While some researchers place negative numbers on a so-called extended mental number line to the left of positive numbers, others claim that negative numbers do not have mental representations but are processed through positive numbers combined with transformation rules. We measured spatial associations of negative numbers with a modified implicit association task that avoids spatial confounds present in most previous studies. In two lab-based magnitude classification experiments (each including 24 participants) and two online replications (with 74 and 77 participants, respectively), positive and negative numbers were combined with two spatial contexts: either directional symbols (left- or right-pointing arrows) or rectangles of varying sizes. In all experiments, we found a robust distance effect for negative numbers. However, there were no consistent associations of negative numbers with directional or size contexts. In the context of directional symbols, holistic processing was prevalent only in the small negative number range (-9, -8, -7, -6) when ensured by the stimulus set, supporting an extended mental number line. In the context of rectangles, however, large negative numbers from -4 to -1 were perceived as small, thus supporting rule-based processing. For negative number processing in the context of size, we further suggest the Semantic-Perceptual Size Congruity Cuing model (SPeSiCC model). We show that associations of size with negative numbers underly more complex processing mechanisms than mere recruitment of a transformation rule. In general, we conclude that associations of negative numbers with space and size are situated in the context, as they depend on the presented number range and differ for spatial direction and size.
Through schooling, individuals become familiar with the Arabic numbers from 0 to 9 and their combinations into multi-digit numbers, decimals, and fractions. However, while humans easily use the label "3" for a set of three apples, three stones, or any other triplet of objects, they hardly can apply its negative counterpart "minus 3" to any tangible set of things. Given that negative numbers lack real-world referents, they are perhaps more abstract cognitive concepts than positive numbers (e.g.,
There are several lines of research on the cognitive status of negative numbers. Firstly, developmental studies show how the concept of negative numbers is acquired as part of numeracy development (see, for example,
Speeded reaction time tasks are widely applied to gain insights into the cognitive processing of negative numbers. Adults process negative numbers considerably slower than positive numbers (e.g.,
In the next paragraph, to contextualize these research aims, we briefly review and explain key signatures of numerical processing: the size effect, the spatial-numerical associations of response codes (SNARC), and the distance effect (for a more detailed review, see
When asked to identify the larger number in a number pair, given that the numerical distance between pairs of numbers is held constant (e.g., "2 or 4", "9 or 7" – in both cases, the distance is 2), participants need more time for pairs of larger numbers – in our example decisions are slower for the pair "9 or 7" than for the pair "2 or 4". This effect of number magnitude on processing speed is called
When comparing two numbers, participants respond slower to numerically close pairs (e.g., "4 or 6") than numerically distant pairs (e.g., "4 or 8"). This is called the
Most relevant in the present context are associations between numbers and either spatial direction or size that were inferred from speeded responses of participants’ hands or eyes. Considering first the domain of space, participants respond to smaller numbers faster on the left side and to larger numbers faster on the right side, thus demonstrating Spatial-Numerical Associations of Response Codes, or the Similarly, spatial associations between left space and odd numbers and between right space and even numbers also extend across both positive and negative number ranges. This so-called linguistic markedness effect was found for positive (
Size, SNARC, and distance effects have together inspired the influential conceptual metaphor of a mental number line, according to which people represent number concepts in a spatial layout where smaller numbers are located to the left of larger numbers (e.g.,
Initially, the
The extended mental number line hypothesis postulates that negative numbers are mentally represented left of zero. Instead, the rule-based account claims that negative numbers have no own conceptual representations, so that positive number representations must be augmented with a polarity mark whenever negative numbers are processed. These two competing hypotheses have different names in the literature: "ontogenetic" vs. "phylogenetic" (
To illustrate, let us calculate "minus one minus three." According to the extended mental number line (MNL) hypothesis, one starts on the left side of zero, at minus one (
Previous studies on the cognitive representation of negative number concepts (reviewed in
However, participants in the same study apparently ignored the sign component when signed positive and negative numbers were mixed within an experimental block. Instead, they focused on the number component alone, as indicated by their duration estimates, which reflected absolute magnitude meaning. Valid inferences about negative number representation, therefore, require a method that ensures both holistic number processing and the need to pay attention to the sign. The present study set out to accomplish this requirement by presenting positive number foils among negative numbers in a magnitude classification task, thus requiring participants to attend also to the numbers' signs. We will explain this rationale further below (see also
Another methodological issue that remained unnoticed in previous studies is the use of lateralized button responses to measure spatial associations of negative numbers. Lateralized responses introduce extraneous spatial features that contaminate the intended measurements by activating pre-existing spatial-numerical associations. Note that regardless of the nature of the stimuli, i.e., simple pictures in the context of numbers, this was not a developmental study: On average, participants in that study were 22.5 years old (range: 19–27).
