MovingImage

In this one hour and forty-minute unedited video, the fourth-grade class was divided into pairs to work on a Towers problem on February 6, 1992. At the beginning of the session, there were two sheets of paper posted on the board with the following statement:
“Building Towers
Your group has two colors of Unifix cubes for building towers. Work together and make as many different towers as you can that are five cubes high. See if you and your partner can plan a good way to find all the towers that are five cubes high and decide a way to record what you find.”

Researcher Maher introduced the problem to the group, who were seated in pairs and had bags of Unifix cubes at their desks. The camera in this video follows Dana and Stephanie as they worked on the problem and then a whole class discussion about the solutions of different groups moderated by Researcher Maher.
At the beginning of the problem solving, Dana built one tower and Stephanie built the other with opposite colors in each position (i.e., all the yellows in the original tower become red and vice versa). Furthermore, Dana noticed another relationship between four of the towers: two towers are color opposites of each other and if you invert these two towers you can build two more different towers, which will be color opposites of each other also. They pursued grouping towers that had both a color opposite and an inverted opposite. They find three sets (e.g., The first set were four towers with exactly one red cube in the bottom position and exactly one red cube in the top position are inverted opposites, and exactly one yellow on the bottom and exactly one yellow on the top are the color opposites of the pair aforementioned, respectively), but abandoned this organization in the fourth set to pursue grouping by color opposites only. After grouping in these various ways and attempting to build a new tower for at least ten minutes, they began to check if each tower is different from the others by comparing them one by one.
Dana and Stephanie claimed to have 28 towers to researcher Maher who prompted them to explain how they know they have them all. Stephanie showed four towers. Dana took one of the towers and explains that they built a “duplicate” by taking that tower and flipping it upside down to create “a new design”. Maher mentioned that another group, Jeff and Michelle, had more towers and Dana noted to Stephanie that their group had 32. Researcher Maher suggested that they check to see if the other group made duplicates. When visiting the other group Dana finds one original tower that she said they did not have.
Next, researcher Maher asked the class to think about how they will explain their findings to the rest of the class and gave them a few minutes to prepare. She asked groups, one by one, to say how many towers they had found, at which point there are answers of 32 and 34. Robert and Sebastian said they had found 35. She asked the class if it is possible to have an odd number of towers. Some students say no. One student, Michael, explained that once you build a tower, it must have a color opposite. Another student said it makes sense to have an odd number of towers because a person has the choice to make or not make the color opposite. Milin explained that you must “duplicate” each tower, which meant one must create the color opposite.
The students who thought they had 32 towers were invited to check for duplicates of the group that got 35. The class finds the three duplicates. Students get time to work on their own tower collections to check for duplicates and many groups find 32. Again, Stephanie pursued grouping by the towers of four with color and inverted opposites; however, it was again abandoned to be grouped by color opposite pairs only. Stephanie and Dana find 28 and Stephanie explains her certainty to researcher Martino by describing how Dana listened to other groups’ results and encouraged Stephanie to build more than the original set of 28. The towers that Stephanie and Dana built were duplicates and the result was again 28 towers. Stephanie explained to Martino and Maher at different moments that they would never be certain of the total of towers for two reasons: first, another person can come up with a new pattern, given a larger amount of red and yellow colored cubes and, secondly, this problem was not like the Shirts and Pants task where one can picture the different outfits in their minds. Stephanie and Dana continued to build and found 4 more, resulting in 32 different towers.
The students then were invited to listen to Ankur’s method for finding all of the patterns. Ankur and Joe had a different method for generating towers with awareness of potential duplicates. They described a “staircase” pattern, beginning with all red, then having one yellow, two yellows, three yellows, four yellows, all five yellow, and then one red, two reds, three reds, four red. They did not include one all-red tower because it was in the beginning of the line for this group. Researcher Maher discussed how many groups, including Jeff’s and Ankur’s, made towers with exactly one red on each “floor” (or each position) but in the case of Ankur’s group, their group of towers with exactly one red did not include the red on the top or bottom floor because they were already in their “staircase” pattern. Maher compared how Jeff’s group also did this and noticed duplication; therefore, both groups continued their new patterns to the extent of no duplication from the first patterns.
She then asked the students to study their towers with exactly two reds or two yellows, what they look like, and how many there are in total. At this point Stephanie was utilizing the scheme of moving the two red cubes down a position along the tower to create the next tower; however, she misses one with the two reds on the two bottom positions. Maher encouraged her to find all the towers with exactly two reds cubes adjacent to each other or “together”, at which point Stephanie realized the other towers she selected with exactly two red cubes were not adjacent from one another.
Researcher Maher posed the class the problem to find a way to convince her that they have all of the towers by using similar schemes for towers with exactly two, three, and four of the same color cubes. She provides them with an idea to think about the different possibilities of exactly two reds together and how they might know that there cannot be any more ways, imagining the cubes moving up one floor starting from the bottom. They find 4 towers with two reds together. The researcher points out that these are not the only towers with exactly two red cubes. Together as a class, the researcher asks if there can be towers separated by one, two, three, and four floors, at which point Stephanie says no to the latter case. They discuss why there are no more cases. Stephanie and others suggest it is not possible to have four or more yellow cubes separating the red cubes because there are only 5 cubes in total. As a class they do a case based organization exploration for exactly two reds together, one yellow, two yellow, and three yellow cubes apart.
They obtain ten total towers with exactly two reds, which researcher Maher points out to the whole class. She asked if there is a set of towers that they can imagine in their heads and know the total immediately. Some students stated that they can make the opposites with exactly two yellows. researcher Maher asked how many they would have in total. The class claimed 20 towers total. She then asked them to imagine the towers with exactly one red and how many there might be. Students realized there were 5 and 5 more with exactly one yellow, the opposites. She asked what the remaining 2 towers are? Robert pointed out that it is the tower with all red or no yellow and its opposite. The video ends with the class determining there were 32 towers in total when selecting from two colors.

Social science

Mathematics education

APA citation Robert B Davis Institute for Learning(B59),Combinatorics

Observational data

Raw data

Research data

Educational interventions (small group)

Longitudinal data

Qualitative research

Field

doi:10.7282/T30V8HFS

The 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.