Blending across modalities in mathematical discourse
Nathaniel J. Smith
To appear in: L. D. Edwards, F. Ferrara, & D. Moore-Russo, Emerging
perspectives in gesture and embodiment in mathematics. Charlotte, NC:
Information Age Publishing.
When people engage in face-to-face discourse about mathematical topics, they deploy a variety of communicative modalities, including at least speech, gesture, and various forms of symbolicand diagrammatic marking on paper or blackboards. These modalities are not parallel streams: tobe parallel means to never meet, but speakers mix these streams constantly, coordinating distinct
representational resources in a fine-grained way within a single, shared, and meaning-laden space,and it’s in this space that communication and thinking are performed. There are even cases where
the apparent boundaries between modalities break down entirely. To understand mathematical discourse, then, it’s not enough to examine individual meaningful actions – single utterances, gesture
strokes, and so forth. We also need to understand the structure of the space in which they occur,and how this structure is built, maintained, and manipulated through communicative acts. In thischapter, I’ll examine this question through a series of examples drawn from a corpus of naturalistic
video recordings of upper-division and graduate mathematics students working together in small
groups to complete their assigned homework in topology and abstract algebra (Smith, 2003).

1

Real-space blends

The theory of real-space blending was originally developed for the analysis of sign language(Liddell, 2003), but has proven useful in the study of both gesture (Parrill & Sweetser, 2004)
and complex conceptual artifacts like maps and compasses (Hutchins, 2005). This theory builds inturn on the general theoretical framework of conceptual integration theory (Fauconnier, 1997; Fauconnier & Turner, 2002). The core idea of conceptual integration theory is that humans constantly
engage in the creation of new conceptual structures, and that they very often do this by following aparticular recipe: they take two or more existing concepts, identify mappings between their parts,and then blend them together to create a new, indivisible concept whose structure is drawn fromthe input concepts, adjusted as necessary to produce a coherent whole. Such a process seems to beneeded to explain phenomena as diverse as adjective/noun combinations (Sweetser, 1999), if statements (Dancygier & Sweetser, 2009), moral judgements (Coulson, 2001), and the development of
abstract mathematical concepts (Lakoff & Núñez, 2000; Edwards, 2009). Here, though, we will
focus on real-space blends, i.e., those which instantiate a mapping between abstract conceptual1

Figure 1: The logarithmic spiral. . .
structure, and the physical structure of the real environment.1 For example, consider the diagram
drawn in (1):2(1) the logarithmic spiral. . .
draws a spiral on the board, Fig. 1
In physical terms, all the speaker is doing is placing some bits of colored dust on the surface of awhiteboard. But of course the reason he does this is because he is setting up a coherent mapping
between this dust’s particular spatial arrangement, and a certain mathematical object. This object– the logarithmic spiral – consists of those complex numbers which can be written as eθ eiθ forsome real number θ , and if we conceptualize the set of all complex numbers as forming a twodimensional plane (itself a complex many-input blend), then the points which satisfy this formula
will form the shape of a spiral around the origin. It is this conceptual spiral which is then mapped
onto the physical markings on the whiteboard – a second blend, and the one that is the focus of ourattention here. Of course a number of accomodations must be made: the whiteboard is not infinite
in extent; his mark does not extend infinitely in either direction; his drawing is not accurate in its
proportions. Nonetheless, a great deal of internal structure does map between these domains: thevarious locations on the drawn spiral correspond to different points on the abstract spiral, the blankspace around the drawn line maps to other complex numbers, and the center of the spiral maps tothe plane’s origin. The speech provides a critical cue for what the marking is intended to mean,
and together with the shape of the drawn spiral, allows us to construct the appropriate mapping.
1 Since co-present interlocutors have perceptual access to their shared physical environment, we can treat their

cognitive representation of this environment as an especially detailed and concrete kind of concept, and use it as an
input to the blending process; see Liddell (2003) and Hutchins (2005) for further discussion of this point.
2 For transcripts, I use ellipses to represent silent pauses, and “<XXX>” to denote inaudible speech. When two
speakers interact, they are labeled ‘L’ and ‘R’, corresponding to their position on either the left or right of the video
frame, according to the reader’s viewpoint. Gestures are represented by [brackets] to indicate the timing of preand post-stroke holds, bolding to mark the gestural stroke itself, and italicized text describing the motions which ishorizontally aligned to the start of the stroke. Underlining marks periods in which the speaker is writing or drawing
simultaneously with their speech.

