previous table of contents appendix B table of contents next
6.3 Seren pauses for a moment then tells the teacher, "If Preston finishes first I'm not surprised. Because you (the teacher) helped him."
6.4 Preston: "Actually, no she just did - -"
6.5 Teacher: "I'm not helping at all."
6.6 Seren: "I know - - you just did - -"
6.7 Preston: "She only did the first thing (completed the diagram for number 105 while showing the entire group) and I already knew it."
7.0 Preston is looking at the charts and predicting how many blocks will be used for each number. " 4, 5, 6, 7, 8, 9, 10. - - 10 right here (118), then 1 (119)." (3:40)
7.1 The teacher questions Preston's prediction about 1.
7.2 Preston: "Maybe, cause see 100 (he points to the 100 in the chart). No, um, four because there's all different ways you could do it. . . Right now, right here you could do 100 and 10 and a 1 or just - - yeah, probably that." EWT Induction
8.0 Preston continues on with his charts.
8.1 Preston: "When it gets to 113, it can kinda get hard." (3:44)
9.0 Joline correctly models the numbers 105 - 109 with the base ten materials but when recording the number of blocks she used, she does not count the hundred block as a block. (3:46)
9.1 Joline goes back and changes all her answers by one. When asked why, she points to the hundred block.
10.0 The teacher asks Seth when he thinks the number of blocks he must use will go down again. He has currently finished completing the chart for 113. (3:47)
10.1 Seth spends some time looking over the charts.
10.2 Seth does not respond.
10.3 NOTE: Seth, Rhona and Joline are the only ones modeling each number at this point. Both Seren and Preston are simply drawing their representations without modeling them.
11.0 Preston has completed his charts for 105 - 119. He has the recorded that he will need 9 blocks to create both numbers 117 and 118. (He forgot to draw in a ten for 118.) The teacher asks him to look at the pattern created by the number of blocks that were used to create each number and make any comments about the things he notices, especially anything unusual. (3:52)
11.1 Preston: "Right there (the two 9's for 117 and 118) it's the same two times."
11.2 The teacher asks Preston to model the two numbers 117 and 118 for her with blocks.
11.3 Preston correctly models both numbers with the base ten blocks.
11.4 The teacher asks Preston to count how many blocks he used for each number. He gets 9 for 117 and 10 for 118. He questions himself when he counts ten, "Ten. That's strange!" Preston counts the ones to make sure he has eight. Then he counts all of the blocks over again, starting with the hundred and the ten block. He gets ten again and sits back in his chair.
11.5 Preston does not change his answer.
11.6 The teacher redirects Preston to his drawings for 117 and 118.
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11.7 Preston: "Hey! I forgot to put a ten in there (118)." (3:56)
11.8 Preston draws in a ten on his chart for 118 and counts, "It's still nine!" (Preston is not counting one of the ones that he has drawn.) "See, look!" Preston counts the blocks in his diagram 3 times and concludes, "It's still nine when there's a ten."
11.9 Preston shows the teacher again that he gets nine blocks with the ten added in.
11.10 Preston erases the nine on his sheet but does not put in a new number.
12.0 The teacher questions Seren about any patterns that she notices, now that she has completed the charts for 100 - 124. (3:58)
12.1 Seren: "As soon as it hits ten, then it changes."
12.2 Teacher: "Why would that happen?"
12.3 Seren: "Because ten is the last part of - -the first part of the teens."
12.4 Teacher: "Why would that make it change?"
12.5 Seren: "It would make it change because you needed - - some more blocks. But it wouldn't work." EWT Induction
13.0 The teacher asks Seren to make a guess about how many base ten blocks she would need to use to make 128. (3:59)
13.1 Seren: "You'd use 4." Seren gets the blocks and checks her answer. She discovers that the number of blocks needed is 11.
13.2 The teacher asks Seren to make a guess about how many base ten blocks she would need to use to make 130. (4:01) "That's simple, no problem. I know, I know. That's trying to trick me but I know."
13.3 Seren writes down four as her guess and the teacher questions why, "Because it does take four." EWT
13.4 Seren models the number 130 and confirms her answer.
13.5 Seren makes a guess for the number 135.
13.6 Seren: "I wrote down nine and I counted with my fingers, so I think it's going to be nine." EWT Simple Deduction
14.0 The teacher asks Preston to make a guess about how many base ten blocks he would need to use to make 128. (4:02)
14.1 Preston states eleven as his answer and explains how he reached his answer when the teacher asks, "I guessed. I knew what you did."
14.2 Seren: "You copying mine!"
14.3 Preston: "No! No! I knew you that you had to take a ten, take a one hundred, two tens and add on eight." EHN Simple Deduction
14.4 Preston moves on to figure out the blocks needed for 130.
14.5 Preston: "It's just four for that one I guess."
14.6 Preston ads the digits in 135 to figure out the number of blocks for 135.
15.0 The teacher asks Preston to think of a number that would use three base ten blocks.
15.1 Preston: "Um, you could use three one hundreds."
15.2 Preston: ". . .three ones blocks makes three."
15.3 Preston: "Three tens blocks makes thirty."
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15.4 Preston is asked by the teacher to figure out a number that uses three blocks that are not all the same.
15.5 Preston: "Um, you could take a 100, then you could do a 10, then a separate 10. . . Or you could do all different things." EHT Simple Deduction
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Activity: Students are presented with mastermind games that have already been played, but are not shown the code. The students attempt to break the code from the information that they are given.
Gameboards:
These three gameboards are the games that were used for all students (some may have only played 1 game, while others may have played all three.) The last column indicates the "correctness" of each guess that was presented. White means that the colour was correct but not in the correct spot. Black indicates that both colour and position were correct.
Students were asked to make their guesses about the code after seeing all of the "clues" available for a particular game.
Interesting Points:
I found game 4 to be very interesting because David saw (at least it appears from the video and his actions) part of the code but it was in the reverse order which he failed to realize. All throughout the game, he seems to be influenced by what he thought he saw as the answer, regardless of what the clues indicated or what his partner Alicia would say.
