Skip to main content
### Number and algebra

### Geometry and measure

### Probability and statistics

### Working mathematically

### For younger learners

### Advanced mathematics

Article by Ruth Trundley# The Value of Two

*This article was first published in the Association of Teachers of Mathematics journal 'Mathematics Teaching' in November 2008. *

**Dr Ruth Trundley is a Primary Maths Adviser (team lead) for Babcock LDP supporting school improvement, delivering training, leading action research projects and writing materials to support teachers. Here she writes about the research she carried out for her doctorate, which focused on the development of counting.**

My doctoral research involved a small-scale case study with a personal flavour - the children in the study were my twin daughters (called Emily and Alice for the purpose of the research). Observations of their mathematical development began at 18 months and continued through to five years plus.

As part of the research, I identified 50 different elements involved in the development of counting and arranged these into a framework. I then mapped the development of the two children onto this framework. This revealed significant differences between the children; differences that have implications for early mathematics and which I will explore here.

At age two, both children were joining in with number songs and rhymes, attaching rhymes to actions, joining in with reciting number names and showing some understanding of the relationship between one and two. Emily consistently started her counts at one whilst Alice often missed out one and started at two. Emily had understood there was something special about the last number spoken in a count, and would indicate this by exaggerated movements and making her voice rise, which Alice did not do consistently. These were minor differences and seemed to indicate that, at the age of two, Emily's development was slightly in advance of her sister.

However, the more striking differences were between how the two children were engaging with numbers and counting and what they were attending to. These revealed much more about how the children were thinking and were the basis of the much bigger differences that emerged by the age of five.

**The first two years - Emily**

At 18 months Emily is observed playing at spooning an imaginary substance from one cup to another, saying a word each time she moves the spoon across from one cup to the other. At the same age she repeatedly goes up and down some steps in the garden, saying “Two, three, four, four, four, four ...” each time she does it, looking at her feet as they move from one step to the next, linking the saying of the numbers to the action. Emily enjoys learning songs and at the age of 19 months she has a particular favourite, a counting song which starts “One, two, three, four, Mary at the cottage door”. When I say, in another context, “One, two” she spontaneously adds “Three, four”. At this age she knows the names of the numbers to four and understands that the order one, two, three, four has some sort of special importance that fits with songs we sing. Learning the counting numbers as part of a song is so influential to her at this age that at first she does not distinguish between the different numbers. When counting three sheep in a picture, I count the first two, pointing at them as I count and then she continues “Three, four” with the same rhythm and intonation as the song, even though there is only one more sheep to count. Two months later she is again observed counting to match actions, this time bringing me a pile of books and passing them to me, saying a number each time she passes me something, matching the count to the movement rather than the books. By 21 months she regularly chooses to count in a variety of situations, with going up and down stairs one of her favourite activities for counting, and she will include counting in just about any context, such as playing with sachets of sugar in a restaurant.

Emily has become interested in, and good at, memorising the language of counting. She responds positively to songs and rhymes and finds these easy to learn, connecting with the rhythm and the pattern of the rhymes. She is often to be found singing counting songs or saying counting numbers along with her actions, matching movement to the language. Emily's early movements are often quite big and exaggerated, involving her whole body, for example, moving around the room swaying and stretching her arms around a table as she circles it. Her interest is in the oral/aural dimension of counting which focuses on the ordinal and with the kinÃ¦sthetic dimension, using movement as an accompaniment to the reciting of numbers.

**The first two years - Alice**

Between 18 and 19 months, Alice is observed exploring what two looks like in a variety of situations. She has become familiar with two as the number of sweets I will give her and she has started asking for “two” in other situations. Following this she has started to use “two” as the response to questions of the nature “How many have you got?” and also started to test out her understanding of two by creating sets of two at different times and checking that she has two. This includes two socks, two balls and two bottles - each time she holds the items up, one in each hand, and says “Two?” On one occasion she asks me to pass her some balls and I ask her how many, to which she replies “Two”. When I pass her only one she gets very cross and when I ask her if she has two she says “No”. I then give her another ball and she is happy. She appears to be connecting up her experiences (through a variety of contexts) with the language and images of the number two, in order to gain a full understanding of what 'two' means. Alice is far more interested in understanding what it means for a set to be two than in memorising and playing with the counting rhyme. She does use movement in her counting but her movements tend to be much smaller than Emily's, often choosing to move her fingers and combine movement with a visual image. Rather than accompanying the count, Alice's movements are often about accessing the maths of the situation and making visual the numbers involved. Alice's interest is in the cardinal value of numbers, building up a sense of the numbers and what they look like.

