Preparing to teach mathematics

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Preparing to teach mathematics
4/14/2017 7:49 AM Preparing to teach mathematics 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

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 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

Demonstrate good subject and curriculum knowledge
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.

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

Arithmetic fluency: Learning mathematical procedures and skills and using this knowledge to solve problems Reasoning: learning to reason about the underlying relations in mathematical problems they have to solve Learning and remembering skills and procedures Calculating efficiently Remembering mathematical vocabulary Remembering facts Eg Knowing how to add Arguing, communicating Problem solving Investigating Thinking mathematically Understanding ideas or concepts Knowing when to add

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 To reason 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/att achment_data/file/184064/DFE-RR178.pdf These three aims should not be lost in the detail of the programmes of study. The new draft seeks to strengthen these aims. These aims come from research carried out by the DfE into high performing jurisdictions https://www.education.gov.uk/publications/standard/publicationDetail/Page1/DFE-RR178 Children’s varying pre-school experiences of mathematics mean they start school with different levels of knowledge of number and shape. For too many pupils, this gap is never overcome While the best teaching developed pupils’ conceptual understanding alongside their fluent recall of knowledge, and confidence in problem solving, too much teaching concentrated on the acquisition of disparate skills that enabled pupils to pass tests and examinations but did not equip them for the next stage of education, work and life.

Conceptual Understanding
BALANCE Procedural Fluency Conceptual Understanding INTEGRATION 7

What’s the difference between mathematical fluency and conceptual understanding? Discuss
Speed Accuracy Efficiency Flexibility Clearly children develop mathematically at different rates, and progression in calculation is far from linear; rather, as connections are made, a child’s understanding of calculation may be considered more akin to a web of linked ideas. To build these links effectively the emphasis has to be on conceptual understanding of addition and subtraction as opposed to being able to calculate procedurally but without understanding.

Make a list of the mathematical concepts children need to know in order to calculate
32 –

Many secondary teachers
Pedagogy Classroom practice Many secondary teachers Many primary teachers The best teachers Subject knowledge Lisa OfSTED (2008) Understanding the Score

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.

Progression in counting
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

Phases of learning mathematics
Singapore Maths Bruner’s phases of learning Concrete Pictorial Abstract Enactive Iconic Symbolic 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

Consider….. 5

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 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 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.

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 Be Nasty

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) See next slide for commentary

Misconceptions and errors in place value:
Twenty eight, twenty nine, twenty ten Writing 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,

Using Place Value Charts
Using gattegno charts or place value cards is like segmenting and blending in phonics!

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

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?

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?

Early Mental Strategies for Addition and Subtraction
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 = Bridging through 10, using known facts to 10 e.g = (6 + 4) + 5; 15 – 9 = (15 – 5) - 4 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

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

31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH
31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

So why is the understanding of place value so important?
Calculate Using Dienes Using Numicon Using Place Value Counters

So why is the understanding of place value so important?
Calculate Numicon Using Dienes Using Place Value counters

Calculate these mentally (and record your strategies on paper):
156 – 99 Then compare strategies with a friend.

Target Board 0.3 1.6 7.2 4.6 0.2 10.5 5.7 2.3 6 5.3 8.3 0.1 2 5.2 7.3 2.7 1 0.7 4 1.9 9.2 3.9 9.8 6.2 2.6 3 6.1 10 1.7

Choice of Mental Strategies
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.

Mental and Written Calculations
We may break the calculation into manageable parts eg 248 – 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 Written We usually begin with the least significant digit
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

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.

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); x 5 as (20 x 5) – (1 x 5); 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 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

Calculation laws Commutative law - axb = bxa and a+b = b+a
eg 3 x 4 = 4 x 3 = 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)

Choice of Mental Strategies
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.

Some strange calculation methods!

Introduction to reasoning

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

Developing higher order thinking
Higher order thinking skills are not just about mathematical content knowledge. Just as it is possible to engage in very hard questions that involve a high level of content knowledge but very problem solving skills it is also possible to engage in activities that only need a low level of mathematical content knowledge. In the former you are going to need well tuned knowledge level skills and in the latter HOTS Bloom, B.S. (Ed.) (1956) Taxonomy of educational objectives: The classification of educational goals: Handbook I, cognitive domain. New York ; Toronto: Longmans, Green.

Multiplication Puzzle
x 32 40 49 22 15 27 24 42 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?

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

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? Simplification Give a restriction to the number of animals Eg Noah had between 20 and 30 animals (answer = 25) Have only two sorts of groups eg; Noah had up to 15 animals when he counted them 2 (even number) he had none left over when she counted them in 5 he had 2 left over Challenge Leave number of animals an open value and find all possibilities – 25, 55, 85 do you notice a pattern (2x3x5 = 30) Set other contexts.eg How many animals were there if they were grouped in fours and one remained sevens and five remained nines and seven remained (61) Ask pupils to make up their own problems for Noah using different numbers

Finding rules and patterns - NIM
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. Nim probably originated in China, the current name of this game is a loan word from German verb Nimm – to take).The theory of the game was discovered by a maths professor at havard Uni 1901. Nim is a simple game with finite possibilities, however there is a tremendous variety in the games implementation. Versions of Nim can be played with from one to at least a dozen row and the number of counters in a row can very from one to as many as two dozen. Some versions require winner takes last object, others involve winner avoids taking last object. My version (from Carol Vorderman’s book) Look for number pattern 1,5,9,13, 17 – if you leave your partner with this number of counters you are sure to win.

How can we encourage higher order thinking skills?
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. Skemp instrumental and relational understanding (1976) Ryan & Williams surface and deep understanding (2007) Hiebert ( 1993)

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 ( )

Sudoko As a group on the floor with laminated shape cards
Or in pairs with small cards Lisa, I have a bag of resources which I will bring so at this point we can share ideas. Can you bring your one is a snail book please.

1 2 3 4

Always Sometimes never
Shepway Teaching Schools Alliance, Unit 31a Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH

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

Can children engage with proof?
Add together three consecutive numbers = 6 And three more What do you notice? Will it always happen? Can you convince yourself it will always happen? Can you convince a friend? Can you convince an enemy?

Planning for progress

Child A Learning Intention: To know how to write and represent
2-digit numbers using knowledge of place value

Child B Learning Intention: To know how to write and represent
2-digit numbers using knowledge of place value

Child C Learning Intention: To know how to write and represent
2-digit numbers using knowledge of place value

Recording Assessments for a Focus Group: Poor
Learning Intention: To know how to write and represent Date: 2-digit numbers using knowledge of place value Name Assessment Comments Target Date Child A Didn’t concentrate very well. Took a long time to get started. Could write some numbers well. Work on concentration and doing more work. Child B Worked hard. Achieved the learning intention for smaller numbers but not for bigger ones. Work on bigger numbers Child C Tried hard. Got most of the answers right with support. Wasn’t sure about the last 3 questions. Practice some more in next lesson Recording Assessments for a Focus Group: Poor

Recording Assessments: Improved
Learning Intention: To know how to write and represent Date: 2-digit numbers using knowledge of place value Name Assessment Comments Target Date Child A Can accurately write two-digit numbers up to 30 for a provided representation but in own representations does not recognise the base 10 structure used in place value. Understanding the need to group in tens plus remaining ones for 2 digit numbers Child B Succeeds with writing teen numbers but creates incorrect additional columns for writing numbers with more than 1 ten Understanding that all the tens need to be combined and written in tens column Child C Can accurately write most two digit numbers up to 30 for a provided representation but does not understand the need to use zero as a place holder for empty columns. Understanding of zero as a place holder in 2-digit numbers Recording Assessments: Improved