1. Effective Mathematics Practise Workshop Whangarei 31 March 2011 Dianne Ogle d.ogle@auckland.ac.nz

2. Target Try to make today’s target in each of these ways -- 10 or 31 1.Adding two numbers. 2.Finding the difference of two numbers. 3.Multiplying two numbers. 4.Dividing one number by another. 5.Adding three numbers. 6.Multiplying three numbers. 7.Multiplying and subtracting. 8.Using a decimal. 9.Using a fraction. 10.Doing it an unusual way.

3. Write all the ways…. How many ways can you make 36 Show as many different ways as you can to make 36 – use materials, words, word stories, digits… After 1 minute you will pass your paper to the next person.

4. Objectives for today Investigate the Frameworks Develop an understanding of Strategy and Knowledge Develop an understanding of big mathematical ideas. Identify some addition and subtraction strategies Investigate Assessment tools -NumPa & GLoSS Learn some games for use in the classroom.

5. 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.

6. Big ideas Numbers are related to each other through a variety of number relationships - more than, less than, composed of “Really big” numbers possess the same place- value structure as smaller numbers. Best understood in terms of real- world contexts Whole numbers can be described by different characteristics, even and odd, prime and composite, square. Understanding characteristics increases flexibility when working with numbers

7. 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

8. Key Mathematical Ideas Developing Meanings for the operations Addition and subtraction are related. Addition names the whole in terms of the parts, subtraction names a missing part Multiplication is related to addition Multiplication involves counting groups of like size and determining how many there are in all. Multiplicative thinking Multiplication and Division are related. Division names a missing factor in terms of the known factor and the product. Models can be used to solve contextual problems for all operations, regardless of the size of the numbers. They can be used to give meaning to number sentences. Van de Walle & Louvin Teaching Student Centred Mathematics,

9. NZ Curriculum Objectives and Number Framework -What to teach Effective Pedagogy -How you teach it -How you respond to students and their misconceptions

10. The Purpose of the Numeracy Project Improve the achievement of students in Number and in the other strands. Develop the pedagogical and content knowledge of teachers to enable them to meet the learning needs of all their students. To promote the dimensions of quality teaching.

11. The Number Framework Embodies the achievement aims and objectives in Levels 1 - 4 A distinction is made between strategy and knowledge Progress through the stages indicates an expansion in knowledge and the range of strategies that children have available to them.

12. Strategy The mental processes children use to estimate answers and solve operational problems with numbers. Knowledge The key items of knowledge that children need to learn.

13. StrategyKnowledge creates new knowledge through use provides the foundation for strategies Three operational domains Add/Sub Mult/Div Prop/Ratios Four content domains Number Identification Number Sequence and Order Grouping/Place Value Basic Facts

14. Fifty and some more Say a number between 50 and 100. Children respond with “50 and ____. For 63, the response is “50 and 13” Use other numbers that end in fifty such as 350, 650 or 0.5

15. Number Knowledge Book One - Page 14 What are the key messages about knowledge from the framework. Work with a partner to create a thinking map Share back one key point.

17. Written Recording Written recording can be seen as a: Thinking tool Communication tool Reflective Tool Important that children are not using algorithms until they are able to use part-whole mental strategies. Why is this?

18. Stages and Expectations 8 Advanced ProportionalEnd of Year 9 7 Advanced Multiplicative Early Proportional End of Year 8 6 Advanced Additive Early Multiplicative End of Year 6 5 Early Additive End of Year 4 4 Advanced Counting End of Year 2 3 Counting from one by imaging 2 Counting from one on materials 1 1-1 Counting Emergent

19. Time to think! 10 + 4 83 - 28 Knowledge Strategy

20. Knowledge and Strategy are equally important 8 + 7 = 15 Strategy: I did 8 + 2 = 10, then 10 + 5 = 15 What knowledge did I need? 8 +2 = 10 (basic facts) 7 - 2 = 5(basic facts) 10 + 5 = 15 (place value) How to read and write the numbers 8, 7 and 15

21. 83 28 I have $83 in the bank. My new shoes cost $28. How much money is left in my bank account now? How would you solve this problem? Talk with your partner about what you did? Record on a piece of paper. (Thinking Map)

22. 83 - 28 = 55 What strategies can we use to solve this? Equal Addition Place Value Partitioning Reversibility Tidy numbers

23. 83 - 28 = 55 Reversibility don’t subtract, add Equal Additions 28 8583 +2 30 - 55 288330 +53 +2

24. 83 - 28 = 55 Tidy Numbers 538355 -30 +2 Place Value Partitioning 558363 -20 -3-5 60

25. Exploring Strategy It’s not just if I get the answer right. It’s how I solved it that’s important

26. Strategy Use We want children to be using strategies that are efficient and effective across a range of situations. Sometimes children will demonstrate strategies across stages. This is usually due to to gaps in key knowledge e.g. A child can use doubles to solve 8+7 as 7+7 + 1 but counts on to solve 9+5 because the teen number code is not known.

