Module 1: Counting
In this module you will:
 learn more about the development of young children’s early counting skills
 identify when children are learning securely through effective assessment
 learn more about the mathematics involved in counting 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 counting in mathematical development and a sense of how it underpins future mathematical competence
 have a greater depth of mathematical subject knowledge
 be better placed to support and guide the development of counting through effective, problem  based activities
 assess progression more accurately through a deeper understanding of the processes involved.
You may find it useful to read some of the research on counting and number. The following suggestions are by no means an exhaustive list, though each of these books would 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.
 Aubrey, C. (1997). Mathematics Teaching in the Early Years: An Investigation of Teachers' Subject Knowledge. London, Routledge
 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.
 MontagueSmith, A. (2002) Mathematics in Nursery Education. Oxford, David Fulton.
 Thompson, I (ed). (2008) Teaching and learning early number, (2nd ed). Maidenhead. Open University Press.
For more general research on Early Years education, you might like to look at:
The Effective Provision of PreSchool Education (EPPE) Project, and EPPE Research.
Pre–task: Observational activity
Plan a short activity focusing on counting 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.
Reflect on what actually happened. The following questions may help you:
 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 Selfevaluation Tools Early Years section. Explore questions 1 and 2 in Numbers as labels and for counting.
Ian Thompson is a Visiting Professor at Northumbria University. Read his article, The principal counting principles in Issue 7 of the NCETM Early Years Magazine.
Now return to the Mathematics Content Knowledge Selfevaluation Tools, Early Years section, and explore questions 3 to 5.
Many researchers in the field of early number have identified counting as the key to understanding number. Revisit your observations and reflections on your session in the light of what you have just read. For each child, consider which counting principles they have a secure understanding of and what they need to work on next.
Record your reflections in your Learning Journal within your Personal Learning Space.
Activity 1
Consider each of the counting principles. What errors would you expect a child to make if their understanding of a particular counting principle is not yet secure? Draw up a chart of examples. You might like to extend your chart to show what you could do to address each of the errors you identify. Here is an example of a partly completed chart. Discuss your ideas with colleagues and develop the chart into a useful aidememoire. Share the final document with colleagues.
You might like to use questions 8 to 10 in the Mathematics Content Knowledge Selfevaluation Tools, Early Years section, to help clarify your thinking.
Consider the barriers to learning
Barriers to learning
A cognitive barrier (thinking) might exist because the child doesn’t know the mathematics, skill or knowledge to use in a particular activity. You could support the development of thinking skills through carefully planned activities and questions to support connections to be made. You may for example, model how to count objects using the numbers as labels, touching each one as you do. You could ask, “Tell me how you might count these dinosaurs ...?” “How do you think you will know you have counted them all?” “Will you need to check to be sure you have the right number?” You could also stimulate the child’s thinking by asking, “Now can we count the number of red dinosaurs?” How do you know there are..?” “How can you be sure?” “How many yellow dinosaurs do you think there are? Why?”
A metacognitive barrier (thinking about thinking in order to solve a problem) might exist because a child gets lost in a particular train of thought and can’t see a way out of it. You could try asking questions which will help to guide them through the difficulty, for example, “Have you seen anything like this before?” “What did you do when you had to count the dinosaurs in the cave?” “Where did you begin?” “How did you know how many there were?”
An affective barrier (feelings and attitudes towards learning mathematics) might exist because the child has difficulty engaging with the context of the problem. This is particularly true for children with English as an additional language. This could be overcome by providing real/realistic activities based on their own experience or designing activities based on the same mathematical focus in different learning areas of the classroom.
Holton, D. (1999) Teaching Problem Solving. Chichester, Kingsham Press
Key aspects to consider when planning effective mathematical learning experiences for counting
All the mathematical learning experiences you plan for children will be more effective if you consider how children learn and design tasks and activities which take this into account. Much has been written about how children learn and there are many different theories of learning developed by researchers such as Piaget, Vygotsky, Bruner, Rogers and Lave, but in order to ensure children learn with understanding, most of them agree that mathematical experiences should include certain interrelated elements in order to make the connections upon which understanding depends.
Liebeck (1990) has summarised these connections as:
 experiences
 language
 symbols
 pictures.
This diagram illustrates how mathematical understanding is built up through making a network of connections:
Reproduced by permission of SAGE Publications, London, Los Angeles, New Delhi and Singapore, from Derek Haylock and Anne D Cockburn, Understanding Mathematics for Young Children  A Guide for Foundation Stage and Lower Primary Teachers, Third Edition, © Sage Publications, 2008.
These interrelated elements support and develop children’s mathematical thinking (cognition) and the way they think and approach mathematical problems (metacognition). When planning both child and adult initiated activities you should try to ensure that children are given the opportunity to make connections through and across each of these representations in order to develop and support their mathematical thinking and help them to communicate their understanding to others.
Digging Deeper
For further research on how children learn, you might like to read one or more of the following:
 Hiebert, J. et al, (1997), Making Sense: teaching and learning mathematics with understanding. Portsmouth NH. Heinemann.
 Liebeck, P (1990), How Children Learn Mathematics: A Guide for Parents and Teachers. London, Penguin.
 Haylock, D. and Cockburn A.(2009) Understanding Mathematics for Young Children. London, Sage.
 Mathemapedia entry: Learning  what does learning mathematics look like?
 Primary Module: Making Connections.
