# Linking Conceptual Subitising to Mental Computation

When children are first shown two groups to combine they often ‘count all.’ If they are not presented with suitable activities ‘counting all’ might become the default strategy for later computational tasks. Being able to subitise supports later mathematical abilities. It is important that children are supported in developing their ability to subitise as not all are able to, especially those children who may suffer from dyscalculia

### Developing Conceptual Subitising

Once children are able to perceptually subitise standard dot configurations to six (those on die) they can be supported to develop the ability to ‘count on’ then subitising two or more groups.

### Activities using Standard Dot Cards

#### Begin with ‘Counting On’ Skills

• Provide students  with a 2 sets of standard dot cards 1-6.
• Use a set of dot cards 1-6 and an extra 1, 2 and 3. The extra cards will be used to practise ‘counting on.’
• Begin with two cards, for example a 5 and a 2. Identify the cards with the students using perceptual subitising.
• Turn over the larger card and leave the ‘count on’ card face up. Model how to ‘count on.’

“We know that the face down card is showing 5 dots. Let’s count on from 5 to find out how                          many dots altogether. (Touch the face down card saying) 5 then count on 6, 7. There are 7 dots                  altogether.”

• Repeat the process with two more cards.
• Make sure when you check how many that you do not count the dots on the face down card. It should be ‘seen’ by subitising.
• Have students work in pairs, one choosing the larger number 4, 5 or 6 and the other the ‘counting on’ card 1, 2 or 3.

#### What’s the Difference

• Begin with two cards. One card is 4, 5 or 6. The other card is 1, 2 or 3.e
• Place the largest card face up, while the other card stays face down.
• Say the total of the two cards and ask what might be on the hidden card. Encourage students to ‘cont on’ to work out the difference.
• Turn the hidden card over and check the answer by ‘counting on.’
• With a bit of practise, students could play with a partner.

#### More or Less

• Provide students with cards 1 to 6.
• Lay the cards out in front of each child.
• Ask, “Which card is one more than ______? Which card is two less than _______?
• Lastly, have students generate their own questions.

### Activities Using Non-Standard Dot Cards

Here are a few ‘oldies’ to use along with the activities for standard dot cards. If you need the instructions just click on the name of the game as it has a link to a version of the instructions.

#### Snap

If you haven’t already discovered Natural Maths, Ann and Johnny Baker have produced two resources, Subitizing: Laying the Foundations and Conceptual Subitizing: Laying the Foundations for Mental Computation. There is a purchase cost for both books but I would highly recommend them. The later focuses on using subitising as a springboard for counting on, doubling and facts to ten knowledge.

Yet, subitising doesn’t only assist addition and subtraction facts. It can also support the ability for student’s to develop visual images of multiplication facts. Graham Fletcher provides an explanation on his blog.

Often, teachers do not fully appreciate the need for students to be able to subitise. Yet, it is a vital component in developing their ability to mentally computate.

Until next time,

Carole

# Subitising – Moving from Perceptual to Conceptual

Conceptual subitising is the ability to recognise a whole quantity as the result of recognising smaller quantities. Sayers and others (2016) suggest that conceptual subitising may lead to the later development of arithmetic as it leads to decomposition or partitioning of numbers, the commutativity of addition, and part-whole knowledge. Sayers article is below and is an interesting and easy read.

Conceptual subitising builds upon the ability to subitise perceptually. The previous post explains perceptual subitising and provides activities to support students.

Once children are competent with perceptual subitising they should be gradually introduced to conceptual subitising by adding 1 or 2 more to a standard pattern, as on a die.

Maths Coach’s Corner blog about Subitizing: Moving from Perceptual to Conceptual here.

### What Does the Research Tells Us?

The Role Of Conceptual Subitising in the Development of Foundational Number Sense by Judy Sayers, Paul Andrews, and Lisa Björklund Boistrup, Stockholm University (2016)

Subitising Through the Years by Valerie Faulkner,  North Carolina State University, USA and Jennifer Ainslie, Wake County Public Schools, USA (2017)

Early Spatial Thinking and the Development of Number Sense by Janette Bobis, Australian Primary Mathematics Classroom 13 (1) 2008

### Different Ways to Subitise

Video from CE Kindergarten talks about subitising using dot patterns (familiar and unfamiliar), tens frames and Rekenreks.

