The spirit of four: Metaphors and models of number

construction

Maria A. Droujkova

Objectives

Researchers have noted the need for investigation of relationships between different types of

reasoning and number construction models (Confrey & Smith, 1995; Olive, 2001; Pepper &

Hunting, 1998; Steffe, 1994). The goal of this study was to look at the development of number

construction through the lens of metaphor.

In particular, the study investigated the interplay between a largely multiplicative environment and

the development of reasoning within this environment that was significantly different from scenarios

from other studies.

Conceptual framework

Observing young children makes a strong case for viewing mathematical thinking as fundamentally

metaphoric (R. Davis, 1984). Metaphor is the recursive movement between a source and a target

that are structurally similar, both changing in the dynamic process of learning (B. Davis, 1996; R.

Davis, 1984; English, 1997; Lakoff & Johnson, 1980; Lakoff & Nunez, 1997, 2000; Pimm, 1987;

Presmeg, 1997; Sfard, 1997).

For analyzing number construction, I used the counting scheme (Olive, 2001; Steffe, 1994) and the

splitting conjecture (Confrey & Smith, 1995; Lehrer, Strom, & Confrey, 2002). The metaphor that

connects sources of sharing, folding or similarity, and the target of multiplicative one-to-many

actions can be considered the basis of splitting as a cognitive scheme. The metaphor that connects

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the source of counting and the target of the number sequence is the basis of the counting scheme. In

the splitting world multiplicative reasoning develops via grounding metaphors with sources such as

sharing. In the counting world multiplicative reasoning is based on the linking metaphor which

connects interiorized, reversible counting with iterable units (Olive, 2001; Steffe, 1994).

Modes of inquiry

This paper presents a longitudinal case study of reasoning in a child up to the age of five, whose

home environment was restructured to incorporate more multiplicative activities. Researchers often

consider metaphor to be private, unformulated and difficult to study (Presmeg, 1997). Additional

access issues came from the need for a very young subject necessary to trace the beginnings of

number concept development, and from the longitudinal nature of the study. These considerations

pointed to the necessity of a close relationship between the subject of the study and the researcher,

and I invited my daughter “Katya” to be the subject of the study. As a parent, I was in a privileged

position of access to the majority of the details of Katya’s day-to-day life, as well as to the meaning

of her utterances and gestures.

Data sources and evidence

Data for the study came from fieldnotes of observations as a participant-observer; videotapes and

audiotapes of unstructured and semi-structured interviews; photographs of activity settings; and a

collection of artifacts used in activities.

Results

The non-sequential order in which conventional number names first appeared in Katya’s speech

corresponded to multiplicative, rather than counting, actions. For example, the utterance “two twos”

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appeared about eight months earlier than the word “four,” and also earlier than the word “three.”

Appearance of “two threes” in games preceded the use of the words “four,” “five” and “six,” and

appearance of “two fours” preceded the use of numbers greater than four.

In constructing numbers from one to four, Katya used individual (Presmeg, 1997) metaphors based

on instant recognition of the quantity. In these metaphors, the source was an image with a quantity

intrinsically embedded in it, such as “dog’s legs” for “four.” Katya mostly used mixed references for

multiplicative situations, for example, “two dogs” to signify “two times four.” This availability of

two systems of signifiers provided a language necessary to address the asymmetrical nature of the

multiplication models Katya used. For example, in the case of “two dogs” the words underlined the

distinction between sets and set members in the set model of multiplication. Lack of signifiers for

this asymmetry of multiplication models may be problematic and may hinder development of

multiplicative reasoning. Confrey and Smith (1995) note that “a counting number is typically used

to name the result or outcome of a split” (p.75, italics mine).

If learners see the splitting and counting worlds as isomorphic (Confrey & Smith, 1995), they can

understand structures of one world by making parallels with the corresponding structures of the

other world. Children’s structure transfer attempts become especially visible when they differ from

accepted standards. For example, researchers often focus on children inappropriately applying

additive strategies to multiplicative situations (Post, Behr, & Lesh, 1986). Katya frequently tried to

use multiplicative relationships instead of additive. For example, when asked to continue a pattern

of arrays made out of circles: 2 by 1, 2 by 2, 2 by 3, ___ she attempted to iterate the previous array

twice, drawing a 2 by 6 array instead of the expected “2 by 4”. Upon my explanation that a pair of

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circles is added to the array in each step, Katya said, somewhat angrily, that these pictures “are not

real.” Multiplicative relationships were more “real” to her.

In another unexpected example, a square was split into four equal squares, and then each of the

small squares was split into four tiny squares. Katya used words signifying size gradients, such as

“large, small, and tiny,” and “babies and adults,” consistently across different multiplicative worlds.

This metaphor of “growth” united different multiplicative worlds and allowed Katya to compare

their structures, working on what a mathematician would call “powers” or “base systems.” Katya

used the word “spirit” to denote the action in each world, for example, talking about “the spirit of

four” in the split square above. She claimed that if we cut the 4-square piece in four, the result

would be zero. Upon cutting, she was surprised that the result was one square. However, in repeated

activities with the same picture, or with pictures based on other powers from other split worlds,

Katya consistently said that the result of splitting the power base picture would be “zero”, or

“nothing,” even after observing again and again that it turned out to be one.

I hypothesized that these names were expressions of metaphors for the origin, and I told Katya that

researchers call the entity in question “the origin.” We compared the origins of additive and power-

based structures, and Katya felt validated to discover a “real” zero at least at some origin. This

instance of isomorphism between additive and multiplicative worlds helped Katya to build her idea

of the origin as a “superordinate construct” (Confrey & Smith, 1995), whereas the idea was

problematic while she stayed within the multiplicative world.

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Deeper understanding of connections between additive and multiplicative reasoning can benefit

further theory construction in areas such as number construction, ratio and proportion, or

exponential functions. Practitioners can draw on possible uses of metaphors for working with deep

mathematical ideas throughout the mathematical curriculum. Since the majority of studies of young

children are done in additive environments, research of cases developed in a predominantly

multiplicative environment can provide a valuable vantage point for theory development.

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