A Natural History of Mathematical Gesture

Laurie D. Edwards

St. Mary’s College of California

The spontaneous gestures exhibited by fourteen undergraduate students taking a mathematics course for prospective elementary school teachers were analyzed, utilizing the work of David McNeill as a starting point. The analysis determined that the gestures fell into four types: iconic-physical, iconic-symbolic, metaphoric, and deictic. McNeill’s hypothesis that mathematical gestures have distinctive, semi-conventional forms received mixed support, with gestures related to specific fractions having the most structure in common.

Introduction

Within the field of cognitive science, the pursuit of the understanding of human nature

was initially based on certain assumptions. These included the assumption that cognition

consists of the manipulation of symbols according to formal rules, that the mind is

fundamentally a computational entity, and that the surest path to understanding human

thinking is to build computational models that also process symbols according to formal

rules. Underlying these assumptions was a more foundational one, inherited from

Descartes: that the mind is separable from the body. That is, in order to understand

human cognition, the fact that human brains reside in bodies is irrelevant. Equally

unimportant is the fact that these bodies have evolved over hundreds of thousands of

years, by coping with a range of environmental challenges within certain fixed physical

constraints (the existence of gravity, the basic bilateral symmetry of mammalian bodies,

etc.). In the “classical” cognitive science view, the senses function only as conduits for

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bringing “input” to the mind’s programs, and the motor system functions simply to carry

out the results of the mind’s processing of information.

During the past two decades, an alternative paradigm has emerged that challenges

the Cartesian assumptions of early cognitive science, and that instead acknowledges the

embodied nature of cognition (Varela, Thompson & Rosch, 1991). This new paradigm

sees the sensorimotor system as an essential aspect of cognition; as Varela states:

“Embodied entails the following: (1) cognition dependent upon the kinds of experience

that come from having a body with sensorimotor capacities; and (2) individual

sensorimotor capacities that are themselves embedded in a more encompassing biological

and cultural context...sensory and motor processes, perception and action, are

fundamentally inseparable in lived cognition, and not merely contingently linked as

input/output pairs” (Varela, 1999, p. 12).

Recently, research into the relationship between physical gesture and language

has added a new dimension to the embodied cognition paradigm. According to work in

this area, not only is it the case that one must consider the connections between the body,

the mind, and the world, but that human gestures form an integral part of language and

thought. Indeed, there is one school of thought that holds that gesture preceded and

scaffolded speech in human evolution, and evidence from neuroscience indicates that the

same areas of the brain are involved in the expressive use of gesture and oral language

(Corballis, 1999).

Foundational work on gesture and language has been carried out by psychologist

and linguist David McNeill (1992, 2000). McNeill has proposed that spoken language

and gesture form an integrated system of communication; that is, that “gesture and speech

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arise from a single process of utterance formation” (McNeill 1992, p. 29). In contrast to

theories that see gesture as in some way epiphenomenal to speech, as embellishments, or

even as “translations” of previously conceptualized verbalizations, McNeill holds that

speech and gesture are produced in concert in the mind, and act together to express the

same intended meaning. McNeill offers several kinds of evidence for this hypothesis,

including the neurolinguistic findings described above, but also the fact that gesture and

speech are closely synchronized in time and meaning, that gesture occurs only during

speech, and that gesture and spoken language develop together in children (ibid.).

McNeill points out that spoken language and gesture have different yet complementary

characteristics that allow them to synergistically support effective communication of

meaning. Speech, on the one hand, is linearly segmented; it takes place one word at a

time, with the order of the words playing an important, if not essential, role in the

meaning of the utterance. Meaning is broken down into discrete segments, which are

organized hierarchically (a discourse, a sentence, a phrase, a word, a phoneme); viewed

another way, utterances are constructed by putting together these linearly organized

segments, which combine to form more complex structures. In general, language is

analytic, with one meaning (or a cluster of related meanings) attached to a given word,

and a fairly arbitrary mapping between the form of words and their meanings. In sum,

oral language is linearly-segmented, hierarchical, and analytic.

