Journal of the Neurological Sciences 208 (2003) 31 – 38
Calculation impairment in neurodegenerative diseases
Casey Halpern, Corey McMillan, Peachie Moore, Kari Dennis, Murray Grossman*
Department of Neurology-3 Gates, Hospital of the University of Pennsylvania, 3400 Spruce Street, Philadelphia, PA 19104-4283, USA
Received 9 April 2002; received in revised form 26 September 2002; accepted 22 October 2002
We examined oral calculation in patients with corticobasal degeneration (CBD; N = 17), frontotemporal dementia (FTD; N = 17), and
Alzheimer’s disease (AD; N = 20), as well as 17 healthy seniors matched for age and education. Our calculation model involves at least three
components: numerosity, combinatorial processes, and executive resources such as working memory. We assessed addition, subtraction,
multiplication, and division involving small numbers (small, single-digit answers) and large numbers (larger, single- and double-digit
answers). We also assessed dot counting for small numbers (2 – 5) and large numbers (6 – 9), as well as a measure of working memory. All
patient groups differed from healthy seniors in oral calculation. CBD (36% correct) and FTD (65% correct) demonstrated a significant overall
impairment in oral calculation relative to AD (76% correct). CBD (66% correct) had more difficulty counting dots overall relative to AD
(94% correct) and FTD (86% correct), consistent with our hypothesis that the calculation deficit in CBD is due in large part to a numerosity
deficit. FTD had more difficulty relative to AD in their performance of reverse digit span, consistent with our hypothesis that FTD patients’
executive resource limitation contributes to their pattern of calculation impairment.
D 2002 Elsevier Science B.V. All rights reserved.
Keywords: Number; Calculation; Working memory; Alzheimer’s disease; Frontotemporal dementia; Corticobasal degeneration
1. Introduction patients with neurodegenerative diseases are relatively
impaired with calculation [12,17 – 20]. Moreover, as we
Mental calculation reflects our knowledge and ability to note below, there is a reason to believe that different groups
use numbers. Despite our daily use of numbers, the neural of demented patients will have deficits with components of
basis for calculation remains unclear. Several studies have this calculation model that are qualitatively different.
begun to assess calculation and the collaboration of cerebral A sense of numerosity supports knowledge of small,
processes that it requires [1,2]. Much of the work concerned single digits, and possibly combining small numbers with
with calculation difficulties has come from occasional ob- elementary operations such as addition via counting.
servations of single case studies [3 –5] and rare reports of Patients with impaired number knowledge thus may expe-
calculation performance in groups of patients, particularly rience difficulty on all but the very simplest of calculations
those with lateralized disease due to stroke [6– 8]. The present with small numbers. Operations such as division, by com-
study investigates oral calculation performance compara- parison, often involve more complex combinations of num-
tively in groups of patients with neurodegenerative diseases. bers that may also require retaining an intermediate solution
Our model of calculation involves three components: in mind, particularly when a memorized set of number facts
numerosity, or number meaning; combinatorial processes, or is not involved [1,9]. Executive limitations are likely to
the various arithmetic operations we apply to numbers; and contribute difficulty in performing these more demanding
executive resources such as working memory that support calculations. Poor working memory may also limit the
complex calculation by holding in mind the results of processes involved in any calculations with larger numbers,
intermediate computations. We hypothesize that groups of even for simpler calculations such as addition [10,11]. We
examined these hypotheses in patients with neurodegener-
ative disease affecting these components of calculation.
* Corresponding author. Tel.: +1-215-662-3361; fax: +1-215-349-
Clinical descriptions of acalculia have been noted in
8464. CBD , but there have been no formal investigations of
E-mail address: email@example.com (M. Grossman). number knowledge or calculation impairment in these
0022-510X/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved.
32 C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38
patients. We expected CBD patients of which we are aware Table 1
Means and standard deviations for age, years of education, MMSE, and
to have a profound impairment in calculation with all but the
disease duration in patient groups and healthy seniors
smallest numbers and the simplest operations due in large
AD FTD CBD WNL
part to their limited number knowledge. We also assessed
their number knowledge by analyzing their performance on N 20 17 17 17
Age (year)a 72.20 F 8.53 66.88 F 8.70 66.94 F 6.67 67.41 F 8.38
a dot-counting task. We predicted significant difficulty on
Gender 10 M; 10 F 6 M; 11 F 5 M; 12 F 8 M; 9 F
this task, consistent with their number impairment and Education 14.30 F 3.18 14.56 F 2.51 14.82 F 3.76 14.74 F 2.68
difficulty in performing simple oral calculations. (year)a
Impairment on tasks requiring working memory is evi- MMSEb 21.70 F 4.35 21.41 F 6.77 16.94 F 6.92 29.06 F 1.20
dent in FTD patients [13 –15], but we are not aware of score
(max = 30)
deficits in number knowledge in FTD. Indeed, one clinical
Disease 69.35 F 35.20 43.65 F 15.79 50.69 F 14.15 NA
report of a patient with semantic dementia, a subgroup of durationc
FTD, describes isolated preservation of number knowledge (months)
in the face of considerable difficulty with other domains of a
T tests showed that patient groups and healthy seniors did not differ at
knowledge . In this context, we expected that FTD the p < 0.05 level for age or education except for AD and CBD patients who
patients, except those with semantic dementia, would have differ in age. According to Pearson’s correlations, age and education do not
calculation difficulty with more demanding operations such correlate with any patient groups’ oral calculation performance.
