=Paper=
{{Paper
|id=Vol-1419/paper0128
|storemode=property
|title=The Cognitive Contribution of Spatial Representation to Arithmetical Skills
|pdfUrl=https://ceur-ws.org/Vol-1419/paper0128.pdf
|volume=Vol-1419
|dblpUrl=https://dblp.org/rec/conf/eapcogsci/Pinna15
}}
==The Cognitive Contribution of Spatial Representation to Arithmetical Skills==
https://ceur-ws.org/Vol-1419/paper0128.pdf
The cognitive contribution of spatial representation to arithmetical skills
Simone Pinna (simonepinna@hotmail.it)
Department of Pedagogy, Psychology, Philosophy, University of Cagliari
via Is Mirrionis 1, 09123 Cagliari, Italy
Abstract Verifications of the SNARC effect are based on simple nu-
merical tasks such as parity judgement (Dehaene et al., 1993)
The aim of the present paper is to show that, contrary to the
main current approach to numerical cognition, the link be- and magnitude comparison (Brysbaert, 1995). In parity
tween space and numbers cannot be reduced to the concept of judgement tasks, where subjects are asked to classify a num-
a mental number line (MLN). A distinction between low-level ber as odd or equal by pressing either a right or a left-hand po-
numerical skills, that involve MLN processing, and higher-
level arithmetical skills, which are related to algorithmic pro- sitioned key, left-hand responses are faster for smaller num-
cessing, is needed in order to understand the different contri- bers and right-hand responses are faster for larger numbers.
bution given by visuo-spatial skills. Magnitude comparison consists of judging whether a num-
I suggest that cognitive skills related to symbolic manipula-
tion should be analyzed on their own in order to better under- ber is smaller or larger than a given one. Here, when left-
stand the importance of spatial representation for arithmetical hand and right-hand keys stand for, respectively, “smaller
processing and, to this purpose, I propose a model of algo- than” and “larger than”, responses are faster than the inverse
rithmic skills that could be useful to study some typical fea-
tures of symbolic manipulation, such as the influence of spatial response-key configuration.
schemes and the role for external resources in working mem- These cognitive effects led to the hypothesis of a MLN,
ory offloading. where numbers are ordered from smaller to larger accord-
Keywords: Numerical cognition; Extended cognition. ing to a left-to-right or right-to-left orientation, depending
on the writing direction (Restle, 1970; Seron, Pesenti, Noël,
Introduction Deloche, & Cornet, 1992; Dehaene et al., 1993). A recent
An important line of research in the field of numerical cogni- animal cognition study (Rugani, Vallortigara, Priftis, & Re-
tion is focused on the link between numbers and space. Many golin, 2015) brings a strong evidence toward the existence of
empirical data suggest that spatial representation and visuo- a left-to-right oriented MLN in newborn chicks, suggesting
spatial skills are involved in various number related cognitive that experiential factors or even, in humans, cultural conven-
activities, as magnitude representation, magnitude compari- tions (as the writing direction) may intervene in modulating
son, simple mental arithmetic and multidigit calculations. or modifying the innately left-to right oriented MLN.
A very influential explanation of space/number interaction, Evidence of the existence of a MLN come also from cogni-
which focuses on the concept of a mental number line (MLN), tive neuropsychology. In line bisection tasks, in which sub-
is based on the fact that numerical and spatial representa- jects are asked to indicate the midpoint of a line, hemi-neglect
tion are mediated by common parietal circuits (Dehaene, patients have the tendency to move the midpoint towards the
Bossini, & Giraux, 1993; Hubbard, Piazza, Pinel, & Dehaene, portion of the line opposite to the controlesional (usually left)
2005). However, this kind of explanation does not seem to fit portion of the space (Driver & Vuilleumier, 2001). Interest-
with other types of space/number correlations, like those im- ingly, when tested on number bisection tasks, where subjects
plied in the performance of multidigit arithmetical procedures are asked to specify the midpoint number of various numer-
(Szucs, Devine, Soltesz, Nobes, & Gabriel, 2013). ical intervals, these patients show a bias toward larger num-
The aim of the present paper is to show that the link be- bers, which suggests that spatial neglect affects also the MLN
tween space and numbers cannot be reduced to the concept (Vuilleumier, Ortigue, & Brugger, 2004; Zorzi, Priftis, &
of a MLN. We should recognize different kinds of cognitive Umiltà, 2002).1
contribution of space representation above numerical skills, Some researchers argue that several cognitive strategies for
according to the level of cognitive activity considered. A number processing have been developed to take advantage
distinction between low-level numerical skills, that involve from the interaction between numerical and spatial represen-
MLN processing, and higher-level arithmetical skills, which tations. Strategies based on the use of the MLN are involved
are related to algorithmic processing, is needed in order to in some mental operations, like subtractions, more than in
understand the different contribution given by space-related others, like multiplications, which rely mostly on verbal facts
skills. retrieval (Dehaene & Cohen, 1997; Ward, Sagiv, & Butter-
worth, 2009). Also, performances on visuo-spatial tasks in
Space and magnitude representation preschool children are positively related to the development
A set of robust empirical data, mostly connected to the study 1 This topic is currently debated. A thorough study involving 16
of the Spatial Numerical Association of Response Codes right brain-damaged subjects shows a dissociation between devia-
(SNARC) effect, suggests that numerical representations trig- tions in physical and number line-bisection tasks, suggesting that the
navigation along physical space and number lines is governed by dif-
ger spatial ones, with smaller numbers connected to the left ferent brain networks (Doricchi, Guariglia, Gasparini, & Tomaiuolo,
side, and bigger numbers to the right side of the space. 2005).
