Neural Substrates of Mathematical Reasoning

Neuropsychology
2001, Vol. 15, No. 1, 115-127
Copyright 2001 by the American Psychological Association, Inc.
0894-4105/01/$5.00 DOI: 10.1037//0894-4105.15.1.115
Neural Substrates of Mathematical Reasoning: A Functional Magnetic
Resonance Imaging Study of Neocortical Activation During
Performance of the Necessary Arithmetic Operations Test
Vivek Prabhakaran, Bart Rypma, and John D. E. Gabriel!
Stanford University
Brain activation was examined using functional magnetic resonance imaging during mathematical problem solving in 7 young healthy participants. Problems were selected from the
Necessary Arithmetic Operations Test (NAOT; R. B. Ekstrom, J. W. French, H. H. Harman,
& D. Dermen, 1976). Participants solved 3 types of problems: 2-operation problems requiring
mathematical reasoning and text processing, 1-operation problems requiring text processing
but minimal mathematical reasoning, and 0-operation problems requiring minimal text
processing and controlling sensorimotor demands of the NAOT problems. Two-operation
problems yielded major activations in bilateral frontal regions similar to those found in other
problem-solving tasks, indicating that the processes mediated by these regions subserve many
forms of reasoning. Findings suggest a dissociation in mathematical problem solving between
reasoning, mediated by frontal cortex, and text processing, mediated by temporal cortex.
Problem solving of mathematical word problems is a
complex task that requires numerous cognitive operations
such as comprehension, reasoning, and calculation. Lesion
and functional imaging studies provide evidence about brain
regions involved in performing basic arithmetic calculations
or rote retrieval of arithmetic facts (e.g., 7 + 3 = 10, 5 X
4 = 20, 9 — 7 = 2) and in performing complex calculations
requiring intermediate steps (e.g., 273 - 45, 387 X 53).
These studies link simple calculation processes to parietal or
parieto-occipital regions (Appolonio et al., 1994; Burbaud
et al., 1995; Dehaene, Spelke, Pinel, Stanescu, & Tsivkin,
1999; Dehaene et al., 1996; Kahn & Whitaker, 1991; Levin
et al., 1996; Warrington, James, & Maciejewski, 1986;
Whalen, McCloskey, Lesser, & Gordon, 1997). More complex mathematical tasks that require multiple calculations
with intermediate steps are dependent also on frontal regions (Burbaud et al., 1995; Gerstmann, 1940; Grafman,
Passafiume, Faglioni, & Boiler, 1982; Jackson & Warrington, 1986; Roland & Friberg, 1985; Warrington et al.,
1986). This may be due to an increase in working memory
demands necessary for maintenance and manipulation of
intermediate products while problem solving.
The Necessary Arithmetic Operations Test (NAOT; Ekstrom et al., 1976) provides a measure of mathematical
reasoning because it is composed of word problems that
require the participant to determine the arithmetic calculations necessary to solve a given problem, but it does not
require the actual execution of these calculations. This test
allowed us to isolate brain regions involved in mathematical
reasoning while minimizing involvement of arithmetic calculations. In our study, we had participants solve word
problems drawn from the NAOT during functional magnetic resonance imaging (fMRI) scanning that required either one or two mathematical reasoning operations.
Psychometric analysis of NAOT performance provides
some basis for hypotheses about the neural substrates involved in mathematical reasoning. The high correlation
between mathematical (NAOT), visuospatial (Raven Progressive Matrices; Raven, 1962), and verbal (Sternberg Verbal Analogies; Ekstrom, French, Harman, & Dermen, 1976)
tasks despite differences in content (Snow, Kyllonen, &
Marshalek, 1984) suggests that processes important for reasoning may be common to all of these tasks.
Functional neuroimaging provides additional evidence
for speculation regarding the brain regions involved in
reasoning. In a study using the Raven Progressive Matrices
(Prabhakaran, Smith, Desmond, Glover, & Gabrieli, 1997),
there was greater activity in frontal regions compared to
posterior regions when participants performed high-level
reasoning problems (analytical) versus low-level reasoning
problems (figural). Because the Raven and the NAOT tasks
are highly correlated (Snow et al., 1984), we expected to
observe frontal lobe activations during NAOT performance.
Whereas prefrontal cortex may be involved in mathematical reasoning processes, parietal cortex may be involved in
mathematical calculation processes. The NAOT has shown
a low correlation with tasks requiring simple calculations
(e.g., arithmetic tests of addition, multiplication, and sub-
Vivek Prabhakaran, Program in Neurosciences, Stanford University; Bart Rypma and John D. E. Gabrieli, Department of
Psychology, Stanford University.
The research reported herein was supported by Office of Naval
Research Grant N00014-92-J-184 and National Institutes of Health
Grant NIAAG12995. We thank John E. Desmond for the analysis
software, Gary H. Glover for the pulse sequence, and Richard
Snow for advice and materials.
Correspondence concerning this article should be addressed
to Vivek Prabhakaran, Program in Neurosciences, Department
of Psychology, Stanford University, Jordan Hall, Building 420,
Stanford, California 94305. Electronic mail may be sent to
[email protected].
115
116
PRABHAKARAN, RYPMA, AND GABRIELI
traction) that have been linked to the parietal region. Lesion
evidence also suggests that although patients with parietooccipital lesions are impaired in performing rote calculations, they are still able to reason about problems and
estimate answers so long as frontal regions are intact (Jackson & Warrington, 1986). Therefore, we expected performing the NAOT to elicit minimal activation in parietal
regions.
