Learning to make it count

Counting is such a basic tool to us that it is easy to assume that children know what they are doing as they count, or what we mean when we ask an apparently simple question like "How many?" However, counting is a complex skill.

Generally teachers of young children are very conscious of the skills which children need to learn before they read. Pre-reading activities are well understood and a wide range of materials are commercially available to help develop constituent skills for reading.
Most maths schemes on the other hand give only a fairly cursory acknowledgement to the constituent skills of learning to count and they move very quickly to using counting as a tool for mathematical and arithmetic processes.

To attain level 1, pupils need to have mastered the skills of counting and moved on to the use of numbers as tools.

The Parts of Counting

An obvious manifestation of counting happens as children learn to speak and use sounds that refer to quantity.There is however much more to counting than repeating sounds.

In the Language of Psychology
"When counting the child must co-ordinate the production of two continuous active sequences, saying the number words and producing points, while concurrently co-ordinating the points with a set of spatially distributed objects.
These requirements, accurate production of number words, plus their co-ordination in time with pointing and in space with objects allows considerable scope for error.
McEvoy J. (1989) "From Counting To Arithmetic". British Journal of Special Education. Vol16. No 3. Research Supplement, pp 107-110.

Knowing the number names

The Initial Acquisition of number names

Though they may use number names referring to small quantities at an earlier age, children usually begin to acquire the sequence of number names around the age of two. (Fuson et al. 1982)

Evidence about how children acquire words from the context of their use suggests that initially they may not perceive the sequence as separate words, but as a relatively meaningless string of rhythmic sound.

One-two-threefourfiveonceIcaughtafishalive

Nevertheless this is a phase of acquisition, and their learning is assisted by the continuity and rhythm of the sound sequences, which cue and prompt sound recall, and helps to fix the order of sounds.
Elaborating on the list
Separating the sound flow into separate words, and establishing the order of those words comes as a later phase of elaboration. A further part of the elaboration phase is when children begin to use the words as number names and the words become "objects of thought" that symbolise quantity, and can be used for counting things. When this knowledge is absorbed the child will begin to recognise that there is a fixed order on which they can progress up and down, they will become able to say number names onwards or backwards from a given point.
To complicate matters different aspects of number sequences − teen structure 13 to 20 and the decade structure 20 to 90 may be being learned at the same time but be in different phases of acquisition. E.g. when one to ten has moved into the phase of elaboration, the teens may just be starting to be acquired as a sound sequence. The whole process takes a number of years and the rate at which children develop the skills is varied.

Learning the words of counting sequences
  1. Acquisition
  2. Learning the sequence connected in a stream, of rhythmic sound.
  3. Beginning to separate the individual words, maintaining their order.
  4. Elaboration
  5. Confirming the order of occurrence.
  6. Knowing the order backwards
  7. Knowing the sequence onward or backward, from a given point.
  8. Confirming the connection of individual words to a related quantity.

Counting objects

Initially children are influenced by perceptions. They make a visual approximation of the quantity e.g.

If a child wants as many sweets as another, they will take a handful that seems to match the quantity.

If asked to take as many blocks, they will line them up side-by-side until the line is about the same length, without regard to the actual number of blocks.

In order to count a group of objects children must be able to itemise them and tag each with a number name. Schaffer et al (1974).
Most young children achieve this by pointing, and linking the word to the object as they point.

It is easy for them to make a mistake whilst:

  1. Controlling the physical act of pointing and controlling their attention to space.
  2. The timing of saying the number words at the right moment as they point.

For young children touching is an important part of this process of itemisation, and provides physical prompt to help with timing the saying of the naming word.

Eventually children must move from physical partitioning and pointing and be able to partition mentally. There is evidence that as they do this they still use physical and rhythmic tactics such as finger counting and tapping.

Easy Counting Mistakes
  • Fail to correspond their pointing to individual objects.
  • Fail to correspond the sound with the pointing action.
  • Miss an object.
  • Itemise an object more than once.
  • Missing a number name.
  • Applying the same name twice.
  • Confusing the order of names.
  • Lose track of what has been counted and what remains to be counted.
  • Don't stop the verbal sequence at the last object, keeping on because of the rhythm.
  • Don't realise the last number is cardinal.
  • Miss some objects because they don't think they should be included in the count because of their colour, shape, position etc.

The Counting Principles

As they co ordinate the physical pointing and the verbal naming, to count accurately there are a number of rules that children need to apply. Five principles necessary for accurate counting were described by Gelman and Gallistel, (1978). The first three principles relate to "How to Count" and the last two are about "Applying Counting".
The One to One Principle
Understanding and ensuring that each item receives one tag only
This requires:

  • Physically keeping track or mentally partitioning - which items have already been counted and those which remain.
  • It also requires tagging - summoning up and applying distinct names one at a time.

It is necessary to realise that the name tags are specially for counting with, they are nothing to do with other characteristics of the items being counted.

Now the child will line the blocks up one-for-one, matching one new block for each of the original ones.

