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table of contents appendix
B table of contents
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11.0 Maurice (red) and Saul (black) start a new game. (3:45)
11.1 Saul takes his time placing #12.
11.2 Maurice blocks with #17 and shows Saul where he could have won.
11.3 Maurice wins.
11.4 Teacher, "What did you think he was trying to do Saul?"
11.5 Saul, "I thought he was trying to go up." EHT Induction
11.6 Maurice: "I'm so good at this because I got at home." EWS Simple Deduction
Game 12
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12.0 Maurice (red) and Ira (black) played. (3:47)
12.1 After move 16, Maurice puts his arms behind his head and says, "He got me."
12.2 The teacher asks why he says that.
12.3 Maurice whispers to the teacher the two possible ways that Ira can win.
EWT Simple Deduction
12.4 Ira wins.
Game 13
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13.0 Maurice (red) and Ira (black) played. (3:49)
13.1 After #12 the teacher says, "This is looking interesting."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-59
To Beginning of Section | To End of Section
13.2 With #19 Maurice blocks Ira in 2 ways. (12, 10, 8) and (14, 18, 16)
13.3 Maurice wins by creating a diagonal.
Game 14
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14.0 Maurice (red) and Ira (black) played. (3:51)
14.1 At the beginning of the game Maurice looks to the camera and says "I am winning"
14.1
Maurice quickly wins.
14.3 Ira, "I could have put it there and then I'd won." (Right above 15 on the next turn.)
EHN Simple Deduction
Game 15
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15.0 Before started a new game, Maurice states, "the pros are me and Laura."
15.1 Maurice (red) and Ira (black) play. (3:53)
15.2 On turn 25 Maurice points out to Ira where Ira could have won if Maurice did not block him. Maurice, "You could've won."
15.3 Ira, "Where could I've won?"
15.4 Maurice, "You could've won 1-2-3 and that was free (Pointing to 25)"
EWS Simple Deduction
15.5 Ira, "My turn?"
15.6
Maurice, "Yup."
15.7
Ira places 26.
15.8 Maurice, "I won!"
Note Maurice could have won if he had placed #25 in 27 instead of using the turn to block Ira.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-60
To Beginning of Section | To End of Section
Game
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16.0 Maurice and Ira (red) play against the teacher (black). (3:57)
16.1 Teacher, "I have a challenge - you can work as a team against me. Now I know what types of things Maurice likes to do."
16.2 The boys whisper intensely after #7.
16.3 After #10 is placed the boys say, "Doh!"
16.4 The boys seem to realize they made a mistake after #13. They say, "No, no, no!" as the teacher places her marker in #14.
16.5 The teacher wins.
Game 17
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17.0 Kirsten (black) and Maurice (red) play. (3:58)
17.1 When Maurice places 5 he shouts, "I won, I won."
17.2 Kirsten: "No three." She points to the three red along the bottom.
EWS Simple Deduction
17.3 Maurice: "Huh, I can win four or four." he points to both ends of the three red.
EWS Multi-Step Deduction
17.4 Maurice: "Game over. I know I'm gonna win."
Game 18
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The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-61
To Beginning of Section | To End of Section
18.0 Kirsten (black) and Maurice (red) play. (3:59)
18.1 Maurice, "I won again, 4 or 4. Game over!" EWS Multi-Step Deduction
18.2 Maurice wins.
Game 19
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19.0 Kirsten (black) and Maurice (red) play. (4:00)
19.1 Maurice quickly wins by repeating game 17.
Game 20
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20.0 Maurice (black) and Laura (red) play. (4:00)
20.1 Cynthia comes over to offer Laura help but she refuses it.
20.1 Maurice wins using a diagonal.
20.3 Maurice, "I win! I am the champion!"
21.0 The teacher asks Maurice how he wins.
21.1 Maurice: "I try to get three either way or I just go in an angle." EWT Induction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-62
To Beginning of Section | To End of Section
Today's activity:
The teacher read the book The Doorbell Rang by Pat Hutchins and had the students solve simple problems based on the story. The story is about a mother who has just baked a dozen cookies for her two children. The children are told they can share the cookies between them. Each time they figure out how many cookies each person will get, the doorbell rings and more children come in. The children then have to share the cookies out again.
1.0 The teacher begins by reading the book to the students. (12:19)
1.1 The story states that each child (there are two of them at the beginning of the story) will get six cookies each. The teacher asks the children to tell her how many cookies in all are there.
1.2 Jared: "Twenty because - -"
1.3 Alicia: "Twelve because 6 and 6 is twelve."
1.4 Cynthia: "There's twelve if there's six put together
1.5 Jared: "Six and six is twelve - - cause it said there is six each."
1.6 Cynthia: "6 plus 6 equals 12." EWT Simple Deduction
2.0 In the story, two more people come to share the cookies. The teacher asks the students how many children are there now.
