Moving Students On AC to EA/EA to AA The Journey to Part-Whole Dianne Ogle 13 July 2011

1. Moving Students On AC to EA/EA to AA The Journey to Part-Whole Dianne Ogle 13 July 2011

2. Overview of today • What are the key pieces of knowledge and strategy for our Cause for Concern Children? • Develop our conceptual understanding of key knowledge and strategy • Develop an understanding of ways to help children who are not achieving at expected level

3. Part Whole thinking – the prize! • Involves splitting numbers into parts (partitioning) in order to solve problems more easily. • Depends on knowing how the parts make up a whole number • Think about 10 • What about 18 • What about 72 • What key knowledge do children need?

4. Why Part-Whole? • At counting stages, the size of numbers is severely restricted, and there is normally only one way to solve problems. • Counting represents a relatively low level of thinking. • Part-whole thinking opens up the world of large numbers and multiple strategies.

5. Early Additive: • Students at this stage have begun to recognise that numbers can be split into parts and recombined in different ways. This is called part-whole thinking. Strategies used at this stage are most often based on a group of ten or use a known fact, such as a double. For example:            38 + 7 as (38 + 2) + 5            24 – 9 as (24 - 10) + 1           7 + 8 as (7 + 7) + 1 • Students working at this stage will be solving number problems in each of the three operational domains. How do EA children solve multiplication and division problems? Proportions and Ratios problems?

6. Advanced Additive • Students at this stage are familiar with a range of part-whole strategies and are learning to choose appropriately between these. They have well developed strategies for solving addition and subtraction problems, for example:367 + 260 as (300 + 200) + (60 + 60) + 7 135 – 68 as 137 – 70 703 – 597 as 597 + ? = 703 • They also apply additive strategies to problems involving multiplication, division, proportions and ratios. For example: 6 x 3 = (5 x 3) + 3 = 15 + 3 = 18 One quarter of 28 as 14 + 14 = 28 so 14 is one half, 7 + 7 = 14 so 7 is one quarter

7. Advanced Multiplicative • Students at this stage are able to choose appropriately from a range of part-whole strategies to solve problems with whole numbers. They are learning to apply these strategies to the addition of decimals, related fractions and integers. For example, 5.5 + 6.8 = 5.5 + 7 – 0.2 = 12.3. • Students are learning to manipulate factors mentally to solve multiplication and division strategies. For example, instead of partitioning 5 x 12 additively as (5 x 10) and (5 x 2) students use strategies such as doubling and halving, renaming 5 x 12 as10 x 6.

8. Profile of the child who is not moving… • Think about the child/children you teach who are not moving. • Discuss with a partner what you notice and observe when that child is engaged in mathematics. • What are some common reasons for lack of progress?

9. What knowledge to children need? • Counting sequence and how to read and write numbers • Place Value • Basic Facts

10. 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

11. 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!

12. Early Number relationships • Spatial relationships: children can learn to recognise sets of objects in patterned arrangements and tell how many without counting.

13. One and two more, one and two less • The two more than and two less than relationships involve more than just the ability to count on two or count back two. Children should know that 7, for example is 1 more than 6 and also 2 less than 9.

14. Anchors or “benchmarks” of 5 and 10 • An understanding of ten is vital in our numeration system and because two fives make up 10, it is very useful to develop relationships for the numbers 1 to 10 to the important anchors of 5 and 10

15. Part – Part – Whole Relationships • To conceptualize a number as being made up of two or more parts is the most important relationship that can be developed about numbers.

16. 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,

17. Strategies Maraea has $37. She spends $9. How much money does she have left now? Caleb has saved $165. He banks another $23 dollars. How much money does he have saved now? Dianne has $72. She spends $28 on a pair of shoes. How much money does she have now? Jody has scored 284 goals in netball this season. She gets another 67. How many goals has she scored altogether? Anaru has 312 tropical goldfish in his aquarium. He sells 198 of them to the pet shop. How many tropical fish does he have now? The electrician has 5.33 metres of cable. He uses 2.9 metres on a job. How much cable is left? Solve these problems independently. When you have your answers talk about how you solved them, are there some key strategies, can you give them a name?

18. Strategies Maraea has $37. She spends $9. How much money does she have left now? Caleb has saved $165. He banks another $23 dollars. How much money does he have saved now? Dianne has $72. She spends $28 on a pair of shoes. How much money does she have now? Jody has scored 284 goals in netball this season. She gets another 67. How many goals has she scored altogether? Anaru has 312 tropical goldfish in his aquarium. He sells 198 of them to the pet shop. How many tropical fish does he have now? The electrician has 5.33 metres of cable. He uses 2.9 metres on a job. How much cable is left? How could we use materials to demonstrate strategies used, help build conceptual understanding.

19. Reading - • Read the article by Young-Loveridge and Mills • Key points • Next steps for you

20. Book Five/Planners • Key ideas in Book Five • Use of diagnostic questions • Required knowledge • Problem progression

21. Core ideas of Place Value • Zero as a place holder Canon of Place Value – Ten for one, one for ten. • Children must understand that as a result of an addition or multiplication the numeral in any column in a place-value table exceeds nine, ten of these must be exchanged for a one that is ten times more. Unique Symbols • The numerals 0 to 9 are unique symbols that are used to represent numbers. They have been adopted universally around the world. 130 should be said one, three, zero not one, three, (letter) o. Irregularity of English language number words • The ‘teen’ and ‘ty’ code can be extremely confusing for children. It is hard for children to hear the difference between the number words when they are said aloud. Seeing the written number word provides the visual cue that ‘teen’ is one ten and “ty” is lots of ten. Therefore sixteen is six ones and one ten while sixty is six lots of ten. “lots of” is multiplicative, sixty = 6 x 10

22. Bundling Bundles of ten board, Ice block sticks, Dice Rubber band 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.

