Preparing to teach mathematics

1. Preparing to teach mathematics

2. We are preparing you to teach mathematics by : • Discussing the importance of subject knowledge and pedagogical knowledge in the teaching and learning of mathematics • Considering the importance of early counting for all learners • Considering the aims of the National Curriculum

3. Standard 3 • Demonstrate good subject and curriculum knowledge • have a secure knowledge of the relevant subject(s) and curriculum areas, foster and maintain pupils’ interest in the subject, and address misunderstandings • demonstrate a critical understanding of developments in the subject and curriculum areas, and promote the value of scholarship • if teaching early mathematics, demonstrate a clear understanding of appropriate teaching strategies.

4. Using the digits 1- 9 arrange them in the 3 x 3 grid so that each row, column and diagonal adds up to the same amount. Magic squares

6. The aims of the National Curriculum • To become fluent in the fundamentals of mathematics, including through varied and frequent practice with increasingly complex problems over time, so that pupils have conceptual understanding and are able to recall and apply their knowledge rapidly and accurately to problems • Toreason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language • To solve problems by applying their mathematics to a variety of routine and non-routine problems with increasing sophistication, including breaking down problems into a series of simpler steps and persevering in seeking solutions. https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/184064/DFE-RR178.pdf

7. BALANCE Procedural Fluency Conceptual Understanding INTEGRATION

8. What’s the difference between mathematical fluency and conceptual understanding?Discuss

9. Make a list of the mathematical concepts children need to know in order to calculate 32 – 3 32 - 29

10. Many secondary teachers The best teachers Many primary teachers Subject knowledge OfSTED (2008) Understanding the Score http://www.ofsted.gov.uk/resources/mathematics-understanding-score

11. Principles of Counting Gelman and Gallistel (1986) • One to one principle – giving each item in a set a different counting word. Synchronising saying words and pointing. • Stable order principle - Keeping track of objects counted knowing that numbers stay in the same order. • Cardinal principle – recognising that the number associated with last object touched is the total number of object. The answer to ‘how many?’ • Abstraction principle - recognising small numbers without counting them and counting things you cannot move or touch. • Order irrelevance principle - counting objects of different sizes and recognising that if a group of objects is rearranged then the number of them remains the same.

12. Ordering numbers • More than, less than • Counting out a given number • Counting from a given number • Reciting number names in order and becoming consistent, including through decade and hundred changes • Reciting number names with decimals and fractions • Ordering numbers including with fractions and decimals Progression in counting

13. Singapore Maths Bruner’s phases of learning Phases of learning mathematics • Enactive • Iconic • Symbolic • Concrete • Pictorial • Abstract

14. 5 Consider…..

15. Common errors in counting in KS1 • Counting one, two, three then any number name or other name to represent many • Number names not remembered in order • Counting not co-ordinated with partition • Count does not stop appropriately • Counts an item more than once or not at all • Does not recognise final number of count as how many objects there are • Counting the start number when ‘counting on’ rather than the intervals (jumps) when ‘counting on’ on a number line.

16. Common errors in counting in KS2 • when counting on or back, include the given number in their counting rather than starting from the next or previous number or counting the ‘jumps’; • Difficulty counting from starting numbers other than zero and when counting backwards; • understand the patterns of the digits within a decade, e.g. 30, 31, 32, ..., 39 but struggle to recall the next multiple of 10 (similarly for 100s); • Know how to count on and count back but not understand which is more efficient for a given pair of numbers (e.g. 22-19 by counting on from 19 but 22-3 by counting back 3); • Not understanding how place value applies to counting in decimals e.g. 0.8, 0.9, 0.10, 0.11 rather than 0.8, 0.9, 1.0, 1.1; • Counting upwards in negative numbers as -1, -2, -3 … rather than -3, -2, -1…

17. Draw a grid big enough for digit cards Be Nasty

18. Be Nasty • Rules • Shuffle the number cards place face down in a stack • Take turns to pick up a number card. You can place your number card on your own HTU line or on your partner’s HTU line. • The aim is to make your own number as close as possible to the target – and to stop your partner making a number closer to the target. • Take it in turns to go first.

