The Involvement of Hands in Counting

It is well established that pointing to, touching, or moving items during counting is an integral part of the development of children’s number knowledge (Graham, 1999). Children use such gestures spontaneously and several independent studies

have confirmed that this facilitates the counting accuracy (Schaeffer et al., 1974;

Gelman, 1980; Saxe & Kaplan, 1981; Gelman & Meck, 1983; Fuson, 1988; Graham, 1999; Alibali & DiRusso, 1999; Carlson, Avraamides, Cary & Strasberg, 2007). Ac-cording to Schaeffer et al. (1974), preventing the children from pointing severely disrupts their counting: in such case a child most often either emits an indefinite stream of numbers or does not count at all. The studies by Gelman (1980) and Gelman and Meck (1983) provide further evidence for the importance of the actual physical contact with the counted items. The children in these studies experienced more difficulties after the objects being counted have been put behind a transparent cover, which allowed them to point to the objects but not to touch them. Alibali and DiRusso (1999) in turn addressed the issue of active and passive gestures. The former refers to the pointing performed by the child itself and the latter to the point-ing performed by somebody else, for example an experimenter-controlled puppet.

As it turns out, both active and passive gestures significantly improve the children’s counting accuracy over the situation in which the gesture is prevented. The count-ing competence in children develops over a significant period of time (see section 2.3.2), and accordingly, the contribution of the counting gestures has a clearly de-velopmental character. This conclusion is supported for example by the results of Saxe and Kaplan (1981) who showed that 4-year-old children significantly benefit from pointing, in contrast to 2- and 6-years-olds.

Following the ample experimental evidence for the supportive role of the counting gestures in learning to count, a number of hypotheses concerning the specific nature of this contribution have been put forward in the literature. Among these, three main themes can be distinguished.

The first group of hypotheses views gesturing as a way to overcome the limitations in the available cognitive resources. For instance, it can be argued that the gestures may ‘externalise’ some of the contents of the working memory. According to one of the earliest proposals, pointing while counting is a way to keep track of the counted items (Schaeffer et al., 1974). In order to adhere to the one-one principle (Gelman &

Gallistel, 1978, see section 2.1.2), one has to separate the objects that have already been counted from those that still remain to be counted, otherwise some items may be counted more than once while others may be omitted. A hand pointing toward an object can fulfil the role if an ‘external memory register’, identifying the current object being counted, and, indirectly, all the objects counted so far. This is especially plausible when the counted set is arranged in a way that can be followed by a smooth hand trajectory (such as a single row) and is consistent with the observation that the children find certain arrangements of items easier to count than others (Beckwith &

Restle, 1966). The study by Alibali and DiRusso (1999) provides however evidence that keeping track is not the only function of the counting gestures. Should that be the case, the children would count most accurately in the passive gesture condition, that is when they follow a flawless pointing performed by somebody else. The results of Alibali and DiRusso did not confirm this prediction — they found no statistical difference between the active and passive gestures in terms of counting accuracy.

The second category of the hypotheses focuses on the possible coordinative role the counting gestures may play in synchronising the production of the number words and matching them with each counted item so that the one-one counting principle is preserved. As pointed out by Fuson (1988), counting gestures combine in a natural way two correspondences: a correspondence in space (between the gesture and the objects), and a correspondence in time (between the gesture and the recited number words). This enables the gesture to perform the role of a ‘cognitive hub’ between the recitation of the number words (characterised only by the temporal aspect) and the objects being counted (characterised only by the spatial aspect). The evidence in favour of this proposal was found by Alibali and DiRusso (1999) in the patterns of the counting errors made by the children in the active and passive gesture conditions.

The children in the study committed less coordination errors when they gestured themselves than when the pointing was done by somebody else, suggesting that the active gestures help the children to coordinate the reciting of the number words with assigning them to the objects being counted. A related issue is the observation

that the rhythmical nature of the counting gestures may help the children to better control the shifts of attention and facilitate the correct segmentation of the counting task. In turn, this would help to treat the counted items as separate entities and therefore highlight more prominently the correspondence between the items and the number words. This proposal is consistent with the fact that touching the objects is more effective than merely pointing to them (interestingly, also in the passive gesture condition), as the former is a less ambiguous indication (see Alibali & DiRusso, 1999, p. 52).

Finally, the social aspect of gestures should not be neglected. Pointing in par-ticular is an example of a gesture that plays important communicative roles very early in the development (Behne, Liszkowski, Carpenter & Tomasello, 2012). Sev-eral studies could be named that looked at the gestures from the perspective of social learning in various contexts (see Graham, 1999, pp. 352–353). For in-stance, the lack of correspondence between the children’s gestures and speech can be taken as an indication that their understanding of a matter undergoes change and therefore that they are ready to learn and would benefit from additional instruc-tions (Breckinridge Church & Goldin-Meadow, 1986; Perry, Breckinridge Church &

Goldin-Meadow, 1988; Goldin-Meadow, Nusbaum, Garber & Breckinridge Church, 1993). Such gesture-speech mismatches can be directly observed by the tutor, providing the latter with the feedback about the child’s learning progress (Goldin-Meadow, Wein & Chang, 1992; Goldin-(Goldin-Meadow, Alibali & Breckinridge Church, 1993). The plausibility of this proposal is reinforced by the studies that show that the adults indeed understand the information conveyed by the children’s gestures and that they subsequently use them to adapt the provided instructions (Goldin-Meadow et al., 1992; Perry, Woolley & Ifcher, 1995).

While there is no doubt that the counting gestures allow the children to improve their counting accuracy, as of today the exact mechanism of this contribution has not yet been pinned down, as demonstrated by the variety of the hypotheses presented above. Importantly, two of the four research questions posed in section 1.2 are

directly related to these considerations. By answering the research question 2, I aim to contribute toward the better understanding of the nature of the phenomenon in general. Research question 3 is in turn linked directly with the second group of the hypotheses presented above, as it focuses on the observations about the spatial correspondence between the gestures and the counted items.

Another motor activity that is omnipresent during the development of the math-ematical skills in humans across virtually all geographic areas and cultures is finger counting (Fuson, 1988; Dehaene, 1997; Butterworth, 2000). Children around 4 or 5 years of age commonly use strategies involving fingers (such as counting of raised fingers or recognition of the hand configuration) in solving simple arithmetic tasks (Bisanz, Sherman, Rasmussen & Ho, 2005; Siegler & Shrager, 1984). In adults, there is evidence for the involvement of the cortical circuits responsible for finger motor control in numerical processing (see Andres, Seron & Olivier, 2007). Brain imaging studies report an overlap between the areas involved in number processing tasks and in finger movements (Kaufmann et al., 2008), and numerical deficits are often observed together with finger agnosia (Roux, Boetto, Sacko, Chollet & Tr´emoulet, 2003). These and related findings suggest that finger counting is a prominent ex-ample of grounded and embodied cognition, where the construction of abstract in-ternal representations is tightly related to motor actions (Fischer & Brugger, 2011).