Advanced training materials for autism; dyslexia; speech, language and communication; emotional, social and behavioural difficulties; moderate learning difficulties

Counting principles

• One-to-one

  This principle is based on only assigning one count word to each individual counted object or item. It involves three processes:

  • Itemising, where the pupil singles out each item that they want to count.
  • Tagging, where they correlate words with each of these items.
  • Partitioning, where they distinguish items that they have already counted from those that still need to be counted.

  The process of itemising is the first to emerge in children, while tagging and partitioning are often supported by gesturing, particularly in children between the ages of four and five, who will touch or point at objects to count.

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• Stable order

  To count, children initially need a stable sequence of words that can be used in a fixed, repeatable order. Understanding counting as a form of progression is an inherent feature of initially learning to count, which is further supported by an early capacity to grasp serial order.

  Learning stable order can come before learning traditional sequencing. By the age of 2 ??, children tend to use stable sequences of words as tags in counting routines, but may at first, for example, count 1, 2, 4, 9. So a child who counts 1, 2, 3, 4 for one set of objects and 2, 1, 3, 4 for another has not grasped the stable order principle. However, a child who consistently counts 2, 1, 3, 4 has ??? without yet understanding traditional sequencing.

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• Cardinality

  The cardinal number is the number a child reaches at the conclusion of a counting sequence. This count word represents the total property of the sequence, and tells the child how many objects have been counted overall.

  It is not simply about ???naming??? the last object in a sequence, but knowing how many items have been counted. A child who recounts a collection when asked how many items they have counted has not grasped the principle. It is also not just a case of repeating the last number they said in a sequence; for example, a child counting five items starting with 3 may say they have counted 7 items.

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• Abstraction

  To understand this principle, children must move past needing tangible, movable objects to count.

  The principle states that the one-to-one, stable order and cardinality approaches can be used on any collection of objects, regardless of whether they are tangible.

  It focuses on children grasping that they can count non-physical things such as sounds, imaginary objects or even counting words.

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• Order irrelevance

  To understand this principle, children must know that objects can be counted in any order; it doesn???t matter about sequencing, just as long as everything is counted once and only once. So, for example, a child that understands this principle could start in the middle of a row of objects and count to the end, and then from the beginning to the middle.

  Children generally start to show at least some understanding of this principle when they are about three years old, although it is not very strongly established at this point. A study by Arthur Baroody, published in 1993, found that some 4-year-olds did not realise that counting the same row of objects twice from different sides would result in the same outcome (Wiese, 2003).

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