2But once the blend is established, it remains indefinitely, lingering long after the speech is gone; infact immediately after this spiral is drawn, the group turned to discuss a tangential topic for several
minutes before mentioning or interacting with this drawing again.
We’ll follow their example, and return to the spiral below, after examining a gesture example.

2

The life-cycle of real-space blends

In (2), the speaker on the right describes how she constructed a new set (let’s call it S2 ) by performing a series of operations on some original set (S1 , say), with reference to a third set she dubs U.
Overall, she has two points she wants to make: the first is that S2 has a simple relationship to S1 ; itis obtained by taking all elements of S1 that are not in U, or in set arithmetic terms, S2 = S1 −U.
The second is that the new set S2 has a certain topological property. In topology, sets can be open,
closed, both, or neither; and these properties are defined in part by how they propagate throughthe two operations of set complement and set union. Here, she demonstrates that if S1 is closed
and U is open, then S2 is also closed. To accomplish this, she describes a complex series of complement and union operations, which together show that S2 is closed; simultaneously, she uses
gesture to build a complex real-space blend to track these operations, and show that this sequenceof operations is equivalent to a simple set difference operation.
Because they are doing topology, the sets are conventionally conceptualized as sub-regions ofsome larger region of space. She begins with the set S1 , blended onto her right hand (Fig. 2a).
Notice that this is essentially similar to the process described above, with the physical structure ofher hand replacing the whiteboard marker pigment3 :(2) so [1 my original closed set,
puts right hand up in air, holds, Fig. 2aright, was [the complement of some open set.]
scoops left hand around to indicate open set, Fig. 2b
now I’m [taking that open set,]
repeats left hand scoop, quicker this time, Fig. 2cunion [another open set. . . right?]
left hand hold, Fig. 2d[that open set]left hand scoop again, Fig. 2eunion [2 U,left hand hold again, Fig. 2f
and taking the ]1 [complement of thatright hand moves away from left, Fig. 2g]]2 . . .
both hands dropFirst we see that by initially placing the right hand, she has created a blend that extends beyond it– she is also structuring the space around her hand, so that left hand gestures she performs in the
3 Subscripts in brackets indicate the beginning and ending of two overlapping gestures.

3

(a) so [my original closed set(b) right, was [the complement (c) now I’m [taking that open (d) union [anotherof some open set]
set,]
set. . . right?](e) [that open set](f) union U