Activity 2
Choose one counting principle and one barrier to learning. Plan an activity which would help you to more effectively support the children’s learning and overcome the identified barriers.
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.
Reflect on what actually happened. The following questions may help you:
 what did you notice?
 how did you move the learning on? What questions did you ask?
 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?
Record your reflections in your Learning Journal within your Personal Learning Space.
Counting Competency
Read The Notion of Principle: The Case of Counting, by Gelman and Meck. They comment that while the counting principles specify characteristics that a correct performance must have, they do not provide a recipe for generating a plan for correct performance. They discuss conceptual, procedural and utilisation competence. Read the chapter with the following questions in mind:
 what is the difference between principles and procedures?
 what is procedural competence?
 which puppet actions were used to test understanding of each principle?
Record your reflections in your Learning Journal within your Personal Learning Space.
Activity 3
Choose a different counting principle to that explored in Activity 2. Revisit your notes on Gelman and Meck’s puppet testing for that principle. Use the same method of testing with at least six children.
Reflect on what actually happened. The following questions may help you:
 what did you notice?
 did the children respond in the same way as described in the reading?
 what did you find out about these children?
 what conclusions can you draw from what happened?
 are your conclusions the same as Gelman and Meck’s?
 how will what you have observed influence your future practice?
Digging Deeper
For further ideas on Counting Competency, read Conceptual Competence and Children's Counting by James G Greeno and others.
The key elements of effective planning
Conditions for learning  a numeracy rich environment
Anything and everything in the indoor and outdoor environment of the early years setting can provide a mathematical learning opportunity. The trick is to ensure that the opportunity for developing and extending mathematical thinking is fully exploited.
Using and applying mathematics, that is, the opportunity for oral and written communication through challenging and motivating problem solving activities and assessment should be an intrinsic part of your planning.
Activity 4
Take a really good look at the learning environment of your setting. How could you develop the environment to ensure that each child has access to the learning experiences necessary to develop their counting skills? Can you identify how and where specific child and adult initiated counting activities could arise or be engineered? How are these fully supporting and developing mathematical thinking?
Return to the Mathematics Content Knowledge Selfevaluation Tools, Early Years section and explore questions 6 and 8 to 11 to help you.
You might like to explore the Mathematics Specific Pedagogy Selfevaluation Tools too. Although the first section on Planning has been superseded by changes in government policy, the rest is still extremely useful.
Take a realistic look at each area of your setting and consider what is normally provided in that space and what the children choose to access in the area.
Now ask yourself the following questions:
1. The learning environment
Have you ensured that all areas of the learning environment, both inside and outside, have been fully exploited for mathematical learning? Have you considered both adult and child initiated activities? Have you considered the key principles of counting and the barriers to learning?
2. Using and applying mathematics
Oral communication
 how will you know when to spot a learning opportunity that will enable the child to reflect/conjecture/ hypothesise about counting and justify/convince themselves and others that they have counted correctly?
 how might you or other adults intervene with an appropriate and well targeted question such as:
‘What if ?’
‘What if not?’
‘How do you know?’
‘What is the same, and what is different?’
‘Can you finish this pattern?’
 what language might children use together to develop and support their thinking?
Written communication
 do you provide the opportunity for children to record and support their thinking in a range of learning areas? For example in the role play area, painting area, outside area.
Problem solving
 are tasks and activities challenging children to learn something new or solve a problem or are they simply reinforcing or practicing existing skills?
 what are the children seeing, both indoors and outside that support their mathematical thinking and language? Are there pictures, written symbols and manipulatives, which should include a range of real life, realistic and representational resources and materials?
 how are children engaged in tasks? Do they work independently, with others and with adults?
 does the learning environment, indoors and outdoors, support and promote problem solving activities?
3. Adult interventions, both in the setting and at home
 do you exploit every learning opportunity?
 can you support other adults to intervene with appropriate and well targeted questions?
 do you know how to extend /support/scaffold mathematical learning?
4. Assessment for learning
 do you know what the children already know and can do?
 what are you intending the children to learn? How will you make this clear to the other adults?
 how will you know when the child has learnt  what should the child be able to say and do?
Now that you have reflected on your learning environment, ask yourself whether you fully exploit the learning environment for mathematics in your setting. The following resources will help you to ensure that planning meets the learning needs of all children:
 sample planning grid  Incey Wincey Spider
 blank planning grid
 sample activities – Making Six; The Charity Shop
 Tucker, K. (2010) Mathematics Through Play in the Early Years. London, Sage
 portal resources:
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 counting in mathematical development and a sense of how it underpins future mathematical competence
 have a greater depth of mathematical subject knowledge
 be better placed to support and guide the development of counting through effective, problem  based activities
 be able to assess progression more accurately
References:
 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
 Holton, D. (1999) Teaching Problem Solving, Chichester, Kingsham Press
 Hiebert, J. et al, (1997). Making Sense: teaching and learning mathematics with understanding. Portsmouth NH, Heinemann
 Liebeck, P (1990), How Children Learn Mathematics: A Guide for Parents and Teachers. London, Penguin
 MontagueSmith, A. (2002) Mathematics in Nursery Education. Oxford, David Fulton
 Thompson, I (ed). (2008) Teaching and learning early number. (2nd ed). Maidenhead. Open University Press
 Tucker, K. (2010) Mathematics Through Play in the Early Years. London, Sage.