768
of more advanced arithmetical skills (LeFevre et al., 2010). of common processing mechanisms, rooted in our need for
This fact is a further sign of the importance of space-numbers information about the spatial and temporal structure of the
interaction. However, some aspects of the cognitive mecha- external world” (Walsh, 2003). According to Walsh, the pari-
nisms underlying this interaction remain unexplained. etal cortex represents environmental information about dif-
ferent kinds of magnitudes—number, space, time, size, speed
Neurocognitive explanations etc.—whose interaction are supposed to have a specific—and,
Since the last decade of the past century, many researchers to date, neglected—meaning for the guidance of action. In
have inquired into the neural basis of numerical cognition. Walsh’s view these cognitive interactions are due to overlap-
An influential model is Dehaene’s triple-code model of num- ping sets of parietal neurons that share a common metric for
ber processing (Dehaene, 1992). According to Dehaene, magnitude representation, and this fact is at the base of the
number representation involves different neural substrates in SNARC effect. ATOM predicts that the SNARC effect is
which numerical information is encoded visually (as strings an instance of a broader SQUARC (Spatial QUantity Asso-
of arabic digits), verbally (as number words, sets of number ciation of Response Codes) effect, “in which any spatially
facts etc.), and analogically (as magnitide). The latter kind or action-coded magnitude will yield a relationship between
of numerical representation is mediated by a neurocognitive magnitude and space” (Walsh, 2003). In some cases this pre-
system called “number module”, or Approximate Number diction has, indeed, been verified (Bueti & Walsh, 2009).
System (ANS), which is shared by different animal species
and makes for the representation of approximate quantities Space and arithmetical skills
(Dehaene, 2011; Landerl, Bevan, & Butterworth, 2004; Pi- So far, ATOM represents the most convincing explanatory
azza, 2010). account of the link between numerical and spatial represen-
The typical problem in which the ANS comes into play is the tations. However, the question about the function of non-
comparison between different sets of objects to decide which symbolic magnitude representation in the development of
is the largest. Some researchers suppose that the ANS is the arithmetical skills remains opened.
only preverbal representational system needed for the devel-
opment of basic numerical concepts (Piazza, 2010). Spelke Developmental dyscalculia and spatial skills
(2011) combines the work of the ANS with the information Developmental dyscalculia (DD) is a learning disability
gained by the Object Tracking System (OTS), which pro- which specifically affects the acquisition of arithmetical skills
vides the capacity to recognize at a glance the number of in otherwise normal subjects. In its pure form (i.e., without
objects in sets less than or equal to 4 items (subitizing). In co-morbidity with other learning problems, such as dyslexia
this account, the OTS supports the representation of inte- or attention-deficit-hyperactivity disorder) it is estimated that
gers, which is needed for symbolic arithmetic processing. On it affects 3 − 6, 5% of school-age population. As other learn-
the contrary, Leslie, Gelman, and Gallistel (2008) propose ing disabilities, it is thought to have a neural basis (Shalev &
that the representation of exact quantities does not involve Gross-Tsur, 2001).
the OTS. Instead, the natural number set is recursively con- Given the essential function in number processing accorded
structed through the successor function on the base of some to the ANS—which is neurally located in the bilateral intra-
innate concept of integer number—the concept of exactly one, parietal sulci—it is natural to suppose that DD is related to
at least—and, then, is mapped on the approximate magnitude impairments to this neurocognitive module (Dehaene, 2011;
representations given by the ANS. Piazza, 2010).