2.1 Alicia: "Four and that means three each."
2.2 The teacher asks why she thinks it would be three each.
2.3 Alicia: "Because 3 plus 3 is 6."
2.4 Cynthia: "No, I think there's five each."
2.5 Alicia: " . . . and 6 plus 6 is twelve. So there's four threes." EWT Multi-Step Deduction
3.0 Two more children come in the story, making six children altogether. The group discusses how many cookies each child in the story will get now. (3:22)
3.1 David: "Now they gets two each."
3.2 Jared: "No, one each."
3.3 David explains why he says two each by pointing to the children in the illustration and saying, "two for her, two for her . . ." The illustration does not show how many cookies each child will have. EWX Induction
3.4 Jared: "One each. Cause there was 2 for - -" EWX Simple Deduction
3.5 Cynthia: "No, one each is the last one." (Cynthia noted earlier that she has heard this story before.) EWS Simple Deduction
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-63
To Beginning of Section | To End of Section
3.6 Alicia: "I know. They get two each. There's two sixes and there's only one six in one six people." EWX Simple Deduction
3.7 Cynthia: "No, I think they should have three each. There was 6 (cookies) and 6 (cookies) what makes 12 and I think they all get three." EWS Simple Deduction
3.8 Jared: "No just one."
3.9 The teacher reads on the find out that each child will get two each.
4.0 The teacher continues to read the story. Six more children come to the door in the story, making a total of twelve children to share the twelve cookies. The teacher asks how many children are there now. (12:24)
4.1 Jared: "Twelve."
4.2 The teacher questions Jared about why he thinks there are twelve people.
4.3 Jared: "Cause I counted 'em up (in the illustration). " EHT Simple Deduction
5.0 The students begin to predict how many cookies each child will get.
5.1 David: "They gets one each." David counts each person in the illustration, "1,2,3,4,5,6,7,8,9,10,11,12."
5.2 Other members of the group keep repeating 1, while Cynthia begins to state 2.
5.3 David: "There's twelve people and twelve cookies so they get 1."
EWX Simple Deduction
5.4 The teacher questions Cynthia about why she feels that each person will get 2 cookies each.
5.5 Cynthia: "Because it's (the book) not down almost to the last page. Because almost at the last page they only got one and then some more people came in and then they had none." EWT Simple Deduction
5.6 David: "Look! That's the last bit of people!" (Turning the page)
5.7 The teacher reads on and finds out that each person will get one cookie each.
6.0 The teacher begins to make a chart noting how many people there were throughout the story and how many cookies each person got. (12:26)
6.1 When the group gets at 4 people on the chart, they have a discussion about how many cookies each person would get.
6.2 The teacher passes each child some paper and a pencil and offers them counters to figure out how many cookies each person will get if there are four people and twelve cookies. (12:29)
6.3 Alicia: "So do we just write it down?"
6.4 The teacher tells the students they can figure it out however the students wish. Cynthia announces that she is drawing Oreos, a tray for them and a woman holding the tray.
6.5 Alicia writes: 6 + 6 = 12, 3+3+6 and if there is 2 sixes that means there's 4 threes.
EWT Multi-Step Deduction
6.6 Seth used counters. He split twelve counters up into four equal groups.
7.0 The students continue to work out the problem.
7.1 The teacher asked David to show the group he how he figured out the answer. (12:36)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-64
To Beginning of Section | To End of Section
7.2 David: "I got four people." (He places four counters down on the table away from each other.) Then he shares out twelve counters to these four "people" by giving them one of the twelve counters at a time until they are all gone.) "Three each."
EHT Simple Deduction
7.3 The teacher points out that he has four counters in each pile but he said that each person got three each. Why?
7.4 David: "Yeah, but one's a person!"
8.0 Alicia explained her solution to the group. (12:36) She used the same reasoning as she did during her explanation in 2.0 - 2.5. She doesn't use any counters.
8.1 The teacher continues to complete the chart. (12:38)
8.2 The teacher gives the students a chance to work out how many cookies each person would get if there were twelve cookies and 6 people.
8.3 Cynthia: "I don't have enough (counters). I have twelve of them but I'm putting the people up."
8.4 Cynthia gets six more counters but does not do anything with them.
8.5 Alicia explains to the teacher that the answer is two. "I found out by counting by twos." (12:40)
8.6 Alicia counts out loud by twos to twelve. "So I counted by twos six times."
EHN Simple Deduction
8.7 Cynthia is questioned about whether she has figured out the solution yet but she indicates that she still has not solved the problem. (12:41) She seems just to be playing with the counters.
8.8 David explains how he solved the problem using the counters. He placed six counters out for people then he said, "I started by giving out one each."
EHN Simple Deduction
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9.0 The teacher pointed to the chart that had been created so far and asked the students if they noticed anything about the numbers. (12:41)
9.1 The teacher: "What's happening to the numbers on this side (number of people)?"
9.2 Jared: "They're going up - - and these (number of cookies each person got) are going down."
9.3 Alicia: "It's like they're counting by twos." EWT analogy
9.4 The teacher completes the bottom row of the chart with 12 and 1. David tries to find a pattern in the chart. He says (while pointing) "1, 2, 6, 12" then "3, 2, 6" and finally, "1, 2, 3, 4".