MMSE—Mini Mental Status Exam.
as division, and would also be impaired for computations c
No correlation was found between disease duration and oral
involving large numbers since these appear to be relatively calculation performance.
dependent on working memory.
Perhaps the most common neurodegenerative condition is
Alzheimer’s disease (AD). Acalculia has been demonstrated
previously in AD [17 – 20], but the basis for this difficulty fluent aphasia (PNFA), and a third group consisting of non-
has been examined rarely . AD appears to compromise aphasic FTD patients with primarily behavioral and execu-
executive functioning . Impairment on tasks requiring tive limitations (BEHAV). This was based on consensus
working memory has been noted frequently in AD patients evaluations of semistructured interviews, detailed mental
[23 –25]. However, we are not aware of deficits involving status evaluations, and clinical neurological assessments
number knowledge. Thus, we also expected some difficulty . We are not aware of published consensus criteria for
for AD patients with resource-demanding calculation prob- the clinical diagnosis of CBD although several experts in the
lems and problems involving large numbers. area have recommended clinical features important in the
diagnosis of CBD, as reviewed in Riley and Lang .
Based on a review of the literature concerned with clinical-
2. Methods pathological diagnosis in CBD [31 –35], the criteria we have
developed include cortical sensory deficit, extrapyramidal
2.1. Subjects features such as asymmetric rigidity but little resting tremor,
unilateral or asymmetrical alien hand, apraxia, visual per-
We examined 54 patients with neurodegenerative dis- ceptual-spatial difficulty, and anomia that are insidious in
eases and 17 healthy seniors who were right-handed English onset and gradual in progression. Other causes of dementia
speakers. These patients were diagnosed with Alzheimer’s were excluded by history, physical exam, serum studies, and
disease (AD), frontotemporal dementia (FTD), and cortico- structural brain imaging. Structural and functional neuro-
basal degeneration (CBD). All groups were matched for imaging in all patients were consistent with the clinical
both age and education except AD patients who were diagnosis. The participants were not taking sedative medi-
significantly older than CBD patients. Demographic fea- cations at the time of assessment, although most of the
tures are summarized in Table 1. All patients were identified patients were taking cholinergic-supplementing agents (e.g.
in the outpatient clinic at the Department of Neurology in rivastigmine) and some were taking serotonin-specific reup-
the University of Pennsylvania Medical Center. The clinical take inhibitor antidepressants (e.g. sertraline and paroxe-
diagnosis was made by a board-certified neurologist with tine). This study was approved by the IRB of the University
expertise in the diagnosis of dementing conditions. The of Pennsylvania, and all patients and responsible caregivers
basis for these clinical diagnoses was the National Institute participated in the Informed Consent procedure.
of Neurologic and Communicative Diseases and Stroke – Overall dementia severity was assessed with the Mini
Alzheimer’s Disease and Related Disorders Association Mental State Examination (MMSE) , a general assess-
(NINCDS –ADRDA)  criteria for AD, and Lund-Man- ment of cognition that evaluates orientation, anterograde
chester criteria  for FTD modified by McKhann et al. memory, language, executive functioning, and visual con-
. In order to account for the different behavioral dis- struction on a 30-point scale. CBD patients had a signifi-
orders among the FTD patients, we divided this group into cantly lower MMSE score than AD patients [t(35) = 3.04;
three subgroups: semantic dementia (SD), progressive non- p < 0.05] and FTD patients [t(32) = 2.55; p < 0.05] although
C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38 33
FTD patients did not differ from AD patients in their MMSE 2.2.2. Dot-counting task
scores. To assess oral calculation in a manner that minimizes To assess numerosity, we presented 16 arrays of dots
the potential confound associated with differences in overall arranged side-by-side in two columns of eight. Each array
dementia severity, we used analyses of covariance (ANCO- contained large, black, filled circles, where half included a
VAs) in our group-wise comparisons, covarying for MMSE. small number of dots (two to five) and half included a larger
In a recent study evaluating diagnostic criteria in FTD and number of dots (six to nine). Half of each set of dots was
AD patient groups that were equated on overall MMSE arrayed in a regular pattern and half in an irregular pattern.
score, the authors found that the impairment on MMSE can These arrays were presented in a fixed, random order.
be diagnostically useful . However, the MMSE may not Patients were asked to determine the number of dots in
account for aspects of disease that contribute to overall each array and we report patient accuracy.
severity in FTD and CBD, even though this test is generally
thought to reflect the severity of cognitive impairment in 2.2.3. Reverse digit span 
AD. For example, it is limited in its assessment of executive This is a brief assessment of working memory. For
functioning, and its simple memory and language tasks are reverse digit span, we give each patient a sequence of digits
relatively insensitive to the early stages of FTD . beginning at a sequence length of two digits and gradually
Unfortunately, there are no well-validated brief measures increasing up to seven digits, or until an error is made.
of disease severity for FTD and CBD. It is beyond the scope Subjects are asked to immediately repeat the sequence in
of this study to determine whether the MMSE equally reverse order. We report the longest string of digits that can
reflects overall disease severity across these patient groups. be repeated accurately in reverse order.