19. Be Nasty Largest number Smallest number Nearest to 500 Nearest to a multiple of 10 Nearest to a multiple of 5 Nearest to a square number Nearest any century Lowest even number Nearest odd number to 350

20. What is Place Value • Positional- the quantities represented by the individual digits are determined by the positions that they hold in the whole numeral. The value given to a digit is according to the position in a number • Base 10: the value of the position increases in powers of 10 • Multiplicative; the value of an individual digit is found by multiplying the face value of the digit by the value assigned to its position. • Additive: the quantity represented by the whole numeral is the sum of the values represented by the individual digits (Ross 1989)

21. Misconceptions and errors in place value: Twenty eight, twenty nine, twenty ten Writing 10016 for 116 Writing £1.6 for £1.06 Placing 1.35 as larger on a number line than 1.5 Lining numbers up incorrectly in column addition Writing the sequence 1.7, 1.8, 1.9, 1.10, 1.11..

22. Using Place Value Charts 1 2 3 4 5 6 7 8 9 10 20 30 40 50 60 70 80 90 100 200 300 400 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000

23. Conceptual understanding and procedural fluency : what do they mean in relation to calculation?

24. Tom had two sweets and John had three sweets how many did they have altogether? Tom had two sweets and bought three more. How many sweets does he have now? What’s the difference between…..?

25. Conceptual structures for addition • Aggregation - combining of two or more quantities (How much/many altogether? What is the total? Tom had two sweets and John had three sweets how many did they have altogether? • Augmentation – where one quantity is increased by some amount (increase by) Tom had two sweets and bought two more. How many sweets does he have now

26. Conceptual structures for subtraction • Partition/change/take away - Where a quantity is partitioned off in some way and subtraction is required to calculate how many or how much remains. (Take away, How many left? How many are/do not?) Tom had five sweets, John ate three sweets. How many sweets did Tom have left? • Comparison – a comparison is made between two quantities. (How any more? How many less/fewer? How much greater? How much smaller? Tom had 5 sweets, John had three sweets. How many more sweets did Tom have than John?

27. Counting forwards and backwards One more than, one less than Counting on or back in steps of 2,5 and 10 Counting on or back from the larger number Partitioning numbers into 5 and a bit e.g. 5 + 7 = 5 + 5 + 2 Bridging through 10, using known facts to 10e.g. 6 + 9 = (6 + 4) + 5; 15 – 9 = (15 – 5) - 4 Early Mental Strategies for Addition and Subtraction

28. Bridging through multiples of 10 e.g. 25 + 7 = (25 + 5) + 2; 22 – 5 = (22 – 2) – 3 Reordering numbers in addition e.g. 6 + 2 + 4 = 6 + 4 + 2 Find differences by counting up e.g. 10 – 6 by counting ’7, 8, 9, 10’ Using inverse operations e.g. 13 + 7 = 20 so 20 – 7 = 13 Special cases: Using doubles facts to derive near doubles facts e.g. 6 + 6 = 12 so 6 + 8 = 14 and 6 + 5 = 11 Early Mental Strategies - continued

30. Calculate 25 + 47 • Using Dienes • Using Numicon • Using Place Value Counters So why is the understanding of place value so important?

31. Calculate 72 - 47 • Numicon • Using Dienes • Using Place Value counters So why is the understanding of place value so important?

32. 26 + 57 • 25 + 24 • 65 + 29 • 73 - 68 • 82 - 26 • 156 – 99 Then compare strategies with a friend. Calculate these mentally (and record your strategies on paper):

33. Target Board

34. Different children prefer different mental calculation strategies • Choice of strategy may vary for different pairs of numbers • The choice of mental strategy for a particular pair of numbers is influenced by a range of factors: • size of the numbers, • personal preferences, • size of the difference between the numbers, • proximity of numbers to 10s or 100s numbers, • special cases etc. Choice of Mental Strategies

35. Mental We may break the calculation into manageable parts eg 248 – 100 + 1 instead of 248 – 99 We say the calculation to ourselves and so are aware of the numbers themselves eg 2000 – 10 is not much less than 2000 Written We never change the calculation to an equivalent one, 248 – 99 is done as it is We don’t say the numbers to ourselves, but talk about the digits instead saying 8 – 9 and 4 - 9 Mental and Written Calculations