Game A:
|
1 |
green |
red |
blue |
Pink |
3 white |
|
2 |
black |
green |
red |
Blue |
2 white |
|
3 |
blue |
pink |
black |
Red |
3 white |
|
4 |
pink |
blue |
red |
Yellow |
2 black, 2 white |
|
5 |
pink |
yellow |
red |
Blue |
4 white |
|
Guess |
|
|
|
|
|
|
Code |
red |
blue |
pink |
Yellow |
|
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Game B:
|
1 |
yellow |
pink |
green |
Black |
2 white |
|
2 |
pink |
yellow |
blue |
Red |
2 white |
|
3 |
black |
blue |
red |
Green |
1 black, 3 white |
|
4 |
black |
green |
blue |
Red |
4
white |
|
5 |
red |
black |
blue |
Green |
1 black, 3 white |
|
6 |
black |
red |
blue |
Green |
2 black, 2 white |
|
Guess |
|
|
|
|
|
|
Code |
blue |
red |
black |
green |
|
Game C:
|
1 |
red |
green |
blue |
black |
1 black, 1 white |
|
2 |
yellow |
green |
black |
blue |
2 white |
|
3 |
green |
yellow |
blue |
pink |
3 white |
|
4 |
red |
pink |
green |
yellow |
2 black, 2 white |
|
5 |
red |
yellow |
pink |
green |
2 black, 2 white |
|
6 |
green |
yellow |
pink |
red |
4 white |
|
Guess |
|
|
|
|
|
|
Code |
red |
pink |
yellow |
green |
|
1.0 Jerome is explained the rules of the game. He is told that there are six possible colours. (12:19)
1.1 Jerome is shown the first clue in game A. (12:20)
1.2 The teacher remind Jerome that he does not need to make a guess until all the clues have been shown to him.
1.3 Jerome: "I'll try to think about my guess while you're putting them (the clues) out."
1.4 After seeing clue 1 and 2, the teacher asks Jerome if he knows anything about any of the colours in the code at this point. He shakes his head no.
1.5 The third clue is presented.
1.6 Jerome: "All I know is that they kept changing the red." He points to the red in turns 1-3.
1.7 Teacher: "Where did they have the most colours?"
1.8 Jerome: "There and there." He indicates turn 1 and 3.
1.9 The teacher asks Jerome to look at the colours they used in each turn and see if he can learn anything about the colours that may be in the code. She asks him to note what colours were changed in each turn. (12:21)
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1.10 Jerome: "Pink (he points to the pink in the first clue) and there's a black there (he points to the black which has replaced the pink in the second clue.)" EWT Simple Deduction
1.11 Teacher: "So what happened?"
1.12 Jerome: "They changed the colour."
1.13 Teacher: "What happened to the code?"
1.14 Jerome: "They were a bit farther away (from the code on clue 2 than on clue 1)."
1.15 Teacher: "Do we know any new information?"
1.16 Jerome says he does not know anything about any of the colours yet.
1.17 The teacher lays down clue number 4. (12:22)
1.18 The teacher asks Jerome if he knows any new information about the code.
1.19 Jerome: "I think those two (red and yellow) are in the right spot."
1.20 The teacher asks Jerome why he thinks so.
1.21 Jerome: "Any other colours (he means clues), there weren't a yellow - - (he pauses then, puts to the pink and blue instead). I think these two are. "
1.22 The teacher asks him why.
1.23 Jerome: "There's no other blue there and there's no pink there." (He points to the columns in which each colour is in.) EWT Multi-Step Deduction
1.24 The teacher lays down the last clue in game A, clue number 5. (12:23)
1.25 Jerome: "He got that one right - - and that one." He points to the red and pink that are in the same positions as before.
1.26 The teacher reminds Jerome that there were no colours in the right spot in the last clue. She then asks Jerome to figure out which colours must have been correct in clue 4.
1.27 Jerome: "Two of these (the yellow and the blue)." Jerome explains why when the teacher questions his answer. "Because they changed around." EWT Multi-Step Deduction
1.28 The teacher directs Jerome into using the collared circles he has, to try to make a guess at the code.
1.29 Jerome places the blue and yellow in the fourth and second positions respectively (which is what occurred in guess 5).
1.30 The teacher asks Jerome if there were any in the correct spot when the person tried to place the blue and yellow as he just did.
1.31 Jerome checks clue 5 and states no. Then he moves the card containing clue five to the side saying, "they didn't have any right there, so I'll just put it right there." EWT Simple Deduction
1.32 Jerome places the pink and red in the correct positions. (12:25)
1.33 The teacher asks Jerome to tell why he did so.
1.34 Jerome: " Because it wasn't right there (in clue 4) and these two (blue and yellow) are in the right spot." EWT Multi-Step Deduction
1.35 Jerome is shown the code and is excited about getting it correct.
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2.0 Jerome begins Game B. (12:26)
2.1 The teacher quickly lays down the first two clues.
2.2 Jerome: "Blue isn't the right colour."
2.3 The teacher asks why he thinks that.
2.4 Jerome: "Cause it wasn't there (in clue 1) and it was still two (correct colour)." EWT Multi-Step Deduction
2.5 The teacher asks Jerome which colours he thinks may be right.
2.6 Jerome pauses for a while, looking at the clues. Then he shouts confidently, "Pink and yellow." The teacher asks him to explain his answer. "Cause that (green) wasn't there (in clue two), and there was still two and black wasn't there (in clue two) and there was still two." EWT Multi-Step Deduction
2.7 The teacher lays down the third clue. (12:27)
2.8 Jerome studies the cards, mumbling things to him self.
2.9 The teacher lays down clue number 4. (12:28)
2.10 Jerome: "So blue can't be there (in the third position). - - So, red can't be there (in the last position). So green can be there, there or there. (First, third or fourth positions)." EWX Multi-Step Deduction
2.11 The teacher asks why green should be there. Jerome says, “ . . .they changed green around and all of them all wrong (in clue 4)." EWT Simple Deduction
2.12 The teacher lays down clue 5. (12:29)
2.13 Jerome: "I think I know where green is supposed to be. They got one right there (clue three) and green was right there (position 4). They got another one right (clue 5) and green was right there (position 4). Blue wasn't there and red wasn't there and black wasn't there." When he talks about the blue and red, he indicates the column that each are in.