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-65
To Beginning of Section | To End of Section
9.5 Seth: "I think I know something. Twelve take away eleven make one (he demonstrates with his counters) - - and there will be one for everybody - - Everybody will have to have one cookie." (12:43) EWT Multi-Step Deduction
9.6 Teacher: "Who did you give the eleven cookies to?"
9.7 Seth: "I just took away."
10.0 The teacher asks the students to imagine that there were 24 children and asks how many cookies would each person get if there were twelve cookies. (12:44)
10.1 David and Alicia: "They get a half each."
10.2 David is asked to explain his answer.
10.3 David: "They'd probably get one little piece. They'd have to cut it into four."
10.4 David is asked to show his answer on paper. He draws 12 cookies and draws a line down the middle of each one. He counts the pieces and says, " . . .24 and there's 24 people." EHT Multi-Step Deduction
11.0 The teacher asks the students to figure out how many scoops of ice cream each group member would get if she brought in ten scoops. (5 people, 10 scoops) (12:48)
11.1 Jared: "I don't understand this."
11.2 Cynthia: "I'm bored. I don't understand."
11.3 The teacher explains the problem again.
11.4 Alicia quickly states that each person would get 2 each because "6 plus 6 is the same as 5 plus 5." EWX analogy
11.5 With guidance from the teacher, Jared, Cynthia and Seth draw 10 scoops of ice cream on their paper. Then they draw 5 people and begin to share the ice cream by drawing lines to the scoops of ice cream.
11.6 Cynthia explains her drawing: "Two for him, two for him, two for him, two for them two. All of them gets two. Yeah see cause this - - he gets two, he gets two, he gets two right there, he gets two and he gets two.- - they could only get one each and there would be some left for the freezer." (12:54) She begins counting again but gives up after 2. EWX Simple Deduction
12.0 David is busy figuring out how much each person would get if there were 48 people and 12 cookies. (12:48)
12.1 David: "Got it! It goes like this." he shows the teacher twelve cookies that are divided up into quarters. (12:51)
12.2
David explains to the teacher how he found his answer. "First I started
with a line there like that (indicating a line down the middle of a cookie)...
(Then he indicates that he drew another line through to make quarters) then I
counted em like that."
EHT Multi-Step Deduction
13.0 The teacher asks David and Alicia to create their own problems for each other to solve. (12:52)
13.1 Alicia and David exchange problems. (12:56)
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-66
To Beginning of Section | To End of Section
13.2 David's problem stated if there were 10 people and 12 cookies, how many cookies would each person get.
13.3 Alicia's problem was "If we had 8 cookies and 4 people how many cookies would everybody have?"
13.4 Alicia unsuccessfully tried to solve this problem using counters.
13.5 David quickly solved Alicia's problem and began to work on his own.
13.6 David (talking about his problem): "They'd only get one cookie each."
13.7 Teacher: "Are there any left over?"
13.8 David: "Only two."
13.9 David begins to split up the two cookies on his sheet that are left over.
13.10 Alicia speaks to David: "David, there's two left over. (She holds up her extra counters.)"
13.11 David (speaking about his problem): "Done!" (1:00)
13.12 David has divided each of the two cookies up into six pieces. (They are not are equal.)
13.13 The teacher asks David to explain to Alicia how he solved the problem. (1:01)
13.14 David: "I cut em up like this. (He shows her his paper.) Like this, like bars." EHT
14.0 The teacher asks Cynthia, Jared and Seth to figure out how many cookies each three people would get if there were 9 cookies. (12:54)
14.1 Cynthia and Jared figure the problem out with drawings, while Seth uses counters.
14.2 Seth: "I say there'll be three cookies left."
14.3 Teacher: "Three cookies left?"
14.4 Seth: "Look there's nine here. (He shows nine counters.) And there's three left. (He points to three other counters - perhaps from the original 12 cookies.)"
14.5 The teacher repeats the problem.
14.6 Seth looks at his counters and says, "these (three) will be my people."
14.7 Seth starts to reorganize his counters. (12:57)
14.8 Cynthia explains what she has found: "You get three people - -
14.9 Jared: "They'll have to have three each."
14.10 Cynthia: "- - And three each. Arghhhh!" Cynthia puts her hands on her head.
14.11 The teacher asks Cynthia to explain her answer again.
14.12 Cynthia: "There was three people and they got three cookies. He said what I was gonna say." (12:58)
14.13 Seth joins in: "Three people will get three ice-creams." He has made three piles of three with his counters."
15.0 The teacher begins to ask Jared, Seth and Cynthia to determine how many cookies two people would get if there were three cookies. (12:58)
15.1 Before stating how many cookies there were, Jared says, "You'd have to have four each."
15.2 The teacher explains the complete problem.
15.3 Seth: "They only get one and there's one left over."
15.4 Cynthia, Jared and Seth continue to work on the problem.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-67
To Beginning of Section | To End of Section
15.5 Cynthia: "I think there should be four cookies." (12:59)
15.6 The teacher explains that there are only three cookies and asks, "what are we going to do?"
15.7 Cynthia: "We should split the middle cookie in half so they each -- half a one and a full one." Cynthia begins to draw her solution.
15.8 Seth: "I know. If there's one extra, take this one (the extra one) back."
15.9
Cynthia explains her drawing: "See the middle cookie's in half and she
gets the one side and he gets the other side and then they both get a full
cookie. Then they get two each."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-68
To Beginning of Section | To End of Section
Activity:
The Doorbell Rang by Pat Hutchins
Group:
Notes:
In scene 11, it was very difficult to follow exactly what was happening because it is impossible to see the diagrams that the students are pointing to on the video. I did collect in all of their sheets but because they were well marked up by the end, it was hard to figure out what they were referring to at certain times.
