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Early Years CPD Modules - Module 2: Number

This page has been archived. The content was correct at the time of original publication, but is no longer updated.
Created on 31 March 2011 by ncetm_administrator
Updated on 13 April 2011 by ncetm_administrator

Module 2: Number

In this module you will:

• identify when children are learning securely through effective assessment
• learn more about the mathematics involved in early number skills and where this can underpin future mathematical knowledge and understanding
• plan effective learning experiences to ensure the secure development of these skills.

At the end of this module you will:

• have a better knowledge and understanding of both the importance of early number skills in mathematical development and a sense of how they underpin future mathematical competence
• have a greater depth of mathematical subject knowledge
• be better placed to support and guide the development of number skills through effective, problem - based activities
• assess progression more accurately.

Any of the books mentioned in Module 1 will continue to be a useful reader for this module. You could also search the NCETM portal and use a search engine to search the internet for ideas.

Consider where and when you have seen numbers during the last few days. Try to work through one whole day, noting when you encountered the numbers and what they were being used for. For example, the time on your alarm clock; on weighing scales; on coins or notes when you were shopping; car number plate etc. Group the experiences into categories. What do you notice?

You may have grouped the experiences into a number of different categories, such as counting and labelling. If you have more categories, further reflect on the decisions you have taken when categorising, but be aware that there is no one agreed list of categories. You might like to look at the Mathemapedia entry What’s in a number? to help you.

Numbers as labels

Numbers are not only used in counting but are also commonly used as labels, from the number 23 bus to your car number plate.  In Module 1, you read Ian Thompson’s article in Issue 7 of the Early Years magazine, The principal counting principles. In particular Ian comments:

For example, there is usually no connection between the number 88 and the number 89 bus – letters or words could be used just as well as a means of distinguishing between them. Nor do the actual numbers allocated to specific buses have any connection with the route – although a notable exception to this is the AD122 bus that passes my house on its way to Hadrian’s Wall!

Activity 1: Observing children

Plan a short activity focusing on numbers as labels for a small group of children. The exact content of the session is up to you, but you may find it helpful to consider the following questions:

• what do I want the children to learn?
• how will I know that this has happened?
• what will this look like?
• what is my intended outcome?

Carry out the session as planned and closely observe what the children do. If you are able to film the session, watching it again will allow you to pick up on things you missed in real time.

• what did you notice?
• did the children learn what you intended them to? How do you know?
• did the children demonstrate the outcome as you intended or in some other way?
• what did you find out about these children?
• what kind of learning took place? Was it new learning, practice or something else?

Now visit the Mathematics Content Knowledge Self-evaluation Tools Early Years section. Explore questions 12 to 14 in Numbers as labels and for counting.

Ordinal Number

The order of the numbers in our number system is somewhat more complicated than in many other languages. In many cases, the numbers one to ten are followed by ten one, ten two, ten three etc, whereas in English we introduce new idiosyncratic words, only reverting to the original numbers once we reach the next decade. In addition, the words used for the second decade, ending in -teen, are easily confused with the decade numbers ending in -ty. For more information on this aspect of number, read Ian Thompson’s article in Issue 13 of the early years magazine, The structure of counting word systems; Counting with Miss Count in Issue 14 is also relevant.

Plan a short activity using a puppet such as Miss Count to identify the children’s understanding of ordinal number. How will you know that a child has a good understanding of this concept? What questions will help you to find out?

• what did you notice?
• what did you find out about these children?
• what do you need to do next?

Cardinal value

Knowing the last number you counted is the number (or label) for the whole collection is the cardinal aspect of numbers. This is known as the numerosity of a set. This ability is dependent on proficiency in  both the one to one and the stable order counting principles as children need to touch objects and count one to one consistently using  the number names in order. In short, they should be able to count with understanding. They must also be able to stop on the last number of the count and recognise that as how many there are in that set.

If a child has securely grasped the cardinal principle they are less dependent on seeing and touching the objects in the set. The operations of addition and subtraction are dependent on a secure understanding of cardinality. In order for children to understand the link between counting and the development of number concepts, it is vital that they have the opportunity to count and calculate in many different real and realistic contexts.

In order for you to assess the children’s grasp of cardinal value try this activity:

Activity 3

Sing ‘Five Little Ducks’ using plastic ducks on water tray. The five ducks can then be randomly left to float. Ask “Can you count the ducks?” “Are there five?” “How do we know?”

Put another five different coloured ducks in the water. Ask “How many blue ducks are there?” “How do you know?” “How can you check?”

Sing 'Five Currant Buns in the Bakers Shop'. Say “Here we have some currants and buns.” Ask “How many buns have we got?” “How do we know?” “How can we be sure?” “How many cherries do we need to put one on the top of each bun?”