Figure 2

4

open(g) and taking the] [complement of that]physical region where her right hand is not located refer conceptually to the points which are notin the set S1 – i.e., the set complement of S1 (Fig. 2b, c).
She then introduces the set U, which she conceptualizes as partially overlapping the original
set S1 .4 This set is also blended onto the left hand (Fig. 2d), and the set overlap is represented byplacing the corresponding gestural hold physically into the region between her right hand index
finger and thumb, so that the convex hulls of the two hands representing S1 and U physicallyoverlap.
In Fig. 2e and f, she alternates between invoking the complement set and the new set U; since
she has only two hands, she cannot perform them simultaneously, while also maintaining the righthand hold as a reference point anchoring the blend. The speech co-timed with both of these gesturalpairs c-d and e-f tells us what the relation between these sets is: she is taking the union between
them (i.e., the set which contains all points that are in either or both of the original sets). Interestingly, her syntax here does not follow the conventional rules of English grammar; while it isembedded in a standard English sentence, the phrasing X union Y instead follows the conventions
for reciting mathematical formulas like X ∪ Y . According to these conventions, we read symbols
from left to right, and this rule is what ultimately provides the temporal ordering on her gestures– the order of her gestures, first the complement, second the new set U, make clear that the two
gestures correspond to the two parts of such a formula.
Once this is established, yet another new set enters the discourse: the set which is the union of
the complement and of U, and which we know by the definition of a union should, in the blended
space, consist of the region which overlaps precisely with the sets which were used to construct
it. Starting in Fig. 2g, then, the left hand hold becomes ambiguous: it’s been established that it
occupies the same space as U, and U occupies the same space as the union set; therefore, the left
hand hold also gives us the blended position of (part of) the union set.
Finally, she takes the complement of this union set. For depicting this final set she uses theright hand, finally releasing it from the position where it has been held since the beginning of thisdescription – it moves slightly from the S1 position, to a new position which is almost identical,
except that it no longer overlaps with the left hand representing U. It’s this motion that encodes
what the speech does not: that this set is exactly the set we originally described, S2 = S1 −U, andthat this sequence of operations together has performed a set difference.
This motion would not be meaningful without real-space blend that it occurs within, and thisblend required the whole sequence of preceding speech and gesture to construct and elaborate. But
once the rich blended space has been established, the simple motion in Fig. 2g is enough to convey
a proof of an interesting theorem: that a closed set minus an open set is a closed set.5 The purposeof the preceding speech and gesture is, in some sense, to put this blend in place, and allow thisfinal gesture.
If we look at this sequence of speech and gesture, we see that it constantly interacts with the
4 Interestingly, the overlap occurs only in the gesture, not in the speech. This corresponds to the fact that formally,

the degree of overlap between S1 and U is irrelevant: the proof goes through just as well even if S1 is a subset of U,
or if they don’t intersect at all. But in either of these cases, the result is trivial (because either S2 is the empty set, orelse S2 = S1 ). Her speech describes the general case, just as a formal written presentation would; her gesture not only
helps by tracking the entities involved, but also by depicting the particular case where the proof is most interesting andrequires real thought.
5 In fact, immediately before the portion of the video analyzed here, she uses exactly this theorem to justify anargument she is making; this sequence follows when she realizes that she must first justify the theorem.

5blended space in complex ways. First, the blended space is first established in Fig. 2a. Then,the left-hand gestures and the final right hand gesture all depend on the existing blend to depict
new content (e.g., the complement set gestures, which are interpretable only by comparing their
spatial position to the spatial position of S1 ), while simultaneously elaborating it further (e.g., byplacing U so that it partially overlaps S1 ). Simultaneously while this is occuring, she devotes someenergy to maintaining the blend. This is accomplished through an elegant dance of hold gestures:
whichever hand is not currently elaborating the blend is held steady, anchoring whichever portionof the blend it is currently assigned – first she uses the right hand, then both (briefly) in Fig. 2f,
and finally the left hand takes over for Fig. 2g (Enfield, 2003, 2009). These holds both signal thecontinued relevance of the blend, and provide a visible spatial reference point for it. And, finally,
once the blend has served its purpose, she drops both hands, signaling the end of this blend’s utility,and freeing up the physical space to whatever content is needed next.
These operations – construction, maintenance, exploitation/elaboration, and destruction – form
the fundamental life-cycle of real-space blends in discourse, which interlocutors must systematically manage in order to accomplish their goals. Different strategies may be available and used
depending on the specific details of how each blend is instantiated, the current goals, and trade-offs
between conflicting ones. For example, a whiteboard drawing remains until erased, but gestures
are more transient, and thus ambiguity is more likely to arise about whether an earlier blend is still
relevant; the speaker in (2) finds one solution to this problem using hold gestures, but this solution
creates a new problem of how to simultaneously maintain and elaborate the blend given her limited
number of hands, which she in turn solves with the switch in Fig. 2g. The point of this schemais not to allow us to predict exactly what interlocutors will do in any particular situation, but toprovide a conceptual framework to understand these competing constraints on the structure of thepersistent, meaningful spaces that discourse participants navigate to succeed – and to encourage us
to consider all the work going on outside of the individual gestural strokes.

