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Year 2 of Programme - Supporting Early Mathematical Development (2023-2024)

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Year 2 of Programme - Supporting Early Mathematical Development (2023-2024)

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These templates are all set to print as one per page. To reduce the size select 'multiple' in the print options and ensure 'landscape' is selected.

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These templates are all set to print as one per page. To reduce the size select 'multiple' in the print options and ensure 'landscape' is selected.

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Supporting Learners with the Early Foundations of Counting - Session 4

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Realising the Ambition: Being Me builds upon the original principles and philosophy of Pre-Birth to 3 and Building the Ambition. The new guidance retains the relevant content from the previous guidance which it replaces, extending and strengthening it in line with current research and evidence about how children develop and learn.

This refreshed early years national practice guidance for Scotland presents key information about the characteristics of child development based on research and evidence.

In essence, 'Realising the Ambition: Being Me' increases expectations of high quality but still provides the necessary support for all who work in the early years sector and beyond.

Explore this resource

High quality provision consists of many different interconnected aspects. The imagery of tartan has been used in the guidance to remind us of the uniqueness of our Scottish context - with each section seen as an interconnected thread contributing to the whole piece. Although each section has been developed so it can be used in any order to meet different needs and different contexts, we must always be mindful of this bigger picture.

The terms of 'when I am a baby', 'when I am a toddler' and 'when I am a young child' have been used to illustrate developmental needs over time - but these should not be seen as rigid lines of progression. All babies and young children develop their individual skills, knowledge and attributes at different rates according to who they are. The guidance can be used to support children with more complex needs by taking this into account.

The guidance can be used individually, or in short informal group settings where practitioners have time to come together, or as part of a more substantial CPL programme. We would recommend that at points of transition between ELC and primary school, the guidance can be used by designated staff working together to promote a clear understanding of curriculum progression and continuity in children's learning and development.

Please note that the link to the Leuven scales on page 50 is no longer live. Alternative links are provided in the 'Related links' section below.

Challenge questions

The Challenge Questions PowerPoint contains suggested challenge questions for each section of Realising the ambition: Being me.

Webinars

Our live Realising the Ambition sessions have already been delivered to thousands of early years educators across Scotland. We now have a recorded version available to enable you to share or recap on the themes.

1. Realising the Ambition overview part 1 discusses the guiding principles and the importance of the imagery used in the guidance.

2. Realising the Ambition overview part 2 looks at pedagogy in practice and transitions.

3. Realising the Ambition Overview for Childminders was produced in partnership with the Scottish Childminding Association.

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Education Scotland is a Scottish Government executive agency responsible for supporting quality and improvement in Scottish education.

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Math Games | Educational Games for Kids | Toy Theater

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Fun & Free online math games for kids K-5, featuring addition, multiplication, subtraction, line plots, pictographs, virtual manipulatives, and more.

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Counting Series - Session 2 - Subitising

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Explore this resource

This package of professional learning resources aims to enhance knowledge and understanding of effective learning and teaching progression, whilst impacting positively on numeracy attainment. It includes the following resources:

Guidance document

Number and Number Processes

Fractions, Decimal Fractions and Percentages

Time

Data and Analysis

Ideas of Chance and Uncertainty

Money (New 2022)

Glossary

These resources, produced as part of the National Improvement Framework, aim to support schools and establishments to identify opportunities for targeted learning and enhance the effective teaching of numeracy concepts and associated skills. They also aim to help practitioners to:

identify methods for enhancing learners' progression and skills in numeracy and support application;

plan how to develop their own practice to ensure the progression of skills when teaching numeracy.

Within each document there is progressive guidance for each level from early through to fourth. Each level includes the following:

Effective learning and teaching approaches;

Things to consider;

Links to other curricular areas;

Reflective questions.

In addition to supporting practitioners to review and reflect on the current progress of their learners, this resource also supports practitioners to identify development themes required for the reinforcement of learning. Furthermore, this resource can support practitioners to review and reflect strengths and improvement areas within their own professional learning.

School leaders can use this resource to support curriculum development and improvement. Local authority leaders can use this resource to support consideration and enhancement of centrally provided professional learning offers to practitioners.

Downloads

Guidance documents
PDF file: Guidance document (510 KB)PDF file: Guidance document (Gaelic version)(480 KB)
Professional Learning resource documents
PDF file: Number and Number Processes (2.7 MB)PDF file: Number and Number Processes (Gaelic version) (3.2 MB)PDF file: Fractions, Decimal Fractions and Percentages (1.9 MB)PDF file: Fractions, Decimal Fractions and Percentages (Gaelic version) (2.2 MB)PDF file: Time (2.1 MB)PDF file: Time (Gaelic version)(2.2 MB)PDF file: Data and Analysis (1.8 MB)PDF file: Data and Analysis (Gaelic version)(2 MB)PDF file: Ideas of Chance and Uncertainty (652 KB)PDF file: Ideas of Chance and Uncertainty (Gaelic version) (900 KB)PDF file: Money (2.6 MB)
Money - Gaelic Version coming soon.

