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TitleB62, Stephanie's and Milin's second of three interview sessions and Michelle's second of two interview sessions revisiting five-tall Towers and other heights (work view), Grade 4, Feb 21, 1992, raw footage

PublisherNew Brunswick, NJ: , , c1992-02-21

Persistent URLhttps://doi.org/doi:10.7282/T3X3523K

DescriptionThis raw footage consists of three separate interviews in one video. The first with Stephanie, the second with Michelle, and the third with Milin.

The interview with Stephanie and Researcher Maher was the second of three individual interviews that occurred in the 4th grade on February 21, 1992. Stephanie brought with her papers of the work she did at home on the six-tall Towers problem: How many different six-tall towers can be made selecting from two different colored Unifix cubes? The interview began with her describing each page of her work, which consisted of a specific group of towers that she drew and labeled with an invented name, a total count, and a total count that includes the opposite set of towers (where an opposite tower was one that has the opposite color in every position as its counter pair). Each cube depicted in her pictures had either a B for blue or an O for orange. The pictures of the towers were drawn in a table format with horizontal and vertical lines delineating the cubes and the towers.

The first page she showed to the researcher depicted a group of six towers that she called “one at a time” with one blue in each position and five oranges in the other positions. She justified this set by directly reasoning about the conditions of the problem, that she “can only make six blocks high towers” and if she built with the blue cube any further down she would “have to add another block”. Above the set of towers was the label “1 at a time = 6 Double opsite [opposite] total = 12” referring to the 6 towers that had one blue and their six opposites, which made a total of 12 towers.

In her second page she had a group of 5 towers that she called “two at a time” and added the word “together” in conversation with the researcher to indicate that the two blue cubes in each tower must be “stuck together”. She described that they were created by starting with two blues at the top and then “cross[ing] over one [position down]” which referred to the next tower with two blue cubes together that are one position down from the previous tower (e.g., the tower following BBYYYY is YBBYYY). Again, Stephanie indicated that the opposite set would give a total of ten and on her paper was written “double opsite [opposite] total = 10”. Similarly, the third page had a group of 4 towers that she labeled as “3 at a time” and their four opposites, totaling 8 towers. She added them all together to get 30 so far.

Stephanie systematically argued about the one, two, three, and four consecutively blue cube groups of towers that the blue cubes cover every possible position in the tower. However, the researcher redirects her attention to a possible duplication in the group of “four [blue cubes] at a time [together]” since four blue cubes together is equivalent to towers with two orange cubes together. Stephanie realizes this saying that “two at a time is also four at a time” and “five at a time…this is the one at a time” and adjusts the total amount of four and five consecutive blue cubes “at a time” towers and their opposites from originally what she thought was 6 and 4 towers, respectively, to 2 and 0 unique towers, respectively. For example, BBBBOO was a duplicate removed from the four consecutive blue cubes towers group because it was also counted in the “opposites” of the two consecutive blue cubes tower group, namely the opposite of OOOOBB. However, Stephanie indicated the tower OBBBBO is not a duplicate because it was “the only one [of the three earlier groups that was created earlier] where the two of them aren’t stuck together” referring to the two orange cubes at the top and the bottom of that tower. Since she found 2 duplicates of the 3 towers that have four consecutive blue cubes, she was left with only one unique tower and its opposite, resulting in 2 different towers with four consecutive cubes of one color total. She used the same reasoning for the towers of five consecutive cubes of one color to show that all 4 of them duplicated the group of towers of one cube of a color (e.g., BBBBBO is in both the “5 [blue] at a time” and the opposite of the “1 [blue] at a time” groups.). Lastly from her system, she had the “six cubes six at a time” totaling 1 and its opposite totaling 2, with the overall six-tall tower count to be 34 thus far.

Moreover, she used other patterns and gave them names, such as “patchworks”, which were two towers with consecutive cubes alternating in color, and “2 separated”, which were 3 unique towers that had a blue cube fixed at the top of the tower and the second blue cube moving up one position from the bottom. The discussion of the patchworks was quick, but the discussion of the “2 separated” included Stephanie recognizing duplicates and a mistakenly drawn row of cubes that made the towers seven-tall. In the end the patchworks and its opposite (2 of them), the two blue cubes separated with the top blue cube fixed and their opposites (6 of them), and the two blue cubes separated with the bottom blue cube fixed and their opposites (6 of them) resulted in 48 towers total thus far.

