Selected item

TitleB61, Stephanie revisits the five-tall Towers problem (work view), Grade 4, February 6, 1992, raw footage

PublisherNew Brunswick, NJ: , , c1992-02-06

Persistent URLhttps://doi.org/doi:10.7282/T3MG7SQ3

DescriptionThis one-on-one interview between Researcher Carolyn Maher and Stephanie is a 48-minute discussion that occurred in the 4th grade on the day after Stephanie and her partner, Dana, worked in their classroom on building towers of height five, selecting from two colors of Unifix cubes. The session opens with Researcher Maher asking Stephanie how she and Dana worked together on the problem. Stephanie describes how she would build a tower, and Dana would build its “match,’ and vice versa. The “match” was the other tower with opposite colors. Researcher Maher prompts Stephanie to create a convincing argument for why she knows it is possible to create 32 towers with no duplicates. Stephanie explains that she is “building backwards,” and starting with a tower with one cube of a certain color, then two cubes of that color, then three, and so on, referring to it as the “one to five pattern.”

Stephanie creates groups of towers as she moves on to the next type, which has two blue cubes together that start at one level, and move up one spot with the creation of each new tower. Stephanie also says that she would replace each of these so that a yellow cube takes the blue’s place. Stephanie talks about recognizing how some of her towers were duplicates, so she had more than 32. She explains a trial and error approach to try to convince someone that there are no more towers that can be made, because they are either duplicates, or the towers would have to be taller. Stephanie discusses “types” of towers, including towers that had “three in the middle,” “one in the middle” and then reversing those towers. She then says that she took one yellow cube that moved places, from the “top floor” and down one position each time, while checking to make sure she eliminated a tower if it repeated the pattern of a tower that she had already made.

Using Unifix cubes and written representations of the towers, Stephanie demonstrates their previous work, sharing the patterns she noticed, how they resulted in duplicates, and what they did to eliminate them. Stephanie confirms the solution of 32 towers in total by reiterating that they made sure they had no missing or duplicate towers by checking each tower against the other patterns. The researcher advises Stephanie, next time, to develop an “absolute” way of being certain that she has found all of the towers, with no duplicates. Researcher Maher also questions Stephanie about what would happen if the towers were a shorter height, specifically four high. Stephanie hypothesizes that if you add a cube to the height, you may have gotten more towers but since she was subtracting a cube from the height of the tower, you would “probably get less.” She says that some tower types would be impossible to build, like the “one in the middle” since four is an even-numbered tower height, and thus fewer towers are possible. Researcher Maher prompts Stephanie to try to come up with a “nice” way to find the patterns to make it easier without creating all of the towers, where she will have no doubt in her mind that she has all towers and is not repeating or missing towers. The camera view focuses on Stephanie’s work. At the end of the session, Researcher Maher asks Stephanie if this problem reminded her of anything they had done previously. Stephanie described a problem she and Dana had worked on previously, called the “Shirts and Pants Problem” where she attempted to find all possible outfits given three different colored shirts and two different pants options. Stephanie spends a bit of time explaining her process for solving that problem before the session ends.

Throughout the session, Stephanie built towers according to different “categories” of towers. For example, one category of towers was keeping two blues together and moving the “stuck together” cubes down one position for each new tower. She also had towers with reverse coloring that were in the same category, just with opposite cube coloring. She also creates categories of towers for when the two blue (or yellow) cubes are separated from each other. Her solution to the total number of tower that could be created choosing from two different colored cubes, making towers of height five was built from her categorization of the different towers and elimination of towers that fell into more than one category, and so were duplicated.

Stephanie creates groups of towers as she moves on to the next type, which has two blue cubes together that start at one level, and move up one spot with the creation of each new tower. Stephanie also says that she would replace each of these so that a yellow cube takes the blue’s place. Stephanie talks about recognizing how some of her towers were duplicates, so she had more than 32. She explains a trial and error approach to try to convince someone that there are no more towers that can be made, because they are either duplicates, or the towers would have to be taller. Stephanie discusses “types” of towers, including towers that had “three in the middle,” “one in the middle” and then reversing those towers. She then says that she took one yellow cube that moved places, from the “top floor” and down one position each time, while checking to make sure she eliminated a tower if it repeated the pattern of a tower that she had already made.

Using Unifix cubes and written representations of the towers, Stephanie demonstrates their previous work, sharing the patterns she noticed, how they resulted in duplicates, and what they did to eliminate them. Stephanie confirms the solution of 32 towers in total by reiterating that they made sure they had no missing or duplicate towers by checking each tower against the other patterns. The researcher advises Stephanie, next time, to develop an “absolute” way of being certain that she has found all of the towers, with no duplicates. Researcher Maher also questions Stephanie about what would happen if the towers were a shorter height, specifically four high. Stephanie hypothesizes that if you add a cube to the height, you may have gotten more towers but since she was subtracting a cube from the height of the tower, you would “probably get less.” She says that some tower types would be impossible to build, like the “one in the middle” since four is an even-numbered tower height, and thus fewer towers are possible. Researcher Maher prompts Stephanie to try to come up with a “nice” way to find the patterns to make it easier without creating all of the towers, where she will have no doubt in her mind that she has all towers and is not repeating or missing towers. The camera view focuses on Stephanie’s work. At the end of the session, Researcher Maher asks Stephanie if this problem reminded her of anything they had done previously. Stephanie described a problem she and Dana had worked on previously, called the “Shirts and Pants Problem” where she attempted to find all possible outfits given three different colored shirts and two different pants options. Stephanie spends a bit of time explaining her process for solving that problem before the session ends.

Throughout the session, Stephanie built towers according to different “categories” of towers. For example, one category of towers was keeping two blues together and moving the “stuck together” cubes down one position for each new tower. She also had towers with reverse coloring that were in the same category, just with opposite cube coloring. She also creates categories of towers for when the two blue (or yellow) cubes are separated from each other. Her solution to the total number of tower that could be created choosing from two different colored cubes, making towers of height five was built from her categorization of the different towers and elimination of towers that fell into more than one category, and so were duplicated.

Math ToolUnifix cubes

Math StrandCombinatorics

Math ProblemTowers

NCTM Grade Range3-5

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)

SettingClassroom

Student GenderFemale

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 Captured1992-02-06

Local IdentifierB61-19920206-KNWH-WV-INT-GR4-CMB-T5T-RAW