# Dear Family, your Operation Maths guide to Counting and Numeration

## Dear Family, your Operation Maths guide to Counting and Numeration

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Dear Family, given below is a brief guide to understanding the topic of counting and numeration, as well as some practical suggestions as to how you might support your children’s understanding at home. Also below, are a series of links to digital resources that will help both the children, and you, learn more about counting and numeration. The digital resources are organised according to approximate class level:

#### Understanding Counting and Numeration

Counting and numeration is about the counting words we use to tell the amount in a group, and the numbers we write for those counting words. And counting is not just about chanting a series of numbers …’one, two, three, four, five…’. It is about using these numbers with meaning, for example, understanding that the word ‘five’ can be written as 5, that it comes after four and comes before six, and that it can be used to describe the amount in a group of five items (and not just the label for the last item in the count). So, even though your child may know how to count to 10 or 20 or more, by the time they come to primary school, this does not necessarily mean that they understand the meaning of each number, or its place in the counting sequence. This is often described as the difference between rote counting (chanting a sequence of numbers) and rational counting (counting with understanding); for more on this, please check out the one minute video below.

Counting and numeration is a strand unit in Primary Maths for children in junior infants to second class only. Children in the senior classes will still do a lot of counting and numeration activities, but mainly as part of the strand unit place value. In the younger classes, the type of learning activities are very similar at every class level; the main difference is that each class level will have different number limits. In school, we expect that most children should be able to read, write and use numbers:

• 0-5 by the end of junior infants, and be able to count to 10
• 0-10 by the end of senior infants, and be able to count to 20
• Up to 99 by the end of first class
• Up to 199 by the end of second class

That is not to say that you should limit your child to only counting up to the number limit for his/her class level. In reality, children will encounter much larger numbers in the real world, than they will encounter in their maths books, so feel free to include larger numbers when you meet them. But, bear in mind that, even if a child can read or say a large number, it doesn’t necessarily mean that they understand it.

#### Practical Suggestions for Supporting Children

• Count, count and count some more! Count out the plates at the table, count out sweets or treats, count steps as you go up and down stairs, count down the days left to a birthday; use every opportunity for your child to hear you count, and when ready involve them in the counting.
• Say rhymes and sing songs that involve numbers or counting, for example, One, Two Buckle my Shoe, Five Fat Sausages, Ten Green Bottles, etc.
• Watch Numberblocks and Numberjacks. Many of the episodes from these two award-wining series from the BBC are available on-line and may also be available on your TV if you have BBC.
• Develop counting skills though play:
• Play tea-time with the toys, where each toy gets one cup, one plate, one bun etc. No toy should have more than one, and no toy should have none. These activities help to reinforce the one-to-one correspondence required in counting correctly.
• Play counting games at home (for example throw items into a basket/box and count them as you throw) and ordering games, for example where you layout playing cards in order. Or guessing games, where you estimate (guess carefully) the number of items in a container, bag etc., and then count to check.
• Play board games where the child has to throw a dice, recognise the number of dots shown, and then move on a counter that number of places.
• Play games where each number in a sequence (e.g. 1 to 10; 45 to 55; 103 to 113) is written on piece of paper/card and placed face-down. The child must turn over every piece in turn and read aloud the number. Then, he/she should put the numbers in a line in order. Finally, you could play hide and seek: remove a number from the line and your child has to tell you the missing number.
• Play counting games on car journeys, e.g. each child in the car picks a colour and counts every car of that colour that they see or meet on the road. The winner is the person who hits the highest number before the driver’s patience wears out!
• When your child starts to write numbers, you will need to monitor their number formation very carefully; it it very important that they don’t get into the habit of writing a number incorrectly.
• Draw your child’s attention to numbers around your home and in the wider environment, e.g. numbers on signposts, car registrations, phone numbers, the number of pieces in a jigsaw, page numbers on catalogues, the numbers on houses or hotel rooms. When you spot a number, ask them to read it out.
• With older children, when you are talking about numbers be careful to use the correct language e.g. for 125 say ‘one hundred and twenty five’ not ‘one-two-five’
• It’s an unfortunate convention, but the way we talk about numbers every day, can often be mathematically incorrect and/or confusing. For example, when calling out a mobile number, that starts with 08….. we will likely say ‘oh eight‘…… Yet 0 is a digit called zero, whereas O (said as ‘oh’) is a letter of the alphabet and not a number at all! So, when verbalising numbers with zero, try to get into the habit of saying ‘zero’ instead of ‘oh’.
• Numbers that end in ‘-teen’ or ‘-ty’ can be difficult for some children. In particular, some children can have difficulty hearing the difference between numbers ending in ‘-teen’ and ‘-ty’ when they are spoken out loud, e.g. ‘fifty’ (50) sounds very like ‘fifteen’ (15) when spoken, yet their values are very different. Try to say these type of numbers clearly, and encourage your child to say them clearly also, so that they appreciate the difference between these similar-sounding numbers.
• For more help and tips, check out this parents’ resource Topmarks: Learning Numbers

#### Digital Resources for Junior and Senior Infants

Underwater Counting: Count the underwater sea creatures and choose the matching numeral. Has different levels: numbers up to 5, up to 10.

Teddy numbers: Learn to count by giving Teddy the correct number of buns. Has different levels: numbers up to 5, up to 10 and up to 15.