In the early stages of learning to count children may be vague or imprecise about their pointing, they wave their fingers in the general direction, but let the rhythm of the verbal counting sequence dominate the speed at which they count, and consequently lose correspondence. Later they become more aware of the importance of co-ordinating the itemising and tagging, and they develop strategies for keeping track, and noticing if they have double counted or missed items. Gelman, R. Meck,E. (1983).
The Stable Order Principle
The name tags must always be used in a stable order
This presents the child with the problem of remembering a long list.The valuable role of intonation and rhythm offer prompts and connections that make memorable "chunks" and so help learning the number sequence.

The Cardinal Principle
The final number represents the size of the set
When the child understands this principle they recognise that earlier numbers were temporary steps towards the last number tag, which is special, because it is the Cardinal Number and represents "how many" items have been counted. Appreciating the importance of Cardinality is an important milestone in a child's mathematical development, it is a keynote in understanding that the process of counting has a meaningful and useful purpose. Fully grasping the cardinal principle depends on understanding the previous two principles, it therefore matures after them. There are three phases in its development (Fuson & Hall 1983 ).

  1. Reciting the last number with no clear idea that it relates to quantity, but because they realise it is the response the adult expects.
  2. Understanding that the last number of the count relates to the quantity.
  3. Understanding the progressive nature of cardinality i.e. if they are stopped in the middle of a count they can say how many they have counted so far, then carry on.

It is necessary for the child to grasp the cardinal principle before they will be able to:

  • Understand that the next number in the sequence represents a larger quantity.
  • Use the technique of counting on.
  • Use counting to determine and compare the equivalence of sets.

To demonstrate the difference between one-to-one correspondence and cardinality, here’s a favorite story:
There was a little boy who was eating cookies with his mom. He counted them in the plate: 1, 2, 3. After he ate one his mom asked, “How many are there now?” “Three!” he replied. Mom asked how could that be, if he ate one? He said that he ate “1”, but not “2” or “3”.
This boy understood that there’s a word for each cookie, but he didn’t understand cardinality. The words for each cookie change when the quantity changes.

Conservation of number is the understanding that the number of objects remains the same when they are rearranged spatially. Piaget proposed that number conservation develops when the child reaches the stage of Concrete Operations at around 7 years of age. Around this time, children also develop an understanding of other forms of conservation (e.g., weight, mass). However, number conservation is often the first form of conservation to develop. Before the stage of Concrete Operations, children may believe that the number of objects can increase or decrease when they are moved around.

The next two principles are about what can be counted and applying counting.
The Abstraction Principle
Counting can be applied to any collection − real or imagined
Adults realise that they may count physical or non- physical entities, objects that are not present, or even ideas. Young children on the other hand count physically present items and they group things in accordance with how they see their immediate relationships. Variations in material properties or position may affect their view as to whether an item should be included in a count. What children of different ages or stages of development conceive as allowable within a counting sequence raises important considerations.

  • What they might think about including or leaving out of the count on grounds of physical properties, position etc.
  • Are they able to understand they can count objects they cannot see.
  • Use counting to determine and compare the equivalence of sets.
  • Can they count events as they happen, and events that occur elsewhere.
  • Can they count ideas.

The Order Irrelevance Principle
The order in which items are counted is irrelevant, the same cardinal value will be reached
This principle requires knowledge about the previous four principles grasping it entails understanding that:

  1. Each counted item is still a "thing" not a "one" or "two" etc.
  2. The name tags were temporarily given and do not necessarily adhere to the objects once the counting is finished.
  3. Whatever order the objects are counted in the same cardinal result occurs.

It is necessary to grasp this principle of abstraction in order to be able to generalise the use of counting as a tool.

The Counting Principles
How to count
  1. The One to One Principle
  2. The Stable Order Principle
  3. The Cardinal Principle
Applying counting
  1. The Abstraction Principle
  2. The Order Irrelevance Principle

Counting skills

•Knowing the number names in order

•Synchronising saying words and pointing or moving objects

•Keeping track of objects counted

•Recognising that the number associated with the last object touched is the total number of objects

•Recognising small numbers of objects without counting them

•Counting things you cannot move or touch or see, or objects that move around

•Counting objects of very different sizes

•Recognising that if a group of objects already counted is re-arranged then the number of objects stays the same

•Recognising that if objects are added or removed the number of objects changes

Summary

Looking back over the skills and processes described, it is easy to regard the course of learning as a linear sequence where skills develop in a specific order and we should teach one skill at once. However such a view is too simple, children have surprising abilities at ages much earlier than a rigid view of the sequence would lead us to expect for example:

  • There is evidence that children have appreciation of quantities at very early ages. They are sensitive that three objects are more than two as early as four months. (Starkey et al 1983) i.e. long before they count or even itemise.
  • Infants have an intuitive perception that addition increases and taking away decreases, they naturally use their body parts for keeping tally, and supporting their own systems of counting small groups (Geary 1994).
  • Vygotsky (1978) described children's spontaneous use of mathematical ideas and processes as they related directly to their own lives, e.g. using numbers which had personal relevance, or inventing their own ways of writing numbers

The stages will overlap, depending on the number of objects. A child who can count to 5 might use counting when there are only a few objects, one-to-one correspondence when there are more than a few, and global quantification when there are many objects.

Many people assume that children do not learn to add or subtract until they can count, but counting itself is a process of addition, and counting backwards is subtraction.