First, we presented both positive and negative numbers randomly within the same procedure. Following
Related to this last point about the nature of spatial inducers, we also took this opportunity to distinguish between contributions of directional features and size features in those spatial inducers. The association between magnitude and left-right space could be driven by either directional or size features or both (cf. We refrained from using circles as inducers because their shape might have instantiated the critical number concept of zero.
We used a magnitude classification task where numbers were categorized relative to the reference value -5. The extended mental number line hypothesis predicts that numerically smaller negative numbers (-9 to -6) are associated with relatively more left space than numerically larger negative numbers (-4 to -1). Processing those numerically smaller numbers should then be facilitated when responding to them under congruent instructions (e.g., "respond when the number is numerically smaller than -5 or the arrow points to the left"; or “respond when the number is numerically smaller than -5 or the rectangle is small”). Similar facilitation would be expected for numerically larger negative numbers (-4 to -1) under conditions that are congruent with an extended mental number line ("respond when the number is numerically larger than -5 or the arrow points to the right" or respond when the number is numerically larger than -5 or the rectangle is large”).
Facilitation should be reflected in reaction times of the magnitude classification task. Thus, we expected faster processing in blocks with response rules compatible with an extended mental number line compared to incompatible conditions. For example, numerically small negative numbers (-6 to -9) relative to -5 should be processed faster in the context of left-pointing arrows or small rectangles than in the context of right-pointing arrows or large rectangles. Instead, the rule-based account predicted a reverse spatial mapping relative to -5, such that, for example, numerically smaller negative numbers (-6 to -9) would be represented progressively to the right of -5. This should induce a reversal of the compatibility effect when numerically smaller negative numbers are processed in the context of either left-pointing arrows or small rectangles, leading to slower responses compared to the other contexts.
We conducted four experiments. The first two experiments were lab-based, and the remaining two were online replications of the first two experiments with increased sample sizes. In Experiment 1 (lab-based,
Consider first the lab-based experiments. Data of 24 participants were collected in Experiment 1. Participants did not receive monetary compensation; most of them were students who received course credit. Due to an administrative error, nineteen participants' personal information was not recorded. The remaining five female participants were between 18 and 23 years old (mean age: 19 years and 9 months); one was left-handed. Twenty-four new individuals participated in Experiment 2 (mean age: 23 years and 3 months, range: 19 – 32 years). One participant was left-handed, and five were male. All participants in Experiments 1 and 2 were from Western cultures, read and counted from left to right, and were naïve about our hypotheses (based on a survey after the experiment, see Appendix I in the
Consider now the online experiments. Seventy-eight participants took part in Experiment 3 (mean age: 24 years, range: 18 – 56 years), 13 were left-handed, and 16 were male. Four participants were excluded from the analysis because of accuracy below 80% in at least one of the experimental blocks. Data of 74 participants were included in the study. Finally, 82 individuals participated in Experiment 4 (mean age: 24 years and 2 months, range: 18 – 47 years). Ten participants were left-handed, and 16 were male. Five participants were excluded from the analysis due to accuracy below 80% in at least one of the experimental blocks. Data of 77 participants were included in the analysis. All participants of Experiments 3 and 4 were from Western cultures, and all were left-to-right readers and counters (assessed via a post-questionnaire in the online Experiment, see Appendix I in the
This study was conducted following the guidelines of the Declaration of Helsinki (
All stimuli in lab-based Experiments 1 and 2 were presented on a 13’' MacBook with 1440 x 900 pixels screen resolution placed centrally on a desk in front of each seated participant. Reaction times were recorded on the space bar of the QWERTY keyboard, and the experiment was controlled with the OpenSesame software (
Stimuli were eight negative numbers (-1, -2, -3, -4, -6, -7, -8, -9), four positive numbers (+6, +7, +8, +9, presented with the plus-sign), and either two arrows of 30 x 128 pixels size which differed in pointing direction (left vs. right; Experiment 1 and 3) or two rectangles which clearly differed in size (small: 64 x 32 pixels; large: 1024 x 128 pixels; Experiment 2 and 4). All stimuli were presented centrally in black on a white background.