In this context, we supplemented our MMSE covariant by
covarying the duration of the disease as well. AD patients 2.3. Procedures
[t(32) = 2.75; p < 0.01] had significantly longer disease
duration than FTD patients, as summarized in Table 1, The entire calculation protocol was administered in one
although CBD patients differed from neither FTD patients session, requiring about 30 min to complete. The compo-
[t(31) = 1.35; ns] nor AD patients [t(31) = 1.98; ns]. In order nents of this protocol were administered in a fixed order,
to confirm the broad patterns of cognitive performance that beginning with the reverse digit span procedure, followed
are consistent with the clinical diagnosis in these patients, by oral calculations and finishing with dot counting. The
we also administered measures of anterograde memory, background procedures were obtained on average within 3
confrontation naming, and visuospatial constructional skills. months of the calculation battery.
Patterns of performance on these tasks were entirely con-
sistent with the clinical diagnosis.1 2.4. Statistical analysis
2.2. Materials All statistical tests are designated significant at a p value
less than 0.05. Nonparametric statistical tests were used to
A pencil and paper formatted task was presented to each test the null hypothesis that differences in oral calculation
subject as part of a larger protocol. This included measures and dot counting do not exist between neurodegenerative
aimed at assessing our calculation model including oral patients and healthy control subjects. We used nonparamet-
calculation, dot counting, and reverse digit span. ric statistics for analyses involving control subjects because
they produced so few errors in our calculation and dot-
2.2.1. Oral calculation counting tasks. In order to assess qualitative differences in
For the oral calculation task, patients were presented with calculation performance between patient groups that assess
32 calculation problems aurally, one at a time. Calculation our hypotheses and take into account the overall disease
problems consisted of eight addition (e.g. 2 + 4=), eight severity, we used parametric, multivariate analyses of cova-
subtraction (e.g. 3 À 1=), eight multiplication (e.g. 2 Â 3=), riance (MANCOVA), covarying for MMSE and disease
and eight division (e.g. 9 H 3=) problems. Four examples of duration, for all contrasts across patient groups.
each operation tested each subject’s ability to calculate with
small numbers and orally produce a response (small, single-
digit answers). We also tested each subject’s ability to 3. Results
calculate with large numbers and orally produce a response
(larger, single- and double- digit answers) with four addi- 3.1. Oral calculations
tional examples of each operation. Patients were given a
maximum of 10 s for each problem, and failure to respond Fig. 1 summarizes the mean ( F S.D.) scores for overall
in the allotted time was scored as incorrect. oral calculation accuracy in patient groups. A Kruskal –
Wallis analysis of variance (ANOVA) by ranks revealed that
According to Pearson’s correlations, performance on these measures patient groups differ from healthy seniors in overall oral
did not correlate with oral calculation performance. calculation [v2(3) = 28.92]. According to Mann – Whitney U
34 C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38
t test to show that CBD patients have more difficulty with
addition using large numbers relative to addition with small
numbers [t(16) = 4.39]. CBD patients also were more
impaired than AD patients for addition using small numbers
[ F(1,29) = 5.31]. CBD patients thus appear to have relative
difficulty with all but the simplest calculation operation,
addition involving only small numbers.
FTD patients were impaired in division relative to their
own performance on all other operations [addition: t(16) =
4.05; subtraction: t(16) = 4.00; multiplication: t(16) = 2.13].
These patients as a whole thus tended to have the greatest
calculation difficulty with the operation making the greatest
Fig. 1. Mean ( F S.D.) oral calculation accuracy. WNL = healthy seniors,
AD = Alzheimer’s disease, FTD = frontotemporal dementia, and CBD = cor- working memory demand. An analysis of FTD patients
ticobasal degeneration. demonstrated differences in relative accuracy between sub-
groups, including SD, PNFA, and BEHAV patients. PNFA
patients (% correct: 50.63 F 28.76) were significantly less
tests, each differed significantly from the healthy seniors at accurate in overall oral calculation performance than SD
least at the p < 0.05 level of significance. patients (% correct: 77.23 F 22.44) [ F(1,8) = 9.24]. Neither
To assess differences in oral calculation between patient SD patients nor PNFA patients differed from BEHAV
groups, we used a mixed-model repeated-measures analysis patients (% correct: 63.13 F 28.93).
of covariance (MANCOVA) with a between-subject factor An ANCOVA covarying for MMSE and disease duration
of group (3—AD, FTD, CBD) and within-subject factors of showed that FTD patients are impaired in their overall oral
operation (4—addition, subtraction, multiplication, division) calculation performance relative to AD patients [ F(1,30) =
and number size (2—small, large), covarying for MMSE 5.69]. AD patients were less impaired in their performance
and disease duration. We found a main effect for group with division relative to FTD patients [ F(1,30) = 6.96]. AD
[ F(2,45) = 4.92] as well as a group Â operation interaction patients were impaired in division relative to their own
effect [ F(6,135) = 2.19]. These findings are summarized in performance with all other operations except multiplication
Table 2. [addition: t(19) = 2.63; subtraction: t(19) = 3.46]. AD pa-
An ANCOVA covarying for MMSE and duration of tients thus were modestly impaired overall, tending to only
disease showed that CBD patients are more impaired in have mild difficulty with more demanding operations such
their overall oral calculation performance than AD patients as division and possibly multiplication.