The next 10 minutes of conversation dealt with two blues separated in other ways than fixed on the bottom or the top. After a difficult time of realizing this, with the help of the researcher she tried to find another two-blue separated tower that is different; however, she was only able to find OOBOBO. After the 10 minutes of discussion, the researcher suggests to Stephanie to explore a similar argument as the one she gave earlier for the one, two, three, four, and five blue cubes together. The researcher provides her with red and white Unifix cubes and Stephanie begins by building WRWRRR, where the white is separated by one red (W and R representing the two blue cubes-separated or four red cubes-separated case). Holding the top W fixed, she moved the second W in each position till the bottom position, resulting in 4 towers. Holding the W in the position right below the top (the 2nd position from the top), she varied the second W in each position starting with a one red separation, resulting in 3 different towers. She continued to control the first W in a position and to vary the second W so that it was at least one red apart, resulting in 2 different towers when W is fixed in the 3rd position and 1 tower when W is fixed in the 4th position. She accounted for 10 total towers and their opposites, recognizing that W in the 5th position would violate the condition that the white cubes must be at least one red cube apart. Towards the end of the interview, she reviewed the following cases of six-tall towers and their respective quantities (these were the cases that she was certain of based on what was negotiated and accepted as a complete argument by her and the researcher): 12 one color cube “at a time,” 10 “twos together,” 20 “twos separated,” and 8 “threes together,” noting to the researcher that the “four and fives together” had duplicate issues that was discussed earlier on in the interview and that there probably was the case of “threes separated” that she didn’t account for yet.

The researcher suggested that she use her “new plan” to solve the problem of four-tall, then five-tall, and then six-tall, knowing that the first two solutions are 16 and 32, respectively, and that the six-tall solution may be in the “fifties”.

On the same day, Michelle was interviewed by researcher Maher following Stephanie’s interview. She was asked to recount the towers problems of various heights. She explained that the four-tall problem solution was less than 32 (the solution for the five-tall problem) and six-tall problem solution was more than 32, but she was not totally sure of either solution. She was given black and white Unifix cubes and was asked to recreate what she remembered from the four-tall Towers problem. In this interview she built six towers in no particular observable order, silently, except to mention that she previously built “designs” and the “opposite of them.” Looking closely at the tower collection she built one could notice that each tower had an opposite pair in the collection. She estimated that she found “like twelve or something” at home and when asked how she was sure she found all possible “designs”, she stated that if she attempted to make more, she would make the same towers she already had. This part of the session is quickly concluded.

Then she was asked to build two-tall towers. She built the correct four different towers selecting from black and white Unifix cubes. She was asked a follow up question on how the two-tall Towers problem was similar to the Shirts and Jeans problem. During the rest of the interview Michelle explained how she thought of the similarities. First, she imagined “two black bottoms and two black tops.” The researcher directed her attention to the reality that her mother probably would not buy the same colored pants, and Michelle agreed. Michelle then changed her explanation that the bottom cubes of the four towers represented black and white bottoms and the top cubes represented black and white tops, where the black and white towers were mixed outfits of the two colors. The researcher (and researcher Alston was observing) then posed her three questions about adding two black and white hats, two feathers, and two belts, which Michelle solved using the three-tall, four-tall, and five-tall towers, respectively. She concluded from the four shirt and pants outfits, there would be 8 when you add two different hats to each of the four shirts and pants outfits, 16 different outfits when you add two different feathers to each outfit, and 32 different outfits when you add two different belts to each outfit, respectively. Her justification for the doubling of the three-item outfit was supported by a drawing depicting a black and white hat (which was represented by two squares, one colored in black and another white) and two sets of lines crossing from each “hat” to each tower. Some of the lines were physically drawn onto the paper and others were gestured by her pen as she was explaining where each hat would be placed. Finally, she was asked how many four-tall, five-tall, and six-tall towers would there be. She used the 8 three-item outfit (recall, the outfits were represented on her paper as towers) and added either to get 16. Continuing in the same fashion, she added sixteen to the 16 four-tall towers to get 32 and thirty-two to the 32 five-tall towers to get 64. The session ended with praise from the researchers that her idea was new and interesting, and she was left with the task to think about how to explain this the next time they meet with her classmates.

