previous table of contents appendix B table of contents next
Game 10: Laura
Needed to win: 3 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
|
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
8 |
5 |
|
|
|
|
xxx |
xxx |
|
xxx |
|
|
7 |
4 |
6 |
10 |
|
|
|
xxx |
|
|
|
|
3 |
2 |
1 |
9 |
|
|
|
|
10.0 Laura decides to change the shape of the game board and try again. She still keeps the number of markers needed to win at three. (3:36)
10.1 Laura is unaware that she can win in move 9. (If she placed the marker next to the 3, she would have made a diagonal of three.)
10.2 The computer wins.
Game 11: Saul
Needed to win: 3 in a row
Winner:
Saul
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
|
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
|
|
|
|
|
|
xxx |
xxx |
|
xxx |
|
|
|
|
|
|
|
|
|
xxx |
|
|
|
2 |
4 |
3 |
1 |
5 |
|
|
|
|
It is interesting to note that Saul was watching Maurice very carefully while he played earlier. Saul uses the same strategy that Maurice used.
11.1 Saul begins a new game. (3:40)
11.2 After Saul places number 1, "You see in the middle there is um - -"
11.3 Kirsten: "You just put two more there. (She points to the empty spaces on both sides of the 1.)" Simple Deduction
11.4 When the computer places 2 Saul says, "See, I'll show ya."
11.5 Kirsten after move 4: "Yes, he's gonna win. See - - look." She points to position 5.
11.6
Saul wins.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-102
To Beginning of Section | To End of Section
Game
12: Kirsten
Needed to win: 3 in a row
Winner:
Kirsten
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
Xxx |
|
xxx |
xxx |
xxx |
|
|
|
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
|
|
|
|
|
|
xxx |
xxx |
|
xxx |
|
|
|
|
|
|
|
|
|
xxx |
|
|
|
2 |
4 |
3 |
1 |
5 |
|
|
|
|
NOTE: This game was played out identically to Saul's previous game.
12.0 Kirsten sets up the board for a new game. (3:41)
12.1 Saul tells Kirsten: "Now, first you do the middle."
12.2 Saul: "I'm gonna try four in a row cause I'm so good at this."
12.3 Kirsten: "He won't even let me go. Saul won't even let me go (on my own)."
12.4 As the computer places number four, Kirsten excitedly shouts, "Yes!"
Game 13: Saul
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
11 |
|
|
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
10 |
15 |
|
|
|
|
xxx |
xxx |
|
xxx |
16 |
14 |
8 |
12 |
13 |
|
|
|
9 |
xxx |
|
|
7 |
6 |
4 |
2 |
1 |
|
|
|
5 |
3 |
13.0 Saul begins a new game but decides to try four in a row. (3:42)
13.1 When asked if four in a row will be harder, Saul says no.
13.2 When the computer places number 2, Saul shouts, "shoot!"
13.3 As Saul is placing move 5, the teacher states that this is going to be trickier.
13.4 Saul: "No! Not at all tricky!"
13.5 After the computer places 6, Saul places 7 stating, "not your tricky ways mister!" As he blocks the computer he laughs and says, "he tried, he tried." then he adds in a sarcastic tone, "Oh, I'm scared!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-104
To Beginning of Section | To End of Section
13.6 After move 10, Saul asks the computer, "Oh you wanna win do ya. Well I'm gonna tell you to just uh, uh. You are not gonna win! Ha, ha."
13.7 On move 13, Saul states, "He's not gonna win easily on me!"
13.8 After Saul places 15, Kirsten quickly points out a diagonal that he can make on the next turn if he places a marker next to 1. Simple Deduction
13.9 The computer wins on move 16.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-105
To Beginning of Section | To End of Section
Activity: Tic Tac Drop (Similar to Connect Four. It was played on the computer.)
Notes: There was no videotape available for Seren and Veronica's games.
