previous table of contents appendix B table of contents next
Note: Initially, I instructed these students to find sets that were different in three ways and included all the cards except the shaded ones. Cards that were outlined in green or red were considered to be white.
0. Brief directions given to students by teacher.
0.1 Students discuss and clarify instructions. (3:23)
0.2 Alison "That are either the same but they have don't have to be exactly the same."
0.3 Alison "they have to be different or to be same." (3:24)
0.4 Cynthia "Does it have to be the same number?"
1.0 Nine cards are laid down and students are asked to collectively find a set. (3:24) Teacher: “Let’s see if we can find one. Is there 3 cards here that are different in shape, number and colour?”
1.1 Jared and Cynthia quickly state “Yes!”
1.2 Cynthia tries to point out cards for a set while Jared picks out the following cards: 1 white oval, 2 white ovals [unclear what the third card is].
1.3 Teacher - "Are they all different shapes?"
1.4 Cynthia “Yes!” Alison “No.”
1.5 Teacher- "so they are not a set."
1.6 Cards are replaced and students to again to find a set.
2.0 Jared and Cynthia both quickly point out pairs of cards.
2.1 Jared "This one and this one and this one and this one." (3:25)
2.2 Teacher "We need three cards."
2.3 Jared and Cynthia continue to point out cards.
2.4 Alicia indicates three cards that she thinks are a set (3 red squiggles, 2 green ovals, and 1 white diamond).
2.5 Teacher goes through the three attributes questioning whether in each case all are different. Teacher – “Okay now, let’s see. Alison says these three – now, are they all different numbers?”
2.6 Alison – “yes”
2.7 Teacher (pointing) “1,2,3”.
2.8 Cynthia – “Hey, this is like counting, 1,2,3!” EWP, Analogy
2.9 Teacher - "are they all different colours?"
2.10 Jared – “No.”
2.11 Teacher – “this is really a white (indicating the 1 oval outlined in red) so we have white, green, red.
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Psychology of Students' Reasoning in School Mathematics
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2.12 Teacher – “are they all different shapes?”
2.13 All – “Yes!”
2.14 Teacher – “Okay, Alison found the first one. I think that was awful tricky. Alison gets to keep these cards.”
3.0 Cards are removed and not replaced and students are asked to find another set. Teacher- “Is there another set here?” (3:26)
3.1 Jared – “yeah” and he immediately points out 3 cards ([not all clear] white , 3 green diamonds, white).
3.2 Teacher Let's check to see if they are a set.
3.3 Alison " No see these two here are the same number." EWT Simple Deduction
3.4 Teacher – “So they are the same number. Are they the same colour?
3.5 Jared – “yes”. Cynthia – “no”.
3.6 Teacher – “So they are not all the same colour so they be a set. Keep looking. Cynthia do you see one, Cynthia, a set?”
3.7 Cynthia " I can't see another one."
3.8 Teacher – “Alison, can you see another set? They have to be all different or all the same.”
3.9 Alison – (points to 3 cards: 1 red oval, 2 white squiggles and 3 green diamonds).
3.10 Teacher- “Oh, now, Alison you picked out 1 red oval, 2 white squiggles and 3 green diamonds”.
3.11 Cynthia " I think they are different."
3.12 Teacher – “And you think they’re a set. Why do you think those are a set Alison?”
3.13 Alison " Because they're all different shapes, all different colours and all different numbers." EWT Simple Deduction
3.14 Teacher – “Is that right?” (to Cynthia)
3.15 Cynthia – “Um, yeah.”
3.16 Alicia's set is removed leaving just three cards on the table.
4.0 Students are questioned as to whether or not the remaining cards (1 white oval, 2 green ovals, white oval ) will make a set. (3:27)
4.1 All three students firmly state no in unison.
4.2 Jared is asked why he thinks the three remaining cards will not make a set.
4.3 Jared "Cause one is white and two of the ones are green."
4.4 Alison "They're all the same shape."
4.5 Jared – “They’re all the same colour and size”.
4.6 Cynthia (interrupting) “That’s white, that’s green and that’s white.”
4.7 Alicia – “Two the same colour and they’re all the same shape.
4.8 Teacher "so why can't they make a set?"
4.9 Alison "Cause there are two things that are the same." EWT Simple Deduction
5.0 The teacher indicated that she would lay down more cards and asked how many cards would be needed. (3:28)
5.1 Alison "We used nine. We have three so we have to use six." EWT Simple Deduction
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Psychology of Students' Reasoning in School Mathematics
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5.2 Cynthia " We need more cards to like make another set."
