Contents
Mathematical Understanding: An Introduction
2
Section 1 - Understanding Place Value: A school Based Progression
4
Section 2 - Understanding Addition: A school Based Progression
Mental Calculation Strategies: Progression into Key Stage 2
7
10
Section 3 - Understanding Subtraction: A school Based Progression
Mental Calculation Strategies: Progression into Key Stage 2
12
16
Section 4 - Understanding Multiplication: A school Based Progression
Key Concepts
Prior Learning
A school Based Progression
Mental Calculation Strategies: Progression into Key Stage 2
18
Section 5 - Understanding Division: A school Based Progression
Key Concepts
A school Based Progression
Mental Calculation Strategies: Progression into Key Stage 2
18
18
19
21
24
24
25
26
Section 6 – Assessment: Use of the Spreadsheet – Notes to teachers
28
Appendix 1: Mathematical Vocabulary
Appendix 2: The Instant Recall of Times Tables
Appendix 3: Examples of Misunderstanding
31
32
34
1
Mathematical Understanding
An Introduction
It is a strange truth but true nevertheless that children can undertake quite complex calculations well into Key stage 2 and yet have
little understanding of the mathematical processes involved or the numerical principles underpinning them. Whilst in the short term
this might appear satisfactory the reality is that without the foundations in place children become vulnerable when new concepts are
overlaid on to insecure understanding.
Two examples of this will suffice; (see appendix for other examples)
At the end of a series of lessons a group of more able Year 6 children were happily calculating 28% of 324 with great accuracy,
however there progress was somewhat derailed when they were asked to explain what a percentage was. Two instantly responded that
they did not know and the closest we came were that two girls knew it had something to do with 100.
Similarly the Year 6 child this year who could calculate TU x TU came a little unstuck when confronted with the calculation 36x38.
He calculated 36x30 which he found equalled 1080, but then realising that he still needed to multiply the 8 proceeded to calculate
1080x8. Apart from the fact that he was not able to readily see that his answer was woefully inaccurate in terms of an estimate, it
materialised upon further discussion that he had no idea what the sum 36x38 meant. He could not “see” that it could be interpreted as
38 people all having 36 pence and was totally at a loss as to what numerical operations he would need to undertake to gain the correct
answer.
This does not bode well for his individual mathematical development and it is imperative that the school’s Maths curriculum delivers a
true understanding of mathematical principles alongside a simple ability to calculate and number crunch.
In many ways therefore this Mathematical understanding document is more important than its calculation counterpart. It seeks to
break down the development of calculation into its constituent parts and outlines clear areas of understanding that children need to
take on board before they can truly make well grounded progress in Mathematics. It is not enough for us to teach that “If you multiply
by 10 then you add a nought on”, because it is mathematically weak and in year 5 the children will find that adding a zero to 2.3x10
does not deliver the right answer. The key is that we engage children in the true, pure principles of Mathematics, which in this case is
that the zero acts as a placeholder. This dovetails into the school’s work on Learning Objectives and the attempts to tease out at
planning level what the “true” learning is for each lesson, rather than satisfying ourselves with the rather bland and meaningless
learning objective of “Divide a two digit number of 100 by 10”. Whilst at first glance this lesson would appear to be devoted to
division, the true learning will probably hinge around place value and the relationship of numbers, this purer learning objective should
be expressed in the teacher’s daily planning.
2
The document outlines the stages of mathematical cognitive development children need to attain before moving on and this document
should be read and used alongside the associated assessment spreadsheets that the school has developed. This shows an individual
children’s progression in the four rules and will allow the teacher to develop clear personalised learning based on areas of relative
weakness, thereby plugging the gaps in their understanding which can easily be unwittingly built upon.
The document fulfils two main purposes:
1. Primarily it provides a systematic programme of mathematical understanding that children can progress through in the early
primary years. In the first instance this will be focused heavily on the early years of the National Curriculum in Key Stage 1 and
then into Key Stage 2 where understanding needs to be consolidated for certain children. Whilst there is no a direct bridging point
between this document and its “Calculation” counterpart teachers should be wary of developing calculation strategies too early
before secure understanding is in place.
2. However the document fulfils another purpose for children who have progressed on to the calculation document. In the
multiplication and division strand the progression provides a framework for the development of a range of mental strategies that
the children should be fluent in alongside the more formal written methods they are being introduced to through the Calculation
document. Whilst the school acknowledges the need to focus on one core calculation strategy throughout the school (the number
line), there was also the recognition that we did not want to lose the wealth and richness of calculation strategies that the original
Numeracy framework introduced. Therefore this document seeks to track children’s progress in these without, hopefully,
detracting from the central core strategy of the number line.
The key to all this is the school’s contention that in its purest and highest form Mathematics ceases to be based around the ability to
undertake basic number calculations but instead becomes increasingly focused around patterns, relationships and the development of
theorems to explain these. Therefore if children are to make good mathematical progress in the latter years of their education they
need to leave their primary schools with a secure foundation in Mathematical understanding not simply an ability to calculate using
large numbers.
3
Section 1
Understanding Place Value: A School Based Progression
Understanding to Acquire
Stage
Errors and Misconceptions
Assessment Task
Order consecutive cardinal numbers
up to 10 orally
Children misplace the order of the
numbers
Orally state the numbers
2
Order consecutive cardinal numbers
up to 10 practically
Children struggle to order the numbers
Practically demonstrate the concept of
consecutive numbers
Order consecutive cardinal numbers
up to 10 written
The inability to decode the symbols of
the numerical language
Write the numbers under 10
consecutively
4
Add 1 to a given number
The inability to know 4+1 may
indicate poor understanding of
consecutive numbers
Be able to explain why 5+1 and 8+1
are “easy sums” to know
5
Order cardinal numbers
Ability to run off the numbers 1-10 in
order may mask a true understanding
Order the following numbers 7, 3, 6, 1
and 4.
