1 Counting, Understanding Number and Place ValueMark Harris 2017/18

2 Objectives of this SeminarTo identify the key steps in progression for the teaching of counting and place value in KS1 and KS2 To develop an understanding of the CPA approach to teaching new concepts in mathematics To explore the representation of counting and place value using models and images To share problems, contexts and rich tasks to support creativity in the teaching of counting To identify areas in which you need to develop your knowledge and understanding of mathematics as a learner yourself and as a teacher.

3 Teaching Standards TS 3 Demonstrate Good Subject and Curriculum Knowledge TS3a: Have a secure knowledge of the relevant subject/s and curriculum areas, foster and maintain pupils’ interest in the subject and address misunderstandings TS3b: Demonstrate a critical understanding of developments in the subject and curriculum areas, and promote the value of scholarship TS3d: If teaching early mathematics*, demonstrate a clear understanding of appropriate teaching strategies. (* early mathematics refers to mathematics taught in both the early years and primary age phases)

4 Research Gelman, R. & Gallistel, C. (1978) The Child's Understanding of Number Munn,P. (1997) Children’s beliefs about counting Ian Thompson (2003) Rules Ok? ATM The National Strategies (2011) Research Project – children who get ‘stuck’ at level 2C in mathematics

5 Numbers are Abstract Practise counting in Jack and Jill as the new number system – forwards and backwards Ask the following questions: What is one more than Jill? One more than fetch? What is one less than hill? One less than pail? Discuss difficulties so far in answering these questions. What are the trainees relying on? Ask: What is went + hill? a What is to subtract the? Jill Hide the number grid. Ask: What is up plus Jill? The What is pail take away up? Hill What is double Jill? Up What is a half of hill? Went Share progression grids with the trainees. Ask them to identify which year group the questions that were asked relate to (reception). What does this mean you need to consider when teaching early mathematics?

6 Some Challenges in MathematicsMaths uses familiar words in an unfamiliar context Numbers are abstract ideas…all we can show children are number representations Numerals are arbitrary symbols…

7 Purpose of Counting Dr. Penny Munn (1997) observed that:during joint counting activities between very young children and adults the process of saying words seems important, but the actual aim of finding quantity is often not emphasised the strength of children’s own natural concentration on the physical aspects of counting activities, touching and handling, often obscures the intended mental function of finding quantity children are dependent on adults to provide for them, they have no urgent real reason to check and tally counting words often occur as parts of games that are not to do with quantity, e.g. One, two, three – go! Research by Penny Munn (1997) has shown that whilst young children may appear to be able to count, many do not understand the purpose of counting. It shows pre-school children do not see counting as a way of "knowing how many", but as an enjoyable social activity or one that pleases adults. The idea of counting for quantification is most often not appreciated or understood. Counting is an essential skill which underpins many areas of later mathematical knowledge and understanding. Whilst on the surface counting may appear to be a relatively simple, straightforward skill, it is complex and consists of a number of different principles.

8 Structured Images Some resources can be used to develop an understanding of counting. Share the resources on the table and ask trainees to consider how they develop counting: numicon, cuisinaire,

9 Misconceptions and Errors in Counting?Counting 1, 2, 3 then any number name or other name to represent many Number names not remembered in order Counting pattern not stable, counting names out of sequence Count does not stop appropriately Counts one item more than once or not at all Does not recognise final number of count as the cardinal value Thinks that because objects are spread out there are more Counting backwards is harder than counting forwards Bridging boundaries (e.g. tens and hundreds) can be very challenging What misconceptions do the trainees think they may encounter when teaching counting? List on board……

10 Developing Counting SkillsTo count accurately and with meaning there are five principles that must be understood and applied, as outlined by Gelman and Gellistel in 'The Child's Understanding of Number' (1978) These may be described as: one to one principle stable-order principle cardinal principle abstraction principle order irrelevance principle Gelman, R. and Gallistel, C.R. (1978) Discuss what each might be and link to previous misconceptions…..before introducing 5 principles

11 Principles of CountingPrinciples about how to count: The one-to-one principle The stable order principle The cardinal principle (how children count) 1:1 Matching counting word to item In order to be able to count, a child needs to be able to understand that each object in the set should be counted once (partitioning) and that once an object has been counted, it cannot be counted again. In addition to this, a label or tag should be applied to that object, and the label can be used only once (tagging). Stable-order principle The labels or tags given to each object need to be in a stable, repeatable order. Counting words in order • Rote • Problems with decades especially 11 – 20 • Rhymes and stories Cardinal principle (How children count) • Final number in the count = cardinal number of the set • Stop at the last one • Recognise how many Because the cardinal principle can develop only after the 1:1 and stable order principles, it is often more difficult for children to use and it develops later.

