FROM CONCRETE REPRESENTATIONS TO ABSTRACT SYMBOLS
Elizabeth B. Uptegrove and Carolyn A. Maher
Rutgers University
## email not listed ## and ## email not listed ##
We report on the transition from personal representation to formal notation for a group of five
students from a community of students engaged in doing mathematics for several years. The
group was
introduced to standard combinatorial notation after they had already investigated
concepts in counting using personal representations that they built over several years. We
describe the strategies used by the students to make sense of their ideas and report on how they
came to represent their ideas using standard notation as they worked together to share and
connect representations.
THEORETICAL FRAMEWORK
It is an expectation that students learning mathematics eventually become proficient in the
use of conventional mathematical notation. Standard notation offers a common language for
communicating mathematically; appropriate notation can be helpful in recording the important
features of a mathematical problem. Davis and Maher (1997) observe that students who are
provided with varied mathematical experiences build repertoires of representations. These
representations are used for building new mathematical ideas. Given rich and challenging
investigations and ample time to explore and revisit ideas, students have an opportunity to
construct new representations and connect these representations to other knowledge.
According to Muter and Maher (1999), in the process of revisiting earlier ideas, learners
extend and refine their representational strategies, moving from objects to symbols. In studying
the problem-solving behavior of a group of five students who apply their earlier representations
and ideas to make sense of a general solution to a problem using standard combinatorial notation,
we explore the following question: How do students use personal representations in developing
an understanding of standard notation?
METHOD OF INQUIRY AND DATA SOURCES
This research uses archived data from a longitudinal study (Maher, 2002) that has followed
the mathematical thinking of a group of public school students (Ankur, Brian, Jeff, Michael, and
Romina) from first grade through high school (1988-2000) and new data following the same
students through university (2002-2003). All sessions were videotaped, most with two cameras,
one following the movements of students and the other following their written work.
Videotapes, student work, and researcher notes provide the data for the analysis. Summaries
were made of all sessions, and they were coded for critical events (events related to students’
representations and use of standard notation). All critical events were transcribed and reviewed
for accuracy.
Students worked on the following three problems (with variations and extensions) over
several years. Limitations in space prohibit a report on students’ initial work on these tasks. For
detailed reports of their initial experience with these investigations, see Maher and Martino
(1996), Maher and Kiczek (2000), Kiczek, Maher, & Speiser (2001), and Powell (2003). This
paper focuses on how these students recalled and continued to extend their earlier work for some
time after the after-school sessions were concluded.
1. The Pizza Problem: Students were asked to find how many pizzas it is possible to make
when there are various numbers of toppings to choose from. C(n,r) (the r
th
entry in the n
th
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Convention