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A dominant hypothesis on how the brain processes numerical size proposes a spatial representation of numbers as positions on a 'mental number line'. An alternative hypothesis considers numbers as elements of a generalized representation of sensorimotor-related magnitude which is not obligatorily spatial. Here we show that individuals' relative use of spatial and non-spatial representations has a cerebral counterpart in the structural organization of the posterior parietal cortex. Inter-individual variability in the linkage between numbers and spatial responses (faster left responses to low numbers and right responses to high numbers; SNARC effect) correlated with variations in grey matter volume around the right precuneus. Conversely, differences in the disposition to link numbers to force production (faster soft responses to low numbers and hard responses to high numbers) were related to grey matter volume in the left angular gyrus. This finding suggests that numerical cognition relies on multiple mental representations of analogue magnitude using different neural implementations that are linked to individual traits.
Different brains process numbers differently:
structural bases of individual differences in spatial and
non-spatial number representations
Florian Krause1, Oliver Lindemann2, Ivan Toni1, Harold Bekkering1
1Radboud University Nijmegen, Donders Institute for Brain, Cognition and Behaviour, The Netherlands
2Division of Cognitive Science, University of Potsdam, Germany
Florian Krause (corresponding author)
Address: P.O. Box 9104, 6500 HE Nijmegen, The Netherlands
Phone: +31 24 36 11027
In press: Journal of Cognitive Neuroscience.
This manuscript may not exactly replicate the final published version.
It is not a copy of record.
A dominant hypothesis on how the brain processes numerical size proposes a spatial
representation of numbers as positions on a 'mental number line'. An alternative hypothesis
considers numbers as elements of a generalized representation of sensorimotor-related
magnitude which is not obligatorily spatial. Here we show that individuals' relative use of
spatial and non-spatial representations has a cerebral counterpart in the structural organization
of the posterior parietal cortex. Inter-individual variability in the linkage between numbers
and spatial responses (faster left responses to low numbers and right responses to high
numbers; SNARC effect) correlated with variations in grey matter volume around the right
precuneus. Conversely, differences in the disposition to link numbers to force production
(faster soft responses to low numbers and hard responses to high numbers) were related to
grey matter volume in the left angular gyrus. This finding suggests that numerical cognition
relies on multiple mental representations of analogue magnitude using different neural
implementations that are linked to individual traits.
Dealing with numerical information is an integral part of our modern society. Numbers
occur throughout all aspects of every day life; they depict information about prices and values
and allow us to count occurrences and entities. During a single day we probably process
several thousand numbers (Butterworth, 1999). Yet, our ability to deal with them varies
greatly across individuals (Butterworth, 2010; Adams, 2007). It is therefore important to
understand how individuals represent numbers and how their brains process this information.
The most influential model of number processing, the triple-code model, dissociates between
three different numerical representations: An Arabic code for digit processing, a verbal code
for retrieval of arithmetic facts and verbal counting, and an analogue magnitude code for the
processing of numerical size (Dehaene, 1992). While the first two representations are
notation- and modality-dependent, an analogue magnitude representation is thought to be
abstract in nature and is thus assumed to be independent of notation and modality (Cohen
Kadosh & Walsh, 2009). The current study seeks to investigate two different manifestations
of this analogue magnitude representation of numerical size.
Over the last decades, several studies on number cognition have provided abundant
empirical support for the hypothesis that numerical size is spatially represented in the brain
(Dehaene, 2009; de Havia, Vallar & Girelli, 2008; Hubbard, Piazza, Pinel & Dehaene, 2005).
This hypothesis assumes that we derive the size of a number from its position on an ordered
'mental number line' on which small numbers are represented on one side and large numbers
on the other (Moyer & Landauer, 1967). For instance, the so-called effect of
Spatial-Numerical Association of Response Codes (SNARC) shows a linkage between
numerical information and spatial responses (Dehaene, Bossini & Giraux, 1993). When
participants are asked to judge the parity of Arabic digits between 1 and 9 by a left or right
response, the numerical size of the digit interferes with the execution of the spatial responses,
with faster left responses to small numbers and faster right responses to large numbers. This
spatial number-response interference effect has been interpreted as evidence for a shared
representation of spatial response codes and the ordinal position of the number representation
in mental space.
More recently, the assumption that spatial codes become obligatorily activated when
processing numerical size has been questioned by several authors (e.g. Fischer, 2006; Santens
& Gevers, 2008). It has been argued instead that information about numerical size are
mapped onto representations of other size-related sensorimotor information, within a system
that processes generalized analogue magnitude (Walsh, 2003). According to this hypothesis,
number meaning is conceptualized by recruiting the same mechanisms that allow us to
experience and control other behaviorally relevant magnitudes in daily life. Evidence for this
notion comes from several studies showing associations between numbers and other types of
magnitude information in action and perception, like physical size (Tzelgov & Henik, 1992),
temporal duration (Oliveri, Vicario, Salerno, Koch, Turizziani, Mangano, Chillemi &
Caltagirone, 2008), grip aperture (Lindemann, Abolafia, Girardi & Bekkering, 2007), object
graspability (Badets, Andres, Di Luca, & Pesenti, 2007) and tactile sensation (Krause,
Bekkering, & Lindeman, 2013). Crucially, the associated sensorimotor magnitudes can be
entirely non-spatial in nature. For instance, a link between numerical information and force
production has been reported (Vierck & Kiesel, 2010), which we will refer to as
Force-Numerical Association of Response Codes (FoNARC). When participants are asked to
judge the parity of Arabic digits between 1 and 9 by a soft or hard response on a single
button, the numerical size of the digit interferes with the execution of force responses, with
faster soft responses to small numbers and faster hard responses to large numbers.
Importantly, the procedure to quantify a FoNARC effect is identical to the procedure to
quantify a SNARC effect except that required motor responses for the latency measurement
do not differ spatially. Due to the homogeneity of all spatial response components, it can be
excluded that the number-response interference effect observed under these condition is
driven by a spatial representation of numbers on a mental number line. The observation of a
FoNARC effect consequently has to be interpreted as a within-magnitude inference between
numerical information and the control of motor force, which in turn suggests the existence of
non-spatial sensorimotor-related representations of numbers.