2.14 The teacher shows Jerome the last clue. (12:30)
2.15 Jerome: "This one (red) and this one (green) are perfect. I found it out!"
2.16 The teacher asks him to explain why he believes he "found it out."
2.17 Jerome: "None there. (He points to the column that red is in, and indicates with his finger that red has never been there before.) Two there. (He points to the green in clue 5 and 6.) And I got one perfect there (He points to clue 5.) - - I won another game!" EWT Multi-Step Deduction
2.18 The teacher asks Jerome to make the code.
2.19 The teacher asks Jerome why he placed the black where he did.
2.20 Jerome: "Cause two was perfect and two weren't. And I know these two are perfect because red wasn't anywhere else. Blue was - - gone all the time, so blue goes right there." EWT Multi-Step Deduction
2.21 The teacher shows Jerome that his code is correct.
3.0 Jerome begins game C. (12:31)
3.1 Jerome is shown the first two clues.
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3.2 Without being prompted, Jerome states, "Green isn't the perfect colour."
3.3 The teacher asks Jerome to explain his reasoning.
3.4 Jerome: "Cause there's no one perfect there (in the second guess). So blue, or black or red might be perfect." EWT Simple Deduction
3.5 The teacher presents Jerome with the third clue.
3.6 Without being asked, Jerome states, "Blue isn't in the right spot."
3.7 The teacher asks Jerome why he thinks that.
3.8 Jerome: "Because it said perfect there (guess 1) but it said no perfect there (guess 3) and they put that (the blue in the third position) there and there's no perfect. Black might be it. " EWT Multi-Step Deduction
3.9 The teacher lays down the fourth clue.
3.10 Jerome quickly states, "It has to be red. - - Look two are perfect. Green is changed from all the other ones. Now let me think about this - -yellow is going down like that (he points to the diagonal it has created in turns 2-4). - - It's a tricky one!" EWT Multi-Step Deduction
3.11 The teacher lays down clue number 5. (12:33)
3.12 Jerome very quickly says, "Red is the right colour in the right place. There are all changed except - - "
3.13 Teacher: "Do you know where one of them go now?"
3.14 Jerome: "Red goes right there (first position). Is there any next one (clue)?"
3.15 The teacher shows Jerome the last clue.
3.16 Jerome: "Red HAS to go there (in the first position). No perfect (in clue 6)."
3.17 Jerome studies the clues for a moment then says, "This one (red in clue 5) and this one (green in clue 5) were in the right spot here. That one's in the right spot. (Yellow in clue 4). EWT Multi-Step Deduction
3.18 The teacher repeats what Jerome has just stated.
3.19 Jerome: "Wait, that doesn't make any sense!"
3.20 Teacher: "Why not?"
3.21 Jerome: "Green is the colour but it's still right."(He means that green is in a different position - he points to the green in clues 4 and 5.)
3.21a Jerome: "These two (clues 4 and 5) are a little bit tricky."
3.21b Teacher: "What makes those so tricky?"
3.22 Jerome begins to make the code. "I know that the red is right here (in the first position in clues 4 and 5). But here (he points to clue 4) it is two perfect." (12:35)
3.23 The teacher suggests to Jerome that he should choose a colour and see if he can figure out where it goes by using the clues he has.
3.24 Jerome: "Blue isn't in this thing! (He moves the blue piece to the side.) - - And black isn't in this thing."
3.25 Jerome: "So here's the colours. All I need to find is what order they go in."
3.26 Teacher: "Can you figure it out by looking at this (the clues)."
3.27 Jerome reviews all the clues. As he does this, he uses his hands to cover up the colours that do not belong in the code. (12:37)
3.28 Jerome: "None perfect! (He is referring to clue number 6.) That doesn't give any help!"
3.29 Jerome continues to "size up" the clues.
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3.30 Jerome: "I'm not going to use that one yet." He moves clue 5 to the side.
3.31 Teacher: "Why aren't you going to use that one yet?"
3.32 Jerome: "I'm not going to use that one cause it sort of like fools me up."
3.33 The teacher once again suggests that he chooses one colour and tries to figure out where it should go. The teacher gives Jerome a pink marker.
3.34 Jerome: "Two perfect there (clue 4) and no perfect there (clue 3). So I know that the pink doesn't go there (position 4). It doesn't go there. (He places the marker in the third position on card 6). No perfect (in clue 3) . . . red doesn't go there. It (the pink) goes there (in the second position)." Jerome lays his pink marker next to the red one he has already placed in the first position. (12:38)
3.35 Jerome holds up clue 4 for a moment. "I figured it out!"
3.36 Teacher: "What do you mean, you figured it out?"
3.37 Jerome: "Look! I now that those (red and pink in clue 4) are two perfect but those (green and yellow) aren't cause look - - none perfect (in clue 6) but I know that red is supposed to go there (in position 1) and pink doesn't go there ... it has to go there. Check it!" Jerome is anxious to see the code revealed. EWT Multi-Step Deduction
3.38 The teacher reveals the code and Jerome is indeed correct.
4.0 David and Alicia are explained how to play Mastermind. Alicia has already played before. David appears to remember playing the game at some point. David, "You have to crack the code." (12:40)
4.1 The teacher lays the first two clues down on the table to game A. (12:41)
4.2 The teacher asks the students if they know anything about the code based on the first two clues.
4.3 Alicia: "When that person took away the pink, he got two correct colours. So it might be the yellow maybe. Um when he did that - -" EWT Simple Deduction
4.4 The teacher repeats what Alicia has just said.
4.5 David: "He should keep the pink." David then tries to peek at the code. (12:42)
4.6 The teacher lays down the third clue.
4.7 Alicia: "So the pink has to stay there because when he took it away, when he put it back it was right. - - Maybe the yellow would be a right colour?" EWT Simple Deduction
4.8 David lifts up the answer. He lifts it high enough to be able to see the colours of the code. What he does not realize is that the answer he sees is reversed. He sees yellow, pink, blue and red but the code is actually red, blue, pink and yellow.