12 Principles of CountingPrinciples about what can be counted The abstraction principle (what children count) The order irrelevance principle Abstraction principle (What children count) A child knowing that he is five years old or that he has his tea at five o’clock appears different to that child from counting five counters, and so the abstraction principle raises the question about what is countable to a child. Adults know that anything, real or imagined, can be counted. If a child does not fully appreciate how a counting procedure can be used to represent number then they have not grasped the abstraction principle. Children need experience in counting any set that they put together. Order Irrelevance principle It doesn’t matter where you start the count. Children need experience in counting the same set starting in different places. A child who demonstrates this principle is able to count an object as a ‘thing’ rather than associate it as a particular number. This demonstrates how the abstraction principle is inherent within the order-irrelevance principle.

13 Counting Skills and IdeasIn 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 Subitising - the capacity to recognise a small number of objects without counting them This is the ability to recognise the size of a set, its cardinality, from the pattern or structure, without having to count the number of objects (usually 5 objects). Dot patterns on dice, dominoes and playing cards are helpful for this, as well as small groups of randomly arranged shapes stuck on cards Making a reasonable estimate of a number without counting As adults we estimate all the time, children need lots of practise in this essential skill. You could have a constantly changing 'How many in the jar?' activity as part of your routines Hierarchical inclusion - understanding that numbers build by exactly one each time they progress – and that they nest within each other by this amount Compensation – which entails: appreciating the balance of increase and decrease applying understanding of whole and part relations within numbers understanding that different combinations can make up the same number…ie number bonds to 10 Conservation of Number – use video (click on image) Conservation is the understanding that something stays the same in quantity even though its appearance changes. To be more technical conservation is the ability to understand that redistributing material does not affect its mass, number, volume or length.

14 Subitising How many circles are there on the slide?

15 Subitising Perceptual subitising (Clements, 1999)Know the number of circles without needing to use another mathematical process. ‘Subitising’ or ‘dot enumeration’ is considered a key pre-requisite skill from numeracy by Butterworth (1999). The number of dots that a person can see and just know ‘how many’ with certainty, is around 4 or 5.

16 Subitising Conceptual subitising (Clements, 1999)Subitised two or more parts and then recombined Research shows conceptual subitising has a major influence on the development of number sense (Sayers et al, 2014) Children with dyscalculia will perform below the levels expected.

17 Progression in Counting, Reading and Writing NumbersYear 1 Year 2 Year 3 Count to and across 100, forwards and backwards, beginning with 0 or 1, or from any given number. Given a number, identify one more and one less. Count in different multiples including ones, twos, fives and tens. Count in steps of 2, 3, and 5 from 0, and count in tens from any number, forward or backward. Count from 0 in multiples of 4, 8, 50 and 100; finding 10 or 100 more or less than a given number. Read and write numbers to 100 in numerals. Read and write numbers from 1 to 20 in digits and words. Compare and order numbers from 0 up to 100; use <, > and = signs. Read and write numbers to at least 100 in numerals and in words. Compare and order numbers up to 1000. Read and write numbers up to 1000 in numerals and words. Identify how counting develops by altering the steps in which the children are counting in.

18 Progression in Counting, Reading and Writing NumbersYear 4 Year 5 Year 6 Count in multiples of 6, 7, 9, 25 and 1000. Find 1000 more or less than a given number. Count backwards through zero to include negative numbers. Count forwards or backwards in steps of powers of 10 for any given number up to Interpret negative numbers in context, count forwards and backwards with positive and negative whole numbers through zero. Use negative numbers in context, and calculate intervals across zero. Order and compare numbers beyond 1000.  Read Roman numerals to 100 (I to C) and understand how, over time, the numeral system changed to include the concept of zero and place value. Read, write, order and compare numbers to at least and determine the value of each digit.    Read Roman numerals to 1000 (M) and recognise years written in Roman numerals. Read, write, order and compare numbers up to and determine the value of each digit.

