 r |
6 b |
8 b |
9 r |
14 r |
15 r |
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1 r |
3 r |
7 r |
2 b |
4 b |
5 r |
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Note: Seth and Jared took two consecutive turns (14 and 15), therefore you'll notice red becomes odd numbered and black becomes even numbered.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-41
To Beginning of Section | To End of Section
9.0 Seth says to the teacher: "We're smart as this game. We're smarter than you!"
9.1 The teacher suggests that she (black) play against Seth and Jared (red). (3:53)
9.2 The game falls and is reset after 6 markers were placed.
9.3 Teacher, "Now, what am I going to do?"
9.4 Jared, "You could put one here (C2), you can be (B3)." EHT
9.5 Teacher, "Oh, you're helping me out."
9.6 After 8 is placed Seth warns, "Uh, uh, uh."
9.7 On turn 11 Jared says to Seth, "Look she's gonna put it up there (where 21 r was later placed) but it won't work. Oh, yes it will." EHN Simple Deduction
9.8 Seth: "Okay, I can block her." Seth places his marker in 11 r.
9.9 Teacher, "You're trying to block me are you?"
9.10 Jared, "Yup."
9.11 Teacher, "I gotta size it up."
9.12 Jared: "Oh, don't put one right there." He indicates to the position where 20b is later placed. EHN
9.13 Seth, "I won't, I'll be careful … I think you're gonna get a bad choice."
9.14Teacher, "Why was it a bad choice?"
9.15 Jared, "If you put one here (24) [the rest is unclear].
9.16 Seth, "Your turn. Put one right there." (G1)
9.17 Teacher, "Why do you want me to put one there?"
9.18 Jared and Seth giggle.
9.19 Seth, "So we can win."
9.20 As the teacher places 16 Seth says, "Bad choice."
9.21 Jared puts 17 r in place and explains that if he didn't do that the teacher could have won. EWS Simple Deduction
9.22 Jared, "There, there, there."
9.23
Seth, "Okay."
9.24 Jared, "Oh no, no, no, no, no, no." (As he sees where she could have won.)
9.25 Teacher, Why did you do that?"
9.26 Jared, "Because you would win if you put it in there." … "Don't put it in there remember." (20) EWT Simple Deduction
9.27 Jared: "We're gonna win."
9.28 Teacher: "You're gonna win?"
9.29 Jared: "We might win."
9.30 Teacher, "I see 3 in a row so I better stop it."
9.31 Jared and Seth, "No!"
9.32 Jared, "We can still win."
9.33 Seth, "If I do that?"
9.34 Jared, "No, if we put it there then she'll put it there" (Seth had pointed to B5 and A6). EHN Simple Deduction
9.35 Jared, (Whispering) "Yeah, put it right there, that's good."
9.36 Teacher, "Now I see something." (She places 24)
9.37 The teacher blocks Seth and Jared's possible four in a row (24b).
9.38 Jared: "You should put it wherever you wanted."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-42
To Beginning of Section | To End of Section
9.39 The two boys try to convince the teacher to place a checker in the bottom corner once again, telling her that she would win if she did that.
9.40 Teacher, "I'm looking to see where I have 3 in a row."
9.41 Seth tries to place but Jared says no to the first few choices.
9.42 On turn 27, Jared says," If you put one right there, then we can put one right there and there and then we could win." (Note: I can not tell from the video where the student is pointing.) EHN Simple Deduction
9.43 The teacher places 28b with comments from the students, "Don't place it there, you won't win, we're telling ya. It's 3 not 4."
9.44 The teacher places 30 b to win the game.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-43
To Beginning of Section | To End of Section
Connect 4
Notes:
When doing the tables, I did not indicate red or black. Odd numbers are always red and even are always black. Today's group decided that red would always go first.
Interesting points??
I found it very interesting that many of the students today seemed to think that if you were red, you would probably win.
Scene 12 is interesting. It is a discussion among Seren, Jocelyn and Veronica about a section in the rules called, "How to win."
Game six and game eight are strikingly the same.
When discussing things to do to help when, Veronica mentions that if anyone covers her up, she finds another way to win. I think she was referring to blocking her vertically (11).