Still other lesion research has provided evidence for the
involvement of the temporal regions (in addition to prefrontal regions) in solving word problems (Fasotti, Eling, &
Bremer, 1992; Fasotti, Eling, & Houtem, 1994). Other studies involving comprehension and reasoning of textual material have also shown prominent activations in temporal as
well as frontal regions (Goel, Gold, Kapur, & Houle, 1997;
Haier & Camilla, 1995; Just, Carpenter, Keller, Eddy, &
Thulborn, 1996; Nichelli et al., 1995; Partiot, Grafman,
Sadato, Flitman, & Wild, 1996). Thus, we hypothesized that
the frontal and temporal regions, but not parietal regions,
would be involved in performing the NAOT and sought,
therefore, to characterize the contribution of these different
systems.
Zero-operation problem
Jim wants to buy 1000 shares of a stock. He
multiplies 1000 by $30.50 - the price of a share.
How much money does Jim need in order to buy
the shares?
1. multiply
3. subtract
2. add
4. divide
One-operation problem
A wholesale meat dealer sells steak for $2.19
per pound. One day he sold 76 pounds. How
much money was taken in?
l.add
3. subtract
2. multiply
4. divide
Two-operation problem
Method
Participants
The participants were graduate students (3 men and 4 women)
from Stanford University. All were right-handed and between the
ages of 23 and 30 (M = 26). Each participant provided written
consent that was approved by the Institutional Review Board at
Stanford University.
Procedure
Prior to entering the scanner, participants were given instructions and shown three sample problems to familiarize them with
the task. During each trial, a word problem (drawn from the
NAOT; see Figure 1) was presented on the screen for 30 s with
four answer choices below it. In the last 5 s, one of the answer
choices was highlighted. The answer choices consisted of either
one or two mathematical operations (e.g., addition, division). The
participant squeezed a squeeze ball to indicate if the highlighted
choice was the correct answer to the problem. Half of the highlighted choices were the correct answers.
Task Design
The NAOT consists of standard word problems in mathematics
that utilize basic addition, subtraction, multiplication, and division
operations. The task is to indicate which arithmetic operations
would be used to solve the problem. The original task was modified for the scanner.
Three types of problems were created with increasing levels of
difficulty. Problem difficulty was operationalized by the number of
mathematical operations required in the problem.
Two-operation problems required simultaneous processing of
two operations (e.g., multiplication and addition) and thus required
extensive mathematical reasoning and text processing. The text
processing demands of the two-operation problems are greater
than those of the one-operation problems, which are greater than
those of the zero-operation problems, because of the necessity of
A farmer has his home and barn insured for
$152,000. The yearly premium rate is $2.07 per
$100. How much does this insurance cost him
each year?
1. divide,add
3. divide,multiply
2. add,multiply
4. subtract,divide
Figure 1. Examples of problem types used in the experiment:
zero-operation problem (top), one-operation problem (middle),
and two-operation problem (bottom). Participants were instructed
to select one of the four response alternatives that would determine
the operation(s) required to solve the given problem.
deriving the two relations (see Figure 1) that are necessary to solve
a two-operation problem from the sentence constructions rather
than the single relation necessary in a one-operation problem and
the absence of relational construction in a zero-operation problem.
One-operation problems required only one operation (e.g., multiplication) and thus required minimal mathematical reasoning but
extensive text processing in comparison to the zero-operation
problems. The one-operation problems were used to control for
cognitive, sensory, and motor activation (e.g., reading the problem,
visual inspection of stimuli, eye movements, motor response) that
was irrelevant to the cognitive factors of interest.
Zero-operation problems consisted of word problems in which
the solution was provided in the text of the problem and the
participants were asked to identify the solution from the choices
provided below and thus required minimal mathematical reasoning
or text processing. The zero-operation problems were used to
control for sensory and motor activation (e.g., visual inspection of
stimuli, eye movements, motor response) that was irrelevant to the
cognitive factors of interest.
Three sets of problems were presented to participants in the
scanner. The first set contained 12 problems or 6 cycles of alter-
NEURAL SUBSTRATES OF REASONING
nating problem types of a one-operation followed by a zerooperation problem (the one-operation/zero-operation condition).
The second set contained 12 problems or 6 cycles of alternating
problem types of a two-operation followed by a zero-operation
problem (the two-operation/zero-operation condition). The third
set contained 12 problems or 6 cycles of alternating problem types
of a two-operation followed by a one-operation problem (twooperation/one-operation condition). Presentation of the three sets
of problems was counterbalanced across participants.
