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TitleB64, Stephanie third of three interview sessions when she used a case-based method for all heights below and including four-tall Towers problems (work view), Grade 4, March 6, 1992, raw footage

PublisherNew Brunswick, NJ: Robert B. Davis Institute for Learning, 1992-03-06, c1992-03-06

Persistent URLhttps://doi.org/doi:10.7282/T3SF30S5

DescriptionThis one-on-one interview between Researcher Carolyn Maher and Stephanie was an 85-minute discussion that occurred in the 4th grade about two weeks after Stephanie’s second interview with researcher Maher in that same school year. Prior to this interview and after the second interview Stephanie was given a home assignment to solve the four-tall towers problem, selecting from two colors of Unifix cubes, using her “new method” by cases that she had used to partly solve the six-tall towers problem in the prior interview. The session opened with Researcher Maher asking Stephanie what they had done in the previous interview. Stephanie said she was asked to work on the four-tall Towers problem. She stated that her solution was 20 towers using her method. She began to explain her reasoning with the first case of exactly one white. Her explanation contained four parts: 1) the heuristic that she used for generating the towers in this case was by a recursive method of moving the white block one position down from the previous position till she reached the bottom most position; 2) She counted a total of four towers in this particular case; 3) She provided a complete argument by contradiction that if there were more than four towers in this case then the conditions of the problem statement that there could be only four positions for the two colors would be violated; 4) She named the case as “one moving down” (25).

The next three cases were “2 white glued together moving down with two black” (33), “3 together whites with one black” (45) and “4 whites together” (53; names in quotes are in their original form from her written work with transcript line references to each case). She found three towers of exactly two white cubes together, two towers of exactly three white cubes together, and one tower of exactly four white cubes together. For the case of exactly two white cubes together she stated there are no more “because if you take the last one [BBWW] you can’t move them [the two together white cubes] down another one because you’re only using four blocks” (45). She also explained that when she used her case method she found all of the cases with white cubes “stuck together [and] then went back and went two white apart” (41).

For the case of exactly two white apart, she found WBWB and WBBW as one set and BWBW as its own set. She described an opposite strategy (reversing WBWB upside down to get BWBW) to find BWBW because it was the “reverse” of WBWB. When questioned she used a recursive method to generate these three towers to justify there were no more and no missing from the case of exactly two white apart. The next cases they discussed were the towers with exactly four white and exactly no white, which Stephanie immediately found to be a total of one and one, respectively.

Earlier Stephanie described using a color opposite strategy to find the cases of exactly one, two, three, and four black, which led her to the solution of 20 (10 total for the cases of the white cubes, and 10 total for the cases of the black cubes). She did not realize yet that some of these opposite cases were duplicates, since a case of a exactly X number of white cubes is equivalent to the case of exactly four minus X number of black cubes. After completing the cases with white cubes, Stephanie summarized that she found sixteen; one “no white,” four “one white,” six “exactly two whites,” four “exactly three whites,” and one “exactly four whites” (lines 137-149). She still wanted to use the color opposite strategy to find the cases for black. In discussion with the researcher about this, she realized the color opposite towers are already accounted for in the case-based argument.

Then the researcher asked her to use the same argument for the three-tall Towers problem. First Stephanie guessed it would be a total of six, comparing it with the four-tall arrangement of towers, and then seven, conjecturing that not every tower would have an opposite. When she constructed it using the case-based method she found eight. Then she used the same case-based method for two-tall and one-tall Towers problems. Noticing a doubling pattern, she conjectured the solution of the five-tall Towers problem would be 32. She also recalled that this was the number of towers she found in a previous class activity (on February 6, 1992 with Dana as her partner).

The researcher introduced an inductive method to show the growth from one- to two-tall towers. Stephanie showed how from the one-tall towers, two towers two cubes high emerge where the bottom color cubes matched. For example, she explained that black, white (BW) and white, white (WW) are two towers that can be built from a white bottom (W). Then she showed how four three-tall towers (namely BBW, BWW, WBW, WWW) come from a white bottom one-tall tower (W) and four other three-tall towers come from a black one-tall tower (B). She did not consider these sets of four three-tall towers in any particular order. In other words, she did not consider the relationship between the second floor of the two-tall towers and the third floor of the three-tall towers. The researcher noted that she did not pay attention to the second floor and they waited in silence for a few seconds until the researcher posed another problem.

The next two tasks Stephanie worked on was to relate the towers problem to a problem that she was reminded of, namely Shirts and Pants (which she did in the first grade), and to predict how many towers for six- and ten-tall Towers problems using the doubling rule she observed earlier. In comparing and using towers and using her doubling rule, she was able to solve versions of the Shirts and Pants problem up to five-tall. When she found sixty-four six-tall towers, she conjectured that she could use that solution to find out ten-tall without multiplying by two multiple times. She conjectured that she could multiply sixty-four by eight, which is 512 to get the solution for ten-tall. Then she compared this with multiplying by two, multiple times starting with 64. She found 1,024 ten-tall towers. The researcher concluded with praising her idea and to think about it more for next time. The next session Stephanie would be participating in is a group assessment, known as the “Gang of Four,” that can be found in the Video Mosaic Collaborative.

