Title: The sensory integration theory: an alternative to the Approximate Number System
*Wim Gevers; Center for Research in Cognition and Neurosciences (CRCN), Université Libre de Bruxelles and UNI – ULB Neurosciences institute, Brussels, Belgium, CP122, avenue F.D. Roosevelt 50, 1050 Bruxelles; e-mail:
[email protected]; phone number +32 26 504228 Roi Cohen Kadosh; Department of experimental psychology, South Parks Road, Oxford, OX1 3UD, United Kingdom; e-mail:
[email protected]; phone number: +44 1865 271385; fax number +44 1865 310447 Titia Gebuis; VU Amsterdam, Department of Molecular and Cellular Neurobiology, Center for Neurogenomics and Cognitive Research, Neuroscience Campus Amsterdam, De Boelelaan 1085, 1081 HV Amsterdam, The Netherlands; e-mail:
[email protected]; phone number: +31 20 5987111
*Corresponding author
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ABSTRACT The proficiency and neural underpinnings of human and non-human animal ability to estimate or compare different sets of items has been investigated in different fields of research such as evolution, development and education. The general consensus holds that these abilities are supported by the so-called approximate number system (ANS). In this chapter we will question the methods used in the ANS studies, challenge the existence of the ANS to some degree and present an alternative sensory integration theory.. In a first step, it is explained how our performance in numerosity judgment tasks can be explained on the basis of a mechanism weighing or integrating the different visual cues. A parallel is drawn between this integration mechanism and conservation abilities. In a second step, it is discussed how such a integration mechanism can be used to explain the observed relation between performance in numerosity judgment tasks and math achievement.
KEYWORDS: Numerosity, conservation abilities, math achievement, sensory integration theory
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INTRODUCTION In everyday life we are using symbolic numbers to understand and communicate numerical information. From computers to shopping, from sports events to banking, symbolic numbers play a fundamental role for these and many other activities. While adults can learn about the magnitude of new symbolic numbers without associating it with numerosity (Tzelgov, Yehene, Kotler, & Alon, 2000), infants and children would rely on an innate non-symbolic number system (Cantlon, Brannon, Carter, & Pelphrey 2006). Such a non-symbolic system is suggested to be composed of 2 subsystems: 1) a system that represents large numerosities (larger than 4 or 5 items) and 2) a system that subserves the exact representation of a smaller number of objects (Cantlon, Platt, & Brannon, 2009; Feigenson, Dehaene, & Spelke, 2004). In the first subsystem, numerosities are approximated while in the second subsystem a rapid and more accurate estimate is performed that has been termed subitizing (Kaufman, Lord, Reese, & Volkmann, 1949; Trick & Pylyshyn, 1994). In the present chapter, we focus on the first subsystem, which has also been termed the approximate number system (ANS) (Halberda, Mazzocco, & Feigenson, 2008a; Park & Brannon, 2013). The ANS is believed to extract large numerosities independent from the visual input like total surface area or density of the display (Burr & Ross, 2008; Piazza, 2010). This means that the estimate or the comparison of numerosities will not be biased by the size, density, surface or other sensory cues present in the image. Furthermore, the ANS is also considered to be the foundation of more complex mathematical skills. Studies showed that participants who perform better in estimating or comparing numerosities tasks are also better in more advanced mathematics (Halberda et al., 2008a; Libertus, Feigenson, & Justin Halberda, 2011; Park & Brannon, 2013; Piazza, 2010 but see De Smedt & Gilmore, 2011; Gilmore et al., 2013; Holloway & Ansari, 2009; Rousselle &
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Noel, 2007; Sasanguie, De Smedt, Defever, & Reynvoet, 2012; Soltesz, Szucs, & Szucs, 2010) . As we will argue below, the processing of numerosity depends on a variety of cognitive processes that are required for successful task performance. Central to our argument is that non-symbolic number stimuli (e.g. arrays of dots) are by definition confounded with sensory cues. However, instead of treating this confound as a problem to study pure numerosity processes, we would like to stress its potential role in numerosity processing. Our hypothesis is that large numerosities can be estimated or compared by integrating their different sensory cues. This idea that the judgment of larger numerosities relies on sensory information has been proposed before (e.g. Allik and Tuulmets, 1991; Dakin, Tibber, Greenwood, Kingdom, & Morgan, 2011) Although this view has been acknowledged as a plausible alternative for the ANS theory (Barth et al., 2003; Dehaene, 1992; Izard & Dehaene, 2008; Piazza et al., 2004; Stoianov & Zorzi, 2012), it did not yet receive large support. We additionally present the assumption that non-numerical abilities such as conservation (Piaget, 1965) can explain the variability in performance on numerosity tasks (i.e. the integration of the sensory cues) and might be a powerful tool to improve numerosity performance and hence more complex mathematical abilities.
