The processing and representation of fractions within the brain

Abstract

Little is known about how adults process fractions and how fractions are represented in the brain. The intraparietal sulcus is assumed to host an analogue representation of number magnitude. It is unknown, however, how the magnitude of fractions is represented in the brain. Fraction magnitude might be represented by the numerical value of the fraction as a whole or might involve separate representations of the values of the fractions' denominator and numerator. The present fMRI study investigated brain areas involved in fraction comparison. As a diagnostic for fraction processing, the numerical distance effect (reaction times and error rates increase as the distance between the numbers being compared decreases) was evaluated. If fractions are represented as their numerical value, a distance effect is expected for the distance between the numerical values of the two fractions being compared. If fractions are represented in parts, however, that is, as the separate values of their denominators and numerators, a distance effect is expected for the distances between the two fractions' numerators and denominators, respectively. Although both types of distance effects were observed in the behavioral data, only the distance between the numerical values of the two fractions was observed to modulate activation within the intraparietal sulcus (IPS). This indicates that, within the IPS, a fraction might be represented by its numerical value as a whole, rather than by the numerical values of its numerator and denominator.

Introduction

As the old joke goes, five out of four people have trouble with fractions. Although a number of recent studies have investigated the neural basis of numerical processing, not much is known about how educated adults process fractions. One of the reasons why fractions are so difficult to understand might be, for example, that fractions are represented differently from other numbers or quantities in the brain. The present study aims to elucidate how fractions are processed and how fractions are represented within the intraparietal sulcus, a brain area assumed to host an analogue representation of number quantity.

Fractions are frequently encountered in daily life. They are used in part–whole relations and in measurements (e.g., half an hour), as well as in proportions (e.g., 15%) and chances (e.g. 1:4). Some fractions are even denoted by monomorphemic words (quarter, half). A failure to understand the basic concept of fractions can therefore cause difficulties in qualifying for any level of skilled labour.

Learning fractions poses major difficulties for students, especially for children with mathematical learning disability (Hecht et al. 2007). Acquiring a conceptual understanding of fractions and the required procedural skills presents difficulties to students that were, so far, only acquainted with the manipulation of natural numbers or positive integers (Hecht et al., 2007, Mack, 1995). Many properties that students have learned about natural numbers are no longer valid for fractions (Stafylidou and Vosniadou, 2004). When counting is acquired, 'one' is the smallest number. Natural numbers can be ordered ascendingly and have a unique predecessor and successor. Fractions, however, can be infinitesimally small and an infinite number of other fractions can enter between any two fractions. Differences between natural numbers and fractions are also pronounced in arithmetic. Whereas addition and subtraction of integer numbers is easily understood on the basis of counting, the addition or subtraction of fractions has to be solved counterintuitively using a series of multiplications and divisions. Another relationship that no longer holds for fractions is that multiplication (division) with natural numbers leads to results greater (smaller) than both operands. The main difficulty in understanding fractions and in learning to manipulate them successfully therefore seems to consist in going beyond the concept of natural numbers and in being able to adapt the heuristics, rules and procedures acquired so far (Mack, 1995).

One means to shed light on the representation of numbers and fractions is to investigate the influence of distance effects in comparison tasks. The numerical distance effect denotes the increasing difficulty to compare two numbers as their distance decreases (Moyer and Landauer, 1967). In numerical cognition, the presence of a numerical distance effect has been interpreted as being indicative of an analogue mental representation of number magnitude, also referred to as the ‘mental number line’ (Dehaene and Cohen, 1995). The intraparietal sulcus was repeatedly observed to be activated in comparison tasks (Cohen Kadosh et al., 2005, Kaufmann et al., 2005) with activation being inversely correlated with numerical distance (Pinel et al., 2001, but see, for a critical evaluation, Göbel et al., 2004). The IPS has been put forward as a candidate area to host the mental representation of number magnitude within the brain (Eger et al., 2003, Pinel et al., 2001). It can be therefore be expected that numerical distance will modulate brain activation in this area also in fraction comparison tasks.

Whereas some of the difficulties of young students in learning and understanding fractions have been described (Hecht et al., 2007, Mack, 1995), little is known about the representation and processing of fractions in highly educated adults. So far, only one behavioral (Bonato et al., 2007) and one explorative imaging study (Schmithorst and Brown, 2004) have been published on the processing of fractions in skilled adults. In the behavioral study by Bonato et al., subjects were asked to compare different types of fractions to a standard value. The authors observed that only the separate distances between the numerators and denominators of the fraction and the standard value influenced reaction times and error rates but not the distance between the numerical values of both fractions. They concluded that educated adults preferentially process the constituents of fractions in comparison tasks and not the numerical value denoted by the fraction. It should be noted, however, that Bonato et al. (2007) used a very limited set of fractions in a blocked design (e.g., 1/3, 1/4, 1/6, 1/7, ⋯ compared to a standard of 1/5) so that subjects might have focussed exclusively on only that part of the fraction that was relevant for the task. This study therefore can only yield limited information about how subjects process and represent fractions in less simple and predetermined contexts. In the imaging study by Schmithorst and Brown (2004), brain activation in an active condition (solving arithmetic problems with fractions) was compared to a low level control condition (viewing numbers). The authors aimed at identifying the general network of brain areas involved in arithmetic processing of fractions. However, because of their study design, their results do not allow any more detailed conclusions on the processing and representation of fractions. Thus, no study so far has explored how fractions are processed and how they relate to the analogue magnitude representation of numbers in the brain. It is therefore still unknown, whether adults, despite many years of experience, still perceive a fraction as separate values of numerator and denominator linked by an operation sign, or whether they access the numerical value a fraction represents as a whole.