Game 1
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20 |
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17 |
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24 |
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19 |
27 |
16 |
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23 |
26 |
15 |
14 |
12 |
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22 |
25 |
11 |
8 |
10 |
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21 |
13 |
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7 |
9 |
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4 |
3 |
1 |
6 |
5 |
18 |
1.0 Leonard and Preston (black) played a game against the teacher (red). (3:22)
1.1 Preston says as he places his team's first checker, "I always got an idea to win."
1.2 Leonard after turn 4 says, "I boxed you there."
1.3 As Preston is figuring out where to put 6, Leonard `guides' him into blocking the teacher's chance of getting connect 4. "Put that in there. No, wrong one, wrong one! In here - - No! Over one! No! In there."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-44
To Beginning of Section | To End of Section
1.4 Leonard: " . . .She would've won had she gone in there." EWS Simple Deduction
1.5 Teacher: "That was good watching. Now, what am I gonna do?"
1.6 Leonard: " I told him to put it in there and he did."
1.7 Preston: "I knew, I was just - - fooled up Leonard."
1.8 Leonard: "Yeah, I told ya to put it down there."
1.9 Preston: "Let her play."
1.10 On turn 10, Leonard tells Preston where to place the checker and he does so immediately. (3:22)
1.11 On turn 11 Preston whispers to Leonard and points.
1.12 Leonard, "Ah, you told her!" (as the teacher places here marker).
1.13 Preston, "Darn!" (as Preston places #12) He should have put it right there. (G1)"
1.14 Teacher, "Why should he have put it there?" (No answer.)
1.15 Leonard, "Your turn."
1.16 Teacher, "I have to look and see what's happening here."
1.17 Leonard, "Now go here and fill up (14). Now go here."
1.18 Preston, "What will happen?"
1.19 Leonard, She's got 2 here. Block her. That one there." EWS Simple Deduction
1.20 Teacher, "Now what are you trying to do over here?"
1.21 Preston, "We're not telling."
1.22 Leonard, "We're not telling ya cause then you'll get it."
1.23 Preston whispers something very quietly into Leonard's ear and then places 14.
1.24 Leonard places #18.
1.25 Preston, "Leonard, No! Leonard, you should've put it there (E5)"
1.26 Leonard, "Why?"
1.27 Preston, "Then we could've had her.
1.28 On move 20, the teacher is blocked. Leonard states, "We're the hottest team here."
1.29 Leonard, "We got ya there, we got one there."
1.30 Leonard, "We'll block you if you put it down there." (They do.) "Yes, we blocked you there!" EHN Simple Deduction
1.31 On move 26, the teacher's possible four in a row (3, 13 and 25) is blocked however the teacher still wins in another location.
Game 2:
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33 |
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30 |
29 |
27 |
22 |
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26 |
28 |
19 |
21 |
16 |
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25 |
12 |
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20 |
15 |
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24 |
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11 |
14 |
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6 |
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4 |
13 |
32 |
2.0 Leonard and Preston (black) begin a new game against the teacher. (3:26)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-45
To Beginning of Section | To End of Section
2.1 Leonard and Preston whisper about something and tell the teacher to go first. Leonard says they have an idea. Preston says, "why did you have to tell her that?"
2.2 Teacher, "Well, I don't know what idea you have."
2.3 On move 6, Leonard shows Preston where to place the checker to block the teacher's chance of getting four in a row.
2.4 Leonard: "we got ya. You thought you had it, didn't you?!" Preston shows Leonard where #8 should go.
2.5 After move eight, Preston tells the teacher, "We want you to take a double turn this time." (Presumably so she would place markers in #11 and #20 so they could win with #21)
2.6 Teacher, "Why did you want me to take a double turn?"
2.7 Preston instructs the teacher on her 11th move, "I think you should put it down right there (11)."
2.8 Leonard: "I think she sees our idea."
2.9 Preston: "She probably would."
2.10 On move 13 the boys try to get the teacher to play either 23 or 20. Teacher plays 13.
2.11 Preston, "No!"
2.12 On move 15 Preston tries again to get the teacher to play 20.
2.13 On move 20 the boys decide to place it themselves.
2.14 Preston, "Darn it!" after teacher places 21.
2.15 On turn 22, Preston explains what he was going to do if the teacher hadn't placed checker 21 where she had, "I was gonna make it go straight up like 1,2 3, . . .4."
EHN Simple Deduction
2.15a Leonard: "Come on, just hurry up."