In order to obtain estimates of the amount of time participants
would take to perform these word-problem types, we had a different set of participants perform them in a behavioral study
outside of the scanner using a self-paced participant design. A total
of 16 participants (age range = 20-29, mean age = 25) were
tested on the 36 problems (12 two-operation, 12 one-operation,
and 12 zero-operation) that were seen in the scanner. Participants
were instructed to proceed at their own pace and select a number
between 1 and 4 corresponding to the four answer choices for each
problem. These problems were presented in a randomized fashion.
fMRI Methodology
Imaging was performed with a 1.5T whole-body MRI scanner
(General Electric Medical Systems Signa, Rev. 5.5, Milwaukee,
117
WI). For functional imaging, two 5-in. diameter local receive coils
were used for signal amplification. These coils were placed closely
abutting each side of the participant's head, with each coil placed
so that the center of the coil was immediately dorsal to the
temporal mandibular joint on an imaginary line between the corner
of the eye and the ear canal. Head movement was minimized using
a bite-bar formed with each participant's dental impression. A T2*
sensitive gradient echo spiral sequence (Lee, Glover, & Meyer,
1995; Meyer, Hu, Nishimura, & Macovski, 1992), which is relatively insensitive to cardiac pulsatility motion artifacts (Noll, Cohen, Meyer, & Schneider, 1995), was used for functional imaging
with parameters of repetition time (TR) = 720 ms, echo time
(TE) = 40 ms, and a flip angle of 65°. Four interleaves were
obtained for each image, with a total acquisition time (sampling
interval) of 2.88 s per image. Tl-weighted, flow-compensated
spin-warp anatomy images (TR = 500 ms; minimum TE) were
acquired for all sections that received functional scans. Eight
6-mm-thick slices were acquired in the horizontal plane of the
Talaraich and Tournoux atlas (Talaraich & Tournoux, 1988) starting from 7.5 mm below the anterior-commisure-posterior-commisure (AC-PC) plane, with a 1.5-mm interslice interval (see Figure
2). Stimuli were generated from a computer and back-projected
onto a screen located above the participant's neck via a magnetcompatible projector. Visual images were viewed from a mirror
Figure 2. The locations of eight axial slices analyzed in this study are depicted as white lines on
a sagittal localizer.
118
PRABHAKARAN, RYPMA, AND GABRIELI
mounted above the participant's head. The sequence of the presentations of the stimuli was synchronized with the imaging sequence of the scanner.
vidually for all horizontal sections. Following transformation, the
average z value for each pixel in a section was computed across
participants, and pixels that reached a statistical threshold of p <
.05 or lower were displayed on each map.
Data Analysis
Image analysis was performed off-line by transferring the raw
data to a Sun SPARCstation. A gridding algorithm was used to
resample the raw data into a Cartesian matrix prior to two-dimensional fast Fourier transform processing. Once individual images
were reconstructed, a time series of each pixel was obtained, and
correlation methods that take advantage of periodically oscillating
paradigms were used to analyze functional activation (Friston,
Jezzard, & Turner, 1994). Because a considerable amount of
artifactual signal that occurs over time is due to events that are
random with respect to the timing of the activation paradigm (e.g.,
pulsatile effects from blood, cerebrospinal fluid, or brain movement), correlations of the pixel responses over time with a reference function that represents the time of the expected activation
(based on the timing of stimulus presentation) were used to remove
artifacts [3, 23]. As described by Friston et al., the reference
function was computed by convolving a square wave at the task
frequency with a data-derived estimate of the hemodynamic response function. The frequency of the square wave was computed
from the number of task cycles divided by the total time of the
experiment. For the experiments, one task cycle consisted of a
control block and an experimental block, each of equal duration.
There were six cycles present over a 360-s scan (frequency
—0.0166 Hz). Correlations between the reference function and the
pixel response time series were computed and normalized (Friston
et al., 1994).
Functional activation maps were constructed by selecting
pixels that satisfied the criterion of z > 1.96 (representing
significance at p < .025, one-tailed). This map was then processed with a median filter with spatial width = 2 voxels to
emphasize spatially coherent patterns of activation. The filter
was used on the assumption that pixels with spuriously high z
values (i.e., false positives due to Type I errors) are less likely
to occur in clusters than pixels with genuinely high z values,
and thus clusters of pixels with high z values are more likely to
reflect an active region. The resulting map was overlaid on a
Tl-weighted structural image.
The procedure used to obtain composite maps of activation over
all participants was as follows: Average functional activation maps
were created by transforming each section from each participant to
a corresponding standardized horizontal section (Talaraich &
Tournoux, 1988) at the same distance above and below the AC-PC
plane (Desmond et al., 1995). This transformation was done indi-
Results
Behavioral Performance
Table 1 shows the accuracy of participants' performance
on the NAOT task within the scanner environment. Participants performed with high accuracy on the zero-operation
problems (M = 97%), slightly less well on the one-operation problems (M = 93%), and least well on the twooperation problems (M = 75%). Performance on each problem type did not differ significantly across scans. Therefore,
scores for each problem type were combined across scans
and examined in a repeated measures analysis of variance
(ANOVA). Scores differed significantly for the three problem types, F(2, 5) = 24.11, p < .0001. Participants performed significantly better on zero-operation than two-operation problems, ?(5) = 11.76, p < .0001 (one-tailed) and
on one-operation than two-operation problems, f(5) = 4.61,
p < .0058. No significant difference was found between
one-operation and zero-operation problems, t(5) = 1.18,
p < .29.
Table 2 shows the average time that it took participants to
solve the different problem types as well as the accuracy on
a participant-paced design of the task performed outside the
scanning environment. Scores differed significantly for the
three problem types, F(2, 15) = 35,48, p < .0001. Participants performed significantly better on zero-operation than
two-operation problems, t(l5) = 6.99, p < .0001 (onetailed) and on one-operation than two-operation problems,
t(l5) = 5.73, p < .0001. No significant difference was
found between one-operation and zero-operation problems,
f(15) = 0.70, p < .50. Reaction time differed significantly
for the three problem types, F(2, 15) = 48.31, p < .0001.