The concept When one needs to compare the numerical quantity of two sets of dots, large differences in the accuracy of different observers is evident, especially when numerosity increases (Gebuis & Reynvoet, 2012d; Izard & Dehaene, 2008). This ability to compare or estimate larger numerosities is suggested to be supported by a mechanism referred to as the approximate number system (ANS). The theory suggests that the number of items would be estimated or compared independent of the sensory properties such as the size or density of the dots that are present in the visual scene (Dehaene & Changeux, 1993; Stoianov & Zorzi, 2012; Verguts & Fias, 2004). On the basis of arguments outlined in detail elsewhere (Gebuis, Cohen Kadosh, & Gevers,
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submitted) we reasoned that the numerosity of a visual display of dots can be estimated, but this would not be done independently from sensory cues such as size, brightness, circumference, loudness, pitch or frequency (Gebuis, Gevers, & Cohen Kadosh, 2014). It is important to examine whether what is assumed to reflect numerosity processing, is indeed numerosity, rather than sensory cues that may be processed by similar cognitive and neural mechanisms (Bueti & Walsh, 2009; Cantlon, Platt, & Brannon, 2009; Cohen Kadosh, Lammertyn, & Izard, 2008; Lourenco, 2015; Walsh, 2003). This similarity at the mechanistic level is not surprising as the abovementioned visual or auditory properties are confounded with numerosity in everyday life. Take for instance the situation of two cues at the airport passport control. Here, you want to quickly estimate the number of people in each line to pick the one with the least people to save as much time as possible. Usually, the longer line holds the larger number of people, and it is therefore not unlikely that this information is used to guide our behavior. Such a confound between numerical quantity and other visual features is not new and was already described by Piaget on the basis of the conservation paradigm (Piaget, 1965). In the conservation of number task, a child is typically shown 2 rows of buttons, each consisting of the same number of buttons. When one row of buttons is distributed more widely, the child at a certain developmental age is likely to say that the longer row now contains more buttons. Piaget claims that childrens’ thoughts in the preoperational stage (roughly age 2 until 5-6 years of age) are pre-logical, as the children are only able to focus on one feature of a problem at a time and are dominated by their immediate perception of things. For instance, during a conservation task with liquids, a child will typically say that one glass contains more
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water, because the water in that glass is higher. Or, in a number conservation task, a child will say that the row of more widely distributed buttons contains more buttons than the more densely packed row of buttons, simply because it is longer. In other words, at this stage a child is seemingly not able to differentiate between the visual features of a stimulus (e.g. length, height or density) and the numerosity associated with that stimulus (more or less items). Only later in development, a child becomes able to differentiate between the abstract notion of numerosity and visual features such as length, density and size. Once a person is able to differentiate between the perceptual cues and the abstract notion of numerosity, the important question is how a person derives the numerosity estimate. One possibility is that the person normalises all visual cues, filtering out the numerosity information per se. This normalisation process is one of the necessary steps in models that aim to explain how we represent numerosity and on forms the basis for the ANS theory (Dehaene & Changeux, 1993; Stoianov & Zorzi, 2012; Verguts & Fias, 2004). According to this theory then, pure numerosity is derived independently from or in parallel with the perceptual information. Another possibility is that the person will actually use the sensory information to perform some sort of integration across the different visual cues to derive a numerosity estimate. In recent work (Gebuis, Cohen Kadosh, Gevers, submitted), we outlined in detail why we believe that the second option is more likely to be the case. Therefore, in the following only a rapid overview of this argumentation is provided. Subsequently, assuming integration of perceptual cues is the process resulting in a numerosity estimate or comparison, we will make a proposal of how this integration process develops and how it could be studied. Finally, we will provide with some ideas on how the integration process could be related to arithmetic performance.