With the apparent coexistence of both spatial and non-spatial representations of
numerical size the question arises in which way the two representations contribute to
numerical cognition. It has been suggested that multiple representations of the same
numerical information rely on different neural implementations and that the weights of their
contribution are simply determined by the requirements of the situation or task at hand
(Dehaene, Piazza, Pinel & Cohen, 2003). A numerical task with a spatial component would
lead to a stronger activation of posterior superior parietal lobe. In contrast, a number task
without any spatial component (e.g. force production) is expected to engage primarily inferior
parietal regions (cf. Dehaene et al., 2003).
However, task demands might not be the only factor to determine how numerical
information are processed. For instance, the general disposition to associate numbers with
space has been shown to vary strongly between individuals (for a review see Wood, Nuerk &
Willmes, 2008) and might even depend on personal preferences to code numerical
information (Fischer, 2006). The same might hold for linking numbers to non-spatial
sensorimotor-related magnitude, as this disposition might be related to the individual's bodily
competence and experience of dealing with magnitudes and sizes in everyday life
(Lindemann, Rueschemeyer, & Bekkering, 2009; see also Barsalou, 2008). Here we assess
whether those inter-individual differences reflect stable individual traits, rather than
stochastic noise or task demands.
This issue was addressed by combining a double dissociation approach with the method
of Voxel-Based Morphometry (VBM; Ashburner and Friston, 2000). The rationale of this
approach is to isolate differential structural variances across two behavioural indexes of
numerical cognition, intrinsically controlling for confounds correlated with both indexes. We
tested whether inter-individual variation in anatomical brain structure explains individual
differences in spatial (SNARC) and non-spatial (FoNARC) number-response interference
effects – reflecting a spatial and non-spatial representation of numerical size, respectively.
VBM was used to measure variability in local grey matter volume in the posterior parietal
cortex, a site consistently associated with numerical representations (see Arsalidou & Taylor,
2011 for a review). Functional Magnetic Resonance Imaging (fMRI) was used to map the
spatial distribution of task-related activity across the posterior parietal cortex.
A total of 33 students (20 female) between 18 and 34 years of age (mean age 21.33,
SD=3.28) participated in the experiment in return of 20 Euro or course credits. All of them
had normal or corrected-to-normal vision and were of general health, with no known
neurological or psychological disorders. The study was approved by the local ethics
committee and participants gave their written consent prior to the experimental procedure.
Stimuli consisted of the Arabic digits 1 to 9, except 5, depicted in white colour (visual
angle: ~1.26 degrees vertical & ~0.53 degrees horizontal) centrally on a black background.
Participants viewed the screen via a mirror attached to the Magnetic Resonance (MR)
scanner's receiver head-coil.
Responses were recorded using MR-compatible button boxes with either spatially
arranged buttons that had to be pressed with the right index and right middle finger, or with a
single isometric force-transducer button which had to be pressed with the right thumb. The
force sensitive button box was a cylinder grasped between the thumb and the remaining
The data collection was done in the context of a larger functional Magnetic Resonance
Imaging study and was thus performed while subjects were lying inside the MR-scanner.
Participants were engaged in two consecutive number parity judgement tasks in which the
presented digits had to be classified as odd or even. Importantly, both tasks differed only in
the required responses. In the spatial task, number parity had to be indicated by a right index
finger (“left”) or middle finger (“right”) response. That is, each response involved the flexion
and extension of either one of the two fingers. In the non-spatial task, responses were given
with the right thumb and involved applying either a small force (>500 N, “soft” responses) or
a large force (>2500 N, “hard” responses).
Each trial started with the presentation of a white fixation cross for 500 ms, followed by
the target stimulus. Participants had to respond within 1000 ms after stimulus presentation. If
it took them too long to respond, or their response was incorrect, an auditory error signal was
played back to them via headphones. After the response was given, a dark grey fixation cross
was presented for a variable time between 2000 and 4000 ms, before the next trial started.
Before the actual experiment, participants were given verbal instructions and practised
the task for about 5 minutes outside the MR scanner. The response mapping to indicate the
parity (i.e., left or right response for odd numbers and soft or hard response for odd numbers)
was reversed in the middle of each task block. The order of mappings as well as the order of
the spatial and non-spatial task were balanced between participants. Eighteen participants
performed 320 trials, 15 participants performed 288 trials. The order of trials was
MRI data acquisition
For each participant a high-resolution anatomical MR image was recorded using a
T1-weighted MP-RAGE sequence with a GRAPPA acceleration factor of 2 (TR/TE =
2300/3.03 ms, voxel size = 1x1x1 mm). Anatomical images were recorded directly after both
tasks were conducted. Due to technical problems, MR images of two participants were
acquired 12 and 5 weeks after the behavioural test, respectively. For one participant, a 7
months older MR image was used. All images were recorded on the same 3 Tesla Siemens
Magneton Trio MR scanner (Siemens, Erlangen, Germany) in combination with the same
32-channel receiver head-coil.
Functional images were acquired using a multi-echo gradient echo planar T2*-weighted
sequence sensitive to blood oxygen level-dependent contrast (TR = 2390 ms; TE = 9.4, 21.2,
33.0, 45.0 and 56.0 ms; FA = 90°; field of view = 224 x 224 mm; number of slices = 31; slice
thickness = 3 mm; resolution = 3.5 x 3.5 x 3.0 mm).
Behavioural data analysis
The behavioural data of each participant were analyzed separately for the spatial and the
non-spatial task, to estimate effect sizes for both, a SNARC effect in the spatial task, as well
as a FoNARC effect in the non-spatial task. Only trials with correct parity judgments within
1000 ms were included in the analysis. Effect sizes were calculated as suggested by Fias,
Brysbaert, Geypens and Ydewalle (1996). First, the difference in the median reaction times
between left and right responses (spatial task) and soft and hard responses (magnitude task)
was calculated for each presented digit. Then, individual linear regressions between these
response time differences and the digits were calculated. The resulting regression coefficients
were used to characterize the size of the SNARC or FoNARC effect in each participant.
VBM analysis
MR image preprocessing and statistical testing was done using Statistical Parametric
Mapping 8 (SPM8, and the integrated DARTEL toolbox
(Ashburner, 2007).
Each anatomical image was segmented into grey and white matter images and resampled
to 1.5 mm isotropic resolution. Afterwards, nonlinear deformations for warping all grey and
white matter images to each other were determined by iterative template creation (7 steps;
Ashburner, 2007). Modulated warped grey matter images were created by smoothing with a
Gaussian kernel of 10 mm and normalizing to the Montreal Neurology Institute (MNI)
coordinate space.