4.9 The teacher asks Alicia why she thinks yellow might be the right colour.
4.10 Alicia: " Because he never choosed it yet." EWT Simple Deduction
4.11 The teacher lays down the fourth clue.
4.11a David points to the red and blue when the teacher states that there were two that were perfect on this clue card.
4.11b The teacher tells the students that she will give them one more clue before they need to make their guess.
4.11c David quickly picks up clue 4 and turns it around so that the yellow is in the first position and states, "Probably that one (is the code)." (12:43)
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4.12 Alicia: "It's probably these colours (pink and blue) because he used it in almost all of them. And he got - - and every time he used the pink and the blue he got three." EWX Simple Deduction
4.13 The teacher explains to the students that the last clue tells us that the person has discovered all of the four colours needed in the code.
4.14 David takes out markers for each of the colours in the code. (12:44)
4.15 The teacher reminds the students that they must now figure out the order of the colours.
4.16 Alicia begins to move the markers around to figure out the code. David takes the yellow from where Alicia places it and puts it in the first position.
4.17 The teacher lays down the fifth clue.
4.18 Alicia chooses a colour other than yellow again to put in the first position (I am not sure what it is) and David quickly removes it and puts yellow first.
4.19 The teacher asks David why he thinks yellow should go first.
4.20 David: "I don't know."
2.21 The teacher reviews the clues with the students.
4.22 David is not looking at the clues or at what Alicia is doing. Alicia makes the following code: red, pink, blue and yellow. The teacher asks David to look at the code and see if he agrees with the code that she created.
4.23 David looks at it and quickly shakes his head. Alicia moves the yellow to the first position, then David shouts, "Yeah, that's it!"
4.24 The teacher asks one again why he thinks yellow should be first.
4.25 David: "I dunno. Cause it wasn't first there (in clue 5)." EWT Simple Deduction
4.26 Now blue is in the last position. The teacher asks, "So you think blue should be in the last position?"
4.27 Alicia: "NO! Because none of them were in the right spot (clue 5 where blue was in the last spot)." Alicia holds up clue 5 to David, then moves yellow back to the fourth position. EWT Simple Deduction
4.28 David takes the yellow and places it back to position one.
4.29 Alicia rearranges the markers again. This time she places them in the following order: pink, red, blue and yellow.
4.30 The teacher begins to say the colours of the code that Alicia has created, "So you think pink should go first - -"
4.31 David: "Yeah but it's already first there (clue 5) and all the colours aren't right. And that (yellow) wasn't in front yet!" David reorganizes the markers to be in the following order: yellow, red, pink and blue. EWT Simple Deduction
4.32 Alicia promptly switches the red and the pink. (12:47)
4.33 The teacher asks the students to look carefully at the clues where they were told that colours were in the correct spot.
4.34 David: "Nope!"
4.35 Alicia: "Yes! Right here, two perfect. (She points to clue 4). Now - -which two perfect was that? - - The two perfect was the yellow and the." EWT
4.36 David cuts Alicia off, "No the two perfect was the."
4.37 Alicia and David argue about what colours were in the perfect spot. David refuses to believe that the yellow could have been in the correct spot.
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4.38 Alicia moves the pink to the third spot and explains why when the teacher asks her, "Because last time (clue 5) the red was there and none of them were in the correct spot." EWT Simple Deduction
4.39 Alicia is still unsure of which colours were correct in clue 4. She continues to move the markers around and once again places the yellow in the last position. "That's the only way we can do it David."
4.40 David takes the yellow and places it in position one again.
4.41 Alicia questions, "Why would we put yellow there?"
4.42 David says, "Look two correct right there." (He points to the red and the blue in clue 4.) EWT Simple Deduction
4.43 Alicia and David continue to move the markers around. Alicia appears to be getting frustrated. David says, "We're ready." Alicia argues, "No we're not!" Alicia continues to work on the code while David simply watches. (12:50)
4.44 Finally, David and Alicia show present their guess: yellow, blue, pink and yellow. (12:52)
4.45 The code is revealed.
4.46 Alicia states, "I didn't say I was ready!"
5.0 The teacher quickly goes over how to play the game with Maurice and Saul. (12:53)
5.1 The teacher lays down the first two clues to game A.
5.2 Saul indicates after the teacher has asked, that red, blue and green are "probably the colours." (Note that these three colours were used in clue two but only two of the four colours used were correct.)
5.3 The teacher reviews the information presented with each of the two clues, questioning how many colours were correct in each.
5.4 Maurice: "Blue and green are in."
5.5 Teacher: "Do you know any other colour that should be in (the code)?"
5.6 Maurice: "Red."
5.7 Saul: "But they got three here (clue 1) and two here (clue 2), so that's a hard decision."
5.8 Maurice: "Yeah."
5.9 The teacher lays down the third clue.
5.10 Maurice: "Blue, pink, red."
5.11 Saul: "No but look, no pink there! " Saul points to the card containing clue 2.
5.12 Maurice: "I Know cause they only got two there... They put a pink there again and they got three. They took away the pink they got two." EWT Simple Deduction
5.13 Saul: "Very well."
5.14 The teacher lays down clue four. (12:56)
5.16 The teacher lays down the last clue, clue five.
5.17 Maurice takes out the colours needed to make the code.
5.18 Maurice creates a code: pink, blue, yellow and red.
5.19 The teacher asks if Saul agrees with his code. "Do you think that pink could be first?"
5.20 Saul: "Yes. But in . . .one there's not a pink, but it really doesn't matter - - where they are. - - I think he's right."
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5.21 The teacher asks Saul why he thinks that Maurice is right.
5.22 Saul: "Because most of these colours have been used in all, well - - everyone has been used except for the green and the pink in one." EWT Simple Deduction
5.23 Teacher: "So has the red been used in that spot (fourth position) before?"
5.24 Maurice: "Yep!"
5.25 Teacher: "Did they have it in the right spot?"
5.26 Maurice: "Maybe."
5.27 The teacher goes over the meaning of the information presented with each guess again. Then asks again, "can red be in the right spot?"