5.3 Teacher " So how many more cards do we need?"
5.4 Alison "Six."
5.5 Teacher "How did you know?"
5.6 Alison "Cause 3 + 3 + 3 is 9." EWT Simple Deduction
6.0 Six additional cards are placed down on the table. Students are asked to looks for new sets. (3:28)
6.1 Jared "I know one." He points to 3 green squiggles, 3 purple diamonds, 3 white diamonds.
6.2 Alison jumps in and states "They're all the same number and two..."
EWX Simple Deduction
6.3 Teacher asks Jared to explain his choice.
6.4 Jared – “They're all the same number and”
6.5 Cynthia - "The colours are all different"
6.6 Jared - "The colours are different."
6.7 Teacher – “They’re all different colours.”
6.8 Alison – (excitedly) “I have a set.”
6.9 Cynthia – “But they’re not all different shapes.”
6.10 Teacher – “Now you said they’re all the same number. SO they have to have all the same or all different – so are they a set?”
6.11 Jared – “No.”
6.12 Teacher – “Why aren’t they a set?”
6.13 Jared – “Because this one is green. They're all the different colours but this is the same shape.”EWT Simple Deduction
7.0 Both Alicia and Cynthia are repeatedly shouting out "I gotta set." (3:29)
7.1 The teacher asks Cynthia to show her set.
7.2 Cynthia points out the following cards : 1 green oval, 2 green ovals and 2 white ovals.
She is asked to explain if it is a set and why.
7.3 Jared quickly shouts "No!"
7.4 Cynthia "I know but it's just trying.."
7.5 Teacher "Why are these 3 a set?
7.6 Cynthia "Because I know it's not a set but I was just wanted to get a turn."
7.7 Teacher “How do you know it's not a set?"
7.8 Cynthia "Those two there are the same colours and that's white."
EWT Simple Deduction
8.0 Cards are placed back with the others and Alicia picks out three cards (3 green ovals, 1 white squiggle and 2 purple diamonds) (3:30)
8.1 Teacher "Tell us if it's a set."
8.2 Alison "Okay. Um, it's a set because they're all the same, different colours. They're all different numbers. They're all different shapes." EWT Simple Deduction
8.3 Teacher “Are they all different colours?”
8.4 Jared "No." He points to the squiggle that is outlined in purple.
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8.5 Teacher "No this one is white."
8.6 Jared "I can't do mine now.” (He points to the card with 3 green ovals on it.) (3:31)
8.7 Jared "I was going to use these two (2 white diamond and 3 green ovals) and put um with that one... and this one, this one and this one.”
9.0 Alicia's set is removed and students are asked to make a set with the remaining 6 cards.
9.1 Jared attempts to find a set. He points to two at a time then pauses and starts over again.
9.2 Teacher - "They need to be all different or all the same."
9.3 Cynthia " I think we need more cards. Three more cards. I think so. Because."
9.4 Jared picks out 3 purple diamonds, white ovals, 2 white(?).
9.5 Jared - "They're all different colours and they're all different numbers and shapes."
Simple Deduction
9.6 Alison "No, these two here aren't the same number."
9.7 Alison "They're not even different in colour, see." She points to the two white cards.
EWX Simple Deduction
9.8 Cynthia "I think we need three more cards."
9.9 Alison "We do. We do."
9.10 Alison "Cause there are no matches."
9.11 Teacher "How do you know there are no more matches?"
9.12 Alison "Okay because."
9.13 Jared "These two are the same colour. These three are the same colour and this one is different. And this one and this one is the same shape and ... Multi-Step Deduction
9.14 Alison "The only ones are these, the only twos are these and the only threes are these. (She indicates two cards each time.) EWT
9.15 Alison stops and puts her hands on her head and tries to find a way to explain why there can't be another set.
9.16 Alison " Um, see."
10.1 The teacher decides to lay down three more cards to make the total nine. (3:32)
10.2 Alison "Um, there's no more ones. I can't make a set." EWX Simple Deduction
10.3 Cynthia "I think I can make a set."
10.4 Alison "I can't make a set. The only twos are these."
10.5 Jared "No, we can't make a set."
10.6 Alison "The only ones are these and the only threes are these."
10.7 Jared begins to point out pairs of cards and Alicia interrupts with "wait, wait, wait."
10.8 Alison "Well we can't make them all the same. They have to be like exactly the same thing."
EWX Multi-Step Deduction
10.9 Teacher "They would have to be the same colour, same number and same shape."
10.10 Jared "I got one." (He points to three cards.)
10.11 Alison "Three the same number. Two the same colours. Two the same shapes." EWX Simple Deduction
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10.12 Jared "No that won't work."