The next three cases were “2 white glued together moving down with two black” (33), “3 together whites with one black” (45) and “4 whites together” (53; names in quotes are in their original form from her written work with transcript line references to each case). She found three towers of exactly two white cubes together, two towers of exactly three white cubes together, and one tower of exactly four white cubes together. For the case of exactly two white cubes together she stated there are no more “because if you take the last one [BBWW] you can’t move them [the two together white cubes] down another one because you’re only using four blocks” (45). She also explained that when she used her case method she found all of the cases with white cubes “stuck together [and] then went back and went two white apart” (41).

For the case of exactly two white apart, she found WBWB and WBBW as one set and BWBW as its own set. She described an opposite strategy (reversing WBWB upside down to get BWBW) to find BWBW because it was the “reverse” of WBWB. When questioned she used a recursive method to generate these three towers to justify there were no more and no missing from the case of exactly two white apart. The next cases they discussed were the towers with exactly four white and exactly no white, which Stephanie immediately found to be a total of one and one, respectively.

Earlier Stephanie described using a color opposite strategy to find the cases of exactly one, two, three, and four black, which led her to the solution of 20 (10 total for the cases of the white cubes, and 10 total for the cases of the black cubes). She did not realize yet that some of these opposite cases were duplicates, since a case of a exactly X number of white cubes is equivalent to the case of exactly four minus X number of black cubes. After completing the cases with white cubes, Stephanie summarized that she found sixteen; one “no white,” four “one white,” six “exactly two whites,” four “exactly three whites,” and one “exactly four whites” (lines 137-149). She still wanted to use the color opposite strategy to find the cases for black. In discussion with the researcher about this, she realized the color opposite towers are already accounted for in the case-based argument.

Then the researcher asked her to use the same argument for the three-tall Towers problem. First Stephanie guessed it would be a total of six, comparing it with the four-tall arrangement of towers, and then seven, conjecturing that not every tower would have an opposite. When she constructed it using the case-based method she found eight. Then she used the same case-based method for two-tall and one-tall Towers problems. Noticing a doubling pattern, she conjectured the solution of the five-tall Towers problem would be 32. She also recalled that this was the number of towers she found in a previous class activity (on February 6, 1992 with Dana as her partner).

The researcher introduced an inductive method to show the growth from one- to two-tall towers. Stephanie showed how from the one-tall towers, two towers two cubes high emerge where the bottom color cubes matched. For example, she explained that black, white (BW) and white, white (WW) are two towers that can be built from a white bottom (W). Then she showed how four three-tall towers (namely BBW, BWW, WBW, WWW) come from a white bottom one-tall tower (W) and four other three-tall towers come from a black one-tall tower (B). She did not consider these sets of four three-tall towers in any particular order. In other words, she did not consider the relationship between the second floor of the two-tall towers and the third floor of the three-tall towers. The researcher noted that she did not pay attention to the second floor and they waited in silence for a few seconds until the researcher posed another problem.

The next two tasks Stephanie worked on was to relate the towers problem to a problem that she was reminded of, namely Shirts and Pants (which she did in the first grade), and to predict how many towers for six- and ten-tall Towers problems using the doubling rule she observed earlier. In comparing and using towers and using her doubling rule, she was able to solve versions of the Shirts and Pants problem up to five-tall. When she found sixty-four six-tall towers, she conjectured that she could use that solution to find out ten-tall without multiplying by two multiple times. She conjectured that she could multiply sixty-four by eight, which is 512 to get the solution for ten-tall. Then she compared this with multiplying by two, multiple times starting with 64. She found 1,024 ten-tall towers. The researcher concluded with praising her idea and to think about it more for next time. The next session Stephanie would be participating in is a group assessment, known as the “Gang of Four,” that can be found in the Video Mosaic Collaborative.

Math ToolUnifix cubes

Math StrandCounting, Combinatorics

Math ProblemTowers

NCTM Grade Range3-5

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

Forms of Reasoning, Strategies and HeuristicsDirect reasoning, Controlling for variables, Recognizing a pattern, Indirect reasoning, Referencing a previous problem, Making conjectures

Grade Level4

Student ParticipantsStephanie (student)

SettingClassroom

Student GenderFemale

Student EthnicityWhite

Camera ViewsWork view

Date Captured1992-03-06

Local IdentifierB64-03-07-1992-KNWH-WV-INT-GR4-CMB-T4T-RAW