### Using Number Talks to Teach Addition and Subtraction

Not sure what a Number Talk is? The Brown Bag Teacher provides a quick explanation here.

Quick Images from the Teaching Channel

Video of a Number Talk by Jo Boaler

### Activities for Conceptual Subitising

John Van de Walle talks about subitising in his book Teaching Student-Centered Mathematics: Grades K-3.  He provides blackline masters of dot patterns that can be used with your students.  Click here for the blackline masters.  Maths Coach’s Corner has enlarged versions of Van de Walle’s cards so they would be easier to use, and you can grab them here.

Find It – Add and Count from Mathematical Thinking

Find It – Bingo from Mathematical Thinking

Subitising Freebie from Natalie Lee Kindergarten

Dog House Dots from A Blog from the Pond

Early Years Focus: 10 Subitising Activities by James A. Russo (Monash University, Melbourne)

Domino Maths Games from Fun Games 4 Learning

Gone Dotty‘ Dominoes – a non-standard version of dominoes

### Conceptual Subitising Flash Cards on the Whiteboard

Ten Frame Fun Flash

Fun Flash to 20

### Interactive Whiteboard Resources

Illuminations offers interactive resources for both five and ten frames. Students can use these to answer ‘How Many?, Build, Fill and Add.’

### iPad Apps (These are great resources but, unfortunately, not free)

Subitising Flash Cards from Little Monkeys

Subitize Tree from Doodle Smith Ink

Quick Images 11 from Tom Patten

Number Flash from Mitchell Mark

Conceptual subitising is one of the most important foundational maths skills children need to support them in developing mental images for addition and subtraction. Its development assists students in moving away from ‘counting on’ and ‘counting back’ and from using their fingers as support.

It’s important children receive lots of repetition and visualising it in different ways. For students who are having difficulty, stick to the traditional arrangements, gradually introducing non-traditional and ensuring the new arrangements are explicitly taught.

There are lots of resources out there. Fortunately all the resources, except for the iPad apps, are free!!

Until next time,

Carole

# Subitising

Subitising refers to being instantly and automatically able to recognise small numerosities without having to count (Clements, 1999; Jung et al., 2013; Moeller et al., 2009; Clements & Sarama, 2009).

‘Some suggest that subitising may well be the developmental prerequisite skill necessary to learn counting. If so, we should examine subitising more closely and determine if reinforcing this skill in children will help them learn counting easier’ (Sousa, 2008). There are also ‘strong suggestions that all later mathematics is built on the ability to subitise (Baker, 2015).

Clements (1999) describes two types of subitising: perceptual and conceptual.

### Perceptual Subitising

Perceptual subitising involves recognising a number without using other mathematical processes. It assists children to separate collections of objects into single units and connect each unit with only one number word, thus developing the process of counting.

### Conceptual Subitising

Conceptual subitising allows one to know the number of a collection be recognising a familiar pattern, such as the spatial arrangement of dots on the faces of dice or on domino tiles. Other patterns may be kinesthetic, such as using finger patterns to figure out addition problems, or rhythmic patterns, such as gesturing out one “beat” with each count.

### What Does Research Tell Us?

Subitizing: What Is It? Why Teach It? Douglas H. Clements (1999)

A seminal article which is often referred to!!

Subitizing: A Critical Early Math Skill Subitizing Paula Hartman, Myoungwhon Jung and Greg Conderman, Northern Illinois University (2012)

An easy to read article that contains activities and apps.

Again, an easy to read article that contains activities to support children in learning to subitize.

### Activities to Support Perceptual Subitising

Dice games – simply playing board games with a 6-sided subitised die will assist children in developing perceptual subitising.

Making Patterns – show a quick image or flash card of a subitised pattern. Children then use counters or coloured glass stones to replicate the pattern. This activity assists children to visualise patterns and embed the image in their minds.

PowerPoint of Quick Images

Display each dot pattern for only 3 seconds. Click ‘enter’ to go to a blue screen then click again to reveal the answer.