11.0 The teacher adds three more to give a total of twelve cards laid on the table. (3:35)
11.1 Jared "I think I got a set."
11.2 Alison " I got a set, I got a set."
11.3 Jared "Give her a turn."
11.4 Alison " Um, um, um, I don't have a set."
11.5 Jared " I might have a set."
11.6 Jared points to 2 red ovals, 3 purple diamonds and 2? white ovals."
11.7 Alison " Two the same shapes and two the same numbers." EWX Simple Deduction
11.8 Jared "No".
11.8 The teacher adds three more cards to make a total of 15 cards. (3:36)
11.9 Jared " I got one."
11.10 Alison " I got a set. I got a set."
11.11 Alison " That one, that one and that one."
11.12 Jared " No, that won't make a set."
11.13 Teacher " How come that won't make set?"
11.14 Jared "cause this one has three, two, one." EWT Simple Deduction
11.15 Teacher " So they're all different numbers."
11.16 Alison " no cause two are the same colours. No I never meant."
EWT Simple Deduction
11.17 The teacher picks up all the cards.
12.0 The teacher takes the sets that Alicia created and asks the students to explain why each set was a set.
12.1 Cynthia is asked to explain why 1 red oval, 3 green diamonds and 2 white squiggles are a set.
12.2 Cynthia " They're all...different. They're all different shapes. They're all different colours.
12.3 Alison " They're all different numbers."
12.4 Cynthia " Numbers!"
12.5 Jared " Different colour number and shape."
12.6 Teacher "Okay, Jared, tell me why are these three a set?"
12.7 Jared "Because these aren't the same shape and they're not the same colour."
12.8 Teacher "Alison, explain the last set."
12.9 Alison (rhymes it off) "Okay, different colours, different shapes and different numbers." EWT Simple Deduction
Note: Because of the difficulty the students had with the cards that were outlined, I removed them from the deck this time. Therefore, the only cards in use were solid coloured cards.
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Also, please note that I instructed the students to find sets that were different in three ways.
Set
13.0 The teacher asks the group if they have ever played Set before. The students say they have played but they don't remember how to play. The teacher lays down 8 cards and asks the students to identify how many different colours there are. (3:40)
13.1 David " purple, red, green, purple."
13.2 Teacher " so how many different colours are there?"
13.3 David " 1,2,3,4."
13.4 Teacher " What's the four colours?"
13.5 Seth " Purple, red, green (he examines the remaining cards) four, three colours!
13.6 Teacher " How many shapes do we have?"
13.7 Seth " 1, 2, 3. Three!"
13.8 Teacher "How many different numbers do we have on these?"
13.9 Seth " 1, 3, 1, 1, 3, 2, 3,...
13.10 David " 1, 2, 3. Three!"
13.11 Seth "And there's three colours, three shapes and three numbers."
13.12 The teacher briefly explains how to make set.
13.13 Seth exclaims, "They're all threes!"
14.0 The teacher lays down 9 cards and asks students to find a set. (3:43)
14.1 Two of the students quickly point out cards.
14.2 David " I choose that one, that one and that one. No, that one, that one and that one."
14.3 Seth " I choose this one, this one and this one."
14.4 Jerome Quietly points to three cards.
14.5 David " No, that one, that one and that one."
14.6 Teacher " Seth do you see one? All different numbers. All different colours. All different shapes."
14.7 Seth chooses 1 green oval, 2 green squiggles, 1 purple diamond.
14.8 Teacher "Why do you think that one is a set?"
14.9 Seth " Because they are all different shapes." EWT Simple Deduction
14.10 Teacher "Are they all different numbers?"
14.11 Seth " No, only one." He points to the card with 2 green squiggles.
14.12 The teacher places the cards back with the others.
15.0 David quickly states he has a set. (3:44)
15.1 David " That one (1 purple diamond) that one (2 green ovals) and ... no."
15.2 David " That one (1 purple diamond) that one (2 green ovals) and ...
15.3 David " That one (1 purple squiggle), that one (2 green ovals)." He then puts back the card with the 2 green ovals.
15.4 Teacher " Why did you put back that one (the 2 green ovals)?"
15.5 David " That's not the one I'm thinking of."
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15.6 David " That one (1 purple oval), that one ( 2 green squiggles) and that one ( 3 purple diamonds).
15.7 Teacher " Now, David, are they all different...
15.8 David " No." He changes the 2 green squiggles for 2 red ovals.
15.9 Teacher " Why did you switch that one?"
15.10 David " Because." Then he puts his hand on his head. "I don't know".
15.11 Teacher " Let's see. Are they all a different number?"
15.12 David " Yes."
15.13 Teacher " Are they all a different colour?"
15.14 Seth " No."
15.15 David " they're not different shape." he switches the 2 red ovals for 2 green squiggles.
15.16 David " And they're all different shapes."
15.17 Teacher " So, do you think they are a set?"
15.18 David " No."
15.19 Teacher " Why?"
15.20 David " No because they are the same." He picks out the two cards that are both purple in colour.
EWT Simple Deduction
15.21 Teacher " Is there any way you can pick a card to make it a set?"
15.21 David chooses two cards containing ovals on them, one with one and one with two and puts them together to make a card with three ovals!
15.22 Teacher " What are you going to switch to make a set?"
16.0 The teacher places the two cards back and returns the original card. (3:46)
16.1 Teacher "Let's see. He has three different numbers. He has three different shapes but he doesn't have three different colours."
