1. Preparing to teach mathematics
4/14/2017 7:49 AM
Preparing to teach mathematics
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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
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3. 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.

4. 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

5. 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

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
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.
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
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.

7. Conceptual Understanding
BALANCE
Procedural Fluency
Conceptual Understanding
INTEGRATION 7

8. 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.

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

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

13. Phases of learning mathematics
Singapore Maths
Bruner’s phases of learning
Concrete
Pictorial
Abstract
Enactive
Iconic
Symbolic
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14. Consider….. 5

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 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. Be Nasty Draw a grid big enough for digit cards Player Hundreds Tens
Units (ones)
Player 1
Player 2

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
Be Nasty

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)
See next slide for commentary

21. 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,

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

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

24. 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?

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. 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
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28. 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

29. 31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH
31b Folkestone Enterprise Centre, Shearway Road, Folkestone CT19 4RH. Website:

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

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

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

33. 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

34. 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.

35. 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

36. 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

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); 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
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40. 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)

41. 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.

42. Some strange calculation methods!

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
of-maths-capabilities-and-confidence-in-primary-school

47. 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.

48. Multiplication Puzzlex 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?

49. 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
( )

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?
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

51. 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.

52. 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)

53. 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 ( )

54. 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.

55. 1 2 3 4

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

58. 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

59. 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?

60. Planning for progress

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

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

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

64. 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

65. 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