1. Developing Early Number Sense 16 March 2011

2. Number Sense Having a good intuition about numbers and their relationships. Develops gradually as a result of exploring numbers, visualising numbers, forming relationships Grows more complex as children learn more.

3. Key Mathematical Ideas Early number sense Counting tells how many are in a set. Ordinality leads to Cardinality Numbers are related to each other through a variety of number relationships more than, less than, connection to ten Number concepts are intimately tied to the world around us. Application to real settings marks the beginning of making mathematical sense of the world. Van de Walle, Karp & Williams Elementary & Middle School Mathematics: Teaching Developmentally Allyn & Bacon 2010

4. Early number sense develops when Children make connections Are able to instantly recognise patterns See relationships related to more, less, after, before, Are able to anchor numbers to five and ten

5. Subitizing The ability to recognise and name small quantities without counting – links directly to cardinality Use dot cards, dot plates, tens frames, slavonic abacus to provide opportunities every day for children to practise

6. Dot Plates Hold up a dot plate for 2-3 seconds, ask “How many? How did you see it? Children copy the pattern they saw on the plate on a piece of paper with counters Children need to be able to recognise the pattern without counting More, less, same – find a dot plate with the same number, more, less, one more, two more

7. Tens Frames Crazy Mixed up Numbers – Read the activity page 46 A diagnostic task – give your children a blank piece of paper and ask them to draw a tens frame and show a number on it In groups – discuss useful activities for tens frames for children at your level

8. War Deal out the dot cards Players turn over a card each. The winner is the player with the highest card. The winner takes the two cards. Keep playing until all the cards have been turned over. Winner is the player with the most cards.

9. Double War Deal out the dot cards Players turn over two cards each and adds them together. The winner is the player with the larger total number.

10. Difference War A set of Dot cards, a pile of 30 to 40 counters. Deal out the cards to two players Each player turns over a card. The player with the greater number of dots wins as many counters from the pile as is the difference between the two cards. The players keep their cards. The game is over when the counter pile runs out. The player with the most counters wins the game.

11. Number Sandwiches Select a number between 5 and 12 Find combinations of two dot cards that total that number. Place the two cards back to back with the dot side outside. When at least ten pairs have been found the next challenge is to name the number on the other side. The cards are flipped over to confirm the answer. The same pairs can be used again to name the hidden part.

12. Counting Principles Gelman and Gallistel (1978) argue there are five basic counting principles: One-to-one correspondence – each item is labeled with one number name Stable order – ordinality – objects to be counted are ordered in the same sequence Cardinality – the last number name tells you how many Abstraction – objects of any kind can be counted Order irrelevance – objects can be counted in any order provided that ordinality and one-to-one adhered to Counting is a multifaceted skill – needs to be given time and attention!

13. The counting sequence Learning the counting sequence is essential and will precede what counting one to one achieves. It is a rote process that is needed to lighten mental load. Knowing the word sequence pattern comes before understanding why the pattern occurs.

14. Counting one to one A critical piece of understanding is that ordinality – position in a sequence – is intimately linked to cardinality – the number in a set. In order to make the crucial linkage children need to be able to: –Say the number words in the right order starting at one –Point at objects one-by-one –Co-ordinate saying the correct words with identifying the objects one-by-one Need to spend time on this, do not expect it will happen quickly

15. Counting from ten to twenty In English the number words from ten to twenty have no regular pattern from a child’s point of view. Learning to count from ten to twenty there is a heavier load: –Eleven bears no relationship to ten and one –Twelve is not linked to ten and two –Thirteen is not decoded by knowing “thir” means three and “teen” means ten –Fourteen is not decoded by it means four and ten, which logically should be ten and four Learning to count from one to nineteen is a rote process

16. Counting to a hundred The next number after nineteen is twenty It’s difficult for children to understand that “twen” means two and “ty” means tens. Then the numbers follow the rote by ones count – to twenty-nine… Understanding the meaning of thirty, not twenty-ten, is a place value issue. Therefore counting to one hundred needs to be rote first and place value understanding must be given time to develop.

17. Counting on Counting on is useful to solve addition problems. But it is complex. To do 19 + 4 children need to: –Start the count at 20, not 19 –Say the next four numbers after nineteen and then stop –Understand the last number they say is the answer. –Have a reliable way to check four numbers have been said Place Value is the critical understanding here.

18. What do we need to do with counting? Talk with children about the counting process. Help them to make links with one more and one less. Connect number words with objects Make sets and count, reorganise the same set, do we need to count. Watch how children operate – it tells us a lot about what they know.

19. Tiles Roll a dice – (1-6) – 5 is rolled Count out number of counters, cover a tile as you count Last number you say tells you how many

20. Tiles You have five marbles. Your friend gives you three more. How many marbles do you have now?

21. Tiles You have nine marbles. You loose three to your friend. How many marbles do you have now?

22. Place Value The most difficult concept for children to master. Why? It is hard! Children need to understand the canon – ten for one Zero as a place holder Language issues – what does “ty” mean? -what does “teen” mean? -Write “sixty” on the board – ask How many bundles of ten do we need to make sixty?