16.2 Seth "I think we need another set. I think we need another set."
16.3 David changes the cards to have 1 green oval, 2 red ovals and 3 purple diamonds.
16.4 David " This is going to be hard."
16.5 Seth points to the 1 green oval and states "no."
16.6 Teacher "Why do you say no to that one Seth?"
16.7 Seth "That's the same as that" (he points to the two cards with diamonds).
EWT Simple Deduction
16.8 David rearranges the cards to have 1 green squiggle, 2 red ovals and 3 purple diamonds.
16.9 Seth "Okay, I think he has it."
16.10 David " There!"
16.11 Teacher " Why do you think you have it now?"
16.12 David " Cause that's a different colour, that's a different colour and that's a different colour. They're different numbers and they're different shapes." EWT Simple Deduction
17.0 The teacher removes the set leaving six cards and asks if any other sets can be made. (3:47)
17.1 Seth " No, we don't got a red."
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17.2 Teacher " Why does that matter?"
17.3 Seth " Cause there won't be a different colour." EWT Simple Deduction
18.0 The teacher lays down three more cards to bring the total number of cards back up to nine. (3:47)
18.1 Seth and David work together taking cards out and returning cards one at a time while Jerome watches.
18.2 Seth " I know, I know, I think. "
18.3 Seth "I need... hmm… different shapes."
18.4 Seth " I don't see any pairs cause there's no different shape." EWX Simple Deduction
18.5 Teacher " Could you change something. Would that help? He had laid down 1 purple diamond and 2 red squiggles.
18.6 Seth changes the squiggles for 2 green ovals.
18.7 Jerome " I got one."
18.8 David " No, no, you can't tell him!"
18.9 Seth rearranges his cards, ending up with 1 purple diamond, 2 green squiggles and 3 red ovals.
18.10 David " He got it!"
18.11 Teacher " Do you think this is a set?"
18.12 Seth " It's different shapes, different colours and"
18.13 David " Different numbers!"
18.14 Seth " Different numbers." EWT Simple Deduction
19.0 The teacher lays down three more cards and makes the total equal nine again. She asks Jerome to try to find a set. (3:49)
19.1 Jerome quickly chooses 1 green diamond, 2 red squiggles, and 3 purple ovals.
19.2 Teacher " Why do you think that is a set?"
19.3 Jerome " Because they're different numbers, different shapes and different colours."
EWT Simple Deduction
20.0 The teacher lays down 3 more cards. (3:50)
20.1 David immediately chooses a set.
20.2 Teacher "Now you've got a big smile on your face - why do you have a big smile?
20.3 David "Different colour, different shape and different number." (David punctuates each statement with a swing of his arm.) EWT Simple Deduction
21.0 The teacher adds 3 more cards to the table once again and Seth after a brief period of time also correctly identifies a set and explains himself as David previously did. (3:50)
21.1 Seth chooses 1 purple oval, 2 red diamonds and then adds 3 green ovals and quickly says "No!"
21.2 Teacher "why do you say no?"
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21.3 Seth "Because those are" (he points to the two cards with ovals and then quickly selects the 3 green squiggles).
21.4 David "Yes, he got it!"
21.5 Seth "They're different shapes, different colours and different numbers"
EWX Simple Deduction
22.0 The teacher adds 3 more cards and gives Jerome a turn. After less than a minute looking over the cards, he correctly chooses a set (1 red diamond, 2 green ovals and 3 purple squiggles). (3:51)
22.1 Seth " I think he got it."
22.2 Teacher " I don't know."
22.3 Seth "Yes he does. I tell ya, he does."
22.4 Teacher " Do you think you have a set?"
22.5 Jerome nods yes.
22.6 Teacher " Why?"
22.7 Jerome " Different shape. Different number." (Seth helps him recite the answer with a smile on his face)
EWT Simple
Deduction
22.8 Seth " Nothing to make it hard for us!"
23.0 Teacher "Okay, we've got 9 cards left."
23.1 David "This is going to be easy."
23.2 David chooses 1 purple squiggle, 2 red diamonds and 3 green ovals.
23.3 Teacher "Now, I 'm not sure about those. Explain to me why you took those cards."
23.4
David "Different shape, different shape, different shape (pointing to
each card in turn) different colour, different colour, different colour and
different number, different number, different number." He collects his
cards EWT
Simple Deduction
24.0 Jerome "I have one in mind." (3:52)
24.1 Teacher "Do you have one in mind Seth?"
24.2 Seth nods.
24.3 Teacher "Okay, let's see."
24.4 Seth chooses "one" 1 red squiggle, "two" 2 purple ovals, "and three" 3 green diamonds.
24.5 Seth "They're different colour, different shape and different number." He takes his time figuring this out.
EWX Simple Deduction
24.6 Seth "It's Jerome's turn."
25.0 There are only three cards remaining (1 green oval, 2 purple squiggles and 3 red diamonds)
25.1 Teacher "Now Jerome, will they make a set?"
25.2 Jerome "They're all different colours, all different numbers and all different shapes."
EWX Simple Deduction
25.3 The students determine how many sets they each "won". (3:53)
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26.0 The teacher asks the students to explain any special "tricks" they knew to help others get really good at this game. (3:54)
26.1 Seth " Oh yes by'. Different colours, different shapes and different numbers." (sing-song)
26.2 Teacher "Some people, when they play, have a hard time. What can I write down to tell them that you guys, special tricks that you guys have noticed when you played?"