19 Progression in Counting ResourcesBead bar/string Bridges between a track and a line Still able to count and move individual beads Extends to moving tens Develops understanding of place value

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21 Progression in Counting ResourcesLabelled Number Lines Can be used for both positive and negative integers. Aids counting forwards and backwards. Supports calculations.

22 Progression in Counting ResourcesPartially labelled and empty number lines Use of landmarks to support counting Supports development towards reading scales Used as a bridge for calculation strategies

23 Counting activities Chanting activities – variations (forwards, back, starting points, intervals, voice, volume, speed) Counting rhymes Incy wincy spider - NCETM Counting stick activities

24 Counting stick Lets learn to count in 17s.

25 Take a break

26 What is Place Value? Royal Society’s Advisory Committee on Mathematics Education (ACME, 2008) explanation of place value is: ‘In our written representation a number like 305 means that the 3 is equivalent to 3 hundreds (and also to 30 tens), the zero indicates no remaining sets of ten but the 5 shows 5 remaining ones.’ Primary Mathematics Teaching Theory and Practise, 7th edition, 2014 Mooney et al ‘Place value is used within the number system to allow a digit to carry a different value based on its position, i.e. the place has a value.’

27 Common Difficulties for Children Working at Level 2CAlmost all children were able to count aloud in tens from 0 to 100 though there was some confusion when distinguishing between the ‘ty’ and ‘teen’ numbers. However, many children were unable to apply the counting to practical contexts. For example, once a group of objects had been arranged into groups of 10, almost all children still needed support to help them count how many objects there were altogether.

28 Common Difficulties for Children Working at Level 2CMost children were not able to recognise and state that there are 52 objects altogether in a set containing 50 objects that were arranged in 5 groups of tens alongside another 2 individual objects. In a test question, none of the children working at level 2C could identify that 37 has 3 tens; this compared with 60% of the children working at level 2B. Source: Research into children who get ‘stuck’ at level 2C in mathematics, National Strategies (2011)

29 What is place value? (Ross 1989)Positional Base 10 Multiplicative Additive (Ross 1989) Discuss what you think each of these terms mean in the context of place value Positional- the quantities represented by the individual digits are determined by the positions that they hold in the whole numeral. The value given to a digit is according to the position in a number Base 10: the value of the position increases in powers of 10- regrouping to make a numbers efficient Multiplicative; the value of an individual digit is found by multiplying the face value of the digit by the value assigned to its position. Additive: the quantity represented by the whole numeral is the sum of the values represented by the individual digits i.e 346 = partitioning

30 KEY Principles of Place ValueDigits – there are only ten digits in the system Position – the columnar position of the digits determines its value Base 10 – in our system we use base 10. Columns represent increasing / decreasing powers of 10 Zero – we use zero to represent an empty column (0 as a place holder) Grouping and exchanging – once we have ten objects in a column, we can exchange them for one object in the next column to the left and vice versa. Reinforce here accurate use of ‘digit’ and ‘number’ If we stop to consider that here are so many principles to learn and understand, it enables us to reflect upon challenges that this presents for children. The first obstacle faced by young children is that numerals are abstract concepts. When children learn to count, they usually do so by being introduced to concrete objects that can be moved, touched and seen. Numerals themselves bear no relation to the objects with which children are familiar.

31 Hindu-Arabic Number SystemLarge whole numbers are constructed using powers of the base ten: ten, a hundred, a thousand… Ten 10 = 1 x 10 =101 A hundred 100 = 10 x 10 =102 A thousand 1000 = 10x10x10 =103 Easy to remember as there is 1 zero in ten, 2 zeros in hundred, etc. Powers of 10 are not limited and can continue indefinitely

32 Concrete, Pictorial, AbractBruner’s model: Concrete-Representation-Abstract Singapore Maths: Concrete-Pictorial-Abstract This model has been shown to be particularly effective with students who have difficulties with mathematics (Jordan, Miller & Mercer, 1998; Sousa 2008)