Sketchnotes
PDF file: Early Level Sketchnote - Number and Number Processes (267 KB)
Glossaries
PDF file: Glossary (436 KB)PDF file: Glossary (Gaelic version) (480 KB)PDF file: Numeracy Glossary (Gaelic version - alphabetical) (479 KB)

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This package of professional learning resources aims to enhance knowledge and understanding of effective learning and teaching progression, whilst impacting positively on numeracy attainment.

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Numeracy and Mathematics : Thematic Report

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Numeracy and Mathematics : Thematic Report

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You can find songs and rhymes to suit different moods or times of day. Try a range of songs and rhymes and discover your child’s favourites!

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Article by Dr Sue Gifford University of Roehampton

Download pdf
Young children have a remarkable skill: they can recognise numbers of things without counting. This is called subitising, and it develops from a very early age. Very young babies can not only tell the difference between one and two but also between large numbers of dots when there are twice as many in one group, as with 16 and 8 (Sarama and Clements, 2009). Young children also have powerful visual memories and some may find it easier to remember images than words: three-year-olds can recognise three things, although they may not say the word. Subitising can help children to build images for numbers, to visualise and to learn number facts. For instance, most four-year-olds readily learn to recognise five dots on a dice, which helps them to understand the cardinal value or 'howmanyness' of five, which they can link to the word and symbol for 5. Structured images like this also help children to begin to see numbers inside numbers, for instance seeing four and one within five.

How do you teach subitising?
Children's ability to subitise can be developed by teaching. Games provide repeated experience of interpreting the dot images, either to decide how many spaces to move, as in our activity Playing Incey Wincey Spider, or how many things to collect, as in our activity Dice, where the dotty number determines how many animals to put in your pen. (To find both of these activities, scroll down the collection of number based EYFS tasks.) For children who are unfamiliar with dot images, dice can be adapted to just show numbers up to three. Large dotty dice are good for outdoor games, like jumping along the track. Spinners with dots on provide an alternative, as with the child in the picture, who has his own spinner on a lanyard: he spins it and then makes that many jumps around the circuit track, then does the same again and again and again..! Having your own dice or spinner is a good option for younger children and avoids waiting for turns.

Large floor dominoes invite children to match number patterns, indoors or out.

Fingers also provide subitisable images for numbers, with the added advantage that they are embodied in muscle memory. Showing all-at-once finger numbers is the key skill here: Marton & Neuman (1990) found that older children with maths difficulties tended to count fingers one at a time, rather than using all-at-once 'finger numbers'. You can ask young children to first 'grow', then 'show' and finally to 'throw' finger numbers.

Using a range of objects and apparatus in different arrangements helps to make children familiar with different images for numbers, while also emphasising different number properties. For instance, Numicon presents numbers as even or odd, whereas linked cubes show numbers as lengths: connecting all these images helps children build a rich network of understanding to link to number words and symbols.

Subitising games can involve less common arrangements of numbers of objects. Games which involve revealing hidden objects will prompt young children to say how many they see (see the NRICH activity Hidden Jewels). One game involves upturned dishes hiding three, four and five objects, rapidly swopping them around and challenging children to keep track of the dish with four. When you stop and briefly lift the dish, children intuitively subitise the number revealed. This activity is taken from Clements and Sarama's very useful website, which presents a learning trajectory for subitising, supported by videos of activities for each stage.(If you have not already done so, you will need to create an account before accessing the free resources.)

https://www.learningtrajectories.org/math-activities/fantastic-five-tricky-two

Activities include auditory subitising, or recognising numbers of sounds, such as claps or drum beats, linking with movement and music.

Conceptual subitising
Subitising can also help children learn addition and subtraction facts. In some of the images below the total number of five is recognised by adding subgroups, like three and two, or four and one (Back, 2014 - see the NRICH article Early Number Sense). This involves part-whole awareness, which enables children to simultaneously see the whole number and numbers which make it up. The ability to think about numbers as compositions of other numbers was described by Resnick (1983) as 'probably the major conceptual achievement of the early school years'.

Recognising how many there are by combining numbers of smaller subgroups is called conceptual subitising: this is more advanced than perceptual subitising, which involves instant recognition (Clements, 1999). Seeing numbers as combinations of other numbers helps children to develop part-whole awareness and build familiarity with number combinations. It is also much more effective than giving children number sentences to complete by counting cubes and hoping they will remember the numbers involved!

Number talks
Showing children 'quick images' and asking them how many they see is one way of developing subitising. An alternative way of developing conceptual subitising is to show children a number of dots and ask them to say how they see the number (see the NRICH activity Number Talks). Teacher Janine Davenall engages young children by showing an image and asking, 'What numbers can you see hidden inside this number?'

They may reply, 'I see three and four', or 'six and one' or 'two and two and two and one'. This is an inclusive activity with many 'right' answers and the opportunity to discover that others see the same thing differently. Children also learn about all the different combinations which can make the same number. An alternative is to use giant magnetic buttons on a tray, which you turn over to secretly rearrange and show different combinations of the same number.