and FTD patients [ F(1,46) = 4.57]. Within-group compar-
isons, using paired sample t tests, showed that CBD patients 3.2. Number knowledge
are impaired in all calculation operations relative to their
own performance of addition [subtraction: t(16) = 4.24; We performed between-group comparisons and within-
multiplication: t(16) = 3.57; division: t(16) = 2.88]. To test group correlations of dot-counting performance to help
our hypotheses that CBD patients are impaired on all but establish the role of numerosity in the calculation perform-
addition with the smallest numbers, we used a paired sample ance of CBD, FTD, and AD patients. Fig. 2 summarizes the
Means (% correct) and standard deviations for calculation operations and reverse digit span in patient groups and healthy seniors
AD FTD CBD WNL
Addition 82.50 F 23.08 72.79 F 36.51 50.00 F 33.37 99.26 F 3.03
Subtraction 82.50 F 21.61 75.74 F 29.47 32.35 F 36.46 97.79 F 6.61
Multiplication 71.88 F 36.02 60.29 F 33.14 36.03 F 35.32 93.38 F 15.38
Division 66.88 F 32.26 46.32 F 33.88 32.11 F 31.45 88.97 F 21.60
Reverse digit span 3.45 F 1.00 2.53 F 1.37 2.71 F 0.99 4.53 F 1.07
We performed an error analysis of incorrect responses to multiplication and division problems. This error analysis involved only multiplication and
division to minimize ambiguities associated with interpreting the nature of the error. We examined table errors (e.g. solutions taken from cells adjacent to the
true product, such as 6 Â 3 = 21), operation errors (e.g. misinterpretation of the operation indicated by the sign, such as 6 Â 3 = 9), and random (nontable) errors
(e.g. solutions that appear to be unrelated to a table error or misinterpretation of an operation, such as 6 Â 3 = 14). This examination of error types was derived
from past studies [5,9]. Mann – Whitney U tests demonstrated that all patient groups made more operation errors than healthy seniors. While Mann – Whitney U
tests showed that CBD patients [U = 85] make more random errors than healthy seniors, consistent with their impoverished number knowledge, FTD patients
[U = 68] appear to make more table errors. We cannot definitely state that table (and operation) errors were not random in nature, but we assume that any of the
table and operation errors are likely to include some random errors in proportion to their occurence in each of the groups.
C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38 35
with the oral calculation performance in CBD [r = 0.65],
FTD [r = 0.76], and AD [r = 0.54].
Our analyses of patients with neurodegenerative diseases
showed that they are impaired in oral calculation relative to
healthy seniors. Moreover, in line with our expectations, we
observed some qualitative differences between the oral
calculation performance of patient groups due to relative
impairments associated with number knowledge and exec-
Fig. 2. Mean ( F S.D.) dot-counting accuracy. CBD patients have the greatest overall calculation im-
pairment. CBD patients demonstrated considerable diffi-
culty with all calculation conditions except the simplest
patients’ dot-counting accuracy. A Kruskal – Wallis ANOVA operation, addition. Indeed, they were relatively successful
by ranks revealed that patient groups generally differ from at addition only for calculations involving the smallest
healthy seniors in counting dots regardless of array size numbers. Our analyses also showed that only CBD patients
[v2(3) = 28.83]. According to Mann – Whitney U tests, CBD are relatively impaired on the dot-counting task. These
patients [U = 21.00] and FTD patients [U = 53.50] differed observations imply that CBD patients have a profound
from healthy seniors in their dot-counting accuracy while numerosity deficit that contributes to their oral calculation
AD patients [U = 119.00] did not. To assess differences in difficulty. Our findings also suggest that ‘‘subitizing’’ (enu-
dot counting between patient groups, we performed a merating a small group of four or fewer objects rapidly by a
MANCOVA with a between-subject factor of group (3— process such as deconstructing the arrays of objects into
AD, FTD, CBD) and a within-subject factor of array size familiar patterns) small arrays of dots (1– 4) and ‘‘counting’’
(2—small and large numbers of dots) covarying for MMSE (an error prone and slower process of serially counting five
and disease duration. We found a significant main effect for or more objects) large arrays of dots (5 –9) appear to rely on
group [ F(2,45) = 3.28]. separate neural mechanisms, consistent with several pre-
An ANCOVA covarying for MMSE and disease duration vious studies [40 –43]. However, it is beyond the scope of
showed that the CBD patients are less accurate in counting this study to definitively state that these processes are
dots overall compared to AD patients and FTD patients distinct . The correlation between dot counting and
[ F(1,46) = 4.50]. FTD patients did not differ from AD calculation further suggests that limited number knowledge
patients [ F(1,30) = 3.52]. We also analyzed FTD subgroups. contributes to calculation difficulty in CBD.
