Mind the gap between both hands: Evidence for internal finger-based number representations in children's mental calculation

Abstract

At a certain stage of development, virtually all children use some kind of external finger-based number representation. However, only little is known about how internal traces of this early external representation may still influence calculation even when finger calculation ceases to be an efficient tool in mental calculation. In the present study, we provide evidence for a disproportionate number of split-five errors (i.e., errors with a difference of ±5 from the correct result) in mental addition and subtraction (e.g., 18 − 7 = 6). We will argue that such errors may have different origins. For complex problems and initially also for simple problems they are due to failure to keep track of ‘full hands’ in counting or calculation procedures. However, for simple addition problems split-five errors may later also be caused by mistakes in directly retrieving the result from declarative memory. In general, the present results are interpreted in terms of a transient use of mental finger patterns – in particular the whole hand pattern – in children's mental calculation.

Introduction

Children of all cultures use some kind of external finger representation for counting and calculation during some stage of their development (for an overview see Butterworth, 1999, and references cited therein). They do so even if this has been forbidden. Moreover, children's finger gnosia was found to be a specific predictor of their numerical abilities (Fayol et al., 1998, Noël, 2005) and numerical abilities may improve after finger gnosia has been trained (Gracia-Bafalluy and Noël, 2008, this issue). This is reminiscent of the close association between finger agnosia and acalculia in neuropsychological patients suffering from ‘Gerstmann syndrome’ (Gerstmann, 1940) or its developmental variant (Benson and Geschwind, 1970, Suresh and Sebastian, 2000). Still, whether the association between finger and number representation is functional, neuronal, or both is not yet resolved definitely (Noël, 2005). However, recent neuroimaging (Kaufmann et al., 2008, Simon et al., 2002) and transcranial magnetic stimulation (TMS) research (Rusconi et al., 2005) points to shared or neighbouring neuronal substrates of finger and number representations.

With respect to possible functional relationships, there are some obvious advantages of using fingers to acquire numerical representations. They are readily available, provide multisensory input (e.g., visual and proprioceptive), and are easy to manipulate. In contrast to number words, Arabic digits or dot patterns, finger representations support the establishment of ordinal as well as cardinal meaning. As fingers are concrete entities, they facilitate the establishment of a one-to-one relationship to the things counted or represented. However, the latter advantage only holds in the number range up to 10. Whereas finger-based strategies prove to be very useful during the acquisition of single-digit addition and subtraction (Baroody, 1987, Fuson and Kwon, 1992, Svenson and Sjöberg, 1983), they have to be abandoned or dramatically modified (losing the one-to-one correspondence property) when it comes to more complex calculation. It is this very transition that the present investigation addresses. We will provide evidence that during this transitional phase at least some children still try to use finger representations even though their use becomes more and more inadequate. This evidence is based on the anatomical peculiarity that the base-10 number system as represented on fingers is structured by the gap between both hands, thus forming chunks of five.

Indeed, the pattern of five fingers on one hand plays a particular role in young children's simple addition and subtraction. After an initial phase of counting every single finger,¹ children typically start to recognise finger patterns without counting (Baroody, 1987, Brissiaud, 1992, Fuson and Kwon, 1992, Svenson and Sjöberg, 1983) whereby typical patterns provide an processing advantage over atypical ones (Noël, 2005). At this point, finger representations of numerosities larger than five invariably include a full hand, i.e., children decompose those numbers into 5 + x (Brissiaud, 1992, Fuson and Kwon, 1992, Marton and Neuman, 1990). For example, the numerosity of seven is usually not represented using four fingers of one hand and three fingers of the other, but with a full hand and two fingers of the other one. Introducing a substructure five into the base-10 system may bring the perceptual advantage that every single-digit number can be recognised at a glance, i.e., subitised (Marton and Neuman, 1990). When sums >10 have to be calculated, a full hand may be reused to represent the two-digit result (Fuson and Kwon, 1992, p. 293). However, as sums >15 involve the reuse of a second hand, this imposes an increasing need to keep track of the number of full hands being reused. To cope with this problem, some children develop strategies of folding and unfolding their fingers (indicating numbers smaller or larger than five on the same hand, respectively) or of touching a surface for each full hand reused (Fuson and Kwon, 1992). However, if no such cues are available, specific errors may occur as a ‘fingerprint’ of failure to keep track of the reused full hands. These errors should deviate from the correct result by five (‘split-five errors’) or in some cases even by a multiple of five.

Traces of full hand representation may also be involved in complex mental calculation with no overt use of fingers at all. For instance, Scott, a child reported by Thompson (1999), described his mental strategy for correctly solving the problem 6 + 7 as follows: ‘…I took 5 out of the 6 and 5 out of the 7 and I was left with 3…’. Obviously, the problem of keeping track of the fives involved is even amplified with mental compared to external representations. As in the example of Scott, mental manipulations leading to the correct result may closely parallel external finger-based strategies applied at an earlier stage of development. However, if the use of finger patterns indeed provides a path to the acquisition of number facts (Marton and Neuman, 1990) it even seems possible that internal representations used in more abstract mental operations like the direct retrieval of arithmetic facts may still inherit features of early external finger representations – most notably their structuring into chunks of five.

In the light of the findings reviewed so far, the main question of the present investigation can be summarised as follows. Do the specific characteristics of the external number representation used by children during the acquisition of counting, simple addition, and subtraction – in particular the special status of the five pattern – influence their later mental calculation? Mental operations with three different degrees of abstractness may be affected during the course of development (Baroody, 1987, Carpenter and Moser, 1984, Fuson and Kwon, 1992, Resnick, 1983). (1) While the procedure itself is performed externally on fingers, only the number of five-patterns reused is stored mentally. (2) A procedure very similar to the former use of external finger representations is now performed mentally – including the temporary storage of five-patterns reused. (3) Results are retrieved from long-term memory, the representations of which have inherited some of the features of finger representations, in particular their structure in chunks of five. The first two types of operation can be expected for simple and complex addition and subtraction at different stages of development including external or internal finger pattern manipulation. Furthermore, these stages may be reached at different points in time by different children for different operations and they may last for different time periods. Yet, finally they will be superseded by a purely symbolic base-10 system which is enforced by the system of number words and Arabic digits which do not have any five-structure. According to the Triple Code Model (Dehaene et al., 2003), the third type of operation can be expected for small addition problems after the transition from procedural, algorithmic calculation to direct retrieval from long-term memory. This transition typically occurs after the introduction of multiplication during second grade and is paralleled by a temporary decrease of overall accuracy (Miller and Paredes, 1990). Developmental dyscalculia may lead to a delayed or absent transition from procedures to fact retrieval (Geary, 2004).

For all three types of operation traces of finger-based number representations would be witnessed by a disproportionately large number of errors that deviate exactly by the number of fingers on one hand from the correct result – i.e., split-five errors. For complex addition and subtraction such errors should show no clear-cut increase (given that they may occur in the context of different strategies), but a slow decrease when chunk-five representations are gradually superseded by purely symbolic base-10 representations. For simple addition problems, on the other hand, a sharp increase of split-five errors may be expected after multiplication problems have been introduced during second grade, followed by a fast decrease when the retrieval of arithmetic facts becomes increasingly accurate. In sum, split-five errors may provide a window into the restructuring of mental number representations during development.