An anatomical mask was created using the SPM8 Anatomy Toolbox (Eickhoff, 2005),
including portions of the superior parietal cortex (areas 7A, 7PC, 7M, 7P; Scheperjans,
Hermann, Eickhoff, Amunts, Schleicher, & Zilles, 2008a; Scheperjans, Eickhoff, Hömke,
Mohlberg, Hermann, Amunts, & Zilles, 2008b), the inferior parietal cortex (areas PFop, PFt,
PF, PFm, PFcm, PGa, PGp; Caspers, Geyer, Schleicher, Mohlberg, Amunts, & Zilles, 2006;
Caspers, Eickhoff, Geyer, Scheperjans, Mohlberg, Zilles, & Amunts, 2008), as well as the
intraparietal sulcus (areas hIP1, hIP2, hIP3; Choi, Zilles, Mohlberg, Schleicher, Fink,
Armstrong, & Amunts, 2006; Scheperjans et al., 2008a; Scheperjans et al., 2008b).
This anatomical mask, based on regions previously involved in numerical cognition
(Cohen Kadosh, Lammertyn, & Izard, 2008; Wu, Chang, Majid, Caspers, Eickhoff, &
Menon, 2009), was combined with a functional mask including posterior parietal voxels
activated during either one of the experimental tasks (see fMRI analysis below).
The preprocessed images entered a multiple regression general linear model (GLM) with
SNARC and FoNARC effect size estimates as regressors of interest. Two additional
covariates were added to the GLM: median overall reaction times, aggregated over both
tasks, to control for general performance differences between participants, and total
intracranial volume, to control for general overall size differences of grey matter, white
matter and cerebro spinal fluid (Good, Johnsrude, Ashburner, Henson, Friston & Frackowiak,
The statistical threshold was p < 0.05 at voxel level, corrected for multiple comparisons
by means of the family-wise error (FWE).
fMRI analysis
Functional image preprocessing and statistical testing was done using SPM8
For each volume, the five multi-echo images were combined into a T2*-weighted
average image (Poser, Versluis, Hoogduin, & Norris, 2006). All weighted average images
were spatially realigned to the first image and corrected for differences in slice-time
acquisition. The T1-weighted anatomical image was co-registered with the mean functional
image, segmented and normalized to the MNI standard space, and resampled to a 2x2x2 mm
resolution. The resulting normalization parameters were applied to the functional images,
which were subsequently spatially smoothed using a Gaussian kernel of 8 mm.
The preprocessed images entered a GLM with 4 sessions, describing two response
mappings for each of the two experimental tasks. For each session, task effects were modeled
using a combination of compatible and incompatible trials for four groups of numerical
stimuli (1, 2; 3, 4; 6, 7; 8, 9), resulting in 8 regressors, each describing the onset of the
response to the stimulus. An additional regressor was used to model erroneous responses. All
task-related regressors were convolved with a hemodynamic response function. Three
translational and three rotational head motion parameters and their first and second
derivative, resulting in 18 regressors, were added as covariates.
To capture posterior parietal voxels activated during the experimental tasks, the t-contrast
of each task compared to implicit baseline was evaluated for the whole brain on the group
level (thresholded at 0.05, uncorrected). The union of the results of both contrasts, restricted
to the entire posterior parietal cortex (i.e., areas 5L, 5M, 5Ci, 7A, 7PC, 7M, 7P, PFop, PFt,
PF, PFm, PFcm, PGa, PGp, hIP1, hIP2, hIP3), served as a functional mask (see VBM
analysis above).
We also assessed whether the regions showing structural variations as a function of
SNARC/FoNARC performance were also functionally engaged in performance of those
tasks. That is we tested whether the fMRI data showed increased BOLD signal (p<0.01,
Family-wise error corrected for search volume) within a search volume defined by two
spherical VOIs centered on the two local maxima of the VBM analyses, with a radius
matched to the FWHM of the VBM results (10 mm).
Behavioural results
On average, participants made 5 % errors in the spatial task and 10 % errors in the
non-spatial task. The average reaction times were 539 ms (SD=60) and 585 ms (SD=62) for
the spatial task and non-spatial task, respectively. SNARC effect sizes of all participants
differed significantly from zero, t(32) = 6.70, p < .001, as did FoNARC effect sizes, t(32) =
8.61, p < .001. There was a weak, but non-significant positive correlation between the
individual SNARC and FoNARC effect sizes , r = 0.31, p = .07. Median overall reaction
times correlated with the size of the SNARC effect , r = 0.36, p < .05, and were therefore
included as an additional covariate in the GLM for the VBM analysis (see Method).
Importantly, there were no correlations between age and SNARC effect sizes, r = -0.10, p = .
57, age and FoNARC effect sizes, r = 0.03, p = .88, or gender and SNARC effect sizes, r =
0.11, p = .55, and gender and FoNARC effect sizes, r = 0.16, p = .37. Therefore, and since
any shared variance with age and gender is common to both regressors of interest in the
GLM, age and gender were not added as explicit covariates into the VBM analysis.
VBM results
Figure 1 shows the main findings of the VBM analysis (thresholded at 0.001,
uncorrected, for illustrative purposes). The multiple regression analysis on the posterior
parietal cortex revealed that SNARC effect size predicted local relative grey matter volume in
the right precuneus (area 5M; peak at MNI coordinates x = 7.5, y = -49.5, z = 52.5; t(28) =
4.97, Z = 4.17, pFWE < 0.05). The stronger the individual SNARC effect (i.e. the disposition to
associate numbers with a spatial response), the more relative grey matter was present in this
particular region. Furthermore, FoNARC effect size predicted local relative grey matter
volume in the left angular gyrus (area PGa; peak at MNI coordinates x = -45, y = -57, z =
37.5 ), t(28) = 5.37, Z = 4.42, pFWE < 0.05). The stronger the FoNARC effect (i.e. the
disposition to link numbers to force production), the more relative grey matter in this region
of the individual's brain. Figure 2 illustrates the differential correlations between grey matter
volume in each of the regions and number-response interference effects, corrected for average
reaction time and total intracranial volume. Importantly, grey matter volume in right
precuneus correlated significantly more with the spatial than with the non-spatial
number-response interference effect, Z = 3.55, p < 0.01, while grey matter volume in left
angular gyrus correlated significantly less with the spatial than with the non-spatial
number-response interference effect, Z = -3.89, p < 0.01, demonstrating a double-dissociation
between the behavioural and the structural cerebral effects.
fMRI results
SVC analysis on the regions identified in the VBM analysis revealed a significant
activation of right precuneus during both the spatial task, t(32) = 5.31, Z = 4.47, pFWE < 0.01,
and the non-spatial task, t(32) = 4.57, Z = 3.98, pFWE < 0.01. Likewise, left angular gyrus was
significantly activated during the spatial task, t(32) = 5.05, Z = 4.30, pFWE < 0.01, as well as
during the non-spatial task, t(32) = 6.97, Z = 5.40, pFWE < 0.01.