**From ages two to five**

There are significant differences in the development of the two children across the next three years. Despite the fact that her development, at age two, looked slightly behind her sister's, Alice's learning around counting accelerates and she reaches a substantial number of the 50 elements of counting between six months and a year in advance of her sister. It is Alice's attention to the cardinal value of numbers that underpins this acceleration. This is particularly relevant to the following:

*Understanding the relationship between numbers*

Alice's attention to the cardinal value of numbers, supported by a variety of images both visual and physical, makes her aware of the connection between consecutive numbers. She sees the '2+1ness' of three and the '3+1 ness' of four by age two years nine months. At this age, when out cycling, Alice is on the back of my bike and Emily on the back of her dad's. Another bike comes the other way and Alice says “There are three bikes now”. Going swimming, Alice says she needs the toilet and we establish that there are four toilets in the changing room. When we get into the changing room Alice looks at the door for one toilet and asks “Where are the other three?” Later, Alice's dad is saying he has four leaflets but has only one in his hand and Alice again asks “Where are the other three?” This is what Sarnecka et al (2005) refer to as the successor function (see below).

**Subitising and partitioning**

This recognition of the connection between consecutive numbers is one way of partitioning numbers. Alice also builds up a picture of numbers as partitioned in different ways and uses this in her counting. At age four, when asked to get five stones, she announces “I'm not going to count them... I picked up two and two more and then one.” To make it ten on the plate she says, “I'm getting five more”. Alice's attention to the value of numbers and images related to those numbers prompts her to use both her ability to subitise and to partition to make counting more efficient. At four years nine months she is asked to count ice creams in a book. She looks at the picture says “Six” quietly to herself and then moves her finger along the last four saying “Ten” at the end. Asked to explain what she had done, she says she started at six and when asked about this further explains: “I started on six because I saw three...” (covering the first three ice creams with one hand) “...and three” covering the next three with her other hand. She had successfully used her ability to subitise three and a known fact, that six partitions into three and three, to make her counting more efficient.

**Keeping track**

Alice uses both the images of small numbers and her understanding of the value of numbers to help her keep track. In the above example she keeps track of counting out five pebbles by getting two, two and one (all of which she can subitise). At three years 11 months she is asked to imagine two people on a bus and two more getting on. She says there will be four and explains that she was thinking of dice.

**Understanding conservation of number**

Understanding the cardinal value of a set involves understanding that the value doesn't change unless something is added or taken away, an element of conservation of number. It also involves understanding that the value of a set will be the same, however you count it. This is something Alice chooses to test out for herself at the age of three years five months. Sitting round the table at home, Alice counts and finds there are five people. She then experiments with starting the count with different people and finds there are five people wherever she starts. A month later, looking at a book with a picture of three fish fingers, Alice chooses to count the fish fingers three times, making a different one the first one she counts each time. This mimics her actions at 18 months, testing out what two is. She seems to be testing out what Gellman and Gallistel refer to as the 'order-irrelevance' counting principle.

All of these elements are clearly linked; they are all about a developing understanding of cardinality. Alice's examination of and interest in the cardinal value of small numbers means she builds up an understanding and an awareness of how numbers can be viewed and this includes partitioning them. The partitioning helps her to use her ability to subitise and also helps her to understand the relationship between numbers, as she often sees and describes small numbers as the previous number 'plus one'. By linking visual images of numbers and viewing them in different ways she also builds up an understanding of what it means to attach a number to a set, and how the set retains that number so long as nothing is added to or taken from the set. This focus on images, which also supports subitising, leads Alice to consider what she already knows when looking at a set that needs counting. She readily partitions the set, seeking out subsets that she can subitise and using known facts to help make the count quicker and more efficient.

This description of understanding of cardinality involves more than may have previously been recognised. Cardinality was one of the five counting principles listed by Gelman and Gallistel (1978). They described cardinality as knowing that 'the last “tag” is also the number name for the whole set'. More recently Sarnecka et al (2005) suggest that evidence shows some children are able to apply a 'last-word rule' without really understanding the cardinal principle and suggest that it involves more than a 'last-word rule':

I describe cardinality as being built on an ability to subitise and attach number names to small numbers. It includes recognition of the fact that these small numbers can be partitioned in a variety of ways, including previous number+1, leading to an understanding of the relationship between successive numbers ('successor function') as well as being able to say the number name for the whole set.

