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

An example - Division – a key idea

Sharing

Sharing by counting out

Sharing by chopping up

Sharing by chopping up

Sharing by folding

Sharing by pouring

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

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

Division by counting out

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?

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.

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

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.

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

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

Abstract number: the long division algorithm 2223÷13 222222 ÷ 13 2222223 ÷ 13 1/(1 + ½) 171 17094 will it go exactly?

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

Anne Watson anne.watson@education.ox.ac.uk Key Understandings in Learning Mathematics by Nunes, Bryant and Watson www.nuffieldfoundation.org