5.28 Saul: "Because they got three correct colours (clue 3) but ... not, I don't know how to explain."EWT Simple Deduction
5.29 Teacher: "But not in the right spot?"
5.30 Saul: "It couldn't be there, it couldn't be right there (last position), no way."
5.31 Teacher: "Could it (red) go here (third position)?"
5.32 Saul: "Well it could."
5.33 The teacher redirects the students’ attention to clue 5.
5.34 The teacher asks Maurice if blue can be the last colour.
5.35 Maurice: "Probably."
5.36 Saul says no, and explains when asked by the teacher. "It's used in codes that have no correct." EWT Simple Deduction
5.37 The teacher asks the students to concentrate on figuring out where the blue would go and then asks them, "Could it go in the third spot?"
5.38 Saul: "Yes it could."
5.39 Teacher: "Could it be in the second spot?"
5.40 Saul: "I don't think the blue is in this code." (1:00)
5.41 Maurice: "It is. Four correct colour." Maurice points to clue 5.
5.42 Maurice is unsure if blue could go in the second spot.
5.43 Maurice and Saul's code is yellow, blue, red, and pink.
5.44 The teacher reveals the code: red, blue, pink, and yellow.
5.45 Maurice wanted to know if anybody got it right before he left the teacher.
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David A Reid
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Activity: Students are presented with mastermind games that have already been played, but are not shown the code. The students attempt to break the code from the information that they are given.
Gameboards:
These three gameboards are the games that were used for all students (some may have only played 1 game, while others may have played all three.) The last column indicates the "correctness" of each guess that was presented. White means that the colour was correct but not in the correct spot. Black indicates that both colour and position were correct.
Students were asked to make their guesses about the code after seeing all of the "clues" available for a particular game.
Game A:
|
1 |
green |
red |
blue |
pink |
3 white |
|
2 |
black |
green |
red |
blue |
2 white |
|
3 |
blue |
pink |
black |
red |
3 white |
|
4 |
pink |
blue |
red |
yellow |
2 black, 2 white |
|
5 |
pink |
yellow |
red |
blue |
4 white |
|
Guess |
|
|
|
|
|
|
Code |
red |
blue |
pink |
yellow |
|
Game B:
|
1 |
yellow |
pink |
green |
black |
2 white |
|
2 |
pink |
yellow |
blue |
red |
2 white |
|
3 |
black |
blue |
red |
green |
1 black, 3 white |
|
4 |
black |
green |
blue |
red |
4
white |
|
5 |
red |
black |
blue |
green |
1 black, 3 white |
|
6 |
black |
red |
blue |
green |
2 black, 2 white |
|
Guess |
|
|
|
|
|
|
Code |
blue |
red |
black |
green |
|
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-135
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Game C:
|
1 |
red |
green |
blue |
black |
1 black, 1 white |
|
2 |
yellow |
green |
black |
blue |
2 white |
|
3 |
green |
yellow |
blue |
pink |
3 white |
|
4 |
red |
pink |
green |
yellow |
2 black, 2 white |
|
5 |
red |
yellow |
pink |
green |
2 black, 2 white |
|
6 |
green |
yellow |
pink |
red |
4 white |
|
Guess |
|
|
|
|
|
|
Code |
red |
pink |
yellow |
green |
|
Notes:
On the previous day that I did this activity, I found that when the students worked by themselves, they were more focused, so for today's activity I took students one at a time. Unfortunately, I did not seem to see the same thing happen on this day.
I found Jared’s reasoning in 5.5 interesting because out of all the students who played game B, he was the only one who came up with the idea that two colours must have been used twice each.
It was interesting that students choose colours because it was a favourite colour (2.5 and 4.5) instead of using the information that they had available to them.
Out of all the students today, Kyla seemed the most focused (3.0 - 3.30).
--------------------------------------------------------------------------------------------------------------
1.0 Cynthia and the teacher review how to play the game. (3:15)
1.1 Cynthia is shown clues one and two to game A.
1.2 Cynthia: "They just need to get one more in the right position."
1.3 The teacher explains that there are no colours that are in the correct position in any of the clues.
1.4 The teacher asks Cynthia if she knows anything about the colours in the code based on these two clues.
1.5 Cynthia indicates that she does not know anything about the code.
1.6 The teacher shows Cynthia the third clue. (3:17)
1.7 The teacher asks Cynthia if she knows about any of the colours now.
1.8 Cynthia points to the black and says that she thinks it is the right colour. When asked why, she is unable to say why.
1.9 The teacher shows Cynthia the fourth clue. (3:18)
1.10 The teacher asks Cynthia if she has any idea what colours might be in the right spot.
1.11 Cynthia: "I think that one (pink) is in the right spot." The teacher asks her why. "Because they're just in the front there (in the other clues) cause if they were all in a row and the pink one was in the front. The pink one might have been right." EWT Induction
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David A Reid
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1.12 The teacher asks if there is any way to find out if the pink is the right colour.
1.13 Cynthia: "It's just a lucky guess."
1.14 The teacher lays down the last clue to game A. (3:19)
1.15 Cynthia: "Well I think that one (yellow) goes right there (in position 3)."
1.16 The teacher asks why? Cynthia does not respond.
1.17 The teacher tells Cynthia that she can work with the "markers" she has to try to figure out the code.
1.18 Cynthia: "I think that one went right there. (Yellow in the third position). Red one went right there (Last position)."
1.19 The teacher asks Cynthia why she thinks the red would go in the last position.
1.20 Cynthia: "Cause then I could put that one (pink) right there (position 2) and that one (blue) right there (position 1)." EWT Simple Deduction
1.21 Cynthia begins make the code. She places blue in the first position. The teacher asks Cynthia if she thinks blue could be right in the first position and Cynthia nods yes. The teacher then asks her if it was ever tried there before.
1.22 Cynthia identifies where blue was used in the first position and then nods her head no when asked if the person had any in the correct spot on that turn.