Game 1: Preston
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
|
22 |
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
13 |
23 |
|
21 |
24 |
20 |
xxx |
xxx |
|
xxx |
|
9 |
12 |
15 |
17 |
14 |
19 |
16 |
18 |
xxx |
|
|
7 |
4 |
2 |
3 |
1 |
5 |
6 |
11 |
10 |
8 |
1.0 Preston begins a game. (3:09)
1.1 Leonard: "We can always beat the computer. It's easy! If Preston can't, I'll try it."
1.2 Preston as the computer takes turn 2: "Now watch, he's gonna go over me - -"
1.3 Leonard: "Block `em."
1.4 Preston: "No, Leonard - -"
1.5 Leonard: "Sometimes when red (the computer is blue) wins, it still doesn't go on and we win."
1.6 Preston shouts "darn" when the computer places 10 to block him.
1.7 The teacher asks Preston what his plan is going to be now.
1.8 Leonard: "Block him!"
1.9 On Preston's 11th turn, Leonard indicates where he should put the marker.(3:10)
1.10 Preston: "Leonard! I know where to go! I'm gonna put it right in the middle there."
1.11 Leonard: "Put it up there! Put it up there."
1.12 Preston: "I know what I'm doing. This is MY game."
1.13 Leonard: "We're going to try to go straight up."
1.14 Preston after move 17: "Too bad when you gotta block like that, they don't count." Preston points to the block of four he has created with 1, 3, 15, and 17.
1.15 Leonard after turn 18, "Oh Preston, we gotta go in there - - yeah if we don't he's gonna win." (He indicates position 19.) Simple Deduction
1.16 The computer places 20.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-106
To Beginning of Section | To End of Section
1.17 Leonard: "Ooo, don't put it there (position 24) cause then he'll fall." Simple Deduction
1.18 The computer wins on move 24.
Game 2: Leonard
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
13 |
|
|
20 |
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
16 |
12 |
|
|
18 |
|
|
xxx |
xxx |
|
xxx |
15 |
11 |
10 |
17 |
|
14 |
19 |
|
|
xxx |
|
1 |
3 |
5 |
6 |
7 |
4 |
2 |
9 |
8 |
|
|
2.0 Leonard starts a game. (3:12)
2.1 Leonard after move three, "I'm gonna go straight over. He might never go."
Simple Deduction
2.2 Preston: "Yeah but Leonard, he can maybe go (he points to a possible diagonal that Leonard can create starting at the 3)."
2.3 After Leonard places 5, Preston states, "You're in big trouble!"
2.4 Leonard, "Why? He's not gonna block me."
2.5 Preston: "Yeah he is."
2.5a The computer blocks Leonard.
2.6 Leonard: "Yeah, so I'll block him." Simple Deduction
2.7 Preston on move 15, "You should put one right there." (Position 15)
2.8 Leonard places 15 and says, "I think I'm gonna win!"
2.9 The computer blocks him.
2.10 Leonard: "So! I'll block him." He places 17.
2.11 Preston points out position 20 on Leonard's 19th turn.
2.12 Leonard places 19.
2.13 Leonard: "Ohhhh! What did I do? I meant to go down that one. (The position that Leonard just pointed out.) Simple Deduction
2.14 The computer wins.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-107
To Beginning of Section | To End of Section
Game 3: Preston
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
Xxx |
|
xxx |
xxx |
xxx |
xxx |
|
7 |
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
6 |
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
17 |
|
|
4 |
|
19 |
|
xxx |
xxx |
|
xxx |
11 |
13 |
|
|
2 |
20 |
16 |
18 |
|
xxx |
|
3 |
9 |
5 |
10 |
|
1 |
8 |
15 |
14 |
12 |
|
3.0 Before beginning, the teacher says, "somebody has to beat this computer." (3:15)
3.1 Leonard: "But we can't!"
3.2 The teacher asks Preston if he has any plans to help him beat the computer.
3.3 Preston: "Some people, what they do is make a row going up like red, blue, red, blue, red, blue. " (He points to 1 and 2.) Induction
3.4 Leonard: "Try that."
3.5 Preston places 3.
3.6 The teacher asks Preston why he put three "way over there."
3.7 Preston: "Watch, now just let him get three and I will block him."
Multi-Step Deduction
3.8 After move 7 Leonard says, "there he's blocked!"
3.9 Before move 9, Leonard points to the place between 3 and 5.
3.10 Preston places 9.
3.11 Leonard: "I don't think he's going to see it."
3.12 Preston: "Shh!"
3.13 The computer blocks.
3.14 Leonard: "Ah! He blocks us every time."
3.15 Preston: "Leonard you know sometimes he does something else."
3.16 After placing 11, Preston points to the screen to indicate that he wants to make a diagonal, starting with 3. Simple Deduction
