This is Archie, our then 3 year old grandson, after spontaneously reassembling a sawn up branch and being asked to pose with it for his delighted grandfather.

Seriation is defined as *“the** forming of an orderly sequence”* and was apparently first used in this sense in the 1650s. (Dictionary.com 2015)

The peak usage of the term appears to have been in 1977. Seriation was one of the key tasks used by Piaget to investigate the development of children’s thinking. (Inhelder & Piaget 1958)

Piaget studied seriation by presenting children with a disordered collection of sticks and asking them to arrange the sticks into an ordered series. (Inhelder & Piaget 1958) (Piaget 1964)

Piaget’s results illustrated above can be summarised as follows.

**Stage I:** children (aged up to about 4) make no attempt to line up the sticks or just move them about randomly.

**Stage II:** children (aged about 5) are unable to arrange a complete series, but can combine the sticks in local ordered pairs or triplets.

**Stage III:** children (aged about 6) can construct a complete series but only by a process of trial and error.

**Stage IV:** children (aged about 7) can *‘operationally’* complete the series, that is select and place say the smallest first followed by the next smallest etc. They are also able to insert a rod taken at random from the pool into its correct place.

Later research has generally supported these conclusions, but with some qualifications. The key stages are now thought to be less sharply defined and the ages a little earlier than those stated above. The ability to seriate also seems to depend on the children’s motivation and their ability to understand and carry out verbal instructions.

Performance is improved if there is greater differentiation between adjacent sizes. The number of sticks also seems to be important, with younger children only being able to serialise smaller numbers of sticks and five or six being the maximum number before Stage IV is reached.

## Transitive Inference

Transitive inferences is an ability that has long been supposed to differentiate humans from animals.

Given the propositions R(a, b) and R(b, c) then R(a, c) can be inferred. This is true for symmetric relations like *“equals”*, *“similar”* etc. and asymmetric relations like *“bigger than”*,* “smarter than”* etc.

According to Inhelder and Piaget, transitive inference is a late developing ability and 8 year old children can fail to solve transitive problems based on size. It appears even later, at up to 9 or 11, for problems based on weight or volume etc. (Inhelder and Piaget 1958, 1964)

They suggest that the Stage IV ability to seriate operationally at around 7 years of age results from children beginning to understand relational *reversibility* and *relativity*.

Reversibility of relations means understanding that if I have a *“brother”* called John, then John also has a *“brother”* – me, and that if I am *“taller than”* John then he is also *“shorter than”* me.

The relativity of relations means understanding that relations are not absolute attributes of objects, so John can be *“taller than”* me and *“shorter than”* Paul.

Another form of transitive inference is given by constructions like if *“A is shorter than B”* and *“B is shorter than C”*, *“which is the shortest?” *This type of reasoning would seem to be important in putting sets of objects into some sort of order.

There is a “spatial model” theory which suggests that transitive inferences are made on the basis of a mental spatial model constructed in the process of understanding the premises and this is easily understood diagrammatically.

In fact, unless one is a trained logician, it is difficult to understand how one can answer this type of question verbally in one’s head without constructing a visual spatial model of some sort.

## Relational Competency in Animals

Many species display what McGonigle and Chalmers call *relational competency*, that is similar to those of 5 year old children. (McGonigle & Chalmers, 1996) See also What Counts. and Counting Ants

They trained squirrel monkeys and five year old children to associate box size with colour, the smallest size with red, the next smallest size with green etc. so that when the monkeys were presented with a randomly arranged row of say green boxes they would ‘know’ into which box food had been placed, the second smallest box, fourth from the left in the example above.

So the uniform colour of the presented boxes was a sort of instruction to select a particular size of box.

The results for each size of box is summarised above, with little difference between squirrel monkeys and 5 year old children. There are good results for the extreme sizes (white and red) and poorer results for the intermediate ones (blue, black and green), probably because these sizes are easier to confuse in a random presentation.

By changing the absolute size of the boxes but conserving their relative sizes it was shown that relative size was being encoded rather than absolute size.

They also found that it was difficult to successfully extend the experiment to more than 5 or 6 boxes, suggesting that this was because of the larger number of possible random arrangements. But given the data above it seems more likely that the problem was differentiating between increasingly similar sizes when they were randomly presented.

If this is the case, it is an example of Weber’s Law because the *just discernible difference* between adjacent box sizes is inevitably reduced when there are more boxes.

## Pointing

To reduce the amount of dexterity required to seriate a set of objects the task was modified to one of sequentially pointing, on a computer screen, to the box sizes in ascending or descending size order.

Feedback was given via bleeps (positive) and buzzes (negative) but the boxes were not moved and remained in their original place on the screen as indicated above. A record was kept of the number of trials required to reach a criterion of getting 8 out 10 attempts correct.

7 year old children had little difficulty achieving the criterion with 5 or 7 items but 5 year old children had increasing difficulty. Again supporting the notion that children achieve Piaget’s Stage IV, operational seriation, at around 7 years of age when they are becoming capable of transitive inference.

## Comparison of Procedures

Whilst the two procedures are superficially similar there is at least one significant difference. Moving sticks or boxes allows one-to-one comparisons to be made on the fly, for instance by moving a candidate box across the other boxes. Pointing in-order on the other hand offers no such opportunities.

## Bubble Sort

When one-to-one comparisons can be made, a bubble sort seems to be a way of overcoming the difficulties, experienced by both young children and animals, in ordering collections with more than five or six items.

This is because it only requires the repeated comparison of two adjacent elements to see if they need to be swapped. It is also a procedure that can be carried out manually, by simply swapping the position of two adjacent objects in the collection, leaving the rest in place.

Importantly it also avoids the need for any form of transitive inference.