On the same day, researcher Alice Alston interviewed Milin for the second time of a three-part series, as with Stephanie. At the onset the researcher asked Milin if he had determined the answer to the question of how many possible different four-tall towers can be made selecting from two colors. He took out the towers and a paper with some of his prior work. He stated there were 16 total and claimed to have changed his mind from the previous answer of 24 in the prior interview. In response to the prompt “let’s organize them in any way that…” he rearranged the towers into opposite pairs. He was asked “How am I going to know that we have every one and that they’re all different?” In addition to justifying the opposite pair arrangements that “all of them are different, but which every way they are the same,” he used cases to show he had them all. He took towers with one yellow and showed how they form a “staircase.” A staircase is a group of towers in which one color of cubes from one tower moves up or down one position in each consecutive tower. He used the staircase pattern to show the existence of 8 towers, four for the case of one red and four for the case of one yellow. He used the staircase pattern as his argument for the case of towers with two red cubes next to each other. He was aware that finding the opposite of this case (two juxtaposed yellows cubes) would create duplicates because he pointed out that the two towers with two red cubes on top (RRYY) and two red cubes on bottom (YYRR) are the same case as the two towers with two yellow cubes on bottom (RRYY) and two yellow cubes on top (YYRR), respectively. Therefore, he only created one tower for the case of two yellow cubes (RYYR) since it was the only unique one in this case. The final case of towers described as “two of each color but they are separated [by a different colored cube]” had two unique towers that he showed were also opposite pairs (YRYR and RYRY). This totaled 6 towers for the case of two red/two yellow cubes. Lastly, the case of “all of one color” had 2 unique towers, equaling a total of 16.

He was then asked to explore the work that was written on his paper from a prior activity. This exploration led to him explaining and justifying how many towers two-tall, one-tall, three-tall, and predicting towers six-tall. Milin guessed “around forty something” for the six-tall Towers problem, explaining that there would be a “bigger staircase” to build. The discussion continued to explore building from towers of one cube to towers of two cubes. Here Milin only showed taking the one yellow cube and putting it on top of the one red cube and vice versa, recognizing that there will also always be two solid (all of one color) towers. Milin voluntarily and spontaneously argued that towers of three colors would be different than two because of the extra color that would create an extra solid tower. After following what he called a “lucky guess” of 8 more towers for the four-tall three-color towers than the four-tall two-color towers, he was prompted to explore the one-tall, two-tall towers, and three-tall towers selecting from three colors. His primary strategies were the use of cases, grouping “pairs of three,” and construction of staircases. With prompts from the researcher to compare the results of the two-tall selecting from both two and three colors and the three-tall towers selecting from two colors, he predicted 17 towers for the three-tall towers selecting from three colors. Finally, he was asked to write and record the work they were doing together in this interview and to work on two tasks at home (the six-tall two color and the three-tall three color tower problems).

The interview with Stephanie and Researcher Maher was the second of three individual interviews that occurred in the 4th grade on February 21, 1992. Stephanie brought with her papers of the work she did at home on the six-tall Towers problem: How many different six-tall towers can be made selecting from two different colored Unifix cubes? The interview began with her describing each page of her work, which consisted of a specific group of towers that she drew and labeled with an invented name, a total count, and a total count that includes the opposite set of towers (where an opposite tower was one that has the opposite color in every position as its counter pair). Each cube depicted in her pictures had either a B for blue or an O for orange. The pictures of the towers were drawn in a table format with horizontal and vertical lines delineating the cubes and the towers.

The first page she showed to the researcher depicted a group of six towers that she called “one at a time” with one blue in each position and five oranges in the other positions. She justified this set by directly reasoning about the conditions of the problem, that she “can only make six blocks high towers” and if she built with the blue cube any further down she would “have to add another block”. Above the set of towers was the label “1 at a time = 6 Double opsite [opposite] total = 12” referring to the 6 towers that had one blue and their six opposites, which made a total of 12 towers.

In her second page she had a group of 5 towers that she called “two at a time” and added the word “together” in conversation with the researcher to indicate that the two blue cubes in each tower must be “stuck together”. She described that they were created by starting with two blues at the top and then “cross[ing] over one [position down]” which referred to the next tower with two blue cubes together that are one position down from the previous tower (e.g., the tower following BBYYYY is YBBYYY). Again, Stephanie indicated that the opposite set would give a total of ten and on her paper was written “double opsite [opposite] total = 10”. Similarly, the third page had a group of 4 towers that she labeled as “3 at a time” and their four opposites, totaling 8 towers. She added them all together to get 30 so far.