2.16 On move 28 Leonard shows Preston where to go.
2.17 Leonard, "You can't get 4 there."
2.18 Teacher plays 29.
2.19 Leonard, "Ah, block it!" (30)
2.20 Teacher, "What did you block?"
2.21 On turn 30 Leonard says, " . . . We blocked you. We thought you were gonna get four there" (27, 29, 30 and above 23) EHT Simple Deduction
2.22 On move 33, Leonard states without realizing that the teacher just connected four in a row, "She can't get it, she can't get it. Block it there (he points to where 34 was placed)."
2.23 Teacher: "Connect four."
2.24 Preston: "No. - - darn."
Game 3
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The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-46
To Beginning of Section | To End of Section
3.0 Leonard and Preston (red) play a new game against the teacher (black). (3:32)
3.1 There is a discussion about why red always goes first. Leonard says that in his rule book red always goes first.
3.2 Preston speaks into the microphone before the game begins saying that it is game three that is about to begin. "The first two games, she (the teacher) won. We lost. She's killing us out there."
3.3 Leonard, "Oh, she blocked me!" (with #6)
3.4 Preston, "How many rounds do we have?"
3.5 Leonard yells loudly on turn 17, "Block it there!"
3.6 Preston blocks the move.
3.7 Leonard states: "She would've got 1,2,3,4 if we didn't block it there." (4, 12, 16, 17)
EWS Simple Deduction
3.8 Leonard: "We keep on blocking you every single way."
3.9 Teacher, "I wonder if there's any other way I can get 3 in a row?"
3.10 Leonard, "Look!" (As teacher makes 3 in a row with #18)
3.11 Teacher, "I've got 3 in a row right now.
3.12 Preston: "Leonard you know she can kill us."
3.13 Leonard: "Ya. But she's not gonna in this one."
3.14 Preston: "Yes she is. Watch."
3.15 Teacher, "Why do you think I'm going to beat you?"
3.16 Preston, "Cause. See?"
3.17 After 21 is played Leonard yells, "Ah man!"
3.18 Preston, "I told you she was gonna beat us."
3.19 The teacher connects four in a row on move 22.
Game 4
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16 |
17 |
20 |
24 |
30 |
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23 |
29 |
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28 |
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31 |
33 |
4.0 Seren and Leonard (red) play a new game against Preston and the teacher (black). (3:36)
4.1 Leonard, "We knows exactly what to do!"
4.2 Turn 8, Seren: "Come on! Hurry it up!"
4.3 Preston was questioned why he placed 16 where he did. He pointed to 16, 10 and 14. (Preston was going to place 16 in 20 until the teacher said, "Wait now."
EWT Simple Deduction
4.4 When Preston placed 20, Seren said, "Ah, you took my place."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-47
To Beginning of Section | To End of Section
4.5 Preston placed 22 and said, "Got it! (He pointed to 16, 10 and 14 and realized that he did not have four in a row.) Darn it!"
4.6 Seren, "I only have three [markers] left."
4.7 On move 26, Leonard: "Man! No one's ever gonna win."
4.8 Seren: "Sometimes you can get the whole board filled out without winning. Pass me [a marker] No, it's your turn Leonard."
4.9 Leonard makes four in a row. "We finally won a game!"
It is interesting to note that at the beginning of this game, Seren and Leonard spent a great deal of time making sure they had an even number of checkers. After 29, Seren had no more checkers left but Leonard had plenty, yet they were taking turns. Seren told Leonard, "You get the rest of the board to yourself." Seren often took three turns in a row before letting Leonard have a single turn. Very little collaboration went on between the two students. Seren placed 1, 5, 7, 9, 13, 15, 17, 21, 25, 27 and 29 while Leonard placed 3, 11, 19, 23, 31, 33, and 35.
The game was actually won after #25 with 9, 13, 19 and 25 but no one seemed to notice.