3

Real-space blends as shared spaces

Real-space blends, by definition, map physical structure and relationships onto conceptual structure and relationships, and vice-versa. The previous examples demonstrated real-space blends in
two different modalities separately – drawing and gesture – together with a third modality, speech.
But, crucially, real-space blends are not modality specific. While these modalities each have theirown unique conventions and affordances (Edwards & Robutti, this volume), their physical components all take place within the same physical environment and a shared spatial and temporal frame.
A real-space blend lives in this shared physical space, and is thus accessible to all modalities simultaneously. At this level of analysis, hand positions, drawn lines, and written text are simply
different ways of locating meaning in space; and hand motions, acts of drawing and writing, andspeech utterances are ways of locating meaning in time (Hiraga, 2005; Sweetser, 2006). Spatial
and temporal relationships can hold between any and all of them, and are used to link together
these disparate representational tools into a coherent, meaningful blended space.
Familiar examples of this phenomenon include the co-timed production of speech and gesture– a temporal relationship indicating a shared meaning – or a pointing gesture directed at a diagram– a spatial relationship used to indicate some conceptual entity. In (3) and (4), our interlocutors
return to the diagram drawn in (1), and produce an elaborate complex of representations around it
6

involving speech, gesture, writing, and drawing.
The problem they are working on requires that they find a function that sends all the points on
the complex plane onto the logarithmic spiral, and that also satisfies some other conditions.6 One
participant begins by proposing a candidate function:(3) I’m thinking we’re gonna send all the [circles]
both hands up and rounded to form circle,
Fig. 3b. . . like e to the t e to the i theta are all going to go to thesewrites “et eiθ ”
writes “−→” pointing at “eθ eiθ ”, which is already on
the board, Fig. 3c
you know what I’m sayin?
moves to other side of board
like. . . all the circles[. . . ]traces a circle on board over spiral with pen, but doesn’t draw, Fig. 3d
are gonna go to[. . . ]the point on the circle thatpoints at “eθ eiθ ”, Fig. 3epoints in the air with the tip of the pen
is [e to the theta times e to the i theta]jabs the air four times, approximate 2-d positions shown in Fig. 3f
In 3-d, it’s clear the positions are laid out 12 34 , schematically similar to eθ eiθ .
His proposed function maps all the points in each circle centered on the origin to the point wherethat circle intersects the logarithmic spiral. To explain this idea, he produces a flurry of different
representations. I count six representations of the circles:• the phrase all the circles;• an iconic circle gesture (Fig. 3b);• the phrase e to the theta e to the i theta, co-timed with:• writing et eiθ , the formula for a circle with radius t;• the phrase all the circles again;• tracing a circle in the air over the spiral (Fig. 3d, again taking advantage of a visible entity toanchor a larger blend; here the presence of the spiral implies the complex plane around it).
Three representations of the mapping:• the phrase are all going to go to,• co-timed with an arrow drawn on the board;• the same phrase again but reduced, are gonna go to.
6 Specifically, it must be a “retraction”; they eventually prove that no such function exists.

7

(a) all the circles(b) e to the t e to the i theta are all going to
go to these(c) all the circles. . .

