At a glance: Subitising

A curious pendant to the approximate number sense is the way people can recognise small numbers at a glance. Some experiments seem to show this as a separate ability from estimation; collections of one, two, three and four items appear to be distinguished immediately, without the effects of size and ratio that characterise the approximate number sense. Babies just a few hours or days old, for instance, have been observed to tell apart collections of two and three objects but not collections of four and six. At a few weeks, they can tell apart sets of one, two, three or four items, all equally fast and equally accurately. But their performance drops sharply if the number of objects becomes larger. For adults, the limit of this ability – ‘subitising’ – seems consistently to be four, with the difference between three and four recognised just as reliably as that between one and four, and the ratio effects characteristic of the approximate number sense appearing only when the numbers are larger than this.

Like the approximate number sense, subitising is cross-modal: heard or even felt stimuli can be shown to be distinguished in a similar way. (Many readers will have experienced the difference between counting musical beats and ‘feeling’ them, which surely has something to do with the limits of subitising in the rhythmic domain.)

Subitising. You know there are three sheep without thinking ‘one … two … three’.

imageBROKER.com GmbH & Co. KG / Alamy Stock Photo.

But, in another difference compared with the approximate number system, subitising seems to stop working when your attention is distracted. While judgements of approximate number are formed ‘in the background’ of other tasks, subitising requires you to focus on the objects in order to know how many of them there are.

Subitising, in fact, seems to be very much about tracking and paying attention to the individual objects. Many researchers have suggested it works something like a set of slots or pigeonholes in your working memory: you can fill them one at a time, but when all the slots are full, you lose track and fall back on estimation. For that reason, this particular ability is sometimes called the ‘object file system’.

Within its range it can even provide for some basic arithmetic. In a classic quip, two tigers go into a cave and one comes out. Is the cave safe? If you are keeping track of the tigers as individuals, you know that there is still one in the cave, even if you have never learned such a thing as two minus one equals one.

It is fair to emphasise again that subitising is controversial, and little about it is really settled. One intriguing question is whether subitising actually replaces estimation for small numbers, or whether instead both systems work alongside one another for object collections numbering from one to four. The object file system can be turned off by distracting the attention, and in that case most studies show that estimates of numerosity are still successfully formed, and that the ratio and size effects characteristic of the approximate number system can be discerned again. Does that mean that approximation is the more fundamental system, overlain in some situations by an ability to track up to four objects precisely? Or, instead, does the approximate number system normally cease to function when the object file system is engaged? The answer is not yet clear.

Another question is whether subitising is possessed by any animal species. It would be surprising if such an ability existed in humans only, but the evidence for it in animals is patchy and in some cases problematic. Some would argue that for certain species the data are best explained by a subitising-like process. Bees, for instance, have been reported to distinguish successfully between sets of up to three objects, while dropping abruptly to the level of random guessing when the sets are any larger. That is the signature of an object file system, not an approximate number system. The same is true for salamanders, while the abilities of certain fish species look more like the two systems of humans: high accuracy for collections of up to three objects, succeeded by an increasingly fuzzy ability to approximate for larger numbers. The evidence for birds and mammals points in various directions. Subitising has been reported for robins, chicks, pigeons and parrots, for dogs, monkeys and chimpanzees, and some scientists certainly regard it as widespread in the vertebrate family tree, and therefore presumably of some evolutionary age. But retesting has not always succeeded in replicating these effects, and much uncertainty remains.

Recent reviews of the evidence emphasise that nearly all animal species tested show signs of an approximate number sense, whereas the evidence for a second system remains limited and inconsistent. Some would say that approximation, which is at its quickest and most accurate for small numbers, can account for all the observations yet published. Others would add that it is not straightforward to think of factors in the environment that would create selection pressure specifically for a subitising-style recognition of small numbers instead of – or as well as – an approximate system. One researcher speaks of an ‘impasse in the literature’ on these questions.

So, can animals count? Yes and no: but mostly no. Animals do not count in the sense in which (most) human beings do. Despite its capabilities, and despite its great age, the approximate number system is not a counting system. There are no exact numbers here: just approximations that become rapidly more fuzzy. This is a system in which one and two are distinct, but three and four less so; and in which one hundred and one hundred and one are for all purposes identical. Unlike counting, when every step from a number to the next is the same size, the approximate number sense is a system in which the relationship between two and three is very different from that between, say, fifteen and sixteen.