BEHAV patients [ F(1,8) = 6.08] and PNFA patients CBD affects several cortical regions, but a review of the
[ F(1,7) = 4.20] were more impaired in counting dots than literature emphasizes core involvement of parietal cortex
SD patients. We also found that dot counting correlated with based on the frequent presence of clinical features such as
oral calculation in CBD [r = 0.51] and FTD patients spatial difficulty, cortical sensory loss, anomia, and apraxia
[r = 0.51]. [30,31,35,45,46]. Case studies with neuroanatomic correla-
tion [5,47] and functional neuroimaging work in healthy
3.3. Executive resources adults [48,49] implicate parietal cortex in number knowl-
edge. Functional imaging data and lesion studies in brain-
Table 2 summarizes the results of reverse digit span damaged patients have suggested that bilateral defects in the
performance. An ANCOVA was used to evaluate group region of the intraparietal sulcus play a central role in
differences in reverse digit span (4—CBD, FTD, AD, number processing [48,49]. We hypothesize that this dis-
WNL), covarying for MMSE and disease duration. We tribution of disease in CBD contributes to a deficit in
found a main effect for the group [ F(3,61) = 3.44]. T tests number knowledge and the profound calculation impair-
showed that all patients differ from healthy seniors in their ment seen in CBD.
reverse digit span performance. An ANCOVA covarying for It is possible to attribute difficulty with the calculation
MMSE and disease duration demonstrated that CBD and dot-counting tasks in part to a deficit in visuospatial
patients do not have difficulty relative to AD patients skills in CBD [12,33]. Our observations suggest that visuo-
[ F(1,29) = 0.67] or FTD patients [ F(1,29) = 2.41] in reverse spatial difficulty cannot fully explain the calculation deficits
digit span. FTD patients were more impaired than AD in CBD. No correlation was found between visual con-
patients for reverse digit span [ F(1,30) = 7.24]. FTD sub- struction and CBD patients’ oral calculation performance.
groups did not differ on this task [ F(2,12) = 1.68]. Accord- The calculation task was administered aurally, minimizing
ing to Pearson’s correlations, reverse digit span correlated errors due to spatial factors such as lining up written
36 C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38
numbers in a spatially inappropriate manner. Difficulty with striatal disease results in difficulty with overlearned calcu-
spatial relations may contribute to their dot-counting impair- lations such as multiplication tables . We did not find
ment, particularly for large arrays. However, the patients disproportionate evidence for errors with overlearned cal-
were impaired for small arrays and as impaired with irreg- culations. One possible reason for this discrepancy is that
ular arrays as the regular arrays that have a familiar our patients have a neurodegenerative disease also affecting
configuration less dependent on a spatial analysis. In sum, parietal cortex, whereas Dehaene and Cohen’s patient 
it appears that a spatial deficit cannot fully explain dot- suffered from a chronic, left-hemispheric capsulo-lenticular
counting difficulty although additional work is needed to hemorrhage that is likely to have led to significant reorgan-
confirm this. ization of cognitive function. Additional work is needed to
Some aspects of number knowledge have been hypothe- establish the precise role of subcortical areas in calculation.
sized to depend on verbal mediation implying that aphasia FTD patients are also impaired in their oral calculations.
can result in calculation difficulty [5,44]. Although some We demonstrated that their greatest impairment is on
CBD patients have been reported to be aphasic [50 – 52], our resource-demanding calculation operations such as division.
CBD patients have only mild naming difficulty without We hypothesized that their impairment on these relatively
gross aphasia or impaired comprehension. Moreover, sev- difficult calculation operations is due to an executive
eral case studies dissociate language and calculation [7,53 – limitation [13 – 15]. We found that FTD patients’ reverse
56]. Patients with aphasia have frequently been shown to digit span performance was impaired relative to healthy
have preserved number knowledge and unimpaired calcu- seniors and AD patients and this correlated with their oral
lation skills, and patients suffering from acalculia and a calculation accuracy. Impairments in tasks requiring work-
profound loss of number knowledge due to parietal disease ing memory have been noted in FTD patients [13 – 15]. We
have been reported to be not aphasic. These observations cannot completely rule out that their deficit may be due in
suggest that aphasia is unlikely to completely explain the part to degraded knowledge of the division operation itself.
acalculia that we observed in CBD. These patients did not show more operation errors than
A related possibility is that the CBD patients’ overall other patient groups. However, their production of many
dementia severity may contribute to their calculation diffi- table errors suggests that they were implementing the
culty. Unfortunately, there are no well-validated brief meas- appropriate operation but may have had difficulty holding
ures of overall dementia severity that allow us to covary intermediate operations in working memory so that the
performance in comparisons of CBD patients with AD correct solution could be selected reliably from the table.
patients and FTD patients. It is unlikely that overall demen- Functional neuroimaging studies show that complex
tia severity can fully explain the differences in oral calcu- calculation tasks increase activation in prefrontal areas that
lation that we observed and, indeed, correlations with are known to contribute to working memory functions
factors commonly contributing to measures of overall [43,58,59]. FTD patients have disease in this neuroanatomic
dementia severity that we assessed independently—such distribution [60 – 62]. Other clinical  and functional
as anterograde memory, confrontation naming, and visual- neuroimaging work  has associated more complex
constructional performance—did not correlate with oral calculations with prefrontal and inferior frontal cortices. In
calculation performance. We also supplemented MMSE this context, we expected that FTD patients would have
with disease duration, an important reflection of overall calculation difficulty with more resource-demanding oper-
disease severity. Nevertheless, a shortcoming of our study is ations.
that we cannot rule out some contribution of overall FTD has been associated with progressive aphasia .
