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
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.
*
* 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.
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
Arguing, communicating
Calculating efficiently
Remembering mathematical
vocabulary
Problem solving
Investigating
Thinking mathematically
Remembering facts
Understanding ideas or concepts
Eg
Knowing how to add
Eg
Knowing when to add
*
 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
*
Procedural
Fluency
Conceptual
Understanding
INTEGRATION
*
*
32 – 3
32 - 29
Many secondary
teachers
Classroom
practice
Subject
knowledge
The best teachers
Pedagogy
Many primary
teachers
OfSTED (2008) Understanding the Score
http://www.ofsted.gov.uk/resources/mathematics-understanding-score
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.
*
* 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
*
Singapore Maths
Concrete
Pictorial
Abstract
Bruner’s phases
of learning
Enactive
Iconic
Symbolic
*
5
*
 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.
*
 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…
*
Draw a grid big enough for
digit cards
Player
Player 1
Player 2
Hundreds
Tens
Units (ones)
*
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.
*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
*
*
• 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)
*
* 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..
*Using Place Value Charts
1
2
3
4
10
20
30
40
5
50
6
60
7
70
8
80
9
90
100 200 300 400 500 600 700 800 900
1000 2000 3000 4000 5000 6000 7000 8000 9000
*
*
* 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
*
•
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?
*
* 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 10
e.g. 6 + 9 = (6 + 4) + 5; 15 – 9 = (15 – 5) - 4
*
* 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
*
*Calculate 25 + 47
* Using Dienes
* Using Numicon
* Using Place Value Counters
*
*Calculate 72 - 47
* Numicon
* Using Dienes
* Using Place Value counters
*
*26 + 57
*25 + 24
*65 + 29
*73 - 68
*82 - 26
*156 – 99
Then compare strategies with a friend.
*
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
2.3
9.8
6.2
2.6
3
6.1
10
1.7
*
*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
*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
• We usually begin with the
Written
*We usually begin with
• We choose a strategy to fit
the least significant
digit
*We always use the same
method
• We draw upon
*We draw upon the
most significant digit
the numbers eg 148 – 99 is
not calculated in the same
way as 84 – 77 although
they are both subtractions
mathematical knowledge
such as properties of
numbers or ‘number
sense’, learned facts etc
memory of a procedure
although we may not
understand how it works
*
• 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
*
• 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);
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
*
*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)
*
*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!
http://www.ness.uk.com/maths
/Guidance%20Documents/Tea
ching%20children%20to%20c
alculate%20mentally.pdf
*
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/developmentof-maths-capabilities-and-confidence-in-primary-school
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.
*
x
32
* Fit into the dark
40
blue boxes:
2,3,4,5,6,7,8,9,10
,11,12.
49
* One number has
22
15
27
24
42
to be used twice!
Which one?
Why?
*
Can you think of a different calculation
(which does not use the digit 6) to give the
same answer?
*32 + 16
*58 - 26 (32+15+1)
*48 x 6
*126 - 58
*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?
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.
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.
Hiebert ( 1993)
*
“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)
*
1
2
3
4
*
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
*
* Add together three consecutive numbers
1+2+3=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?
*
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
Learning Intention: To know how to write and represent
2-digit numbers using knowledge of place value
Date:
Name
Assessment Comments
Child A
Didn’t concentrate very well. Took a Work on
long time to get started. Could
concentration and
write some numbers well.
doing more work.
Child B
Worked hard. Achieved the learning
intention for smaller numbers but
not for bigger ones.
Child C
*
Target
Work on bigger
numbers
Tried hard. Got most of the answers Practice some
right with support. Wasn’t sure
more in next
about the last 3 questions.
lesson
Date
Learning Intention: To know how to write and represent
2-digit numbers using knowledge of place value
Date:
Name
Assessment Comments
Target
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 2digit numbers
*
Date