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Sensory cues remain to influence numerosity processes even when they are seemingly controlled Studies that manipulate sensory cues show that changes in visual properties affect numerosity estimation (Allik & Tuulmets, 1991; Dakin, Tibber, Greenwood, Kingdom, & Morgan, 2012; Frith & Frith, 1972; Gebuis & Reynvoet, 2012a; Ginsburg, 1991; Ginsburg & Nicholls, 1988; Sophian, 2007; Sophian & Chu, 2008; Szucs & Soltesz, 2008; Szucs, Soltesz, Jarmi, & Csepe, 2007) while studies that tried to control the different sensory cues still found effects of the sensory cues such as congruency effects (Gebuis & Reynvoet, 2012b; Gilmore, Attridge, & Inglis, 2011; Halberda & Feigenson, 2008). This implies that the sensory cues influence numerosity processing even when sensory controls are applied. This can be explained as follows: studies that controlled sensory cues to investigate numerosity processing neglected the most important fact, namely that a perfect control for sensory cues is practically impossible. Two sets of items that differ in numerosity always differ in one or more sensory cues. Consequently, changing the sensory cues, i.e. by making for instance dots larger or denser, etc. does not prevent reliance on these sensory cues. Instead, numerosity estimates could become less accurate, possibly because the integration of the different sensory cues becomes more difficult (Gebuis & Reynvoet, 2012c). Few studies investigated the effects of sensory cues. These studies showed that congruency effects change when the sensory cues change (Gebuis & Reynvoet, 2012b; Gebuis & Van der Smagt, 2011; Hurewitz, Gelman, & Schnitzer, 2006). Some of these changes cause the cancellation of a congruency effect, or even the reversal. A likely explanation for these results is that when multiple sensory cues exist, they tend
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to compete with each other in their weight given to the numerosity estimate. For instance, if there is a stimulus that is incongruent in surface but congruent in diameter, more weight could be given to diameter if this cue is more salient (e.g. differs to a larger extent between the two stimuli that need to be compared). In this case better performance for surface incongruent trials can be observed. According to this explanation two or more sensory cues can also cancel each other’s effect (Gebuis & Reynvoet, 2012b). This interesting interplay between the different sensory cues to reach a numerosity estimate shows striking resemblances with the process of conservation, explained above.