3.17 Before placing move 15, Leonard advises Preston that he should put the marker between 8 and 14. (3:16)
3.18 Preston: "See - - no -- Leonard"
3.19 Leonard: "YES! He'll win if you don't!"
3.20 Preston: "Leonard, it doesn't work see." (He points to the empty places next to 14 and 12.)
3.21 Leonard: "Yeah but he'll still block you!" Simple Deduction
3.22 Preston finally agrees to place 15 where Leonard suggests.
3.23 Before Preston places 19, Leonard points out position 20 to him, "Right there, right there."
3.24 Preston places 19.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-108
To Beginning of Section | To End of Section
3.25 Leonard: "Oh, he's gonna win."
3.26 Preston: "Oh, I thought I blocked."
3.27 Leonard: "Oh Preston, you went in the wrong one."
3.28 Preston: "Leonard!"
3.29 Leonard: "It's my turn. I'm gonna beat him."
Game 4: Leonard
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
24 |
15 |
26 |
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
23 |
22 |
14 |
25 |
|
|
|
xxx |
xxx |
|
xxx |
13 |
21 |
16 |
12 |
20 |
17 |
9 |
|
18 |
xxx |
|
3 |
5 |
6 |
1 |
2 |
19 |
7 |
4 |
11 |
10 |
8 |
4.0 Leonard begins a new game. (3:17)
4.1 Leonard after he places 3, "I'm gonna beat him."
4.2 While the computer pauses for a second, Leonard talks to it saying, "Come on boys, just go!"
4.3 Preston: "Leonard, just let him get three and --"
4.4 Leonard: "THAT'S WHAT I'M DOING PHILIP!"
4.5 The computer places 6.
4.6 Leonard: "Look! I just tried to win and he blocked me."
4.7 Leonard: "What if we get them all filled out and we don't have any winners?"
4.8 The teacher tells Leonard that is a sign of two very good players.
4.9 After move 11, the teacher asks Leonard what he's going to do to win. (3:18)
4.10 Leonard: "I'm not sure. Go all over the place maybe." Simple Deduction
4.11 After having placed 17, Preston tells Leonard he should have placed the marker in position 23 so that he can work on making a diagonal from 3. Simple Deduction
4.12 Leonard: "No!"
4.13 After placing 21, Leonard says, "Ha! Look what I did." He just blocked the computer from winning. Simple Deduction
4.14 Leonard: "This is gonna be an easy game." (3:19)
4.15 The computer wins with move 26. (3:20)
4.16 Leonard: "Ah man! I can't win `em."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-109
To Beginning of Section | To End of Section
Game 5: Seren (This game is not on video tape!!!!!)
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
8 |
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
7 |
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
5 |
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
|
|
3 |
|
|
|
xxx |
xxx |
|
xxx |
|
|
|
|
2 |
11 |
|
|
|
xxx |
|
|
|
|
|
9 |
1 |
4 |
12 |
10 |
6 |
|
5.0 Seren begins a new game.
5.1 Teacher: "What are we going to do to win?"
5.2 Seren: "Drop it down the middle."
5.3 The teacher asks why.
5.4 Seren: "So the computer won't get it." Simple Deduction
5.5 Teacher: "Is the middle a good spot?"
5.6 Seren: "Yep!"
5.7 The teacher asks why the middle is a good spot.
5.8 Seren doesn't answer.
5.9 Veronica: "I know how he's trying to win!"
5.10 Veronica: "You better be careful cause he'll go up like that." (I'm assuming from the time that has elapsed in the game so far, Veronica is indicating to 2 and 4.)
Simple Deduction
5.11 Seren: "Oh no he won't."
5.12 Veronica after Seren places 7, "All you need is one more to fill it in."
Simple Deduction
5.13 Veronica: "you'll never win. He's gonna block it off right there." Simple Deduction
5.14 Seren: "No he won't."
5.15 Veronica after Seren places 11, "Oh no! The computer is going to win."
5.16 Seren: "No he isn't."
5.17 The computer wins with move 12.
5.18 Seren: "Yes he is."
5.19 Veronica: "I told ya."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-110
To Beginning of Section | To End of Section
Game 6: Veronica (NOTE: THERE IS NO VIDEO TAPE OF THIS GAME.)
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
16 |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
|
|
14 |
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
11 |
|
|
13 |
18 |
|
xxx |
xxx |
|
xxx |
|
|
10 |
3 |
15 |
12 |
17 |
20 |
|
xxx |
|
1 |
|
5 |
4 |
2 |
7 |
6 |
9 |
19 |
8 |
|
6.0 Veronica begins a new game.
6.1 On move 3, Seren tells Veronica to place the marker next to 2 but Veronica places it on top of 2.
6.2 Seren explains why she thinks Veronica should have placed it next to 2 instead of on top: "He'll get four in a row."
6.3 Veronica: "I know what to do! It's easy."
6.4 Seren: "he's gonna drop it down there - -
6.5 Veronica: "Now I have to drop it down the middle." (Move 7)
6.6 Seren: "Yes! So it won't get it." Simple Deduction
6.7 Seren: "Oh. I can see how he can win!"
6.8 Veronica: "I can see how he can win. But it won't win. " (I believe that they are referring to 6 and 8 along the bottom at this point.)