23. Number words seventeen one ten + seven ones 17 Fourteen one ten + four ones 14 Fifty five tens 50 Seventy seven tens 70 Children need to have multiple opportunities to work with teen/ty numbers to develop their understanding. Begin with materials.

24. Advanced Counting - Early Additive Crucial number knowledge - understanding of place value concepts - “teen, ty” Need plenty of opportunities to bundle to ten etc to develop understanding of canon rule of place value. Need to know basic facts to 10 - all facts that make 10 and those below. Using tens frames, fly flips, hands etc to instantly recall facts to 10

25. Place Value EA - AA Understanding of zero as a place holder Knowing how to explain place value for 98 + 5 = what happens and why Tell the story - using exchange. 1003 - 7 = where are the ones? Knowing how many ones, tens, hundreds in a number (not by using a rule but because 10 ones are exchanged for 1 ten, one hundred can be exchanged for ten tens.

26. 10 003 - 6 You have ten thousand and three dollars. You owe your friend six dollars. Do you have 6 dollars to give to your friend. Imagine place value money. Write the story of how you will get enough ones to be able to give your friend what you owe. Where do you start?

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

28. Equipment for developing place value concepts

29. Stage 4 – numbers from 10-99 • 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.

30. 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

31. 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

32. Stage 5 – Part-whole • Essential that students can move quickly through each representation of a number. • The Slavonic Abacus becomes an essential piece of equipment to help students understand place value. • Ask the children to identify numbers on the abacus by recognising the patterns in ten and ones using the quinary (five) patterns. • Imaging what a number will look like on the abacus. • Materials • Tens frames • Place value play money

33. Stage 6 • Children must have automatic making and breaking of numbers • What distinguishes early part whole from advanced part whole thinking is the number of mental steps needed. • It is not the size of the numbers but the number of mental steps needed. • EA: 199 + 56 • AA:56 - 37, 567 + 78 • Being able to quickly know how many hundreds, tens, ones there are in 4 digit numbers and beyond • Materials – number lines (introduced here) • Continue with place value money and blocks where necessary

34. Stage 7 • Children must understand that 3 or more digit numbers have multiple forms • Peter works in a cake factory: packing ten cakes into a packet. His job is to pack 265 cakes. How many packets does he have? How many loose ones? • He packs ten packets into a box for shipment. How many boxes, packets and loose ones?

35. Difficulties at AA • Understanding the nested place values in 4, 5, 6 digit numbers • Recognising the canonical and non-canonical forms. • Fifty five thousand is non-canonical • Sixteen hundred is non-canonical

36. Read, Say, Do • Children need to be able to explain what makes up a number • Use a variety of Place value materials to demonstrate how a number can be made up • Place value model – Peter’s biscuit factory.

37. 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

38. 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.

39. Place Value Questions • Diagnostic Interview – to find out what misconceptions children have. • Place Value Questions – use as diagnostic questions to find out what children need further help with • Allow children to discuss their thinking and explain how they know.

40. Basic Facts The place value system is universally adopted because all calculations can be performed by knowing correct procedures and the basic number facts. Knowing the addition facts from 1 + 1 to 9 + 9 will enable addition and subtraction problems to be solved, includes decimal fractions. Knowing the multiplication facts from 2 x 2 to 9 x 9 will Enable all multiplication and division problems to be solved, including decimal fractions. A lack of instant recall of basic facts, along with not understanding place value are the two key reasons children are not making progress in number.

41. Basic Facts A lack of instant recall of basic facts, along with not understanding place value are the two key reasons children are not making progress in number. • It is important that children are learning their basic facts when they need to be using them. • Addition and subtraction facts learned first • Times tables follow, when children are using multiplicative strategies.

42. 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

43. Basic Facts 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. Stage 4: Addition and subtraction facts up to ten Doubles Optional: – Addition and subtraction facts from 1 + 1 to 9 + 9 - Derive and learn the two times tables from doubles.

44. Basic Facts – 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 three times tables from 3 x 3 to 3 x 9 using repeated addition and the reverse facts. Optional: - Four and Five times table

45. Basic Facts – Stage 6 Instant recall of times tables with 100% reliability is needed for stage 7 so regular teaching and practising of tables must occur at this level. Failure to know times tables is a major obstacle in children ever becoming multiplicative in their thinking. Recall of multiplication facts can begin with a focus on the commutative principle for multiplication Stage 6: Essential- Derive and learn, connect to division 4 times table from 4 x 4 to 5 x 9 5 times table from 5 x 5 to 5 x 9 6 times table from 6 x 6 to 6 x 9 7 times table from 7 x 7 to 7 x 9 8 times table from 8 x 8 to 8 x 9 Derive and learn 9 x 9, connect to 81 ÷ 9 Use the 0 and 1 principles

46. 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

47. Basic Facts – only 36 facts to learn

48. Basic Facts – from understanding to rote Van de Walle Mastery of the basic facts is a developmental process, students move through stages, starting with counting, then to more efficient reasoning strategies, and eventually to quick recall. Instruction must help students move through these phases, without rushing them to memorisation. Page 167 , 2010

49. Approaches to fact mastery • Explicit strategy instruction – designed to support student thinking – show students possibilities and let them choose strategies that help them get the soltion without counting • Guided invention – using strategies children have, guiding them to the efficient ones. Teacher’s job is to design tasks and problems that will promote the invention of efficient strategies

50. What not to do • Don’t use lengthy timed tests • Don’t use public comparison of mastery • Don’t proceed through facts in order – (knock out the ones you know) • Don’t move to memorization too soon • Don’t use facts as a barrier to good mathematics – mathematics is about reasoning, give children real mathematical experiences.