Also, this group did not seem ready to try to figure out multiple level problems, so I did not do so with them.
1.0 The teacher begins to read the story The Doorbell Rang by Pat Hutchins. (12:22)
1.1 At the beginning of the story, it states that the cookies will have to be shared among two children giving them six each. The teacher asks the students to tell her how many cookies in all there must be.
1.2 Preston: "Three each. Three plus three equals six." EWT Simple Deduction
1.3 Jocelyn is busy shouting, "Twelve, twelve, twelve" as Preston explains his answer.
1.4 Teacher: "So you think there are six cookies?"
1.5 Preston: "Well no but there's twelve cookies."
1.6 Teacher: "So how do you know there's twelve cookies?"
1.7 Preston: "Six plus six equals twelve." EWT Simple Deduction
1.8 The teacher asks Jocelyn if she agrees. She nods her head and explains, "3 + 3 is 6 and 6 + 6 is 12." EWT Multi-Step Deduction
2.0 The teacher continues to read the story. She stops to ask the students how many cookies each person will get in the story now that there are four people that must share the twelve cookies. (12:24)
2.1 Jocelyn: "Three each."
2.2 The teacher asks Jocelyn why she thinks they will get three each.
2.3 Jocelyn: "Because they had six each but they never ate them yet so they can still share it. And if they had six each it wouldn't be fair . . . and if they had three each it would be equal." EWT Multi-Step Deduction
2.4 Veronica: "Maybe they should wait until they eat their six cookies."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-69
To Beginning of Section | To End of Section
2.5 Preston: "Everyone could maybe have two and - -"
2.6 The teacher lets the students get some counters to solve the problem. (12:26) Jocelyn just watches Joline work and Seren is playing with the large pile of counters. Veronica and Preston are both working with their own counters.
2.7 Joline quickly puts her counters into four piles of four and shouts out, "I know, I know!"
2.8 Preston and Veronica are unsure about how to share them out and Seren has lined hers up into two rows of six.
2.9 The teacher asks Preston if he would like Jocelyn to help him. He then begins to work out the problem, "You'd say two for each person. (Preston creates four piles of two on the table.) And you would add on one more. (He distributes the remaining four counters to the four piles.) So each person would get three." EHT Multi-Step Deduction
2.10 The teacher asks Joline to explain to the group what she did. (12:28)
2.11 Joline explains the four piles she has in front of her: "See there's twelve there and there's four children and um there's four children and I got four cookies so there's three cookies each." EWT Multi-Step Deduction
2.12 The teacher asks Joline how she knew that she had to put three in a group.
2.13 Joline puts all the counters back in a pile and begins to explain: "I went like this (she moves three counters to the side). I put three then I put three (she moves some more counters) and then I went like that (she splits the remaining six counters into two equal groups)." EHT Multi-Step Deduction
3.0 The teacher continues to read the story. (12:29)
3.1 The teacher stops the story once again when two more people enter the story. There are now six people and twelve cookies and the students are asked to figure out how many cookies each person will get now.
3.2 Joline quickly moves her counters into six groups of two and raises her hand.
3.3 Preston then places his counters into groups of two while Jocelyn raises her hand with the answer.
3.4 Seren has not appeared to do anything yet but she states, "Two, two cookies. Because there are six people and the last time there was three so it has to be two. And also because it says two right there. (Seren points to a chart that the group has been keeping noting how many children there are and how many cookies they each get. She points to the 2 representing two children and not the answer to this problem.) EWX Simple Deduction
3.5 Preston: "If you have twelve cookies and if you just give everyone one then there would be enough for one for each person. So you could give them one more to each person and that would make two." EWX Multi-Step Deduction
3.6 The teacher asks Jocelyn how she figured it the solution.
3.7 Jocelyn: "Cause the last time it was three." EWT Simple Deduction
3.8 Preston: "It goes like 2, 4, 6, 8, 10." (He points to the chart which now has 2, 4 and 6 in one column indicating the number of children in each scene of the story." EWX Induction
3.9 There is a brief discussion about the number of children going up by twos and the group feels that in the next scene there will be 8 children.
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-70
To Beginning of Section | To End of Section
4.0 The teacher continues to read the story. (12:32)
4.1 The next time the doorbell rings in the story it states, "It was Joy and Simon with their four cousins." The students are asked to identify how many children there are now.
