Brandon’s Aha! Fourth grader explains an isomorphism between problem solutions for the Pizza and Tower problems

PurposesStudent elaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionGreer and Harel (1998) state that: "We recommend that awareness of structure, including specifically the recognition of isomorphisms, should be nurtured in children as part of the general development of expertise in constructing representational acts" (p. 5).

In this RUanalytic, Brandon, a 10-year old fourth grader, shares his reasoning on how he relates his solution of The Pizza Problem (below) to his solution of The Towers Problem.

The Pizza Problem: A local pizza shop has asked us to help design a form to keep track of certain pizza choices. They offer a cheese pizza with tomato sauce. A customer can then select from the following toppings: peppers, sausage, mushrooms and pepperoni. How many different choices for pizza does a customer have? Find a way to convince each other that you have accounted for all possibilities.
The Towers Problem: Your group has two colors of Unifix Cubes. Work together and make as many different towers four cubes high as is possible when selecting from two colors. The students were asked to develop a way to convince themselves and their classmates that their solution was complete and included no duplicates.

This Analytic depicts Brandon’s discovery of the isomorphic relationship between the solution structure of The Pizza Problem and The Towers Problem.. The interviewer, Researcher Martino, poses questions that ask Brandon to share his reasoning and to accurately portray his representations. Isomorphic reasoning between mathematical problems exists when one problem’s approach can be structurally applied (mapped) to another problem to achieve the solution. Events 1 and 2 showcase Brandon’s thought process on how to approach the Pizza problem (Event 1) and the Towers problem (Event 2). In Events 3 and 4, Brandon explains how he discovered his choices for the pizzas (Event 3) and the towers (Event 4). In Events 5, 6, and 7 Brandon demonstrates his isomorphic understanding; when ask by Researcher Martino to tell how the pizzas relate to the towers, Brandon utilizes an argument by cases where he finds all the pizza and tower choices and then structurally maps the Pizza choices solution to the Towers solution. Brandon’s isomorphic perspective of pizza and tower choices are that: a pizza with no toppings represents a tower with 4 cubes of one color and no cubes of the other color, a 1 topping pizza represents towers with 1 cube of 1 color and 3 cubes of the other color, a 2 topping pizza represents towers with 2 cubes of one color and 2 cubes of the other color, a 3 topping pizza represents towers with 3 cubes of one color and 1 cube of the other color, then ultimately a pizza with all 4 toppings represents a tower with 4 cubes of one color and no cubes of the other color. The chart (created and named a "graph" by Brandon) and the Unifix cube models that Brandon build help him to have visual representations to support his reasoning. As Researcher Martino inquires about Brandon’s thoughts, Brandon gains more insight into the isomorphic relationship of The Pizza and Towers problems. At the end of Event 7, Brandon spontaneously describes (previously incorrect) the isomorphic relationship between a tower of all red cubes (tower of all 1 color) and a pizza with no toppings. This realization gave concise clarity to Brandon’s isomorphic understanding of the two problems, as indicated by Greer and Harel in (1998): "Brandon’s insight into the isomorphism between the two problem situations was not the result of a sudden recognition, but rather the culmination of a lengthy process of construction, mediated by the notational system he had devised, and by explanations of his thinking" (p. 6).
This Analytic exhibits Brandon’s mathematical ability to reason isomorphically, a feat that, in a sense, allows Brandon to be considered a mathematician in his own right. This conviction is supported by Harel and Greer (1998) when they assimilate Brandon’s isomorphic cogitation to Poincaré’s (a French mathematician) mathematical discoveries in: “These are two examples of mathematicians – the first a great French genius, the second a fourth grade New Jersey student – experiencing insights about structural identity underlying what, on the surface, appear to be different situations.”

Greer, B., & Harel, G. (1998). "The Role of Isomorphisms in Mathematical Cognition". Journal of Mathematical Behavior 17 (1), (pp. 5 - 24).

Maher, C. A. & Martino, A. (1998). Brandon’s proof and isomorphism. In C. A. Maher, Can teachers help children make convincing arguments? A glimpse into the process (pp. 77-101). Rio de Janeiro, Brazil: Universidade Santa Ursula (in Portuguese and English).


Pizza problem selecting from four toppings, Clip 1 of 1: Brandon and Colin solve the problem and compare their solutions [video]. Retrieved from Robert B. Davis Institute for Learning. (1993)

B43, Interview with Brandon about the Towers and Pizza problem (student view), Grade 4, April 5, 1993, raw footage [video]. Retrieved from
Robert B. Davis Institute for Learning. (1993)
Created on2015-11-16T13:08:56-0400
Published on2016-08-02T09:52:10-0400
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