Gesture contrasts with oral language on all these dimensions. Rather than

presenting meaning through a linearly segmented stream, gesture is what McNeill calls

“global-synthetic”; the meaning comes from the motion as a whole, rather than being

built up from simpler parts. In addition, one gesture can convey an entire complex of

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meanings, including aspects such as spatial location, physical shape, speed, and rhythm

that are difficult to convey through words. Gestures are also non-hierarchical and non-

combinatoric; that is, more complex gestures are not built up from simpler ones; instead,

“With gestures, each symbol is a complete expression of meaning unto itself” (ibid., p.

21).

McNeill points out that that gesture can play a mediating role between internal,

subjective imagery, and shared, conventional speech. His central hypothesis is that

“Speech and gesture are elements of a single integrated process of utterance formation in

which there is a synthesis of opposite modes of thought — global-synthetic and

instantaneous imagery with linearly-segmented temporally extended verbalization.

Utterances and thoughts realized in them are both imagery and language” (ibid., p. 35).

Since mathematics itself has a dual identity comprising both an internal, individual

experience based at least in part in imagery, as well as a shared, conventional symbol-

based language, gesture has the potential of illuminating its nature within the human

conceptual system.

Research on Gesture in Mathematics and Science

In recent years, a body of empirical research into the role of gesture in doing, teaching

and learning mathematics has emerged. These studies address topics ranging from the

way young children use gesture in counting (Alibali & diRusso, 1999; Graham, 1999);

teachers’ gestures when providing instruction in problem-solving (Goldin-Meadow, Kim

& Singer, 1999); and how students working together make sense of various kinds of

graphs (Moschkovich, 1996; Reynolds & Reeve, 2002). Similar studies have examined

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gesture in science talk (Crowder, 1996; Roth & Welzel, 2001; see Roth, 2001 for a more

thorough review). The results of these studies have suggested that when there is a

mismatch between gesture and speech, it can indicate readiness to learn a concept, that

students may be able to convey an understanding through gesture before they can do so in

words, and that appropriate (matched) gesture can enhance the effectiveness of spoken

instruction (summarized in Roth, 2001). Additional research has investigated bodily

movement as well as gesture in the context of technology-based tools for understanding

physics, mathematics, and the representation of both through graphing (Nemirovsky &

Noble 1997; Nemirovsky, Tierney & Wright 1998).

Given that the study of gesture in mathematics is still in its infancy, it seems

appropriate at this stage to undertake a descriptive analysis of kinds of gestures observed

in use by mathematics learners and teachers. In the study described in this paper,

prospective teachers (who are, in a sense, both teachers and learners) were interviewed

about fractions. The objective of the work is to describe the type and range of gestures

utilized by the students during these interviews, in a sense, to begin to develop a “natural

history” of gesture in this particular social and mathematical context. A further goal is to

investigate a hypothesis put forward by McNeill in regards to mathematical gestures,

specifically, that within a given mathematical domain, gestures may have a shared, semi-

conventional structure (McNeill, 1992).

Mathematician’s Gestures

McNeill analyzed a videotape of two mathematicians in conversation, and determined

that, unlike other spontaneous gestures, the hand movements that corresponded to various

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mathematical concepts were similar between the two speakers. In addition, the same

gestures were used for the same concepts at different times in the conversation, and a

given gesture appeared to correspond to a single concept (rather than representing a

complex of concepts or events). For example, when discussing the concept of a

mathematical dual, both mathematicians used a gesture in which the hand was rotated

between two distinctive positions. The gestures accompanying the word “limit” are

described by McNeill as “straight-line trajectories followed by ‘end-marking’ (a tensed

stop)” (ibid., p. 166). McNeill stated that similarly distinctive gestures were employed for

concepts such as quotients, factoring, and compactness. His characterization of

mathematical gesture was that it occupies a space between idiosyncratic, spontaneous

gesture and true gesture languages, such as sign language, which share the characteristics

of spoken language.