4.2 Joline quickly states, "Now it's twelve."
4.3 Teacher: "How do you it's twelve people there."
4.4 Joline: "Cause there's lots of people there." EWT Simple Deduction
4.5 Preston: "Yeah and look!" (Preston points to the illustration and counts the number of children.
4.6 The students quickly agree that the children will now get one each.
4.7 Veronica: "They can have one each. . . now if they have two there won't be enough for all of them." EWX Simple Deduction
4.8 Jocelyn: "One each. Because the numbers like um sort of counting backwards. First there was six them there was three, then there was two and it's like, I know it isn't but . . .like three, two, one." EWX Analogy, Induction
4.9 Joline: "One each. There was twelve of these (counters) right and there's twelve people so I got out one because they'd have one each." Simple Deduction
4.10 Preston: "If you took two then you'd have to take one away from one person and then it wouldn't be fair because six people wouldn't have cookies. And six people would have two cookies. So it has to be one." EWX Multi-Step Deduction
5.0 The teacher asks the students what they think is going to happen in the story next. (12:35)
5.1 Veronica: "Next they can all have zero cookies!"
5.2 The teacher continues to finish the story.
5.3 Preston: "Maybe they can crack them in half."
6.0 When the story is finished, the teacher asked what would have happened if there were a total of twenty-four children at the end of the story. (12:37)
6.1 Preston: "None, actually - -"
6.2 Preston: "They'd get two cookies each."
6.3 Teacher: "They'd get two cookies each?"
6.4 Preston: ". . .twelve plus twelve equals twenty-four and . . . now they have one so now they should have two. Each person should have two."
6.5 Teacher: "Each person should have two?"
6.6 Preston: "If you add on one more for each person - -" Induction
6.7 Teacher: "There's only twelve cookies."
6.8 Preston: "But Grandma brings back..."
6.9 The teacher explains that she wants them to see how many each person would get if there were still only twelve cookies.
6.10 Preston: "Then no one would have a cookie."
6.11 Preston: ". . .what they could do is crack them in half or they could not eat them or just give them back to the grandma." EWT Simple Deduction
6.12 The teacher discusses the possibility of cracking the cookies and whether or not they would have to crack all of the cookies.
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6.13 At this point a parent comes into the room and gives the students the answer!
6.14 Seren suggests that this would be like the problem she would face if she brought in muffins to class tomorrow but she only had enough muffins for the girls. (12:39)
7.0 The teacher asks the students to create their own problem for other people to solve. (12:40)
7.1 The students created "problems" that were not readily solved by the others. The problems were not of the same type as those that were just discussed throughout the story.
7.2 Jocelyn was the only person to create a similar problem. She wrote, "I have 3 apples but I have 6 sisters. What can I do?" On the back she wrote, "Yes because I can slept (split) them up."
7.3 Joline's question: "Joline bought 6 ice cream cones she gave one to Seren how much did she have?"
7.4 Preston explains his question to the teacher, "First you have ten people. Then another ten people. Then another ten people. Then another ten people. And then fifty people came. So how many trays of cookies would they have to have?"
7.5 Teacher: "So how many are they going to have each?"
7.6 Preston: "Well first I'm going to have them ask how many each - - . . . one cookie or two."
7.7 Teacher: "So how many cookies are they going to have?"
7.8 Preston: "One of the questions that I have too is how many trays of cookies will I have."
7.9 Teacher: "How many fit on a tray?"
7.10 Preston: "Um - - twelve cookies on each tray."
7.11
8.0 The teacher decides to present a problem to the group. (12:49)
8.1 The teacher asks the students to figure out how many scoops of ice-cream each person would get if there were 12 scoops of ice-cream and four children.
8.2 Joline quickly states, "One!" and Seren immediately repeats Joline's answer.
8.3 Joline is unable to explain her answer.
8.4 Veronica says the answer is three. When questioned as to why she says, "I looked in the book."
8.5 Veronica: "Look it's right here." (She turns to the pages in the story that states the four children would get three cookies each.) EHT Simple Deduction
8.6 Seren insists that there will be two for each person.
8.7 Veronica shows the book to Seren and counts the number of children in the picture and reads how many cookies they each receive.
9.0 The teacher poses a new problem: There are 15 scoops of ice cream and there are five children. How many scoops will each child get? (12:52)
9.1 Joline states that there will be three for each person.
9.2 The teacher asks Joline why she thinks it will be three each.
9.3 Joline: "Cause I counted."
9.4 The teacher asks Joline to explain how she counted.
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9.5 Joline: "I counted three, three, 1, 2, 3, 4, 5." EHT Multi-Step Deduction
9.6 Veronica: "It is three."
10.0 The teacher poses a new question. She states that she found three chocolate bars and she wanted to share them with a friend. The children are asked to figure out how the teacher will share the bars. (12:53)
10.1 Veronica: "Cut one in half? Cut one in half."
10.2 Veronica: "They'll get one each."
10.3 Teacher: "They'll get one each?"
10.4 Veronica: "Yeah cause three chocolate bars - -"
10.4 The teacher questions Seren about what she thinks.
10.5 Seren: "They'd get two each."
10.6 Teacher: "Veronica said they'd get one each."
10.7 Seren: "One each. I think one each. No! Um, one and a half each."
10.8 Teacher asks why she says one and a half.
10.9 Seren: "Cause you can cut one bar in half and you can share one half and the other half." EWT Simple Deduction
10.10 Seren demonstrates what she would do pictorially.
11.0 The teacher asks how three chocolate bars could be shared among four people. (12:55)
11.1 Veronica begins to repeatedly shout, "Divide it in half!"
11.2 Veronica is unable to state how much each person would get.
11.3 The teacher draws a diagram representing three chocolate bars and asks Seren and Veronica to see if they could equally share the bars among four people. Multi-Step Deduction
11.4 Veronica and Seren quickly draw a line down the middle of one bar and point out that there are now four pieces.
11.5 The teacher reminds them that the sharing should be fair.
11.6 Seren: "That wouldn't be fair cause she'd just have one for herself."
EWT Simple Deduction
11.7 Preston attempts to explain what he would do to Jocelyn and Joline.
11.8 Preston: "what you can do is one piece for each person. Cut this bar into four halves and then it will be fair." EHT Simple Deduction
11.9 Joline divides one of the bars on her paper into four.
11.10 Preston says that is what he meant for her to do.
11.11 Joline tries to explain how the pieces of bar would be shared. "Two, two and two. . ."
EHT Simple Deduction
11.12 Veronica counts the pieces they have created and notes that there are nine pieces. "You'd need nine people," she says.
