1. Anne Watson, Barking & Dagenham, 2012
Onward and upward
How deep connections in mathematics override the written curriculum and assessments
Anne Watson, Barking & Dagenham, 2012

2. An example -
Division – a key idea

3. Sharing

4. Sharing by counting out

5. Sharing by chopping up

6. Sharing by chopping up

7. Sharing by folding

8. Sharing by pouring

9. Division by measuring How many 3s go into 6? How many 4s go into 20?

10. Division: inverse multiplication
How many 6s go into 3?
How many 20s go into 4?
How many 3s go into 1 ½ ?

11. Division by counting out

12. Division by Counting in groups Dealing out one by one
Division by sharing out equally (what is that the opposite of?)
Division by repeated taking away groups (opposite of repeated addition)
Other actions for division?

13. Rods, tubes and sweets
How many logs of length 60cm. can I cut from a long log of length 240 cm?
How many bags of 15 sweets can I make from a pile of 120 sweets?
I have to cut 240 cm. of copper tubing to make 4 equal length tubes. How long is each tube?
I have to share 120 sweets between 8 bags. How many per bag?
Understanding div in everyday life and secondary school: continuous/measure; discrete/counting; one at a time, or deal with the whole thing. Subtraction or dividing the whole.

14. Three equal volume bottles of wine have to be shared equally between 5 people. How can you do this and how much will each get?
Three equal sized sheets of gold leaf have to be shared equally between 5 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?
Need for measuring device
Need for fraction notation

15. 98 equal volume bottles of wine have to be shared equally between 140 people. How can you do this and how much will each get?
98 equal sized sheets of gold leaf have to be shared equally between 140 art students, and larger sheets are more useful than small ones. How can you do this and how much will each get?
I would argue that setting these up as fractions is more useful than diving into doing a learnt method.

16. Two paths to division
knowledge of quantities and counting develop separately
thinking about relations is key to later success
interacting with objects and stuff
sharing out
pouring
cutting up
stretching/scaling
fitting 16 16

17. Relations between quantities
focus on the connections between informal knowledge (e.g. of pouring) and formal learning
experts use multiple representations to explore relations
include the study of relations explicitly to enable children to use their knowledge to understand calculations 17

18. Abstract number: the long division algorithm
2223÷13
÷ 13
÷ 13
1/(1 + ½)
171
17094
will it go exactly?

20. Inspired to .... Find out more about division
Explore division with colleagues
Explore division with children
Take another hard-to-learn aspect of maths
Find out more about it
Explore with colleagues
Explore with children

21. Anne Watson
Key Understandings in Learning Mathematics by Nunes, Bryant and Watson