Domino: Games with Dots from Lessons Learnt Journal

Printable Dominoes from First-School.ws

Printable Dominoes from Helping With Math

Subitizing with Dot Plates from Mathematics for the Curious Pre-K-K  Click on the owl to download

Flower and Flower Pot Match from Lovely Commotion

1-6 Playing Cards for Go Fish and Concentration

Monster Dice Match from The Measured Mom

Ice Cream Count and Match from The Measured Mom

– yes, there is a cost involved but this book by Ann and Johnny Baker provides activities and games, problematized situations and mental routines. The mental routine includes closed, open and flipped questions, that encourage a deeper level of thinking.  great resource!!

Until next time,

Carole

# Number Recognition

Numeral Recognition involves a variety of skills including:

• Numeral identification (recognising all 10 numerals from 0 through 9)
• Knowing each numeral’s name
• Tagging a numeral to a quantity
• Correctly writing the numerals

## Numeral Identification

At its most basic level, numeral identification is a form of shape recognition, which can result in a simple association of the word “two” with the symbol ‘2’ without a cardinal meaning (Mix, Sandhofer, & Baroody, 2005). This means that numeral identification can develop at a different rate to number knowledge.

Visual discrimination, or distinguishing a numeral by sight, is an important part of developing numeral recognition. Some numerals have a similar appearance, like 6 and 9, 1 and 7, or 2 and 5 are often confused or written backwards. Children need to be supported to identify and read them in their everyday environment and provided with numerous visual and tactile experiences.

Learning to identify, recognise and write numerals is an important part of early arithmetical development. When a young child learns the name of a numeral it sows the idea that a symbol can stand for a whole word (Mix, Huttenlocher, & Levine, 2002).

When children acquire the skills of identifying numerals, they are ready for the next step, which is understanding the amount each numeral represents.

### Practice

Colouring activity from The Measured Mom

• Point out and name numbers on street signs, houses and buildings while you are out and about.
• Find numbers around the house on appliances, telephones, remote controls (remove the batteries and let him play), clocks and thermometers.
• Give your child a calculator and let them to play with the numbers. Ask if they can find the number that shows how old they are and show them the number that shows how old you are. Have them type in the numbers 0-10 in order.
• Purchase a package or two of magnetic numbers. Allow your child to match up pairs of the same number and put the numbers in order. Take a cookie sheet and numbers in the car for on-the-go learning.

Number Line Counting from Learning 4 Kids

Farm Fence Number Ordering Mats from Fairy Poppins

Count to 20 Song from ABC Education

Numbers Car Park Game from B-Inspired Mama

Bingo 1 to 5

Bottle Top Count and Match from The Imagination Tree

Counting and Number Recognition from Learning 4 Kids

### Online Games

Bingo from ABCya.com

Numbers to 10 Balloon Pop from education.com

Numbers 11 to 20 Balloon Pop from education.com

Numbers 1 to 10 Matching from education.com

Number Demolition 1 – 10 from education.com

Number Demolition 11 – 20 from education.com

## Tagging a Number to a Quantity

Assigning a number/numeral to a quantity can be a difficult concept for many children as a group of objects is tagged to one or two digits. In other words, the one-to-one relationship does not exist.

This a particular difficulty for children who have the learning difficulty known as dyscalculia.

### Practice

Apple Number Quantity Activity from Jady A

Dinosaur Count and Clip 1-10

Numbers 1 to 5 Cut and Paste

Numbers 1 to 10 Cut and Paste

Numbers 1 to 20 Cut and Paste

Number Recognition 4-in-a-Row

### Online Games

Hungry Spider from Fuel the Brain

Number Pictures Matching from education.com

Number Trains: numbers 1-10 from FUSE

Number Trains: numbers 1-20 from FUSE

Underwater Counting from Topmarks UK

## Correctly Writing the Numerals

I’m sure we’ve all experienced children who write their numerals backwards, particularly, 2 and 5 or 1 and 7. It is important that they are provided with numerous visual and tactile experiences to help develop their muscle memory. Once bad habits are established it’s often difficult for children to change.

Providing resources can be difficult as different education systems use different fonts. I have simply offered a range of free resources hoping that there will be something to suit your purpose.

### Practice

When using these resources, think a little more broadly as to how you can extend the visual and tactile aspects. Some ideas include:

Writing the numerals in the air. I usually use a rhyme here so children also have a verbal connection. I’ve provided some rhymes further down. Hopefully there’ll be one that suits your purpose.

