Shepway Teaching Schools Alliance 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RHTelephone Number: 01303 298266 e-mail: office@shepwayts.co.uk, website: www.shepwayts.co.uk

Preparing to teach mathematics

*A game for two players

*Start with 20 counters*Each player can remove 1,2,3, counters in turn*The loser is the person who picks up the last counter.

Finding rules and patterns - NIM

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 how arithmetic can be taught through using and applying activities

New standards: Standard 3Demonstrate 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.

* The Aims of The New Curriculum

The latest draft National Curriculum for mathematics aims to ensure all pupils: • 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

• reason mathematically by following a line of enquiry, conjecturing relationships and generalisations, and developing an argument, justification or proof using mathematical language

• can 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.

BALANCE

Procedural Fluency

Conceptual Understanding

INTEGRATION

*Fluency

*The government wishes to continue to emphasise fluency, but this should not be understood to mean “rote learning without understanding”.....conceptual understanding is clearly important and ..any emphasis on practice needs to be a part of achieving that understanding.*Stefano Pozzi Mathematics in School May 2013 p2

Many secondary teachers

Many primary teachers

The best teachersSubject knowledge

OfSTED (2008) Understanding the Score http://www.ofsted.gov.uk/resources/mathematics-understanding-score

* Make a list of things

1) Children need to know in order to calculate

32 – 3 32 - 29

Principles of Counting Gelman and Gallistel (1986)

1. One to one principle – giving each item in a set a different counting word. Synchronising saying words and pointing.

2. Stable order principle - Keeping track of objects counted knowing that numbers stay in the same order.

3. Cardinal principle – recognising that the number associated with last object touched is the total number of object. The answer to ‘how many?’

4. Abstraction principle - recognising small numbers without counting them and counting things you cannot move or touch.

5. 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.

http://webarchive.nationalarchives.gov.uk/20110202093118/http:/nationalstrategies.standards.dcsf.gov.uk/node/84889

0 10 20 30 40 50 60 70 80 90 100

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98 99 100

0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17 18 19

20 21 22 23 24 25 26 27 28 29

30 31 32 33 34 35 36 37 38 39

40 41 42 43 44 45 46 47 48 49

50 51 52 53 54 55 56 57 58 59

60 61 62 63 64 65 66 67 68 69

70 71 72 73 74 75 76 77 78 79

80 81 82 83 84 85 86 87 88 89

90 91 92 93 94 95 96 97 98 99

0-99 or 1-100 Midge Pasternack http://www.atm.org.uk/journal/archive/mt182files/ATM-MT182-34-35.pdf

1-100 rules OK Ian Thompson http://www.atm.org.uk/journal/archive/mt185files/ATM-MT185-14-15.pdf

http://www.teachfind.com/national-strategies/mathematics-itp-number-grid

*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.

*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…

Be Nasty*Draw a grid big enough for digit cards

Player Hundreds Tens Units (ones)

Player 1

Player 2

*Be NastyLO: To use knowledge of place value to order

numbers up to 1000

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.

*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

*From Counting to Addition: 2 + 3 = 5

Count allCount on from firstCount on from largerCount on from eitherKnown factDerived facts

1,2...1,2,3..1,2,3,4,53,4,5

4,5 3,4,5 or 4,5 5

2+3 =5 so 3+3 =6 and 5-3=2

Carpenter and Moser (1983)

*What’s the difference

between…..?*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?

*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

*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?

* From Counting to Subtraction: 9 – 4 = 5

*Count out:

*Count back from:

*Count back to:

*Count up to:

*Recall of known number facts*Use derived facts:

*Hold up 9 fingers (1, 2, 3, ...., 9) and fold down 4 (1, 2, 3, 4). Count the remaining 5 fingers (1, 2, 3, 4, 5).*Count back 4 numbers from 9: ‘8, 7, 6, 5’

*Count back from first number to second saying: ‘8, 7, 6, 5, 4’ and tallying the numbers said (1, 2, 3, 4, 5)*Count on from 4 up to 9: ‘5, 6, 7, 8, 9’*to 10, 20 …, ways of making 100, 1000 e.g. 20 - 5 = 15 so 20 - 6 = 14

Ian Thompson (1997) Teaching and Learning Early Number, OUP

*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

*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.