Game 5
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5.0 Seren and Preston (red) play a new game against the teacher and Leonard. (3:40)
5.1 Seren and Preston have difficulty remembering to take turns for their team.
5.2 Seren tells Preston he should put 11 where 12 is eventually placed because they could block Leonard.
5.3 Preston states, " I got a better idea. Let's save that for later." Preston places 11.
5.4 Leonard gets four in a row. Preston could have blocked if 15 was placed in 16.
5.5 Preston states that they could have got four in a row if they placed one on top of 16.
Simple Deduction
5.6 Leonard: "But we did it before you!"
Game 6
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The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-48
To Beginning of Section | To End of Section
6.0 Veronica and Jocelyn (red) play against Seren and the teacher. (3:44)
6.1 The game begins.
6.2 Teacher, "Now, Seren, what should we do? Where should we go? What do you find works well?"
6.3 Veronica: "I usually wins against Seren."
6.4 Teacher: "Don't forget that Jocelyn gets a turn too."
6.5 Veronica: "I know but I'm just showing her how to win."
6.6 Teacher, "Okay, you guys talk about it, where Jocelyn's gonna put it."
6.7 Seren places 6.
6.8 When questioned why Seren placed 6 in the spot she did, she states, "Because we might get four in a row." (She point to the three that are now in line.)
EWT Simple Deduction
6.9 Veronica: "No - - but - - I'm gonna win anyway."
6.10 Veronica and Jocelyn connect four.
6.11 Seren; "I know. It's only a game."
Game 7
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7.0 Veronica and Jocelyn (red) play against Seren and the teacher. (3:46)
7.1 Teacher, We gotta keep an eye on them this time Seren. We gotta make sure they don't have 3 in a row."
7.3 Veronica, "4 in a row!"
7.4 After placing 2, Seren says, "See, I already blocked you."
7.5 Veronica: "Sorry! We win AGAIN!"
7.6 Veronica: ". . . you never kept a sharp eye on us cause look
7.7 Seren: "I kept as sharpest eye as I could on you."
7.8 Veronica, "It's a bad idea, I think, to play with us!:
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-49
To Beginning of Section | To End of Section
Game 8
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8.0 Veronica and Jocelyn (red) play against Seren and the teacher. (3:48)
8.1 Teacher, "Red goes first."
8.2 Seren, "Again?! They won two games in a row. Maybe we can go first because they always win."
8.3 Veronica, "You decided that red goes first."
8.4 Teacher, Yes, we'll have to change markers next time."
8.5 Teacher: "Let's watch them carefully Seren."
8.6 Seren: "Okay you tell me where to put them."
8.7 The teacher explains that they must decide together.
8.8 After the first move, Seren points out with the teacher's prompting the possible ways the other team can get four in a row. Teacher, "Let's see which ways can they get 4 in a row from that position?"
8.9 Veronica, "We can go from up here, we can go up and over." EHT Simple Deduction
8.10 Seren decides to block one of the ways by placing 2.
8.11 On turn 4, Seren: "Maybe we should block this one too."
8.12 Teacher asks Seren why she placed 6 where she did.
8.13 Seren: "Oh that wasn't a smart idea - - because she's gonna win there." (She indicates along the bottom." EWT Simple Deduction
8.14 Veronica and Jocelyn win.
8.15 Seren, "Now we trade markers, pass them over."
8.16 Veronica: " We're really good at this game. Even if we're black, we might still win this game."
8.17 Teacher, It doesn't matter what colour you are. Maybe we'll see. Maybe black will win."
8.18 Seren: "I was thinking that black would always win but I tried - -"
8.19 Veronica: "Maybe cause it's because we're always first and we can get more in."
VERIFYING Induction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-50
To Beginning of Section | To End of Section
Game 9
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9.0 The teacher and Seren (red) play against Veronica and Jocelyn. (3:50)
9.1 On turn 8 Veronica shows Jocelyn where to place the marker.
9.2 Jocelyn hesitates over 18.
9.3 Seren, "You don't want to be found guilty do you?"
9.4 Veronica, "Jocelyn don't even know where to put it. Don't put it there, she'll go that way."
9.5 After turn 23 Seren says, "Oh!" (As she sees 3 in a row - 9, 15, 23)
9.6 Seren gets excited after placing 25.
9.7 Seren placed 27 to get four in a row. "I won, four in a row."
Note, she had actually won a turn earlier but no one noticed (17, 19, 23, 25).