Participants were significantly faster on zero-operation than
two-operation problems, ?(15) = 7.38, p < .0001 (onetailed); on one-operation than two-operation problems,
f(15) = 6.72, p < .0001; and on zero-operation than oneoperation problems, t(l5) = 3.52, p < .0031.
Table 1
Performance (Percentage Correct) on Word Problems in the Scanner
Scan
Zero-operation/one-operation
M
SD
Two-operation/zero-operation
M
SD
One-operation/two-operation
M
SD
Zero-operation
Problem type
One-operation
97.2
6.9
91.7
13.9
97.2
6.9
Two-operation
75.0
17.4
94.3
75.0
17.4
119
NEURAL SUBSTRATES OF REASONING
Table 2
Self-Paced Performance on Word Problems Outside the Scanner
Performance measure
% accuracy
M
SD
Reaction time (in milliseconds)
M
SD
Zero-operation
97.9
4.8
11,229
5,193
fMRI Scans
The two-operation/zero-operation scan yielded a number
of cortical activations that were all greater for two-operation
than for zero-operation problems (see Figure 3 and Table 3).
Major foci of activity occurred bilaterally in the frontal
lobes in superior, middle, and inferior frontal gyri and
premotor areas (Areas 6, 8, 9, 10, 44, 45, and 46) and in the
temporal lobes in inferior, middle, and superior temporal
gyri (Areas 21, 22, 37, and 39). Minor foci of activity
occurred in left parietal areas in supramarginal and angular
gyri (Area 39) and bilaterally in superior and inferior parietal areas (Areas 7 and 40) and in early visual areas—
precuneus, medial occipital gyri, and lingual gyrus (Areas 7,
18, and 19) as well as bilateral anterior cingulate (Area 32).
The two-operation/one-operation scan yielded activations
that were greater for two-operation than one-operation problems (see Figure 3 and Table 3). Major foci of activity
occurred bilaterally in the middle, inferior frontal gyri and
premotor areas (Areas 6, 8, 9, 10, 44, 45, and 46). Bilateral
activation was seen in inferior and middle temporal gyri
(Areas 37, 21, and 19) and in precuneus, medial occipital,
and lingual gyri (Areas 18, 19, and 37). Minor foci of
lateralized activity were seen in left versus right superior
parietal, inferior parietal, angular, and supramarginal gyri
(Areas 7, 39, and 40) and bilaterally in the anterior cingulate
(Area 32).
The one-operation/zero-operation scan yielded activations that were greater for one-operation than for zerooperation problems (see Figure 3 and Table 3). In contrast
to the two-operation/zero-operation activations, one-operation/zero-operation activations were fewer and less pronounced when occurring in the same region. Minor foci of
activity occurred bilaterally in the middle, inferior frontal
gyri and premotor areas (Areas 6, 8, 9, 10, 44, 45, and 46),
bilateral supramarginal-angular gyri, and superior and inferior parietal regions (Areas 7, 39, and 40). Inferior, middle, and superior (Wernicke's area) temporal gyri (Areas 21
and 37) also showed activation.
Discussion
Solving word problems yielded fMRI activation of an
extensive, but specific, network of cortical regions. There
were major bilateral frontal and temporal activations and
minor left-lateralized parietal activations. Lesion studies of
word-problem solving have also highlighted the contribu-
Problem type
One-operation
96.9
6.0
14,023
4,630
Two-operation
78.1
10.0
30,658
13,018
tions of frontal and temporal regions, with posterior lesions
impairing comprehension of sentences and frontal lesions
impairing the formation of internal constructions necessary
for solving the word problems (Fasotti et al., 1992, 1994;
Lhermitte, Derousne, & Signoret, 1972; Luria, 1966). The
frontal and temporal involvement in mathematical reasoning contrasts with and complements the essential involvement of parietal lobes seen in prior studies of simple
calculation.
In the present study, the parietal, temporal, and frontal
activations had different characteristics across conditions
(see Figure 4). Parietal lobe activations showed minimal
change as participants performed zero-, one-, or two-operation problems. Temporal lobe activations increased proportionately as participants performed zero-, one- or twooperation problems. Frontal lobe activations changed minimally from zero to one operation and increased equally
from zero to two operations and from one to two operations.
Thus, major frontal lobe activation occurred specifically for
two-operation problems.
The prefrontal activation for two-operation problems accords with a proposal that the prefrontal cortex plays a
critical role when reasoning demands integrating two or
more relations (Waltz et al., 1999). Patients with a frontal
variant of fronto-temporal dementia, where hypometabolism was initially localized in the prefrontal cortex, had
impaired performance only on two-relation verbal and spatial problems. Performance on one-relation problems was
intact. In the present study, one-operation word problems
required reasoning about one mathematical relation. Twooperation problems required the simultaneous processing of
two mathematical relations. The fact that substantial prefrontal activation occurred only for two-operation problems
supports the hypothesis that prefrontal regions are essential
for reasoning when two or more relations have to be
integrated.
