A:
So you got to add a blue.
R3:
So you’re going to add a blue to that one.
A: Uh-huh.
R3:
OK. So this one combines with this one in the sense that-
A:
You add all blues to this one [second term] and all reds to that one [first term]. A
red to all three of those and a blue to all three of those and that’s how you get-
that’s why the bottom number’s X plus 1.
The researcher asked Ankur to follow up this reference to the general case:
R3:
So tell me again with the general one. All right, here’s an N-tall with X reds. [R3
indicates the first term in Figure 2.] And how are you going to get down there?
[R3 indicates the last term in Figure 2.]
A:
You’re going add a red.
R3:
And you go from there [the second term in Figure 2] to there [last term]?
A:
By adding the other color.
3
1
+
3
2
=
4
2
Figure 4: Generating 4-Tall Towers With Two Red Cubes
CONCLUSIONS
Davis and Maher (1997) suggest that students can learn new mathematics by building on
powerful representations (mental or written) with which they are already familiar. Many of the
abstract ideas with which mathematics is involved have concrete early origins, such as building
towers, making pizzas, and finding taxicab routes. Over the years students had opportunity to
build and extend their early ideas and to extend and refine them. The earlier ideas became the
building blocks for the more abstract and sophisticated concepts about counting illustrated in this
report. Michael’s use of notation across interviews over the years suggests that he was not just
recalling a memorized formula but that he was using notation that made sense to him. Michael
talked about “making things into problems.” In doing mathematics, Michael connected symbols
to problem situations with which he was already familiar. He represented these situations with
symbolic notations that had meaning for him. His use of formal notation expressed Michael’s
generalizations of his ideas.
The students located familiar numbers from their work with building towers and making
pizzas in Pascal’s Triangle. They investigated Pascal’s Triangle to explain whether those
problems were related. They explained Pascal’s Identity in terms of the rules for generating
successive answers to the towers and pizza problems. When a standard notation was introduced
and its relationship to Pascal’s Triangle was observed, they expressed a connection between their
personal representations and the standard notation using Pascal’s Triangle. The meaning derived
from their earlier work seemed to guide their exploration into Pascal’s Identity and the
generalization of their earlier ideas seemed to facilitate their use of standard notation. A year
later, when they encountered the Taxicab problem, their understanding of Pascal’s Triangle
became an important representation to detect the structural similarity in spite of the surface
differences. Finally, some years after their last investigation into Pascal’s Triangle, they
continued to maintain an impressive ability to explain and generate its numbers.
Our research is abundant with examples showing that, over the years, this community of
students built ideas that came from extensive personal experience. The personal experience was
Convention