The present study provides evidence for a contribution of both spatial and non-spatial
representations of numerical size when processing Arabic digits and demonstrates that the
weights of this contribution rely on stable individual traits. We show that inter-individual
differences in the dispositions to link numbers to either space or non-spatial sensorimotor
magnitude can be directly related to structural variance in two distinct regions in the superior
and inferior posterior parietal lobes.
Structural bases of spatial and non-spatial representations of numerical
There was a relation between the strength of the SNARC effect and the structure of a
parietal region (area 5m) in the right precuneus. Increased grey matter in this region predicted
stronger interference of numerical size with spatial responses, but not with the force of a
response. Although little is known about the specific functionality of area 5m in humans, its
cytoarchitecture suggests that it is comparable with area PE in the macaque brain
(Scheperjans, Grefkes, Palomero-Gallagher, Schleicher, & Zilles, 2005). Macaque PE has
been involved in somatosensory integration and in creating a spatial representation of limbs
during movement (Bakola, Passarelli, Gamberini, Fattori, & Galletti, 2013; Lacquaniti,
Guigon, Bianchi, Ferraina, & Caminiti, 1995; Jones, Coulter, & Hendry, 1978; Mountcastle,
Lynch, Georgopoulos, Sakata, & Acuna, 1975). In humans, the right precuneus has
repeatedly been shown to be important for spatial processing, such as shifting attention in
visual space or visual imagery (for a review see Cavanna & Trimble, 2006), but not in the
processing of numerical magnitude information (for a review see Dehaene et al. 2003). In
fact, the present findings indicate that this parietal region is involved in the spatial
representation of numbers, suggesting that the association between numbers and space is
closer to general cognitive-spatial processing than to numerical magnitude per se.
The strength of mapping numbers to motor force ( FoNARC effect) correlated with the
structure of the left angular gyrus. Increased grey matter in this region predicted stronger
interference of numerical size with the force of a response, but not with the laterality of a
response. In contrast to the precuneus, the angular gyrus – the left side in particular – has
been consistently related to the processing of numerical information (Arsalidou et al., 2011).
For instance, a lesion in the left angular gyrus can lead to arithmetical deficits (Gerstman
Syndrome; Gerstmann, 1940). Dehaene et al. (2003) concluded that the left angular gyrus is
involved in the retrieval of linguistic arithmetic facts from memory. Evidence for this notion
comes, for instance, from neuroimaging studies showing that activity in this region is
modulated by arithmetic training (Ischebeck, Zamarian, Siedentopf, Koppelstätter, Benke,
Felber & Delazer, 2006; Delazer, Domahs, Bartha, Brenneis, Lochy, Trieb & Benke, 2003).
Further support for memory-based processes is provided by the observation that neuronal
activity in angular gyrus is higher when arithmetic problems are solved by fact retrieval,
compared to calculations (Grabner, Ansari, Koschutnig, Reishofer, Ebner & Neuper, 2009).
In contrast, it has been argued by several authors that the angular gyrus is involved in the
processing of number symbols and their numerical magnitude information (Rusconi, Walsh &
Butterworth, 2005; Göbel, Walsh & Rushworth, 2001). For instance, transcranial magnetic
stimulation (TMS) over angular gyrus is known to disrupt parity as well as magnitude
comparisons (Rusconi et al., 2005). An involvement of the left angular gyrus has been
furthermore demonstrated while processing numerical symbols in the absence of any
arithmetic demands (Price & Ansari, 2011; Holloway, Price & Ansari, 2010). Ansari (2008)
hypothesized therefore that the left angular gyrus mediates the mapping of numerical symbols
onto magnitude representations. The present finding now adds further empirical evidence for
this idea and demonstrates an involvement of the left angular gyrus in a non-linguistic and
non-spatial analogue representation of numerical size. We therefore interpret our findings as
support for the notion that this region is involved in the mapping of number symbols onto
magnitude information.
Spatial and non-spatial number-response interference effects
The current findings shed new light on the nature of spatial and non-spatial
number-response interferences effects and might also have practical implications for
investigating numerical representations. The present study is one of the first to demonstrate a
direct brain correlate of the well known association between numbers and spatial responses
reflected by the SNARC effect. The results of a recent functional near-infrared spectroscopy
study show a functional activation of bilateral intraparietal sulcus and left angular gyrus when
participants are engaged in a spatial number-response interference task (Cutini, Scarpa,
Scatturin, Dell'acqua, & Zorzi, 2012). The present study now extends these findings and
shows that the existing individual preferences in the association between numbers and space
(e.g. Fischer, 2006) relate to structural variance in right precuneus. Importantly, the precuneus
is known to be involved in the processing of various types of spatial information in different
domains and modalities, but the brain region is typically not assumed to play any specific
crucial role in the processing of numerical magnitude information.