Sensory integration and ANS tasks As discussed earlier in this chapter, integrating the sensory cues when performing an ANS task like the numerosity comparison task seems to hold strong similarities with the well-known process of conservation (Piaget 1952). Piaget divided children in different stages on the basis of their performance. The stage termed pre-operational representation (beginning around 18 months of age and lasting until 6 or 7 years of age) is related to our argument. This is the period in which a child makes the famous conservation errors. A classic example is the liquid conservation problem where two identical glasses of water (e.g. tall and narrow glasses, filled up to the same level) are presented to a child. Subsequently, while the child is watching, the water is poured from a tall and narrow glass into a lower and wider glass. If a child is not able yet to conserve, he/she will typically indicate that the narrow tall glass contains more water. Another conservation example relates to numerosity, here the same number of buttons is placed in two parallel lines of equal length. Then the experimenter moves the buttons of one line such that this line becomes longer than the other and again asks
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the child which line contains most buttons. A child that cannot conserve yet will typically say that now the longer line contains more buttons (Piaget, 1965). These are clear examples of situations or tasks where a single perceptual characteristic interferes with the correct numerical answer. An important factor to explain the difficulties observed in a conservation task is the notion that children have to understand that an increase in one dimension is compensated for by a decrease in another dimension (Piaget, 1952). This means that in the liquid conservation task children should understand that an increase in height is compensated for by a decrease in width while in a numerosity task, children have to understand that a group of large dots scattered over a large area generally contains fewer dots than a group of small dots scattered over a smaller area at a higher density. The understanding that all these sensory variables are related to each other and all together give information about the numerosity presented is the basis for what we termed the ‘integration procedure’. In this view, one individual cue cannot inform about numerosity but instead an integration of different visual cues is required. In other words, the integration is not derived from a calculation on the parts but a system on its own that receives input from the different sensory streams (for a similar position, see Anobile, Cicchini, & Burr, 2014; Arrighi, Togoli, & Burr, 2014). The integration of visual cues seems to relate to the concept of an inverse relationship between dimensions as observed in the conservation task. Indeed, in conservation tasks, children are tested on their understanding that number can remain the same even when physical appearances change. One could interpret this such that representations of number exist as well as its physical properties. In our sensory-‐based theory, we do not dispute that we have a concept of numerosity. What we do dispute is that this concept is derived independent from the different sensory properties via an ANS system. We believe
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that the children’s inability to perform the Piaget task at an early age could indicate that they think of numerosity in terms of sensory cues and are not equipped with an innate number system that judges number separate from the visual cues. It is only at a later stage that they start to understand the concept of conservation. The children now understand that even though one row of buttons is longer, it does contain more space in between the individual buttons, and therefore might consist of an equal number of buttons as the more compressed row. In other words, understanding conservation would be similar to understanding that numerosity is comprised of several sensory cues that together give an insight in the number presented. Conservation abilities are acquired throughout development. Therefore, if a parallel can be drawn between performance in numerosity comparison tasks and conservation tasks, it could be expected that children become better with age in comparison tasks. In other words, one would expect that the size of the congruency effect observed in numerosity comparison tasks decreases with increasing age. Such a decrease in the size of the congruency effect with age has indeed been observed (Szucs, Nobes, et al., 2013; Gebuis et al 2005). Furthermore, Szucs et al. (2013) showed that performance of adults and children was the same for congruent but largely differed for incongruent trials. The authors concluded that the difference in the size of the congruency effect was related to inhibition abilities, arguing that younger children have more difficulty to inhibit irrelevant information compared to adults. This notion fits nicely with the theory of Piaget (1953) that children have to learn to inhibit the false heuristic of choosing the longer line or the taller glass, and instead base their judgment on numerosity. Indeed, very bad performance in numerosity tasks is visible at ages where inhibition is not yet fully developed. For instance, it has indeed been observed that children of 3 (Rouselle & Noël, 2008; Rouselle et al.,
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2004) or 4 (Soltezs, Szucs, & Szucs, 2010) years of age were unable to perform the numerosity comparison task if visual cues were manipulated inconsistently. Performance is above chance starting from the age of 5 (Gebuis, Cohen Kadosh, de Haan, & Henik, 2009). However, the ability to judge numerosity does not only depend on the general ability to inhibit false heuristics (Houde et al 2011) but also on the ability to weigh the different perceptual dimensions (Defever, Reynvoet, & Gebuis, 2013). Defever et al. (2013) compared the size of the congruency effect in a numerosity comparison task across different ages (1st, 2nd, 3rd and 6th grade of primary school). Surprisingly, the congruency effect increased with age. Closer inspection of the data demonstrated that visual cues were important for all age groups but that younger children relied on a subset of the sensory cues. Not all of these children relied on the same subset of sensory cues (for similar heterogeneity in congruency effects see: Halberda & Feigenson, 2008). In the youngest age group, about half of the participants associated a large visual cue with a larger numerosity while the other half associated the larger visual cues with smaller numerosities. This response pattern resulted in opposite congruency effects, which lead to the cancellation of the overall congruency effect. These opposite congruency effects diminished with increasing age, most likely because the older children stopped responding to a single subset of sensory cues but instead adopted a more diverse strategy of taking into account the full range of sensory cues. That this was a more effective strategy was visible in the overall performance as it increased with increasing age. New research could investigate the direct link between conservation abilities, knowledge of the inverse rule and numerosity comparison performance across different ages by disentangling the congruency effect with respect to different sensory
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manipulations. More work is needed on this topic. For instance, it would be interesting to know whether the developmental trajectory for a given child on conservation tasks is exactly the same as his trajectory for the congruency effect in typical ANS tasks like comparing larger numerosities.