6.9 Veronica places 15 (I think.)
6.10 Veronica explains why she does this: "It might go through there and go across."
Simple Deduction
6.11 Veronica: "Don't worry. I know I'm gonna win."
6.12 Veronica is asked what she is going to do to win, "We're going to make sure that we have all four in a row and it’s just - -" Simple Deduction
6.13 Seren: "VICTORIA!"
6.14 Veronica: "I'm trying to think."
6.15 Seren makes a suggestion on where to place 19. (Position 19.) "She should put it right there. It would be blocking off it and it could go up." Simple Deduction
6.16 The computer wins with 20.
6.17 Seren: "Wrong idea."
6.18 Veronica: "Okay. Wrong idea."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-111
To Beginning of Section | To End of Section
Game 7: Seren (THERE IS NO VIDEOTAPE FOR THIS GAME.)
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
12 |
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
11 |
20 |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
|
15 |
9 |
19 |
22 |
xxx |
xxx |
xxx |
|
xxx |
xxx |
|
|
14 |
5 |
8 |
21 |
|
xxx |
xxx |
|
xxx |
|
|
13 |
10 |
4 |
7 |
18 |
|
|
xxx |
|
|
|
|
3 |
6 |
2 |
1 |
16 |
17 |
|
|
7.0 Seren begins a new game.
7.1 The teacher asks Seren what she's going to do to win.
7.2 Seren and Veronica spend time arguing about their time on the computer.
7.3 Seren gets back to the game.
7.4 Seren: "Oh. I'm lucky."
7.5 Veronica: "You're lucky you can get right there."
7.6 Seren: "Oh, I have a better idea than that."
7.7 Veronica: "He's going to block you off playing."
7.8 Seren: "No I'm not. I'm claiming it."
7.9 The Seren must block Seren because she says, "Oh man."
7.10 Veronica: "You better go right there."
7.11 Seren: "The computer always thinks of a good way to win."
7.12 The teacher says that they need to work on a plan to win.
7.13 Veronica: "Don't worry, we have a bunch of diagonals."
7.14 Seren: "I blocked off the computer so it wouldn't get somewhere."
7.15 Seren: "I have a good idea." She doesn't elaborate on her idea.
7.16 After Seren places 21, Veronica starts shouting, "I see one."
7.17 Veronica: "I see win that you are going to win."
7.18 Seren: "Yes!"
7.19 The computer wins with move 22.
7.20 Seren: "Vicky! You told me a bad idea!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-112
To Beginning of Section | To End of Section
Game 8: Joline
Needed to win: 4 in a row
Winner:
Computer
|
xxx |
xxx |
xxx |
xxx |
xxx |
|
xxx |
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
xxx |
|
|
|
Xxx |
xxx |
xxx |
xxx |
|
xxx |
xxx |
xxx |
17 |
15 |
|
|
|
xxx |
xxx |
xxx |
|
xxx |
xxx |
21 |
16 |
14 |
|
|
|
|
xxx |
xxx |
|
xxx |
|
7 |
13 |
12 |
22 |
20 |
18 |
|
|
xxx |
|
1 |
3 |
6 |
5 |
4 |
2 |
9 |
8 |
19 |
11 |
10 |
8.0 Joline begins a new game. (3:37)
8.1 The teacher asks Joline how she plans to beat the computer.
8.2 Joline: "I don't know."
8.3 Joline: "I got a great idea the way I'm going to win." (After 5 is placed.)
8.4 Jocelyn: "The next time if you go there the computer might not." Induction
8.5 Joline: "They won't win because - - we need five, right?"
8.6 The teacher informs them that they only need four in a row.
8.7 The computer wins.
Game 9: Jocelyn
This game is started but is not completed because centre-time ends.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-113
To Beginning of Section | To End of Section
Activity: The teacher read a book called The 512 Ants on Sullivan Street by Carol A. Losi. The students made predictions about the number pattern that was present in the book and later had the chance to create their own number patterns. The pattern presented in the book was a doubling pattern.
