Digging Deeper into … Counting and Numeration
Category : Uncategorized
Counting and numeration are listed as strand units in the strand of number for Junior Infants, Senior Infants, First and Second Class in the Primary Maths Curriculum (1999) and counting and numeration at each of these classes require similar skills, although the range of numbers will differ. However, while counting and numeration is specified as strand units only in infants to second, the understanding required is just as relevant and as important in the higher classes e.g. counting with larger numbers, counting fractions, decimals, percentages, etc.
Learning to count: rote versus rational counting
You are probably all familiar with the scenario: a parent declares that their pre-school age child can count because they can rattle off numbers to ten! As we all know, counting involves much more that just listing off numbers (rote counting). Watch this one minute video, which synopsises the difference between rote and rational counting.
While rote counting is relevant when learning to count, to count with understanding (i.e. rational counting) depends on the child developing an appreciation of rational counting, via the five counting principles, (briefly outlined in the video above); each of these counting principles are explained further in these follow-on videos from Origo Education:
- Teaching the Stable-Order Counting Principle
- Teaching the Cardinal Counting Principle
- Teaching the Order-Irrelevance Counting Principle
Apart from rattling off numbers, a child’s main interest in counting is to identify the quantity of objects in a set. “How many cars do you have? I have six cars”. Cardinality is using counting to find out “how many”. And, since most of the sets that children will encounter, and will want to count, will be randomly arranged, then teaching the order-irrelevance principle will probably be most relevant to the children themselves. Therefore, the children must develop some strategies to ensure that they count every object, once only:
- Count and tag: as each item is counted it is touched (this works quite well if the set to be counted is already in a line, or a rectangular array, but doesn’t work as well with scattered sets .
- Count and push/put: as each item is counted it is pushed to the side or put into a pot, tray etc.
- Count and mark: put a mark beside each item as it is counted; this works well for pictorial representations that cannot be physically moved.
- Count and group: in the case of large collections (for example in first and second classes), rearrange the objects into “friendly” groups (eg two, tens or fives) that the children can easily skip-count. Using the Operation Maths frames and structures to help to reorganise the objects can be of particular benefit.
This ability to demonstrate one-to-one counting should not be taken for granted; while it seems quite a simple concept, many children can struggle. Therefore, when the focus is on the cardinality of counting (establishing how many), all counting activities should be counting something; lining toys up and counting how many by tagging each one, etc.
When observing children as they count, check:
- Do they “tag” each object as they count (eg pushing them aside)?
- Can they count regular arrays or rows?
- Can they count random groups in some sort of systematic way so that they don’t miss or double up on objects?
- Can they count the same set several times, starting with a different object each time?
- Can they show how rearranging the objects does not change the quantity?
Counting without counting!
When can you count without counting? When you subitise! Subitising is the ability to recognise a quantity at a glance, without counting. When you throw a five on a die, usually it is not necessary to count the individuals dots; we recognise that there are five dots from their shape. So, while it is very important that we spend significant time practising one-to one counting initially, this is not the most efficient approach, and we do want the children to progress to a point where they do not need to count each item/object individually.
Ways to promote subitising:
- Play lots of dice and domino games; the Operation Maths TRBs have game suggestions and station activities in every STP plan, many of which are based around dice etc.
- Use the Operation Maths frames: the visual layout of various numbers in the frames (see image below) encourages the children to internalise a picture of how the numbers look and to recognise this in other situations.
- Play dot flash: briefly show the children dot cards in various arrangements and ask them to tell you what they saw. There are photocopiable dot cards at the back of the Operation Maths TRBs for this purpose.
- Use other structures that have a definite layout eg rekenreks (or maths rack) can also be used. This visual structure features quite strongly in the Number Talks presentations for junior infants, senior infants and first class, all available at the link above.
- Arrangements of Base Ten blocks, bundled sticks and/or place value discs can also be used.
- Use online games (eg Number Flash from Fuel the Brain) and/or suitable apps (such as this free one)
Numeration involves the children being able to match a numeral and its matching number word to each other and to various different arrangements of objects (both identical and non-identical) of that amount eg 3 = three = 🏀⚾⚽ = 🚗🚗🚗.
As the children move into first and second classes, numeration will move beyond the numbers to ten, through the teen numbers and all the way up to 199. Numeration in these classes involves much more than just matching a quantity to the numeral and to the number word:
- The children need to appreciate the visual pattern of numbers in sequence: 20, 21, 22, 23, 24, 25, 26…
- The children need to recognise the patterns in the number word sequence when spoken: “twenty one, twenty two, twenty three, twenty four…”
- From this understanding the children should be able to count forwards and backwards from various starting points. They should also be able to identify the number before or after a given number.
Visual structures, such as the Operation Maths 100 Square e-Manipulative (see below), can be very useful, as:
- they provide the numbers in order
- the patterns can be easily identified
- individual squares and/or large sections can be hidden and then revealed for the children to test their ability to identify preceding and subsequent numbers in a sequence.
- Principles of counting
- Subitizing: What Is It? Why Teach It? By Douglas H. Clements
- The Power of Subitising by Christina Tondevold, The Recovering Traditionalist
- Aspects of Cardinal Value, an article from the Primary and Early Years magazine of the NCETM (National Centre for Excellence in the Teaching of Mathematics
- A Sense of ‘ten’ and Place Value from nrich.maths.org
- Maths strategies you can count on: a blog post with teaching ideas
- Ideas for teaching counting from the Early Math Collaborative
- Movement Counts: a brief video showing how to reinforce counting via physical movement
- Counting Activities from the PDST
- Two other informative blog-posts are Count on Me and Show Me.