Children’s Mathematical Thinking, number development and problem solving.

1. Children’s Mathematical Thinking, number development and problem solving.

2. Objectives • To have an understanding and overview of how children understand number and develop number skills. • Explore what children's mathematical thinking might look like. • Have a back ground understanding of theory behind children's mathematical thinking and learning. • Look at problem solving opportunities in the Early Years

3. Research • Under three’s mathematical learning. European Early Childhood Education Research Journal 2015. Vol 23 43-54 Routledge. • Playing with maths: implications for early childhood mathematics teaching from and implementation study in Melbourne, Australia. Education 3-13 8th Jan 2014 Routledge.

4. What do we mean by Mathematics in Early Years?

5. Maths used to be known as PSRN (problem solving, reasoning and number) • Development Matters now has ‘Number’ and Shape Space and Measure’ • However the content and requirements have not changed.

6. Let take a moment and look at those pages in Development matters. • Read numbers from birth through to learning goal. • Discuss in your group • What can you see that you understand as progression? • Is it what you expect? Is there anything that surprises you? • What do you think some children might find hard?

7. Maths in Early Years can be done with in pretty much anything and is to do with numbers, patterns, shapes, measures and spaces. This means that maths is everywhere. As an Early Years Teacher you will teach maths session to the whole class, you will take small groups that will focus on specific areas. But most importantly you will need to be able to see the children demonstrating their mathematical thinking in their play. This is when you can really see what they have understood and are using independently in order to plan effectively for their next steps.

10. The Theory Bit…… Developing a knowledge and understanding of number and counting skills is more complicated than it might appear to an adult for whom counting and number are now second nature and require no attention. Learning number and counting are intertwined and require children to be able to perform a number of different cognitive operations. “Early number consists of a network of inter-related skills and knowledge broadly divided into cardinal and ordinal aspects…. Both are commonly associated with counting….. However, the simpler forms of counting – reciting a number string just as a verbal activity….(is) independent of both cardinality and ordinality” (Bruce,B. and Threlfall, J. 2004 p.3)

11. A bit more……… • It is argued that children enter the world able to ‘subitise’. Subitising is the ability to “determine the cardinality of a small number of objects without overt counting” (Bruce,B. Threlfall, J. 2004 p4) • If we accept that children have this ability then we have to acknowledge that they come into school with some cardinality, which is the aspect of number which concerns the quantifying of a set of objects using a number name. • In contrast it seems that ordinal number names are not so easily learned and will come some time after conventional counting. • Number conservation requires children to understand and know that if two groups have been counted and found to be the same, rearranging the group without addition or subtraction will not change the size of the group and so re-counting is not necessary.

12. Early Learning Goal “ Children count reliably with number from one -20, place them in order and say which number is one more or one less than a given number. Using quantities and objects, they add and subtract two single-digit numbers and count on or back to find the answer. They solve problems, including doubling, halving and sharing.”

13. Evidence All evidence suggests that the better mathematical understanding young children have, the greater their achievements later on. However, that “does not mean teaching ‘school maths’ earlier” (Montague-Smith A. 2012:11) there is evidence to suggest that this has a negative effect on learning. Rather it is appropriate to teach maths in the early year as long as it is the right maths taught in the right way.

14. Counting When children begin counting they are copying counting behaviours from the adults they observe. Dr Penny Nunn (1997) observed that during joint activities with adults and very young children the process of saying the words seems important and the actual aim of finding quantity is not emphasised. Gelman and Gallistel (1986) identified through their research 5 principles that are used when counting which can be divided into two groups.

15. Gelman’s 5 principles How to count • The 1:1 principle. • The stable – order principle. • The cardinal principle. What to count • The abstraction principle. • The order-irrelevance principle.

16. Counting Skills and Ideas In conjunction with these five principles is the development of other significant skills and ideas: • Subitising • Making a reasonable estimate of a number without counting • Hierarchical inclusion • Compensation • Conservation of Number

17. It’s worth remembering that number is an abstract concept. • We know that very young children are born with a sensitivity to quantity. (Subitization) • In Early years we often use numbers as adjectives to describe small quantities. • Gradually children recognise what is the same about the groups, 2 cups, 2 socks, 2 stones etc. and attached the word 2 to the concept. • Cardinal number is the use of a number to label a set, to say how many, number conservation is the understanding that a counted set, rearranged will not change. • Ordinality describes the order that things come 1st, 2nd • Reciting numbers is separate again, and all areas develop separately. • Using actions and songs will help reinforce children’s learning.

18. This short clip shows what Piaget describes as the pre-operational stage in development, where the child does not yet have a concept of conservation.

19. (Kamii et al. 2004) argue in their study, that 1-4 year olds building with large and small bocks can be seen as using logico-mathematical knowledge for problem solving. But what does that mean…….. Logico-mathematical knowledge was identified by Piaget as one of three types of knowledge

20. 1 & 2 Physical Knowledge Identified learning through senses, the sun is warm, the dog is brown, chocolate is tasty. Social knowledge Discovered through the sense but also has to be told to us, my name is Rebecca, we drive on the left.

