1. Liz Woodham, NRICH Project & Michael Hall, Open University 2014 - 2015
Developing teachers’ mathematics subject knowledge in primary schools to improve the attainment of all pupils
Liz Woodham, NRICH Project
&
Michael Hall, Open University

2. In between face-to-face days …
Don’t forget

3. Day 7 – 5 February 2015
Welcome Working on mathematical tasks together Tackling tasks and reflecting on them in terms of subject knowledge and pedagogy Break Working on more mathematical tasks together Tackling one or more tasks and reflecting on them in terms of subject knowledge and pedagogy Sharing experiences since last time Curriculum priorities Lunch Update on various project strands Curriculum development work In pairs, planning for at least one task back at school Working on more mathematical tasks together – geometry follow-up Reflection

4. Common themes from mathematical needs identified on day 1
The following were flagged up by at least two schools:
Fractions/decimals/percentages
Problem solving
Place value
Time
Algebra
Word problems
Application of calculation strategies
Subtraction
Indicate that we will be taking further suggestions later in day – today focus on algebra

5. What springs to mind when you hear the word ‘algebra’?
Give delegates time to jot down ideas, then chat to neighbour

6. “algebraic and pre-algebraic ideas … can be embedded throughout the primary curriculum”
Anne Watson

7. Number Balance http://nrich.maths.org/4725
How could you record what you’re doing?
Show interactive on site, put 9 on one side, 3&4 on the other. If had one weight, where hang it to make it balance?
6 on one side, 4 and 9 on other. If had one weight, where hang it to make it balance?
9&6; 4&10. What if had two weights?

8. The Equals Sign
How else could you write = 9? For example … = = = 5 = 11 – = 3 x 3
Cuisenaire very useful resource here

9. importance of ensuring that = does not signify a calculating instruction ('makes') but does mean 'equivalent to’ the idea of both sides of an equation being worth the same is a fundamental concept

10. Missing Number Problems and Balance7
☐ + 2 = 7
= ☐
9 - ☐ = 7
☐ = 6 + 9
☐ - 4 = 5 + 5
☐ + 8 > 7
What do you notice?
Write 5 missing box statements that you feel show progression in thinking.
How would you start missing box with youngest children?
Write a balance statement at the level of your class.
What’s important about a balance statement?
When do we move from empty boxes to coloured in symbols?
When can empty box be anything and symbol something specific?

11. Heads and Feet http://nrich.maths.org/924
On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there?
Idea of equivalence important in missing number problems, which might start off as e.g. 6+?=10, where about number bonds, then move on to examples where solution not immediately obvious
Here is an example of a missing numbers problem which encourages children to represent mathematics in their own ways, possibly using symbols

12. Shape Times Shape http://nrich.maths.org/5714
The coloured shapes stand for eleven of the numbers from 0 to 12. Each shape is a different number. Can you work out what they are from the multiplications below?
Help children get more confident in using symbols
Also good for chains of reasoning and giving opportunity for children to communicate their reasoning

13. Colour Wheels http://nrich.maths.org/2220
What will we see on the ground?
Can you predict the colour of the 18th mark? 19th? 31st? 59th? 299th? 3311th?
How did you work it out?
Show interactive without saying much first
Idea of repeating pattern not in itself algebraic, but second and third questions are as they related position number to value r/b/g

14. Ip Dip http://nrich.maths.org/7185
If you were playing a game with one friend and you wanted to be chosen to be 'it', would you start the rhyme pointing at yourself or your friend? If there were three of you, how would you position yourself so that you were sure you'd be chosen?   How about with four of you?  Five ...? Six ...?  Seven ...? Eight ...? Nine ...?  Ten ...?  And so on?   How would you predict where you should stand to be chosen for any number of players?
Ip dip sky blue, who’s it? Not you!
This is about noticing a pattern and then generalising it

15. Up and Down Staircases http://nrich.maths.org/2283
One block is needed to make an up-and-down staircase, with one step up and one step down.
4 blocks make an up-and-down staircase with 2 steps up and 2 steps down.
How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?
Explain how you would work out the number of blocks needed to build a staircase with any number of steps.

16. The Roots of Algebraic Thinking
Balancing and the equals sign
Using symbols then letters
Generalising from a number of examples and stating a rule
Lynne McClure

17. The Algebra Strand Pupils should be taught to:
use simple formulae expressed in words
generate and describe linear number sequences
find pairs of numbers that satisfy number sentences involving two unknowns
enumerate all possibilities of combinations of two variables
In Y6

18. Algebra in Other Strands
number bonds in several forms (e.g = 16; = 9; 7 = )
commutativity and inverse relations (e.g. 4 × 5 = 20 and 20 ÷ 5 = 4)
relation between arrays, number patterns and counting
mental methods: commutativity and associativity, etc. to derive methods
distributivity can be expressed as 3 × (5 + 2) = (3 × 5) + (3 × 2)
write statements of equality of expressions (e.g. 39 × 7 = 30 × × 7)
perimeter of rectangle expressed algebraically as 2(length + breadth)
angle sum facts and shape properties as missing number problems
missing coordinates using properties, e.g. (a, b) and (a+d, b+d) being opposite vertices of a square.
linear number sequences, including those involving fractions and decimals, and find the term-to-term rule
combinations of operations; explore order of operations; meaning of the equals sign.
Thanks to Anne Watson

19. Algebra in the New Curriculum Feature http://nrich.maths.org/10941
Includes:
Article - Making Algebra Rich by Lynne McClure
Article - What’s x Got to Do with It? by Anne Watson
A selection of tasks to support the development of algebraic thinking

20. NCETM Website http://www.ncetm.org.uk/
Charlie’s Angles (thoughts from NCETM Director, Charlie Stripp)
Curriculum support
Primary and Early Years Magazine archive
Mention need to be logged on for many resources (free to register)

21. Shanghai Project Part of new Maths Hubs programme
English teachers visited Shanghai in Sep 2014
Shanghai teachers visited English schools in Nov 2014 – pairs linked to one or two primary schools
Another group of Shanghai teachers due to come here in late Feb 2015
Shanghai approach – mastery teaching – entails, among other things, keeping the whole class together on the same material, effective use of high quality textbooks, and communicating the expectation that all pupils will achieve to a high standard
Interim report:
Video of teachers talking about their experiences

22. Before we meet again …
Continue to try out rich tasks with your learners and colleagues
Ask your pupils to complete the questionnaire and bring copies next time
Complete the subject audit in the back of the ‘Mathematics for Primary and Early Years’ book
Take a look at the NRICH Algebra feature
Explore the NCETM website

23. Liz Woodham emp1001@cam.ac.uk Michael Hall wichaelhall@gmail.com