1.23 Cynthia moves the blue away. She now places red first and yellow second.
1.24 The teacher asks Cynthia if yellow could be correct in the second place and she nods her head yes. (3:21)
1.25 Cynthia notices yellow in the second position in clue 5 and takes the yellow out from position two.
1.26 Cynthia takes a black marker.
1.27 Teacher: "Why did you choose black?"
1.28 Cynthia: "Cause it's never been used in the second row." EWT Simple Deduction
1.29 The teacher asks Cynthia how many colours are in the code and she quickly states four. She nods yes that black is one of the colours.
1.30 Cynthia then chooses green as the third colour (in the third position) and explains that green has never been used in the third position. EWT Simple Deduction
1.31 After a brief pause, Cynthia states, "I don't think green goes there. I think yellow goes there." Cynthia removes the green and replaces it with a yellow. “Cause yellow was never there.” EWT Simple Deduction
1.32 The teacher asks Cynthia if any card shows her that some colours were in the right spot.
1.33 Cynthia notes clue 4.
1.34 Cynthia shows her code: red, black, yellow and green.
1.35 The teacher reveals the code: red, blue, pink and yellow. (3:24)
1.36 Cynthia: “I got one in the right spot.”
2.0 Veronica and the teacher briefly go over the rules of the game. (3:25)
2.1 The teacher lays down the first and second clues in game A.
2.2 The teacher asks Veronica if she knows anything about the colours the person should have used.
2.3 Veronica: "Pink."
2.4 The teacher asks Veronica why she thinks pink should have been used.
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2.5 Veronica: "It's my favourite colour."
2.6 The teacher asks if there are any other reasons why pink should have been used and she responds by saying she doesn't like black. EWT Aesthetic reasoning
2.7 The teacher goes over the information on cards 1 and 2 again and questions whether the clues can give us any help with the code.
2.8 Veronica: "They need the pink. It's up there (in clue 1)." EWT Simple Deduction
2.9 The teacher asks why but Veronica does not give any reason.
2.10 The teacher lays down clue 3 and 4. (3:27)
2.11 Veronica asks the teacher which two are in the correct spot in clue 4. The teacher explains that she needs to try to figure that out from all of the clues.
2.12 Veronica decides that yellow and pink are in the right spot. The teacher asks her to explain why. "They're both my favourite colours." EWT Aesthetic reasoning
2.13 The teacher lays down the last clue in game A. (3:28)
2.14 The teacher reviews each clue with Veronica.
2.15 Veronica creates the following code: red, blue, yellow and green.
2.16 The teacher reveals the code: red, blue, pink, and yellow.
3.0 Kyla and the teacher review how to play the game. (3:31)
3.1 The teacher lays down clues number 1 and 2 in game B. (3:32)
3.2 When asked, Kyla identifies that black has been changed to blue, and green has been changed to red.
3.3 Kyla: "These two are the right colours (pink and yellow)."
3.4 The teacher asks why she thinks that.
3.5 Kyla: "Cause they got this (2 correct colour) there too." EWT Simple Deduction
[This
is a tricky code because 2 colours are changed and the response is still 2
correct. Kyla interprets this as the unchanged colours must be right whereas
the two that were exchanged were actually the two that were correct.]
3.6 The teacher asks Kyla how many colours are needed for the code, how many possible colours are there and then what colours does she think will go with the pink and yellow. Kyla shrugs, indicating she has no idea what colours will go with pink and yellow.
[This is an interesting question from the teacher because all 4 of the colours have already been tried in the first two clues. If pink and yellow were in the solution it would have to be two of each colour. But because yellow wasn’t perfect in either of the first two spots and neither was pink there would have to be different colours involved – therefore pink and yellow are not in the solution. Compare Kyla’s remarks here with Jared’s in episode 5]
3.7 The teacher lays down clue 3. (3:33)
3.8 Kyla studies the cards.
3.9 The teacher lays down clue 4. (3:34)
3.10 Kyla clarifies the clue for herself: "So they have all the right colours but not in the right spot?"
3.11 The teacher points out that all the correct colours were used in clue 3 as well, with one being in the correct spot.
3.12 Kyla studies the cards again. Then she says, "It's the black one."
3.13 The teacher asks, "So you think the black one's in the right spot?"
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3.14 Kyla states, "No." the teacher asks why. "Because it would have been one perfect right there (in clue 4)." EWT Simple Deduction
3.15 The teacher lays down clue number 5. (3:35)
3.16 Kyla examines all the cards. Then says, "It's the green one."
3.17 The teacher asks why.
3.18 Kyla: "Cause this one is right here (red in clue 5) and if that was right there in every one . . . it would’ve been all perfect. (She points to the first position in each clue.)" Kyla goes on to explain with the same reasoning for black and blue, pointing to the columns in which they are in. EWT Multi-Step Deduction
3.19 The teacher lays down the last clue. (3:37)
3.20 The teacher asks if the green can be in the right spot in clue 6. Kyla nods yes but says she does not know why.
3.21 The teacher asks her if any of the clues tell her any information about the green in the last position.
3.22 Kyla: "Right here (clue 3), it says one perfect and the green's right here."
3.23 The teacher asks if green was last in any other clues. Kyla notes cards 5 and 6.
3.24 The teacher asks if Kyla thinks the blue is perfect in clue 6. Kyla nods her head no. The teacher asks her to explain.
3.25 Cause it would've been right here, two (clue 5) and here (clue 2). EWT Simple Deduction
3.26 The teacher suggests to Kyla to start modeling her answer. (3:38)
3.27 Kyla, without being asked says that she thinks red is in the perfect spot in the last clue. The teacher asks her to explain why. Kyla responds by shrugging her shoulders.