dementia severity to CBD patients’ calculation performance. However, language difficulty is unlikely to account for their
CBD patients may have disturbed working memory  impaired calculation performance. This is because aphasia
and this may contribute to their poor calculation skills as presumably would interfere equally with all calculation
well. CBD patients had only mild difficulty on reverse digit operations, but these patients were more impaired in divi-
span, a measure of executive functioning. Their preserved sion than in other operations. Other evidence dissociating
reverse digit span performance relative to AD patients, language functioning and number processing comes from
despite profoundly impaired oral calculations, further sug- stroke patients [6– 8]. SD patients have been reported to be
gests that their calculation impairment is not entirely significantly impaired in their knowledge of semantic facts
dependent on limited working memory. We did find a involving numbers, such as the number of years in a
correlation between executive functioning and oral calcu- millennium and the number of corners in a cube . This
lation in CBD. It would be important to establish in the kind of measure does not assess number knowledge per se,
future whether this executive resource limitation is material- such as the meaning of the number ‘‘4’’. Instead, our
specific and related to the CBD patients’ particular difficulty observations are consistent with another report  that
with tasks dependent on their impoverished number sense, found number knowledge to be relatively preserved in
or is a broader, material-neutral deficit in executive func- patients with SD, emphasizing dissociation between knowl-
tioning possibly related to the striatal disease seen in CBD edge of semantic facts involving numbers and knowledge of
[31,40]. In this context, we note one study showing that number meaning per se.
C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38 37
AD patients were impaired in calculation relative to  Hittmair-Delazer M, Semenza C, Denes G. Concepts and facts in
calculation. Brain 1994;117:715 – 28.
healthy seniors. Unlike Crutch and Warrington , how-
 Dehaene S, Cohen L. Two mental calculation systems: a case study of
ever, we did not find that AD patients are the most impaired severe acalculia with preserved approximation. Neuropsychologia
dementia group in their calculation performance. We attrib- 1991;29(11):1045 – 74.
ute their impairment in part to a deficit of working memory.  Jackson M, Warrington E. Arithmetic skills in patient with unilateral
Several reports have described significant deficits in AD cerebral lesions. Cortex 1986;22:611 – 20.
patients on measures of executive function [23 – 25]. This  Dehaene S, Cohen L. Cerebral pathways for calculation: double dis-
sociation between rote verbal and quantitative knowledge of arith-
condition can compromise prefrontal brain regions  metic. Cortex 1997;33:219 – 50.
leading to difficulty with executive measures [30 – 32] as  Grafman J, Passafiume D, Faglioni P, Boller F. Calculation disturban-
well as more demanding aspects of calculation. Thus, we ces in adults with focal hemispheric damage. Cortex 1982;18:37 – 50.
also expected some difficulty for AD patients with resource-  Dellatolas G, Deloche G, Basso A, Claros-Salinas D. Assessment of
demanding calculation problems. Lafleche and Albert  calculation and number processing using the EC301 battery: cross-
cultural normative data and application to left- and right-brain dam-
indicate that AD patients’ impairment is unlikely to be a aged patients. J Int Neuropsychol Soc 2001;7:840 – 59.
result of distractibility. Our results also support the claim  Roselli M, Ardila A. Calculation deficits in patients with right and left
that AD patients are impaired on tasks that required working hemisphere damage. Neuropsychologia 1989;27:607 – 17.
memory relative to healthy seniors and this is correlated  McCloskey M, Aliminosa D. Facts, rules, and procedures in normal
with their oral calculation accuracy. The results of dot calculation: evidence from multiple single-patient studies of impaired
arithmetic fact retrieval. Brain Cogn 1991;17:154 – 203.
counting suggest that AD patients do not have limited  Oberauer K, Demmrich A, Mayr U, Kliegl R. Dissociating retention
number knowledge. and access in working memory: an age-comparative study of mental
AD has been associated with language difficulty . arithmetic. Mem Cognit 2001;29(1):18 – 33.
However, this would not fully account for their selective  Takayama Y, Sugishita M, Akiguchi I, Kimura J. Isolated acalculia
difficulty with only some operations but not the others. due to left parietal lesion. Arch Neurol 1994;51:286 – 91.
 Caselli RJ. Focal and asymmetric cortical degenerative syndromes. In:
Cummings et al.  found that despite language abnor- Litvan I, Goetz CG, Land AE, editors. Corticobasal degeneration.
malities, AD patients did not differ from controls in number Advances in Neurology, vol. 82. Philadelphia: Lippincott Williams
repetition and counting. & Wilkins; 2000.
In the present study, calculation impairment is evident in  Jagust WJ, Reed BR, Seab JP, Kramer JH, Budinger TF. Clinical-
physiologic correlates of Alzheimer’s disease and frontal lobe demen-
patients with certain neurodegenerative diseases relative to
tia. Am J Physiol Imag 1989;4:89 – 96.
healthy seniors. Calculation difficulty appears to be most  Pachana N, Boone K, Miller BL, Cummings JL, Berman N. Compar-
pronounced in CBD due to a severe deficit in number ison of neuropsychological functioning in Alzheimer’s disease and
knowledge and possibly a limitation in executive resources, frontotemporal dementia. J Int Neuropsychol Soc 1996;2:505 – 10.
sparing their performance only in addition with small  Johanson A, Hagberg B. Psychometric characteristics in patients with
numbers. A pattern of impaired calculation involving the frontal lobe degeneration of non-Alzheimer type. Arch Gerontol Ger-
iatr 1989;8:129 – 37.