23. Stages 0-1 Emergent and Stage 1- counting activities, recognising the number, word and set. Grouping activities. Not yet ready for ten for one rule. Grouping with five – instantly knowing patterns for five Moving to patterns with ten Ability to subitise is important here.

24. Stage 2 - Numbers 10 – 19 Core place value idea is to understand that number from 10 to 99, whether spoken or written have two different mental representations (count by ones and groups of tens and ones) Stage 2 - Key idea is that children practise bundling and connecting to symbols - 17 is read as seventeen, one ten and seven ones, modeled as one ten and also seventeen ones. Children need lots of opportunity to group materials into tens and ones - beans and containers, sticks and pipe cleaners Bundles ofLoose tenones 17

25. Stage 3 As for stage 2 but with largest number increasing from 19 to 99. Children need experience with grouping materials so that eventually it is understood that one ten and ten ones are the same. Plenty of experience with bundling, beans and containers See 24 and say it in two ways - 2 tens and four ones and twenty-four, do twenty-four sticks, do two bundles of ten and four loose sticks.

26. Materials for Stage 2 - 3 Materials need to be representational. Children need to be able to see concrete representations of ones and tens. Ten ones must be able to be bundled or organised into tens Children must develop their understanding that the bundle is ten individual ones and one ten at the same time. Materials: –Sticks and pipe cleaners – bundling to ten –Beans and photo canisters –Counters and plastic bags –Unifix and wrappers

27. Stage 4 – numbers from 100-999 Numbers increased to 999 Children need experience with grouping materials so that eventually it is understood that one ten and ten ones are the same. Ten tens is exchanged for 1 hundred. Place value money can be used here. Continue experience with bundling, beans and containers, place value money, place value houses, digit cards Explore “teen” meaning one ten, “ty” meaning lots of ten. Give children number words. Provide a mix of numbers and words is a powerful indicator of understanding. Sixty - 20, 60 + twenty seven, twenty three + fifteen 200 plus 4 hundred

28. Stage 4 – numbers from 100-999 Materials: –Sticks and pipe cleaners –Beans and photo canisters –Counters and plastic bags –Unifix and wrappers –Tens frames –Place value play money –Place-value blocks –Open abacus Mental method for addition not expected – use materials. Continually practising the ten for one swap. Using a mix of numbers and words is a powerful indicator as to whether children are understanding place value.

29. Stage 5 – Part-whole Crucial that children are given opportunities to solve problems where one number is a tidy number. Using tens frames to solve 28 + 2 = Use place value money where swap is involved Mentally solve 4 + 46 Move to problems where part whole thinking is required and the numbers move through a decade. Move through teaching model –tens frames, imaging, mentally solve –Place value money, imaging, mentally solve

30. Stage 5 – Part-whole Essential that children can engage in the internal talk of place value. For example 56 + 78 would be: Six ones and eight ones equals fourteen ones Swap this for one ten and four ones One ten plus five tens plus seven tens makes thirteen tens Swap this for one hundred and three tens The answer is one hundred and thirty-four

31. See, say, do - Peter Hughes Say the numeral one way, e.g. 13 is thirteen Write the numeral e.g. 13 Model the numeral as ones e.g. as 13 ones Model the PV form of the numeral e.g. 13 is 1 ten and 3 ones Say the numeral in the other way, e.g. 13 is ten and three

32. Repetition - see, say, do There needs to be extensive repetition of problems using see, say, do in no particular order Show 14 on the board. Say “Get me this number of blocks from the box” Show a plastic bag and say “If you put ten in this bag how many will be left? I have a packet of ten lollies and I have seven loose lollies. How many children in the class can have a lolly? Get out 28 sticks. You are going to bundle them into tens. How many tens will there be? How many loose? Check by doing the bundling Peter Hughes

33. Basic Facts By stage five instant recall of basic addition facts is required. There is plenty of time to learn them. A framework for learning basic facts: Stage 2: Addition and subtraction facts to five Stage 3: Essential to recall addition and subtraction facts to five Optional – Addition and subtraction with sums up to ten, doubles to ten

34. Stage 4 Essential for part-whole reasoning that comes in stage five is the instant recall of basic addition and subtraction facts with answers no more than ten. Addition and subtraction facts up to ten Doubles – to ten Optional: – Addition and subtraction facts from 1 + 1 to 9 + 9 - Derive and learn the two times tables from doubles.