Writing the numeral on the back of someone else. Again, using a rhyme.

Rainbow Writing Write the numeral using one colour, then change colour and repeat. I usually do this between 5 and 10 times and encourage children to say the rhyme as they write.

Playdough Numbers Use playdough and pre-made templates to create numbers. This is my favourite playdough recipe, no saucepans but you do need boiling water!!

Bumpy Writing Similar to Rainbow Writing but write on a plastic grid or something with a similar texture. I’ve also purchased a plastic place-mat from my local \$2 Shop. Much cheaper than the plastic grid. Once made, the Bumpy Writing sheet can be used for children to trace over with their finger.

Sand Trays I really like this idea from Mama Miss.

A similar one –  a Montessori version (Don’t use salt just in case a child has a cut or a scratch).

Sandpaper Numerals

Sandpaper Numerals for tracing (Be careful of the coarseness of the sandpaper as some children may not like the ‘feel’.) Use one of the numeral outlines, provided below.

Victorian Modern Cursive Numbers 0-9 Templates

Playdough and Number Outlines from First Grade Garden. Look at Station 6.

Number Outlines from Stimulating Learning

Number Tracing Cards from The Measured Mom

### Numeral Formation Rhymes

Chant and Write from Rourke Classroom

Number Rhyme Cards from Communication 4 All

Writing Numbers Rhyme from K-3 Teacher Resources

From ABC123tv

### Handwriting Practice Sheets

Writing Numbers: Free Pages for 1-20 from This Reading Mama

Preschool Handwriting Practice for numbers 0-9 from The Measured Mom

Number Formation Teaching Resources from Sparklebox

### Other Ways to Practice

Printable Lacing Cards: Numbers 0 – 10 from Recentlessly Fun, Deceptively Educational

Race Car Highway Numbers from Make Learning Fun Use small cars on the tracks. You will need to make sure the children are correctly forming the numbers.

Hopefully you’ve found a resource that will be beneficial for your children.

Until next time,

Carole

# Abstraction Principle – Everything and Anything Can be Counted

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

# Order-Irrelevance Principle – The Order of the Count Doesn’t Matter

The order-irrelevance principle refers to the understanding that the order in which objects are counted is not important. It doesn’t matter whether the counting procedure is carried out from left to right, from right to left or from somewhere else, so long as every item in the collection is counted once and only once.

For example:
In a collection of three buttons there may be a red, green and blue one. I may start counting with a blue button the first time but the next time I may begin with the red button. Whichever button I begin, with I will always arrive at the same number.

This seems like a simple concept to understand but children often have to recount the set if you ask them to begin counting from a different object.

## Difficulties

• When counting, some children may assume that the number they tag an object with stays with that object and is not ‘temporary’
For example:
If we go back to the previous button example. Children may start with counting the blue button and         tag or label it as ‘one’. Some children will insist that it is ‘one’ and that’s where they must start                   counting from.

Source: Origin One

Recent research by LeFevre and others, 2008, indicates that from Foundation through to about Year 2 the application of conventional counting rules (left to right and top to bottom) was related to higher numeration skill. At this age, the order-irrelevance principle appeared to impact negatively on children’s working memory and therefore, their ability to count proficiently.

However, from Year 3 onward, children’s numeration knowledge was unrelated to whether they had acquired order-irrelevance. Although the order-irrelevance principle is an important understanding for children to have, some children do not fully develop this understanding until they are around eleven years of age.

The research suggested that only when counting becomes automatic might there be the opportunity for children to accept that it is not necessary to count a set in a strict order.

Thus, for children who are still developing their counting skills, the principle of order irrelevance
might be logical, but it is not necessarily practical.

## Practice

Have children count objects in their every day environment in a variety of different ways. Ensuring that they touch the object as they do so.
Constant exposure to counting helps develop this skill as well as making a game out of it by “mixing up” objects in a set to see if the numbers change.

•  Count left to right and right to left with the same row of objects
•  Count the same set but starting with a different “one” each time, for example, the middle object
•  For a greater challenge, ask the children to count all the objects, making the middle object ‘5’
•  Re-arrange the objects so they are no longer in a row and ask the child to count from a particular object
• Ensure that sets for counting contain, not only sets with the same objects but also sets that have miscellaneous objects without any commonality.