Methodology

The participants in the research were fourteen prospective elementary school teachers,

sophomores at a small liberal arts college. All were women, approximately 20 years of

age. The students were enrolled in the first semester of a required two-semester sequence

of mathematics courses taught by the author; the content covered in the first semester

included number systems and algebra. The participants were volunteers, and received

extra credit points for participating. Each pair of participants was interviewed twice by

the author, once before and once after the topic of fractions was addressed in class. The

interviews lasted approximately 30 minutes, and were videotaped. The participants spent

between fifteen to twenty minutes of each interview working together to solve a set of

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problems involving addition, subtraction, multiplication, division and comparing

fractions. In the remaining time, the students answered the following questions, posed

orally by the interviewer:

How were you first introduced to the idea of fractions?

Do you remember anything that was particularly difficult about learning

fractions? What about adding, subtracting, multiplying or dividing

fractions?

Have you ever used fractions in everyday life, or in other classes?

How would you introduce fractions to children?

How would you define a fraction to children?

The current analysis focuses on gestures displayed while answering these questions since,

with one exception, the only gestures evident during the problem solving consisted of

pointing to the written work. The goals of the analysis were to examine and categorize

the variety of gestures associated with the students’ discourse about fractions, and to test

the hypothesis that there are distinctive gestures for mathematical objects and processes.

Data Analysis

The data analyzed in this report were taken from the six half-hour interviews carried out

before the course instruction on fractions (the analysis of the remaining six interviews is

in progress). A total of 86 gestures were analyzed. Gestures (also referred to by some

researchers as gesticulations) are defined by McNeill as “idiosyncratic spontaneous

movements of the hands and arms while accompanying speech” (McNeill, 1992, p. 37).

The gestures were classified using a scheme established by McNeill and other researchers

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(McNeill, 1992, p. 76). This scheme has five basic categories; the terms and definitions

below are from McNeill (1992):

Iconic gestures: “bear a close formal relationship to the semantic content of speech” (p. 12)

Metaphoric gestures: “the pictorial content presents an abstract idea rather than a concrete object or event” (p.14)

Beats: “indexes the word or phrase it accompanies as being significant...for its discourse-pragmatic content” (p. 15)

Cohesives: “serve to tie together thematically related but temporally separated parts of the discourse” (p. 16)

Deictics: “pointing movement [that] select a part of the gesture space” (p. 80).

Iconic Gestures

All of the 86 gestures could, in a general sense, be classified into one of three of the

above categories: iconic, metaphoric, or deictic. However, the particular discourse

context seemed to call for an additional category or subcategory. As noted above, iconics,

which refer to concrete objects or events, are distinguished from metaphorics, which refer

to “an abstract idea.” In analyzing the gestures used by the students while talking about

fractions and fraction operations, there were certain gestures that occurred in association

with speech about entities that had both concrete and abstract characteristics, namely,

written mathematical algorithms or procedures. When describing how they learned about

operations with fractions, a number of students exhibited gestures that I came to think of

as “algorithms in the air.” That is, they used their fingers or hands to indicate spatial

positions that would correspond to the relative position of different aspects of a written

algorithm. For example, in describing learning how to add fractions, one student stated,

“I remember learning that you put one under the other.” The still image corresponding to

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this phrase is shown in Figure 1 (the underlined portion of the text indicates the “stroke,”

or most active, phase of the gesture).

Figure 1: “I remember learning that you put one under the other...”

Similarly, when discussing multiplication of fractions, another student said, “I just

remember crossing”. These word were accompanied by a gestured “X” drawn in the air

with one finger (although not the focus of the current analysis, it is to be noted that the

student’s words indicate a certain confusion between multiplication of fractions and

taking the cross-product of two ratios).