11.13 Preston: "No! What I meant - - there's four people so there's one half for this person, one half for this person, one half for this person and one half for this person. And you take this one, and cut it into four halves. Then you give this half (quarter) to this one,
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this half (quarter) to this one, this half (quarter) to this one and this half (quarter) to this one." EHN
11.14 The teacher asks if that would be fair for everyone.
11.15 Joline: "No. Two for each person. Two for one, two for one, two for one, two for one."
11.16 Preston: "Yeah that's what I meant! Two for each person...one big one and one small piece." EWS Simple Deduction
11.17 Veronica: "Yeah but that wouldn't be fair!"
11.18 Preston: "Yeah but everyone would get the same size."
11.19 Preston shows that there are four big pieces (halves) and four little pieces (quarters).
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Activity: Problem solving based on The Doorbell Rang by Pat Hutchins.
Notes:
1.0 The teacher begins to read the story called The Doorbell Rang, by Pat Hutchins. (3:19)
1.1 At the beginning of the story, the mother states that her two children can share the cookies between them. It indicates that each child will get six each. The teacher asks the students how many cookies are there all together if each child will get six each.
1.2 Some students say six while others say twelve.
1.3 Laura: "Six, cause she said three each - - six each." (She self-corrects.)
EWT Simple Deduction
1.4 Teacher: "So do you say there's six cookies."
1.5 Laura nods yes first then states, "No twelve."
1.6 The teacher asks Laura why she changed her mind.
1.7 Laura: "Cause I got mixed up. "
1.8 Teacher: "Why did you say twelve?"
1.9 Laura: "Cause it said six each." EWT Simple Deduction
1.10 Maurice, "There's 12 cause 6+6=12." EWX Simple Deduction
2.0 The teacher continues to read the story. In enters two more children in the story. (3:20)
2.1 Laura quickly predicts that the children will get three each before being prompted to do so by the teacher.
2.2 Saul agrees and explains his reasoning, "because three plus three would be um six and another two threes would be six and because three plus three is six and another three plus three would be another six. So it's three." EWS Multi-Step Deduction
2.3 Kirsten is asked whether she agrees with Saul. (3:21)
2.4 Kirsten: "There's four people and they get two each. "
2.5 Teacher: "you think they would get two each. Is there any way we could check it?"
2.6 Saul: "It would have to be eight if it were two because two plus two is four and another two and two are four." EWT Multi-Step Deduction
2.7 Ira notes that there are four people at the table (actually there are six including the teacher). He suggests that he could pretend the teacher made the cookies and share them out.
2.8 Saul begins to point to each person, counting by twos.
2.9 Saul and Kirsten both count 2,4,6,8,10.
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2.10 The group seems confused.
2.11 The teacher offers the students some counters. (3:22)
2.12
Maurice shares out the cookies (one at a time) to four of the group members
and figures out that each person gets three each.
3.0 The story continues. Next two more children come to create a total of six children to share the twelve cookies. The group quickly begins to anticipate the question of how many cookies each person will get. (3:24)
3.1 Kirsten: "We have to count you (the teacher) to make six." EHN Simple Deduction
3.2 Maurice: "Or I'll be two people."
3.3 Maurice shares the counters out evenly among the six people at the table.
3.4 The group agrees that everyone will get two each.
4.0 The teacher continues to read. Six more children come in making a total of twelve children in the story who must share the twelve cookies. (3:26)
4.1 Teacher: "how many people are there now?"
4.2 Maurice: "Twelve."
4.3 Teacher: "Why do you say twelve?"
4.4 Maurice: "Cause I added it up." EHT Simple Deduction
4.5 The group quickly decides without sharing out any counters that each person will get one cookie.
5.0 The teacher finishes reading the story. (3:27)
5.1 The teacher poses some new problems.
5.2 The teacher tells the group that there were 2 people and six cookies.
5.3 Ira and Laura both indicate three very quickly.
5.4 Laura holds up three fingers on each hand.
5.5 Ira is asked why he says three. "Because three plus three is six." Simple Deduction
5.6 Kirsten repeatedly says eight. She adds two plus six on her fingers.
6.0 The teacher tells the students that there are now two people and 10 cookies. (3:29)
6.1 Ira and Laura quickly indicate five.
6.2 Laura holds up five fingers on each hand.
6.3 Laura: "Five plus five equals ten." EWX Simple Deduction
7.0 The teacher says that she has 100 jellybeans to share between two people. (3:29)
7.1 Maurice quickly states 50.
7.2 Maurice: "Cause 50 plus 50 is a hundred." EWX Simple Deduction
8.0 The teacher indicates that there are three people and nine cookies. (3:30)
8.1 The students in the group quickly get the counters.
8.2 Kirsten is the first to raise her hand.
8.3 Kirsten: "Eleven."
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8.4 Kirsten counts the counters that she has taken out. She has three for the people and 8 for the cookies. She counts them all together.