Similarly, subitisation – if it is real – seems to have more in common with pattern recognition than with counting: it is a form of perception in which a group of two objects and a group of three are sharply distinct, but which has nothing whatever to say about the difference between groups of four objects and five. If it is right to think of it as an object file system – a set of mental slots to be filled – then it really contains no explicit representations of how many, and the fact that you can get a number from it is a mere side effect.

These innate abilities are roots (seeds, even?) of real use for constructing counting practices, but they are not the same as counting. They constrain what counting can be like; they constrain which species can acquire it. But there remains a real gap between these abilities and counting. Humans are not born able to count, then; they have to learn to do it. Cultures have to invent ways of counting and transmit them from one generation to another, with the real possibility that ways of counting may change out of recognition over time, or die out or be replaced.

2

Counting before writing

: Africa and beyond

If inherited abilities are one root of human counting, another is those things in the environment that have sometimes been called the ‘wild number lines’. Things like pebbles that you can use as counters, butchery marks that you can repurpose as tallies, or the set of fingers and toes that (nearly) all humans have. Each can be used to keep track of a sequence and thereby make counting possible. Each can serve the functions of a set of numbers, recording and communicating the outcome of a count. And so, of course, can words, once humans have started to use them.

Each of these four wild number lines is grounded in Africa, like the human species itself. For counters and for tally marks, African archaeology provides some of the earliest evidence, and therefore a tantalising glimpse of these roots of counting in the Stone Age. For counting on the fingers, the best early evidence that survives is from north of the Mediterranean. For words, there is no really direct evidence at all, although much can be deduced from counting words in living languages. But the story starts with beads.

Blombos: Counting with beads

Blombos Cave in South Africa, 75,000 years ago. A woman takes a shell, collected from the coast. Much the same size and colour as a large human tooth, it was once the home of a small sea snail. Now it will be a bead.

She pierces the lip of the shell with a bone point, lays the shell aside and takes up another. When she has enough, she threads them on a strip of hide or plant fibre; ties it. A necklace.

The human and chimpanzee lineages split around seven million years ago. Over a dozen early hominid species are now recognised, and the shape of the family tree changes as new fossil material is uncovered. About eight species, at different times, made up the genus Homo, which evolved in Africa over two or three million years. These species committed to living on the ground and grew a proportionately enormous brain compared with their ancestors. Brains do not fossilise, but there is some evidence from the shape of fossil skulls that the parts that grew the most included the interparietal sulcus, used for the approximate number system, among many other functions. More visible in the archaeological record are changes in behaviour: butchery marks appeared on bone as early as 3.2 million years ago, and stones modified for cutting first appeared in the Kenyan and Ethiopian Rift Valley 2.6 million years ago. Fire was used, perhaps, as early as 1.5 million years ago.

Beads from Blombos.

Heritage Image Partnership Ltd / Alamy Stock Photo.

The period called the Middle Stone Age began perhaps a quarter of a million years ago in Africa. It was distinguished by a new, more sophisticated suite of stone tools: flakes and blades, cores and points. This was a more variable set of tools than its predecessors, but still it lasted for many tens of thousands of years without a great deal of change. The archaeological record is coarse-grained, though, and the scales of time and distance involved are enormous. There were anatomically modern humans – Homo sapiens – in Africa by 200,000 years before the present, perhaps by 300,000 years; by 100,000 years ago, the species could be found from South Africa to the Levant. Lifestyles were based on hunting and gathering. Shelter was in caves, many of which are still visible.

Blombos Cave is located in the southern Cape, South Africa; it is about 300 kilometres east of Cape Town. Today, the Indian Ocean is just 100 metres away. The cave site was occupied in three or four phases, starting upwards of 100,000 years ago and ending perhaps 70,000 years ago; the occupations may have been relatively short in relation to the millennia that separate them. Beach sand seals the last Stone Age layer.

Those who last lived here made a range of artefacts including stone points and bone tools. They made much use of marine resources such as seals, dolphins, fish and shellfish, although the sea level was lower in their time and the coast certainly further away: at times, dozens of kilometres. The climate was relatively warm and moist during the final period of occupation seventy-odd millennia ago; the landscape a patchwork of open areas, shrubs, trees, waterways and woodland, constantly changing under pressure from plants, animals and slowly shifting rainfall patterns. The people also moved around, to forage, to hunt and to sleep.

