The school report of Emily, our 4 year old grandchild, said that she could subitise up to the number 6, and I had no idea what this meant.
Subitising is a technical term that comes from the Latin root subito meaning suddenly or immediately. It describes the ability to just glance at a small group of objects and without effort be immediately aware of how many objects are in the group. For instance you can look at a dice and realise that three is showing without having to count the number of dots.
The strong Gestalt of the arrangement of the pips (spots) on dice almost certainly aids subitisation due to the high degree of symmetry and the gridded nature of their location.
The task gets a little more difficult with randomly located spots. These were located with a Dart Throwing Poisson Disk distribution to avoid overlapping pips (see Find Your Own Space for details https://grahamshawcross.com/2012/10/12/find-your-own-space/)
And perhaps even more difficult if the pips are arbitrarily colour coded.
Or allowed to overlap, with their location just randomly selected with the proviso that each pip must be completely within the bounding square; pips that would overlap the boundary just being rejected. This ignores the possible pathological cases were randomly selected locations are more or less coincident.
It should not be surprising that this arrangement looks rough and clustered (see The Aesthetics of Aperiodic Tiling https://grahamshawcross.com/2012/11/02/aesthetics-of-aperiodic-tilings/ )
Subitising ability is usually measured by recording how quickly the number of objects in a group is recognised. My assumption with the examples above is that with each example it progressively takes longer to identify the numbers represented, but that it is still possible to do so.
The objects to be counted can obviously be anything at all, all be different, and not have the strong Gestalt form of primary coloured discs, as used above. These strong forms probably aid feature recognition as thought of in Attention Theory. On the other hand the effect of a regular canonical arrangement of pips is summarised in the graph below. (Piazza et al. 2002)
As perhaps might be obvious, with 1-4 spots there is little difference between random and canonically arrangements but with 6-9 spots there is a marked difference, showing that the canonical arrangement aids subitisation.
Both reaction time and accuracy are subject to a subitisation effect. The graph below summarises the effect on accuracy were the number of spots presented ranged from 1 to 200, and in this case where subjects were prompted to carry out the task as rapidly as possible. When subjects were asked to concentrate on accuracy the results were very similar. (Kaufmann et al. 1949)
Accuracy is high, and reaction times low, when the number of dots range from 1 to 5, 6 or 7, the subitising range. Accuracy falls off rapidly above 10 spots with larger numbers startlingly underestimated. Within the subitising range response times increase by approximately 40-100ms per extra item whilst outside the range they increase by 250-350ms per item. These rates are somewhat higher for children but similarly separated.
The subitising range also increases with age, 3 week old children can subitise 1-3 objects and 7 year old children 1-7. (Dehaene & Cohen 1994) Subitising is thus a learnt automatic response. That is a skill that requires no conscious effort but needs to be learnt.
There seems to be evidence that subitisation is a separate but overlapping neurophysiological feature of enumeration. Part of the evidence for this is that people with Ballint’s syndrome, who are unable to perceive visual scenes properly, or position objects in space, cannot accurately enumerate objects outside the subitising range but within the range can subitise normally. The disorder is associated with an area of the brain responsible for spacial shifts of attention, something that is thought necessary for counting. Some research has questioned this, suggesting that attention is also required for subitisation. (Piazza et al. 2002)
There is evidence that new-born children have an innate subitising ability and that this is shared with other genera such as fish, indicating that subitisation has a deep primitive evolutionary basis.
Dominoes and Cards
With the addition of a blank half tile the same regular pip arrangement familiar from dice is also used in the design of dominoes.
A slightly different arrangement is used with playing cards where a different and pictorial mnemonic strategy is used for what might be considered the numbers above 10, the Jack, Queen and King. The 7, 8, 9 and 10 are also more difficult to subtise than the lower numbers and the numerical cues in the corner of the cards are probably made more use of with these cards.
