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The Mental Number Line February 9, 2007

Posted by Johan in Cognition, Mathematical Cognition, Neuroscience.

If you close your eyes and imagine the numbers 1 through 9 on a line, what does the image that appears in your head look like? Most people will say that they imagine a horizontal line, with 1 on the left, and an orderly progression to 9 on the right. Naturally, this finding could be an effect of cultural convention in societies that use arabic numerals, but research indicates that there may be more to it than that. In this post I will outline some research in the field of mathematical cognition (sometimes known as numerical cognition), which suggests that this anecdotal finding may reveal a fundamental characteristic of how numbers are represented in the brain.

If you ask participants to press a key to the left for numbers larger than a reference number, and to press a key to the right for smaller numbers, an interesting pattern appears, after counterbalancing with the converse condition. On average, participants respond faster to smaller numbers with the left key, and to larger numbers with the right key. This is known as the Spatial-Numerical Association of Response Codes (or as its more catchy acronym, SNARC) (Dehaene et al, 1993). This finding has been interpreted as evidence that numbers are coded spatially in the brain. The coding appears to be relative: the same number will show a left-side advantage if it is smaller than the reference number in one condition, and a right-side advantage if it is larger than the reference in the next condition. The effect appears when numbers are presented as arabic digits or as written words, suggesting that the effect is mediated by abstract representations, rather than any characteristics of the visual form of the stimuli (Fias & Fischer, 2005).

At this point, you might think that the SNARC effect has something to do with lateralization: perhaps the left hemisphere is simply better at small numbers, and the right hemisphere is better at larger numbers. Aside fomr the difficulty in joining such a notion with the previously described relative coding, Dehaene et al (1993) showed convincingly that the effect is not a mere consequence of slower processing when information has to cross the corpus callosum: when participants responded with their hands crossed over, the left-side advantage for smaller numbers still appeared, even though the participant was now responding with the right hand, and vice versa. Other investigators have found SNARC effects when participants respond with a single hand (Fias & Fischer, 2005).

Turning to the brain, the hunt for a tidy line of neurons arranged in a line from left to right, has, unsurprisingly, been fruitless (Fias & Fischer, 2005). Still, some evidence suggests that the intraparietal sulcus is involved in number comparison tasks (Göbel et al, 2004). However, this area also responded to the spatial control task that was used in this particular experiment. The same pattern emerged when repeated transcranial magnet stimulation was applied to a nearby area, in order to disrupt neural activity. The SNARC effect was diminished, and so was performance in a visuospatial search task (Göbel, Walsh & Rushmore, 2001).

The failure to find specific neural activity for number comparison tasks in contrast to visuospatial tasks is not necessarily bad news for the idea of a mental number line: if number comparison and visuospatial search tasks activate the same areas, it could be viewed as converging evidence for a spatial organization of numbers. Same areas, and thus, the same form of processing. This sounds good, but runs into conceptual problems, as you are essentially trying to prove your hypothesis with a null result. The core of the problem with current paradigms appears to be that the control conditions which show no SNARC effect (e.g., visuospatial search) still produce activation patterns in fMRI that are not significantly different from that produced by the number comparison task.

Regardless, this area of neuroscience is in its infancy. It will be interesting to see if a neuroimaging approach can inform mathematical cognition in general and the SNARC paradigm in particular, beyond providing a probable brain location.

If you’d like to learn more about the mental number line, the chapter by Fias and Fischer (2005) is a good starting point.


Dehaene, S., Bossini, S., & Giraux, P. (1993). The Mental Representation of Parity and Number Magnitude. Journal of Experimental Psychology: General, 122, 371-396.

Fias, W., & Fischer, M.H. (2005). Spatial Representation of Numbers. In J.I.D. Campbell (ed.) Handbook of Mathematical Cognition. Hove: Psychology Press.

Göbel, S.M., Johansen-Berg, H., Behrens, T., & Rushworth, M.F.S. (2004). Response-Selection-Related Parietal Activation during Number Comparison. Journal of Cognitive Neuroscience, 16, 1-17.

Göbel, S., Walsh, V., & Rushworth, M.F.S. (2001). The Mental Number Line and the Human Angular Gyrus. Neuroimage, 14, 1278-1289.

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1. raghavgupta - February 12, 2007

Intriguing. It would be interesting to find out the response of participants who have grown up solely in the left-to-right or vertically written language environment.
Also, another question that pops into mind is, why do most cultures(even the right-to-left ones) write multiple-digit numbers with the most significant digit at the left and not the other way round?

2. Johan - February 12, 2007

The chapter by Fias and Fischer touches on the issue of cross-cultural evidence. As far as I recall, what little research has been done seems to suggest that the same pattern appears in cultures were written language is not done mainly left-to-right.

In any case, it’s hard to see why a left-to-right convention in writing should result in faster processing of smaller numbers to the left and larger numbers to the right.

As for your second question, I have no good answer. :) This is an area of cognitive science that is not well researched.

3. chchatham - June 5, 2007

There’s also some fascinating developmental evidence relevant to the mental number line, in which kids of a certain age (below 6, I believe) can correctly mark “3″ and “7″ on a number line going from 1 to 10, but will actually place “250″ beyond the three-quarters mark on a number line going from 1 to 1000. it turns out these errors correspond to a logarithmic mental number line, thought to be some kind of innate scalar system that humans share with many other mammals. Cool stuff!

4. Das Beste der skurrilen Wissenschaft @ wissen + konzeptewissen + konzepte - October 8, 2012

[...] unsere Einschätzung von Größen oder Mengen. Grund für diese Annahme ist die sogenannte „mental-number-line“-Theorie – ihr zufolge verknüpfen wir kleine Werte mit der Richtung links, große Werte entsprechend mit [...]

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