Sensory integration and arithmetic Previous studies suggested that performance on a numerosity comparison task is related to different mathematics abilities (Gilmore et al., 2011; Halberda, Mazzocco, & Feigenson, 2008b; Libertus, Feigenson, & Halberda, 2011; Libertus, Feigenson, & Halberda, 2013; Lourenco, Bonny, Fernandez, & Rao, 2012; Mundy & Gilmore, 2009). Three reasons for this association have been put forth: 1) the acuity of the ANS system; the higher the acuity the better the performance on ANS-like tasks and consequently mathematics (e.g. Halberda…Libertus..etc), 2) inhibition; the better the ability to inhibit responses, the smaller the congruency effects in number-size congruency tasks, and the better math performance (e.g. Szucs et al…. Gilmore et al..), and 3) conservation ability or the integration of different sensory cues; the better both abilities the better math performance (e.g. Defever et al 2013). In our view, both inhibitory processes and conservation or integration abilities are required for numerosity tasks and consequently could both relate to math ability. More specifically inhibition is required to suppress a direct response to the for instance the most prominent sensory cues and in a subsequent step the participant has to be able to integrate the various sensory cues to make a numerosity judgement. Gilmore et al (2013) suggested a different role for inhibition ability. Instead of suppressing the most prominent sensory cues to allow integration of the various sensory cues, they suggested that inhibition is necessary to inhibit all sensory cues to
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allow an ANS-‐based numerosity judgment independent of the sensory cues. In their first experiment, children performed a dot comparison task indicating which of two stimuli contained the largest number of dots. Children showed a congruency effect with worse performance on incongruent compared to congruent trials. The size of this congruency effect was taken as a marker of inhibitory skills and correlated with math ability. In the view of Gilmore et al. (2013) inhibition is required to suppress a response to the visual cues so that in the end participants can respond on the basis of their ANS. Freely translated, inhibition can be taken as the functional processing mechanism to accomplish normalization of the visual cues. The better the inhibition, the less influence of the visual cues, the smaller the congruency effect. The idea now would be that better inhibitory functions (i.e. ability to suppress visual information) would relate to math achievement. Interestingly however, Gilmore also observed that the correlation between math achievement and inhibitory skills resulted from the incongruent trials only. This is not in line with the idea that inhibition is applied to all trials. Rather, inhibition would be applied to incongruent trials only, and it would be exactly this inhibition, which is related to math ability. One can ask the question how the child is able to selectively apply inhibition to incongruent trials only. In other words, how does the child know that a trial is incongruent if he did not apply inhibition on the visual cues yet? In our proposal, inhibition is an important feature, but it would be needed to suppress the initial tendency to respond to the most salient visual feature, and this regardless of whether a trial is congruent or incongruent. The difference in performance between congruent and incongruent trials would result simply because weighing (i.e. the integration) of the different sensory variables is more difficult on incongruent compared to congruent trials.