In separate blocks of their respective experiments, participants responded with a single button when a specific response rule applied: numbers were either numerically larger than -5 (i.e., numbers -4, -3, -2, -1, +6, +7, +8, and +9) or numerically smaller than -5 (i.e., numbers -6, -7, -8, and -9). Inducer stimuli were either arrows (left- vs. right-facing) or rectangles (small vs. large). This resulted in four different instruction blocks for each participant: In Experiments 1 and 3 with directional inducers: smaller + left; smaller + right; larger + left; larger + right; in Experiments 2 and 4 with size inducers: smaller + small; smaller + large; larger + small; larger + large). Each block contained 50% go trials and 50% no-go trials. As previously demonstrated by
Each experimental block consisted of 96 trials: 64 with negative numbers, 8 with positive numbers, and 24 with context objects. There were 12 small and 12 large rectangle context trials per block. However, depending on the response rule, the arrow context trials were divided into 8 left and 16 right arrows, or vice versa. Their purpose was to keep an overall equal balance between go and no-go trials (50%) per block. For example, under the rule "respond to numbers numerically larger than -5 or arrows facing left", all positive numbers were go trials, and therefore only 8 left-facing arrows were presented. Trial lists were generated randomly before testing, and positive numbers were distributed approximately equally throughout each list. Participants completed all four instruction blocks in a counterbalanced order (Latin square).
In the two lab-based experiments, participants were instructed in English, German or Dutch (depending on their native language) and provided their informed consent at the beginning of the experiment. Each block started by stating the response rule. Participants performed a practice block until they reached at least 80% accuracy. Each stimulus was presented for 2,000 ms or until response. Participants were free to use their preferred hand to press the space bar. Trials were separated with a blank screen of 500 ms. The entire experiment took approximately 30 minutes. After the experiment ended, participants were asked to fill out a demographic questionnaire that assessed handedness, reading direction, counting direction, cultural background, and knowledge about our hypothesis (see Appendix I in the
Analyses were conducted in Microsoft Excel 365 and JASP statistics, version 0.16.0.0. All practice trials, no-go trials, and trials with positive numbers were discarded. 25,472 trials remained (3,072 trials in Experiment 1; 3,072 trials in Experiment 2; 9,472 trials in Experiment 3; 9,856 trials in Experiment 4). Error trials ( We inspected the data visually to identify the lowest and highest thresholds. Such thresholds can clearly be seen on a histogram, especially the lower threshold. Another source of our decision was information from previous studies: the average reaction time in various number processing tasks is around 600 ms (see
Given that the distance effect is a hallmark of semantic number processing, we first examined the effect of numerical distance on decision speed in our magnitude classification task. Individual linear regression slopes were computed for each number magnitude range (numerically small: -9, -8, -7, -6; numerically large; -4, -3, -2, -1; see
Slope Coefficients |
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Descriptive Statistics |
One-Sample |
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Experiment / number range | |||||
Lab-based | |||||
Experiment 1 ( |
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large numbers | -8.130 | 10.195 | -3.906 | 23 | < .001 |
small numbers | 9.452 | 9.012 | 5.138 | 23 | < .001 |
Experiment 2 ( |
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large numbers | -8.167 | 13.246 | -3.020 | 23 | .006 |
small numbers | 5.617 | 10.214 | 2.694 | 23 | .013 |
Online | |||||
Experiment 3 ( |
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large numbers | -9.828 | 16.442 | -5.142 | 73 | < .001 |
small numbers | 12.339 | 16.246 | 6.533 | 73 | < .001 |
Experiment 4 ( |
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large numbers | -11.085 | 10.568 | -9.204 | 76 | < .001 |
small numbers | 10.117 | 11.528 | 7.701 | 76 | < .001 |
As shown in
Next, to measure the effect of Mental-number-line compatibility across Experiments, we calculated individual compatibility scores based on reaction times. The average reaction time was 545 ms in Experiment 1, 584 ms in Experiment 2, 548 ms in Experiment 3, and 555 ms in Experiment 4. We subtracted individual reaction time means in mental-number-line compatible blocks from individual reaction time means in mental-number-line incompatible blocks to receive Mental-number-line compatibility scores in milliseconds. If this calculation resulted in a positive compatibility score, the effect was as expected because faster answers in mental number line compatible blocks reflect processing facilitation congruent with the extended mental-number-line account. Given that Experiments 3 and 4 replicated Experiments 1 and 2, data were aggregated across the lab-based (1, 2) and the online experiments (3, 4) In our analyses, we aggregated data across lab-based and online experiments because of the limited sample size of the lab-based versions of the experiments. In both the lab-based and the online-setting, the experiments were identical. Nonetheless, future studies might research different outcomes of lab-based versus online-experiments. For our research, this is not applicable due to the highly varying sample sizes of the lab-based and online-experiments. We aggregated data across number range. Nonetheless, also when calculated for the large and small number range separately, the effect reached significance:
First, we conducted one-sample
As
One-sample
In addition, when negative numbers were presented together with size-inducers (Experiment 2, 4), we found a
The present study aimed to replicate and extend findings by
The reliable distance effects we obtained for negative numbers in magnitude classification indicate semantic processing of the numerical value of negative numbers. Our findings are in line with several previous experiments. For instance,
Moreover, our results extend previous findings by
In contrast to our hypothesis, we did not detect an overall congruency effect of spatial-numerical associations across experiments. We used a method with non-spatial responses first introduced by
We found spatial-numerical associations in the expected direction in Experiments 1 and 3, reflected by positive Mental-number-line compatibility scores, but only in the small negative number range (-9; -8; -7; -6), where also positive counterparts (+6; +7; +8; +9) of the negative numbers were present. It shows that our experimental design worked: presenting positive counterparts to negative numbers ensured holistic processing (see also
In contrast, in Experiments 2 and 4, we found a processing advantage for numerically large negative numbers (-4, -3, -2, -1) combined with small objects, reflected by negative Mental-number-line compatibility scores. This result suggests that these numbers are perceived as “small” and are thus processed more easily when shown together with small objects. Therefore, we suspect that participants may have co-activated, together with their numerical value, a linguistic label shared with the small rectangle inducers. The fact that this was not apparent in the directional inducers illustrates that different mechanisms might work when mapping number magnitude onto space (cf.
In a recent publication,
Note that the authors used positive numbers and spatially aligned response keys in their experiment. Thus, we cannot compare our results to these findings in a one-to-one fashion. One strength of our study is that we removed the spatial component of responses. Consequently, we were not able to measure a direct link between response location and other parameters such as number magnitude and physical number size. However, these previous results enable us to explain our results in Experiments 2 and 4 by similar mechanisms. In our experiments, in contrast to
As displayed in
Our experiments contrasted two contradicting accounts of processing negative numbers: the extended mental-number line and the rule-based account. Overall, our findings cannot conclusively support either of these accounts. Instead, we showed here that processing of negative numbers is situated in the context: when holistic processing was ensured and when numbers were presented together with directional inducers, we found evidence for the extended mental-number line account. Nonetheless, we found tentative evidence against the extended mental number line account in Experiments 2 and 4: Numerically large negative numbers were not processed according to their numerical value but their absolute value. Here, the rule-based account applies better to explain the results. We explain both sets of results in the framework of the novel SPeSiCC model. Thus, we suggest that a simple rule-based processing strategy is not the only mechanism that drives negative number processing. This proposal awaits further examination. Moreover, we found a consistent distance effect across experiments that indicates similar semantic processing for positive and negative numbers. Taken together, our results show that associations of negative numbers with space and size are highly situated in their context.
This study reports an extensive assessment of the cognitive representation of negative numbers with several methodological improvements over previous work. In contrast to the study conducted by
The findings from four experiments suggest that the distance effect is robust across negative numbers. At the same time, associations of numbers with space and size are not inherent but are elicited by the task and experimental context (see
The authors have no funding to report.
We thank Sam Shaki and Stefan der Vries for their valuable contributions to the development of the study. We thank Stefan de Vries for assisting with data collection.
The authors have declared that no competing interests exist.
Hereby we confirm that the research has been carried out in accordance with relevant ethical principles and standards. Participants gave their informed consent at the start of the experiment, in accordance with the principles specified in the Declaration of Helsinki.
The Supplementary Materials contain online appendices for this study (for access see