Notes: This activity did not lead to the insights that I hoped it would have. Alicia’s explanations of her mental arithmetic are neat though! (2.8, 5.1)
Chart created to go with story:
# of ants How many more ants joined in?
|
1 |
------ |
|
2 |
1 |
|
4 |
2 |
|
8 |
4 |
|
16 |
8 |
1.0 The teacher begins to read the book The 512 Ants on Sullivan Street. The only student who had heard the story before was Leonard. (3:27)
1.1 The teacher stops reading the story after the following numbers have been presented in the story: 1, 2, 4, and 8. She creates a chart displaying these numbers and asks the students to predict what number they think will come next. (In the story, the number is a group of ants.) (3:29)
1.2 David: "It all depends on how heavy the food is." (In the story, on each page, a group of ants come along carrying a particular item of food.)
1.3 Shelley: "Three." When asked why she thinks three she shrugs her shoulders.
1.4 Leonard is busy repeatedly saying Shelley is wrong.
1.5 Sally makes a guess of eight.
1.6 Alicia: "Ten. - - Because it is starting to go by twos." EWT Induction
1.7 Alicia's answer causes the group to start shouting out various responses about what the numbers are increasing by. (3:30)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-114
To Beginning of Section | To End of Section
1.8 Jared: "It was going by 1, then it went by 2 and now it's going by three." EWT Induction
1.9 The group examines the numbers and state that the numbers increased first by 1, then 2 and then 4.
1.10 Veronica makes a guess at how many ants will be coming next, "Ten. Because it's going up by fours. It goes 4, 8, - - then it goes two more ants and it almost goes like twos there (she is pointing to the chart). It goes eight and then you count by twos again cause 9, 10."
1.11 The teacher decides to continue read the story.
2.0 16 ants now enter. The teacher stops the story and adds the new number to the chart. (3:32)
2.1 The class figures out that the number went up by eight this time.
2.2 David: "Look, it's the same pattern as over there." (David is comparing the two columns of the chart.) EHT Induction
2.3 Leonard: "The next one is going to be sixteen."
2.4 Sally: "The next one is going to go up by sixteen." EHT Induction
2.5 Teacher: "If it goes up by 16, how much will that be? How can we figure that out?"
2.6 David: "16 plus 16."
2.7 Some of the students begin to try to figure out how much 16 plus 16 is.
2.8
Alicia: "32. Well, I knew there was two tens in - - 16. So I added ten
plus ten which equals twenty. And I knew there was 5 in 6 so then I added on
two fives, then I added on two ones and that made thirty-two." (3:35)
EWT Multi-Step Deduction
3.0 The teacher continues to read the story, "Then 32 ants..."
3.1 The group counts on from 16 to 32 to determine if the number of ants increased by 16.
3.2 The class then begins to make predictions about what the next number will be.
3.3 Shelley raises her hand. When asked, she states 42 but is unable to give a reason for her response.
3.4 Sally: "I think now it's going to go up by 32." EHT Induction
3.5 Preston: "62." When asked why 62, he says, "just guessed."
3.6 Maurice: "62." He explains that he added up 32 plus 32 by doing the following: "I just got thirty. I counted by tens up by thirty. Then I just got two more." (3:37)
EHT Multi-Step Deduction
3.7 Preston: "Just remember 3 plus 3 equals six, so that would be sixty. And add two and that would be 62." EHT Multi-Step Deduction
3.8 Veronica: "42."
3.9 Leonard: "Um, I say 64!"
3.10 When the teacher asks him why he says 64, someone else shouts out, "that's because he read the book!" Preston nods his head yes and grins.
3.11 Alicia explains why she thinks the answer is 64. "Because I added 30 plus 30 then I added 2 plus 2 which equals 64." EHT Multi-Step Deduction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-115
To Beginning of Section | To End of Section
4.0 The teacher continues the story. 64 ants come along. (3:39)
4.1 Alicia immediately says, "64 plus 64!"
4.2 Leonard: "That would be a hundred or something."
4.3 The teacher questions the group about the pattern again.
4.4 Saul: "It's adding up. It's adding up all the ants."
4.5 The teacher asks Alicia what the pattern is; "I don't know how to say it."
5.0 The group is focused on making predictions about the next number of ants.
5.1 Alicia states that she thinks it will be 128 and explains why, "Because I think 60 plus 60 equals 120 - - Plus 4 plus 4 equals 8. 120 plus 8 equals 128. I'm just adding!" EHT Multi-Step Deduction
5.2 The teacher confirms Alicia prediction by continuing the story. (3:42)
5.3 The group can easily predict that the next number will be 128 plus 128.
5.4 The teacher finishes reading the story.
6.0 The teacher puts up the number pattern 2, 4, 6, 8, 10 and asks the students to tell her the next number and explain the pattern. (3:45)
6.1 Leonard: "Twelve. Cause it's two more. Twos!" EWT Induction
7.0 The teacher puts up the pattern 5, 7, 9, 11 and asks the students to tell her the next number and explain the pattern.
