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# Early Years CPD Modules - Final Reflections

This page has been archived. The content was correct at the time of original publication, but is no longer updated.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

## Final Reflections - What have I learnt?

You have now completed the first two Early Years CPD modules. In this final section, you will have the opportunity to reflect upon the impact that the modules have had on your own mathematical knowledge and understanding, and how your experiences have influenced teaching and learning in your setting. There will also be suggestions for further study in the Digging Deeper sections.

Firstly, consider the learning outcomes for the modules, which were to:

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

Reflection 1

To what extent do you consider that you have achieved these outcomes? To help you assess the impact of your study, look back at your reflections in your Personal Learning Space, and the evidence you have from your class based research to answer these key questions:

To what extent have you been able to develop your own mathematical knowledge using the NCETM Content Knowledge Self-evaluation Tools or other resources?

How has this improved understanding helped you to teach number and counting more effectively?

Activity 1

Answer one of the following questions:

Is this statement always, sometimes or never true?

The effect of multiplying a whole number by 10 is to place a zero at the end of it, e.g. 15  x 10 = 150

Is this statement always, sometimes or never true?

Whole numbers are divisible by nine if the digits add up to nine.

Reflection 2

Sue Gifford (2005) summarised the key thought (cognitive) processes identified by theorists such as Piaget and Vygotsky and endorsed by researchers and educationalists involved in learning mathematics. These processes represent a social constructivist approach:

• imitating and practising
• attending to instruction
• generalising: Piaget’s ‘assimilation’, including making connections
• representing: including visualising, verbalising and symbolising
• restructuring: Piaget’s ‘accommodation’ and cognitive conflict
• meta cognition: reflection on thinking processes
• problem solving: a context for many of the above processes.

Digging Deeper

Read the paper learning theory from infed, the encyclopaedia of informal education. This paper explores and compares the various approaches to learning theory. Follow some of the links.

Relection 3

To what extend have you developed a better understanding about how to plan challenging, problem solving experiences which will help young children to learn effectively and enable them develop secure early number concepts?

Do you understand when, where and how you might ask questions to provoke mathematical thinking?

Are you are better able to assess the progress of young children and to help them to overcome any barriers to learning effectively?

Digging Deeper

The Assessment Reform Group has written extensively about formative assessment , which provides the practitioner with the information about what a child knows and can do in order to plan more appropriately to meet their future learning needs. Visit Mathemapedia to find out more.

Read Thinking Mathematically, Mason, J. Burton, L. & Stacey, K. John Mason explores the development of mathematical thinking in a very readable way.

For further exploration of the barriers to learning, read Teaching Problem Solving by Derek Holton et al.

Find out more about talk for learning in Pratt, N. (2002) Mathematics as Thinking. Mathematics Teaching, 181: 34-7. Mathematics Teaching is the journal of the Association of Teachers of Mathematics (ATM). You may have to pay a small fee to access this article if you are not a member of the ATM.

Activity 2

At the end of Module 2: Number, you visited the Mathematics Content Knowledge Self-evaluation Tools (SET)  to explore how the early number concepts are further developed in Key Stage One and Key Stage Two. Revisit the SET and work through any missed sections. Extend your understanding of progression in number and counting by exploring the Key Stage 3 SET sections Numbers and the number system, and Algebra.

Considering the impact of teaching on learning

As you will have learnt from the Introduction, there are many definitions and views of mathematics. People can have very particular beliefs about mathematics and how and why it is used. Teachers also hold a wide range of beliefs about mathematics. This can often affect the way they teach the subject. For example, if you believe that mathematics is a body of knowledge that needs to be transmitted from one person to another, then you will plan experiences that enable you to transmit the knowledge. If you believe that mathematics is a way of making sense of the world around you, then you will plan experiences for children which enable them to discover and learn mathematics skills through problem solving activities.

In this extract from Relational and Instrumental understanding, Richard Skemp outlines two contrasting views of mathematical understanding. That is, relational and instrumental understanding.
from Skemp, R (1977) Relational and Instrumental understanding. Mathematics Teaching 77, pp 20-6.

Consider the following:

If children rely on instrumental understanding what are the implications for their competence and confidence in the long term?

Discuss your response with a colleague if possible.

Implementing and continuing to learn:

Each of the books listed below has been recommended as course readers, or has been consulted in the writing of these modules. Any of them would help you to further develop your understanding and support you to follow up on particular areas of interest.

• Aubrey, C. (1997). Mathematics Teaching in the Early Years: An Investigation of Teachers' Subject Knowledge. London, Routledge.
• 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.
• Haylock, D. and Cockburn A. (2009) Understanding Mathematics for Young Children. London, Sage.
• Hiebert, J. et al. (1997) Making Sense: teaching and learning mathematics with understanding. Portsmouth NH, Heinemann.
• Holton, D. (1999) Teaching Problem Solving, Chichester, Kingsham Press.
• Liebeck, P (1990), How Children Learn Mathematics: A Guide for Parents and Teachers. London, Penguin.
• Mason, J. Burton, L. & Stacey, K. (2010) Thinking Mathematically (2nd ed). London, Prentice Hall.
• Montague-Smith, A. (2002) Mathematics in Nursery Education. Oxford, David Fulton.
• Skemp, R. R. (1987) Psychology of Learning Mathematics. New York, Routledge.
• 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.

You have now completed the Early Years CPD Modules, Counting and Number. Draw up a plan of action.

Consider what you will aim to do or try out:

• tomorrow
• next week
• next year.

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

At the time of publishing, the Early Years Framework, Primary and Secondary Curriculums were being revised. You might like to monitor the development and implementation of the new curriculums and consider whether or not they resonate with what you have learned.

We hope that these modules have whetted your appetite for further study. There is a wealth of resources to support your further professional development on the NCETM portal and elsewhere.

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