Until next time,

Carole

# Cardinality – Giving Meaning to Numbers

Cardinality is the ability to understand that the last number which was counted when counting a set of objects is a direct representation of the total in that group.

Children will first learn to count by matching number words with objects   (1-to-1 correspondence) before they understand that the last number stated in a count indicates the amount of the set.

A child who understands this concept will count a set once and not need to count it again. They will automatically remember and know how many are represented.

Source: Origo One

Students who are still developing this skill need constant repetition of counting and explicit teaching through modelling so they understand they do not need to count over and over again when it will result in the same number. Students who have difficulty with their working memory may have difficulty with this concept.

So What Can You Do To Help!!

Research by Paliwal and Baroody (2017) stresses the importance of:

•  Labelling the total number of items then counting them (Label-first).  For example, on a page with 3 elephants, saying, “Look there are 3 elephants. Let’s count them.” And counted them as, “one, two,three.” or
• Counting the items, then emphasising and repeat the last word (Count-first). For example, on a page with 3 elephants, say, “One, two, three, t-h-r-e-e. There are three elephants.”
• Researchers indicate that the latter is the preferred method of modelling, suggesting that the first did make a difference compared to Counting Only, where the total number of items was not emphasised.

We can help children develop the understanding of cardinality by involving them in activities where they answer questions about ‘how many’. They need not only to be able to say the counting names in the correct order, but also to count a group of, for example, seven objects and say that there are seven.

This video from the Connecticut Office of Early Childhood provides examples of ways to develop cardinality in the classroom.

Activities

Counting Collections Activities should have some basis in reality, giving a purpose to counting. For example, create a need to count by involving children in food preparation. They will need to know how many people, plates or apples in order to complete the task.

How Many? Provide opportunities for students to count using a variety of objects such as buttons, counters, shells, coins, and dot cards. Objects can be put into jars, counted then draw and recorded.

Order Disorder Place objects to be counted in different arrangements. Firstly, perhaps, a straight line then, the same objects, in a circle then a random arrangement. Always asking children “How many?” If they need to recount the objects, they do not understand the concept of cardinality.

Show Me Provide children with a bag, box, or bucket of objects and ask them to count out a certain number of objects. For example, say, “Show me 5 buttons.” Once the child has counted out the required number of objects, again ask, How many?”

Bugged Out Children roll a number cube and put that many bugs into the jar. If they roll a fly-swatter, they have to remove a bug. If they roll the bug spray, they have to remove ALL of their bugs. The first person to get 10 bugs in their jar wins!! Printable

Count and Graph Worksheets here

Nature Scramble Engage children in activities in the school ground, beach or local park. Ask them to collect different numbers of object, for example, shells, rocks or leaves. Always referring to “How many?”

Rocket to 10 Printable Provides opportunities to talk to children about number and their thinking. Ask children, “How many cubes did you put in the rocket?” and “How many more do you need to fill the tower?”

Spot the goof from Parenting Science

Want to make your own sock puppet for Spot the goof?? The following videos may help. And neither require sewing!!

How Many Snails? a counting book by Paul Giganti Jr from the National Centre for Excellence in the Teaching of Mathematics.

Once a child has a sense of cardinality, then we can involve them in matching activities where a number word is matched to a quantity and the numeral that belongs to it.

Matching Activities (ensuring that they are still using concrete manipulatives)

Match It Provide children with opportunities to match numerals with the number of items in the set they have counted.

Count It Provide children with a numeral card and ask them to read the number. Children then count out that many items to represent the number.

Mouse Match and Thread Printable I recommend that you only use one colour of beads, otherwise children will make coloured patterns instead of thinking about the counting!!

Activities from In My World

Until next time,

Carole

# Stable-Order-Principle -Saying the Number Names in the Same Order Every Time

The stable-order-principle is one of the most basic principles of number and parents often think that this is the only concept a child needs to know.

It is the simple concept that the sequence for how we count always stays the same.

For example, it is always 1, 2, 3, 4, 5, 6, 7, 8, 9, etc.

NOT 1, 2, 5, 7, 3, 4, 6, 9, 8!