7.1 Sally: "I think it's - - twos again. "(The teacher asks what the next number would be.) "The next number would be 13." EWT Induction
8.0 The teacher puts up the pattern 16, 14, 12, 10 and asks the students to tell her the next number and explain the pattern. (3:46)
8.1 Maurice: "You're counting down by twos." EHT Induction
9.0 The teacher puts up the pattern 1, 2, 4, 5, 7, 8, 10, 11, 13 and asks the students to tell her the next number and explain the pattern. (3:48)
9.1 Shelley: "I think the next number will be 16." When questioned why, Shelley says she's just making a guess.
9.2 Leonard: "14! By twos, and you went by 1 then 2. Then that's 2 plus 2. 5 then a 2 and a 1 then a 2 . . ." EWT Induction
10.0 The students are asked to create their own number patterns for others to solve. (3:49)
10.1 Seren: "Can it be just a bunch of mixed up numbers?"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-116
To Beginning of Section | To End of Section
Activity: "Building numbers with pattern blocks”: Students were given a chart consisting of three rows and 25 columns. The rows were used for (1) the number to be made using base ten blocks, (2) a diagram of the base ten blocks used and (3) the number of blocks used. Once students have modeled numbers from 100 to 124, they are asked if they can see any patterns that are created. It was hoped that students would eventually discover that the number of blocks they had to use (if they always traded ten of a smaller unit for 1 of a larger unit) would be the sum of the digits.
After they examine the numbers for patterns, they are asked to make predictions about the number of blocks that would be needed to make particular numbers and then told to test their guesses by modeling them with base ten blocks.
A final aspect of this task, (which most students did not get to) was to come up with several numbers that fit a specific description. For example, students are asked to come up with two different numbers that would use five blocks.
Notes: On this day, there was very little "interesting information" that was observed other than Leonard's comments throughout the activity. This happened because of several difficulties with the activity itself:
- Some students had difficulty using the materials to model the numbers
- Some students had difficulty drawing diagrams to represent their models
Next time, I work in the math centre, I will not restrict the students.
1.0 The students each get a sheet with several charts. The first is a completed chart for the numbers 100 to 104. The teacher goes through this chart with the entire group to help the students understand what it is they are required to do. (3:20)
1.1 All students are able to identify how the number represented by each base ten block.
1.2 Cynthia demonstrates quickly her "lack of understanding" (I don't know what else to call it) regarding the base ten blocks. "If you count both sides, you'd have two hundred." EHN Simple Deduction
1.3 Leonard quickly begins to model each number as the teacher goes through the chart with the group.
1.4 The teacher asks students how many blocks were used to make 102. (3:23)
1.5 Jared, "Two." he does not count the hundred block he used. Ira does the same.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-117
To Beginning of Section | To End of Section
1.6 The group work through modeling the numbers 102 - 104 and looking at the completed chart that they were given.
1.7 Students begin to model the numbers 105 -125 on their own and complete their individual charts. (3:26)
2.0 Jared records on his chart that 105 would use 8 blocks. (3:27)
2.1 Jared is asked to model 105 with base ten blocks.
2.2 Teacher asks Jared: "So how many blocks did you use (to make 105)?"
2.3 Jared points to the 100 block and states "one."
2.4 The teacher questions him about the number he has created with 1 hundred block.
2.5 Jared: "But I have five of these blocks." (He indicates the ones he used.)
3.0 Cynthia has difficulty modeling the first number 105. (3:30)
4.0 Leonard is very focused on the activity. (3:30)
4.1 Leonard tells the teacher without being asked: "This one is easy (111). Hundred, a ten and a one. It's only three! And that's only 3 blocks! A hundred and a ten and a one is only three blocks!" Leonard has not reached this number yet; he is currently working on 107.