Stephanie systematically argued about the one, two, three, and four consecutively blue cube groups of towers that the blue cubes cover every possible position in the tower. However, the researcher redirects her attention to a possible duplication in the group of “four [blue cubes] at a time [together]” since four blue cubes together is equivalent to towers with two orange cubes together. Stephanie realizes this saying that “two at a time is also four at a time” and “five at a time…this is the one at a time” and adjusts the total amount of four and five consecutive blue cubes “at a time” towers and their opposites from originally what she thought was 6 and 4 towers, respectively, to 2 and 0 unique towers, respectively. For example, BBBBOO was a duplicate removed from the four consecutive blue cubes towers group because it was also counted in the “opposites” of the two consecutive blue cubes tower group, namely the opposite of OOOOBB. However, Stephanie indicated the tower OBBBBO is not a duplicate because it was “the only one [of the three earlier groups that was created earlier] where the two of them aren’t stuck together” referring to the two orange cubes at the top and the bottom of that tower. Since she found 2 duplicates of the 3 towers that have four consecutive blue cubes, she was left with only one unique tower and its opposite, resulting in 2 different towers with four consecutive cubes of one color total. She used the same reasoning for the towers of five consecutive cubes of one color to show that all 4 of them duplicated the group of towers of one cube of a color (e.g., BBBBBO is in both the “5 [blue] at a time” and the opposite of the “1 [blue] at a time” groups.). Lastly from her system, she had the “six cubes six at a time” totaling 1 and its opposite totaling 2, with the overall six-tall tower count to be 34 thus far.

Moreover, she used other patterns and gave them names, such as “patchworks”, which were two towers with consecutive cubes alternating in color, and “2 separated”, which were 3 unique towers that had a blue cube fixed at the top of the tower and the second blue cube moving up one position from the bottom. The discussion of the patchworks was quick, but the discussion of the “2 separated” included Stephanie recognizing duplicates and a mistakenly drawn row of cubes that made the towers seven-tall. In the end the patchworks and its opposite (2 of them), the two blue cubes separated with the top blue cube fixed and their opposites (6 of them), and the two blue cubes separated with the bottom blue cube fixed and their opposites (6 of them) resulted in 48 towers total thus far.

The next 10 minutes of conversation dealt with two blues separated in other ways than fixed on the bottom or the top. After a difficult time of realizing this, with the help of the researcher she tried to find another two-blue separated tower that is different; however, she was only able to find OOBOBO. After the 10 minutes of discussion, the researcher suggests to Stephanie to explore a similar argument as the one she gave earlier for the one, two, three, four, and five blue cubes together. The researcher provides her with red and white Unifix cubes and Stephanie begins by building WRWRRR, where the white is separated by one red (W and R representing the two blue cubes-separated or four red cubes-separated case). Holding the top W fixed, she moved the second W in each position till the bottom position, resulting in 4 towers. Holding the W in the position right below the top (the 2nd position from the top), she varied the second W in each position starting with a one red separation, resulting in 3 different towers. She continued to control the first W in a position and to vary the second W so that it was at least one red apart, resulting in 2 different towers when W is fixed in the 3rd position and 1 tower when W is fixed in the 4th position. She accounted for 10 total towers and their opposites, recognizing that W in the 5th position would violate the condition that the white cubes must be at least one red cube apart. Towards the end of the interview, she reviewed the following cases of six-tall towers and their respective quantities (these were the cases that she was certain of based on what was negotiated and accepted as a complete argument by her and the researcher): 12 one color cube “at a time,” 10 “twos together,” 20 “twos separated,” and 8 “threes together,” noting to the researcher that the “four and fives together” had duplicate issues that was discussed earlier on in the interview and that there probably was the case of “threes separated” that she didn’t account for yet.

The researcher suggested that she use her “new plan” to solve the problem of four-tall, then five-tall, and then six-tall, knowing that the first two solutions are 16 and 32, respectively, and that the six-tall solution may be in the “fifties”.

On the same day, Michelle was interviewed by researcher Maher following Stephanie’s interview. She was asked to recount the towers problems of various heights. She explained that the four-tall problem solution was less than 32 (the solution for the five-tall problem) and six-tall problem solution was more than 32, but she was not totally sure of either solution. She was given black and white Unifix cubes and was asked to recreate what she remembered from the four-tall Towers problem. In this interview she built six towers in no particular observable order, silently, except to mention that she previously built “designs” and the “opposite of them.” Looking closely at the tower collection she built one could notice that each tower had an opposite pair in the collection. She estimated that she found “like twelve or something” at home and when asked how she was sure she found all possible “designs”, she stated that if she attempted to make more, she would make the same towers she already had. This part of the session is quickly concluded.

