Stephanie’s Journey with the Towers (Grades 3-8): A Metaphor for Making Connections

PurposesEffective teaching; Student elaboration; Reasoning; Representation
DescriptionDuring this analytic, I hope to show that "learning is primarily metaphoric - we build representations for new ideas by taking representations of familiar ideas and modifying them as necessary, and the ideas we start with often come from the earliest years of our lives" (Davis, 1984, p. 313). Davis’ idea of teaching was centered on the idea that students should be provided with opportunities to build assimilation paradigms. According to him, assimilation paradigms were created when students used something from their past that they already knew (a tool, a representation, a model, an experience) in order to take in and process new information. The “something they already knew” is an assimilation paradigm (Davis, 1996a, p. 8).

The events will follow Stephanie, a student in the Rutgers longitudinal study of children’s reasoning.* As a third grader, she builds towers 4 cubes tall selecting from two colors of unifix cubes, red and yellow. She builds towers again in the fourth and fifth grade. This forms the foundation for this analytic as the latter events will show how Stephanie, now an eighth grader, uses the towers as a metaphor to make sense of combinatoric notation for selecting a specific number of objects from a set. She then connects this notation for tower choices when selecting from two colors to the first 5 rows of Pascal’s Triangle.

Stephanie’s Journey with Combinatorics

3rd Grade - Stephanie’s journey with the towers problem first began in October of 1990 when she was in the third grade. Her class was given the following problem statement: "How many different towers 4 blocks tall can you build when selecting from blocks of two colors?" At this time, Stephanie was paired up with Dana. The girls built towers at random and then checked to make sure they didn’t have any duplicates. Although they hadn’t come to organize their findings in a way that would account for all possible towers, this form of building random combinations and then checking for duplicates is a valid heuristic for problem solving.

4th Grade - Stephanie’s next experience with the towers problem occurred in February of 1992, when she was in the fourth grade. Asked now to build towers that were five cubes tall selecting from two colors, Stephanie and her partner Dana took a somewhat different approach. They built certain patterns and then their opposites. A month later, Stephanie was part of a group interview entitled "Gang of Four" with Jeff, Michelle, and Milin where she had a chance to justify how she knew that she could account for all possible towers 3 cubes tall when selecting from two colors. During this interview, facilitated by Professor Carolyn Maher, Stephanie was able to justify her solution by using an argument by cases showing the towers with zero blues, one blue, two blues, and three blues.

5th Grade - Stephanie next comes across the towers problem in the fifth grade when she worked on "Guess My Towers." In this activity, students were given the following problem statement: "You have been invited to participate in a TV Quiz Show and have the opportunity to win a vacation to Disneyworld. The game is played by choosing one of the four possibilities for winning and then picking a tower out of a covered box. If the tower matches your choice, you win. You are told that the box contains all possible towers three tall that can be built when you select from cubes of two colors, red and yellow. You are given the following possibilities for a winning tower: (a.) All cubes are exactly the same color; (b.) There is only one red cube; (c.) Exactly two cubes are red; (d.) At least two cubes are yellow. Question 1: Which choice would you make and why would this choice be any better than any of the others? Question 2: Assuming you won, you can play again for the Grand Prize, which means you can take a friend to Disneyworld. But now your box has all possible towers that are four tall (built by selecting from the two colors, yellow and red). You are to select from the same four possibilities for a winning tower. Which choice would you make this time and why would this choice be better than any of the others?" Stephanie recalls the "doubling rule" that Milin first introduced the year before, but she doesn’t remember why it works. She works with her partner, Matt, and her classmates to rebuild her knowledge. A detailed view of Stephanie rebuilding the "doubling" argument can be seen in the VMC clips titled "Building Towers, Selecting from two colors for Guess My Tower" (Clips 1-5).

8th Grade - Stephanie, now in the eighth grade, is interviewed by Professor Carolyn Maher. Maher introduces Stephanie to proper combinatoric notation. Throughout the interview, Maher makes references to the towers problem that Stephanie had worked on several times before. As a result, Stephanie is able to use this metaphor to make connections with the notation and see how it relates to Pascal’s Triangle.

Mathematical Practices

The CCSS mathematical practices are deeply embedded within each event I have chosen. Stephanie’s problem-solving activities exhibit reasoning and proof, communication, representation and the ability to make connections.

Make sense of problems and persevere in solving them - Stephanie displays this in the third, fourth, and fifth grades. She and her partner start by explaining to themselves the meaning of the problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt.

Reason abstractly and quantitatively - Stephanie is able to make sense of quantities and their relationships in the towers problem. She and her partner are able to create a coherent representation of why the "doubling rule" works by starting with towers 1 cube tall.

Construct viable arguments and critique the reasoning of others - When Stephanie explains her solution of the number of towers 3 cubes tall selecting from two colors during the "Gang of Four" interview, she analyzes each situation by breaking them into cases. She starts by creating towers with no blue cubes, then one blue cube, two blue cubes, and so on. She justifies her solution in this manner using an argument by cases.

Look for and express regularity in repeated reasoning - Stephanie, now in the eighth grade, is able to predict the sum of each row of Pascal’s Triangle (knowing the sum of the previous row) and she can also generate the numbers that would appear in the next row (given the previous row). She is able to do this by noticing patterns that could be repeated when looking at consecutive rows.

* The Kenilworth longitudinal study of students’ mathematical reasoning sought to show that building new ideas is a process that comes from old ideas that are revisited, reviewed, extended, and connected. The study also sought to show that building new ideas involved the retrieval and modification of representations of existing ideas. This study, directed by Carolyn Maher, Professor of Mathematics Education and Director of the Robert B. Davis Institute of Learning at Rutgers, followed the same group of children from first grade through college.


Davis, R. B. (1984). The ’Paradigm’ Teaching Strategy. Learning Mathematics: The Cognitive Science Approach to Mathematics Education (p. 311-316). Norwood, NJ: Ablex Publishing.

Davis, R. B. (1996a). Classrooms and Cognition. Journal of Education, 178(1), 3-12.

Ortiz, S. (2014). Problem-Solving: Student Reasoning and Invention of Proof-Like Arguments in the 8th Grade. Unpublished Masters Project.

"Standards for Mathematical Practice." Home. N.p., n.d. Web. 24 Apr. 2014.
Created on2014-04-15T21:35:46-0400
Published on2015-06-18T13:56:33-0400
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