The ability to subitise up to 5, plays an important part in the design (or evolution) of the abacus. In the Chinese and Japanese abacus the singles are counted up to 5 and recorded as sets of five as illustrated in the diagram below which also shows the carry mechanism.
The carrying mechanism is illustrated by adding 2 to the displayed number 715408.
Step 1 is to move 2 beads up the singles section (row) of the units wire (column). This leaves 5 beads recording the number 5.
The number 5 can be represented by 5 beads at the top of a singles section or 1 bead at the bottom of the corresponding fives section.
But the bead in the fives section of the units column has already been used to represent the number 8, as 3 singles plus 1 five.
Step 2 is to move all 5 beads down to the bottom of the singles section of the units column.
Step 3 compensates for this by adding 5 to the units fives row. This is done by moving the fives bead to the top of the column preparatory to recording the carry.
Step 4 records the carry by moving 1 bead up the singles section of the tens column.
This is a very visual way of doing maths that relies heavily on being able to to effortlessly recognise up to five objects.
Those of us getting on in years are more likely to encounter subitisation as part of an assessment of cognitive ability as recommended in SIGN 86. Management of Dementia. A National Clinical Guideline. (SIGN 2006) which recommends the use of the Addenbrook’e Cognitive Examination (ACE-R) part of which is illustrated bellow. (Mioshi, Eneida, et al. 2006)
Given the previous discussion these high numbers, displayed without any canonical patterning, represent quite difficult tests of cognitive functioning. The distributions also look suspiciously un-random with very little clustering as should be expected. I have noticed that in psychological experiments very little attention is given to the design of random arrangements.
It is interesting that the primitive ability to subitise seems to form the basis of the design, or design evolution, of cultural objects such as dice, dominoes, playing cards and abacuses. Reinforced in the case of the games by strong Gestalt type patterning.
In the design of playing cards a completely different pictorial mnemonic strategy is used for the face cards and an auxiliary aid added for all cards but of most use for numbers usually considered to be outside the subitising range. The design thus subtly takes advantage of subitising whilst also using other strategies and auxiliary cues where subitising would be unlikely to work.
The design and efficiency of the Chinese and Japanese abacus, relies directly on the ability of an individual to visualise one to five objects without conscious effort.
Neurophysologically subitising seems to relate to enumeration in a way that is similar to the relationship between face recognition and visual perception. That is subitisation and face recognition are both separately identifiable, specific skills that fit seamlessly into a more generalised skill.
A recognition of subitising in architectural design could prove useful in making groups of architectural features and units more easily read and understood by utilising an automatic, unconsious visual response in the observer, with playing card design providing a useful exemplar.
In aesthetics subitisation helps to identify Gestalt groupings particularly those relating to proximity, form or colour.
See https://grahamshawcross.com/2012/01/25/aesthetics/ for what amounts to a discussion of such Gestalt based aesthetics related to housing design.
Dehaene, S., & Cohen, L. (1994). “Dissociable mechanisms of subitizing and counting: neuropsychological evidence from simultanagnosic patients”. Journal of Experimental Psychology: Human Perception and Performance 20 (5): 958–975.
Kaufmann, E. L., Lord, M. W., Reese, T. W. & Volkmann, J. (1949) The Discrimination of Visual Number The American Journal of Psychology Vol. 62, No. 4. pp 498-525
Mioshi, Eneida, et al. 2006 “The Addenbrooke’s Cognitive Examination Revised (ACE‐R): a brief cognitive test battery for dementia screening.” International journal of geriatric psychiatry 21.11 (2006): 1078-1085.
Piazza, M., Mechelli, A., Butterworth, B., Price, C. 2002. “Are Subitizing and Counting Implemented as Separate or Functionally Overlapping Processes” NeuroImage 15 435-446
SIGN 86, 2006, Management of Patients with Dementia: A National Clinical Guideline Scottish Intercollegiate Guidelines Network. NHS Quality Improvement Scotland