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Verbal Working Memory in Sentence Comprehension
Evelina Fedorenko (evelina9@mit.edu)
Department of Brain and Cognitive Sciences, 77 Mass Ave, NE20-437d
Cambridge, MA 02139 USA
Edward Gibson (egibson@mit.edu)
Department of Brain and Cognitive Sciences, 77 Mass Ave, NE20-459
Cambridge, MA 02139 USA
Douglas Rohde (dr@tedlab.mit.edu)
Department of Brain and Cognitive Sciences, 77 Mass Ave, NE20-437
Cambridge, MA 02139 USA
Abstract
This paper investigates the nature of verbal working memory
(WM) in sentence comprehension and provides evidence for
overlapping pools of verbal WM resources between on-line
sentence comprehension and other verbally-mediated tasks.
We report the results of two dual-task experiments. In
Experiment 1, participants simultaneously performed a self-
paced reading task and a self-paced arithmetic addition task in
a 2x2 design crossing syntactic complexity (low, high) and
arithmetic complexity (low, high). In addition to two main
effects, the most interesting result was a significant
interaction between syntactic and arithmetic complexity
during the critical region of the linguistic materials:
participants processed the complex/complex condition more
slowly than would be expected if the two tasks relied on
independent resource pools. To address a potential confound
of shared attentional resources, Experiment 2 was conducted,
where participants simultaneously performed a self-paced
reading task and a self-paced spatial-rotation task in a similar
2x2 design crossing syntactic complexity with the complexity
of the spatial task. As in Experiment 1, there were two main
effects of complexity in the critical region. However, in
contrast to Experiment 1, these effects were strictly additive,
with no trace of interaction. The results of the two
experiments therefore support a WM framework where on-
line linguistic processing and on-line arithmetic processing
rely on overlapping pools of verbal WM resources.
Introduction
A major question in psycholinguistic research concerns the
nature of the working memory (WM) resources used in
language processing. Empirical research has suggested that
different pools of WM resources are used for processing
visuo-spatial information and verbal information (e.g.,
Baddeley & Hitch, 1974; Baddeley, 1986; Vallar &
Shallice, 1990; Hanley et al., 1991; Jonides et al., 1993;
Shah & Miyake, 1996). Some researchers (Caplan &
Waters, 1999; cf. Just & Carpenter, 1992) have
hypothesized that the verbal WM pool can be further
divided into two sub-pools: (1) verbal WM for natural
language comprehension and production; and (2) verbal
WM for non-linguistic verbally-mediated cognitive tasks.
This paper attempts to empirically evaluate this hypothesis.
One way to address this question is via dual-task
paradigms in which participants perform two tasks
simultaneously: (1) on-line sentence processing, and (2) a
non-linguistic verbally-mediated task. The underlying
assumption is that we should observe a super-additive
interaction when the complexity of both tasks is high only if
the two tasks rely on overlapping pools of resources.
Previous dual-task experiments found either no
interaction or only a suggestion of one (e.g. King & Just,
1991; Just & Carpenter, 1992; Caplan & Waters, 1999;
Gordon et al., 2002). In all of the previous experiments,
however, the secondary task involved storage of words or
digits across the sentence-processing task. Although storage
– in a very general sense of keeping track of previously
encountered information – plays an important role in on-line
sentence comprehension (e.g., Chomsky & Miller, 1963;
Kimball, 1973; Gibson, 1991; 1998; Lewis, 1996), it may be
qualitatively different from the kind of storage involved in
the secondary tasks in the earlier experiments.
According to one recent resource-based theory of on-line
syntactic processing, the dependency locality theory (DLT;
Gibson, 1998, 2000), there are two working memory
components to sentence comprehension: storage and
integration. The storage component involves keeping track
of partially processed syntactic dependencies that are still
awaiting their second element in order for the sentence to be
grammatical, whereas the integration component involves
connecting a newly input word into the structure that has
been built so far. Critically, the storage component of on-
line sentence comprehension is unlike the storage involved
in keeping track of a list of unconnected items.
Consequently, it is possible that the lack of on-line
interactions between syntactic complexity and memory load
in earlier studies could be a result of the distinct nature of
the storage processes involved. Moreover, there have been
no previous attempts to explore the potential interaction
between integration processes in sentence comprehension
and secondary verbally-mediated tasks, which involve
similar but non-linguistic on-line integration processes. In
the current paper, we propose a novel paradigm to address
this issue.