These gestures were iconic in the sense that they referred to an actual experience

of the students, and to a written inscription for a mathematical algorithm. However, the

algorithm itself is a symbolic procedure drawn from the abstract world of mathematics.

Thus, I have chosen to divide McNeill’s category of iconics into two subcategories:

iconic-physical and iconic-symbolic. Iconic-symbolic refers to gestures that refer to

written symbolic or graphical inscriptions, and/or to the procedures associated with these

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inscriptions. Iconic-physical corresponds to McNeill’s category of iconics, in which the

referent of the gesture is something concrete or physical.

A number of the instances of iconic-physical gestures arose in response to the

questions about how the students were introduced to fractions, or how they would

introduce them to children. Their responses often referred to concrete hands-on

manipulatives, or everyday objects such as food (“pie” was a favorite). An example of

such a gesture, with the associated speech, is given in Figure 2.

Figure 2: “I think we did, like, just a stick or a rod…”

The student’s gesture in Figure 2 can be seen as indicating the length or extension of a

physical rod; she went on to talk about how she remembered “dividing it in half,” with an

associated cutting gesture (left hand perpendicular to the table). This was a typical iconic

gesture associated with describing fractions in terms of “splitting,” “cutting,” and

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“dividing;” there were 11 instances of this type of gesture. Other iconic-physical gestures

corresponded to objects associated with materials used to teach fractions such as “pie

chart,” “pie pieces,” “boxes,” and “worksheets.”

Metaphoric Gestures

A wide variety of terms and phrases were associated with metaphoric gestures, as might

be expected with gestures corresponding to mathematical abstractions. These terms

included “explanations,” “statistics,” formulas,” “the same as,” and “subtraction,” to

mention only a few. Figure 3 shows an example of one such gesture. It came about while

the student was describing an application of fractions in a high school class, in an activity

called “Fantasy Football.” The gesture pictured came about two seconds into a sequence

that extended over a period of about 15 seconds, when the student described how the

outcome of the game depended on the statistics of the “players” (presumably expressed

as fractions).

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Figure 3: “And according to their statistics and depending

on like whether or not your team was going to win”

One interpretation of this gesture is that the left hand corresponds to a written table of

statistics, with the right hand pointing to it.

An interesting set of metaphoric gestures was found in two students talking about

a situation involving an “extra” amount. In one case, the student was discussing cooking,

and adding “an extra teaspoon.” In the associated gesture, the student tapped the table,

moving from left to right. In the second case, the student was discussing converting an

improper fraction to a mixed number, and said, “If it was more than what the bottom was,

it would become, like, one and – you’d have to figure, there was a lot of subtraction...”

The stroke, or most active, phase of this gesture was a chopping gesture on the edge of

the table, also moving from left to right, with a slight pause on the “one.” The similarity

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of the two gestures suggests the possibility that the left-to-right iteration represents the

idea of “adding on” (a hypothesis that, clearly, would need to be confirmed through

additional research).

Deictic Gestures

There were only three apparently deictic gestures among the 86 analyzed, although none

was the prototypical pointing with a finger or hand. Instead, one gesture consisted of

waving a hand in a circle in the air, while referring to “the other four people.” In the

second case, the student spoke about dividing something “between two different people;”

the associated gesture concluded with her two hands separated at chest height. The final

deictic gesture occurred at the end of a sequence of three gestures, as a student was

describing an initial confusion she had about fractions. The associated speech included

the phrase, “the whole concept of how you can, it can split and split, and this can still be

the same as this.” This phrase was associated with three gestures. The first was an iconic-

physical “chopping” motion, corresponding to the phrase “split and split.” In the second

gesture, associated with the phrase “the same as”, the student’s hands formed open,

downward facing, C-shapes, which she quickly alternated in lifting and dropping onto the

table. This gesture was classified as metaphorical, in that the sameness or equality of two

things was indicated by the same left and right hand shapes, and by the alternation of the

same motion.