26.3 David " Keep you eyes on the cards."
26.4 Teacher " Keep your eyes on the cards?"
26.5 David " And keep the thing in your mind.
26.6 Teacher " What thing in your mind?"
26.7 David " The match for the set."
26.8 Teacher " What do you mean the set? What types of things do you need to keep in your mind?"
26.9 Seth " Ah! My brain holds it in mine. He keeps it in a little tray."
26.10 David looks through the cards he has in his hands and lays down three.
26.11 Jerome "I don't know how I did it. I just thought it up."
26.12
David "There like that. Like keep a set in your mind. Like keep a set in
your mind. Like if somebody doesn't know and it's your turn keep a set in your
mind.
EHT induction
27.0 The teacher collected in all the cards and began a new game. This time, the students were told to raise their hand and state set as soon as they saw one. (3:55)
27.1 All three students started shouting set.
27.2 Jerome chose 1 red oval, 2 green ovals and 3 purple squiggles. As soon as he removed them from the rest of the cards he realized that they were not a set and returned the 1 red oval for 1 red diamond.
27.3 Teacher "Why is that a set?"
27.4 Jerome " Different shapes. Different colours and um...(checks the list) different numbers." EWT Simple Deduction
28.0 The game continued but no cards were added to the ones left on the table. (3:57)
28.1 Seth quickly chose 1 green diamond and 2 purple squiggles but then became "stuck" on choosing a third card.
28.2 When this happened, the teacher placed the two cards back, and let David try to find a set.
28.3 David chose 1 green diamond and 3 red squiggles quickly and then stated out loud, "now two."
28.4 David then changed the 3 for 3 red diamonds and picked up 2 purple squiggles.
28.5 Teacher " Now why do you think those are a set?"
28.6 David (quickly)" Different shape, different colour. Different numbers." EWT Simple Deduction
29.0 The cards were removed from the table, leaving three (1 red oval, 1 green diamond and 3 red squiggles) (3:57)
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29.1 The teacher told the students to decide whether or not the remaining cards could be a set and explain why or why not. Teacher "If you can't make a set you have to tell me why you can't make a set."
29.2 Seth points to the two cards containing 1 item on each and states, " they're not a set cause they're not allowed to match. Look, there won't be any ones. Those two are the same colour" EWT Simple Deduction
29.3 Teacher " You said there won't be any ones, but this is a one isn't it?"
29.4 Seth " And that's a one and that's a three."
29.5 Teacher " So how come you can't make a set? The red oval, the green diamond and the three red swirls."
29.6 Seth "There isn't a two, and they're not different colours and there's no purple." EWT Simple Deduction
30.0 The remaining cards were given to Seth, and nine more were placed on the table.
30.1 David shouted set before all the cards had been placed on the table. (3:58)
30.2 The teacher changed the rules and stated that any card touched must be taken and could not be exchanged until that student was called upon again.
30.3 David " Oh, I got no set."
30.4 David and Seth " Set."
30.5 Seth chose 1 red squiggle, 2 green diamonds and was unable to find a third one.
30.6 The cards were returned and the play continued.
31.0 David chose 3 red ovals, 1 green squiggle and then stated, "And now I needs a two." (3:59)
31.1 David spent some time looking over the remaining cards. At this point, the first group (Cynthia, Jared and Alicia) came over to watch.
31.2 Teacher " So David, can you make a set using those two cards?"
31.3 David took a card from his previous set containing two purple squiggles but quickly put it back when he realized it didn't complete the set.
31.4 The two cards were returned and the game continued.
32.0 Jerome chose 3 purple diamonds, 1 red squiggle and two green diamonds. (4:00)
32.2 Jared began to shout, "No, no, no, that's not one."
32.3 Alison " That's not a set. Two the same shapes." EWX Simple Deduction
32.4 David " Set! Don't tell me." He chose 2 green diamonds, 1 green oval and began to say, "And now I need a three, a three, a three."
32.5 Alison " Bad choice David!"
32.6 Teacher" Why is it a bad choice?"
32.7 Alison " Because the three would be an oval. (She points to the 1 oval already laid down).
EWT Multi-Step Deduction
32.8 Jared " I got a good set. I know but I'm not telling until my turn."
32.9 David chose 1 purple squiggle, 2 green diamonds, and 3 red ovals."
32.10
Jared " No, no, no."
32.11 Alison "Yeah. That's a set."
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... indicates that there was a pause or break in the students speech. Often this is used when the students stop speaking before they complete a thought.
