DescriptionIn 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 are two sheets of paper posted on the board with the following statement:
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 introduces the problem to the group, who are seated in pairs and have bags of Unifix cubes at their desks. The camera in this video follows Dana and Stephanie as they work on the problem and then after some problem solving, there is a full class discussion about the solutions of different groups moderated by Researcher Maher.
At the beginning of the problem solving, Dana builds one tower and Stephanie builds another with opposite colors (i.e., all the yellows in the original tower become red and vice versa). They use various patterns to construct an original tower and then build its opposite. After building an original, they check to see if the tower is a duplicate by comparing the tower to each of the previously built towers. The towers are grouped in pairs by their opposites.
Dana and Stephanie provide an answer of 28 towers to Researcher Maher who prompts them to explain how they know they have them all. Stephanie shows four towers, Dana takes 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 mentions that another group, Jeff and Michelle, have more and Dana says they have 32. Researcher Maher suggests that they check to see if the other group made duplicates. When visiting the other group Dana finds one original tower that she says they did not have.
Researcher Maher asks the students to think about how they will explain their findings to the rest of the class and gives them a few minutes to prepare. She asks groups one by one how many towers they found, at which point there are answers of 32 and 34. Robert and Sebastian say they have found 35. She asks the class if it is possible to have an odd number of towers. Some students say no. One student, Michael, explains that once you build a tower, it must have a color opposite. Another student says 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 explains that you must “duplicate” each tower, which meant one must create the color opposite.
The researcher invites students who think they have 32 towers to check for duplicates in the towers 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. 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 explains to Martino and Maher at different moments that they would never be certain of the total of towers for two reasons: another person can come up with a new pattern, given a larger amount of red and yellow colored cubes and it is not like the Shirts and Pants task where one can picture the different outfits in their minds. Stephanie and Dana continue to build and find 4 more, resulting in 32 different towers.
The students then are 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 describe 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 an all red tower because it was in the beginning of the line for this group. Researcher Maher discusses 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 compares 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 is 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 encourages her to find all the towers with exactly two reds cubes adjacent to each other or “together”, at which point Stephanie realizes the other towers she selected with exactly two red cubes were not adjacent from one another.
Carolyn poses the problem to the class where she wants them 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 gives them 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 get ten total towers with exactly two reds, which Dr. 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 say they can make the opposites with exactly two yellows. Dr. Maher asks how many they have in total so far? The class says 20 towers total. She asks them to imagine the towers with exactly one red and how many there are. Students realized there is 5 and 5 more with exactly one yellow, the opposites. She asks what the remaining 2 towers are? Robert points out that it is the tower with all red or no yellow and its opposite. It ends with the class determining there are 32 towers in total when selecting from two colors.
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