( www.teachingchannel.org/videos/visualizing-number-combinations)

Not only do younger children respond more to real objects than to images on a screen, but they can also see that the number of buttons stays the same despite being rearranged.

More complex images can present greater challenges for some children. Mulligan and Mitchelmore (2009) showed young children a 6 dot triangle and asked to draw it from memory: they found their responses varied significantly, with some just drawing a lot of dots.

Even when asked to copy the image, many five-year-olds produce dotty triangles like the one on the right, apparently finding it hard to focus on the number and the arrangement of the dots (Gifford & Thouless, 2018). This implies that some children will need a lot more practice in looking and talking about number patterns, in order to see how they are organised. Daily 'number talks' can help children to analyse and visualise number images, building an understanding of cardinal values, like the 'sixness' of six.

Reception children enjoy creating their own arrangements for numbers like seven and nine, and seeing which their friends recognise most easily. Older children with mathematics difficulties found the 'lollipop seven' image most memorable, at the same time learning a key number fact, that four and three make seven (Emerson & Babtie, 2010).

In Hungary, children are taught to recognise such numbers as 'five and a bit', using the dice image for 5 and linking with finger numbers (see the NRICH article From Objects and Images to Mathematical Ideas).

In the Netherlands, researchers developed bead-strings and rekenreks (counting frames) using groups of five, as they found these helped children to 'see' numbers more quickly (Gravemeijer, 1991). Being able to subitise groups supports children's understanding of larger numbers and later multiplicative thinking: for instance, recognising larger numbers by counting groups of two, three or five.

(The above images are taken from the Number Talk Images website.)

Subitising with everyday objects is more challenging than with simpler shapes, like dots. However, it can be more engaging for children to discuss images which are of particular interest and relevance to them, or even to create their own images for number talks.

Acknowledgments:

Janine Davenall
Buckinghamshire, Berkshire & Oxfordshire Mathematical Pedagogy Work Group

References:

Back, J. (2014) Early Number Sense
Clements, D.H. (1999) Subitising: What is it? Why teach it? Teaching Children Mathematics 5(7) 400-405
Emerson, J. & Babtie, P. (2010) The dyscalculia assessment: Emerson House Mathematics London: Continuum.
Thouless, H. & Gifford, S. (2018) 'Dotty triangles: two different approaches to analysing young children's responses to a pattern replication activity' in Golding, J., Bretscher, N., Crisan, C., Geraniou, E., Hodgen J. and C. Morgan (Eds). (2018) Research Proceedings
of the 9th British Congress on Mathematics Education (3-6 April 2018, University of Warwick, UK) 175-182 Online at www.bsrlm.org.uk/bcme-9/
Gravemeijer, K.P.E. (1991) An instruction-theoretical reflection on the use of manipulatives in Streefland, L. Realistic mathematics education in primary school. Utrecht: Freudenthal Institute 57- 76
Marton, F. & Neuman, D. (1990) 'Constructivism, phenomenology and the origin of arithmetic skills' in Steffe, L.P. & Wood, T. (1990) Transforming children's mathematics education: international perspectives NJ: Lawrence Earlbaum Associates 62- 75
Resnick, L.B. (1983) 'A developmental theory of number understanding' in Ginsburg, H. (ed) The development of mathematical thinking. London Academic Press 109 - 181.
Sarama, J. S., & Clements, D. H. (2009). Early childhood mathematics education research. London: Routledge.

here.

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In this article for EY practitioners, Dr Sue Gifford outlines what we mean by subitising and how we can teach it.

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Counting Series - Session 1 - An Overview of Counting

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The Count On Races game, played by an adult and a child, helps children learn to count on mentally. A coin or a dice with 1 or 2 on each face is used. Children say the next number when counting on. For example, when starting from 7, for a jump of 2 we say '8,9'.

The Cube Model (4-6) and Cube Model Advanced (4-6) tasks, children recreate the models. This task develops visual-spatial skills and helps children to recognise two parts within the whole.

The Reasoning in Addition Puzzles get children thinking deeply about the value of different shapes. This Mathsbot link allows you to generate new reasoning in addition puzzles.

Visual Representations

This article explains the importance of children's finger discrimination skills and this article gives different ways in which fingers can be used. Here are some Finger Discrimination tasks by Youcubed.

These Base-5 and Base-10 Number Tracks are coloured to help children to see quantities within a number track.

This Online Base-5 Abacus by Math Rack represents quantities 10-100 using a base-5 colouring.

The Coloured 100-Squares help to identify patterns in the 100-square and can support children as they count in 2s and 5s.

The Teens & Tens Cards help children to discern between teen and tens numbers. The Place Value Sorting Task show different representations of teen numbers and challenge children to spot misconceptions.

Progression in Pattern

The Progression in Pattern guidance shows how children's understanding of repeating patterns can be built. The Grids for Repeating Patterns can be used as children look to create circular patterns with a fixed number of spaces.
Number Foundations Assessment Fluency in Calculation Within 20 Spatial Reasoning in Early Childhood, Early Childhood Maths Group
Deconstructing Word Questions KS1 Tasks and Build-Up Presentation

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A range of resources and games for building children's early number sense with small quantities and developing conceptual understanding.