Experiment 1
This experiment had a dual-task design, in which
participants read sentences phrase-by-phrase, and at the
same time were required to perform simple additions. The
on-line addition task is similar to on-line sentence
comprehension in that an incoming element – a number –
must be integrated into (i.e., added to) the representation
constructed thus far: the working sum. Both tasks had two
levels of complexity, resulting in a 2x2 design. Critically,
there was no difference in linguistic complexity between the
easy and hard arithmetic conditions: the complexity of the
arithmetic task was manipulated in terms of the difficulty of
the arithmetic operations (by making the addends larger),
while keeping the linguistic form of the two conditions
identical (number + number + number, etc.). Therefore, if
we observe a super-additive interaction between the two
tasks when the complexity of both tasks is high, then we
may infer that the verbal WM resources that are involved in
performing the arithmetic task overlap with those that are
involved in syntactic integration processes. In contrast, if
language processing relies on an independent verbal WM
resource pool, there should be no such interaction.
Methods
Participants Forty participants from MIT and the
surrounding community were paid for their participation.
All were native speakers of English and were naive as to the
purposes of the study.
Design and materials The experiment had a 2x2 design,
crossing syntactic complexity (subject-extracted relative
clauses (RCs), object-extracted RCs) with arithmetic
complexity (simple additions (low initial addend,
consequent addends between 1 and 3) vs. complex additions
(higher initial addend, consequent addends between 4 and
6)).
The language materials consisted of 32 sets of sentences,
having four different versions as in (1):
(1) a. Subject-extracted, version 1:
The janitor | who frustrated the plumber | lost the key |
on the street.
b. Subject-extracted, version 2:
The plumber | who frustrated the janitor | lost the key |
on the street.
c. Object-extracted, version 1:
The janitor | who the plumber frustrated | lost the key |
on the street.
d. Object-extracted, version 2:
The plumber | who the janitor frustrated | lost the key |
on the street.
As described above, there were only two levels of
syntactic complexity – subject- and object-extractions – but
there were four versions of each sentence in order to control
for potential plausibility differences between the subject-
and object-extracted versions of each sentence. As a result,
no independent plausibility control is needed in this design.
Each participant saw only one version of each sentence,
following a Latin-Square design.
The numbers for the addition task were randomly
generated online for each participant with the following
constraints: (1) the value of the initial addend in the easy-
math condition varied from 1 to 10, whereas the value of the
initial addend in the hard-math condition varied from 11 to
20, and (2) the addends varied from 1 to 3 in the easy-math
condition and from 4 to 6 in the hard-math condition.
In addition to the target sentences, 40 filler sentences with
various syntactic structures other than relative clauses were
included. The length and syntactic complexity of the filler
sentences was similar to that of the target sentences. The
stimuli were pseudo-randomized separately for each
participant, with at least one filler separating the target
sentences.
Procedure The task was self-paced phrase-by-phrase
reading with a moving-window display (Just, Carpenter &
Woolley, 1982). The experiment was run using the Linger
2.85 software by Doug Rohde. Each experimental sentence
had four regions (as shown in (1a)-(1d)): (1) a noun phrase,
(2) an RC (subject-/object-extracted), (3) a main verb with a
direct object (an inanimate noun phrase) and (4) an adjunct
prepositional phrase. The addends for the addition task
were presented simultaneously with the sentence fragments,
above and aligned with the second character of each
fragment. The first sentence region had a number above it
(e.g. “12”) and all the consequent regions had a plus sign
followed by a number (e.g. “+4”), as shown in Figure 1.
Figure 1: Sample frame-by-frame presentation of an item.
Each trial began with a series of dashes marking the length
and position of the words in the sentence. Participants
pressed the spacebar to reveal each region of the sentence.
As each new region appeared, the preceding region
disappeared along with the number above it. The amount of
time the participant spent reading each region and
performing the accompanying arithmetic task, was recorded
as the time between key-presses.
To make sure the participants performed the arithmetic
task, a window appeared at the center of the screen at the
end of each sentence and the participants were asked to type
in the sum of their calculations. If the answer was correct,
the word “CORRECT” flashed briefly on the screen, if the
answer differed by up to 2 from the correct sum, the word
“CLOSE” flashed briefly, and if the answer was off by more
than 2, the word “INCORRECT” flashed briefly on the
screen. To assure that the participants read the sentences for
meaning, two true-or-false statements were presented
Time 1: 12
The janitor --- ---------- --- ------- ---- --- --- -- --- ------.