36. Mental We usually begin with the most significant digit We choose a strategy to fit the numbers eg 148 – 99 is not calculated in the same way as 84 – 77 although they are both subtractions We draw upon mathematical knowledge such as properties of numbers or ‘number sense’, learned facts etc Written We usually begin with the least significant digit We always use the same method We draw upon the memory of a procedure although we may not understand how it works

37. Conceptual structures for multiplication • Repeated addition - ‘so many sets of’ or ‘so many lots of’ This is four lots of two this is written as 2 x 4 • Scaling structure – increasing a quantity by a scale factor (doubling, so many times bigger...so many times as much as). Tom has three times as many sweets as John. John Tom

38. Conceptual structures for division • Equal sharing- (shared between, divided by) There are 8 sweets shared between four children. How many sweets do they get each? • Equal grouping - I want to buy 8 sweets they come in packs of two . How many packs must I buy.

39. Some mental calculation strategies for multiplication and division • Using commutative law e.g. 5 x 9 as 9 x 5 • Repeated operations: e.g. 324 ÷ 4 as (324 ÷ 2) ÷2, or 32 x 8 as 32 x 2 x 2 x 2 • Using associative law: (16 x 2) x 5 as 16 x (2 x 5) • Multiplying and dividing by 10, 100 etc: 3 x 4 = 12 so 30 x 4 = 120 • Using partitioning and the distributive law: 12 x 7 as (10 x 7) + (2 x 7); 19 x 5 as (20 x 5) – (1 x 5); 4 x £1.99 as (4 x £2) – (4 x 1p) • Doubling and halving (for multiplying): 15 x 18 as 30 x 9 • Using factors: 6 x 18 as 6 x 9 x 2; 324 ÷ 18 as (324 ÷ 3) ÷ 6 • Using inverse operations: 100 ÷ 5 = 20 because 20 x 5 = 100

40. Commutative law - axb = bxa and a+b = b+a eg 3 x 4 = 4 x 3 55 + 45 = 45 + 55 Associative law - (axb) x c = a x (bxc) eg 24 x 6 = (4x6) x 6 = 4x (6x6) (5 + 7) + 3 = 5 + (7 + 3) Distributive law or partitioning (a+b) x c eg 12 x 7 = (10 x 7) + ( 2 x 7) and 84 ÷ 7 = (70 ÷ 7) + (14 ÷ 7) Calculation laws

41. Different children prefer different mental calculation strategies • Choice of strategy may vary for different pairs of numbers • The choice of mental strategy for a particular pair of numbers is influenced by a range of factors: • size of the numbers, • personal preferences, • size of the difference between the numbers, • proximity of numbers to 10s or 100s numbers, • special cases etc. Choice of Mental Strategies

42. Some strange calculation methods!

43. http://www.ness.uk.com/maths/Guidance%20Documents/Teaching%20children%20to%20calculate%20mentally.pdfhttp://www.ness.uk.com/maths/Guidance%20Documents/Teaching%20children%20to%20calculate%20mentally.pdf

45. Introduction to reasoning

46. Mathematical reasoning, even more so than children’s knowledge of arithmetic, is important for children’s later achievement in mathematics (Nunes et al 2009 p.3) Nunes, T., Bryant, P., Sylva, K. and Barros, R. (2009) Development of Maths capabilities and confidence in Primary school https://www.gov.uk/government/publications/development-of-maths-capabilities-and-confidence-in-primary-school

47. Developing higher order thinking Bloom, B.S. (Ed.) (1956) Taxonomy of educational objectives: The classification of educational goals: Handbook I, cognitive domain. New York ; Toronto: Longmans, Green.

48. Fit into the dark blue boxes: 2,3,4,5,6,7,8,9,10,11,12. One number has to be used twice! Which one?Why? Multiplication Puzzle

49. 146 ÷ 7 62 x 16 263 – 76 Sixes are banned Can you think of a different calculation (which does not use the digit 6) to give the same answer? (32+15+1) 32 + 16 58 - 26 48 x 6 126 - 58

50. Noah’s Ark Noah sets sail on his ark. How many animals did he squeeze on to the ark? When the animals were paired off in twos, one was left over. When they were grouped in threes, one was left over. But when the animals were grouped in fives, not one was left over. How many animals did Noah have?