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Steve Wyborney\\\'s Blog: I\\\'m on a Learning Mission.

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100 Subitizing Slides & 10 Challenge Patterns - Steve Wyborney’s Blog: I’m on a Learning Mission.

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I have a new resource for you. 100 Subitizing Slides & 10 Challenge Patterns.

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Scottish Book Trust: Books that Support Numeracy

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Scottish Book Trust: Books that Support Numeracy

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Looking for more in Book Lists?

There are so many ways to bring numbers to life for little ones. Sharing stories, songs and rhymes that feature counting or other mathematical concepts can help to lay the foundations of numeracy - and it's never too early to start! Here is a selection of new and classic stories that explore numbers in very different ways.

Pick up these books through our bookshop.org affiliation! As well as supporting independent bookshops, a small portion of the profits helps our programmes to change more lives through reading and writing.

Buy and support

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Bring numbers to life for little ones by sharing stories, songs and rhymes that feature counting or other mathematical concepts

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Sing along and learn all about numbers with the Numberblocks!

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Bookbug Session Plans: Maths Week Scotland

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We are a national charity bringing the benefits of reading and writing to everyone in Scotland.

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These lower primary tasks are all based around the idea of counting and ordering numbers.

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Counting Collections with Older Children

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Counting Collections with Older Children

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Topics: Counting, skip counting, addition, multiplication (optional)
Materials: Paper, pencil, objects of many types (button, beans, stones, pencils, markers, blocks, etc.)
Target Grades: K, 1, 2, 3
Common Core: K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B.4, K.CC.B.5, K.MD.B.3, 1.NBT.A.1, 1.NBT.B.2, 1.NBT.B.2.a, 1.NBT.B.2.b, 1.NBT.B.2.c, 2.OA.C.4, 3.OA.A.1, MP1, MP6

Kids love to count things. This is a simple exercise, but a great way to get kids excited about arithmetic. Counting Collections also seeds ideas about how the place value system helps make counting work. Counting Collections is a foundational structure you can come back to again and again. We'll describe the initial launch first, and then some variations to make the activity more sophisticated.

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Topics: Counting, skip counting, addition, multiplication (optional) Materials: Paper, pencil, objects of many types (button, beans, stones, pencils, markers, blocks, etc.) Target Grades: K, 1, 2, 3 Common Core: K.CC.A.1, K.CC.A.2, K.CC.A.3, K.CC.B.4, K.CC.B.5, K.MD.B.3, 1.NBT.A.1, 1.NBT.B.2, 1.NBT.B.2.a, 1.NBT.B.2.b, 1.NBT.B.2.c, 2.OA.C.4, 3.OA.A.1, MP1, MP6 Kids love to count things.

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This article is a summary of a report that was first published in the Association of Teachers of Mathematics journal 'Mathematics Teaching' in February 2021
Early counting

Families recognise how important it is for children to be able to count but we know that learning to count is complex and takes time. Children need to count regularly and in a range of contexts. There are frequent opportunities for children to count during their day, using verbal counting (saying or singing the counting sequence) and object counting (counting to find how many) as they go about their everyday activities. These are hugely valuable and meaningful counting experiences but, alongside these, moments where children deliberately focus on counting and attend to how to count are important too. They allow children to spend time playing with counting, experimenting with ways to count and to wallow in counting as an enjoyable activity. In this article we explore one such approach, counting collections, detailing how practitioners might develop this approach for use with 2 to 5 year old children (see also the NRICH activity Counting Collections). The examples are drawn from our project work with twenty-one early years settings.

The counting collections approach

The collaborative counting of sets of objects is currently promoted in classrooms in the United States and New Zealand with older children. These counting collections sessions follow a teacher-led, carefully sequenced structure. Children work in pairs to plan, count and record their shared counting of collections of objects (Schwerdtfeger & Chan 2007). The teacher observes and interacts with the children as they engage in paired discussion, counting and representing the structure of their count on paper. The teacher then draws the class together, choosing selected examples of children's counting strategies to share with the class, perhaps modelling a specific approach.

Resources for counting collections in the early years

Counting collections are containers (boxes, jars or bags) with differing numbers of items within them, depending on the children. Some settings organise the collections into number ranges so the pink collections are 0-5, blue are 6-10, green are 10-20, and so on. It is important to have at least one zero collection and some larger collections too, even if these are fewer when working with the youngest children. The objects should be varied with many collections containing non-identical items to support development of the abstraction principle (as described by Gelman & Gallistel, 1978) so children realise they can count like and unlike items as one set. Some items, such as toy cars or balls, work less well because they can distract children from counting.

Alongside the collections, there are tools which the children may select to help them with their count. Bowls, dishes, pots, bags, mats, five and ten frames can be used to move objects into or out of when counting or opportunities to create groups for counting (e.g. 'in 3s' or '10 and 5 more'). Wooden and tactile numerals or writing equipment help children label counted collections.