8.5 The teacher asks her if she feels that all the people will get 11 cookies each.
8.6 Kirsten replies no but is unable to explain why.
8.7 The teacher asks Saul to tell the group his answer.
8.8 Saul: "Because three plus three is six then three people. And there's nine. You take like one, one." (He begins to distribute the nine counters out to each of his three "people counters.") EWT Multi-Step Deduction
8.9 Maurice also uses counters to get 3 each. EHN Multi-Step Deduction
8.10 Laura explains her findings. She tells the group that she used three counters for the people and had nine counters for the cookies. First she tried to give each person four each but she ended up with just one left over for the last person so she tried three.
9.0 The teacher tells the group that there are 3 people and 15 cookies. (3:33)
9.1 Ira quickly shouts "I know" and starts saying, "this is easy."
9.2 Laura works away with her counters but seems to have difficulty getting the groups to be even. She counts the counters and removes one from the pile. (She originally counted out three for the people and thirteen for the cookies.)
9.3 Ira is asked to tell the group his answer.
9.4 Ira: "Five each. Cause five plus five plus five is fifteen." EWT Simple Deduction
9.5 Maurice: "I just pretended I had three people and then I gave cookies (out to each of them)." EHN Simple Deduction
9.6 Laura is asked if she found five.
9.7 Laura: "No four each."
9.8 The teacher asks Laura to count how many cookies she has.
9.9 Laura counts and discovers that she only has twelve cookies. She gets three more counters.
10.0 The teacher tells the students that there are twelve cookies and four people. (3:36)
10.1 Maurice: "That was in the book."
10.2 Ira quickly raises his hand without using any counters.
10.3 When asked to tell the group his answer Ira states, "Two. Because there's four people and there's twelve cookies . . .they would get two each. " EWT Simple Deduction
10.4 Maurice is asked to explain how many he thinks each person will get. He has created three piles of four with his counters.
10.4 Maurice: "Four."
10.5 The teacher goes over the question again and then asks Maurice to look at what he has done. She asks him how many people he has given cookies to which he quickly states three.
10.6 the teacher then asks Maurice how many people were in the problem. He states "four." After a brief pause, Maurice puts his hands on his head and then begins to reorganize his counters into four groups.
10.7 Laura shows the group that she split the counters up into four groups of three. EHN Simple Deduction
10.8 Kirsten states that the answer will be sixteen. She has added 4 plus 12.
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10.9 The teacher asks Kirsten if she thinks that each person will get 16 cookies.
10.10 Laura laughs at Kirsten, so the teacher asks Laura to tell the group what she thinks Kirsten did. Laura is unable to do so.
11.0 The teacher tells the students to share three cookies between two people. (3:39)
11.1 Laura: "It doesn't make sense."
11.2 The teacher asks Laura why she says that and Laura holds up a third counter and says that you can't do anything with it. EWT Simple Deduction
11.3 Maurice: "Three cookies, slice, slice, slice." (Maurice has created 3 cookies using two counters for each one. He pretends to slice each cookie in half. ) "Three halves."
EHN Multi-Step Deduction
11.4 The teacher asks Maurice if there is another way to say three halves.
11.5 Maurice: "One cookie and a half cookie."
12.0 The teacher asks the students to share 4 cookies amongst 6 people. (3:41)
12.1 Maurice: "Ah, I got it. You get one cookie and slice the other ones in half."
EHT Simple Deduction
12.2 The others are still trying to figure it out.
12.3 Ira: "But this is hard. I have six cookies. Look four people. (He takes away four counters, leaving him with two to share.) Arghhh!"
12.4 Ira: "You need eight. You need two more to make eight."
12.5 Ira continues to work on the problem. After another minute he states that he got it.
12.6 Ira has five counters taken out to represent the cookies. He is missing one but is not aware of it. "If there's one more cookie left (he has given out one each to four people) then I slice it into four pieces. Then they could have enough." (3:44) EHN Multi-Step Deduction
12.7 The teacher asks Ira to count how many cookies he has.
12.8 Ira: "You slice it in four and you have eight."
12.9 Teacher: "Before you slice it, how many cookies do you have?"
12.10 Ira says that the fifth cookie is a big cookie. He is unaware that he only has five cookies instead of six.
12.11 The teacher suggests that any student who would like to draw a picture to figure it out can. Ira, Saul and Maurice decide to try to draw a picture.
12.12 Ira completes his drawing of the problem that shows five cookies being shared among four people. (3:48)
12.13 Saul is drawing a picture to solve but his comments are inaudible.
12.14The teacher decides to start something else.
13.0 The teacher tells the students that there were 6 cookies and that each person got two cookies. She asks the group to figure out how many people there were. (3:49)
13.1 Ira quickly shouts out two.
13.2 The teacher restates the problem.
13.3 Saul begins to verbalize the problem, "there were 6 cookies - - three, three people."
13.4 The teacher asks Saul to explain his answer. "Because um, six cookies and each person would get two - - two . . . because two, four, six." EWT Multi-Step Deduction
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13.5 The
teacher uses a diagram to show the rest of the group Saul's explanation.