Time 2: +4
--- ------- who frustrated the plumber ---- --- --- -- --- ------.
Time 3: +5
--- ------- --- ---------- --- ------- lost the key -- --- ------.
Time 4: +4
--- ------- --- ---------- --- ------- ---- --- --- on the street.
sequentially after the sum question, asking about the
propositional content of the sentence they just read.
Participants pressed one of two keys to respond “true” or
“false” to the statements. After a correct answer, the word
“CORRECT” flashed briefly on the screen, and after an
incorrect answer, the word “INCORRECT” flashed briefly.
Participants were instructed not to concentrate on one task
(reading or additions) more than the other. They were asked
to read sentences silently at a natural pace and to be sure
that they understood what they read. They were also told to
answer the math and sentence questions as quickly and
accurately as they could, and to take wrong answers as an
indication to be more careful.
Before the experiment started, a short list of practice
items and questions was presented in order to familiarize the
participants with the task. Participants took approximately
35 minutes to complete the experiment.
Results
Arithmetic accuracy Participants answered the arithmetic
sum correctly 88.7% of the time. A two-factor ANOVA
crossing arithmetic complexity (easy, hard) and syntactic
complexity (easy, hard) on these question-answering data
revealed a main effect of arithmetic complexity
(F1(1,39)=9.45; MSe=0.120; p < .005; F2(1,31)=7.21;
MSe=0.087; p < .02), but no other significant effects.
Comprehension question performance There were two
comprehension questions following each experimental trial.
Participants answered the first question correctly 80.2% of
the time, and the second question 78.1% of the time. The
percentages of correct answers by condition were very
similar for the two questions, so we collapsed the results in
our analyses. A two-factor ANOVA crossing arithmetic
complexity (easy, hard) and syntactic complexity (easy,
hard) on the responses to the two comprehension questions
revealed a main effect of syntactic complexity
(F1(1,39)=9.8; MS=0.1; p < .005; F2(1,31)=4.04;
MS=0.074; p=.05) and a main effect of arithmetic
complexity in the participants analysis (F1(1,39)=4.31;
MS=0.047; p <.05; F2(1,31)=2.9; MS=0.042; p =.10), but
no significant interaction (Fs < 1).
Reaction times Because participants had to answer three
questions (one math, two language) for each sentence, the
odds of getting all three correct were not very high overall
(55.6%). As a result, we analyzed all trials, regardless of
how the comprehension questions were answered. The data
patterns were very similar in analyses of smaller amounts of
data, in which we analyzed (1) trials in which one or both of
the language comprehension questions were answered
correctly, or (2) trials in which the math question was
answered correctly. To adjust for differences in word length
as well as overall differences in participants’ reading rates, a
regression equation predicting reading times from word
length was derived for each participant, using all filler and
target items (Ferreira & Clifton, 1986; see Trueswell,
Tanenhaus & Garnsey, 1994, for discussion). At each word
position, the reaction time predicted by the participant’s
regression equation was subtracted from the actual
measured reaction time to obtain a residual reaction time.
The statistical analyses gave the same numerical patterns for
analyses of raw reaction times. Reaction time data points
that were less than 100 msec in the raw data (indicating
erroneous key presses) or more than 2.5 standard deviations
away from the mean residual RT for a position within a
condition were excluded from the analysis, affecting 3.3%
of the data. Figure 2 presents the mean residual RTs per
region across the four conditions of the experiment.
-600
-400
-200
0
200
400
600
800
The janitor who frustrated the
plumber/ who the
plumber frustrated
lost the key on the street.
Subject / Easy Math
Object / Easy Math
Subject / Hard Math
Object / Hard Math
Figure 2: Reaction times per region in the four conditions
of Experiment 1. The critical region is circled.