In the present study, the linearly increasing temporal lobe
activations may reflect task-specific text-processing demands for word problems that increase steadily from zerothrough two-operation problems. Although all the word
problems contain similar numbers of words, text-processing
demands are likely to increase from zero- to one- to tworelational problems (Halford et al., 1998). The minimal
recruitment of the parietal lobe as the number of operations
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PRABHAKARAN, RYPMA, AND GABRIELI
Table 3
Cortical Activation by Problem Comparisons
Lobe
Region of activation
Hemisphere"Brodmann's area
X
y
z
Z score
Voxels"
Areas of greater activation for two-operation than zero-operation problems
Frontal
Superior frontal
RIO
R9
B8
L10
Superior-cingulate
Cingulate
Cingulate-mid
R32, 10
R24
R31
L31
L23, 24
R29
R23, 30
Cingulate-posterior
R30
R23, 31
R17.31
Superior-middle
Middle frontal
L23
L29
RIO
L10
L9
L9.46
L10, 46
L8
L6, 8
L6
RIO
R9
Middle-inferior
Inferior frontal
R8
R6
L9.44
R9, 44
L10,44
L44
L45.46
Inferior-premotor
Middle-premotor
Parietal
Premotor
Medial frontal
Postcentral
Inferior parietal
RIO
L6.44
L6, 9
L4, 6
L6
L10
R40
L39
L40
R40
Temporal
Angular gyms
Superior parietal
Superior
R39, 40
R39
L7
R22
L21.22
L22
Middle-superior
L22, 39
L21
R21,22
11
14
13
38
45
-2
-14
10
11
1
-7
-19
7
7
18
13
20
-11
-3
23
-19
-28
-37
-32
-44
-39
-45
-43
-44
-36
-33
-34
30
34
38
43
35
33
-45
44
-36
-58
-48
35
-52
-35
-23
-21
-5
62
-55
-50
-38
-42
38
31
34
28
-36
38
-50
-43
-61
-56
-43
47
54
59
57
17
26
19
64
48
30
-66
-62
-18
-40
-51
-48
-46
-62
-38
-37
56
41
44
51
55
15
28
20
36
46
29
11
3
59
45
25
9
29
-4
5
9
55
8
27
49
-4
-1
1
-12
60
-16
-58
-46
-43
-47
-47
-50
-49
-60
-59
5
-2
-13
-50
-54
-27
-17
-4
12
24
24
24
50
1
-4
1
12
12
32
20
20
20
24
12
24
12
-4
-4
-4
12
20
32
32
40
24
20
40
50
50
1
12
32
40
40
50
32
32
1
20
20
1
24
40
50
40
1
20
24
32
32
40
40
50
32
32
50
-4
1
1
12
20
-4
1
36
3.39
3.15
63
30
3.23
3.20
62
3.86
51
4.02
294
44
2.88
4.07
135
3.65
35
2.86
25
3.41
105
3.55
60
4.15
75
3.57
60
3.23
65
2.88
33
3.84
238
3.02
90
2.96
30
6.14
552
67
3.23
3.04
60
5.48
278
3.15
66
3.89
218
3.52
43
5.00
827
3.31
167
5.40
404
3.60
77
3.73
199
3.89
99
3.57
70
5.08
221
3.17
119
4.89
1,016
3.76
42
4.34
179
5.08
882
4.31
483
4.02
35
4.58
65
4.31
72
2.91
28
6.30
294
4.97
639
3.25
102
3.39
53
3.41
33
3.73
90
4.15
94
2.73
39
2.73
39
3.57
607
3.20
35
2.91
47
3.33
76
2.88
46
2.91
34
3.28
73
3.02
67
3.17
47
4.58
382
3.52
30
3.68
41
4.31
257
(table continues)
121
NEURAL SUBSTRATES OF REASONING
Table 3 (continued)
Lobe
Region of activation
HemisphereaBrodmann's area
x
y
z
Z score
44
34
46
50
54
-41
-44
-46
34
45
-36
-38
-22
-28
32
-28
-22
18
14
5
8
14
23
-4
6
12
7
-7
-4
4
11
-25
-18
-17
-21
-19
29
-59
-37
-44
-42
-58
-43
-76
-58
-59
-62
-72
-81
-87
-80
-85
-80
-77
-85
-66
-76
-41
-61
-64
-93
-91
-9
-86
-81
-75
-6
-67
-74
-67
24
-39
1
19
-4
-4
-4
1
12
1
20
-4
1
1
-4
24
24
20
20
-4
-4
-4
-4
1
1
1
1
1
1
-4
12
12
24
24
50
40
50
12
20
20
12
4.66
2.83
3.02
3.62
3.89
3.33
3.94
5.45
3.44
3.76
4.23
4.58
3.02
3.68
2.73
3.02
4.50
3.65
3.41
3.10
3.02
3.86
4.13
3.94
3.12
3.41
4.68
3.12
3.02
3.39
2.88
3.10
3.07
3.25
3.23
3.20
3.70
-4
12
12
50
32
-4
12
20
24
32
40
40
50
50
50
24
40
40
50
3.36
114
3.12
26
2.88
24
4.37
433
4.05
137
2.83
77
2.96
26
4.07
267
3.78
147
4.47
224
5.42
1,930
4.23
307
2.96
33
3.15
56
4.07
163
4.52
253
5.77
1,511
3.44
133
4.00
333
3.36
138
3.44
190
6.98
1,471
4.92
764
3.04
29
3.39
22
37
2.88
2.96
31
184
4.15
3.86
128
69
3.49
3.33
44
26
2.73
(table continues)
Voxels"
Temporal (continued)
Middle
Middle-inferior
R37
R21
R39
L21
L19
L37, 19
R37
Occipital
Occipital-temporal
Occipital
Middle occipital
Inferior occipital
Lingual-inferior
Occipital lingual
R37, 21
L18, 19
L19
L19
L19
R19
L18
L18
R18
R19
Cuneus-lingual
L17, 18
R17, 18
Cuneus
R17
R17, 18
L17, 18
Caudate
L18
R31
R7
L19
L7
L
Claustrum
R
Cuneus—precuneus
Precuneus
Subcortex
Frontal
Areas of greater activation for two-operation than one-operation problems
54
RIO
27
Superior frontal gyrus
Superior-middle frontal gyrus
Middle frontal gyrus
L10
L8
L9
RIO
RIO, 46
R9,46
R9
R8,9
R6
R4, 6
L9,46
L8,9
L8
L6
Middle frontal gyrus-premotor area
Middle-inferior frontal gyrus
Inferior frontal gyrus
Inferior frontal gyrus-premotor area
Premotor area
L4, 6
L10, 46
L44,9
R44, 9
15
-18
-2
-36
34
32
-39
49
42
44
40
27
37
32
-40
-37
-35
-41
-9
37
50
12
-47
56
-1
10
24
20
L45
L44
R44
R44
L4
40
-45
L44, 6
R44.45
28 .