35. Stage 5 Essential for advanced additive thinking in stage six is the instant recall of all addition and related subtraction facts 1 + 1 to 9 + 9 Recall of multiplication facts can begin with a focus on the commutative principle for multiplication Stage 5: Essential – Addition and subtraction facts from 1 + 1 to 9 + 9 -Derive and learn the two times tables from doubles. -Derive and learn the five times table -Derive and learn the three times tables from 3 x 3 to 3 x 9 using repeated addition and the reverse facts. Optional: - Four and Five times table

36. Strategise – Practice - Memorise 1.Start with strategies 2.Plenty of Practice 3.Move on to memorise the basic facts

37. Strategies that help for addition Links to counting – Adding nothing leaves the number unchanged Adding one gives the next number in the counting sequence Adding two corresponds to the skip counting pattern

38. Strategies that help for subtraction Links to counting – Taking off zero (nothing) leaves the number unchanged Taking off one gives the previous number in the counting sequence Taking off two corresponds to the skip counting pattern

39. Strategies - Addition Doubles and near doubles plus or minus 1 plus or minus 2 Make a ten Near groupings to ten 6+5, 6+3 Plus nine – plus 10, take away one

40. Strategies - Subtraction Halves The opposite of addition – subtraction as counting on from, 8-6 as count on from 6 until reach 8 Subtract nine – take away 10, add one Derived from addition using the family of facts

41. Basic Facts Learning of times tables 0 times or times 0 –A principle not a table 1 times or times 1 –A principle not a table 10 times or times 10 –An English language issue, not a table

42. Assessment Once some facts have been learnt regular assessments, both written and oral, help strengthen the learning. It is crucial that all errors be corrected immediately then the correct fact practised If the student got 6+3=8 the teacher ensures that the child writes 6+3 =9, 3+6=9, 9–6 = 3, 9-3=6. The student should repeat saying, seeing, and writing each fact at least 5 times.

43. Connecting oral to written Important that children are given opportunities to practise often. Practise must be correct. Take one or two facts to memorisation at a time. Oral connection to basic facts is important for the brain

44. Missing Number worksheet Begin with circles and ask children what they notice about the numbers Teach the children the circle always has the answer Fill in sheet with two numbers children have to find missing number 62 610 6 4

45. Triplets – Family of Facts Introduce triplets 10, 6, 4 Make chains of number triplets Try 2 out of 3 10 6 4 4 2

46. Tens Frames Hold up a tens frame and have the children say the 10 facts that go with the card. Children need to be able to say the four connected facts that go with each tens frame Seven and three makes ten Three and seven makes ten Ten take away seven is three Ten take away three is seven As children tell the story it is important they see written forms – words and symbols

47. Games to help children with basic facts.

48. Add to ten Two players Deal all cards out between two players. Take turns to turn over one card - state what else makes 10. Also play by taking number off ten. Modify for younger students – make five (remove some cards, use five frames/tens frames Working backwards - subtraction is harder. Children need lots of practise with subtraction

49. 1,2,3 Fists - Paper, Scissors, Rock Two players Play as for Paper, Scissors, Rock One or two hands Count 1,2,3, put down some fingers - add/multiply together

50. Make Ten, Two players Deal out ten cards in a row. First player looks across the row for combinations that make ten. Aim is to collect as many cards as possible, so combinations that require more cards are best. Continue playing until all the cards are used or until there are no more combinations that add to ten. Winner has the most cards.

51. Make Ten again, Two players Deal all cards out in 3x3 grid Take turns to make 10 - Continue playing until all the cards are used or until there are no more combinations that add to ten. Winner has the most cards.

52. Salute You need three players A pack of playing cards (take out 10s and colour cards Two players collect one card each. Without looking at the card they put it on their forehead. The third player calls out the sum of the two cards The two players then call out what card they hold on their forehead by looking at the other player’s cards. The player who calls out first wins those cards. Continue playing until all the cards are used. Variations 10 more or ten less/ one more or one less Multiply Doubles

53. Speed (War) Two players Deal all cards out between two players. Place one card in middle. - e.g. 2 (add this number to card that is turned over) Take turns to turn over one card - both players call out answer. First to call wins both cards. If a tie, turn over another card. Highest card gets to keep all three cards. Also for multiplication

54. Grab Five Grab five sticks Put them in order from smallest to biggest. Winner is the first one to grab the object from the centre of the table. Must have sticks in the right order. Can be made to fit children from Year 1 - 8

55. If I know, then I know To help children make connections with what they know and how it helps them to solve other problems Makes links to knowledge they have Independent and group activity. Can begin as a whole class warm up. With a partner - if I know 4 + 3, then I know… Share with other group at your table.

56. Circle a Fact Place a set of A4 numeral cards zero to nine in a circle on the floor. Children form a circle around cards or make two teams either side of the circle. Two people walk around the outside of the circle, on stop place their toe on a card. Winner is the person who calls out answer first. They can –Add the two numbers together –Double the numbers –Add 10, double plus or minus one or two –Multiply the numbers –Find the difference of the two numbers

57. A thought to leave you with …listen to children’s mathematical explanations rather than listen for particular responses. Fiona Walls in Handling Number p.27s Teaching Primary School Mathematics and Statistics Evidence-based Practice Averill & Harvey (Eds) NZCER 2010