Fuson (1992) suggests that in each culture children must learn a number of things including 'The cardinal or “manyness” significance of counting.' Understanding of cardinality is crucial not only when counting to find out 'How many?' but also when asked to create a set of a given number. An example of this is asking a child to get four spoons. The response in this situation gives a clear indication of whether cardinality has been understood; there is no action to imitate in the way that reciting number names and pointing can be imitated when asked to find out 'How many?' At the age of three and a half, when she was able to match number names one-to-one for up to 40 objects, Emily was asked to put three people on a bus and declared “I don't know what three is”.

In contrast to Alice, Emily's early focus on the language of numbers allowed her to make good early progress with reciting number names in order but because the language is detached from images of number it is more difficult for her to make sense of numbers and what is happening to them when counting. There is a sense that for Emily much early counting has nothing to do with finding 'How many?' and therefore learning the cardinal significance of counting is a much bigger shift for her than for Alice.

This difference between the development in counting for Alice and for Emily has implications beyond counting. Counting is where calculating begins and Alice's earlier development of key aspects of counting has had a knock-on effect in her understanding of calculation; she was able to use deductive strategies built on this understanding of counting and the number system before the age of five.

The implications for teachers and educators are that the development of an understanding of cardinality is an element of counting that can be overlooked but plays a crucial part in linking counting skills to calculation. Currently, children are not always given access to the cardinal significance of counting until they have established the ability to recite the numbers, and this limits understanding that can emerge from learning to count. What teachers need to do is make understanding cardinality a central part of counting, and include many opportunities for children to create sets of a given number as well as count all of a set. This will, amongst other things, prepare them for the different addition situations of 3+4=? and 3+?=7.

And the value of two? Priceless.

**References**

Fuson, K. (1992) Research on whole number addition and subtraction. In D.A. Grouws (Ed.)

Handbook of Research on Mathematics Teaching and Learning 243-275. Macmillan & Co

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

Sarnecka, B., Cerutti, A., and Carey S. (2005) Unpacking the cardinal principle of counting: A last-word rule + the successor function; Poster presented at fourth biennial meeting of the

Cognitive Development Society [Online]. Available at www.cogsci.uci.edu/cogdev/sarnecka/Sarnecka%20&%20Cerutti%202005%20FT.pdf Accessed 12 July 2007

Trundley, R. (2008) One to Many: The development of counting in pre-school children. PhD thesis, Bristol

*This article was published in April 2015*

Or search by topic

Age 3 to 7

Published 2015

My doctoral research involved a small-scale case study with a personal flavour - the children in the study were my twin daughters (called Emily and Alice for the purpose of the research). Observations of their mathematical development began at 18 months and continued through to five years plus.

As part of the research, I identified 50 different elements involved in the development of counting and arranged these into a framework. I then mapped the development of the two children onto this framework. This revealed significant differences between the children; differences that have implications for early mathematics and which I will explore here.

At age two, both children were joining in with number songs and rhymes, attaching rhymes to actions, joining in with reciting number names and showing some understanding of the relationship between one and two. Emily consistently started her counts at one whilst Alice often missed out one and started at two. Emily had understood there was something special about the last number spoken in a count, and would indicate this by exaggerated movements and making her voice rise, which Alice did not do consistently. These were minor differences and seemed to indicate that, at the age of two, Emily's development was slightly in advance of her sister.

However, the more striking differences were between how the two children were engaging with numbers and counting and what they were attending to. These revealed much more about how the children were thinking and were the basis of the much bigger differences that emerged by the age of five.

At 18 months Emily is observed playing at spooning an imaginary substance from one cup to another, saying a word each time she moves the spoon across from one cup to the other. At the same age she repeatedly goes up and down some steps in the garden, saying “Two, three, four, four, four, four ...” each time she does it, looking at her feet as they move from one step to the next, linking the saying of the numbers to the action. Emily enjoys learning songs and at the age of 19 months she has a particular favourite, a counting song which starts “One, two, three, four, Mary at the cottage door”. When I say, in another context, “One, two” she spontaneously adds “Three, four”. At this age she knows the names of the numbers to four and understands that the order one, two, three, four has some sort of special importance that fits with songs we sing. Learning the counting numbers as part of a song is so influential to her at this age that at first she does not distinguish between the different numbers. When counting three sheep in a picture, I count the first two, pointing at them as I count and then she continues “Three, four” with the same rhythm and intonation as the song, even though there is only one more sheep to count. Two months later she is again observed counting to match actions, this time bringing me a pile of books and passing them to me, saying a number each time she passes me something, matching the count to the movement rather than the books. By 21 months she regularly chooses to count in a variety of situations, with going up and down stairs one of her favourite activities for counting, and she will include counting in just about any context, such as playing with sachets of sugar in a restaurant.

