Elsevier

Brain and Cognition

Volume 69, Issue 2, March 2009, Pages 369-381
Brain and Cognition

The exact vs. approximate distinction in numerical cognition may not be exact, but only approximate: How different processes work together in multi-digit addition

https://doi.org/10.1016/j.bandc.2008.08.031Get rights and content

Abstract

Two types of calculation processes have been distinguished in the literature: approximate processes are supposed to rely heavily on the non-verbal quantity system, whereas exact processes are assumed to crucially involve the verbal system. These two calculation processes were commonly distinguished by manipulation of two factors in addition problems: the identity of the target and the distance of the distractor. However, in all previous studies, these two factors were not manipulated independently.

In this fMRI study, we could disentangle the two factors by using a different (two-digit) number stimulus set. Both behavioral and neurofunctional data suggest that the cognitive processes involved could be best explained by the (independent) factors target and distractor distance.

Based on these data we suggest that the exact/approximate distinction does not seem to be as generally valid as previously assumed. We conclude that this study may be a starting point for a closer examination of the experimental, procedural and strategic conditions of when the exact/approximate distinction is valid and when it is not.

Introduction

A distinction has been made between calculation tasks, which are solved by using approximate size (approximate calculation), and those which are processed via retrieval of arithmetic facts from verbal memory (exact calculation): approximate tasks are supposed to rely more heavily on the non-verbal quantity system, whereas exact tasks are assumed to crucially involve the verbal system. This distinction dates back to Ashcraft and Stazyk (1981), but has received much interest in recent years (e.g., Dehaene et al., 1999, Kalaman and Lefevre, 2007, Kucian et al., 2006, Lemer et al., 2003, Stanescu-Cosson et al., 2000).

How are exact and approximate processes commonly distinguished? In studies examining the exact/approximate distinction, typically simple addition problems are presented. The participants are presented with two possible candidate answers to an addition problem. The participants’ task is to choose the (target) answer which is closer to the correct solution of the addition problem and to ignore/reject the answer which is more deviant from the correct solution (further called distractor). For instance, the participant is presented with 1 + 2 = ? with the potential results 4 or 8. In this example, he has to choose “4” because it is closer to the correct response 1 + 2 = 3 and to ignore/reject “8”.

This general task set-up has been used to study two hypothesized types of calculation processes: exact calculation and approximate calculation. These two calculation processes are commonly distinguished by manipulation of two factors in this task: the identity of the target and the distance of the distractor. It is important to note that in all previous studies these two factors were not independently manipulated (see below). This is different in the current study and we will argue in the following why we think this may be a fruitful way to proceed.

In the studies mentioned above, in the so-called “exact” trials (indexing exact processing), the target matches the correct solution exactly and the distractor is very close to the correct solution (e.g., 3 + 4 = 7 or 8). This combination of target and distractor is chosen so that the participants must not only calculate the exact result but also select the exactly correct answer (target) because both target and distractor are numerically (with regard to their magnitude) close to the correct solution. In so-called “approximate” trials (indexing approximate processing), both target and distractor do not match the correct solution of the addition problem. However, in contrast to the exact trials, the distractor is far away from the correct response (e.g., 1 + 2 = 4 or 8). Obviously, these latter trials cannot be solved on the basis of an exact match because the target does not exactly match the correct response. Rather, they are supposed to be solved on the basis of an approximate evaluation of target and distractor magnitude. In the above example the distractor “8” can be rejected, because it grossly deviates from the correct response.

The exact stimulus selection, presentation modes, instructions, and designs can differ between studies (Dehaene et al., 1999, Kalaman and Lefevre, 2007, Kucian et al., 2006, Lemer et al., 2003, Stanescu-Cosson et al., 2000). However, all studies comprise an addition problem and a target and distractor as defined above. And, importantly, in all studies target and distractor distance are both larger in the approximate trials than in the exact trials (in which the target distance is actually zero because the target matches the correct solution to the problem). Also in a patient study by Dehaene and Cohen (1991, see Experiment 2) the precise effect of distractor distance was not reported for the multiple choice addition problems chosen.

The probably most influential and detailed elaboration of the topic is an fMRI study by Stanescu-Cosson and colleagues (2000) which will be considered in more detail now. In that study, approximate trials activated more the bilateral intraparietal sulcus while exact trials activated more the left angular gyrus and other left-hemispheric perisylvian language areas. The authors proposed that the neural systems underlying exact and approximate calculation can be separated, although normally “the networks for exact and approximate processing are not mutually exclusive, but are functionally integrated and are co-activated when solving difficult problems.” (Stanescu-Cosson et al., 2000, p. 2253). In that study, response latencies increased with increasing problem size only for exact calculation while problem size had no effect on behavioral responses during approximate trials. The authors interpreted this as evidence for a categorical difference between exact and approximate calculation. Problem size also modulated brain activation specifically for exact calculation while no modulation was observed for approximate calculation. Although the results reported by Stanescu-Cosson et al. (2000) are very clear regarding the involvement of different neural networks when solving exact and approximate problems, we will argue that the explanation based on the distinction between so-called exact and approximate trials may be not as clear-cut as has previously been assumed. In the following, a detailed inspection of the properties of typical items from exact and approximate conditions will be presented.

Closer examination of the numerical properties of the trials employed in studies investigating exact/approximate calculation processes reveals that numerical confounds may at least in part undermine conclusions about separate systems for exact and approximate calculation. First, let us consider the distances of target and distractor to the correct result and their relation to the exact/approximate distinction: in exact trials both distances are typically small while in approximate trials, both target and distractor distance are relatively large. Consequently, a high correlation between target and distractor distance and the exact/approximate distinction is present when either both the target and distractor distance are small or both are large (e.g., a correlation of r = .84 is obtained from the stimuli used in the study by Stanescu-Cosson et al. (2000); that corresponds to a confound of 70% of the variance). Because these two distances are correlated, it is not possible to separate the effects of target distance and distractor distance as well as of a potential interaction between those two distances on brain activation in a categorical contrast comparison.