_ indicates that I left out unnecessary words.
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Note: I used all the cards that were solid in pattern with these two groups. I asked students to find sets that were all alike or all different in shape, colour and number.
SET
0. Teacher goes over the rules of the game and presents sets of three cards for students to decide whether or not the cards could be a set. ( 3:19 )
0.1 Teacher asks Seren to explain how to make a set.
0.2 Seren "How you make a set is, they have to be all the same, like, the jack of hearts and the jack of diamonds, then that would be a set except they have different symbols." EHT analogy
0.3 Teacher "Okay, if you were making a set with just 2 cards."
0.4 Leonard "One would be a diamond and one would be a heart."
1.0 Teacher begins to start game and indicates that she will lay out nine cards. (3:23)
1.1 There is discussion about how many sets there could be.
1.2 Preston: "So out of nine of all of them, we could end up having two sets of one."
1.3 Leonard: "Yeah we could."
1.4 Preston: "We could have like two sets like in nine the sets have to be three so there could be ... in nine there could be like two sets."
1.5 Leonard: "Yeah, there could be three of them cause 3, 6, 9. There could be three pairs."
EHT Simple Deduction
2.0 Seren states that a set could have just two cards. (3:24)
2.1 The teacher lays out 15 cards and has the students figure out how many different types of cards there are for each attribute.
2.2 Students agree that there are three different colours, numbers and shapes.
3.0 The teacher begins lays down nine cards and asks students to look for a set. (3:25)
3.1 There is a brief discussion about the fact that the game is just being played for fun by Preston.
3.2 Preston: " We don't have to be real, real serious.
3.3 Teacher: " It's just a game."
3.4 Preston: " It's just to have fun. It's just to see what we can do...cause some people have a hard time finding sets."
4.0 The game begins.
4.1 Preston almost immediately indicates that he has a set but then says, "wait."
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David A Reid
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4.2 Both Seren and Leonard indicate that they have found sets.
4.3 Leonard chooses 3 green ovals, 3 green squiggles and 3 red diamonds.
4.4 He is asked by the teacher to explain why he thinks the three cards he chose are a set.
4.5 Leonard: " They're the same numbers. They're the same number and they have to be all different or all the same. But one isn't the same colour." EWT Simple Deduction
4.6 Teacher: " So is it a set?"
4.7 Leonard: "No."
4.8 Teacher: " So why did you say it's not a set?"
4.9 Leonard: "I heard Preston say it so I..."
5.0 Seren questions the conditions to clarify the rules. (3:27)
5.1 Seren: " Do they have to be exactly the same colour, exactly the same shape and exactly the same number?"
5.2 Teacher: "They could be all the same or all different."
6.0 The teacher asks Preston to try to find a set.
6.1 Preston examines the cards and notes that he needs another card to make a set.
6.2 Preston takes out the following cards: 1 green squiggle, 2 green squiggles and 2 red squiggles.
6.3 Preston: "If one of these (pointing to the 1 green squiggle and the 2 red squiggles) were purple see if you add another one of those right there (pointing to the one green squiggle)and that changed (2 red squiggles) to green, it could've been a set. Or if...or if...or if like one could be like purple, red, and green but like the same number." EWX Multi-Step Deduction
6.4 Preston's cards are returned with the others and the game is continued.
7.0 Seren indicates that she has found a set and takes out 1 red squiggle, 1 purple diamond and 1 green oval. (3:28)
7.1 Seren explains her choice: " Because they're all different!"
7.2 Teacher: " They're all different..."
7.3 Seren: "Everything!"
7.4 Teacher: " Let's check and see...They're all a different colour, they're all a different shape and they're..."
7.5 Seren: "same number!" EWT Simple Deduction
8.0 The teacher removes the three cards in Seren's set and replaces them with three new cards. (3:29)
8.1 Preston once again shouts "set," but then quickly changes his mind and says, "wait."
8.2 Leonard is asked to check for a set.
8.3 Seren: "I see a set already, oh no, I might not."
8.4 Leonard chooses 2 green ovals, 2 red squiggles and 2 purple ovals.
8.5 Leonard: "They're all different colours and ….
8.6 teacher: "Okay you have two green ovals, 2 red squiggles, 2 purple ovals. Okay, so you said they're all different colours."
8.7 Preston: "All different shapes."
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8.8 Leonard: "And they're all the same number."
