# Counting at a glance: experiments on the approximate number system

By dkl9, written 2023-271, revised 2023-271 (0 revisions)

Usually, you can look at a set of things briefly, without explicitly counting, and get a rough idea of how many things are there. This is part of what is called the approximate number system (ANS). To measure just how effective this is, I had people look at randomly scattered sets of dots, of random quantity from 1 to 100 dots, and report their guesses as to how many they were shown.

Said people comprised 6 distinct Subjects, convenience-sampled. I was Subject 1, and tried it myself in four separate sessions (2022-312, 2022-313, 2023-023, 2023-024). All other subjects were tested in one session each.

The dot-images were displayed for 1 second (first session) or 0.8 seconds (other sessions) each. I found no significant difference in error in glance-counting between these cases (p = 0.30 by t-test on relative error).

After seeing the image, subjects had some limited interval during which to give their guesses. That interval was variously set between 3.2 and 8 seconds. I found significant differences between relative errors for 3.2 seconds vs 5.6 seconds (p = 0.006 by t-test), 4 vs 7.2 (p = 0.007), and 5.6 vs 7.2 (p < 0.001), but not between any other pairs of response-times. 5.6 seconds for responses gave the lowest average relative error.

Some subjects were very miscalibrated in their counting (e.g. Subject 5 often guessed over 100, and Subjects 2 and 4 tended to guess around half the actual value). Nevertheless, their guesses accurately corresponded to the actual number of dots present, with most error accounted for by a scale factor. I computed, for each subject, the correlation coefficient between their guesses and the actual numbers of dots.

 Subject number Trials Response time (s) Correlation 1, day 1 39 4 0.892 1, day 2 10 7.2 0.951 1, day 3 32 7.2,3.2 0.941 1, day 4 32 3.2 0.952 1, pooled 113 varied 0.900 2 15 7.2 0.860 3 13 7.2 0.908 4 14 5.6 0.847 5 33 5.6 0.961 6 3 8 0.828

Different people glance-count with noticeably different accuracies. It appears from Subject 1's (my) multiple trials that this correlation between the ANS and ground truth can improve with practice.

Further research is needed (but may have already been done — I haven't much looked), and may include:

• more consistency and organisation than what I did
• larger samples (more subjects and longer sessions) than what I did
• further investigation into the effects of practice, including intervals between practice sessions
• keeping track of and checking for effects from time-of-day
• displaying the set of dots differently (e.g. with varying colour or different shapes) and checking for effects from such changes