Emily has become interested in, and good at, memorising the language of counting. She responds positively to songs and rhymes and finds these easy to learn, connecting with the rhythm and the pattern of the rhymes. She is often to be found singing counting songs or saying counting numbers along with her actions, matching movement to the language. Emily's early movements are often quite big and exaggerated, involving her whole body, for example, moving around the room swaying and stretching her arms around a table as she circles it. Her interest is in the oral/aural dimension of counting which focuses on the ordinal and with the kinÃ¦sthetic dimension, using movement as an accompaniment to the reciting of numbers.

Between 18 and 19 months, Alice is observed exploring what two looks like in a variety of situations. She has become familiar with two as the number of sweets I will give her and she has started asking for “two” in other situations. Following this she has started to use “two” as the response to questions of the nature “How many have you got?” and also started to test out her understanding of two by creating sets of two at different times and checking that she has two. This includes two socks, two balls and two bottles - each time she holds the items up, one in each hand, and says “Two?” On one occasion she asks me to pass her some balls and I ask her how many, to which she replies “Two”. When I pass her only one she gets very cross and when I ask her if she has two she says “No”. I then give her another ball and she is happy. She appears to be connecting up her experiences (through a variety of contexts) with the language and images of the number two, in order to gain a full understanding of what 'two' means. Alice is far more interested in understanding what it means for a set to be two than in memorising and playing with the counting rhyme. She does use movement in her counting but her movements tend to be much smaller than Emily's, often choosing to move her fingers and combine movement with a visual image. Rather than accompanying the count, Alice's movements are often about accessing the maths of the situation and making visual the numbers involved. Alice's interest is in the cardinal value of numbers, building up a sense of the numbers and what they look like.

There are significant differences in the development of the two children across the next three years. Despite the fact that her development, at age two, looked slightly behind her sister's, Alice's learning around counting accelerates and she reaches a substantial number of the 50 elements of counting between six months and a year in advance of her sister. It is Alice's attention to the cardinal value of numbers that underpins this acceleration. This is particularly relevant to the following:

Alice's attention to the cardinal value of numbers, supported by a variety of images both visual and physical, makes her aware of the connection between consecutive numbers. She sees the '2+1ness' of three and the '3+1 ness' of four by age two years nine months. At this age, when out cycling, Alice is on the back of my bike and Emily on the back of her dad's. Another bike comes the other way and Alice says “There are three bikes now”. Going swimming, Alice says she needs the toilet and we establish that there are four toilets in the changing room. When we get into the changing room Alice looks at the door for one toilet and asks “Where are the other three?” Later, Alice's dad is saying he has four leaflets but has only one in his hand and Alice again asks “Where are the other three?” This is what Sarnecka et al (2005) refer to as the successor function (see below).

This recognition of the connection between consecutive numbers is one way of partitioning numbers. Alice also builds up a picture of numbers as partitioned in different ways and uses this in her counting. At age four, when asked to get five stones, she announces “I'm not going to count them... I picked up two and two more and then one.” To make it ten on the plate she says, “I'm getting five more”. Alice's attention to the value of numbers and images related to those numbers prompts her to use both her ability to subitise and to partition to make counting more efficient. At four years nine months she is asked to count ice creams in a book. She looks at the picture says “Six” quietly to herself and then moves her finger along the last four saying “Ten” at the end. Asked to explain what she had done, she says she started at six and when asked about this further explains: “I started on six because I saw three...” (covering the first three ice creams with one hand) “...and three” covering the next three with her other hand. She had successfully used her ability to subitise three and a known fact, that six partitions into three and three, to make her counting more efficient.

Alice uses both the images of small numbers and her understanding of the value of numbers to help her keep track. In the above example she keeps track of counting out five pebbles by getting two, two and one (all of which she can subitise). At three years 11 months she is asked to imagine two people on a bus and two more getting on. She says there will be four and explains that she was thinking of dice.