33 concrete, pictorial, abstract?Models concrete, pictorial, abstract? Give time to explore and look at the different representations – need resources for tables Instigate a discussion about progression in representations – where would you start? Manipulatives - Manipulatives are defined as concrete objects (things you can touch and move around) Possibly move onto a discuss about adv and disadv of using manipulatives Benefits of Manipulatives in the Classroom – following are notes to prompt discussion The National Council of  Teachers of Mathematics (NCTM) strongly advocates the use of manipulatives in the classroom. The NCTM believes manipulatives allow students to actively construct their own understanding of math. Through manipulatives, students are able to explore, develop, test, discuss, and apply ideas of math concepts through the use of manipulatives. Manipulatives make math concepts interesting and engaging for students (especially the younger grades). Computers can also be used as a manipulative - contributing to your technology standards. Many resources can be found online and incorporated into lesson plans easily. Manipulatives encourage active learning and class or small group participation. For early elementary students, manipulatives are great for patterning, counting, sorting, and organizing. In middle and upper elementary grades the use integer bars, grid paper, geometric shapes, compass/protractor, solids, 3D objects are all great manipulatives that engage students and aid in instruction. Concerns About Manipulatives in the Classroom While I absolutely agree with the value manipulatives play in the classroom and have used them, I am concerned that manipulatives may prevent students from learning abstract thinking skills. Struggling students are found to benefit from instruction that is explicit and systematic. Not all children learn in the same way or react to physical materials in the same way. What works for one may not work for all.

34 Place Value Chart Place value charts can be used to help support children’s understanding of place value. The use of colour helps children to recognise the link between tens and tenths. Th H T 1s t h th Tens are one place to the left of the ones and tenths are one place to the right of the ones. It leads on easily to hundreds and hundredths, thousands and thousandths, etc. If we draw children's attention to pattern in mathematics it makes sense. If something makes sense then it is remembered easier.

35 Digits and numbers Draw a 4 square gridRoll the die – the score gives the multiple of 1000 that is the target Take turns to roll the dice. Decide which of the four squares to put the score in. Brief activity in pairs Other version of the game: After rolling the dice, you could either decide to record the number on your own or your partner’s board. Another version: Pairs work collaboratively and decide which board to record numbers on in order to create two 4-digit numbers that have the smallest difference. Numbers created could form the basis of questioning – who has largest number? Smallest? 3 hundreds? Etc. Opportunities for public talk: Talk about strategies developed. Could you predict if someone had won before the end of the game? Extend: Use a number line to model how to calculate the exact different between each of the scores and the target number.

36 Abacus Investigation H T 1sDraw an abacus like this and use 6 counters. Place all six counters on the abacus to make different numbers. (e.g. 312 by placing three counting on the hundreds column, 1 on the tens and 2 on the ones) What is the largest/smallest number you can make? Are there more even or odd numbers created? Explore the numbers you can make that are: multiples of 5 multiples of 3 multiples of 4 H T 1s Investigation in pairs Spend some time discussing the mulitples Discuss activity in terms of differentiation and several outcomes? Did anyone develop a system?

37 Partitioning Encouraging the children to partition numbers regularly reinforces their place value understanding. E.g. 784 = (Various models and equipment can be used to reinforce this understanding for younger pupils) Other forms of partitioning can are also used to support future understanding of subtraction and division methods. 78 = 78 = 78 = Ask trainees to partition numbers into HTU.

38 Multiplying and dividing by 10Multiply 23 by 10 Divide 6 by 100 Explain how you did this… Mention article they will be reading later

39 Number Investigation ‘Ah’ the old man said, ‘I must have slept for a million hours.’ Is this possible? How could this be investigated? What other information will be required? Will you need to make any assumptions?

40 Representing DecimalsDecimals are the way in which fractional parts are represented within the place value system. The key ways of looking at decimals are as part of a unit, as a point on a number line and as a result of a division operation. (Barmby et al 2009)

41 Representing decimals1.15 0.3

42 Use of Base 10 (Dienes) It is vitally important to call each piece of equipment by its correct name. Base 10 equipment is proportional equipment that can be used to represent digits in any number based on powers of 10. A small cube represents one: 10 small cubes can be built upwards to form what is termed a ‘long’ or ten. 10 ‘longs’ can be stacked backwards to form a ‘flat’ or a hundred. 10 ‘flats’ can be stacked to form a ‘cube’ or a thousand.

43 Using Dienes for Decimals

44 Partitioning DecimalsReinforces place value of decimals and recognises the use of zero as a place holder: 0.461 =

45 Ordering Decimals Use either the diennes equipment or decimal place value counters to create a representation of these decimal numbers: 0.54 1.27 0.8 1.4 0.92 Common Misconception: Children will associate the value of decimal numbers with the quantity of digits. E.g. They may think 3.74 is smaller than Ask trainees to order the decimals as well.