Counting collections can also be used outdoors using buckets as containers. With these larger outdoor collections, children are able to use features of the outdoor environment such as walls or grass verges to line up their items to count or as a space to count their items into. They can transport collections around the outdoor area by carrying the buckets or taking them for journeys on their ride-on vehicles.
List of possible counting collections and tools
An important aspect is the everyday availability of the collections and tools so that children freely access them throughout the day as part of the continuous provision. They can choose to play, pattern and count with the collections during their child-initiated play, expressing and expanding their mathematical interests.

It works well for children to create their own collections in addition to those provided by adults. They can then choose to count each other's collections and can change the number of items in their collection over time. Families can be provided with bags, jars or boxes to create collections at home. Counting collections is also a lovely activity to do with parents invited into the setting where children engage in paired counting with their parent.

Counting collections sessions in early years

Counting collections sessions follow a general structure where children work in pairs to:

1. Choose a collection

2. Plan how they will count

3. Count the collection

4. Record the count

The structure needs to be appropriate to the children. A session can be very fluid with a practitioner working with two children to count and draw. Equally, a session might be more structured with the practitioner leading four clear sections with a whole class. The important thing is that children talk and communicate as they count. Working in pairs helps with this. The children often count the collection more than once as there is often some recounting to check the total, providing additional counting practice as well as supporting accuracy.
Roman and Maya's Reception class are engaged in a counting collections session. They choose the box of corks from the shelf of counting collections. Maya gets a bowl and Roman places one cork into it at a time. He counts each cork as it is moved into the bowl but miscounts. He empties the bowl and starts again. Maya counts with him this time and they count 10 corks. Roman looks up and states "There's 10 in the bowl", Maya nods "ten". They agree to count them again and this time Maya puts them in a line on the floor. They count to 8 but then the next cork rolls away and Roman has to retrieve it. Maya continues from 8, where she still has her finger poised, and they count together "9, 10".
Counting the collection

Counting collections support young children to develop one-to-one correspondence, giving one number word to one object when counting. Having pots, tray and bags to count into can help children to move one item at a time, knowing that all the objects counted so far are in the container. Small containers, such as muffin cases, can be used to hold just one item each helping younger children to match one object to one container before counting.
Abbie, choosing to count with her toy dog, used upturned clear plastic shot glasses to cover the wooden cubes in her collection. She then lifted each glass one by one high in the air, shouting the numbers as she counted each cube. This deliberate physical activity clearly separated the counting words and attributed one number word to each item.

One-to-one correspondence can also be developed through drawing around the items in a collection when recording the count. As children's counting develops, they might use containers or perhaps 5 or 10 frames for counting. They can count out 5 or 10 and then the remaining objects, emphasising the '5 and some more' and 'ten and some more' relationships.

Older children enjoy counting in unitised groups. They might create groups of 2, 3, 5 or 10, containers or frames. In some of the Reception classes that we worked with, pairs of children continued to fill ten frames until all of their objects were placed, using multiple frames. Others used one frame as a way of creating units of ten, collecting up the ten and settling them in another container safe in the knowledge that the container held ten items. Some practitioners noticed children predicting how many units there would be in their collection. For example, Sarah noted:
"One child estimated how many ten frames they would need before they started counting 34 buttons. They got three and were delighted when they were all filled up and they had to go and get another one!"
Recording the count

It is important that children's recording is their own and that how they represent the collection graphically is left open-ended. This ensures that the children feel ownership of the mathematics, making their own decisions over how to record (further developing mathematical thinking), and making their thinking visible to practitioners. Children's own recording is rich and varied and can include (as described by Carruthers & Worthington, 2005):

Pictographic representations which capture something of the appearance of what is in front of the child

Iconic representations which record one mark (of the child's own devising) for one object

Dynamic representations which are suggestive of action

Symbolic representations which begin to incorporate standard forms of numerals

Children need to see adults using these different forms of representations to know that these are valued and are ways to be mathematical. In the settings we worked with, the practitioners made a conscious effort to use more iconic representations, in particular.

Working in pairs, children make shared decisions about how to record their count. The process of recording provides additional counting and organisation opportunities. Some children prefer to record as they count (for example by drawing around items). Some children (particularly if they have seen an adult modelling this) might use an iconic style of recording, using circles or lines to represent plastic fish, spoons or toy cars, for example.
Macey, a three-year old, recorded her count in an increasingly efficient iconic style, along with her partner, four-year old Eva. They had counted their collection of ten objects in groups of three, using dishes. Having created three groups of three, they discussed what to do with the remaining one item. Macey and Eva initially made one group a group of four, but then moved that final item to create an additional group of one. They recorded this change on a large sheet of paper with two groups of three (left), a group of four (bottom right) then a long line to show the dynamic movement of one object (middle) into the group of one (top right). The practitioner prompted Macey to record the groups they were counting in and Macey frowned for a moment before writing 'M' (top right). When the practitioner looked quizzically at her, Macey exclaimed "I'm three!". She had apparently used her name as her symbol for '3'. Eva looked and exclaimed, "I can't, I'm four!"
Through storing counting objects as collections (sets), children seemed more confident with attributing a number to a collection (typically using numerals), focussing attention on the cardinality (how many-ness) of the set. Children can collect numerals or number cards to label collections or can use dry-wipe pens (on a plastic lid) or sticky notes to write the number for themselves.