3.28 Kyla presents her code: blue, red, black and green.
3.29 The teacher reveals the code. Kyla smiles.
3.30 Without prompting, Kyla offers an explanation about her guess. "The reason why I . . . right here it said two (clue 6), and right here it says one (clue 5), so it got to be that one (green). And here it says one and then I thought that one (red in the 6th clue) was the second one. So, and I just . . . so I say, that can't be right (the black in the first position - she points to the first column indicating that black was unsuccessfully tried there). So I put it right there." EWX Multi-Step Deduction
4.0 The teacher explains how to play the game to Ira. (3:40)
4.1 Ira begins game B. He is shown the first two clues. (3:42)
4.2 Ira is asked if he knows which two might be the right colours. Ira chooses red and blue. The teacher asks why and he says because, "it's at the end." EWT Simple Deduction
4.3 The teacher shows the third clue to Ira. Note: Ira models each clue he when he is presented with it. (3:43)
4.4 The teacher shows Ira the fourth and fifth clues.
4.5 The teacher asks Ira if he knows which colour is in the correct spot. He indicates the green one. When questioned why, he responds, "cause it's Maurice's favourite colour." (3:44) EWT Aesthetic reasoning
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David A Reid
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4.6 The teacher asks Ira if green was ever used in the fourth spot before. Ira locates clue number 3 where green was in the fourth position. The teacher directs Ira to look at the information given next to the clue (1 black, 3 white).
4.7 The teacher shows Ira the last clue.
4.8 The teacher asks Ira if he knows which two colours are in the correct spot.
4.9 Ira indicates that blue and green are correct.
4.10 The teacher asks Ira why he believes blue and green are correct. Ira explains, "Yeah, cause these two (blue and green) are still in the same spot." EWT Simple Deduction
4.11 The teacher redirects Ira to look at the information given with clues 5 and 6. There was only 1 black in 5 but 2 black in 6.
4.12 The teacher questions whether the information given with the clues tells Ira anything.
4.13 Ira: "This one (green) was the same one."
4.14 The teacher tells Ira to start thinking about what the code might be.
4.15 Ira once again states that he thinks blue is correct in the third position. The teacher asks him why and he says, "because it's two perfect (in clue 6)." EWT Simple Deduction
4.16 Teacher: "So you don't think red or black is perfect?"
4.17 Ira: "Nope." When asked why not, he continues, "because . . . these two other guys (red and black) were here last time and this time these guys were here . . . switched around. But not these two (blue and green)." EWT Multi-Step Deduction
4.18 Ira’s code (black, red, blue green) is checked with the actual code. Note that Ira’s code was identical to the sixth clue in which there were 2 black and 2 white.
5.0 The teacher reviews how to plat the game with Jared. Jared has played this game previously with the teacher using the Mastermind board. (3:50)
5.1 Jared begins game B. The teacher shows Jared clue 1 and clue 2.
5.2 The teacher tells Jared that on both of the clue cards, the person had two colours that were correct.
5.3 Jared: " I know which ones. (He points to the pink and the yellow.)"
5.4 The teacher asks Jared why he thinks the correct colours are pink and yellow. "Because they're (pink and yellow) right here (in clue 1) and the same colours (red and blue) aren’t right here (he points to the green and the black in clue 1), so it has to be pink and yellow." EWT Multi-Step Deduction
5.5 The teacher asks Jared which two colours he thinks might go with the pink and the yellow if pink and yellow are correct, knowing that there are only six colours in all. Jared chooses green and black and then replaces them with red and blue. He then takes away red and blue and tries green and red. Jared continues to stare at the collared markers and finally concludes, "AH! I need two more of these (pink and yellow)."
5.6 The teacher shows Jared the third clue. (3:54)
5.7 The teacher asks Jared if he now knows which two were correct in the first two clues. He indicates that black and green were correct in the first clue and pink and yellow were correct in the second clue.
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5.8 The teacher asks Jared if pink and yellow were right in the first clue if they were right in the second clue. Jared responds with a "yeah." The teacher then asks how come the person only had two correct colours in the first one. Jared continues to look at the clue cards but offers no explanation.
5.9 The teacher lays down the fourth clue. Note that Jared always has a "working model" of the possible code as the game is played. He makes changes as new clues are presented. (3:55)
5.10 Red, black, blue and green was the order of his markers before the fourth card was laid. When the fourth card was laid he switched the positions of blue and green. The teacher asked him why he did this and he explained, "because the blue was right here and it wasn't the right spot so I moved it around like that." EWT Simple Deduction
5.11 The teacher lays down the fifth clue. (3:56)
5.12 The teacher asks which colour is perfect. Jared spends some time looking at all of the clues, then he begins to verbalize, "this one (black) stays right here (second position). This one (blue) goes right there (first position). "
5.13 The teacher asks why does the black one need to stay "right there."
5.14 Jared: "Cause it was right here." (He points to clue number five.) (3:57) EWT
5.15 The teacher suggests that maybe the red or blue are correct. Jared states that the blue can't be in the right position in clue five. The teacher asks for an explanation, Jared says, " cause right here (clue 3) the blue is here (second position) and right here (clue 5) the blue is right here (third position), so it can't be the blue."
5.16 The teacher asks which colour is perfect. Jared says the green is perfect in clue 5.
5.17 The teacher asks Jared why, he says, "cause right here (clue 3) the green's in the same spot and 1 perfect and right here (clue 5) the green's in the same spot and it's one perfect." EWT Multi-Step Deduction
5.18 The teacher shows Jared the last clue, number 6. (3:58)
5.19 Jared quickly rearranges his working model to blue, red, black and green which is the correct code but then switches red and blue.
5.20 The teacher asks Jared if red was ever used in the first spot. Jared responds quickly with a "no", he does not notice that it was used in clue five. The teacher repeats what he has just said, "the red was never used in the first spot?"
5.21 Jared looks at the clues and then states, "Something is wrong here." He takes the red marker and switches it with the blue again. When asked why he put the red there he said he didn't know why.
5.22 The teacher asks him why he placed black where he did. He responds, "because it wasn't there before." EWT Simple Deduction
5.23 The teacher reveals the code. (4:00)
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Activity:
Students were given two sheets to do containing examples of "growing patterns." The two patterns that they were asked to make and observe using pattern blocks were the square and the triangle. For both patterns the first three shapes of each pattern were depicted, labelled as 1, 2 and 3. The students were then required to come up with the 4th shape in each pattern on their own and eventually make predictions about the number of pattern blocks needed to make the 5th shape in each pattern.