46
48
41
25
26
10
27
10
7
-4
40
2
31
6
-30
-43
-47
44
-44
-57
-35
-44
44
L44
59
57
18
3
9
13
8
13
11
5
-4
-6
32
32
12
20
20
24
24
32
50
554
56
28
77
106
39
215
1,082
63
137
153
281
134
129
29
59
404
545
76
37
84
67
43
196
88
61
377
41
102
59
38
26
43
56
120
70
81
122
PRABHAKARAN, RYPMA, AND GABRIELI
Table 3 (continued)
Region of activation
Lobe
Frontal (continued)
Cingulate gyrus
Temporal
Inferior-middle temporal
Parietal
Middle temporal
Superior temporal
Insula
Inferior parietal
Occipito-parietal
Superior parietal-precuneus
Occipital
Angular gyrus
Precuneus
Inferior occipital
Lingual gyrus
Cuneus
Middle occipital
Superior occipital
Occipital
Occipital-cuneus
Cuneus-precuneus
Hemisphere3Brodmann's area
L31
B32
B32
L19, 37
R19, 37
L39
L39
R
L40
L40/7
R40/7
R19/40
R7
L39
L7
R7
L18
L17
L18
R17
R18
R17
R17/18
L37
L19
L19
L19
L19
R19
L7
R7
Frontal
Temporal
Parietal
'R = right; L = left.
X
y
z
Z score
Voxels"