We present the analysis of the critical region (Region 2)
first, followed by the analyses of the other regions. The
critical region included the RC (“who frustrated the
plumber” / “who the plumber frustrated”). A 2x2 ANOVA
(easy-math / hard-math, subject-extracted RC / object-
extracted RC) in this region revealed two significant main
effects and a significant interaction. First, the hard-math
conditions were read significantly slower than the easy-math
conditions (F1(1,39)=47.26; MSe=7641827; p < .001;
F2(1,31)=42.58; MSe=5880083; p < .001). Second, the
syntactically more complex object-extracted RC conditions
were read significantly slower than the subject-extracted
conditions (F1(1,39)=38.74; MSe=5283587; p < .001;
F2(1,31)=33.4; MSe=4072481; p < .001). Third, and most
interestingly, there was a significant interaction, such that in
the hard math conditions, the difference between subject-
and object-extracted RCs was larger than in the easy math
conditions (F1(1,39)=4.74; MSe=623599; p < .05;
F2(1,31)=7.15; MSe=526415; p < .02). This interaction is
predicted by the hypothesis whereby sentence processing
and arithmetic processing rely on overlapping pools of
resources, but not by the hypothesis that the pools of
resources are independent.
In Region 1, consisting of the main clause subject (e.g.,
“The janitor”) together with the initial addend, a 2x2
ANOVA revealed a main effect of arithmetic complexity
(marginal in the items analysis), but no other significant
effects. The hard-math conditions were read slower than the
easy-math conditions (F1(1,39)=5.08; MSe=245326; p <
.05; F2(1,31)=3.62; MSe=149836; p = .067). In Region 3,
the top-level verb and its object (“lost the key”), a 2x2
ANOVA revealed a main effect of arithmetic complexity
(F1(1,39)=30.21; MSe=5726294; p < .001; F2(1,31)=33.32;
MSe=3978352; p < .001), but no other effects. Finally, in
Region 4, the sentence-final prepositional phrase (“on the
street”), there was again an effect of arithmetic complexity
(F1(1,39)=72.58; MSe=13066602; p < .001; F2(1,31)=
105.06; MSe=10545386; p < .001), but no other effects.
Discussion
The results of Experiment 1 are consistent with a WM
framework where online sentence comprehension and
arithmetic processing rely on overlapping resource pools.
Most importantly, there was an interaction between
syntactic complexity and arithmetic complexity in the
critical region of the linguistic materials, where syntactic
complexity was manipulated between subject-extracted RCs
(low complexity) and object-extracted RCs (high
complexity). There was no evidence of any interaction of
this kind in any of the other three regions. Critically,
linguistic complexity was not varied in the arithmetic task,
so the observed interaction is not due to an overlap in the
linguistic processes that are involved in the two tasks.
It should be noted, however, that there is an alternative
explanation for the observed pattern of results in terms of
attentional resources required for the simultaneous
performance of the two tasks. In dual-task paradigms,
resources are needed in order to direct attention to one task
or another. It is possible that in the difficult conditions,
more attention switches are required, or the switches
between tasks are more costly. The observed interaction
could therefore be a result of additional task-switching costs
in the high syntactic complexity / high arithmetic
complexity condition. Experiment 2 was designed to
address this issue.
Experiment 2
This experiment used a similar dual-task paradigm as the
first experiment. In contrast to Experiment 1, however, the
secondary task was a spatial-rotation task matched for
difficulty with the addition task used in Experiment 1. In
this task, participants were instructed to visually imagine
adding different-size sectors of a circle and to keep track of
the angle subtended by the combined segments. The most
natural way to solve this task is to mentally rotate each
incoming sector until it abuts the estimated sum of the
previous sectors. The on-line spatial-rotation task is similar
to the addition task in that an incoming element – a sector –
must be integrated into, or added to, the representation
constructed thus far. Critically though, the spatial-rotation
task does not rely on verbal WM resources, and should not
therefore interact with the sentence-processing task if the
cause for the observed interaction in Experiment 1 is an
overlap in the use of verbal WM resources. However, if the
attentional costs are responsible for the interaction, we
should observe a similar interaction, regardless of the nature
of the secondary task.
Methods
Participants Twenty-four participants from MIT and the
surrounding community were paid for their participation.
All were native speakers of English and were naive as to the
purposes of the study. None of the participants took part in
Experiment 1.
Design and materials The experiment had a 2x2 design,
crossing syntactic complexity (subject-/ object-extracted
RCs) with the complexity of the spatial-rotation task (simple
rotations with small-angle sectors/ complex rotations with
larger-angle sectors). The language materials were exactly
the same as those used in Experiment 1.