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tasks, a logical consequence is to expect a relationship between sensory integration or conservation ability and math achievement as well. This is indeed the case: a large number of studies observed strong correlations between sensory processes (Lourenco et al. 2012; Tibber et al., 2013) and conservation ability and arithmetic achievement at the end of first grade (Dimitrovsky & Almy, 1975; Dodwell, 1961; A. S. Kaufman & Kaufman, 1972). The correlation between conservation abilities and math achievement persist even when IQ was controlled for (Taloumis, 1979). Further research was performed trying to establish the value and the meaning of this correlation. One line of research focused on intervention studies. Researchers asked the question whether training children the conservation method would result in better arithmetic performance (Bearison, 1975). The answer was negative. Children who spontaneously achieved conservation earlier benefited more from instructions in arithmetic but this benefit could not be induced by training the children on conservation-like tasks (Bearison, 1975). Can a similar pattern of observations be made in studies relating numerosity processing abilities to math achievement? To establish the association between the age onset of numerosity processing abilities and math achievement later on, a developmental individual differences approach is needed, which to our knowledge, has not been conducted yet. However, a few intervention studies have been conducted that included some form of numerosity training and measures of math achievement (Räsänen, Wilson, Aunio, & Dehaene, 2009; Wilson, Dehaene et al., 2006; Wilson, Revkin, Cohen, Cohen, & Dehaene, 2006). However, none included control conditions and therefore do not allow for strong conclusions. One recent study did use language processing as a control condition (Obersteiner, Reiss, & Ufer, 2013). In line with the results observed in
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studies linking conservation to math achievement, no main effect of numerosity training on math achievement was observed. Caution is needed as this parallel is based on a null finding (e.g. no effect of training numerosity on math achievement). Clearly, more work is needed here. On the basis of an extensive review of the literature, Hiebert and Carpenter (1982) used a different approach. They investigated which task characteristic in conservation tasks would be responsible for the observed association with certain aspects of math achievement. A first relevant observation was the strong relation (beyond IQ) between conservation abilities and math achievement. Importantly however, success on the conservation task was not a prerequisite for success on a number of basic mathematics tasks. In other words, even when children had difficulties with some aspects of the conservation task, mathematic skills could (at least in some children) still be acquired (Hiebert, Carpenter, & Moser, 1982; Steffe, Spikes, & Hirstein, 1976). The ability that seemed most directly related to math achievement was again the notion of an inverse relationship between unit number and unit size. Maybe this observation could provide a clue to answer the important remaining question as to why the ability to compare more accurately larger numerosities, which requires the integration of the different sensory cues is related to math achievement. In our view, such a link, if existent, might be mediated by inhibitory mechanisms (Houdé, 2009; Pina, Moreno, Cohen Kadosh, & Fuentes, 2015). The ability to combine different sensory cues to make an estimate of numerosity seems to be key to strong performance on numerosity studies. However, a parallel reasoning would also have a more pessimistic implication, namely that intervention studies improving numerosity judgments would not easily result in improvements on math achievement. However, to properly address
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this relationship, more intervention studies focusing on the specific strategies used to judge numerosities and relate it to math achievement are needed.
Conclusions In the present chapter, we described a number of important caveats in the theory of the ANS, which describes our ability to estimate and or compare large numerosities. Empirical evidence is not consistent with, or even contradicts, the existence of an ANS. We therefore proposed an alternative sensory integration mechanism that, without invoking a number sense, seems to be able to explain our performance in numerosity judgment tasks. By making a theoretical parallel between the integration mechanism and conservation abilities, some testable hypotheses could be formulated. These hypotheses concern the processing of numerosities itself as well as the observed link between numerosity processing and math achievement. We hope that such insights will open new venues for studies that will be able to take into account the effect of sensory cues, and will allow better theoretical progress with impact not only on basic science but also on numerical deficiencies (e.g. dyscalculia) and intervention studies.
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