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Abstraction Principle – Everything and Anything Can be Counted

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The abstraction principle is the last of Gelman and Gallistel’s Five Counting Principles. The one-to-one correspondence, stable-order, cardinal, and order-irrelevance principles have been addressed in previous posts.

It is vital that children learn the other principles first, because as they get older, the abstract principle will be easier to understand.

The abstraction principle states that the preceding principles can be applied to any collection of objects,
whether tangible or not.

For young children learning to count is easier if the objects are tangible and, where possible, moveable, in order to help them to distinguish the ‘already counted’ from the ‘yet to be counted’ group. To understand this principle, children need to know that they can count non-physical things such as sounds, imaginary objects or even the counting words – as happens when ‘counting on’. 

 

Have you ever had a child who looks at two sets that have the same number of objects but automatically says the set with the larger objects has more? Or a child who sorts the set first so that different sized or coloured objects are together?

Child need to understand that regardless of what is in the set or how it looks, we are only interested in the number within the sets.

 

For example: The number within each set is the same.

Set A                                                                                          Set B

                               

 

Sources: Origo One

Difficulties

  • Sometimes when presented with different objects, children state that larger items have more value, for example, they are worth more than ‘1’.
  • It is more difficult to count non-tangible objects, for example, sounds, actions or words people say (this may impact upon literacy – the numbers of words in a sentence or the syllables in a word).
  • Matching sets of different objects that have the same quantity.

Practice

Provide children with opportunities to count a range of sets, real or imagined, similar or different.

  • Use different arrangements within the sets.
  • Use different colours and encourage them to not just focus on their favourite colour.
  • Ensure sets have different sized objects.
  • Balls Roll or throw them to a partner. Count the number of times a child bounces a ball.
  • Musical instruments Use a xylophone, drum or shake and count the number of times it’s hit or shaken.
  • Counter Drop Have children count the number of counters/pebbles that are dropped one by one into a can.

 

Remember, plan lots of opportunities for students to count whether formally or informally.

Ensure they:

  • touch and move objects, where possible, to keep track of what they have and haven’t counted
  • apply one number name to each object or action
  • say the number words in the correct order
  • say the last number word after they have finished counting then ‘you’ ask ‘how many?’
  • have a variety of things to count – differing sizes and colours in the same set, objects in different arrangements, and tangible and intangible things to count
  • begin counting from different objects within the set

 

Until next time,

Carole