However, a recent paper by Szucs et al. (2013) reviews previ-
A Theory Of Magnitude (ATOM) ous experiments directed at verifying the implication of ANS
Despite the aforesaid differences, all theoretical accounts rec- impairments for DD and denies that we have sufficient em-
ognize a crucial role of the ANS. Moreover, space-numbers pirical evidence to prove this correlation. They conducted an
correlations, as those revealed by the SNARC effect, sug- extensive series of tests and experiments on a population of
gest that there should be some kind of interaction between 1004 DD affected 9-10-year-old children, concluding that the
neural circuitry involved in spatial and numerical process- main cognitive factors that cause DD are visuo-spatial mem-
ing. In fact, functional MRI (fMRI) studies reveal that non- ory and inhibition impairments which, crucially, are related to
symbolic number processing activates neurons of a bilateral IPS activity as well as the ANS. Consequently, they propose a
parietal cortical area, the Intra-Parietal Sulci (IPS), which different approach to the explanation of DD in which findings
have a functional role also in visuo-spatial and manual tasks, about IPS morphological and functional differences between
such as grasping and pointing (Butterworth, 2000; Hubbard DD subjects and controls (Mussolin et al., 2010; Price, Hol-
et al., 2005; Pinel, Dehaene, Rivire, & LeBihan, 2001; Si- loway, Räsänen, Vesterinen, & Ansari, 2007; Rotzer et al.,
mon, Mangin, Cohen, Bihan, & Dehaene, 2002). 2008) are linked to general purpose cognitive processes in-
In 2003 an influential proposal, called A Theory Of Mag- volving the IPS, rather than magnitude representation deficits.
nitude (ATOM), was made in order to “bring together [...] In particular, inhibition impairments “could lead to mathe-
disparate literatures on time, space and number, and to show matical problems because Numerical Operations require the
similarities between these three domains that are indicative temporal and spatial (in imagination) coordination of several
769
processes and the retrieval of several highly similar facts— Different cognitive contributions
impaired inhibition probably interferes with the organization To sum up, we have seen at least two very different cognitive
of these processes” (Szucs et al., 2013). Also, inhibitory contribution of spatial skills to numerical abilities:
processes seem to have a crucial function for the central ex-
ecutive component of the working memory (WM) (Carretti, 1. Numerical and spatial representations are linked by the ex-
Cornoldi, De Beni, & Palladino, 2004). Then, for DD sub- istence of a MLN, where numbers are represented as or-
jects, problem in visuo-spatial memory tasks may be sec- dered from smaller to larger according to a precise orienta-
ondary to the impairment of inhibition processes. tion. This hypothesis is supported by, e.g. , experiments on
the SNARC effect and the number bisection task in hem-
Neuropsychological studies ineglect patients.
2. Visuo-spatial skills are implied in arithmetical processing
Some hints about the correlation of spatial and numerical for monitoring the temporal and spatial coordination of the
skills also come from the field of neuropsychology. Here, many processes needed in order to carry out an arithmetical
some interesting cases have been described that seem consis- operation—selecting the right factor, keeping track of the
tent with the findings of Szucs et al. (2013) reported in the partial results, arranging numbers correctly in the space,
previous paragraph. performing operation steps according to a given schema
Semenza, Miceli, and Girelli (1997) report a case where arith- etc. In this case, the link between space and numbers is
metical difficulties seem to be related to the lack of monitor- explained by resorting to general purpose cognitive capac-
ing of arithmetical procedures. The patient, M.M., 17 years ities, such as inhibition processes, which are crucial for the
old, was affected by hydrocephalus with the posterior portion central executive component of the working memory.
at the right dorso-frontal cortex and the right upper parietal
lobe severely damaged. His mental calculation abilities were The two points sketched above may be put in correspondence
excellent and, in some cases, surprising (he could solve 2 × 2 with different skill levels in which a link between numbers
digit multiplications where the two factors were the same, as, and space is on hand.