Taking into account the current finding and the general function of the precuneus for
spatial processing, one might question whether spatial number-response interference effects
are actually informing us about core mechanisms of number processing, rather than about the
use of more general cognitive coding strategies. For instance, associations with space have
also been observed for a variety of ordinal/sequential information like letters of the alphabet,
months or days of the week (Gevers, Reynvoet & Fias, 2003; 2004), suggesting that spatial
associations are driven by any type of ordinal information and not specifically related to the
representation of numerical magnitude. Our current data are in line with this view, as they
suggest that the nature of the number-space mapping is not magnitude related, but related to a
brain structure involved in general spatial processing. On the other hand, differences in the
strength of the mapping between numbers and force production (FoNARC effect) were
shown to be related to structural variance in a brain region known to be crucial for processing
magnitude-related aspects of numbers. This emphasizes the importance of non-spatial
magnitude representations and the suitability of using non-spatial magnitude-related
number-response interference paradigms to investigate the mechanisms of numerical
Both behavioural and VBM results of the present study showed that the inter-individual
differences in the size of a SNARC and FoNARC effect are uncorrelated and independent of
each other, which strongly suggests that the two effects reflect different aspects of the
cognitive processing and representation of numerical information. The dissociation of spatial
and non-spatial representation of numerical size might have relevant implications for
education. While a number line mapping seems to be a suitable tool for children to visualize
numerical information (Fischer, Moeller, Bientzle, Cress & Nuerk, 2011), the success of this
method might vary tremendously from child to child. If numerical representations vary
strongly between individuals, as the current results suggest, identifying and supporting these
differences might be educationally beneficial. However, at this point in time, the relation
between differences in number representation and differences in number competence remain
largely unclear. While a relation between left angular gyrus activity and mathematical
competence has recently been reported (Grabner, Ansari, Resihofer, Stern, Ebner & Neuper,
2007), future research with a strong focus on inter-individual differences in numerical skills
will be needed to address this open question further.
Interpretational limitations of the current study
The exact nature of the grey matter volumetric differences identified with VBM is still
poorly understood, as they could be related to changes in neuropil, neuronal size, dendritic or
axonal arborisation, as well as cortical folding (Michelli, Price, Friston, & Ashburner, 2005).
This complicates the interpretation of any VBM study with respect to linking structural
variability to its underlying functionality. In the current study, this interpretational issue is
somehow reduced by the observation that the task-related structural variability occurs in
parietal regions functionally engaged during performance of those tasks. Although it is
generally assumed that larger grey matter volume reflects enhanced neuronal processing
(Kanai, Feilden, Firth & Rees, 2011; Mechelli, Crinion, Noppeney, O'Doherty, Ashburner,
Frackowiak & Price, 2004; Maguire, Gadian, Johnsrude, Good, Ashburner, Frackowiak &
Frith, 2000), future studies will need to detail the microstructural and computational
mechanisms associated with the present findings.
The structural findings emerged from an analysis focused on the parietal lobe. The
rationale of this choice was to include in the analysis posterior parietal areas previously
shown to be involved in number processing (Cohen Kadosh et al., 2008; Wu et al., 2009), and
active during performance of the tasks used in the current study. However, the structural
changes reported here could also be observed in a whole-brain analysis (right precuneus: Z =
4.40; left angular gyrus: Z = 4.42).
It is unclear in how far the brain regions identified in the current study are bound to
specific demands of the numerical tasks used. This is especially important for understanding
the role of left angular gyrus during number processing. Here, force production was used as
an instance of non-spatial sensorimotor magnitude. It remains to be seen whether the current
observations generalize from the domain of force production to other non-spatial
sensorimotor magnitudes.
Taken together, the current findings suggest that numerical cognition relies on multiple
mental representations of analogue magnitude using distinct neural implementations that are
linked to individual traits. We showed that the way we represent numerical size is not only
dependent on situational requirements of a given task, but also subject to inter-individual
differences. Importantly, these differences appear to be stable traits as they can be linked to
distinct structural variance in the posterior parietal cortex. Our finding of individual traits
stimulates new research to investigate whether these traits are innate (“nature”) or the result
of external factors and emerge only later during development (“nurture”; see also Dehaene,
1997) – a question whose answer will have wide implications for math education.
We would like to thank Pascal de Water and Paul Gaalman for technical support and
assistance during data acquisition, as well as Roi Cohen Kadosh for his valuable comments
during an early data analysis stage.
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Figure Captions
Figure 1. Relative changes in grey matter volume in the posterior parietal cortex, related to spatial and non-spatial
representations of numerical size. Individuals' disposition to associate numbers with spatial responses predicts local grey
matter volume in right precuneus (blue). The disposition to link numbers to force production predicts grey matter volume in
left angular gyrus (red). Thresholded at p < 0.001, uncorrected, for illustrative purposes.
Figure 2. Correlations between grey matter volume and number-response interference effects, corrected for average
reaction time and total intracranial volume, demonstrating a double-dissociation between brain regions and representations
of numerical size. Significant correlations (as revealed by the multiple regression analysis) are plotted with a continuous
regression line.
... Experiences with temporally and spatially mapped sequences can hence provide a grounding mechanism for ordinality rather than magnitude. The neural processing of ordinality is supported by activity in superior parietal lobes representing spatial aspects [49], such as the precuneus [50,51], in combination with more general memory-related activity in the prefrontal [52,53] and entorhinal cortex [54]. ...
... This suggests that the mental representation of numerical size is related to spatial extent. Moreover, number processing interacts with grip aperture [56], object graspability [57], response force [50], temporal duration [58], perceptual strength in binocular rivalry [59], and visual luminance [60], establishing obligatory multimodal (visuomotor) magnitude processing. ...
... Another clearly prothetic measure of manual responses is their force. For example, adults respond faster to large numbers with a button response requiring more force and to small numbers with little force [50]. Already toddlers spontaneously use more force in response to larger numerosities [115]. ...
Numbers are present in every part of modern society and the human capacity to use numbers is unparalleled in other species. Understanding the mental and neural representations supporting this capacity is of central interest to cognitive psychology, neuroscience, and education. Embodied numerical cognition theory suggests that beyond the seemingly abstract symbols used to refer to numbers, their underlying meaning is deeply grounded in sensorimotor experiences, and that our specific understanding of numerical information is shaped by actions related to our fingers, egocentric space, and experiences with magnitudes in everyday life. We propose a sensorimotor perspective on numerical cognition in which number comprehension and numerical proficiency emerge from grounding three distinct numerical core concepts: magnitude, ordinality, and cardinality.
... This link is bidirectional: perceiving graspable small vs. large objects also affects the processing of small vs. large numbers differently (Ranzini et al., 2011). Further supporting evidence for ATOM comes from the FoNA effect-Force Numerical Association of response codes, the association between response force and numerical magnitude: participants respond faster to small numbers with weak responses and large numbers with forceful responses (Vierck and Kiesel, 2010;Krause et al., 2013). However, no direct automatic association between numerical magnitude and the amount of applied force was found (Fischer and Miller, 2008;Vierck and Kiesel, 2010), as one would expect: the mapping between force and numbers is categorical, i.e., large numbers are associated with forceful responses, but relatively larger numbers do not lead to relatively stronger responses. ...