Two charts were given to be completed, one for each shape pattern which resembled the following:
|
Number |
Number of blocks in bottom row |
Number of blocks in square/triangle |
|
1 |
|
|
|
2 |
|
|
|
3 |
|
|
|
4 |
|
|
Upon completion of these two sheets, students were given the opportunity to create their own growing patterns. Many students did not reach this stage.
Students:
Notes:
You will notice that the entire "transcript" does not go in a linear fashion. I have tried to keep individual student comments together.
---------------------------------------------------------------------------------------------------------------
1.0 The teacher quickly reviews the instructions for the activity and the students individually begin to model each pattern. (3:21)
1.1 Leonard quickly copies the square pattern illustrated on his sheet.
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1.2 Leonard then creates the fourth square that is actually a 4x3 rectangle. He counts the number of blocks in the bottom row of each square without prompting from the teacher, "1, 2, 3, 3 again. But that (the rectangle) is 3 add more." (3:24)
1.3 Teacher: "That is three add more is it?"
1.4 Leonard: "Yeah, but I did that then . . . look see, it keeps going on higher." Leonard points to each successive shape in his pattern, noting that the height is changing. "So it's got to be going up that way."
1.5 The teacher asks Leonard what shapes he has created in each step of the pattern. He states square for the first three then says the last one (the rectangle) as an oval."
1.6 Teacher: "An oval?"
1.7 Leonard: "It's pretty long. And it's not going like that (He points to the third square that he created)."
1.8 Teacher: "What makes these (number 1, 2 and 3 of the pattern) squares?"
1.9 Leonard: "They're square."
1.10 The teacher asks Leonard how can he tell when something is a square.
1.11 Leonard: "Cause it got four pointers." He points to the four corners.
1.12 Teacher: "Well this (the rectangle) has four pointers."
1.13 Leonard pauses for a moment then states, "Cause it's straight."
1.14 Teacher: "Is this (the rectangle) one straight?"
1.15 Leonard: "And it gotta be the same - - and it gotta be the same size on every way and that isn't."
1.16 The teacher goes over the idea of making the squares grow and asks him if he can make number 4 into a square. (3:25) UMT
1.17 Leonard takes the three pattern blocks from the top of the rectangle and moves them to the side of it, still creating a rectangle but with 4 on the bottom instead of three.
1.18 Leonard: "then it looks like a square but the top isn't."
1.19 Teacher: "What could you do to make it a square?"
1.20 Leonard: "Take three away."
1.21 Teacher: "Then it would be the same as this one (number 3 of the pattern)."
1.22 Leonard: "Yeah, but there's no way!" EHT Simple Deduction
1.23 The teacher asks Leonard to think about the ways in which the squares grow. He looks at the squares he has created and then moves the three blocks from the side of his rectangle back to the top. He concludes that there is no way to make the fourth square. "There's no way. . .either way it's an oval." EWT Simple Deduction
1.24 The teacher suggests that Jared and Leonard work together to figure out how to make the fourth square. (3:28)
1.25 Leonard says to Jared: "The only way I can make a square is if I take three away. It's the only way we can do it."
1.26 Jared: " If we put this one back on and this one back on it'll be a rectangle."
1.27 Leonard: "A rectangle? - - there's no way you can make squares with the fourth one, there's no way."
1.28 Jared: "Yes there is. There's gotta be one way to do it."
1.29 Leonard: "The only way it is - -"
1.30 Jared: "I know how to do it."
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1.31 Leonard: "Ohhhh! I got it! I got it!" Leonard places another row of blocks on his rectangle to create a square.
1.32 Jared: "Yep. You got it."
1.33 Leonard: "Add on four here. . . same size each way." EWS Simple Deduction
2.0 The teacher points out the question on the bottom of the sheet to Leonard and Jared asking them to predict how many pattern blocks would be used in the fifth square. (3:30)
2.1 Leonard says twenty and the teacher questions why. "Cause look, add on to sixteen, and add on four more. (Leonard adds four pattern blocks on top of the last square, which creates a rectangle 4 long by 5 high.) Add some more on this way. (Leonard adds five blocks along the left side of the rectangle he has just created to make a square.) It's not twenty! It's more than twenty."
2.2 Teacher: "Why do you say it's going to be more than twenty?"
2.3 Leonard: "I just thought that I'd add on four more but then I saw it's not a square. But now I knows." EWT Simple Deduction
2.4 Leonard removes the extra blocks from the fourth square and makes the fifth square of the pattern. "I got it. I guess 24. (He says this before he completes the fifth square.)"
2.5 The teacher asks Leonard how many blocks are in square 5. (3:33)
2.6 Leonard: ". . . Five at the bottom. 5, 10, 15, 20 - - 25. 25! . . . one over what I said, I said 24."
3.0 Jared and Ira are busy building the fifth square. They have no difficulty building the square but they count all the blocks they have used for the entire pattern to find out how many the last one needs. (3:35)
3.1 Ira to Jared: "We need to put down 54." (3:38)
3.2 The teacher goes over to Ira and Jared and asks them how many blocks are in the fifth square. (3:39)
3.3 How many blocks would you need in the bottom row of the fifth square?
3.4 Jared: "100." When asked why he says he just guessed. (3:40)
3.5 Ira: "You got to count by fives like this, 5, 10, 15, 20, 25, 30, . . .100."
3.6 Teacher: "How many fives did you count?"
3.7 Jared: "- - 100."
3.8 Teacher: "You counted a hundred fives. Why did you count a hundred fives?"
3.9 Jared: "Cause you never told us how many to count so we counted up to a hundred." EHT Induction
3.10 The teacher asks again, how would they make a square, square number five, in the pattern.
3.11 Teacher: " How many fives would be in a square if you had five on the bottom?"
3.12 Ira says to Jared: "We better make 5 to see how many blocks it used up." (3:42)
3.13 Ira: "This is 15, 20, 21, 22, 23- -"
3.14 Jared: "Wait." He starts counting the blocks.
3.15 Ira: "You're going around in circles. Wait, wait, wait." (3:43)
3.16 Ira fixes up the square and begins to count. He counts 28 and Jared continues to 30. (I can not tell but I think they have incorrectly created the square.)
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