Donald Knuth, the father of the formal analysis of algorithms, said that the bubble sort *“seems to have nothing to recommend it, except a catchy name”.* (Knuth 1973) Bubble sorts, however, have been remarkably resistant to his scorn partly because they are very easy to understand and implement. They are also reasonably efficient, at least for partly ordered lists.

## Pseudo Code

procedure bubbleSort( A : list of sortable items ) n = length(A) repeat swapped = false for i = 1 to n-1 inclusive do /* if pair meet swap criteria */ if swap_criteria(A[i-1], A[i]) then /* swap them and remember being swapped */ swap( A[i-1], A[i] ) swapped = true end if end for until not swapped end procedure

The application of this pseudo code, illustrated below for sorting a set of objects in descending order of height, uses repeated iterations. These are necessary to ensure that all possible swaps are carried out, but it might be argued that this is a circular method of creating seriation because it requires a knowledge of number series.

## Manual Version

On the other hand in a manual, non-computer, implementation the swaps can be carried out in any order, without explicit iterations. Just continue to swap objects until no more can be swapped. The procedure shown below relies only on the ability to compare the values of two adjacent squares to see if they need to be swapped and recognising when no more swaps are possible. It is essentially non teleological, one does not have to specify the end result, one simply gets it by carrying out the procedure.

The need for manual dexterity could be reduced by displaying the list on a computer screen and allowing the simian or child subjects to indicate when a swap was required by pointing at, say the left-hand item of, a pair needing to be swapped.

If an allowable swap is indicated, the swap would actually be made on the screen. As before, success would be measured by comparing the number of correct choices with the number of incorrect choices.

## Media Matters

Given that doing a bubble sort does not require a computer, I thought it might be instructive to investigate *sorting* before the advent of the computer. One of the few examples I could find was instructions on how to sort a hand of cards using what would now be called an *insertion sort* but might just be thought of as the natural, if slightly geeky, way of sorting a hand of cards.

An insertion sort, as illustrated above , shows why *sorting* and *searching* are usually treated as opposite sides of the same coin. (Knuth 1993) In order to sort the cards efficiently one needs to be able to search the cards one is assembling on the left in order to insert the next card into the correct position.

Like playing cards, index cards have the essential quality that the cards, like sticks and boxes, can be easily handled and for instance a new card easily pushed into the correct position in an already ordered set of cards. A dropped set of cards can also be reordered by following a procedure similar to that used for ordering the playing cards above.

Edge notched card systems are an example of a mechanical sorting and data retrieval system in use before the advent of computers. They use inverted indexing to allow cards with particular properties to be extracted with a knitting needle like tool.

Finally an IBM Hollerith card, an horrendous example from the SS Race Office, of the card that later became ubiquitous in data processing by computer.

## Discussion

Seriation looks as if it might provide an underlying mechanism for satisfying the first two, and particularly the second, of Gelman and Gallistel’s five principles that define counting. (Gelman & Gallistel 1978)

**1. The one-to-one principle.** Each item in a set (or event in a sequence) is given a unique tag, code or label so that there is a one-to-one correspondence between items and tags.

*2 . The stable-order principle (ordinality). The tags or labels must always be applied in the same order (e.g., 1, 2, 3, 4 and not 3, 2, 1, 4). This principle underlies the idea of ordinality: the label ‘3’ stands for a numerosity greater than the quantity called ‘2’ and less than the amount called ‘4’.*

For a fuller explanation see What Counts.

Unfortunately, unlike Subitising which can be demonstrated in 3 month old children, seriation only appears to be available to children aged 5 or above.

There is a real difference between pointing to objects in the correct order on a computer screen and seriating objects, sticks, boxes or nesting cups, that are ordered by physically moving the objects.

If side by side comparisons can be made, a manual bubble sort method would be an improved experimental paradigm. It allows very small differences to be discriminated and enables the ordering of large sets of objects where, for instance, adjacent boxes may differ by very little in size.

A bubble sort is simple enough to be carried out manually by simians or small children. An insertion sort, on the other hand, requires some knowledge of transitive inference so that cards can be inserted into the correct position in the sorted set of cards. This applies to playing cards and index cards.

It is interesting that cards as media are retained in early mechanical data management systems, like edge notch cards and Hollerith cards.

## Bibliography

Dictionary.com (2015). *Dictionary.com Unabridged*. Random House, Inc. http://dictionary.reference.com/browse/seriation (accessed: January 12, 2015).

Gelman, R. & Gallistel, C. (1978) *The Child’s Understanding of Number. *Harvard University Press, Cambridge M.A.

Inhelder, B. & Piaget, J. (1958) *The Growth of Logical Thinking from Childhood to Adolescence.* London: Routledge and Kegan Paul. 1958. Pp. xxvi + 35t. 32s.

Inhelder, B. and Piaget, J. (1964) *The early growth of logic in the child*, London: Routledge & Kegan Paul.

Knuth, D. (1973) *The Art of Computer Programming, Volume 3. Sorting and Searching. *First Edition Addison-Wesley Reading Massachusetts

Mareschal, D & Shultz, T. R. (1999) *Development of Children’s Seriation: A Connectionist Approach.* Connection Science, Vol 11, No. 2 Pp 149-186

McGonigle, B. & Chalmers, M. (1996) *The ontology of order*. In L. Smith, ed. *Critical Readings on Piaget*. Routledge, pp. 279-311.

Piaget, J. (1965) *The Child’s Conception of Number*. Norton, New York.

A little bit frightening as well as pleasing. Best regards to everyone

Graham

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Hi Graham – Pretty amazing for a then 3 year old, no doubting whose grandson Archie is!

Sandra

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