... In contrast to this view, Krause et al. (2013) interpreted FoNA and SNA effects as signatures of alternative mapping strategies for mapping magnitudes either onto force (FoNA) or onto space (SNA), with different strategic preferences across participants. They found a correlation between gray matter volume in the left angular gyrus and strength of individual FoNA effects, as well as a correlation between gray matter volume around the right precuneus and strength of individual SNA effects. ...
... Hypothesis 5: Individuals have a preference to map numbers either to space or to force. If there are individual differences in utilizing different mechanisms (inter-individual variability, as suggested by Krause et al., 2013), for some participants FoNA effect should be found, whereas for others SNA. At the individual level, a negative relationship between FoNA and SNA can be expected because participants who rely on one kind of representation do not need another one. ...
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The study has two objectives: (1) to introduce grip force recording as a new technique for studying embodied numerical processing; and (2) to demonstrate how three competing accounts of numerical magnitude representation can be tested by using this new technique: the Mental Number Line (MNL), A Theory of Magnitude (ATOM) and Embodied Cognition (finger counting-based) account. While 26 healthy adults processed visually presented single digits in a go/no-go n-back paradigm, their passive holding forces for two small sensors were recorded in both hands. Spontaneous and unconscious grip force changes related to number magnitude occurred in the left hand already 100–140 ms after stimulus presentation and continued systematically. Our results support a two-step model of number processing where an initial stage is related to the automatic activation of all stimulus properties whereas a later stage consists of deeper conscious processing of the stimulus. This interpretation generalizes previous work with linguistic stimuli and elaborates the timeline of embodied cognition. We hope that the use of grip force recording will advance the field of numerical cognition research.
... Fischer, 2017;Sixtus et al., 2023). Signatures of sensorimotor experience are documented in children's and adults' quantity and number processing (e.g., Domahs et al., 2010;Krause et al., 2014;Krause et al., 2019;Sixtus et al., 2017;Sixtus et al., 2018Sixtus et al., , 2020Tschentscher et al., 2012). For example, 2.5-to 3-year-old children spontaneously used more force in response to larger numerosities in a computer game (Krause et al., 2019); adults responded more efficiently to numbers for which they performed the adequate hand posture from finger counting (Sixtus et al., 2017); in an fMRI-study, visually presented numbers from 1 to 5 selectively evoked activity in the cortical motor area contralateral to the hand that would be used for counting the presented number (Tschentscher et al., 2012). ...
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Several lines of research have demonstrated spatial-numerical associations in both adults and children, which are thought to be based on a spatial representation of numerical information in the form of a mental number line. The acquisition of increasingly precise mental number line representations is assumed to support arithmetic learning in children. It is further suggested that sensorimotor experiences shape the development of number concepts and arithmetic learning, and that mental arithmetic can be characterized as "motion along a path" and might constitute shifts in attention along the mental number line. The present study investigated whether movements in physical space influence mental arithmetic in primary school children, and whether the expected effect depends on concurrency of body movements and mental arithmetic. After turning their body towards the left or right, 48 children aged 8 to 10 years solved simple subtraction and addition problems. Meanwhile, they either walked or stood still and looked towards the respective direction. We report a congruency effect between body orientation and operation type, i.e., higher performance for the combinations leftward orientation and subtraction and rightward orientation and addition. We found no significant difference between walking and looking conditions. The present results suggest that mental arithmetic in children is influenced by preceding sensorimotor cues and not necessarily by concurrent body movements.
... With the invention of brain imaging methodologies (e.g., positron emission tomography; PET, functional magnetic resonance imaging; fMRI), pioneering empirical neuroimaging research revealed that the AG is associated with calculation in healthy adults (Dehaene et al. 1996;Gruber et al. 2001;Rickard et al. 2000;Rueckert et al. 1996). Since then, many empirical studies have explored the precise role of the AG in a range of mathematical competencies including the processing of number symbols (Holloway et al. 2010;Rusconi et al. 2005;Sokolowski et al. 2021), the mental number line (Göbel et al. 2001) numerical and sensorimotor associations (Krause et al. 2014), arithmetic (Zamarian et al. 2009), and higher-level mathematical reasoning (Liu et al. 2019). While the AG has been implicated in a range of basic and advanced skills within the domain of mathematical cognition, the association between the AG and arithmetic problem-solving (i.e., mental arithmetic) is the only association with consistent, replicable, and well-documented evidence. ...
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Since the pioneering work of the early 20th century neuropsychologists, the angular gyrus (AG), particularly in the left hemisphere, has been associated with numerical and mathematical processing. The association between the AG and numerical and mathematical processing has been substantiated by neuroimaging research. In the present review article, we will examine what is currently known about the role of the AG in numerical and mathematical processing with a particular focus on arithmetic. Specifically, we will examine the role of the AG in the retrieval of arithmetic facts in both typically developing children and adults. The review article will consider alternative accounts that posit that the involvement of the AG is not specific to arithmetic processing and will consider how numerical and mathematical processing and their association with the AG overlap with other neurocognitive processes. The review closes with a discussion of future directions to further characterize the relationship between the angular gyrus and arithmetic processing.
... To circumvent this possibility, we collected an implicit motor-based report: button pressure. Previous work in number cognition has found that force is affected by number representations [11][12][13][14][15]. From this work, we know that tasks that do not ask for explicit magnitude comparisons (e.g. ...
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Negative numbers are central in math. However, they are abstract, hard to learn, and manipulated slower than positive numbers regardless of math ability. It suggests that confidence, namely the post-decision estimate of being correct, should be lower than positives. We asked participants to pick the larger single-digit numeral in a pair and collected their implicit confidence with button pressure (button pressure was validated with three empirical signatures of confidence). We also modeled their choices with a drift-diffusion decision model to compute the post-decision estimate of being correct. We found that participants had relatively low confidence with negative numerals. Given that participants compared with high accuracy the basic base-10 symbols (0-9), reduced confidence may be a general feature of manipulating abstract negative numerals as they produce more uncertainty than positive numerals per unit of time.
... Also, interindividual differences might be related to neuroanatomical differences among individuals. For instance, Krause et al. [99] showed that grey matter volume in different brain regions correlated with the strength of number-space interactions (small numbers associated to left-sided responses, and vice versa) and number-action interactions (small number associated to soft response, and vice versa) at the individual level. Considering this, it will be important in future studies to clarify the incidence of the effects and the factors contributing to individual differences. ...