Activities that employ the stable-order-principle are most useful when they are simultaneously employed with the one-to-one principle. Children need to understand that one word is said as one object is touched or action is completed. To be able to count also means knowing that the list or sequence of words used must be in a repeatable order.

Source: Origin One

Number language is complicated as it involves rote learning of words that do not have a recognisable pattern.
Initially children may just be chanting words memorised through rhymes and stories with it not
having much meaning. Increasingly, the order of words takes meaning and children will begin to
realise that the order of counting words is always the same and must always be said in this order:
the stable order principle (Montague-Smith and Price, 2012).

## Difficulties

• For children who speak English, learning number words greater than ten is difficult (Fuson & Kwon,
1991; Miller & Stigler, 1987). This is because the number words for values up to the hundreds are often irregular and do not assist children by NOT relating to the base-10 number system.

• Many of these confusions are avoided in East Asian languages, because of a direct one-to-one relation between number words greater than ten and the underlying base-10 system (Fuson & Kwon, 1991; I. Miura, Kim, Chang, & Okamoto, 1988; I. Miura, Okamoto, Kim, Steere, & Fayol, 1993). The Chinese word for twelve is translated as “ten two.” Using ten two to represent 12 has two advantages. First, children do not need to memorise additional word tags, such as eleven and twelve. Second, the fact that twelve is composed of a single tens value and two units values is obvious.

### Teen Numbers

• Usually I talk to children about the numbers after ten, explaining that it would be easier if we said ‘ten and one’ rather than eleven or ‘two tens’ rather than twenty. Introducing the numbers in this way seems to develop some understanding of the structure of numbers.

• Some researchers have suggested that we introduce the numbers eleven, twelve, and the ‘teen’ numbers after they learn to count to 100. I understand why they suggest this but have never been able to reconcile as to how we would teach this way when we have groups of objects above ten!!!!

## Activities

Count Around the Circle  The teacher/adult sets what the children will count to. Children can say either 1, 2 or 3 numbers, for example, child 1 says, “1, 2”, the next child “3” and the next child, maybe “4, 5, 6”. It is up to the child. When the target number is reached that child sits down and the next child starts from 1 again.

Counting books, singing simple number songs, repetitive counting and consistent modelling help students develop this concept of number sense and correct errors that may occur.

Counting Dinosaurs from FUSE Education

online game

online game

Counting Videos

Until next time,

Carole

# One-to-One Correspondence – A Counting Fundamental

Children come to school with varied experiences related to counting. Even if young children can recite the number sequence it cannot be assumed that they can apply this knowledge to counting small sets of objects. Knowing the one-to-one correspondence principle is essential for organised, meaningful counting. This leads to an eventual ability to perform higher-level calculations (McCarthy, 2009).

Source: Origo One

Difficulties

One-to-one correspondence is often difficult for young children to comprehend. In Maths recognising the number “ten,” and being able to count out “ten” items are two separate skills. Linking objects with numbers enables a child to count with understanding (McCarthy, 2009).

Common errors when counting a set of items can be:

• Skipping an item
• Assigning more than one number word to a single item
• Pointing to two or more items while saying one number word (Clements and Sarama, 2014)

Ways to Develop One-to-One Correspondence

Role Play

• Setting the table – For each plate on the table the child needs to place one fork, one knife and one spoon
• Fruit ice-cubes – Chop pieces of fruit, for example, pineapple or strawberries. Place one piece of fruit into each space in an ice-cube tray. Add water or fruit juice. Freeze.
• Teddy Bear’s Picnic – Set out between 3 to 5 small chairs. Place one teddy bear on each chair.

Activities

• Rhythm Movements – Children count the number of claps an adult makes. This can be the number of beats on a drum, taps on a triangle. Children count aloud and aim for rhythm.
• Follow Me – Children make the number of movements given by an adult, for example, clap three times, hop three times, skip five times, nod six times. Children count aloud as the actions are done.
• Bean Drop – An adult drops dried beans into a container. As the beans are dropped the children need to count them.
• Jumping on the Lily Pads (From Young Mathematicians)

• Count Me – Place a group of objects (e.g.: shells, leaves, counters, teddies, boats, cars) on the table. Ask the student to count how many objects there are. Watch carefully and see if you can determine how the student decides how many objects there are (DET, accessed 11/6/2018)
• Activities From The Measured Mom
• Auditory Activities from https://mykidsturn.org/

Source: By the Numbers (ESSDACK)

• Hands-on and Auditory Activities from www.raepica.com

Game Boards

PreKinders is an amazing website with free resources, not just for Maths, but also fine motor skills, literacy, science and more.