Because of this high correlation, one could argue that activation in exact trials is not due to the exact match of the target and the correct response, but is instead due to the more difficult distractor (with a smaller distractor distance). We simply do not know which activation is due to cognitive processes related to the exact target identity match and which activation is due to distractor distance because these two factors are always manipulated together.

Vice versa, the activation in approximate problems may not be due to the approximate rejection of the grossly deviant distractor; rather, it may as well be due to the more difficult larger target distance. It is fairly reasonable to assume that targets with a larger distance to the correct result are harder to select because priming studies revealed systematically faster responses when the magnitude of prime and target was similar, i.e., the numerical distance between prime and target was smaller (Naccache and Dehaene, 2001, Reynvoet and Brysbaert, 1999). So, again because both the distance of the target (to the correct result) and the distance of the distractor are large, we cannot be sure which distance is cognitively related to the functional activation observed.

Finally, we wish to point that there is another possible confound with the exact/approximate distinction. In many (but not all) of the studies presented above (Dehaene et al., 1999, Kalaman and Lefevre, 2007, Kucian et al., 2006, Lemer et al., 2003, Stanescu-Cosson et al., 2000) examining the exact and approximate distinction, the addition problem and the candidate answers are not presented simultaneously, but consecutively. Thus, the solution of the addition problem has to be kept in working memory to solve the task. Therefore, the (parietal) activation in the approximate condition can partially be due to higher working memory demands because in the approximate condition the relations between three different values (correct sum, target, distractor) have to be determined while in the exact condition only the relation between two different values has to be computed (since the correct sum equals the target). In this study, we presented problem and candidate answers simultaneously (see also Kalaman and Lefevre, 2007, Lemer et al., 2003, for the same procedure) because the use of two-digit numbers would probably require even more working memory capacity in consecutive presentation than the use of single-digit numbers.

A further problem with the interpretation of the exact and approximate calculation is sometimes posed by the effects of carry-over and problem size: In the most influential study about the exact/approximate distinction by Stanescu-Cosson et al. (2000), tasks with large problem size were generally carries, while small problem size was combined with no-carries. Carry-over is assumed to differentially affect performance in addition problems (Kalaman & Lefevre, 2007) and requires the understanding and manipulation of digit numbers in the Arabic base-10 system. Such manipulation also seems to be related to parietal activation. In a recent study, Wood et al. (2008) have examined the effect of base-10 processing on fMRI signal in a number bisection task and found a strong increase of activation in intraparietal cortex due to decade crossing (see also Wood, Nuerk, and Willmes (2006) for a discussion on the effect of base-10 processing on fMRI activation). Larger parietal activation has also been found for larger problem sizes (Zhang & Wang, 2005). So, when both carry-over effects and problem size effects are likely to be associated with parietal activation, it is in our view not conclusive when carry-overs and large problems are confounded with approximate trials. Therefore, in our view, disentangling the effects of carry-over and problem size from exact and approximate processes is crucial when (parietal) brain activation is interpreted.

Section snippets

Objectives

The aim of the present study is to examine the generality and the validity of the exact/approximate distinction. Since we have referred to the fMRI study by Stanescu-Cosson et al. repeatedly, we wish to point out that the aim of our study is not to replicate or to fail to replicate the specific experiment by Stanescu-Cosson et al. (2000). We even had to chose a modified experimental approach compared to many previous studies since the aim of our study was to examine the distinction between

Participants

Nineteen male right-handed healthy students participated in this study (mean age: 23.6 years; SD = 3.5 years). They were recruited on a volunteer basis and gave their written consent in accord with the protocol of the local Ethics Committee of the Medical Faculty. One participant was excluded from the analysis due to technical problems with the recording of the imaging data (damaged storage medium).

Stimuli

The stimulus material comprised 192 different two-digit addition problems presented in Arabic

Behavioral data

Reaction time (RT) analysis was based on correct trials only; 17.6% erroneous responses were excluded from the data set. The considerable error rate can be explained by the high difficulty of the calculation problems used. Furthermore, response latencies falling outside the interval between 200 ms and 3500 ms were not considered and in a second step responses outside the interval of ±3 standard deviations around the individual mean were excluded. This trimming procedure resulted in 0.26%

Discussion

In this fMRI study we investigated the neuroanatomical correlates of two-digit mental addition. Previous evidence has been interpreted in light of the presence of two cognitive systems: one dedicated to fast and approximate estimation and another one to exact calculation. In Section 1, we have argued that previous evidence concerning this distinction are studies containing distance confounds which alone can explain the behavioral and functional data. In our study, we have controlled such

Summary and conclusions

In the domain of two-digit numbers no specific fMRI activation difference between exact and approximate calculation was observed after controlling for the impact of target distance, distractor distance, problem size, and carry-over effects. Furthermore, the categorical effect of problem size on exact but not on approximate problems was not replicated in the present study. To the contrary, a parametric effect of target and/or distractor distance was observed in both exact and approximate

Acknowledgments

The research and the preparation of this article were supported in part by grants from the European Commission (RTN “NUMBRA” Grant CT-2003-504927) and from the Interdisciplinary Center for Clinical Research ‘‘BioMAT.’’ within the Faculty of Medicine at the RWTH Aachen University (TV N44 and 53) as well as by a START-programme grant of the Faculty of Medicine at the RWTH Aachen University to Hans-Christoph Nuerk. We are indebted to Martina Graf and Jochen Weber for their assistance. We thank two

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