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Embodied and grounded cognition theories state that cognitive processing is built upon sensorimotor systems. In the context of numerical cognition, support to this framework comes from the interactions between numerical processing and the hand actions of reaching and grasping documented in skilled adults. Accordingly, mechanisms for the processing of object size and location during reach and grasp actions might scaffold the development of mental representations of numerical magnitude. The present study exploited motor adaptation to test the hypothesis of a functional overlap between neurocognitive mechanisms of hand action and numerical processing. Participants performed repetitive grasping of an object, repetitive pointing, repetitive tapping, or passive viewing. Subsequently, they performed a symbolic number comparison task. Importantly, hand action and number comparison were functionally and temporally dissociated, thereby minimizing context-based effects. Results showed that executing the action of pointing slowed down the responses in number comparison. Moreover, the typical distance effect (faster responses for numbers far from the reference as compared to close ones) was not observed for small numbers after pointing, while it was enhanced by grasping. These findings confirm the functional link between hand action and numerical processing, and suggest new hypotheses on the role of pointing as a meaningful gesture in the development and embodiment of numerical skills.
... Nevertheless, what I point to now is more radical forms of neuronal degeneracy, which is the ability of structurally different elements to produce the same function or output (Edelman and Gally 2001). It becomes much more challenging to defend modularity when faced with the evidence of neuronal degeneracy, for example, numerical processing in varying Journal of Cognition and Neuroethics areas of the brains of different subjects (Krause, Lindemann, Toni, and Bekkering 2014), reorganized sensorimotor cortex in people born without particular limbs (Hahamy, Macdonald, Heiligenberg, Kieliba, Emir, Malach, Johansen-Berg, et al. 2017), and significant motor control without a cerebellum (Lemon and Edgley 2010). In addition to serving as considerations in opposition to the modularity theses, the previously stated examples of degeneracy also serve as evidence of the highly interconnected organization of the brain and, possibly, mental states. ...
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Embodied cognition is the idea that cognition is causally related to and/or constituted by bodily activities. In spite of accumulating reasons to accept embodied cognition, critics seem to have a knockdown argument: appealing to locked-in syndrome (LIS). Patients with LIS are said to be at least minimally conscious to fully awake, except they have no motor control of their body and cannot produce speech. LIS seems to undermine embodied cognition: if cognition is embodied, then LIS patients cannot have intact cognitive capacities because they do not have motor control of their body. The present goal is to provide supporters of embodied cognition with a set of three responses when faced with the challenge from LIS. The first is deflationary and highlights the fact that most cases of LIS are not total and that much evidence of LIS are not actually cases of LIS. The second is skeptical and provides reasons to question the evidence of LIS based on neuroimaging data. The third is that the types of pathologies that cause LIS are likely to alter cognition in radical ways. With these responses at the ready, the supporter of embodied cognition need not surrender at the mere mention of LIS.
... In contrast to effects of spatio-numerical associations, within-magnitude interference effects are characterized by a congruency between numbers and the size of visual stimuli (Henik & Tzelgov, 1982;Schwarz & Heinze, 1998) or a compatibility between numbers and size-related motor features such as grip size (Lindemann et al., 2007) or response intensity (Krause, Lindemann, Toni, & Bekkering, 2014). For instance, in the classical size-congruity paradigm, two digits are simultaneously presented in different sizes, and subjects are instructed to indicate the numerically (or physically) larger digit. ...
Many studies demonstrated interactions between number processing and either spatial codes (effects of spatial-numerical associations) or visual size-related codes (size-congruity effect). However, the interrelatedness of these two number couplings is still unclear. The present study examines the simultaneous occurrence of space- and size-numerical congruency effects and their interactions both within and across trials, in a magnitude judgment task physically small or large digits were presented left or right from screen center. The reaction times analysis revealed that space- and size-congruency effects coexisted in parallel and combined additively. Moreover, a selective sequential modulation of the two congruency effects was found. The size-congruency effect was reduced after size incongruent trials. The space-congruency effect, however, was only affected by the previous space congruency. The observed independence of spatial-numerical and within magnitude associations is interpreted as evidence that the two couplings reflect Different attributes of numerical meaning possibly related to orginality and cardinality.
The reliance of number processing on sensorimotor mechanisms involved in hand action has been extensively documented by behavioural studies. Nonetheless, where and how the computations of number and hand action interact in the brain has received limited attention. In this study we investigated the brain networks underlying symbolic number comparison and the hand action of reaching and grasping, capitalizing on functional imaging studies meta-analyzed with the seed-based d mapping with permutation of subject images meta-analytic method (SDM-PSI). The main objective was to test whether and to what extent symbolic number processing recruits the same sensorimotor network involved in the hand action of reaching and grasping. We included 42 studies (641 participants) adopting symbolic number comparison tasks and 58 studies (814 participants) investigating hand reaching and hand grasping. The conjunction analysis of brain networks common to number processing, reaching, and grasping revealed spatial convergence over frontoparietal areas. Specifically, four clusters were identified, in and around the left and right intraparietal sulci, in the left precentral gyrus, and in the supplementary motor area. The degree of overlap was extensive, since the reach/grasp network mostly included the number areas. A qualitative analysis of functional characterization capitalizing on the Neurosynth database depicted a strong multifunctionality of the regions of overlap between numbers and hand action: these brain areas were also associated to a variety of functions within the domains of memory and imagery, visuospatial attention, and language. Overall, these results characterize the neuroanatomical substrate of the interaction between reaching, grasping, and symbolic number comparison.
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In macaques, superior parietal lobule area 5 has been described as occupying an extensive region, which includes the caudal half of the postcentral convexity as well as the medial bank of the intraparietal sulcus. Modern neuroanatomical methods have allowed the identification of various areas within this region. In the present study, we investigated the corticocortical afferent projections of one of these subdivisions, area PE. Our results demonstrate that PE, defined as a single architectonic area that contains a topographic map of the body, forms specific connections with somatic and motor fields. Thus, PE receives major afferents from parietal areas, mainly area 2, PEc, several areas in the medial bank of the intraparietal sulcus, opercular areas PGop/PFop, and the retroinsular area, frontal afferents from the primary motor cortex, the supplementary motor area, and the caudal subdivision of dorsal premotor cortex, as well as afferents from cingulate areas PEci, 23, and 24. The presence and relative strength of these connections depend on the location of injection sites, so that lateral PE receives preferential input from anterior sectors of the medial bank of intraparietal sulcus and from the ventral premotor cortex, whereas medial PE forms denser connections with area PEc and motor fields. In contrast with other posterior parietal areas, there are no projections to PE from occipital or prefrontal cortices. Overall, the sensory and motor afferents to PE are consistent with functions in goal-directed movement but also hint at a wider variety of motor coordination roles.
