The Learning Framework in Number

1. The Learning Framework in Number Julie Rees K-6 Quality Teaching Consultant New England Regional Office Noel Park House

2. A thought to begin • Teacher: ‘Who can tell me what 7 times 6 is?’ • Student: ‘It’s 42!’ • Teacher: ‘Good, and who can tell me what 6 times 7 is?’ • Student: ‘That’s easy. It’s 24!’

4. Levels within parts

5. What research found • Students arithmetical learning develops through two complimentary processes: counting and grouping. • The transition from count-by-one strategies to collection-based strategies underpins the structure of the framework.

6. Underlying the framework is a belief that it is important to observe and take account of children’s knowledge and strategies.

7. 3 components Assessment – what can they do? Framework - ‘where are they now’ Teaching tasks – ‘where to from here’

8. The details of the learning framework: Early Arithmetic Strategies • Where is it on the LFIN? • Let’s break it down…

10. Addition and subtraction • What key concepts are you trying to assess? • To solve the task does the student: • count perceptual items using 1:1 correctly? • rely on perceptual items? • count from one when visualising the groups? • count on from the larger number? • use groups to solve tasks? • use knowledge of combinations of numbers to 10 or 20? CMIT

11. Number/name confusion • Sounding out the letters …b…a…t • Sound images for 1 to 10 • Imagine a young child who learns how to read 13, 15, 16, 17, etc correctly (i.e. sound image starts from the right). How might he or she read 43? • Confusion between the names of the numerals 19 and 90 • What is the next number after 19?

12. Order these! 33 12 13 30 20 32 28

13. Numerals • The clothesline. • Children can all be given the same challenge but this can be differentiated to draw out more sophisticated strategies.

14. 100 50 ? The Clothesline • Emphasis is on the order of numbers rather than the exact position • Can start at any point. Not always zero or one • Models the concept of an empty number line

15. How many times can you subtract 7 from 83, and what is left afterwards? • As many times as you want, and it leaves 76 every time.

16. ? 13 Tape Diagrams • Tape diagrams are developed in the Japanese curriculum from Grade 2 or Grade 3. 7

17. The empty number line • Represents the linear model of numbers • Is used to record student thinking • Can be used flexibly by varying the parameters

18. Using a diagram John had 29 buttons. His brother gave him some more buttons. Altogether John had 73 buttons. How many buttons did John get from his brother?

20. Beyond addition and subtraction What is 14 x 10 = ? Can you use your answer to work out 14 x 12? What is 16 x 25 = ? What is 16 x 24 = ? How many groups of ten can be formed from 611?

21. Review • Multiplication and Division relies on the structure of a composite group. • A composite group is a collection of individual items that can be viewed as one thing. • For example, the student must be able to see a group of 5 items as one group of 5 and for it to be one unit repeated.

23. Elicit as many strategies as possible

24. Multiplication & division • What key concepts are you trying to assess? • To solve the task : • Does the student form equal groups? • Does the student need concrete material? • Does the student use materials to represent each item or each group? • Does the student keep track of the groups while counting? • What counting strategies does the student use to find the total? CMIT

25. The why chart… What does a maths lesson focus on? What will the deep knowledge and deep understandings be? What will a maths lesson sound like? This will be the metalanguage and substantive communication. What will a maths lesson look like? This will be H.O.T.S. background knowledge, problematic knowledge.

26. Learning Plans