(d) go to[. . . ]the point(e) e to the theta times e to the i theta

Figure 38And seven representations of the intersection point:• the anaphoric phrase these co-timed with:• the drawing of an arrow pointing at the text eθ eiθ ;• a gestural point at this text (Fig. 3e);• an iconic gesture co-timed with:• the word point;• the phrase e to the theta times e to the i theta co-timed with:• a schematic representation of the spatial layout of this formula gestured into the air in frontof him (Fig. 3f).
Several of these actions depend on pre-existing real-space blend structure in his environment:
the writing of the formula and its arrow pointing to an existing formula (Fig. 3b), and the gestural
point to this same formula (Fig. 3d); and the gesture over the existing spiral diagram (Fig. 3c).
Notice that these are performed without any regard for the apparent boundaries between modalities.
It’s possible that for some purposes we might want to analyze the lower portion of the whiteboard,
with its equations, as a different space from the upper portion, with its diagrammed spiral, anddifferent again from the iconic circle gesture performed in peripersonal space (Fig. 3a). But sucha division is entirely orthogonal to the division between modalities: in all of these spaces we
see the same principles of spatial and temporal relationships being used to bind meaning acrossmodalities that include speech, gesture, writing, and drawing (J. B. Haviland, 2000; J. Haviland,
2007; Goodwin, 2007).
This explains some key parts of how the many representations we counted above gain theirindividual meanings. But how is it that he produced sixteen representations, and yet communicatesa single coherent idea of circles mapping to points according to a single rule? How does he bindthese different representations together? (Nathan & Alibali, 2011) Some of the linking is done
locally by co-timing, and by the word like which indicates syntactically that the meaning of thephrase e to the theta e to the i theta is an expansion/clarification of the meaning of the phrase
all the circles. But then overlaid on this local structure is a larger repetition (Du Bois, 2010).
The phrase you know what I’m sayin? and his movement from the left to the right half of theframe divide the explanation into two parts. Both parts first describe the circles, and then their
mapping, and then the points they map to. This structural repetition, together with individual
repeated fragments (all the circles versus all the circles, are all going to go to versus are gonna go
to, the repeated use of the written eθ eiθ ) make clear that the two overt descriptions in fact refer tothe same underlying concept, and similarly for the corresponding pieces of the two descriptions.
Note that these are repetitions of physical form – when he repeats a phrase, we notice the repetition
because he literally makes similar sounds with his mouth. That is, this is yet another relationshipbetween forms which is mapped into relationships between meanings via real-space blending. It’sby tracing multiple such formal relationships that we come to understand that the circle gestures inFig. 3a and c refer to the same entity, and the formula in Fig. 3b likewise.
Another interesting possible example of form repetition used to link concepts occurs purely
within the mathematical notation used in this example. The scope and quantification of the variable9θ is quite ambiguous. For et eiθ to be a circle, t must be fixed while θ varies; for eθ eiθ to be a point,θ must be fixed. As a result, there’s no mathematical formalism that lets us define θ so that theformula et eiθ → eθ eiθ makes sense as a mapping of a circle to a point. Yet this doesn’t seem tobother the participants. Perhaps they note that the formulas are identical except for first exponent,
and understand this as defining an identity constraint, just like the speech’s syntactic repetition
creates identity mappings between analogous elements: for each circle with its varying θ , pick thevalue of θ where t = θ .
Returning to the episode illustrated in Fig. 3, thirty seconds later, the original speaker andone of his collaborators begin to find reasons why the mapping he’s suggested cannot fulfill therequirements set out in their homework problem:(4) R

the pre-image of the [origin can’t just be the] origin
points with pen at center of spiral on board, multiple beats...
so that can’t be right, like
we can’t– [we can’t be sending all the circles] to–draws a circle on the board, along same path as had gestured a circle earlier, Fig. 4aLcause it seems like you’re like [nailing down all this stuff here
O hand taps down along spiral, Fig. 4b
and then you’re gonna have to break something apart too] <XXX>
hand stops moving, uses fingers as pincers to “pull”
bits of space toward the spiral, Fig. 4cRwell I was thinking like we could send [all of the circles]
index finger traces previously drawn circle[to the unique point that this thing intersects]
index finger stops on intersection of drawn circle and drawn spiral, Fig. 4dcuz [the spiral] intersects a circle at one and only one point
traces spiral in air in front of him with index finger