*Calculation laws

*Commutative law - axb = bxa and a+b = b+a

eg 3 x 4 = 4 x 355 + 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)

http://dera.ioe.ac.uk/778/

*

Introduction to reasoning

*Magic squares*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.

*What would happen if you added two to each number - would the square still be magic? What could you tell you partner about the magic square now.

*What learning was going on? In pairs or small groups, consider the magic square activity in relation to Skemp’s theory of relational and instrumental understanding.

* Bruner (1961) Discovery Learning • Based on constructivist learning theory and problem solving.• Learner actively constructs knowledge and skills rather than passively

receiving knowledge from a teacher/text book or equivalent.• Learning is more effective when a student is actively engaged in the

learning process• Pupils retain knowledge and have deeper understanding if they

discover it for themselves• learning builds upon prior knowledge and understanding• Pedagogical aims:

Promote "deep" learningPromote meta-cognitive skills (develop problem-solving skills, creativity, independent learning , evaluation)Promote student engagement.

Bloom's Taxonomy is a hierarchy of skills that reflects growing complexity and ability to use higher-order thinking skills (HOTS).

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

Developing higher order thinking

* If children spend most of their time practising paper and pencil skills on worksheet exercises, they are likely to become faster at executing these skills.

* If they spend most of their time watching the teacher demonstrate methods for solving special kinds of problems, they are likely to become better at imitating these methods on similar problems.

* If they spend most of their time reflecting on how various ideas and procedures are the same or different, on how what they already know relates to the situations they encounter, they are likely to build new relationships. That is, they are likely to construct new understandings.

Hiebert ( 1993)

How can we encourage higher order thinking skills?

*Designing mathematical

tasksMathematical 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)

* But surely only the most able children can reason…….

“If teachers consider that tasks involving mathematics thinking are suitable for ‘high attainers’ then the result may be that ‘low attainers’ are given a diet of routine and repetitive tasks on which they have already demonstrated their low attainment. But if all learners are treated as possessing the powers necessary to think mathematically, and if those powers are evoked, developed and refined, the so called ‘low attainers’ can transcend expectations (Mason and Johnston-Wilder (2006 . 41)

*Sudoko

1

2

3

4

x32

40

49

22

15

27

24

42

*Fit into the purple boxes: 2,3,4,5,6,7,8,9,10,11,12*One number has to be used twice! Which one?Why?

Shepway Teaching Schools Alliance, Unit 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH

*Always Sometimes never

*EYFS*How many different patterns of dots can you make with five dots?

*Year 1*When you add two numbers, you can change the order of the numbers and the answer will be the same*You can make 4 different two digit numbers with the digits 2 and 3: 23, 32, 22, 33*When you add 10 to a number the units digit stays the same.

*Year 2*When you subtract ten from a number, the units digit stays the same*You can add 9 to a number by adding 10 and subtracting 1*All even numbers end in 0, 2, 4, 6, 8*If you have 3 digits, and use each one exactly once in a three digit number, you can make 6 different three digit numbers

*Questioning

‘The subtle art of questioning is the art of teaching. In a real sense, learning to teach is learning to ask questions.’

Tanner and Jones (2000)

*Improving your questioning

Examples of open-ended questions that invite children to think include*What do you think…………?*How do you know………….?*Why do you think that………..?*Do you have a reason………..?*Is this always so……….?*Is there another way/reason/idea…………..?*What if………….? What if…….does not…..?*Where is there another example of this…..?*What do you think happens next?

*

* Instead of * Find the perimeter of a 3x8 rectangle

* You could ask* If the area of a rectangle is 24cm2 what is the

perimeter?

Improve these Questions1.A chew costs 3p and a lolly costs 7p. What do they

cost together?

2.What is 6- 4?

3. Is 16 an even number?

4.What are 4 threes?

5.What is this shape called?

* Effective questioning strategies to promote thinking

*Sequencing a set of questions*Pitching appropriately*Distributing questions around the class*Prompting and probing*Listening and responding positively – inviting further questions*Challenging right as well as wrong and underdeveloped answers*Using written questions effectively.