The Gingerbread Man Game: Counting, matching and ordering games with options for numbers up to 5, and up to 10.

Ladybird Spots: Counting, matching and ordering games with options for numbers up to 5, and up to 10.

Toys Counting Game: Place the correct number of toys on the shelf. Counting to 5, 10 and using number words.

Curious George Hide and Seek: Find the number word, the numeral and the matching number of creatures. Numbers up to 10.

Curious George Apple Picking: Pick the number that is missing from he sequence. Numbers up to 9.

Count the Yeti 1-10: Count the number of yetis and shoot the correct number at the top.

Helicopter Rescue: Find on the number path the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers up to 10, up to 20, up to 30, up to 50 and up to 100.

Caterpillar Count: Count and collect the numbers in order up to 15, to watch the caterpillar change into a butterfly.

Treasure Hunt: Help the pirate find his lost treasure by clicking on the island that shows the correct number. Select ‘Find the biggest number’ option and then adjust to set the maximum number.

Chopper Squad: Find a number 1 more/less or 10 more/less than a given number. Has different levels: numbers up to 20, up to 30, up to 50 and up to 100.

Blast Off: In the Find a Number game (red labels) you are asked to find, from 3 options, the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers 10-20,  10-30, 30-60, and 60 to 99.

Caterpillar Ordering: Choose between ordering (where you put the given numbers in order) or sequencing (where you complete the sequence with the correct numbers from those given).  Has various levels including 1-5, 1-10 and 1-20.

Count and match: Count the items and drag over the matching numeral (up to 10)

Number Matcher: Find the matching number and number word.

Happy Numbers Pre-Kindergarten: Work through the activities from Module 1, counting to 5 and/or Module 3, counting to 10. Alternatively, go to Kindergarten, Module 1, numbers to 10

I Know it! – Counting: Scroll to numbers + counting  + place value to do any of the activities.

Splash Learn – Counting Games: An assortment of games organised according to US grade levels; junior and senior infants should choose among the games for kindergarten level.

Counting: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

#### Digital Resources for First and Second Classes

Please note: The digital resources for first and second classes often overlap with the place value digital resources for these classes, as the skills are very related.

Maths Visuals – Counting by one: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Counting above 100: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Maths Visuals – Place Value Concepts: Watch any of the videos and count out loud the numbers and images that are shown. Do you spot any patterns?

Helicopter Rescue: Find on the number path the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers up to 10, up to 20, up to 30, up to 50 and up to 100.

Chopper Squad: Find a number 1 more/less or 10 more/less than a given number. Has different levels: numbers up to 20, up to 30, up to 50 and up to 100.

Blast Off: In the Find a Number game (red labels) you are asked to find, from 3 options, the direct number that the computer asks, or find the number in between two given numbers. Has different levels: numbers 10-20,  10-30, 30-60, and 60 to 99.

Treasure Hunt: Help the pirate find his lost treasure by clicking on the island that shows the correct number. Select ‘Find the biggest number’ option and then adjust to set the maximum number.

Caterpillar Ordering: Choose between ordering (where you put the given numbers in order) or sequencing (where you complete the sequence with the correct numbers from those given).  Has various levels including 1-100.

Coconut Ordering: Hit the numbers in order of size. Select ‘numbers’ and then choose from numbers up to 10, up to 20, up to 100 (in tens) or up to 100 (any number).

Battleship Numberline: Can you blow up the enemy submarines? This game starts very easy, where you must click the correct number on the number line, but then the game progresses in difficulty as the player must work out where a given number would be placed on the blank number line. Choose the whole number game.

Happy Numbers Kindergarten: First class could explore the activities from Module 5, Numbers 10 – 20 and Counting to 100.

I Know it! – Counting: Scroll to counting and number patterns to do any of the activities.

Splash Learn – Counting Games: An assortment of place value games organised according to US grade levels; first class should choose from among the games for first grade, and second class should choose from the games for first and second grade.

Counting: a selection of games from ixl.com. You can do a number of free quizzes each day without having a subscription. (Please note that the class levels given do not always align accurately with the content of the Irish Primary Curriculum)

## Digging Deeper into … Counting and Numeration

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For practical suggestions for families, and links to useful digital resources, to support children learning about the topic of counting and numeration, please check out the following post: Dear Family, your Operation Maths Guide to Counting and Numeration.

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:

HINT: For more information on the Counting Principles, including suggestions on what to look out for and what to ask/do, check out this blog post.

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?

HINT: use relevant number rhymes and stories to reinforce counting and number word sequence. Many of the short-term plans (STPs) in the Operation Maths TRBs list various possibilities; see the Literacy suggestions in section on Integration

### 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)

HINT: For more suggested subitising activities read this blog post Counting With Your Eyes

### Numeration

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.

HINT: Particular attention should be given to the multiples of ten ie the “ty” numbers and a deliberate distinction should be made between the “ty” numbers and the “teen” numbers, especially when being verbalised i.e. there is little difference verbally between eighteen and eighty, but there is a significant difference between these numbers in value . Like the “teen” numbers, “ty” numbers are also widely acknowledged as common hurdles for children and so time spent now will be time well spent for the future.

This is part of the series “Digging Deeper into …” which takes a more in-depth look at the various topics in primary maths. To ensure you don’t miss out on any future posts, please subscribe to the blog via email, on the top right hand of this page.