Did the child have to touch each object whilst counting? Could they say the counting names in order? Could you be sure that the child has fully understood the cardinal principle?

Digging Deeper

If you would like to further develop your own subject knowledge about numbers and counting, one of these books could help:

• Cooke H. (2007) Mathematics for Primary and Early Years, Developing Subject Knowledge. Open University.
• Haylock D. (2010) Mathematics Explained for Primary Teachers.  Sage publications.

Developing Number Sense

Number sense refers to "a well organised conceptual framework of number information that enables a person to understand numbers and number relationships and to solve mathematical problems that are not bound by traditional algorithms" (Bobis,1996). Children need a wide range of experiences to ensure that they develop the network of connections that lead to the range of conceptual developments that we call ‘number sense’. Read Jenni Way’s article Number Sense Series: Developing Early Number Sense to find out what number sense is, how it might begin and which teaching strategies and games promote early number sense.

Equivalence

Equivalence of number refers to situations where two sets of objects have the same quantity value. The sets may contain different items, so a bag of five sweets has the number equivalence of a garage with five cars. It is thought that children learn about equivalence as they count small sets and observe that these have the same number label. Items which come in pairs become ‘two,’ a label which can be applied to any set in any situation. Gelman and Gallistel found that children counted both sets to determine if both sets are equivalent. If their cardinal values were the same, the sets were equivalent and if not, they were not equivalent.

The equals sign represents equivalence, a balance between both sides of the equation, but children generally see the equals sign as an instruction to perform an operation, i.e. to do something.

Activity 4

Carry out an internet search to explore what researchers have found out about how children view the equals sign. There are several interesting articles you might like to browse. Be aware that some sites will charge a subscription before you can read the full article. You will usually be shown an abstract, and sometimes this is sufficient. Consider how you can help children to develop an understanding of equivalence in the early years, long before they encounter the equals sign.

Record your reflections on what you find out in your Learning Journal within your Personal Learning Space.

Inequality

Children then begin to look at the difference between two sets and are able to say which set is greater or smaller. They then move on to exploring what is needed to make both quantities equivalent, as they begin to understand the words more and less. Children find comparative language difficult throughout the primary years and often on into secondary school. Understanding of words such as biggest, smallest and greatest seems to be more easily acquired than those with the -er suffix, e.g. greater, fewer, bigger, smaller.

Activity 5

Carry out a further internet search to see what you can find out about the use of mathematical language in early years settings. Remember, some sites will charge a subscription before you can read the full article, but an abstract is sometimes sufficient.

Try to answer the following questions:

• which type of language is well taught in early years settings?
• which type of language is not well taught in early years settings?

Record your reflections on what you find out in your Learning Journal within your Personal Learning Space.

Subitizing

What is subitising (or subitizing)? You will come across both spellings. When you looked at question 10 in the Numbers as labels and for counting section of the Mathematics Content Knowledge Self-evaluation Tools, Early Years, this may well have been the first time you encountered the word.

Subitizing is knowing ‘how many’ there are in small groups of objects without counting them. Read the Mathemapedia entry What is Subitizing? and Ian Thompson’s article How good are you at subitizing? in Issue 11 of the Early Years magazine to develop your understanding. Follow the link in the Mathemapedia entry to the BBC News Channel feature and take the subitizing test to find out if you can count faster than a chimp.

Estimation

Estimation is a life skill. Read the Mathemapedia entry Developing Estimation Skills. Children’s estimations can be very revealing of their understanding of number. A more exact understanding of cardinal value is needed for small differences. Many children do not understand that each counting number is one more than the previous number. Building staircases using cubes or rods can help to develop this understanding. Children need to develop the language of estimation alongside counting. Words such as nearly, close, in between, almost, just about, not quite, greater than, less than, the same etc are the everyday beginnings of an understanding of estimation.

Numbers without limit

Children are interested in quantity from a very early age. They often surprise us with their understanding of number beyond what we might expect.

When James, aged nine months, was given a small quantity of finger food, he ate some and then seemed to notice Granddad’s rather fuller plate. He leaned towards Granddad, making sounds and gestures to indicate that he was aware that Granddad had much more than James did. He appeared to be comparing quantities long before he could begin to count.

In a Reception class, Sameena chose to count the utensils in the mother bear’s kitchen. When she ran out of utensils, she continued to count all manner of objects, both inside and out. She regularly returned to the practitioner to ask “What comes after...” This continued, off and on, throughout the day. That evening, she carried on counting at home, supported by her parents. The next day, when she returned to school, she told the practitioner, “Number go on forever and ever!” She appeared to be making generalisations about the number system from her experience.

Does your setting allow for these types of experiences? How could you ensure that opportunities to develop deeper and wider understanding are regularly offered? What might the children be seeing and experiencing to ensure that this happens?