• The following board game is easy to prepare, simple for children to understand and doesn’t require many resources.
• Roll and Collect is from Kate in Kinder  and sourced through Teachers Pay Teachers but as a free download. There are a few different versions. The game uses a 6-sided die. This might not give the children enough turns so might need to be adapted, showing the numbers 1, 2 and 3 on two sides each.

Online Games

Underwater Counting from Topmarks.co.uk

Counting Fish from abcya.com

My aim is to provide free resources, where possible, that support the academic research. If you find other resources or relevant information please contact me and I’ll include it on the blog.

Thank you to all the educators who supply free resources to support teachers and their children.

Until next time,

Carole

# Number Sense – What is it? Why teach it?’

Number Sense is ….

“… good intuition about numbers and their relationships.
It develops gradually as a result of exploring numbers, visually them in a variety of contexts, and relating them in ways that are not limited by traditional algorithms.” (Howden, 1989)

“Research indicates that early number Sense predicts school success more than other measures of cognition, such as verbal, spatial, or memory skills or reading ability.” (Watts, Greg, Duncan, Siegler, & Davis-Kean, 2014)

Jo Boaler, a professor at Stanford University explains it in this way:

https://www.youcubed.org/resources/what-is-number-sense/

Van de Walle and others (2018) suggest Four Early Numeracy Concepts and Four Number Sense Relationships necessary for the development of Number Sense, something we will continue to explore.

Early Number Concepts

1. Verbal Counting – to say the numerals in the correct order
2. Object Counting – 1 to 1 correspondence
3. Cardinality – the last object counted in a set tells how many
4. Subitising – the ability to see how many without having to count

Number Sense Relationships

1. Spatial relationships – having a visual to go with a number
2. One or Two More & Less – instantly knowing the amount that is one or two more & less
3. Benchmarks of 5 and 10 – knowing how a number relates to 5 and 10
4. Part-Part-Whole – understanding how a whole can be broken into parts

Counting

Counting is fundamental to later maths development. Early counting predicts later mathematical success (Clements and Sarama, 2014) and even later reading fluency (Koponen, Salmi, Eklund and Aro, 2013).

Many children begin school with the ability to differentiate between sets of certain ratios (in particular, 2:1 and 3:2) enabling them to tell the difference between sets just by looking at them. However, they are unable to tell the difference between sets that are close in number (for example, 10 and 9). It should be noted, that this does not apply to all children, in particular, those that may suffer from Dyscalculia. A term referring to a wide range of life-long learning disabilities involving maths. It includes all types of maths problems ranging from an inability to understand the meaning of numbers, to an inability to apply mathematical principles to solve problems.

Those who study children’s mathematical development explain that counting involves five principles (Gelman and Gallistel, 1978):

1. One-to-one correspondence
2. Stable number word order
3. Cardinality
4. Order irrelevance
5. Abstraction

Sound complicated? It is! Something we adults take for granted as “simple” is actually quite complex developmentally. Although we will explore each principle in future posts the following by the National Center on Intensive Intervention is a brief introduction.

Verbal Counting – Stable Number Word Order

As children engage with nursery rhymes, songs and role-play opportunities they often develop the order of number words. However, just because children are able to rote recite the number words it does not mean that they understand one-to-one correspondence nor the knowledge of differences in the magnitude or size of numbers (for example, knowing that 7 comes after 6 doesn’t mean that the child knows that 7 is more than 6).

Counting rhymes, books and videos are ways of supporting children in developing Verbal Counting. An enabled adult who is able to support and intervene when a child experiences difficulty is significant at this stage.

I usually add the counting rhymes to the classroom library and make activities (available in the counting rhymes section) so that students can retell the rhymes. This assists with developing the rhythm of counting.

What strategies/ideas do you use for Verbal Counting? Please feel free to comment.

Until next time,

Carole