The Relationship Between Arithmetic and Spatial Intuition

15Jun08

Label one end of a straight line 0, and the other end 100. Ask a person to place the number 10 at some point along the line. Where would the person choose to place the number 10?

This study finds that while Western adults tend to place small numbers at the ‘correct’ position along the scale (in the above example, that means 1/10 of the way to 100 starting from the 0 end), the relatively uneducated and uncontaminated (by Western civilization) members of the Amazonian Mundurucu tribe tended to place 10 about halfway between 0 and 100. And Westerners tend to do similar things with large numbers — they tend to place it much ‘higher’ up the scale then we’d expect if we mapped the number line linearly to the spatial line. Remarkably, this systematic ‘error’ is found to closely resemble a log relationship. It’s also been known for some time that children in Western countries commit the same ‘error’.

To eliminate the possibility that the error is caused by some special characteristic of the representation of the stimuli, the researchers repeated the experiments with varying kinds of stimuli — with numbers presented in the subjects’ native language, in the subjects’ acquired language, in pictorial form, etc. The log relationship remained. So came the tempting conclusion that this is in fact an innate characteristic, and Western adults fail to display it for small numbers only because they have been schooled to place numbers linearly along a spatial line (the schooling tends to use examples involving mostly small numbers, so they fall back on their logarithmic intuitions for large numbers).

A possible explanation for this log relationship is the ‘compressed numerosity code’ the brain uses for mental arithmetic:

Research on the brain mechanisms of numerosity perception have revealed a compressed numerosity code, whereby individual neurons in the parietal and prefrontal cortex exhibit a Gaussian tuning curve on a logarithmic axis of number (27). As first noted by Gustav Fechner, such a constant imprecision on a logarithmic scale can explain Weber’s law—the fact that larger numbers require a proportional larger difference in order to remain equally discriminable. Indeed, a recent model suggests that the tuning properties of number neurons can account for many details of elementary mental arithmetic in humans and animals (21). In the final analysis, the logarithmic code may have been selected during evolution for its compactness: Like an engineer’s slide rule, a log scale provides a compact neural representation of several orders of magnitude with fixed relative precision.

Reference:
Dehaene, S., V. Izard, E. Spelke, and P. Pica (2008, May). Log or linear? distinct intuitions of the number scale in western and amazonian indigene cultures. Science 320 (5880), 1217-1220.

Filed under: mathematics, psychology   |  Leave a Comment