most resource-demanding computations such as division is  Cappeletti M, Butterworth B, Kopelman M. Spared numerical abil-
evident in FTD. This appears to be due to an executive ities in a case of semantic dementia. Neuropsychologia 2001;39:
resource limitation. AD patients also appear to have a 1224 – 39.
resource-based limitation in calculation. Different patterns  Deloche G, Hannequin D, Carlomagno S, et al. Calculation and num-
of oral calculation difficulty in patients with neurodegener- ber processing in mild Alzheimer’s disease. J Clin Exp Neuropsychol
1995;17:634 – 9.
ative diseases are consistent with a model of calculation  Carlomagno S, Iavarone A, Nolfe G, Bourene G, Martin C, Deloche
involving partially dissociable components of number G. Dyscalculia in the early stages of Alzheimer’s disease. Acta Neurol
knowledge and executive resources. Scand 1999;99:166 – 74.
 Hirono N, Mori E, Ishii K, et al. Regional metabolism: associations
with dyscalculia in Alzheimer’s disease. J Neurol Neurosurg Psychia-
Acknowledgements try 1998;65:913 – 6.
 Grafman J, Passafiume D, Faglioni P, Boller F. Calculation disturban-
ces in adults with focal hemisphere damage. Cortex 1982;18:37 – 50.
This work was supported by the National Institute of  Crutch SJ, Warrington EK. Acalculia: deficits of operational and
Health (AG17586, AG15116, NS35867). Portions of this quantity number knowledge. J Int Neuropsychol Soc 2001;7:825 – 34.
work were presented at the meeting of the International  Arnold SE, Hyman BT, Flory J, Damasio AR, Van Hoesen GW. The
Neuropsychological Society in Toronto, February 2002 and topographical and neuroanatomical distribution of neurofibrillary tan-
gles and neuritic plaques in the cerebral cortex of patients with Alz-
at the meeting of the American Academy of Neurology in
heimer’s disease. Cereb Cortex 1991;1:103 – 16.
Denver, April 2002.  Lafleche G, Albert MS. Executive function deficits in mild Alzheim-
er’s disease. Neuropsychology 1995;9(3):313 – 20.
 Mack JL, Patterson MB. Executive dysfunction and Alzheimer’s dis-
References ease: performance on a test of planning ability, the Porteus Maze test.
Neuropsychology 1995;9(4):556 – 64.
 Dehaene S. Cerebral basis of number processing and calculation.  Cherry BJ, Buckwalter JG, Henderson VW. Memory span procedures
The new cognitive neurosciences. Cambridge: MIT Press; 2000. in Alzheimer’s disease. Neuropsychology 1996;10(2):286 – 93.
p. 987 – 98.  McKhann G, Drachman D, Folstein M, Katzman R, Price D, Stadlan
38 C. Halpern et al. / Journal of the Neurological Sciences 208 (2003) 31–38
EM. Clinical diagnosis of Alzheimer’s disease: report of the NINCDS –  Lhermitte F, Pillon B, Serdaru M. Human anatomy and the frontal
ADRDA work group under the auspices of Department of Health and lobes: I. Imitation and utilization behaviors: a neuropsychological
Human Services Task Force on Alzheimer’s Disease. Neurology 1984; study of 75 patients. Ann Neurol 1986;19:326 – 34.
34:939 – 44.  Gruber O, Indefrey P, Steinmetz H, Kleinschmidt A. Dissociating
 The Lund and Manchester Groups. Clinical and neuropathological neural correlates of cognitive components in mental calculation. Cereb
criteria for frontotemporal dementia. J Neurol Neurosurg Psychiatry Cortex 2001;11(4):350 – 9.
1994;57:416 – 8.  Cohen L, Dehaene S, Chochon F, Lehericy S, Naccache L. Language
 McKhann GM, Albert MS, Grossman M, Miller B, Dickson D, Tro- and calculation within the parietal lobe: a combined cognitive, ana-
janowski J. Clinical and pathological diagnosis of frontotemporal tomical and fMRI study. Neuropsychologia 2000;38:1426 – 40.
dementia: report of the work group on frontotemporal dementia and  Chochon F, Cohen L, van de Moortele PF, Dehaene S. Differential
Pick’s disease. Arch Neurol 2001;58:1803 – 9. contributions of the left and right inferior parietal lobules to number
 Neary D, Snowden JS, Gustafson L, et al. Frontotemporal lobar de- processing. J Cogn Neurosci 1999;11(6):617 – 30.
generation: a consensus on clinical diagnostic criteria. Neurology  Wenning G, Litvan I, Jankovic J, et al. Natural history and survival of
1998;51:1546 – 54. 14 patients with corticobasal degeneration confirmed at postmortem
 Riley D, Lang A. Clinical diagnostic criteria. In: Litvan I, Goetz CG, examination. J Neurol Neurosurg Psychiatry 1998;64:184 – 9.
Lang AE, editors. Corticobasal degeneration. Advances in Neurology,  Riley DE, Lang AE, Lewis A, et al. Cortical basal ganglionic degen-
vol. 82. Philadelphia: Lippincott Williams & Wilkins; 2000. eration. Neurology 1990;40:1203 – 12.