Let’s compare Fig. 3c to Fig. 4a. Both occur co-timed with the initial portion of schematicallysimilar utterances, all the circles. . . to the point. (In the second case he is interrupted half-way
through this utterance, but we can see what he intended by how he resumes the utterance at the endof the example.) Both involve very similar motions: tracing the pen in a circular motion over the
spiral diagram. The difference is that in the first case, the pen is held just above the surface of thewhiteboard, so that no mark is made; in the second, the pen is lowered slightly, so that a trace isleft behind, and the motion is somewhat slower and more deliberate. Why is this?
Consider that for the students’ purposes, the precise size of the circle is irrelevant – it’s a genericrepresentative of the class of all circles that share the same center, so the motions could just as
easily have been a bit bigger or smaller. And a gesture, being both schematic and transient, leaves
vague the exact size of the circle that is intended. So while the gesture establishes that there is somepoint where the circle and spiral intersect, it does not establish which precise physical, spatial
location that point should be blended onto, and in particular, leaves no way to later determine
whether any particular location corresponds to this special point. The first time, this is irrelevant:while the student immediately produces a reference to this point, his elaboration uses the formula10(a) all the circles(b) nailing down all this stuff here(c) break something apart too(d) the unique point

Figure 4(pointing at eθ eiθ , saying e to the theta times e to the i theta), not the diagram. The second time,
though, when he eventually gets a chance to finish his sentence and say to the unique point, henow elaborates by directing a gesture at the particular location in the diagram where the circle andspiral lines intersect. This reference succeeds only because the circle he has drawn has elaborated
the spatial structure of the blend, creating a stable and unambiguous perceptual anchor for thisintersection point, in a way that the gesture did not.
Clearly his reasoning did not proceed as follows: first, deciding which modality to use (gestureor drawing); second, deciding what kind of gesture or drawing to make. In fact, the drawing has
more canonical ‘gesture-like’ features than the actual gesture, including both a pre-stroke holdand simultaneous production with a co-referential syntactic constituent. Instead, it appears as if
he decided that he wanted to exploit this real-space blend to spatially describe an abstract circle,
and then compared the affordances of these two different methods of tracing the circle: pen up isquicker, more vague, and more transient, thus cluttering the diagram less; pen down is slower, but
having the side-effect of leaving a precise and stable anchor for the elaborated structure. Whichchoice is optimal in any particular situation depends not on the current referential goal – they bothrefer to the same circle via the same real-space blend – but on whether having a stable anchor will
be useful for future actions. And in each case, he does in fact choose the option which matches the
future actions which he does in fact perform. It seems unlikely that this is an accident. Rather, I
would argue, it demonstrates the strategic selection between different strategies for manipulating
the blended space on the basis of current discourse goals, and plans for achieving those goals.
Finally, consider the interruption. For our purposes, the interesting thing about this is thatspeaker L simply moves into the real-space blend spaces and uses speaker R’s diagram as a substrate for making her own statements, in a way reminiscent of what Furuyama (2000) refers to as“collaborative gestures”, in which one person reaches into another’s gesture space to manipulate
11Figure 5: Example (5): pshew pshew pshew
the structure of their blend directly. These blends are shared not just between modalities, but also
between interlocutors.
In (5) we see something that blurs the line between gesture, writing, and drawing to invisibility— what one might call ‘gestural drawing’. Here the speaker starts with markings on her paper
corresponding to a group T that is ‘acting’ on a point. The idea of a group acting on a point isthat each element of the group somehow modifies the point, so that from one point arises a wholecollection of differently modified points:(5) L

you’re taking it [up. . . ]draws an arrow showing the movement of a pointR

you’re moving it somewhere elseLacting on it with T , [. . . pshew pshew pshew pshew pshew pshew]
repeated strokes across paper from T to point, Fig. 5
and then bringing it back.

