Developing teachers mathematics subject knowledge in primary schools

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

In between face-to-face days … Don’t forget https: //nrich. maths. org/haringey London Schools Excellence Fund

Day 7 – 5 February 2015 9. 15 -9. 30 Welcome 9. 30 -10. 45 Working on mathematical tasks together Tackling tasks and reflecting on them in terms of subject knowledge and pedagogy 10. 45 -11. 00 Break 11. 00 -11. 45 Working on more mathematical tasks together Tackling one or more tasks and reflecting on them in terms of subject knowledge and pedagogy 11. 45 -12. 00 Sharing experiences since last time 12. 00 -12. 15 Curriculum priorities 12. 15 -1. 00 Lunch 1. 00 -1. 20 Update on various project strands 1. 20 -2. 20 Curriculum development work In pairs, planning for at least one task back at school 2. 20 -3. 10 Working on more mathematical tasks together – geometry follow-up 3. 10 -3. 15 Reflection London Schools Excellence Fund

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 London Schools Excellence Fund

What springs to mind when you hear the word ‘algebra’? London Schools Excellence Fund

“algebraic and pre-algebraic ideas … can be embedded throughout the primary curriculum” Anne Watson http: //nrich. maths. org/10906 London Schools Excellence Fund

Number Balance http: //nrich. maths. org/4725 How could you record what you’re doing? London Schools Excellence Fund

The Equals Sign How else could you write 4 + 5 = 9? For example … 4+5=6+3 4 = 5 = 11 – 2 London Schools Excellence Fund 4+5=7+2 4+5=3 x 3

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 London Schools Excellence Fund

Missing Number Problems and Balance • 7 ☐ +2=7 5 +4=☐ 9 -☐ =7 4+5+☐=6+9 ☐ -4=5+5 ☐ +8>7 London Schools Excellence Fund

Heads and Feet http: //nrich. maths. org/924 On a farm there were some hens and sheep. Altogethere were 8 heads and 22 feet. How many hens were there? London Schools Excellence Fund

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? London Schools Excellence Fund

Colour Wheels http: //nrich. maths. org/2220 • What will we see on the ground? • Can you predict the colour of the 18 th mark? 19 th? 31 st? 59 th? 299 th? 3311 th? • How did you work it out? London Schools Excellence Fund

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? London Schools Excellence Fund

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. London Schools Excellence Fund

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 Mc. Clure http: //nrich. maths. org/10908 London Schools Excellence Fund

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 London Schools Excellence Fund

Algebra in Other Strands • • • number bonds in several forms (e. g. 9 + 7 = 16; 16 - 7 = 9; 7 = 16 - 9) 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 + 9 × 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 London Schools Excellence Fund

Algebra in the New Curriculum Feature http: //nrich. maths. org/10941 Includes: • Article - Making Algebra Rich by Lynne Mc. Clure https: //nrich. maths. org/10908 • Article - What’s x Got to Do with It? by Anne Watson https: //nrich. maths. org/10906 • A selection of tasks to support the development of algebraic thinking London Schools Excellence Fund

NCETM Website http: //www. ncetm. org. uk/ • • • Charlie’s Angles (thoughts from NCETM Director, Charlie Stripp) http: //www. ncetm. org. uk/resources/42295 Curriculum support http: //www. ncetm. org. uk/resources/41229 Primary and Early Years Magazine archive http: //www. ncetm. org. uk/resources/12691 London Schools Excellence Fund

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: http: //www. ncetm. org. uk/news/46090 Video of teachers talking about their experiences http: //www. mathshubs. org. uk/what-maths-hubs-are-doing/englandchina/shanghai-teachers-in-english-schools-november 2014#video 2 London Schools Excellence Fund

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 London Schools Excellence Fund

Liz Woodham emp 1001@cam. ac. uk Michael Hall wichaelhall@gmail. com London Schools Excellence Fund