(3:51)
14.0 The teacher tells the students that there were 9 cookies and that each person got three cookies. She asks the group to figure out how many people there were. (3:52)
14.1 Saul quickly raises his hand.
14.2 Saul states three. "Because three plus three is six and another three is nine. So there's one, two, three." EWT Multi-Step Deduction
15.0 The teacher asks Saul and Maurice to work on another problem while the other 3 students are given paper and pencil to create their own problems. She tells Saul and Maurice that there were 2 people and that each person had five cookies. The teacher wants them to figure out how many cookies there were in all. (3:53)
15.1 Saul: "Probably ten. Because two people and five cookies. - - Because 5 plus five is ten." EWT Simple Deduction
15.2 Maurice agrees with Saul's answer.
16.0 The teacher has Maurice and Saul figure out the following problem: There were 3 people. Each person had 5 cookies. How many cookies were there in all? (3:54)
16.1 Saul quickly raises his hand and states fifteen.
16.2 Saul: "3 is 5 and 15. 3 people would make 5, 10, 15." EWT Simple Deduction
16.3 Saul is asked to explain how he figured it out to Maurice. Saul: "I counted by fives, 5,10,15." EHT Induction
16.4 Teacher: "Why did you choose to count by fives?"
16.5 Saul: "Cause it was the shortest way and the fastest way I could think of."
16.6 Maurice explains his answer. "I just added one more person and five more cookies." (Maurice points to the previous question completed.) EHN Induction
17.0 The teacher then poses the following problem to Maurice and Saul: First there were 8 cookies. The children got four each. Then some more children came in. Then they got 2 each. How many people came in? (3:55)
17.1 Both Saul and Maurice reread the question several times, which is written on a piece of paper. They try to solve this problem with paper and pencil only.
17.2 Saul circles the number eight in the question. (Note: The videotape stops here!)
17.3 Ira interrupts and states, "Very easy!"
17.4 Saul: "No! Eight take away two is six. - - First there were eight cookies. Then the children got four each. Oh! Let me see. First there were 8 cookies..."
17.5 Saul: “Four people!"
17.6 Teacher: "Did four people come in? How many people were there at the beginning?"
17.7 Saul: "There were two."
17.8 Teacher: "You think there were two people in the beginning and then four people came in?"
17.9 Saul: "No, then two more people came in. That made four. Because eight, - - let me think about that again." EWT Multi-Step Deduction
17.10 Saul: "Um, this is hard."
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17.11 Saul: "I think there were four or six people."
17.12 The teacher asks Saul how is he going to figure out if there were four or six people.
17.13 Teacher: "Are there any parts that you can figure out?"
17.14 Saul: "There were four people in there."
17.15 Saul rereads the question again silently.
17.16 Maurice talks with the teacher and Saul.
17.17 Maurice agrees that there were four people as well. (Note: As the other members of the group get tidied up, the microphone cord gets pulled from the tape recorder, so the tape is not very clear from this point on!)
17.18 The teacher asks once again how many people came in.
17.19 Maurice says, "Two."
17.20 There is a diagram drawn by Saul on the bottom of the page showing eight cookies, two rows of four. He has the two groups of four circled. Then he has a line drawn down through each group of four, creating four groups of 2.
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Note: The game board is set up so that the number of each row indicates the order in which guesses were made. The right hand column indicates how many white pegs (w) and how many black pegs (bk) the guess earned. A white peg indicates that there is a peg that is the right colour but not in the correct spot. The black peg indicates that the peg is the right colour and in the correct spot.
You will also note that there sometimes is a seventh row indicated on the chart even though there are only six places on the board. At times, I removed the pegs from the first attempt and gave students a seventh guess.
Game 1 - Cynthia
|
1 |
o |
Bl |
y |
r |
3
w |
|
2 |
bl
|
O |
r |
y |
3
w |
|
3 |
br |
Y |
bl |
r |
2
w |
|
4 |
y |
R |
o |
y |
1 bk , 1 w |
|
5 |
br |
O |
r |
bl |
2
w |
|
6 |
gr |
Bl |
r |
gr |
2 w |
|
Code |
y |
Gr |
bl |
o |
|
1.0 The teacher explains how to play the game. (3:22)
1.1 The teacher asks Cynthia to make her first guess.
1.2 Cynthia: "The colours that I think you used?"
1.3 Teacher: "Yes."
1.4 Cynthia: "Different colours or the same?"
1.5 Teacher: "I didn't use any (colour) more than once."
1.6 Cynthia: "Like do you have to make them the same colours or different colours?"
1.7 Teacher: "You could do four different colours or four of the same colour."
1.8 Cynthia: "I'll do four different colours."
1.9 Cynthia creates her first guess. (3:24)
1.10 Cynthia clarifies what the three white pegs she receives for her guess means: "There's three in the wrong spot?"
1.11 The teacher explains what the three pegs indicate.
1.12 Cynthia: "Do I like have to find out if - - what ones are the same?"
1.13 Cynthia makes a second guess. (3:25)
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