Game 10:
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10.0 Teams change for a new game. Veronica and the teacher (red) play against Jocelyn and Seren. (3:53)
10.1 Veronica: "But I'm red!"
10.2 Seren: " But red is the winning colour."
10.3 Seren indicates that she'd like to do another activity.
10.4 The game begins and Veronica and the teacher quickly win.
10.5 Jocelyn: "It could be the red or Veronica's just really good."
10.6 Teacher, "How could we figure out if it is red or Veronica?"
10.7 Seren, "Ah, play against her again?"
10.8 Jocelyn questions Veronica about the number of games she's played and how many she has won.
10.9 Veronica has won four out of five games that she has played.
10.10 Teacher, "Why don't we let Veronica have black and see what happens?"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-51
To Beginning of Section | To End of Section
VERIFYING Induction
Game 11:
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11.0 Veronica and the teacher (black) play against Jocelyn and Seren. (3:56)
11.1 It is decided that Veronica's team will use black to see if red is the reason she's winning so often.
11.2 Veronica: "Or maybe I would win because I'm a good player."
11.3 Jocelyn: "It could be the red."
11.4 Seren: "You're not such a good sport. - - Sometimes people can't be good sports and still win. You . . .don't have to brag about if you're good or not."
11.5 Jocelyn: "It could hurt someone's feelings."
11.6 Seren and Veronica talk about some students in another class who brags.
11.7 Jocelyn: "It's not nice to brag cause it could hurt people's feelings."
11.8 Seren is not pleased with where Jocelyn places 7 when she is asked where she should have placed the checker, Seren says she should have put it where 9 is placed.
11.9 Veronica: "I'm gonna win."
11.10 Veronica: "Too late! I win the game."
11.11 Seren: "VICTORIA!"
11.12 Veronica giggles.
11.13 Veronica: "It's me cause I'm a good player. It's not because of the red."
11.14 Seren, "Maybe not Vicky."
11.15 Seren: "It's not nice to brag now."
11.16 Veronica: "I'm not bragging about it."
11.17 the teacher asks Veronica to tell the group things she does to help her play well.
11.18 Veronica: "I watch the game. See - - if you don't watch the game, you might not win." Simple Deduction
11.19 Teacher: "So what do you watch for?"
11.20 Veronica: "I watch for things. . .if they put something on top of me, I tries to find another way to win." EHT Simple Deduction
12.0 Jocelyn begins to read the instructions for the game on how to win. (3:59)
12.1 Jocelyn and Seren read the section titled, "How to win."
12.2 Seren: "That's not!"
12.3 Jocelyn: "This doesn't make sense."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-52
To Beginning of Section | To End of Section
12.4 Seren: "It doesn't even tell you how to win."
12.5 Teacher: "What does it say?"
12.6 Jocelyn reads from the instructions: ". . .If you are the first person to get four checkers in a row, you win the game."
12.7 Seren: "That doesn't tell you how to win."
12.8 Veronica: "Yes it is - - because four in a row - - wins. That's how you play it."
12.9 Seren: "But it doesn't tell you how to like win."
12.10 Teacher: "Do you have any tips on how to win the game Veronica?"
12.11 Seren: "She probably does nut won't tell them."
12.13 Veronica: "Watch the game."
12.14 Jocelyn: "What channel is it on?"
12.15 Veronica: "What channel is what on? . . . There's a commercial that's called Connect 4 and I watch it. And I learned how to win."
12.16 Jocelyn: "What channel is it on?"
12.17 Veronica: "I don't know."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-53
To Beginning of Section | To End of Section
Connect 4
Notes:
Overall, I did not find today's games to provide many insights into how the children were thinking about the game as they played or their feelings about what was going on as they played. I have included many of the tables indicating the moves in the games and some comments where I felt they might provide "something". Very little talking went on today, even with questioning from me.
odd numbers - red
even numbers - black
Interesting points (maybe??):
Game 10 : Even though Saul noted 6, 8, and 10 several times throughout the game ( which his opponent could use to win), his opponent eventually won using these three checkers.
Game 17: Maurice recognizes that he can win after placing three checkers along the bottom, with nothing placed on either side. His opponent, Kirsten, does not seem see how he knows he has won after only having three in a row.
Game 19: Game 17 is repeated again! Identical game...moments later!
Game 1
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1.0 Laura (black) and Kirsten (red) played. (3:25)
1.1 After #6 the teacher says, "I want to know, how come you put this one there a little while ago?"
1.2 Laura, "I don't know."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-54
To Beginning of Section | To End of Section
1.3 Both students got connect four quickly. Neither realized the other had got four in a row.
Game 2
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2.0 Kirsten (black) and Laura (red) played. (3:27)
2.1 Laura, "Now its ME who's red."
2.2 The teacher asked Laura why she placed 9 and 11 where she did. Laura shrugged her shoulders in response as if to say, "I don't know."