In the present study, subjects were asked to compare a pair of two simple fractions. We investigated whether brain activation correlated with the distance between the numerical values of the two fractions (total distance) or the distances between the two fractions' numerators and denominators (partial distance). For example, a fraction pair such as 2/7 and 3/5, has a total distance of 0.314, a numerator partial distance of 1 and a denominator partial distance of 2. If fractions are represented within the intraparietal sulcus as the numerical value they denote, it can be expected that the distance between these two values should dominate the modulation of activation within this brain area (total distance effect). If, on the other hand, fractions are represented as the separate values of their numerator and denominator, then the activation within the intraparietal sulcus should depend on the distances between the two fractions' numerators and denominators (partial distance effect). To investigate these two types of distance effects we constructed fraction pairs with different partial distances between numerators or denominators. The distance between the numerical values of the two fractions (total distance) was calculated for each fraction pair.

An additional aim of this study was to investigate the role played by strategies when skilled adults compare fractions. Although all fraction comparison problems can be solved by comparing the two fractions on the basis of their numerical value, estimating the numerical value of a fraction might be difficult. Skilled adults might therefore prefer short cut strategies to arrive at a correct solution. Such short cut strategies are possible, given suitable fraction pairs. For example, in the case of fraction pairs that share either denominators or numerators (e.g. 2/7 and 5/7, or 3/5 and 3/8) subjects might use a simple strategy based on the comparisons of the non-identical numerators or denominators of the fraction pair. In the following, we will denote any strategy that is based on separate comparisons of the two fractions' numerators or denominators and that does not require an estimate of the fraction's numerical value, as segmental processing strategy. On the other hand, a comparison strategy based on estimates of the two fractions' numerical values will be referred to as global processing strategy.

To evoke the use of different strategies by participants, four conditions of fraction pairs were presented. In three of these conditions, the segmental processing strategy can be successfully applied. In the first of these conditions, the same denominator condition (SD), the two fractions of a pair had the same denominators (e.g., 2/7 and 5/7). In this case the first (or left) fraction is greater than the second (or right) fraction because 5 is greater than 2. In the second condition, the same numerator condition (SN), the two fractions had the same numerators (e.g., 3/5 and 3/8). Fifths are greater than eighths, given the same numerator, so that the first fraction must be greater than the second. In the third condition, the congruent comparison condition (CO), fraction pairs were presented that had different numerators and denominators, but they were selected such that separate comparisons of the nominators and denominators of the fraction pair yield the same answer. For example, when comparing the numerators of the two fractions 2/8 and 3/5, the numerator on the right is greater than the numerator on the left, yielding a higher numerical value for the right fraction. When comparing the denominators of 2/8 and 3/5, the denominator on the right is smaller than the denominator on the left, yielding, again, a higher numerical value of the right fraction. Therefore, the application of the segmental processing strategy, namely, comparing numerators and denominators separately, yields congruent results and is successful in solving these fraction comparison problems. In the fourth condition, the incongruent condition (IC), the segmental processing strategy is no longer successful, because separate comparisons of numerators and denominators lead to different results. For example, when comparing the numerators of the two fractions 3/8 and 2/5, the numerator on the left is greater than the numerator on the right, suggesting, according to the segmental processing strategy, a higher numerical value for the left fraction. However, the denominator on the right is smaller than the denominator on the left, suggesting a higher numerical value of the right fraction. In this case the segmental processing strategy would lead to incongruent results and cannot successfully be applied to arrive at a correct answer for the fraction comparison problem. A full listing of all stimuli is given in Table 1.

On the basis of these experimental manipulations, the following hypotheses can be formulated. If skilled adults employed the segmental processing strategy where it is successful, systematic differences between the four conditions should emerge. Differences between the first three conditions (SD, SN, CO) are expected because of different levels of difficulty, given the segmental processing strategy. The SD condition is the simplest condition, because the larger of the two numerators also indicates the larger fraction. The SN condition is more complex than the SD condition because here the larger of the two denominators indicates the smaller of the two fractions, giving rise to a response conflict (see also, Bonato et al., 2007). Third, the CO condition is more complex than the SN and SD conditions because the numerators and the denominators have to be compared. The results of these comparisons have also to be stored in memory and evaluated in relation to each other. Finally, in the IC condition, separate comparisons of denominators and numerators no longer lead to a correct result and the segmental processing strategy can no longer be successfully applied. In this condition, the numerical values of the two fractions have to estimated or calculated before they can be compared. If, on the other hand, subjects did not use such the segmental processing strategy but instead exclusively solved fraction comparison by estimating the value of each fraction (global processing strategy), no differences between the four fraction comparison conditions are expected. In this case, the pattern of results should only be determined by the distance between the numerical values of the two fractions (total distance effect). With regard to brain activation, we expect that the two distance effects might modulate activation within the intraparietal sulcus, the assumed location of the amodal quantity representation. Specifically, the type of distance effect modulating activation in this region might also be related to the way fractions are represented in the brain of skilled adults. If partial distance modulated activation within the intraparietal sulcus, this would suggest that fractions are represented as the separate values of their numerators and denominators. If, on the other hand, total distance is found to predominantly modulate activation within this region, it indicates that fractions are represented by their real, that is, numerical value.