Understanding the cardinal value of a set involves understanding that the value doesn't change unless something is added or taken away, an element of conservation of number. It also involves understanding that the value of a set will be the same, however you count it. This is something Alice chooses to test out for herself at the age of three years five months. Sitting round the table at home, Alice counts and finds there are five people. She then experiments with starting the count with different people and finds there are five people wherever she starts. A month later, looking at a book with a picture of three fish fingers, Alice chooses to count the fish fingers three times, making a different one the first one she counts each time. This mimics her actions at 18 months, testing out what two is. She seems to be testing out what Gellman and Gallistel refer to as the 'order-irrelevance' counting principle.

All of these elements are clearly linked; they are all about a developing understanding of cardinality. Alice's examination of and interest in the cardinal value of small numbers means she builds up an understanding and an awareness of how numbers can be viewed and this includes partitioning them. The partitioning helps her to use her ability to subitise and also helps her to understand the relationship between numbers, as she often sees and describes small numbers as the previous number 'plus one'. By linking visual images of numbers and viewing them in different ways she also builds up an understanding of what it means to attach a number to a set, and how the set retains that number so long as nothing is added to or taken from the set. This focus on images, which also supports subitising, leads Alice to consider what she already knows when looking at a set that needs counting. She readily partitions the set, seeking out subsets that she can subitise and using known facts to help make the count quicker and more efficient.

This description of understanding of cardinality involves more than may have previously been recognised. Cardinality was one of the five counting principles listed by Gelman and Gallistel (1978). They described cardinality as knowing that 'the last “tag” is also the number name for the whole set'. More recently Sarnecka et al (2005) suggest that evidence shows some children are able to apply a 'last-word rule' without really understanding the cardinal principle and suggest that it involves more than a 'last-word rule':

'We suggest that the missing piece may be understanding of the successor function (the

function describing how numbers are formed: N,N+1, [N+1]+1, ..., etc)... We conclude that researchers should think of children's cardinal-principle knowledge as a last-word rule plus understanding of the successor function.' (p.1)

I describe cardinality as being built on an ability to subitise and attach number names to small numbers. It includes recognition of the fact that these small numbers can be partitioned in a variety of ways, including previous number+1, leading to an understanding of the relationship between successive numbers ('successor function') as well as being able to say the number name for the whole set.

Fuson (1992) suggests that in each culture children must learn a number of things including 'The cardinal or “manyness” significance of counting.' Understanding of cardinality is crucial not only when counting to find out 'How many?' but also when asked to create a set of a given number. An example of this is asking a child to get four spoons. The response in this situation gives a clear indication of whether cardinality has been understood; there is no action to imitate in the way that reciting number names and pointing can be imitated when asked to find out 'How many?' At the age of three and a half, when she was able to match number names one-to-one for up to 40 objects, Emily was asked to put three people on a bus and declared “I don't know what three is”.

In contrast to Alice, Emily's early focus on the language of numbers allowed her to make good early progress with reciting number names in order but because the language is detached from images of number it is more difficult for her to make sense of numbers and what is happening to them when counting. There is a sense that for Emily much early counting has nothing to do with finding 'How many?' and therefore learning the cardinal significance of counting is a much bigger shift for her than for Alice.

This difference between the development in counting for Alice and for Emily has implications beyond counting. Counting is where calculating begins and Alice's earlier development of key aspects of counting has had a knock-on effect in her understanding of calculation; she was able to use deductive strategies built on this understanding of counting and the number system before the age of five.

The implications for teachers and educators are that the development of an understanding of cardinality is an element of counting that can be overlooked but plays a crucial part in linking counting skills to calculation. Currently, children are not always given access to the cardinal significance of counting until they have established the ability to recite the numbers, and this limits understanding that can emerge from learning to count. What teachers need to do is make understanding cardinality a central part of counting, and include many opportunities for children to create sets of a given number as well as count all of a set. This will, amongst other things, prepare them for the different addition situations of 3+4=? and 3+?=7.

And the value of two? Priceless.

Fuson, K. (1992) Research on whole number addition and subtraction. In D.A. Grouws (Ed.)

Handbook of Research on Mathematics Teaching and Learning 243-275. Macmillan & Co

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

Sarnecka, B., Cerutti, A., and Carey S. (2005) Unpacking the cardinal principle of counting: A last-word rule + the successor function; Poster presented at fourth biennial meeting of the

Cognitive Development Society [Online]. Available at www.cogsci.uci.edu/cogdev/sarnecka/Sarnecka%20&%20Cerutti%202005%20FT.pdf Accessed 12 July 2007

Trundley, R. (2008) One to Many: The development of counting in pre-school children. PhD thesis, Bristol