Then she was asked to build two-tall towers. She built the correct four different towers selecting from black and white Unifix cubes. She was asked a follow up question on how the two-tall Towers problem was similar to the Shirts and Jeans problem. During the rest of the interview Michelle explained how she thought of the similarities. First, she imagined “two black bottoms and two black tops.” The researcher directed her attention to the reality that her mother probably would not buy the same colored pants, and Michelle agreed. Michelle then changed her explanation that the bottom cubes of the four towers represented black and white bottoms and the top cubes represented black and white tops, where the black and white towers were mixed outfits of the two colors. The researcher (and researcher Alston was observing) then posed her three questions about adding two black and white hats, two feathers, and two belts, which Michelle solved using the three-tall, four-tall, and five-tall towers, respectively. She concluded from the four shirt and pants outfits, there would be 8 when you add two different hats to each of the four shirts and pants outfits, 16 different outfits when you add two different feathers to each outfit, and 32 different outfits when you add two different belts to each outfit, respectively. Her justification for the doubling of the three-item outfit was supported by a drawing depicting a black and white hat (which was represented by two squares, one colored in black and another white) and two sets of lines crossing from each “hat” to each tower. Some of the lines were physically drawn onto the paper and others were gestured by her pen as she was explaining where each hat would be placed. Finally, she was asked how many four-tall, five-tall, and six-tall towers would there be. She used the 8 three-item outfit (recall, the outfits were represented on her paper as towers) and added either to get 16. Continuing in the same fashion, she added sixteen to the 16 four-tall towers to get 32 and thirty-two to the 32 five-tall towers to get 64. The session ended with praise from the researchers that her idea was new and interesting, and she was left with the task to think about how to explain this the next time they meet with her classmates.

On the same day, researcher Alice Alston interviewed Milin for the second time of a three-part series, as with Stephanie. At the onset the researcher asked Milin if he had determined the answer to the question of how many possible different four-tall towers can be made selecting from two colors. He took out the towers and a paper with some of his prior work. He stated there were 16 total and claimed to have changed his mind from the previous answer of 24 in the prior interview. In response to the prompt “let’s organize them in any way that…” he rearranged the towers into opposite pairs. He was asked “How am I going to know that we have every one and that they’re all different?” In addition to justifying the opposite pair arrangements that “all of them are different, but which every way they are the same,” he used cases to show he had them all. He took towers with one yellow and showed how they form a “staircase.” A staircase is a group of towers in which one color of cubes from one tower moves up or down one position in each consecutive tower. He used the staircase pattern to show the existence of 8 towers, four for the case of one red and four for the case of one yellow. He used the staircase pattern as his argument for the case of towers with two red cubes next to each other. He was aware that finding the opposite of this case (two juxtaposed yellows cubes) would create duplicates because he pointed out that the two towers with two red cubes on top (RRYY) and two red cubes on bottom (YYRR) are the same case as the two towers with two yellow cubes on bottom (RRYY) and two yellow cubes on top (YYRR), respectively. Therefore, he only created one tower for the case of two yellow cubes (RYYR) since it was the only unique one in this case. The final case of towers described as “two of each color but they are separated [by a different colored cube]” had two unique towers that he showed were also opposite pairs (YRYR and RYRY). This totaled 6 towers for the case of two red/two yellow cubes. Lastly, the case of “all of one color” had 2 unique towers, equaling a total of 16.

He was then asked to explore the work that was written on his paper from a prior activity. This exploration led to him explaining and justifying how many towers two-tall, one-tall, three-tall, and predicting towers six-tall. Milin guessed “around forty something” for the six-tall Towers problem, explaining that there would be a “bigger staircase” to build. The discussion continued to explore building from towers of one cube to towers of two cubes. Here Milin only showed taking the one yellow cube and putting it on top of the one red cube and vice versa, recognizing that there will also always be two solid (all of one color) towers. Milin voluntarily and spontaneously argued that towers of three colors would be different than two because of the extra color that would create an extra solid tower. After following what he called a “lucky guess” of 8 more towers for the four-tall three-color towers than the four-tall two-color towers, he was prompted to explore the one-tall, two-tall towers, and three-tall towers selecting from three colors. His primary strategies were the use of cases, grouping “pairs of three,” and construction of staircases. With prompts from the researcher to compare the results of the two-tall selecting from both two and three colors and the three-tall towers selecting from two colors, he predicted 17 towers for the three-tall towers selecting from three colors. Finally, he was asked to write and record the work they were doing together in this interview and to work on two tasks at home (the six-tall two color and the three-tall three color tower problems).

Math ToolUnifix cubes

Math StrandCombinatorics

Math ProblemTowers

NCTM Grade Range3-5

NCTM Content StandardNumber and operations

NCTM Process StandardProblem solving, Reasoning and proof, Communication, Connections, Representation

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Reasoning by cases, Reflecting on past experience, Recognizing a pattern, Considering a simpler problem

Grade Level4

Student ParticipantsStephanie (student), Milin (student), Michelle I. (student)

SettingClassroom

Student GenderFemale

Student EthnicityWhite

Camera ViewsWork view

Date Captured1992-02-21

Local IdentifierB62-19920221-KNWH-WV-INT-GR4-CMB-T5T-RAW