21. 3 Logico mathematical knowledge Is constructed inside our brain, it is the connections constructed our selves, the knowledge of relationships that didn’t exist until we made them. Abstract nouns, fourness, number. This knowledge once constructed is something, where before there was nothing.

22. However this knowledge is unique to each person, what one person will know will be different for another. What we construct as ‘something’ in out minds becomes as tangible to us as if it was real in nature and it is virtual impossible to imagine what it was to know have this knowledge. This knowledge is constructed logically by each individual. School and the experiences we have stimulate this development.

23. Gifford 2005:152 argues that problem solving is ‘a major vehicle for learning’. If logico mathematical knowledge is stimulated through experiences in school and is the knowledge used in problem solving we want to explore how we can stimulate children thinking and understanding, building those connections in the early years to provide children with the right type of mathematical understanding and thinking to engage successfully later on with more complicated mathematical ideas.

24. Problem Solving As children become secure in their counting they will automatically begin to make comparisons. Normally these will focus on fairness which gives us an obvious lead into Sharing – division – fractions - partitioning

25. Activity 1– Tower building 1. On your tables are some lego pieces, can you build a tower. 2. Compare your tower to a neighboring table, is there are problem? Ask these questions • Are they the same? • How many blocks have been used? • Why are they different? • Can you make them the same?

26. Activity 2 – Sharing (partitioning) On your table can you share the duplo equally? If not why not? What can you do to solve this problem? Can they be divided equally at all? Once a set has been partitioned it can then be put back together, combined again – addition.

27. Children need to be secure in ordinal number values for this, and be able to begin to be able to count on from a given number. Key Questions to use with children • How many? • Who has more? • Are there enough?

28. Further Problem solving activities • Sand sorting • Big dice game • Train track • Numicon • Dominoes You have 10 minutes at each activity and then we will feed back our thoughts.

29. How else do children express mathematical thinking….. • We have looked at some pictures of how mathematical thinking can be seen in construction play, and some test that show pre number conservation. • Children also express their mathematical thinking through their mark making and graphics.

30. Children’s mathematical mark making • We know a child's first marks are a major development in a child's steps towards multi-dimensional representations of their world. • Children will refer to their marks as numbers when they are, and so it is our responsibility to investigate what the children are exploring in their marks and not make assumption! • Lets have a look at some examples……

33. In the nursery Joe (age 4yrs 3months) was playing with soft toy spiders. He later chose to draw a picture of a spider (see figure 1). He showed his drawing to the teacher and said, 'My spider has got eight legs.' He had drawn the spider with many more than eight legs. Looking at the spider he saw lots of legs and represented that idea in his drawing. Joe showed a growing awareness of number and quantity and was able to describe it. He knew that a spider has eight legs and you can represent that idea in your drawing. Joe has used numbers in his own meaningful way. There is liveliness in Joe's spider. He was uninhibited in his demonstration of his knowledge of spiders and number. We want to keep this mathematical thinking as Joe progresses through his education. The mathematics he used is what we have termed dimension 4a, 'representing quantities that are not counted'. This early development ensures firm foundations for early addition and subtraction, explored in ways that make sense to the child. One fascinating area that was revealed is the way in which children explore symbols in non-standard ways, by implying or inventing symbols that can be read as though the symbol was there. This understanding leads to calculations that also include multiplication and division. • Primary Mathematics (Mathematics Association). November 2003. Vol. 7. Issue 3. pp. 21 - 25 • Research Uncovers Children's Creative Mathematical Thinking Maulfry Worthington and Elizabeth Carruthers

34. Kamrin (5years 7months) was in a reception class. The children had been introduced to several forms of division and the teacher had modelled the children's ideas of division. Kamrin chose the number eight to work out if it could be shared equally without leaving a remainder. Kamrin represented this in his own unique way (see figure 2). He created the 'tweedle birds', giving each bird four eggs. He then wrote the numeral eight and a question mark. At first he decided that he could not share eight equally and put a cross. He checked this and then decided that he could do so. He scored through his cross and put a tick beside the cross. Kamrin used a combination of symbolic and iconic forms of graphics. He has not only progressed beyond representing quantities but used his knowledge of counting to find out about division. He has also skilfully self-corrected. • Primary Mathematics (Mathematics Association). November 2003. Vol. 7. Issue 3. pp. 21 - 25 • Research Uncovers Children's Creative Mathematical Thinking Maulfry Worthington and Elizabeth Carruthers

35. In the class room Quickly brain storm on your table ways that you could create opportunities for children to express their mathematical thinking? What opportunities could you provide to encourage them to explore mathematical mark making?

36. To finish • We have looked at some of the theory behind how children’s mathematical thinking develops • How and what they are expected to achieve. • What and where you might see this learning. • And how that might look.

37. Further reading • Bruce, R. Threfall, J. One, two, three and counting. Educational Studies in Mathematics. 2004 March vol55 issue 1 (3-26) • Carruthers, E. and Worthington, M. (2006) Children’s mathematics: making marks, making meaning, 2nd edn. London: Sage Publications. • Hughes, M. (1986) Children and number: difficulties in learning mathematics. Oxford: Blackwell. • Munn, P. (1997) Writing and number, in I. Thompson, (ed.) Teaching and Learning Early Number, 1st edn.  Buckingham: Open University Press.