4.2 Leonard once again: "It's only three blocks!" EWP Simple Deduction
4.3 Leonard draws his diagram and continues with modeling the numbers.
5.0 Ira is asked to explain his answer of 5 blocks for the number 105. (3:32)
5.1 Ira counts the ones only.
5.2 The teacher questions him about the big block he has drawn in his diagram.
5.3 Ira: "That's a hundred."
5.4 The teacher asks him if that is a block. Ira looks at the teacher and pauses for a while then, slowly nods yes.
6.0 The teacher holds up each of the base ten blocks and tells the group that each one whether it is a one unit or a hundred is 1 block. (3:33)
7.0 Leonard excitedly shouts out his discovery for 110: "Look! Look! (Leonard pauses and studies his chart.) This is 110. Just a hundred and a one. One hundred and a ten block." (3:33)
7.1 Teacher: "How many blocks did you use?"
7.2 Leonard: "Two. Just two blocks. - - A hundred and ten. (He shows the base ten blocks that he used.)
7.3 When questioned whether he thought that the number 110 would require more than two blocks, he explained that if you used more ones (instead of a ten) you could. EHT Simple Deduction
8.0 The teacher directs Ira to trade ten of one kind of block for 1 of a larger unit. (3:38)
9.0 Leonard holds up his work and asks, "Is this good?" (3:44)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-118
To Beginning of Section | To End of Section
9.1 The teacher asks him to look at his drawings of 115 - 119 again. He has drawn models for the numbers 125-129.
9.2 After looking at the drawings for a brief time, he begins to erase the extra ten he has drawn.
9.3 When asked why he was erasing, he tried to explain by describing the number he had modeled, "20, 2, 200 - - 126!" EWT Simple Deduction
10.0 Leonard has completed his charts for the numbers 100 - 124. The teacher asks him to go back and look at the numbers of base ten blocks he had to use for each and notice whether there are any patterns. (3:51)
10.1 Leonard: "It's (the number of blocks you need to use) going up."
10.2 The teacher directs Leonard to look back over all of his charts.
10.3 Leonard: "It went up and down by a number."
10.4 Veronica joins in with Leonard to look for a pattern. The teacher asks why the pattern goes down in certain places. (3:53)
10.5 Leonard: "Oh I see. Yeah cause here (110) to make a ten - - you could use ten of these (ones units) (instead of using a ten unit)." EHT Induction
10.6 Leonard, Veronica and the teacher continue to examine the pattern and notice that the number of blocks needed goes down again at 120.
10.7 Leonard: "I'm not sure what the pattern is. - - I just figured it out. It keeps going up by numbers." Leonard goes back to his chart and begins to read out the numbers 100 - 124.
10.8 The teacher asks, "This one (the number modeled) keeps going up, how come this one (the number of base ten blocks used) doesn't keep going up?"
10.9 Leonard: "I'm not sure what the pattern is!”
11.0 The teacher works with Kyla, Veronica and Leonard as a small group to examine for any patterns. (3:56)
11.1 Leonard: "The numbers keep going up on these. (The numbers that they are asked to model.)"
11.2 Kyla: "(The numbers) keep going up."
11.3 Leonard adds that he noticed that the number of blocks used does go down. The teacher asks him when will the numbers go down.
11.4 Leonard: "When we gets another ten over."
11.5 Teacher: "What would be the next one (to go down)?"
11.6 Leonard: "It would keep going down. . . when you gets 110, then you needs ten more."
11.7 The teacher goes over Veronica's chart, and questions when would the number of blocks go down again. The chart goes to 124.
11.8 Leonard: "Every time we gets a ten, . . .then um - - . . . Just look. Every time we gets a ten, then it goes down, see (he points to the first page.)" EWT Induction
12.0 The teacher asks Leonard to figure out without modeling it, how many blocks he would use to make the number 128. (4:00)
12.1 Leonard: "You would use 1 hundred, 2 tens and 8 ones."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-119
To Beginning of Section | To End of Section
12.2 Teacher: "how many (base ten) blocks would that be?"
12.3 Leonard: "It would be um - - 11. Cause 8 plus 3 equals 11. " EWT Simple Deduction
12.4 Leonard calls his mother over to ask her if he is right about 8 + 3 = 11. "Mom, just said I got it right!"
13.0 Leonard is asked how many blocks would be needed to make 130. (4:02)
13.1 Leonard's mother tries to help him but he refuses shouting, "I knows!"
13.2 Leonard: "Four!"
13.3 Leonard's mom says, "No you wouldn't."
13.4 Leonard yells, "YES! There's no ones! . . .No ones means 4!" EWP Simple Deduction
14.0 The teacher asks Leonard if he can think of a number that would be made using 5 blocks. (4:04)
14.1 Leonard quickly states: "500!"
14.2 The teacher asks him to think of another number, which would use 5 blocks.
14.3 Leonard quickly responds with: "50!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-120
To Beginning of Section | To End of Section
Activity: "Building numbers with pattern blocks”: Students were given a chart consisting of three rows and 25 columns. The rows were used for (1) the number to be made using base ten blocks, (2) a diagram of the base ten blocks used and (3) the number of blocks used. Once students have modeled numbers from 100 to 124, they are asked if they can see any patterns that are created. It was hoped that students would eventually discover that the number of blocks they had to use (if they always traded ten of a smaller unit for 1 of a larger unit) would be the sum of the digits.