The impact of using counting collections

Using counting collections increases the frequency of counting, range of strategies for counting and the amount of communication of counting (oral and recorded). As Rachel, a Reception class teacher, explained:
"My class now really enjoy counting and have the confidence to talk to others about it".
The paired activity seems to slow down the act of counting. It makes counting actions more deliberate and overt, supporting accuracy. Cooperative counting provides young children with 'a heightened awareness of the need to keep track of one's counting acts' (Wiegel 1998: 222). Keeping track is more explicit as a shared endeavour. There are greater opportunities for error spotting and self-checking with mathematical thinking communicated through both manipulation of the objects, and articulation of strategy and reasoning. It also enables children to learn how to count from each other.

Slowing down the counting process and exposing children's thinking through collaborative activity helps us, as practitioners, to assess children's counting. It provides the opportunity for us to pinpoint the specific aspects of counting or counting principles that each child seems secure with and which are emerging. We can observe and listen in on children's counting as well as initiate purposeful conversations about counting. This is evident in Leah and Alfie's counting in 3s. The recording enables us to assess the accuracy of their counting and also to recognise that the counting has been in ones, despite the grouping in threes (because they would have put the group of two last or miscounted it as a group of three when skip counting). The physical organisation of the count, the dialogue between the two, and the recording, together support the practitioner in identifying appropriate next steps in learning.

In counting collections, children express preferences about the collections, identifying their favourite ones to count. Having a range of different types of objects in the collections is important as some children like to count long objects which they enjoy lining up, some prefer round objects, and some enjoyed larger objects to cover a space. Fuson, Clements and Sarama (2015) explained that repeatedly counting collections of objects is interesting to young children in the same way that they enjoy repeatedly reading a familiar story. Children have favourite stories and favourite counting collections which are comforting and enjoyable to return to again and again, each time rehearsing the familiar alongside attending to previously unnoticed features.

One of the most important things that practitioners and children say about the counting collections approach relates to the pure joy of counting. Clara explained the effect in her Reception classroom:
"Children are excited about counting and show enthusiasm. They are engaged and asking to count!"
The collections gave children something to talk with, something to talk about and someone to talk to...
"Are we doing counting today? I love it" (Jacob, age 4). Dr Catherine Gripton is an Assistant Professor in primary education at the University of Nottingham. Deliah Pawluch is a Senior lecturer in primary education at Nottingham Trent University. This project was funded by East Midlands West maths hub. The work was part of a larger research project into children's mathematical agency in early years.
References

Carruthers, E. & Worthington, M. (2005) Making Sense of Mathematical Graphics: The Development of Understanding Abstract Symbolism. European Early Childhood Educational Research Journal, Vol. 13(1), 57-79.

Fuson, K.C., Clements, D.H. and Sarama, J. (2015). Making early math education work for all children. Phi Delta Kappan, 97 (3), 63-68.

Gelman, R. & Gallistel, C. (1978) The Child's Understanding of Number. Cambridge, MA. Harvard University Press

Schwerdtfeger, J.K. & Chan, A. (2007). Counting Collections. Teaching Children Mathematics, 13 (7), 356-361.

Wiegel, H.G. (1998). Kindergarten students' organization of counting in joint counting tasks and the emergence of cooperation. Journal for Research in Mathematics Education, 202-224.

Further information about counting collections:

DREME TE (2020) Supporting the development of representation. https://prek-math-te.stanford.edu/counting/classroom-videos-counting

Franke, M.L., Kazemi, E., & Turrou, A.C. (Eds.). (2018). Choral Counting and Counting Collections: Transforming the PreK-5 Math Classroom. New Hampshire: Stenhouse Publishers.

Teacher Education by Design (TEDD) (2014) Counting Collections. https://tedd.org/counting-collections/

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In this article for practitioners, Cath Gripton and Deliah Pawluch explore the 'counting collections' approach, which encourages children to spend time playing and experimenting with counting.

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Article by Elizabeth Carruthers and Maulfry Worthington

Introduction

The key concept in understanding and supporting young children's mathematical graphics is knowing the difference between children 'recording' a piece of mathematics after they have done it practically, as opposed to children representing their own mathematical thinking (i.e. children's mathematical graphics).

Children's mathematical graphics is a term we originated to refer to the visual marks and representations (graphics) young children choose to use to explore mathematical meanings and communicate their thinking. It includes a wide range of graphics such as scribble-marks, drawings, writing, invented and standard symbols. In education 'mark-making' is often used as a generic term for children's early explorations such as drawing or writing and there is confusion about its use and application. When discussing specific aspects such as 'drawing', 'writing' or 'children's mathematical graphics', it is preferable to use these terms.