Finally, as shown in Figure 4, the second “this” in the sentence was accompanied

by a “placing” gesture toward the right, indicating the location, in gesture space, of one of

the two equivalent objects (fractions).

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Figure 4: “The same as this”

Distribution of Gestures

Table 1 below indicates the distribution of gestures across the four types found among the

data, as well as the percentage of each type within the total set of gestures.

Gesture Type Frequency PercentageIconic-Physical 33 38%Iconic-Symbolic 7 8%

Metaphoric 43 50%Deictics 3 3%

Total 86 100%

Table 1: Distribution of “Fraction” Gestures

As can be seen, the total number of iconic gestures, whether referring to physical or

symbolic entities, is somewhat less than the number of metaphoric gestures. Iconic-

physical gestures form the majority of the iconic type, but about 8% of all the gestures

were iconic references to symbolic algorithms. As noted above, within the experimental

context of interviews about fractions, there were few deictic gestures.

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An interesting comparison can be made with gestures collected in a different

experimental context. Much of the analysis of presented by McNeill is based on gestures

collected during the narratives of university students who were asked to describe a

cartoon they had just viewed (McNeill, 1992). The distribution of these gestures is shown

in Table 2 (after McNeill, 1992, p. 93).

Gesture Type Frequency PercentageIconic 226 42%Beat 134 25%Metaphoric 12 2%Deictic 25 5%None 146 27%Total 543 100%

Table 2: Distribution of “Cartoon” Gestures

As might be expected, within a mathematical context, a much higher proportion of the

gestures was metaphoric. Within the context of telling the story of what happened in a

rather complicated cartoon, there were more iconics, as well as more beat gestures that

help to mark the progress of a narrative.

The “Mathematical Gesture” Hypothesis

As noted above, McNeill has proposed that mathematicians have distinctive gestures for

mathematical terms, and that these gestures are “somewhere on the road to a gesture

language, but not all the way there” (McNeill, p. 164). That is, these gestures are similar

across speakers and “have a more or less constant form,” yet are “apparently

unconscious, not combining with other gestures, and ...not composed of parts” (ibid., p.

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164). One of the purposes of the current research was to investigate whether people

speaking about fractions exhibit distinctive gestures of this type for this concept.

In order to investigate this hypothesis, three kinds of phrases related to the

concept of fraction were compiled. One type consisted of responses to the question

asking for a definition of fraction suitable for children. The second consisted of phrases

that named a specific fraction (e.g., “one eighth”). And the third were uses of the actual

term “fraction” (the latter were generally in response to the questions about situations that

used fractions or difficulties with learning fractions).

Table 3 shows the speech and associated gestures within each of these categories.

The first column gives the initials of the speaker, the second, the spoken phrase, with the

words associated with the stroke phase underlined. The third column gives an abbreviated

description of the stroke phase of the gesture, and the last, the categorization of the

gesture. (Abbreviations for the gesture descriptions are: RH, LH, BH= Right hand, Left

hand, Both hands; C-, L- and S-shapes=ASL hand shapes).

Who Speech Gesture Description Type

DefinitionKG But it's only a piece of - LH, L-shape, cutting motion, palm toward face IPKG a piece of the wh- LH, open L, parallel to table MKG a piece of whatever we're dealing with

that's wholeBH, symmetric open L-shapes, thumbs up, palms facing body

M

KG it's just a portion of LH toward body, slightly curled S-shape, bounced toward body

M

AT a portion of a pie slide LH fingers along edge of table M

Specific fractionsLR like in 1/8ths RH cutting table, fingers back IPAT cause there's a (1/)5th piece RH cutting horizontally in air, chin height IPKP like dividing it in half RH cutting table, LH open-C IPKG it would become, like, 1 and - LH chopping table, L to R MMS 1/4ths and 4 one fifths LH cutting table IP