 Pillon B, Blin J, Vidailhet M, et al. The neuropsychological pattern of  Ikeda K, Akiyama H, Iritani S, et al. Corticobasal degeneration with
corticobasal degeneration: comparison with progressive supranuclear primary progressive aphasia and accentuated cortical lesion in supe-
palsy and Alzheimer’s disease. Neurology 1995;45:1477 – 83. rior temporal gyrus: case report and review. Acta Neuropathol 1996;
 Boeve BF, Maraganore DM, Parisi JE, et al. Pathologic heterogeneity 92:534 – 9.
in clinically diagnosed corticobasal degeneration. Neurology 1999;53:  Thioux M, Pillon A, Samson D, De Partz M-P, Noel M-P, Seron X.
795 – 800. The isolation of numerals at the semantic level. Neurocase 1998;4:
 Rinne JO, Lee MS, Thompson PD, Marsden CD. Corticobasal degen- 371 – 89.
eration. Brain 1994;117:1183 – 96.  Rossor MN, Warrington EK, Cipolotti L. The isolation of calculation
 Grimes DA, Lang AE, Bergeron CB. Dementia as the most common skills. J Neurol 1995;242:78 – 81.
presentation of cortical-basal ganglionic degeneration. Neurology  Warrington EK. The fractionation of arithmetic skills. A single study.
1999;53:1969 – 74. Q J Exp Psychol 1982;34:31 – 51.
 Schneider JA, Watts RL, Gearing M, Brewer RP, Mirra SS. Cortico-  Whetstone T. The representation of arithmetic facts in memory: re-
basal degeneration: neuropathologic and clinical heterogeneity. Neu- sults from retraining a brain-damaged patient. Brain Cogn 1998;36:
rology 1997;48:959 – 69. 290 – 309.
 Folstein MF, Folstein SF, McHugh PR. ‘‘Mini Mental State.’’ A prac-  Pillon B, Dubois B. Memory and executive processes in corticobasal
tical method for grading the cognitive state of patients for the clini- degeneration. In: Litvan I, Goetz CG, Land AE, editors. Corticobasal
cian. J Psychiatr Res 1975;12:189 – 98. degeneration. Advances in Neurology, vol. 82. Philadelphia: Lippin-
 Rosen HJ, Hartikainen KM, Jagust W, et al. Utility of clinical criteria cott Williams & Wilkins; 2000.
in differentiating frontotemporal lobar degeneration and AD. Neurol-  Rickard TC, Romero SG, Basso G, Wharton C, Flitman S, Grafman J.
ogy 2002;58:1608 – 15. The calculating brain: an fMRI study. Neuropsychologia 2000;38:
 Mathuranath PS, Nestor PJ, Berrios GE, Rakowicz W, Hodges JR. A 325 – 35.
brief cognitive test battery to differentiate Alzheimer’s disease and  Cowell SF, Egan GF, Code C, Harasty J, Watson JDG. The functional
frontotemporal dementia. Neurology 2000;55:1613 – 20. neuroanatomy of simple calculation and number repetition: a para-
 Wechsler D. Wechsler memory scale-revised. New York: Psycholog- metric PET activation study. Neuroimage 2000;12:565 – 73.
ical Corporation; 1987.  Turner RS, Kenyon LC, Trojanowski JQ, Gonatas N, Grossman M.
 Kaufman EL, Lord MW, Reese T, Volkmann J. The discrimination of Clinical, neuroimaging, and pathologic features of progressive non-
visual number. Am J Psychol 1949;62:498 – 525. fluent aphasia. Ann Neurol 1996;39:166 – 73.
 Mandler G, Shebo BJ. Subitizing: an analysis of its component pro-  Lieberman AP, Trojanowski JQ, Lee VMY, et al. Cognitive, neuro-
cesses. J Exp Psychol Gen 1982;11:1 – 22. imaging, and pathologic studies in a patient with Pick’s disease. Ann
 Trick LM, Pylyshyn ZW. What enumeration studies can show us Neurol 1998;43:259 – 64.
about spatial attention: evidence for limited capacity preattentive pro-  Tyrrell PJ, Kartsounis LD, Frackowiak RSJ, Findley LJ, Rossor MN.
cesses. J Exp Psychol Hum Percept Perform 1993;19(2):331 – 51. Progressive loss of speech output and orofacial dyspraxia associated
 Peterson S, Simon TJ. Computational evidence for the subitizing with frontal lobe hypometabolism. J Neurol Neurosurg Psychiatry
phenomenon as an emergent property of the human cognitive archi- 1991;54:351 – 7.
tecture. Cogn Sci 2000;24(1):93 – 122.  Luria AR. Higher cortical functions in man. New York: Basic Books;
 Piazza M, Mechelli A, Butterworth B, Price C. Are subitizing and 1966.
counting implemented as separate or functionally overlapping pro-  Crutch SJ, Warrington EK. Acalculia: deficits of operational and
cesses? Neuroimage 2002;15:435 – 46. quantity number knowledge. J Int Neuropsychol Soc 2001;7:825 – 34.
 Brooks DJ. Functional imaging studies in corticobasal degeneration.  Cummings JL, Benson F, Hill M, Read S. Aphasia in dementia of the
In: Litvan I, Goetz CG, Land AE, editors. Corticobasal degeneration. Alzheimer type. Neurology 1985;35:394 – 7.
Advances in Neurology, vol. 82. Philadelphia: Lippincott Williams &