The cave at Blombos is not a large one, and no more than perhaps thirty people would have stayed there at any one time. But its excavation since 1991 has revealed a much-studied set of artefacts in addition to what is usual for its time and place. There are thousands of pieces of ochre, including some bearing unmistakable signs of engraving: this at a very early date for abstract representations anywhere. And there are beads.

The Blombos beads are, so far, the oldest Stone Age beads to be dated really securely; indeed, they are the oldest securely dated personal ornaments of any kind. They date from the last occupation level at Blombos Cave, that is roughly seventy-two to seventy-five millennia before the present. The beads are made of tick shells: the species is called Nassarius kraussianus and lives only in estuaries. This was an exotic material and an expensive one in terms of time and effort; the shells would have had to be transported from rivers twenty kilometres or more from the cave.

Sixty-eight tick shells have been found at Blombos. Nearly nine in ten have a hole near their lip, made by inserting a bone tool through the main opening and then pressing. This was the action that transformed them into beads. Once pierced, the beads were threaded onto cord or gut; traces of wear on the edges of the holes, and on the outer sides of the shells, show where they rubbed against both the cord and each other. There could be as few as two beads on a necklace, or as many as two dozen; up to about twelve was average. They were strung in symmetrical pairs, one facing left, one facing right. The marks are deep: the strings of shells were worn for months or years.

Shell beads have also been found at a range of other archaeological sites, with some possible dates – less certain than at Blombos – as old as 115 millennia. Most of the sites are in south or east Africa, some in Morocco and one or two in Israel. A common pattern is that the shells were from species exotic to the locations where they were found; they were brought in from some distance away. Particular species and sometimes particular sizes were selected. Sometimes natural holes from wave or beach action were used for stringing; other beads, like those at Blombos, were deliberately pierced. The tradition of shell beads lasted many thousands of years and ranged over five or six thousand kilometres; at one time it may have been common, beadwork one of the everyday items of human life.

In archaeological terms, beads are one of the first visible examples of symbolic behaviour, something like evidence for culture rather than mere survival. Are beads also evidence of the first steps towards human counting?

Here the mystery deepens. The beads are from millennia before writing; if they had a meaning, their archaeological context cannot show what it was. Beads were surely first made as personal decorations, with a purely aesthetic significance. But they are probably also some of the first surviving objects that were meant to communicate information – information about their wearers and their communities – making them some of the first evidence for humans’ manufacture and use of symbols.

Beads also have a suggestive set of properties that have led more than one scholar to describe them as a wild number line – capable, in the absence of number words or gestures or even number concepts, of existing and doing some of the things that counting sequences do. If you see a modern pre-schooler playing with beads on a string, you assume she’s learning important things about sequence, order, and eventually number. About adding-one-more, or creating a new sequence by joining two shorter ones together. About subtracting from a sequence to get a shorter one, or even dividing a sequence into equal portions.

If Stone Age Africans had similar devices in their hands, surely they learned similar things. With beads, you can play, manipulate and contemplate processes like sequencing, adding and subtracting as physical actions – perhaps for years, perhaps for generations – before eventually learning to associate a numerical meaning with them. Not only that, but compared with fingers or the small collections of objects that can be subitised, strings of beads can embody quite large numbers: up to twenty-four, in the case of the artefacts recovered at Blombos.

You don’t necessarily even have to have a set of counting words in order to use the beads for counting: in order to correlate them one-to-one with other objects in the world, or with actions or events. There are, to this day, many situations in many cultures in which a ritual action is done once for every bead on a string: prayers or prostrations, for instance. No counting words need be uttered and no number concepts need be held in the mind; you simply hold one bead and perform the action, then you hold the next bead and perform the action again. When the beads run out you stop performing the action. You don’t have to use or even know any number words in order to use a set of beads like this; indeed, it is not clear that your culture has to feature any other way of counting in order for a string of beads to work for this purpose.

In other words, it is possible to imagine a world in which the beads on strings are the main, the best, and even the only available way of counting. The beads at Blombos and other locations may have existed and been manipulated for many generations without anyone using them to count. But they may, eventually, have been used to keep track of things or objects – to count – even if the culture of which they were part contained no other way of counting. It is not certain that such a world ever existed, but the possibility is a most intriguing one.

Lake Rutanzige to Laussel: Counting with tallies