Speaker L draws multiple lines from the place corresponding to the group towards the place corresponding to the point. These lines are drawn in a very fast and haphazard fashion, which makes nosense in terms of drawing, for a line is the same whether drawn quick or slow – but if one conceptualizes the process as a series of group elements hurtling out of T and smacking into the point, thenthe motion does code appropriate force dynamics and imagistic structure. Furthermore, they are
drawn on top of or nearly on top of each other, which again makes little sense if one’s motivation is
simply to draw a particular picture, but does accurately represent the motion of multiple elements
whose paths share start and end points. It’s even unclear whether this is drawing or writing, sincearrows are such a common conventionalized mathematical notation, and notational arrows can do
all the normal things arrows can do in diagrams. Is this gesture or goal-directed action? In thisinstance, the distinction doesn’t seem useful.
12Figure 6: . . . [. . . ]. . .
Meanwhile, she uses speech to produce co-timed sound effects, which iconically represent
repeated fast metaphoric motion. Here the overall timing is driven by an imagined activity, andboth the hand movements and speech are aligned to and represent it. The standard terminology forreal-space blends seems particularly inadequate here, since ‘space’ is so clearly just one aspect ofthe physical environment that is participating in the blend (a residue of the theory’s sign-language
heritage). Perhaps it’s a real-space-and-sound blend.

4

Thinking in a real-space blend

Interlocutors share a physical environment, but, lacking telepathy, do not directly share a mental
environment. This makes real-space blends, which map rich mental constructs into the shared
physical environment, very useful for accomplishing communicative goals. Theoretically, though,this is not a requirement; real-space blends can and do exist outside of communicative contexts.
In some ways, this is a commonplace observation: of course people draw diagrams when solving
problems alone, and there are many cultural artifacts whose entire purpose is to encode meaningfulstructure into their physical structure: clocks, maps, slide rules, protractors, calculators, and soforth (Hutchins, 2005).
Because real-time interactional communication uses the same cognitive tool of real-space
blending, it is perhaps not a surprise that people also use ad hoc, iconic/metaphoric gestures tosupport their own thinking (Chu & Kita, 2008; Rodrı́guez & Palacios, 2007). In (6), the two students are working independently in their notebooks. Then one of them pauses her writing, raises
her hands in a hold gesture, and remains motionless for thirty seconds, before resuming writing:(6) R. . . [. . . ]. . .
30 second silent hold, Fig. 6

At no point during this does her partner look up or interact in any way. This appears to be an
anchoring gesture, like those we saw in (2), used to maintain a blend that she is using for somecognitive purpose of her own. While the absence of speech makes it difficult for us as analysts
to know what exactly this gesture means to her, it’s intriguing to observe that this gesture is very
similar to one that she produced approximately one minute earlier, while they were still discussing
the problem:(7) Ryeah it’s [basically it’s like taking an open interval
13Figure 7: it’s [basically it’s like taking an open interval
forms a two-handed hold, Fig. 7
rotates both hands back-and-forth, making a beat
While this occurred a minute earlier, it is the last-but-one utterance before they turned to theirnotebooks to work independently, and highly suggestive that the real-space blend that she uses forcognitive purposes in (6) is actually the same real-space blend that she using earlier for communicative purposes in (7). If the very same blend can be used for both purposes, though, then we have a
new analytic puzzle: when we observe people using such representations within a problem-solving
discourse, then how can we tell whether they are being used for communicative (other-directed)
or cognitive (self-directed) purposes? Many meaningful actions might even serve both purposes
simultaneously.

5

Conclusion

A real-space blend is a structure-preserving map between physical and conceptual realities, and isa generic cognitive tool which underlies many different modalities of meaningful action, including
at least speech, gesture, writing, and drawing. These blends often persist through time, and allow for rich, fine-grained interaction between modalities, between interlocutors, and support bothcommunicative and cognitive activity. To be useful, though, they must be managed: blends must
be constructed, maintained, elaborated, and deconstructed, and many discourse actions are systematically chosen to accomplish these tasks simultaneously with accomplishing more immediate
communicative goals.
When looking at a single gesture in isolation, it is the stroke that catches our attention. But inthe context of an ongoing discourse, every word, gesture, and stroke of the pen is performed intoa physical environment whose structure has been richly overlaid with meaning, and every suchaction also further modifies this environment, affecting future actions. In many cases, it may be
more important to understand the space in which a gestural stroke occurs, and the way in which it
modifies that space, than it is to understand the stroke itself.

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