Children enjoy exploring big numbers so we should not put a ceiling on their explorations. Numberlines should extend beyond the early counting range and stories such as How Much is a Million? by David M Schwartz and How Big is a Million? by Anna Milbourne could be shared with the children. Discussions about how many grains of sand in the sand tray, stars in the sky or leaves on a tree will allow children to explore the concept of infinity.

Mark Making – emergent recording in mathematics

Sir Peter Williams raised the issue of early mark making in mathematics in his 2008 report, Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools. Sir Peter recognised that children’s mark making through role play is valuable as a means of communication and to support acting out the activities they have observed, but that it is rare to find adults supporting children to make mathematical marks to help the development of their mathematical thinking. He commented that this lack of recognition “misses a valuable opportunity to encourage early experimentation”. He advised that, “Early years practitioners should encourage mathematical mark-making and open ended discussion (or sustained shared thinking) in children’s mathematical development” and made it the subject of the fourth of his 10 recommendations. Recommendation 4 proposes, “That the DCSF commissions a set of materials on mathematical mark making and children’s mathematical development which can be used to support early years practitioners’ CPD.”

Now visit the Mathematics Content Self Evaluation Tool (SET), Early Years section. Answer question 7 in Numbers as labels and for counting and question 5 in Calculations.

Read Chapter 3 Children’s mathematical graphics and Chapter 4 Numbers as labels and for counting in Children Thinking Mathematically: PSRN essential Knowledge for Early Years Practitioners. In Chapter 4, you will revise some of the ideas from Module 1. For this module, you should focus on pages 22 to 25 of Chapter 4.This document may have been archived so you may need to search to find it. However, many schools and other settings will have hard copies.

Activity 6

Identify opportunities for mark making in your own setting. Are the necessary resources kept in one area or spread throughout the setting? Remember, mark making is not simply paper and pencil. Anything which makes a mark on any surface or leaves a mark within the material can be used – chalk, water, paint, sand etc. Not all of it is easily preserved, so if you want to keep a record, you may need to take a photograph. How can you improve the opportunities for mark making throughout your setting?

Record your reflections in your Learning Journal within your Personal Learning Space. Consider what you will aim to change, do or try out:

• tomorrow
• next week
• next month
• next term.

Activity 7

Now that you have explored children’s early mark making in mathematics, collect some examples from your setting. If necessary, set up an activity such as those described by Ian Thompson in his article Children representing quantity. As you collect, take notes as you ask the child to explain what they have done and what it was for. Have a go at classifying the examples you have collected. You will need to decide which system to use from your readings. For example, will you use Martin Hughes pictographic, iconic, symbolic and idiosyncratic; Carruthers and Worthington’s taxonomy of children’s mathematical graphics; Newburn Manor Nursery School’s matrix, or perhaps a system of your own devising? You might like to browse the galleries on Children’s Mathematics Network to help with your interpretations. Consider what you have learned about the children whose mark making you have explored.

Digging Deeper

If you would like to explore this topic further, any of the following would be useful starting points:

Now visit the Mathematics Content Knowledge Self-evaluation Tools. Explore how the early number concepts are further developed in Key Stage One and further into Key Stage Two by working through both Counting and understanding number sections. What aspects of early mathematical experiences do you think are underpinned by early number concepts?

Implementing and continuing to learn. Draw up a plan of action. Consider what you will aim to do or try out:

• tomorrow
• next week
• next year.

Meet with colleagues online in the Early Years Forum to discuss issues which are of importance to you in the light of your reading. Record your plan of action in your Learning Journal within your Personal Learning Space and set up appropriate reminders.

Now that you have completed this module, you will:

• have a better knowledge and understanding of both the importance of early number skills in mathematical development and a sense of how they underpin future mathematical competence
• have a greater depth of mathematical subject knowledge
• be better placed to support and guide the development of number skills through effective, problem - based activities
• be able to assess progression more accurately.

References:

• Carruthers, E. & Worthington, M. (2006) Children’s Mathematics: Making Marks, Making Meaning. London: Sage.
• Cooke, H. (2007). Mathematics for Primary and Early Years: Developing Subject Knowledge, (2nd ed). London, Sage.
• Gelman, R. & Gallistel, C. (1978) The Child's Understanding of Number. Cambridge, MA. Harvard University Press.
• Gifford, S. (2005) Teaching Mathematics 3 – 5; developing learning in the foundation stage. Maidenhead. Open University Press.
• Montague-Smith, A. (2002) Mathematics in Nursery Education. Oxford, David Fulton.
• Thompson, I (ed). (2008) Teaching and learning early number. (2nd ed). Maidenhead. Open University Press.
• Independent Review of Mathematics Teaching in Early Years Settings and Primary Schools. Final Report – Sir Peter Williams 2008.

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