Term "fraction"KG statistics, a lot …was fractions BH "holding ball", chest high MKG a lot of fractions LH, open-5, fingers up, at table edge MKG take it from decimal and turn it into a

fractionLH slightly open C-shape, wiggled, fingers on table

M

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MB put fractions into percents LH finger pointing to RH C-shape, rotated L to R

M

Table 3: Gestures associated with Fraction Talk

One observation is that most of the students did not use any gestures in giving their

definition of a fraction. In fact, of the five gestures in this category, four were from a

sequence displayed by the same student, and the fifth from a second student. Although

three of the gestures in the first sequence utilized L-shapes for one or both hands in some

way, the gestures were not otherwise similar, nor did they resemble the other two

gestures in this category. At least in the context of defining the mathematical term

“fraction”, it is not possible to say that these data support the hypothesis about distinctive

mathematical gestures.

In the second category, when students are talking about a specific fraction, the

hypothesis received more support. In all five cases, the gestures involved either a static

“cut” perpendicular to the tabletop, or a chopping motion. Finally, when the actual word

“fraction” was used, there were no apparent similarities among the gestures (however,

there were very few instances, and in the third gesture, the stroke phase did not

correspond to the word itself).

Clearly, additional research is needed to more fully explore the hypothesis of

distinctive mathematical gestures in this context. It seems that an iconic physical gesture

of cutting is associated with speech describing specific fractions, but that a variety of

gestures (even by the same speaker) are associated with either the explicit term

“fraction,” or the attempt to define a fraction. There are a number of possible reasons for

the lack of support for the hypothesis. In the first place, the participants in the study

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would probably not be considered “mathematicians.” They were college students, some

of whom were not very confident in their understanding of mathematics. It is possible

that McNeill’s hypothesis applied only or primarily to individuals with more

mathematical expertise, who are accustomed to engaging in discussions about

mathematics. That is, it may be a hypothesis not about mathematical gestures, but about

mathematicians’ gestures. It may also be that the particular interview questions were not

effective in eliciting gestures related to fractions. It would be interesting to examine

further situations in which students worked with and talked about fractions, perhaps

involving collaborative problem solving without the use of pencil and paper. Such a

situation might not only result in more use of gesture, but might lead the participants to

converge, consciously or unconsciously, on similar gestures for the same mathematical

concepts. It is also possible that there were not enough instances of “fraction gestures” to

fully test the hypothesis; the analysis of the remaining six interviews may provide more

evidence on the question.

Discussion

The purpose of the research reported here was to investigate the types of gestures utilized

by college undergraduates in talking about a particular mathematical topic, that of

fractions. The undergraduates were, in general, not strong mathematics students; as

prospective elementary school teachers, however, the topic was important for them to

understand. It is hoped that spontaneous, unconscious gestures can serve as a window

into students’ understanding of particular mathematical concepts and procedures. The

current analysis determined that the gestures fell into four categories: iconic-physical,

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iconic-symbolic, metaphoric, and deictic. Future research will continue to explore

gestures related to fractions as well as other mathematical topics, and will undertake a

deeper analysis of metaphorical gestures in situations involving mathematical talk. A

second goal of the study was to look for evidence for the hypothesis that gestures for

mathematical concepts have distinctive, semi-conventional forms. Although the evidence

for this hypothesis from the current study was mixed, this hypothesis will continue to be

investigated, drawing both on additional gestures from this experiment, as well as data

collected in new situations. The exploration of the meaning and function of gesture in

mathematical thinking, teaching, and learning has only recently begun, and it is hoped

that the “natural history” of gestures related to talk about fractions will constitute a

contribution to this new area of inquiry.

_____________________

The research reported in this paper was supported by the Lasallian Scholars grant

program of St. Mary’s College. I would like to express my appreciation to St. Mary’s

Committee on Teaching and Scholarship for this support, and most particularly, to my

Mathematics 101 students for their participation in the research.

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