Ofsted repeatedly raises concerns regarding children's over-reliance on formal written methods, emphasising the importance of children's informal methods of notation. Annual inspection reports highlight the importance of building links between children's mental and written methods and for greater opportunities for children to use and apply mathematics.

Ginsberg first identified written mathematics as the aspect that young children find most difficult (1977). Later Hughes demonstrated how young children could represent quantities they had counted by using their own, self-chosen representations (1986). Our research has explored this important aspect in depth, charting children's understanding of the written language of mathematics from birth to eight years (e.g. Carruthers & Worthington, 2005, 2006; DCSF, 2008, 2009). The 'Williams Maths Review' features our work on children's mathematical graphics, making specific recommendations about its role in supporting abstract mathematical symbolism (DCSF, 2008: 4 / 34-37).

Recording mathematics

Historically the emphasis has been on children 'doing' mathematics, working something out with practical resources and then 'recording' it afterwards. However our research has shown that this has limited value. Recording places the emphasis on marks, symbols and drawings as a product, and is a lower level of cognitive demand. The National Numeracy Strategy emphasised that children do not need to record their mathematics if they can work something out mentally, neither do they need to record something they have already worked out in a practical context with resources (QCA, 1999).

Exploring differences between 'recording mathematics' and 'representing mathematical thinking' at the MA's 2009 conference in Cambridge, led to rich discussion and raised considerable interest.

Representing mathematical thinking

To differentiate between the early written mathematics that most young children experience and children's mathematical graphics, we use the term 'representing': children represent their internal, mental representations, in effect - exploring their mental methods on paper. Children's own graphics support deepened thinking about the mathematics in which they are engaged, and significantly, about their use of symbols and other visual representations to signify meanings. They enable children to build on what they already know and understand - not as a pre-cursor to abstract written symbolism of mathematics but as a meaningful and continuing process throughout the Foundation Stage and Key Stage 1.

Figure 1: "A cross means you lose."

Figure 2: "This is where you double-lose."

For example, playing ball outside, three boys decided to invent a range of symbols (Figures 1 and 2) including those above - to represent the goals they scored and the progress of their game. This flexible sign use enables children to develop understanding of the abstract written language of mathematics.

Mathematical disconnections

Figure 3: Recording - Leo (aged 4 years 6 months)

Figure 3 shows an example of a child's 'recording'. Leo's teacher showed the children addition using multilink. Giving each child three cubes to count and then three additional cubes, she asked the child how many cubes they had altogether. Leo counted from the first group "$1, 2, 3$" and then continued counting the cubes in the second group, "$4, 5, 6$". His teacher wrote the calculation on the white board commenting "$3$ add $3$ equals $6$". She asked the children to draw how many cubes they had altogether. Some of the children copied what the teacher had written: Leo hesitated so his teacher wrote "$3 + 3 = 6$" beneath the cubes Leo had drawn.

The children's drawings were all virtually the same: the only differences were the ways the children had drawn and coloured the cubes. In effect the teacher had done all of Leo's thinking for him - preventing him exploring possible ways that might make sense to him. An unintended consequence of working in this way leads to children's confusion about written calculations and superficial understanding of the abstract written language of mathematics.

Mathematical connections

Figure 4: Children's mathematical graphics (representing personal mathematical thinking) - Jack (4 years 8 months)

Figure 4 shows an example of addition in another reception class where children were encouraged to represent their own mathematical thinking using their mathematical graphics.

Jack's teacher put a plate of grapes on the table where his group was sitting, adding some blank paper for any of the children who might want to explore their ideas on paper. She invited the children to choose a small quantity of grapes to put in each of two small dishes and work out how many grapes they had altogether. Jack chose to use paper and first drew two separate sets of grapes, leaving a gap that allowed his first calculation to be read as '$4$ and $3$' (we term this an 'implicit symbol'). He confirmed the quantities by writing the numerals and drew a line between these and the final '$7$', using the line to signify an equals sign.

Since Jack represented his mathematical thinking in ways he chose, his calculations were personally meaningful. Two of the children worked out their calculations mentally, whilst others chose to explore their ideas on paper, connecting mental and written methods. They used a range of personal graphics including words, numerals, drawings and invented mathematical symbols. Significantly children's mathematical graphics support children's processes ('using and applying mathematics'; DfES, 2006). The important point is that the children were making their own connections, building on their early understanding of marks and symbols and using them to make mathematical meanings.

Figure 5: "Can $8$ be shared equally between two?" Using his own mathematical graphics, Kamrin solved the problem by inventing 'Tweedle Birds' and sharing eight eggs equally between them.

Key attributes of children's mathematical graphics include:

children's own choice to represent their mathematical thinking

children's own meanings - both independent and co-constructed (rather than copied)

children's own graphics (drawings, writing, symbols, marks)

children's own layouts

children challenge themselves as they explore their mathematical thinking and meanings, communicate their ideas and solve problems

Some aspects of the adult's role:

Adults value children's mathematical graphics

Observing children closely allows adults to understand more about individual's thinking

Annotating children's mathematical graphics allows adults to sensitively uncover children's thinking and meanings

Conclusion

The Williams Maths Review (DCSF, 2008) urges that 'To secure effective pedagogy, local authorities, leaders, managers and head teachers should provide the following key elements of support in order for all settings to develop the conditions for learning ...