8.9 Preston: "Actually, no. Look two the same shapes and one different shape."
8.10 Leonard: " There's none there to make one. None can make one." EWX Simple Deduction
9.0 The teacher asks the students to explain why they think it is impossible to make a set from the cards that have been placed on the table. (3:30)
9.1 Leonard: " Cause all of them are different." Simple Deduction
9.2 Preston: " I know why. There's two that are the shape. That one' purple (2 purple ovals) and that one's green (2 green ovals) and they're the same shape. This one (2 green squiggles) is a different shape. And they have to be all a different shape or they have to be all the same shape. But there's two that are the same shape and two ones (pointing to 2 green squiggles) aren't the same shape." EWT Multi-Step Deduction
9.3 Teacher: "Can you make a set using other cards?"
9.4 Leonard: "Why don't you add in three more and see if that would work?"
9.5 Leonard: "I don't see anything else."
9.6 Seren: "Neither do I."
10.0 The teacher redirects the students to check again and see if it is possible to make any other sets. (3:30)
10.1 Preston: "Too bad there wasn't...this (3 red diamonds) would be this, and I would have this (3 green squiggles) and if I have 3 squiggles (pointing to a card that has 3 green ovals) then I'd be able to have a set."
10.2 Leonard: " No because they're all different but this is red."
10.3 Preston: " No but I said if I had a red one, you said they could be all different colours so I could, if I had a red one, a green one and a purple one then I'd have a set. EHT Multi-Step Deduction
11.0 The teacher adds in three more cards to make the total number of cards laid down twelve. (3:31)
11.1 Preston calls "set" and chooses 2 green squiggles, 2 red diamonds and 2 purple ovals.
11.2 Preston: " All different shapes and all different colours." Simple Deduction
11.3 Leonard: "And all the same number." EWX Simple Deduction
12.0 Preston's set is removed from the table leaving nine cards. Seren immediately shouts set. She chooses 1 red diamond, 1 purple squiggle and 1 green squiggle.
12.1 Preston: " Yeah, Seren got a set. She got a red one."
12.2 Leonard: " It can't be a pair. Two are the same and one's different. It can't be a set. They got to be the same. Yeah, cause two of them, it wouldn't work. If that one (1 green squiggle) was a different one, she would of got a set."
EWX Simple Deduction
12.3 Teacher: " So if that was a ..."
12.4 Seren: "Oval."
13.0 The cards are put back (3:33)
13.1 Leonard: "I think I see a set. (3 green squiggles, 3 red diamonds and 3 green ovals.)
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13.2 Seren helps Leonard move the third card (3 green squiggles). Simple Deduction
13.3 Leonard: "Hey how did you know I was gonna take that card?" Simple Deduction
13.4 Leonard: " They're the same number and they're different but they're not the same colour (he says this while pointing to the 1 red card).
13.5 Preston: "One red, two green, that can't be a set."
13.6 Leonard: "There's no more sets. There can't be. EWX Simple Deduction
13.7 The teacher asks why Leonard feels there can not be anymore sets.
13.8 Leonard: "There's no more sets. I don't see none."
13.9 Teacher: "How can we be sure there are no more sets?"
13.10 Seren: "You can't be sure."
13.11 Leonard: " There is no more sets."
13.12 Preston starts picking up cards (1 green squiggle, 1 red diamond and 1 purple squiggle) "All different shapes. All different colours." Simple Deduction
13.13 Leonard: " Yeah but two are the same." (He states this while pointing to the cards with squiggles on them."
EWX Simple Deduction
13.14 Seren: "Those are the ones I took."
13.15 Preston: "Oh, yeah." Preston quickly put the cards back.
13.16 Teacher: "Wait Preston, now let me see what you did. Why do you say they were no good?"
13.17 Preston: "For a minute I was like, o, yeah, but then I noticed these were both" (points to 2 cards with squiggles)
14.0 Seren states she has found a set and takes the 2 green ovals, 2 red squiggles and 2 green diamonds. (3:34)
14.1 Seren: "I think."
14.2 Preston: " Nope cause two the same colours and one red." Simple Deduction
14.3 Seren: "But they're both different shapes." EWX Simple Deduction
14.4 The teacher comments on the attributes and the cards are put back.
14.5Eric: " I think you should put in three more cards cause it's not working."
14.6 Seren: "Egads! That's hard."
14.7 Teacher: " It is hard isn't it."
14.8 Leonard: " I think you should put out three more cards."
14.9 The teacher states that she will add three more cards if the students can explain why the nine cards on the table will not make a set.
14.10 Leonard: "Cause they're all different."
14.11 Seren: "I feel like the brain."
14.12 Preston: "They're all different like they're - say if we had these there would be 2 of these (he points to 1 purple squiggle, 1 green squiggle) and one of those (1 red diamond). So we wouldn't be able to have a set there. And if we had ...there's only one diamond in three and" (he leaves the diamonds and chooses three completely different cards.)