46 So what…? money measures addition subtraction multiplication division Which other areas of maths are reliant on a secure understanding of place value? Place Value Discussion activity and feedback – share these examples, not exhaustive , develop examples on the board. Mention addition seminar next week…

47 Place Value, Partitioning and Addition Using resources to develop fluency – key aim of the curriculum

48 Common Errors in Place ValueA teacher overhears a child in his class saying to his friend, write down the number 6 and the number 3. That makes sixty-three. The child is using the term ‘number’ incorrectly. In this situation, the number contains two digits. He is yet to understand that 0,1,2,3,4,5,6,7,8 and 9 should be referred to as digits when they are used in numerals. A child believes zero is the lowest number. His/her experience to date may have included no experience of negative numbers, or if they have met negative numbers (such as cold temperatures in winter) they may not have associated the negative numbers as less than zero.

49 Common Errors in Place ValueReversal of digits in the number: Reading and writing whole numbers: I’ve written the number fourteen! 41 That is the number sixty-ine! Reversal of digits: The child has written the numeral 4 first because in the English counting system, we say ‘fourteen’. The teens numbers often cause this difficulty. In addition, for 11 and 12, children must learn two new number names. Using ‘digit’ and ‘number’: The child is using the term ‘number’ incorrectly. In this situation, the number contains two digits. Perhaps the child has heard other children say four add 2 make six and he has borrowed some of the language to explain to his friend how to write 63. He has understood that four and six are numbers, but he has just to understand that 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 should be referred to as digits when they are used in numerals. So, 5 is a one-digit number, written using the digit ‘5’ and ‘5’ is also a numeral is a three-digit number, written using the digits ‘3, 2, 7’. It is a numerals, where the ‘2’ in the tens column has a value of 20. Reading and writing whole numbers: The child has read the digits as 60 and 9. The child does not recognise that the 6 is worth 6 hundreds because it is in the hundreds column. This type of error occurs because the child does not understand that the potion of a digit determines its value. The child may be unaware of the principles of grouping and exchange. Children need to have practical experience of grouping sets of objects into tens and then exchanging each group of ten for another object which represents a ‘ten’ and so on for hundreds, thousands, etc. 609

50 Three Stages of Understanding Place ValueStage 1 – the child can write a number correctly but cannot explain why Stage 2 – the child can recognise when a number has been written down incorrectly Stage 3 – the child can understand what each digit in a number represents Dickson et al (1984) and Ginsburg (1977) The teaching and learning of mathematics that depends on understanding place value is complex. To support child, teachers need to be able to accurately diagnose children’s place value errors. They should not attempt to re-teach poorly understood methods, but return to earlier stages of understanding of place value concepts. Children’s Errors in Mathematics p 24

51 Directed task Read the article PUTTING PLACE VALUE IN ITSPLACE by Ian Thompson Reflect on his findings – have you seen similar issues to those he raises in any lessons you have observed?

52 Further Reading Askew, M (2016) A Practical Guide to Transforming Primary Mathematics. Routledge Barmby, P., Bilsborough, L., Harries, T. and Higgins, S. (2009) Primary Mathematics Teaching for Understanding Maidenhead: OUP McGraw Hill Cotton, T (2010) Understand and Teaching Primary Mathematics. PearsonWitt, M (2014 Primary Mathematics for Trainee Teachers. London: Sage DCSF Numbers and Patterns: Laying foundations in Mathematics: DCSF (2009) Hansen A (2014) Children's Errors in Mathematics (Third Edition) London: Sage Haylock, D. & Cockburn, A. (2013) Understanding Mathematics for Young Children (Fourth Edition) London: Sage. Haylock, D. (2014) Mathematics Explained for Primary Teachers (Fifth Edition) London: Sage Thompson, I. (2008) Teaching & learning early number Oxford: OUP

53 Maths Audit Complete the maths audit in the Pre-Course Reading and Tasks Mark the maths audit (calculations Q9 has an error in the answers) ( ) x (12 – 3) = 135 Complete the Needs Analysis sheet and Confidence with Maths Subject Knowledge form Upload results to Moodle and Mahara (more details about this will be given at the next seminar)