The sectors for the spatial-rotation task were randomly
generated online for each participant in the following way:
the size of the sectors for the easy condition varied from 5 to
90 degrees, whereas the size of the sectors for the hard
condition varied from 30 to 180 degrees. As a result, it was
more likely in the hard condition for the sum of sectors to be
more than 360 degrees, thus “wrapping around” the circle.
Pilot testing of the pie task by itself suggested that the task
is easier to perform with smaller sectors.
As in Experiment 1, 40 filler sentences with various
syntactic structures other than relative clauses were
included, and the stimuli were pseudo-randomized
separately for each participant, with at least one filler
separating the target sentences.
Procedure The procedure was identical to that of
Experiment 1, except for substituting the spatial-rotation
task for the arithmetic task. Above each sentence fragment,
participants saw a small circle. They were instructed to
think of it as a plate for a pie. On each “plate”, there was a
“pie-slice” shown in blue. The size of the “pie-slices”
varied (as described in Materials and Design above), but
they all started at the 12:00 position, as shown in Figure 3.
Figure 3: Sample figure of the spatial-rotation task.
Participants were instructed to visually imagine adding
each new “pie-slice” to the previous one(s) by mentally
“putting” them next to each other. To assure that the
participants performed the task, at the end of each trial a
large blank circle appeared at the center of the screen with a
vertically-pointing radius. Participants were instructed to
drag this radius (by using the mouse) to the end-point where
all the “pie-slices” they just saw would come to when
placed next to each other. If the answer was within 10
degrees of the correct answer, the words “Very Close!”
flashed briefly on the screen; if the answer was within 35
degrees, the words “Pretty Good” flashed briefly; if the
answer was within 90 degrees, the words “In The Ballpark”
flashed briefly; finally, if the answer was not within 90
degrees, the words “Not Very Good” flashed briefly on the
screen. The participants were warned that sometimes the
“pie-slices”, when added together, would form more than a
complete pie. In such cases, they were told to assume that
the slices “wrapped around” and to ignore the complete
portion of the pie.
As in Experiment 1, this task was followed by two
comprehension questions about the content of the sentences.
Results
Spatial-rotation task accuracy On average, participants’
estimates were 30.3 degrees off of the correct answer. A
two-factor ANOVA crossing spatial-rotation task
complexity (easy, hard) and syntactic complexity (easy,
hard) revealed a main effect of complexity of the spatial-
rotation task (F1(1,23)=18.36; MSe=2676; p < .0005;
F2(1,31)=22.28; MSe=3568; p < .0005), but no other
significant effects. It is worth noting that this pattern of
results for the spatial-rotation task accuracy is parallel to
that of the results for the arithmetic task accuracy in
Experiment 1.
Comprehension question performance There were two
comprehension questions following each experimental trial.
The percentages of correct answers by condition were very
similar for the two questions, so we collapsed the results in
our analyses. Across conditions, participants answered the
questions correctly 83% of the time. A 2x2 ANOVA
crossing spatial-rotation task complexity (easy, hard) and
syntactic complexity (easy, hard) on the responses to the
comprehension questions revealed no significant effects or
interactions (Fs<1). This pattern of results differs slightly
from that in Experiment 1 in that there was no effect of
syntactic complexity in Experiment 2. Note, however, that
overall, subjects performed better on comprehension
questions in Experiment 2 (83% across conditions),
compared with Experiment 1 (79% across conditions). This
accuracy difference across the experiments may have
resulted from greater interference of the secondary task in
Experiment 1 with subjects’ memory of the propositional
content of the sentences, due to its verbal nature. The lack
of syntactic complexity effect in Experiment 2 could then be
explained by a possible ceiling effect in the comprehension
performance: without a verbally interfering task, people
perform well on both the subject- and object-extracted
relative clause sentence types.
Reaction times As in Experiment 1, we analyzed all trials,
regardless of how the comprehension questions were
answered. Also, as in Experiment 1, reaction time data
points that were more than 2.5 standard deviations away
from the mean residual RT for a position within a condition
or less than 100 msec in the raw data were excluded from
the analyses, affecting 3.7% of the data. Figure 4 presents
the mean reaction times per region across the four
conditions in the experiment.
-800
-600
-400
-200
0
200
400
600
800
The janitor who frustrated the
plumber/ who the
plumber frustrated
lost the key on the street.
