Student-invented numeration systems: Pattern-analysis and mathematical understanding

School Science and Mathematics, Mar 1997 by Elizabeth S Senger

Thirteen fifth graders were given an assignment to invent their own numeration systems, following a unit on bases and a look at early events in the history of numbers. The task presented options that required the students to make decisions (such as whether to use a base, which base to use, design of symbols, etc.), and build a rationale for the elements of their system. Analyses of patterns embedded in their invented systems provided an assessment of student understanding of numeration. The progression of more and less complex thinking related to the student's choice of a base other than 10, consistency of logic throughout the system in words and symbols, rationale for change, and perception of real life examples that would change if the system was adopted. The invention task is presented as another way to make connections.

Looking at mathematics as a dynamic field that continues to grow as mathematicians conduct further research is a worthwhile route to help students understand invention aspects of the field. Invention and discovery play significant parts in constructing knowledge so that learning can take place (Cobb & Steffe, 1983; von Glasersfeld, 1987). Recent research and reform literature support these notions (Grouws,1992; National Council of Teachers of Mathematics, 1989, 1991,1995; National Research Council, 1989). These "visions" include students working on problems meaningfully and exploring processes that more closely reflect those of mathematicians (Lampert, 1995). To accomplish these visions requires a rethinking of the ecology of the mathematics classroom as a laboratory of opportunity for student design, invention, problem posing, and problem solving.

Reimer and Reimer's (1995) use of historical developments and mathematicians' biographies as a starting point for student creativity in mathematics reflects questions of the fifth-grade teacher in this study. Could the history of counting and the invention of multiple numeration systems similarly inspire student creativity? Would an ill-structured assignment, to invent a fantasy numeration system, help students understand the symbolic nature of numeration and the underlying issue of quantification?

These questions inspired the research objective to analyze students' mathematical understandings in a classroom setting in which the pedagogical practice supported learning mathematics with meaning. The focus of this article is on fifth-grade children's construction of mathematical knowledge as they problem solved through an invention of numeration. This culminating activity of a unit that included place value, number bases, and the history of numeration highlights students' work. The assignment demonstrated the students' creativity, their ability as mini-mathematicians, and their personal visions of mathematics.

This article presents an example of an instructional approach and a qualitative analysis of the student products that resulted. The approach allowed the students to view mathematics as a mathematician, without veering too far from the standard school curriculum. Using a history of numeration focus, fifth graders invented symbol/base systems as products of an open-ended classroom project. Their thinking was analyzed from the perspective of their created projects. The results highlighted the students' understanding of numerical units and number groups.

History of Mathematics Background for the Project

Early civilizations relied on finger counting or the use of pebbles or sticks in a one-to-one correspondence to keep track of flocks and property. More extensive counts were made possible by grouping numbers into convenient unit groups. A typical approach to counting found in early civilizations was to decide an arbitrary group-size, then continue the pattern consistently as quantities increased. Words or "names" for numbers, like the symbols, also took on consistent form or pattern. After some number was selected as a base for counting (for example, some number b), names for numbers larger than b were combinations of the number names already assigned. For example, in a base-10 system, there were 10 symbols (0-9) and these were continually repeated in words and symbols to represent larger numbers. Ten was chosen most often as the base number, probably due to the proximity of fingers (Eves, 1990).

Considering the number words in our system, distinct names exist for the numbers through 10. "When we come to 11, we say eleven, which the philologists tell us derives from ein lifon, meaning `one left over,' or one over ten. Similarly, twelve is from twe lif (two and ten)." (Eves, 1990, p. 12) Thirteen through 19 are formed consistently (3 and 10, etc.). Twenty means "two lOs," and 21 represents "two lOs and 1." The pattern continues till the word hundred that, we are told, means "10 times" [10]. Bases other than 10 are also used as primitive bases, such as 2, 3, 4, 5, 12, and 20 (Mayans), and 60 (ancient Babylonians) (Eves, 1990, p. 12)

Despite the base chosen, what was common to all these ancient systems was the use of patterning or repetition of words and symbols in combinations as the numbers increased (Steen, 1990). Peoples who used a distinct name for each number were rare. Some civilizations did exist where the need for quantifying objects was not a major concern. Even today systems such as "one," "two," and "many" exist that suffice to meet the needs of the culture (Plotkin, 1993). However, in numeration systems of any size, patterning plays a significant role in both understanding the underlying structure itself and in communicating number-meaning within the structure.