e.g., 24 × 24, with the same speed as for table problems). In Point 1 corresponds to low level skills, based on innate
written calculation, however, his performances were dramat- cognitive systems of magnitude representation. These skills
ically poorer. His problems, especially manifest in complex are needed to perform tasks such as non-symbolic magnitude
multidigit multiplications, were of different nature, but the comparison, number comparison and parity judgments. Also,
mostly committed errors were connected to wrong factor se- strategy based on the representation of numbers on a MLN
lection. M.M. often repeated subsequently the same opera- are very likely involved in some types of mental operations,
tion, and did not realized when he had reached the end of the like simple mental subtractions and, to a lesser extent, mental
procedure—the authors report that “[h]e kept asking the ex- additions.
aminer whether the operation was finished” (Semenza et al., Point 2, on the other hand, corresponds to higher level skills,
1997). In some cases, M.M. made errors also in the spatial mostly based on learned abilities. These capacities, which
arrangements of numbers. may be included in the specific concept of algorithmic skills,
In another study, Granà, Hofer, and Semenza (2006) report a are needed in order to apply computing strategies, i.e. to ex-
case of “spatial acalculia”, i.e. a specific deficit in the spatial ecute set of rules for symbolic transformation, and often rely
arrangement of numbers in written calculation. The patient, on the use of paper and pen or equivalent external resources.
PN, had a vast parietal damage as a consequence of a brain
haemorrage. In addition to spatial arrangements errors, PN Algorithmic skills
committed errors in factor selection. According to the au- With the word “algorithm” I refer to any finite set of rules for
thors, “the best explanation for PNs problems is that he might symbolic transformation which can be performed by a sub-
have difficulties in relying on a visuo-spatial store containing ject with the only aid of paper and pen (or equivalent external
a layout representation specific to multiplication. As a conse- resources) and without resorting to any particular insight or
quence, while knowing what, when and how to carry out the ingenuity.
various steps, PN does not know where” (Granà et al., 2006). This informal definition may be dated back to the work of
The procedural problems described in these neurospycholog- Alan Turing (Turing, 1936). Only in recent past (Giunti,
ical reports seem consistent with the functional interpretation 2009; Wells, 2005) the cognitive importance of this work has
of inhibition processes in arithmetical procedures given by been fully recognized.
Szucs et al. (2013). M.M.’s problems with factor selection Algorithmic skills consist on all those cognitive skills that
and operation ordering may be explained as a consequence employ algorithms. The possibility of using external tools
of the interference of previously performed operational steps. such as paper and pen, which is included in the concept of al-
On the other hand, visuo-spatial deficits like those manifested gorithm, seems particularly relevant for the issue about num-
by PN may be also linked to inhibition impairments by a sim- bers/space interaction. Algorithmic strategies (e.g., the stan-
ilar mechanism as that seen in the previous paragraph. dard multiplication algorithm), indeed, make extensive use
770
of spatial schemes in order to subdivide arithmetical opera- Sketch of a model2
tions into simple computational steps and for working mem- A model of algorithmic skills3 should include different parts
ory offloading. Rumelhart, Smolensky, McClelland, and Hin- in order to reflect a set of relevant features that I previously
ton (1986) consider such kind of cognitive tasks as cases of highlighted, i.e.:
online symbolic transformation. Here, visuo-spatial skills are
involved in order to “maintain on-line the spatial layout and (i) a central unit, in which rules of execution and knowledge
digits of an ongoing multi-digit calculation” (Dehaene & Co- of arithmetic facts are embedded;
hen, 1995). (ii) a temporary store, where, at each step of a computation,
The spatial layout may in itself facilitate carrying out compu- the relevant data are hold;
tational steps, e.g., by simplifying the individuation of each
single step or by making easier to recognize the temporal or- (iii) an external store, which contains written symbols;
der of single operations.