After they examine the numbers for patterns, they are asked to make predictions about the number of blocks that would be needed to make particular numbers and then told to test their guesses by modeling them with base ten blocks.
A final aspect of this task, (which most students did not get to) was to come up with several numbers that fit a specific description. For example, students are asked to come up with two different numbers that would use five blocks.
1.0 The teacher briefly introduced/reviewed the base ten blocks with the students to ensure that all were familiar with the materials. (3:24)
1.1 The teacher made numbers using the base ten blocks and asked two questions, "What number has been made?" and "How many blocks have been used?” to help students distinguish between the two questions.
1.2 The teacher models the number twenty-five with the base ten blocks. The group agrees that 7 blocks have been used to make the number. The teacher then asked what number has been made. (3:25)
1.3 Rhona does not give an answer when asked.
1.4 Preston: "I got a guess. 27."
1.5 The teacher counts the blocks with the group and gets 25.
1.6 Preston: "Yeah, but look!" Preston counts the ones, "1,2,3,4,5 - -that's what I guessed but they guessed 27."
2.0 The teacher distributes base ten materials to the students. (3:27)
2.1 The teacher has all students model numbers 100 - 104. The teacher asks the questions, "What number has been made?" and "How many blocks have been used?" for each number.
2.2 The students easily model the numbers.
2.3 Seren uses 10 ten blocks for a hundred, all the others use 1 hundred block.
3.0 The teacher asks the students to tell her how many blocks were used to make each number once again and notes that the number of blocks increases. The teacher then asks if
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-121
To Beginning of Section | To End of Section
the group have any predictions about when the number of blocks needed will decrease. (3:31)
3.1 Preston: "When all the blocks are used up, we'll have to start over again."
4.0 The teacher passes out the sheet containing the charts. Numbers 100 - 104 have been already completed.
4.1 Preston looks at the charts and comments, "These are the charts and you keep on adding on - - 100, 101, 102 . . ."
4.2 The teacher points out the first chart (100-104). She then has the students look at the next number in the second chart that needs to be completed (105) and asks Preston to model this number.
4.3 The teacher shows the students where to record the information that Preston just found out.
4.4 The group begins to model each number given and record their drawings of the number created and the amount of blocks they used. (3:37)
5.0 Preston: "This (107) is (going to be) 8." Preston has just finished drawing a picture for 106.
5.1 The teacher asks Preston why he thinks that.
5.2 Preston: "When it's 5 (105), there's 6 blocks. When it's 6 (106), it's 7 blocks. When there's 7 (107), there's 8 blocks. When there's 8 (108), there's 9 blocks."
5.3 Preston: "When you get to 110, it's really easy. All you have to do is get a hundred and a ten." (3:38)
5.4 Teacher: "You said it gets easy when you get to 110?"
5.5 Preston: "When you get to 110, it goes right down, it's 2 and that's where the numbers go down."
5.6 Preston: "When it comes to ten, that's when the numbers (of blocks used) go down."
5.7 The teacher asks Preston if he knows when the number of blocks used will go down again.
5.8 Preston: "I dunno know. Wait! Every second line, it goes down." (The charts are organized in this way: 100 - 104 on the top, then 105 - 109, then 110 - 114, then 115 - 119 until the numbers reach 124.)
5.9 Teacher: "Every second line goes down?"
5.10 Preston points to the last chart on the first page (114 - 119), "I don't think (it goes down) on this one." Preston studies the charts again.
5.11 Preston: "See look. Wait! Every first one like that see. (He is pointing to the first number in the first and third charts, numbers 100 and 110.) It goes down right there (100) then skip (the next chart) then it goes down right there (110) . . .going down, going up, going down, going up." EWT Induction
6.0 The teacher tells Seth that she notices him counting his base ten blocks to check his answers which is a good idea. (3:39)
6.1 Preston: "I didn't (use blocks) cause I knew what happened."
6.2 Seren: "Preston!"
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-122
To Beginning of Section | To End of Section
previous table of contents appendix B table of contents next