A culture with a significant focus on mathematical mark-making in line with early writing, through, for example, role-play and the use of popular mathematical signage in the environment

A learning environment that encourages children to choose to use their own mathematical graphics to support their mathematical thinking and processes' (p. 37)

At the MA conference, delegates compared examples of children's 'recording' with examples of children's mathematical graphics and their comments included:

"Recording is just neat and tidy, the adult's agenda for a 'nice' display."

"Why do you think some children prefer to use arrows?"

"See how the child did this in a different way the second time; she's understanding that you don't always need to draw - if you can do it a quicker way."

"You can really see the children's thinking in their mathematical graphics."

Some principles of children's mathematical graphics

Young children continually explore and make meanings in their play

They sometimes choose to use their own mathematical graphics to explore, make and communicate mathematical meanings

They represent their mathematical thinking in their own ways

They build on what they already know about the meaning-potential of marks and visual representations to also carry mathematical meanings

They co-construct understanding about mathematical notation through collaborative dialogue

They focus on processes of learning, rather than notation as a product.

Our work is based on the recognition that all children are capable and intelligent and children's mathematical graphics are inclusive.

The problem with mathematics is that children and adults find it difficult because it always has been someone else's way of working. In contrast, children's mathematical graphics unlock a freedom for children to explore their own mathematical thinking and meanings. This is at the heart of understanding written mathematical language. A recent report argues 'Mathematics is a language ... Effective teaching therefore focuses on the communicative aspects of mathematics by developing oral and written mathematical language' (NCETM, 2008: 20).

We suggest a significant conceptual shift is needed if teachers and practitioners are to effectively support young children's understanding of the abstract written language of mathematics: supporting children's mathematical graphics is a highly positive means of deepening children's understanding.

Elizabeth Carruthers is Headteacher of Redcliffe Children's Centre and Maintained Nursery, Bristol. Maulfry Worthington is engaged in research for her doctorate (Free University, Amsterdam). They are co-founders of the Children's Mathematics Network: www.childrens-mathematics.net.

References

Carruthers, E. & Worthington, M. (2005) 'Making sense of mathematical graphics: the development of understanding abstract symbolism' European Early Childhood Education Research Association Journal, (EECERA) Vo 13, No.1 (pp. 57-79).

Carruthers, E. & Worthington, M. 'Children's Mathematics, Making Marks, Making Meaning'. London: Sage Publications, (2nd Ed.).

DCSF (2008) Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools, London, DCSF.

DCSF (2009) Children Thinking Mathematically: PSRN essential knowledge for Early Years practitioners, London: DCSF.

Ginsberg, H. (1977) Children's Arithmetic, New York: Van Nostrand.

Hughes, M. (1986) Children and Number: Difficulties in learning mathematics, Oxford: Blackwell.

National Centre for Excellence in Teaching Mathematics (2008) Mathematics Matters (NCETM)

QCA (1999) Teaching Written Calculations, London: QCA.

This article first appeared in Primary Mathematics Autumn 2009, Volume 13, Issue 3 published by the Mathematical Association.

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Children's Mathematical Graphics: Understanding the Key Concept

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Article by Elizabeth Carruthers and Maulfry Worthington Introduction The key concept in understanding and supporting young children's mathematical graphics is knowing the difference between children 'recording' a piece of mathematics after they have done it practically, as opposed to children representing their own mathematical thinking (i.e. children's mathematical graphics).

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Animated video clips of nursery rhymes and songs for learning the numbers 1 to 10.

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Counting and ordering - KS1 Maths - BBC Bitesize

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KS1 Maths Counting and ordering learning resources for adults, children, parents and teachers.

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Erikson Institute Early Math Collaborative

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Series: Book Ideas

This series of book ideas encourage using stories to infuse the early years with mathematical learning. From the infant years until they are putting together more advanced numeracy concepts in elementary school, books can be a gateway for early math discussions.

Below find all the resources we have placed in our series titled BOOK IDEAS.

September 28, 2022

Counting books that offer children opportunities to participate bring play into math learning. These books engage children physically, cognitively, and emotionally-making them favorites that children will ask to read again and again.

Free videos.

Free newsletter packed with ideas.

Free professional learning modules.

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Book Ideas | Early Math Collaborative at Erikson Institute

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Book ideas for using stories to infuse the early years with mathematical learning and exploring. From birth to age eight.

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Read what others on the training have shared

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Read what others on the training have shared

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Year Two: Supporting Early Mathematical Development 2023 / 2024

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Year Two: Supporting Early Mathematical Development 2023 / 2024

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Numeracy and Mathematics @EducationScotland 

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View the interactive image by EdScot Curriculum Innovation Team