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B66,Combinatorics problem,Grade4,towers five tall,March 6,1992,Workview-Raw [video]. Retrieved from https://doi.org/doi:10.7282/t3-84hw-y657
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TitleB66,Combinatorics problem,Grade4,towers five tall,March 6,1992,Workview-Raw
PublisherNew Brunswick, NJ: , 1992-03-06, c1992-03-06
Persistent URLhttps://doi.org/doi:10.7282/t3-84hw-y657
DescriptionThis summary has been retrieved from Sran (2010):
“During this third individual interview, Milin discovered his “family” strategy. This helped him with the global organization of his towers of various heights and assisted him in making personal meaning for the doubling rule for the number of tower built when choosing from two colors. The same was true for the pattern for the towers when selecting from three colors. Since Milin still appeared to be unsure about his doubling and tripling rule working for towers of all heights, R1 assigned these two problems to Milin to work on again.
Milin started his third interview by recording his results from his earlier work. He recorded his towers of different heights (3, 4, 5-tall) on a piece of paper.
Milin then built four two-tall towers when choosing from blue and black cubes and was able to explain his understanding of the doubling pattern by reasoning that each tower could be turned into two new towers in the next height by adding a blue cube on top for the first tower and then adding a black cube on top for the second tower. He extended this by explaining that any two-tall tower could be made into a taller tower using this approach. He referred to all towers originating from a one-tall black tower as one “family” and those originating from a one-tall blue tower as another “family”. This family strategy provided a global organization scheme for Milin. Milin was able to conclude that this strategy would also work for towers that were four-tall and five-tall. Milin was unsure about this strategy working for six-tall towers. During his work at home on six-tall towers, Milin was only able to construct 50 towers, not 64 towers as suggested by his new strategy. During this interview, Milin made several conjectures as to why the “family” strategy would not work for six-tall towers.
As Milin continued to build six-tall towers from his five-tall towers utilizing his “family” strategy, he seemed to be doubting his earlier conclusion that the pattern of doubling would suddenly breakdown for six-tall towers. This interview concluded with Milin’s summarization of his results from his three-tall towers choosing from three colors.
During this interview, Milin was able to refine his justifications and reasoning even further and was able to provide a convincing argument for the doubling pattern which he had noticed as he was recording all his results in the beginning of the interview. He admitted that this written work had prompted him to recognize the doubling pattern which resulted in the development of his “family” strategy. Even though Milin understood how each tower turned into two new towers, he continued to use his earlier strategy of opposite by color to monitor his work throughout this interview.”
The methods for building towers, monitoring towers and the types of reasoning and justifications used by Milin during his third interview are summarized in Table 5-6 in Sran’s (2010) dissertation.
Reference:Sran, M.K (2010).Tracing Milin's development of inductive reasoning:a case study(Doctoral dessertation ,Rutgers Graduate School of Education).
“During this third individual interview, Milin discovered his “family” strategy. This helped him with the global organization of his towers of various heights and assisted him in making personal meaning for the doubling rule for the number of tower built when choosing from two colors. The same was true for the pattern for the towers when selecting from three colors. Since Milin still appeared to be unsure about his doubling and tripling rule working for towers of all heights, R1 assigned these two problems to Milin to work on again.
Milin started his third interview by recording his results from his earlier work. He recorded his towers of different heights (3, 4, 5-tall) on a piece of paper.
Milin then built four two-tall towers when choosing from blue and black cubes and was able to explain his understanding of the doubling pattern by reasoning that each tower could be turned into two new towers in the next height by adding a blue cube on top for the first tower and then adding a black cube on top for the second tower. He extended this by explaining that any two-tall tower could be made into a taller tower using this approach. He referred to all towers originating from a one-tall black tower as one “family” and those originating from a one-tall blue tower as another “family”. This family strategy provided a global organization scheme for Milin. Milin was able to conclude that this strategy would also work for towers that were four-tall and five-tall. Milin was unsure about this strategy working for six-tall towers. During his work at home on six-tall towers, Milin was only able to construct 50 towers, not 64 towers as suggested by his new strategy. During this interview, Milin made several conjectures as to why the “family” strategy would not work for six-tall towers.
As Milin continued to build six-tall towers from his five-tall towers utilizing his “family” strategy, he seemed to be doubting his earlier conclusion that the pattern of doubling would suddenly breakdown for six-tall towers. This interview concluded with Milin’s summarization of his results from his three-tall towers choosing from three colors.
During this interview, Milin was able to refine his justifications and reasoning even further and was able to provide a convincing argument for the doubling pattern which he had noticed as he was recording all his results in the beginning of the interview. He admitted that this written work had prompted him to recognize the doubling pattern which resulted in the development of his “family” strategy. Even though Milin understood how each tower turned into two new towers, he continued to use his earlier strategy of opposite by color to monitor his work throughout this interview.”
The methods for building towers, monitoring towers and the types of reasoning and justifications used by Milin during his third interview are summarized in Table 5-6 in Sran’s (2010) dissertation.
Reference:Sran, M.K (2010).Tracing Milin's development of inductive reasoning:a case study(Doctoral dessertation ,Rutgers Graduate School of Education).
Math ToolUnifix cubes
Math StrandCombinatorics
Math ProblemTowers
NCTM Grade Range3-5
NCTM Content StandardData analysis and probability
NCTM Process StandardProblem solving, Reasoning and proof, Connections, Communication
Forms of Reasoning, Strategies and HeuristicsInductive reasoning
Grade Level4
Student ParticipantsMilin (student)
SettingInformal learning
Student GenderMale
Student EthnicityWhite
Camera ViewsWork view
RightsThe video is protected by copyright. It is available for reviewing and use within the Video Mosaic Collaborative (VMC) portal. Please contact the Robert B. Davis Institute for Learning (RBDIL) for further information about the use of this video.
Date Captured1992-03-06
Local IdentifierB66-19920306-KNWH-WV-INT-GRD4-CMB-T5T-Milin-Raw
Related Publication
Type: Related publication
Label: Ed.dissertation references video footage that includes combinatorics,Grade 4:Towers five tall.
Date: 2010
Author: Sran, M.K (Rutgers,the State University of New Jersey)
Name: Tracing Milin's inductive reasoning;a case study(Doctoral dissertation)
Type: Related publication
Label: Ed.dissertation references video footage that includes combinatorics,Grade 4:Towers five tall.
Date: 2010
Author: Sran, M.K (Rutgers,the State University of New Jersey)
Name: Tracing Milin's inductive reasoning;a case study(Doctoral dissertation)