Landy and Goldstone (2007) experimentally tested the hy- (iv) a perception/action mechanism, which connects the central
pothesis that symbolic spatial layout is organized to reflect unit to the stores and makes for rules execution.
syntactic relations among symbols. They start from suggest- If we take the cue from algorithmic execution with pa-
ing that “formal notations are diagrammatic as well as senten- per and pen, part (i) and (ii) may be specified as the inter-
tial and that the property conventionally described as syntac- nal side of the model— respectively, the long-term memory
tic structure is cognitively mediated, in part, by spatial infor- and the WM’s slave systems (Baddeley & Hitch, 1974; Bad-
mation. Elements of expressions are bound together through deley, 2000)—, while part (iii) is the external memory (the
perceptual grouping, often induced by simple spatial proxim- paper and its content). However, the model should be flexible
ity.” (Landy & Goldstone, 2007). To test this hypothesis, they enough to include examples where, e.g., rules to be executed
designed two experiments. are found written or symbolic transformation is made men-
In the first, subjects had to write down with paper and pen tally. In the latter case, part (iii) may be omitted.
simple equations they found written on a computer screen. Part (iv) is needed for making rules of execution effective by
In one side of the equation were three single-digit numbers correctly connecting the other parts of the model, and incor-
with two operands (addition and/or multiplication sign); the porates the executive component of the WM. The cognitive
other side contained the same expression, but with one oper- work carried out by this part may help understanding the con-
ation completed (e.g, x × y + z = w + z). The experimenters tribution of spatial representation to algorithmic skills. For
found that subjects tended to leave a tighter space between instance, let us think of any case of algorithmic execution
numbers and the operand in the operations that, according to with paper and pen. A mechanism of perception/action, at
syntactic rules, had to be completed earlier—e.g, the space each computational step, has to complete the following cycle:
between numbers and sign in multiplications was tighter than
in additions—even if they need not perform any operation. • take inputs from the external store, i.e. draw subject’s at-
The second experiment consisted in the production and writ- tention to relevant data among written symbols;
ing (with paper and pen) of formal propositional logic ex- • choose from the central unit a rule to be executed, on the
pressions. Even in this case, the result was that the spac- basis of the symbol or set of symbols on which subject’s
ing around logical connectives reflected syntactic rules (e.g., attention is drawn;
the blank space around principal connectives were wider than
that around secondary connectives). • apply the rule to the right symbol or set of symbols and
These results seem to point toward a confirmation of the cen- elaborate the output or retrieve it from the central unit;
tral function earlier ascribed to spatial representations in al- 2 Anderson, Lee, and Fincham (2014) have recently proposed a
gorithmic skills. In particular, the fact that similar spatial fea- neuro-functional model of mathematical problem solving. Although
tures are found in the production of both numerical and logi- relevant for this issue, it should not be seen in contrast with the
cal expressions is consistent with the definition of algorithmic model sketched here, for it is meant to explain a different set of
cognitive activities, i.e., mathematical problems where the cognitive
skills given above, for this definition does not limit them to subject does not necessarily know in advance what is the best solv-
numerical transformation but refers to all kinds of symbolic ing strategy. The model I propose, on the contrary, deals only with
manipulation. This, interestingly, suggests that we are facing symbol transformation activities with clear-cut sets of rules of com-
putation, such that the subject should be able to access them without
some kind of broad set of cognitive skills, which deserves to particular cognitive effort.
be investigated in its own right. 3 Giunti (2009) proposed a Turing machine-inspired model—the
Bidimensional-Turing machine—of what he calls “Phenomena of
human computation”, a concept that roughly corresponds to the idea
of algorithmic skills proposed above. An implementation of Giunti’s
approach to the study of human computational skills is in Pinna and
Fumera (in press). Here, I will describe intuitively and without deep-
ening into technicalities the main features that a model of algorith-
mic skills should include, in order to help explaining some issues of
numerical cognition as the importance of the spatial component of
the WM for arithmetic.
771
• hold in the temporary store, if necessary, some data relative manipulation task, or the similarity/difference between men-
to the output, e.g., a carry; tal and externalized strategies.
The empirical adequacy of the model may be tested, e.g,
• transcribe the output of actual symbolic operation into the by inspecting to what extent different spatial schemes used
external store; for performing similar symbol manipulation tasks influence
WM load. On the other hand, indirect indications may also
• hold in the temporary store the relevant information needed come from behavioral experiments, e.g., by analyzing re-
to start the next cycle. sponse times to the solution of similar tasks presented accord-
ing to different spatial layouts.
Features of the spatial schema used in a paper and pen al-
Such kind of cognitive analysis may be useful to shed
gorithm must be hold correctly to complete a cycle of compu-
light on the relation between space and numbers and, by a
tation. Spatial features seem to be particularly important, e.g.,
broader perspective, to further investigate the issue of organ-
for choosing the right symbols to which apply a rule of trans-
ism/environment interaction for cognition.
formation, and to transcribe in the right place of the schema
the result of an operation. An incorrect execution of these
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