Subject / Easy Math
Object / Easy Math
Subject / Hard Math
Object / Hard Math
Figure 4: Reaction times per region in the four conditions
of Experiment 2. The critical region is circled.
We first present the analysis of the critical region, Region
2, which included the RC (“who frustrated the plumber” /
“who the plumber frustrated”). A 2x2 ANOVA conducted
on this region revealed two significant main effects. First,
the hard-spatial-task conditions were read significantly
slower than the easy-spatial-task conditions
(F1(1,23)=22.98; MSe=5451605; p < .001; F2(1,31)=40.08;
MSe=6428277; p < .001). Second, the syntactically more
complex object-extracted RC conditions were read
significantly slower than the subject-extracted RC
conditions (F1(1,23)=15.59; MSe=3791349; p < .001;
F2(1,31)=22.94; MSe=4675397; p < .001). Critically, there
was no trace of an interaction between syntactic complexity
and the complexity of the spatial task (Fs<1). Moreover, the
effect of syntactic complexity in the hard-spatial-task
conditions was numerically smaller than that in the easy-
spatial-task conditions. This result rules out the attentional
explanation of the interaction that was observed in
Experiment 1.
In Region 1, consisting of the main clause subject (e.g.,
“The janitor”) together with the initial “pie-slice”, a 2x2
ANOVA revealed no significant effects. In Region 3, the
top-level verb and its object (“lost the key”), a 2x2 ANOVA
revealed a main effect of spatial task complexity
(F1(1,23)=39.36; MSe=9601145; p < .001; F2(1,31)=62.5;
MSe=12710598; p < .001), but no other effects. Finally, in
Region 4, the sentence-final prepositional phrase (“on the
street”), there was again an effect of spatial task complexity
(F1(1,23)=16.1; MSe=2925378; p < .001; F2(1,31)=45.2;
MSe=4061993; p < .001), but no other effects.
Discussion
The attentional account of the interaction between syntactic
and arithmetic complexity that was observed in Experiment
1 predicted a similar interaction between syntactic and
spatial-rotation complexity in Experiment 2. No such
interaction was observed. In fact, the numerical trend was
in the reverse direction. The lack of such an interaction
therefore argues against the attentional account of the
interaction observed in Experiment 1.
In general, the lack of an interaction between the
complexity of two tasks could arise for at least two different
reasons: (1) independent resource pools required for each
task; or (2) ceiling or floor effects on one or both of the
tasks, such that resources are either abundant or insufficient.
Hence, in order to argue that the results of Experiment 2 are
due to independent resource pools for the two tasks, we
need to be confident that the secondary task is neither too
complex nor too simple. It is unlikely that the spatial-
rotation task is too simple, because we observed a highly
significant complexity effect for this task. Neither is it
likely that the spatial-rotation task is too complex for the
following reasons. First, the performance on the spatial-
rotation task was extremely good, averaging only 30.3
degrees off from the target position. Second, the range of
the reaction times across conditions for the two experiments
was almost identical, suggesting that the arithmetic and
spatial-rotation tasks were comparable in difficulty.
Conclusions
In summary, using a dual-task paradigm, we have
demonstrated an on-line interaction between syntactic
complexity and arithmetic complexity in Experiment 1
suggesting that these two cognitive functions rely on
overlapping pools of verbal WM resources. Furthermore, in
Experiment 2, we have ruled out an attentional account of
the observed interaction by showing that a spatial task,
which does not rely on verbal WM resources, does not
interact with on-line sentence comprehension. These results
therefore support a WM framework in which sentence
processing and arithmetic processing overlap in the use of
verbal WM resources. The results are not consistent with
the hypothesis whereby sentence processing relies on an
independent pool of verbal WM resources (Caplan &
Waters, 1999).
An open question that we have not yet addressed is the
exact nature of the overlap in verbal working memory
resources for sentence and arithmetic processing. One
possibility is that both syntactic and arithmetic processes
involve a subservant mechanism for integrating verbal
symbolic information units. In this mechanism, the
difficulty of integrating linguistic elements depends on the
distance between elements to be connected. Relatedly, the
difficulty of adding numbers depends on the distance
between the initial addend and the resulting sum in the
computation on the number line. We leave it to future work
to distinguish this hypothesis from other possibilities.
Acknowledgments
DLT Rohde was supported by NIH NRSA 1-F32-
MH65105-02.
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