-1
-18
6
-57
54
-59
-55
31
-48
-48
-48
-42
-38
-35
-40
-48
-39
37
33
37
-39
-28
26
-22
-7
-2
8
7
15
12
16
-54
-46
-31
-28
-34
-24
17
-7
-8
6
-6
27
18
-52
-61
-58
-58
19
-44
-44
-38
-57
-40
-52
-36
-45
-49
-43
-59
-47
-60
-59
-61
-78
-86
-83
-81
-89
-63
-95
-73
-68
-84
-73
-69
-85
-86
-83
-71
-66
-69
12
32
40
-4
-4
12
24
12
32
40
40
40
40
40
50
50
50
50
40
40
32
50
50
-4
-4
1
-4
1
1
-4
12
1
1
32
40
24
24
24
40
50
50
3.31
3.20
2.83
4.79
4.92
3.39
3.31
3.23
3.78
3.68
3.17
3.68
3.12
3.52
3.20
2.91
3.07
3.49
3.89
3.49
3.60
4.18
3.33
3.04
4.05
3.07
5.50
2.94
2.88
4.05
4.26
4.13
3.47
3.39
3.78
2.94
3.07
2.80
3.52
4.26
3.39
65
43
35
950
524
81
34
114
52
19
43
521
53
320
88
51
112
200
82
149
82
650
67
82
119
48
732
74
25
142
174
48
29
133
278
110
56
32
123
428
25
4
12
20
24
24
32
32
40
40
-4
24
45
45
32
32
24
40
-4
4
-4
32
40
45
45
50
50
3.36
3.02
3.94
3.12
3.36
3.81
3.92
3.97
3.68
3.39
3.31
2.88
3.31
4.23
3.12
3.07
3.39
3.17
3.52
2.94
3.39
68
29
105
26
58
322
243
627
393
116
25
38
182
214
108
21
123
99
55
47
132
114
38
29
35
21
Areas of greater activation for one-operation than zero-operation problems
Middle frontal gyrus
L10
-42
55
L10
-44
51
L46
-50
27
L9,46
-40
31
R46
51
30
L9
-41
17
R9
39
32
L8,9
-46
16
R8, 9
36
12
Superior frontal
RIO
33
40
1
59
L8
-34
14
~4
23
Inferior frontal gyrus
L44
-41
3
R44
41
11
Cingulate
R23
12
-45
R32
3
16
Middle temporal gyrus
R21
55
-87
L21
-62
-42
Inferior temporal gyrus
L19
-54
-41
Angular gyrus
R39
31
-57
Inferior parietal
L40
-44
-45
L40
-43
-45
R40
38
-43
Superior parietal
L7
-35
-59
R7
20
-63
b
1.41 X 1.41 X 6 mm3.
3.33
3.02
3.36
2.91
NEURAL SUBSTRATES OF REASONING
123
Figure 3. Statistical parametric maps of three comparisons: two-operation (two-op) versus zerooperation (zero-op) tasks (top row), two-operation versus one-operation (one-op) tasks (middle
row), and one-operation versus zero-operation tasks. Each column depicts activation from the same
section. Sections A, B, C, and D (Talaraich & Tournoux, 1988) correspond to Slices 1 (4 mm below
the anterior-commisure-posterior-commisure plane), 6 (32 mm above the anterior-commisureposterior-commisure plane), 7 (40 mm above the anterior-commisure-posterior-commisure plane),
and 8 (50 mm above the anterior-commisure-posterior-commisure plane) in Figure 2. Functional
maps are normalized and scaled with the lowest significant correlation magnitudes appearing in
white-gray and the highest in dark black. The left side of the image corresponds to the left side of
the brain. Section A depicts temporal and occipital areas at bottom and frontal area at top. Sections
B, C, and D depict parietal and occipital areas at bottom and frontal area at top. The two-operation/
zero-operation comparison, as well as the two-operation/one-operation comparison, reveals major
foci of activation in bilateral frontal (B-C) and temporal lobes (A) and minor foci of activity in
parietal lobes (B-D). The one-operation/zero-operation comparison reveals minor foci of activation
in frontal lobes (B-C).
increased is consistent with the minor demands placed on
the actual calculation component of word-problem solving
in the NAOT.
The frontal lobe activations observed in the present study
have also been found on other problem-solving tasks that
differ considerably in stimulus content and reasoning demands, such as the Raven Progressive Matrices (Prabhakaran et al., 1997), the Weather Prediction task (probabilistic
classification; Poldrack, Prabhakaran, Seger, & Gabrieli,
1999), the Tower of London task (Baker et al., 1996), and
visual prototype learning (Seger et al., 2000). These tasks
differ in terms of verbal versus visuospatial stimuli, deter-
ministic versus probabilistic outcomes, and whether performance changes with learning. Despite their many differences, these reasoning tasks yield bilateral frontal lobe
activations in relatively circumscribed locations (see Figure 5).
A question of interest is what differentiates the left frontal
and right frontal activations ubiquitously present in reasoning studies. Insights about the asymmetric roles of the
frontal lobes in mathematical reasoning can be drawn from
a word-problem-comprehension model proposed by Nathan
and Kintsch (Nathan, Kintsch, & Young, 1992). This model
is conceptualized to involve textual representation (a text-
124
PRABHAKARAN, RYPMA, AND GABRIELI
Frontal Lobe
Temporal Lobe
2SOO3000O< 2000-
o
o
1.
w
2000-
1-
I
9
1500-
g 10009
^< 500-
1000-
Z
one vs. zero
two vs. one
two vs. zero
Number of operations
one vs. zero
two vs. one
two vs. zero
Number of operations
Figure 4. Activity levels in the temporal (left graph) and frontal (right graph) lobes. The number
of significantly activated pixels (p < .05) in the two lobes for each comparison (one vs. zero, two
vs. one, and two vs. zero operations) is shown.
base), a situation model (a conceptual understanding of the
problem to be solved), and the problem model (a mathematical formalization of the problem to be solved). The
temporal lobe activations in the present study may corre-
spond to the textbase. Representations of situation models
use real-world knowledge and are often spatial in nature,
whereas representations of problem models use formal relations are often abstract, symbolic, and nonspatial in nature
Figure 5. Diagrammatic representation of foci of activation in the dorsolateral prefrontal cortex for
reasoning tasks. The letters represent reasoning tasks: M = mathematical reasoning; T = Tower of
London; P = prototype learning; W = Weather Prediction task; F = Figural Raven's tasks; A =
Analytic Raven's task. The top row depicts a left sagittal, coronal, and axial view of the brain. The
bottom row depicts a coronal, axial, and right sagittal view of the brain. The horizontal line
represents the y-axis and the vertical line represents the z-axis in the sagittal view. The horizontal
line represents the *-axis and the vertical line represents the z-axis in the coronal view. The
horizontal line represents the *-axis and the vertical line represents the y-axis in the axial view.
NEURAL SUBSTRATES OF REASONING
(Greeno, 1989; Kintsch & Greeno, 1985). Word problems require that inferences be made relating the situation
model to the problem model for successful mathematical
reasoning.
Bilateral frontal activation during NAOT performance
may reflect separable right and left unilateral representations, respectively, of the situation and problem models.