2.3 Note that after move 18, Laura was able to connect four but did not do so until much later. Also Kirsten could have won after #20.
2.4 The teacher asked Kirsten why she placed 28 where she did.
2.5 Kirsten: "- - cause it's two blacks." She indicates to 26 and 28.
EWT Simple Deduction
2.6 Laura wins when she places 29.
Game 3
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3.0 Laura (black) and Kirsten (red) played. (3:30)
3.1 Kirsten points to her 3 in a row (7,3,5) but doesn't act on it after #11 is placed.
3.2 After move 13, Kirsten could have connected four but neither her nor Laura notice.
3.3 Laura wins the game.
3.4 Kirsten goes to take another turn, not realizing that Laura has just won.
3.5 At the end of the game the teacher asks, "Did you notice that you had 3 in a row for awhile?"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-55
To Beginning of Section | To End of Section
Game 4
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4.0 Kirsten (black) and Laura (red) played. (3:31)
4.1 Kirsten and Laura argue over who has won.
4.2 Kirsten: "1,2,3,4." EWS Simple Deduction
4.3 Laura: "I won before you."
4.4
Kirsten: "No!"
4.5
Laura: "Yes!"
4.6 Kirsten and Laura continue to argue about who has won. They do not come to an agreement about who won.
Game 5
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5.0 Laura (black) and Kirsten (red) played. (3:33)
5.1 Kirsten wins quickly.
5.2 Kirsten, "I got it before you!"
Game 6
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6.0 Kirsten (black) and Laura (red) played.
6.1 Teacher: "Kirsten why did you put that (2) there?"
6.2 Kirsten: "Cause she won't win. " EWT Simple Deduction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-56
To Beginning of Section | To End of Section
6.3 Laura explains that she put 9 in place because Kirsten had two going across the bottom. EWX Simple Deduction
6.4 Before #13 the teacher says, "Now Laura's sizing it up to see what she's gonna do."
6.4 Laura wins. While Kirsten is building her 3 in a row she doesn't even watch what Laura's doing.
Game 7
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7.0 Saul (black) and Maurice (red) played. (3:37)
7.1 The teacher asked Maurice why he placed 5 where he did.
7.2 Maurice whispered that he likes to make "it" go on an angle. EWT Induction
7.3 Saul, "I heard that." (As he places #6).
7.4 After #8 Saul says, "He tried to fool me. Look, but it doesn't go like that (7, 3,5). He can't do that 1, 2, 3 (10, 8, 6) EWS Simple Deduction
7.5 Maurice looks carefully at the board before making his move (13).
7.6 Teacher, "I like what Maurice's doing. He sizing up the board and seeing it."
7.7 As soon as Saul places #15 Maurice says, "I win."
7.8 Maurice wins.
Note, Saul goes twice in a row (6 and 7) so black becomes odd and red becomes even after #6.
Game 8
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8.0 Saul (black) and Maurice (red) play. (3:40)
8.1 Teacher, "I thought he had you then. Why did you change your mind?" (As Saul almost missed the block at #6)
8.2 Maurice smiles as he blocks with #15.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-57
To Beginning of Section | To End of Section
8.3 Before Maurice places 17, he holds the checker over every column and looks carefully at it.
8.4 Saul, "Rats."
8.5 Teacher, "I didn't think he was gonna put it there did you Saul?"
8.6 Maurice announces that he won as soon as Saul places #22.
8.7 Maurice wins.
Game 9
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9.0 Saul (red) and Maurice (black) play. (3:42)
9.1 Maurice pauses before he places #12 and looks over the board. Up until then they had been going quickly.
9.2
Maurice quickly wins.
Game 10
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10.0 Maurice (black) and Saul (red) play.
10.1 Before placing 11, Saul examines the playing board carefully. He traces possible connections that Maurice can make with his finger. He notices 6, 8 and 10. He does not point to his own "connections".
10.2 Before placing 15, Saul points to 6, 8 and 10 again.
10.3 After Maurice places 24, Saul says, "Now let's see that." He traces with his finger the possible connections that Maurice can make using 24.
10.4 Maurice wins using 6, 8, 10 and 26.
10.5 Maurice: "Bad luck Saul."
10.6 Saul, "I knew that though."
Note: The first 6 moves of game 9 and 10 are the same.
Game 11
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-58
To Beginning of Section | To End of Section
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