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Evidence for an approximate analog system of numbers has been provided by the finding that the comparison of two numerals takes longer and is more error-prone if the semantic distance between the numbers becomes smaller (so-called numerical distance effect). Recent embodied theories suggest that analog number representations are based on previous sensory experiences and constitute therefore a common magnitude metric shared by multiple domains. Here we demonstrate the existence of a cross-modal semantic distance effect between symbolic and tactile numerosities. Participants received tactile stimulations of different amounts of fingers while reading Arabic digits and indicated verbally whether the amount of stimulated fingers was different from the simultaneously presented digit or not. The larger the semantic distance was between the two numerosities, the faster and more accurate participants made their judgments. This cross-modal numerosity distance effect suggests a direct connection between tactile sensations and the concept of numerical magnitude. A second experiment replicated the interaction between symbolic and tactile numerosities and showed that this effect is not modulated by the participants' finger counting habits. Taken together, our data provide novel evidence for a shared metric for symbolic and tactile numerosities as an instance of an embodied representation of numbers.
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Nine experiments of timed odd–even judgments examined how parity and number magnitude are accessed from Arabic and verbal numerals. With Arabic numerals, Ss used the rightmost digit to access a store of semantic number knowledge. Verbal numerals went through an additional stage of transcoding to base 10. Magnitude information was automatically accessed from Arabic numerals. Large numbers preferentially elicited a rightward response, and small numbers a leftward response. The Spatial–Numerical Association of Response Codes effect depended only on relative number magnitude and was weaker or absent with letters or verbal numerals. Direction did not vary with handedness or hemispheric dominance but was linked to the direction of writing, as it faded or even reversed in right-to-left writing Iranian Ss. The results supported a modular architecture for number processing, with distinct but interconnected Arabic, verbal, and magnitude representations. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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Ss evaluated either numerical size or physical size of stimuli varying along both dimensions. Size congruity, distance, and semantic congruency effects were obtained for numerical comparisons of digit pairs and for comparisons of digits with an internal standard (5). Only the size congruity effect was obtained for physical judgments. It was smaller for pairs in which both stimuli were either both smaller or both larger than 5 than for pairs that contained the digit 5. The results are consistent with the notion that intentional processing is mainly algorithm-based, whereas autonomous processing is mainly memory-based. Implications of the results for models of numerical processing are discussed. (PsycINFO Database Record (c) 2012 APA, all rights reserved)
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Interactions between numbers and space have become a major issue in cognitive neuroscience, because they suggest that numerical representations might be deeply rooted in cortical networks that also subserve spatial cognition. The spatial-numerical association of response codes (SNARC) is the most robust and widely replicated demonstration of the link between numbers and space: in magnitude comparison or parity judgments, participants' reaction times to small numbers are faster with left than right effectors, whereas the converse is found for large numbers. However, despite the massive body of research on number-space interactions, the nature of the SNARC effect remains controversial and no study to date has identified its hemodynamic correlates. Using functional near-infrared spectroscopy, we found a hemodynamic signature of the SNARC effect in the bilateral intraparietal sulcus, a core region for numerical magnitude representation, and left angular gyrus (ANG), a region implicated in verbal number processing. Activation of intraparietal sulcus was also modulated by numerical distance. Our findings point to number semantics as cognitive locus of number-space interactions, thereby revealing the intrinsic spatial nature of numerical magnitude representation. Moreover, the involvement of left ANG is consistent with the mediating role of verbal/cultural factors in shaping interactions between numbers and space.
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In recent years, a whole-brain unbiased objective technique, known as voxel-based morphometry (VBM), has been developed to characterise brain differences in vivo using structural magnetic resonance images. The present review provides a brief description of VBM and then focuses on exemplar applications in healthy and diseased subjects. The procedure involves normalising high-resolution structural magnetic resonance images to a standard template in stereotactic space. Normalised images are then segmented into gray and white matter and smoothed using an isotropic Gaussian kernel. Finally, a series of voxel-wise comparisons of gray and white matter in different groups of subjects are performed, using Random Field theory to correct for multiple comparisons. VBM has been useful in characterizing subtle changes in brain structure in a variety of diseases associated with neurological and psychiatric dysfunction. These include schizophrenia, developmental and congenital disorders, temporal lobe epilepsy and even cluster headache. In addition, VBM has been successful in identifying gross structural abnormalities, such as those observed in herpes simplex encephalitis, multiple sclerosis and Alzheimer's disease. Studies of normal subjects, on the other hand, have focussed on the impact of learning and practice on brain structure. These studies have led to the finding that environmental demands may be associated with changes in gray and white matter. For instance, it has been reported that the structure of the brain alters when human beings learn to navigate, read music, speak a second language and even perform a complex motor task such as juggling. We conclude the present review by discussing the potential limitations of the technique.
This paper is concerned with the syndrome, described by me some years ago, of finger agnosia, disorientation for right and left, agraphia and acalculia, appearing as a result of a cerebral lesion located in the transitional area of the lower parietal and the middle occipital convolution.In 1924 I first described the symptom of primary elective disability for recognizing, naming, selecting, differentiating and indicating the individual fingers of either hand, the patient's own as well as those of other persons, and called the condition "finger agnosia." Subsequent to this gnostic disorientation with respect to the fingers, restriction in their separate kinetic realization not infrequently occurs. I also showed that the symptom of finger agnosia is characteristically associated with disorientation for right and left in respect to the patient's own body, as well as that of other persons, with special reference to the hands and fingers. The symptoms tend to appear
Identifying individuals with mathematical difficulties (MD) is becoming increasingly important in our education system. However, recognising MD is only the first stage in the provision of special educational needs (SEN). Although planning the effective remedial support is vital, there is little consensus on the interventions that are appropriate. There are two main reasons for this: first, MD has a variety of manifestations which appear to change with age; and second, there are many potential causes for the difficulties individuals experience. This paper addresses these issues by reviewing research evidence from three ‘domains’ of psychological research (genetic, cognitive, behavioural), all of which appear to offer insights into potential influences on mathematical ability.