EWT Simple Deduction
14.13 Preston takes 2 purple squiggle, 2 green ovals and 2 green diamonds. "You can't say that. You can't say that so...
14.14 Teacher: "Why can't you say that one?"
14.15 Leonard: "We got to put in three more cause there's no more."
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15.0 The teacher adds three more cards to create a total of twelve cards. (3:36)
15.1 Seren chooses three cards (3 green ovals, 3 red diamonds and 3 purple squiggles).
15.2 Preston: "Ah, she got a set."
15.3 Leonard: "Different shape, different colour but the same number." (Checks the list as he explains.)
15.4 Teacher: "And that's a set why?"
15.5 Seren: "Because they're all different but they're the same number."
15.6 Teacher: " They're all different how?"
15.7 Seren: "They're all different colour and shape." EWT Simple Deduction
16.0 Seren's cards are removed, leaving nine cards on the table. Leonard spends some time looking over the cards. He chooses a pair (1 purple squiggle and 1 green squiggle) and put them back and then (1 green squiggle and 1 red diamond). Finally he states, "I can't figure it out. There's no more pairs there." (3:37)
16.1 Leonard: " I don't see any more pairs."
16.2 The teacher informs the group that there is at least one set amongst the cards on the table.
16.3 Seren chose 1 green diamond, 1 red diamond and a purple squiggle.
16.4 Preston: "Two diamonds!"
16.5 Leonard: "And one squiggle."
16.6 Seren: "Two diamonds and ah..."
16.7 Seren: "Not right."
16.8 Teacher: "Why is it not right?"
16.9 Seren: "Because there's two the same shapes."
16.10 Seren returns the cards and the students try to locate a set.
17.0 Seren tries again. This time she chooses 2 green ovals, 2 green diamonds and 2 purple diamonds. (3:38)
17.1 Preston: "There are two same colours and one different." EWX Simple Deduction
17.2 The teacher reminds the group that there is at least one set there. The students make several attempts to find it but are unable to do so.
17.3 The students continue to pick cards, trying to put them into sets. The group quickly point out why chosen cards can not be a set.
17.4 Teacher: "One thing that helps me is, I should pay attention to the cards with the most of a certain thing. And then I look at other things. For instance, if I saw a whole bunch of red I would look at all of the red and then go 'can I make a set with those?' Then if I saw a whole bunch of twos then I would look to see if I could make a set with those."
18.0 Seren: "I see one, I see one. Maybe not." (1 green squiggle, 2 green diamonds and 3 green squiggles.)
18.1 Teacher: "Okay let's see. All the same…"
18.2 Leonard: "No, same and same (pointing to the squiggles) and that one's different." EWT Simple Deduction
18.3 Seren: "Oh, (she looks over the other cards) where is…?"
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Psychology of Students' Reasoning in School Mathematics
David A Reid
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18.4 Leonard: "You don't have 3 ovals."
18.5 Seren: "Actually you're kinda wrong" (she switches to 1 red diamond).
18.6 Teacher: "Let's give Preston a chance."
18.7 Preston quickly chooses 2 green diamonds, 2 green ovals and 1 green squiggle.
18.8 Preston: "All different shapes"
18.9 Leonard silently points to the 1 green squiggle.
18.10 Preston: "Oh."
18.11 Teacher: "Why are you saying oh?"
18.12 Preston: "I don't …" (The cards are put back.)
18.13 Leonard asks if it's his turn. He looks over the cards, shakes his head. "There's no set there."
18.14 Teacher: "There's at least one there."
18.15 Leonard selects the same cards as Preston.
18.16 Preston: "That's the set I just did Leonard."
18.17 Leonard: "Yeah."
18.18 Seren: "That's a set."
18.19 Leonard: "Yeah, they're all the same colour and they're one, two all different numbers."
18.20 Preston: "But look there's 2 twos and 1 one." EWX Simple Deduction
18.21 Seren: "Beat that Leonard."
18.22 Leonard: "I took the wrong one."
18.23 Seren: "I got it." (She picks 3 purple squiggles, 1 green diamond and 2 green ovals.)
18.24 Leonard: "and different numbers."
18.25 Seren: "Oh."
18.26 Teacher: "Why do you think that's a set?"
18.27 Seren: "I don't think it’s a set. I was just, thought" (switches 2 green ovals for 2 red squiggles).
18.28 Seren: "I'm not ready Leonard. Now let's see." (Switches 2 green ovals for 1 red diamond) "No." (Switches back to 1 green diamond.)
18.29 Preston: "Look" (switches to 1 red diamond).
18.30 Seren: "Different colours - can't be that, can't be that or can't be that."
18.31 Teacher: "Why do you think that Leonard?"
18.32 Leonard: "2 red things and 1 green."
18.33 Teacher: "What does that mean Leonard?"
18.34 Leonard: "It's not a set because they have to be a different colour." EWT Simple Deduction
18.35 Seren: "Okay, Leonard!"
18.36 Leonard: "2 the same."
18.37 Seren: "Don't worry Leonard, I know. But now there's one wiggle."
18.38 Preston: "Yeah, but one wiggle, you have a …"
18.39 Seren: "But I can change it, I can change it!" (She tries several more combinations.)
18.40 Preston: "But you have 2 things with one colour."
19.0 Teacher: "I think Leonard should have a chance."
19.1 Leonard: "I see it."
The
Psychology of Students' Reasoning in School Mathematics
David A Reid
Page B-20
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