Text

doi:10.7282/t3-bv69-dj55

Task Statement: Pizza selecting from 4 different toppings.

Students may revise and enhance their solutions and representations when they are observed working on well-defined problem tasks, given opportunities to speak about their ideas, build models to represent ideas, and have opportunities to receive input of their ideas from others. You will see Brandon revise his guess and check method and use organization by case to build towers 4-tall using 2 different color cubes.
Teacher intervention can also play a critical role in children’s successful proof making. This involves encouragement of student ideas, observing how they represent ideas by analyzing students working together and presenting their ideas. The questioning of students and seeking more details about their thinking are also helpful teacher interventions. In this Analytic, you will observe Researcher Martino’s questioning, and encouraging feedback Prompted Brandon to build a convincing representation.
Brandon, a 9-year-old 4th grade student developed a strategy for accounting for the number of all possible pizzas that could be made selecting from 4 different toppings (Maher & Martino, 1998). Brandon used the notation 0 to represent the absence of a topping and 1 to represent the presence of a topping. He used an inclusive organization by cases to account for pizzas with no topping, one, two, three, or all four toppings. Brandon made a connection between the pizza with 4 toppings to the problem of building 4-tall towers selecting from 2 colors, providing an example of a young student recognizing an isomorphism between the two tasks (Greer & Harel, 1998; Maher & Martino, 1998). Brandon and his partner, Justin, initially built towers 4 tall selecting from red and yellow plastic cubes using opposite pairs (a different color in respective position). In a later interview about his pizza solution, Brandon could not remember the name of his partner, however, he did remember how they came up with the solution to the tower problem and was also able to match his tower pairs to pizza groups by categories. He indicated that the structure of the tower and pizza solutions are equivalent. Brandon noticed that he could move from one set of representations to the other, saying, “Just do it in groups, how many ways you can make pizza-how many ways you can make towers.”
Brandon provided a convincing argument that accounted for all possible pizzas selecting from 4 different toppings without any duplicates and without leaving any possibilities out as well as making a clear connection to the all possible 4-tall towers using 2 different colors.
Video References
B43, Interview with Brandon about the Towers and Pizza problem (student view), Grade 4, April 5, 1993, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T34B2ZFZ

B44, Interview with Brandon about the Towers and Pizza problem (work view), Grade 4, April 5, 1993, raw footage [video]. Retrieved from https://doi.org/doi:10.7282/T30K26QM
Pizza problem selecting from four toppings, Clip 1 of 1: Brandon and Colin solve the problem and compare their solutions [video]. Retrieved from https://doi.org/doi:10.7282/T33F4N1B
PUP Math Brandon interview [video]. Retrieved from https://doi.org/doi:10.7282/T3VX0FRD

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