27. Strategy Thinking Stages Revised framework for stages – describes observable behaviours. Provides examples of the different types of problems children will be solving at Stage 5 and Stage 6.

28. Numeracy Interviews Watch the video clips carefully. Think about what you can see and what you hear. What do you notice? Why have the children been put at the stage they are?

29. Combinations Aim: to record as many combinations using up to four cards Deal out four cards. Record as many combinations as you can - use all the operations. E.g. 6, 4, 7, 9 is dealt 9+6+7- 4 =116 x 4 + 7 = 319 - 7 = 6 - 4

30. Data Gathering How do we gather information about children in the Numeracy project? NumPA interview - provides information about both strategy and knowledge. Used initially in Numeracy Project as it provides professional development for teachers concerning children’s knowledge and strategy stage. GLoSS - provides information about Strategy Stage IKAN - provides information about knowledge. After time using numeracy programme teachers are more knowledgeable - use GLoSS and IKAN.

31. GLoSS Tests Strategy only. Questions similar to Strategy section in NumPA Can be accessed from NZ Maths Website. Several versions of GLoSS - children won’t become familiar with questions.

32. Administering GLoSS Record as much information as you can about what children are doing. Make notes on forms. Ask clarifying questions of children if you are unsure about what they are doing. Watch for indicators that show children are using algorithm - key factor. Key question - what happens when children try to answer the question at stage 6 – what response are you getting

33. IKAN Pointpoint – sample of knowledge From Year 3 on – can ask the questions to year 3s and record it but you must have instant response (3 - 4 seconds) Look for gaps in knowledge – teach to fill gaps

34. Summary Sheets Record children’s strategy stages from GloSS and IKAN results on Summary Sheet Plot children on expectation grids – who is cause for concern, at risk or above expectation? Plan for next steps – whole class/group/individual

35. Some key ideas so far Frameworks are the starting point Knowledge is key to development of strategy Important to keep referring to Frameworks Listen to what the children are saying Look towards development of effective strategies - that will help children move forward

36. 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?

37. Place Value Explore – Sixty to ninety Twenty to Fifty Fourteen, Sixteen to nineteen Eleven to Thirteen and Fifteen. Discuss the ten for one rule – bundling to ten Canon of Place Value

38. Read, say, do - Peter Hughes Say the numeral one way, e.g. 13 is thirteen Read 13 as “thirteen” and thirteen as “thirteen” 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

39. Repetition - read, say, do There needs to be extensive repetition of problems using read, 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

40. Extend - read, say, do If we have 243 lollies, how many packets of ten would we have? How many loose ones? If we pack ten packets into a box, how many boxes will we have, how many packets and how many loose ones? Peter Hughes

41. Modeling with Place value money Children need to be able to verbalise the ten for one exchange in problems such as I have $1003 and I owe my friend $7 – must follow ten for one rule. Try the same with problems like $998 + 6 Aim for children to be able to verbalise the ten for one exchange fluently. Peter Hughes

42. Equipment for developing place value concepts StageEquipment Stage 2- 4 Concrete representation of ones Bundling sticks, beans and containers Counters and plastic bags, Slavonic Abacus Stage 5 Non representational Place value money, place value blocks, arrow cards, place value houses Stage 6Number Lines Stage 7Decimal Fraction Mats

43. Important activities for developing place value Reading numbers as words. Move flexibly from sixty to six tens to 60 Making sense of 20 + seventy = ___ tens Explaining where the tens are in 67, 17, 127. Describing how to count – explaining ten for one, one for ten.

44. Listening/Watching to learn Listen carefully when children are counting – forward and back – watch for 32,31,30, 20, 29, 28, Ask children to write numbers such as one hundred and three – look to see if they understand zero as a place holder Count out 6 tens ( tens frames, money, abacus) ask how many tens there are? Listen for answers such as sixty and children who don’t know and have to count again.

45. Bundling To Ten Bundles of ten board, Ice block sticks, Dice Pipe cleaners Roll the dice - put the number of ice block sticks in ones column - in tens frame pattern. Roll again add ice block sticks - what happens when we get to ten? Bundle the 10 put into tens column - Part whole thinking Record the story Introduce to group Play in pairs - first to 100.

46. 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

47. 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

48. 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.

49. 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.

50. 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

51. 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

52. 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

53. 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.

54. 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

55. 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

56. Where to next Gather data on what your children know Respond to gaps – teach knowledge Explore strategy Listen to your children, respond to what you hear.