Support for such a hemispheric asymmetry comes from an
electroconvulsive therapy study (Deglin & Kinsbourne,
1996). In a syllogism task, under conditions of left-hemisphere suppression, participants solved syllogisms using
prior knowledge of real-world facts (as situations). Under
conditions of right-hemisphere suppression, participants
solved syllogisms using formal or logical operations (as
problems). Wharton and Grafman (1998) related this to
other findings and concluded that content-independent reasoning is mediated by the left hemisphere (i.e., problem
model), whereas content-dependent reasoning (i.e., situation
model) is mediated by the right hemisphere. Convergent
evidence for this hemispheric asymmetry comes from an
fMRI study of the Raven Progressive Matrices (Prabhakaran et al., 1997). In that study, participants performed two
types of reasoning tasks. Figural problems required participants to solve matrices on the basis of quantitative differences between adjacent elements of a matrix. These problems could be solved by visuospatial analysis and required
only construction of a situation model with minimal analytic
reasoning. Analytic problems required participants to solve
matrices on the basis of formal operations or rules applied to
elements of the matrix; this required construction of both a
problem model and a situation model. There was right
frontal lobe activation when participants solved figural
problems (i.e., construction of situation model) but bilateral
frontal lobe activation when they solved analytic problems
(i.e., construction of situation and problem models; see
Figure 3). The situation-problem proposal posits that right
frontal activation will occur for all problems, but that left
frontal activation will occur only for problems that demand
formal or logical operations.
Support for the spatial nature of situation models and the
nonspatial nature of problem models comes from studies
examining which frontal regions mediate spatial and nonspatial working memory. There is a close relation between
reasoning (mental models) and working memory (Kyllonen
& Christal, 1990). This relation is evident via behavioral
correlations in child development (Fry & Hale, 1996), normal adults (Kyllonen & Christal, 1990), aging (Salthouse,
1993), and patient studies (Gabrieli, Singh, Stebbins, &
Goetz, 1996) and in similar locations of activations that
occur in reasoning and working memory studies in which
participants maintain and manipulate simple information
over a brief period (Baker et al., 1996; Prabhakaran et al.,
1997). A meta-analysis of working memory studies
(D'Esposito, Aguirre, & Zarahn, 1998) showed predominantly right dorsolateral prefrontal cortex (DLPFC) involvement in spatial representations and bilateral DLPFC involvement in nonspatial representations. Thus, it is plausible that the right frontal regions make use of spatial working
memory resources to construct a situation model and left
125
frontal regions make use of nonspatial working memory
resources to construct a problem model.
In addition to DLPFC activation, regions thought to mediate storage and transformational processes of verbal and
numerical information (e.g., letters, digits, and semantic
information) were activated, including left inferior frontal
operculum, Broca's area (44,45, and 47), premotor area (6),
left parietal regions, supramarginal and angular gyrus, and
Wernicke's area (39, 40; Fiez et al., 1996; Gabrieli, Desmond, et al., 1996; Paulesu, Frith, & Frackowiak, 1993;
Petrides, Alivisatos, Meyer, & Evans, 1993; Prabhakaran,
Narayanan, Zhao, & Gabrieli, 2000; Roland & Friberg,
1985; Rypma, Prabhakaran, Desmond, Glover, & Gabrieli,
1999; Smith, Jonides, & Koeppe, 1996). This provides
further evidence for the close link between reasoning and
working memory.
The present study has a number of limitations. One is the
considerable differences in difficulty of zero-operation, oneoperation, and two-operation problems. This was shown by
the self-paced behavioral study of the NAOT, where there
were significant differences in both accuracy and reaction
time among the different types of NAOT problems (see
Table 2). Ideally, the time and amount of mental processing
would approximate one another across conditions as did the
perceptual and motor demands of the three conditions. Such
equivalence, however, may be logically impossible when
comparing tasks that are aimed to differ greatly in their
demands on high-level reasoning. One approach would be
to increase the number of trials in the easier conditions. This
could equate total mental processing time, but at a cost of
mismatching perceptual and motor demands across conditions. Another approach would be to analyze only an equivalent unit of time from the different trials (e.g., the first 10 s,
given that it took, on average, 11s for the zero-operation
problems, 14 s for the one-operation problems, and 30 s for
the two-operation problems). The problem with this approach is that it does not take into account the unknown
temporal distribution of the cognitive processes of interest
(word reading, text comprehension, and reasoning). The
initial 10-s period of the two-operation problems may consist of text processing (given that it took, on average, 11 s to
perform zero-operation problems that mainly consisted of
text processing). From this perspective, analyzing the initial
10 s in these different problem types may isolate only neural
substrates involved in text processing rather than neural
substrates involved in mathematical reasoning. Event-related designs, or other strategies, may isolate the mental
processes independent of processing duration. An advantage to the present format is that problems were almost
identical to those used in the NAOT that have been carefully
designed to manipulate problem difficulty. A second issue is
that technical limitations prevented scanning of the complete brain, and therefore we did not have the opportunity to
examine activations in potentially relevant structures like
the cerebellum. Furthermore, deep structures such as the
basal ganglia and thalamus may be active but sufficiently
distant from the surface coils that such activation may not
have been measurable. Third, the participants in this study
were graduate students, and it remains to be determined how
126
PRABHAKARAN, RYPMA, AND GABRIELI
these findings extend to other groups. These issues may be
resolved in future positron emission tomography or fMRI
studies.
The present study, nevertheless, provides initial evidence
characterizing the neural network underlying the ability to
reason mathematically. Other studies have confirmed that
arithmetic calculations are highly dependent on left parietal
areas. The present results indicate that solving mathematical
word problems is dependent on a fronto-temporal network
and that the mathematical reasoning component in solving
word problems is largely dependent on bilateral frontal
regions.
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Received May 20, 1999
Revision received July 20, 2000
Accepted July 22, 2000