Order numbers 10 – 20
The children often struggle with the
Orally and written the children should
teens numbers where the fourteen
demonstrate their ability to order the
implies the 4 comes first e.g. 41 not 14 numbers from 0-20
Understanding of the place value of
tens and units
12 = 1 ten and 2 units not vice versa
6
7
Numbers 10-20
3
Numbers 0-10
1
Understand 12 is one ten and 2 units,
explaining the role of the tens and
units column
4
Understand labels of teens numbers
Thirteen is 13 not 31 despite what the
language suggests
9
Understand relationship of numbers in
columns
Explain how and why the 9 in the
A 9 in the units column is not a greater
units column is smaller that the 2 in
number than a 2 in the tens column
the tens column
10
Appreciation of base 10 i.e. an
understanding that when unit numbers
reach a ten a new ten is created
Questions like “What is the next
number after 19?” Explain why?
11
Order number 20-100
Ensure the children have a good grasp
of the pattern in the tens numbers i.e.
1,2,3,4 tens
12
Know the number 10 more than a
given number
Numbers 20-100
8
The children should appreciate that to
add one ten to 27 will create a number
1 ten bigger i.e. 37
Write and correctly label the numbers
13,14,15,16,17,18 and 19
27 + 10 =37
Ability to see 20 as 2 tens
Ability to see the 4 in 43 as 4 tens not
just as an isolated number
14
Ability to see 63 as 6 tens and 3 units
Ability to see the 6 in 63 as 6 tens and
the 3 as 3 units, and also to understand
the relationship between them
15
A secure understanding that place
determines value
Understanding that to take a 9 from
the units and put it in the tens
markedly changes its value
Place a 9 on a tens and units grid and
allow children to interpret its value
16
Understand the role of zero as a place
holder
That the zero in 60 is not superfluous
but “places” the 6 in the tens column.
Without it the 6 reverts to 6 units
Ask the children the value of 6; then
ask the value of 60. What is the
difference and how do they know?
13
5
18
19
Numbers to 1000
17
Ordering numbers up to 1000
The children should be able to order
the numbers up to 1000
Ordering numbers using the highest
denominator.
When the hundreds are identical then
use the tens column to ascertain the
highest number
Take the numbers 245, 236 and 222
and use the tens numbers to order
them.
Calculate to the nearest 10 with
numbers under 100
Ability to round numbers
The children should be able to round
34 down to 30, round up 36 to 40 and
know 35 is halfway inbetween.
6
Section 2
Understanding Addition: A school Based Progression
Stage
Sum
Errors and Misconceptions
1
4+3=
Lack of understanding as to what
happens in the process of addition
The concept of two numbers
forming a new total
Place 4 cubes and 3 cubes on the
table. What will happen when I
add them?
2
8+0=80
8+0=0
Lack of understanding of the role
of zero. Some see the zero as a
number which should make a
larger total, where others see zero
as the dominant number and
therefore the answer is zero
The value of zero in a variety of
contexts (not to be confused with
zero as a place holder)
8+0=
Can they explain the answer?
3
3+1=5
Cannot identify number before or
after a given number
The order of the numbers needs to
be established.
What is 3+1? Can the children
explain why it is 4 i.e. because 4 is
the next number
4+2=
Can only begin counting at one;
Cannot hold the first number as an
initial total
The initial number does not
change it is the addition process
that changes the total
4+2= How did you work the sum
out?
4
Understanding to Acquire
5
10 can be split
Conservation of number
into 6+4
Understanding that with 10 cubes
you can create different addition
sums without changing the total
6
Understanding
of Number
Number bonds to 10
Bonds to 10
Understand that there are a range
of sums that can be created and
that these are in a pattern
Assessment Task
Give the children 10 cubes and see
whether they can make groups
with them whilst understanding
the total remains the same
Put the number 10 in the middle of
a piece of paper. Can they write
down the number bonds to 10
7
7
Number
Bonds
8
Number
Bonds
9
Rapid Recall
of number
bonds
10
+2=5
To build on number bonds of other There is an inverse pattern in the
numbers
bonds i.e. 9+1 and 1+9
Whatever the number used the
To build on number bonds of other
family of number bonds will be
numbers
one more
The children can recall instantly
Children unable to recall number
all the number bonds for the
bonds under 10 with any rapidity
family of numbers under 10
Unable to translate comprehension Introduction to algebraic maths
of “usual addition” into new
such that concepts such as if
context. Inability to use concept of a+b=c then
inverse operations
c-b=a and c-a=b
2+8
Doesn’t use the largest number to
count on from
12
3+4+7
To counteract the feeling that
addition is only ever two numbers
13
8+10=19
Lack of tens to units correlation
14
8+7=22
Has no feel for the answer being
between 10-20
11
Lack of appreciation of specific
number value and the concept of
“adding on”
To introduce the concept as a
precursor to repeated addition that
the children need for
multiplication
Establishing the introduction of
mathematical pattern
A concept of the relative value of
specific numbers
Teacher writes 9+1 and seeks to
elucidate from the child whether
they can “see” that this allows
them easy access to 1+9
“If I choose 8 how many number
bonds will there be?”
Test questions and instant recall
either written or verbal
Give the children the sum +2=5
and ask if they can give the answer
Give the children 2+8 and see how
they calculate the answer. They
should also explain why they use
the bigger number first
Teacher writes sum and asks
whether the sum is a “real one”
8+10=18
Why does it equal 18?
Give the children 8+7 and ask
them to point on a number line
where the answer might lie. They
must not calculate the answer
8
Write the sum and ask “Does it
make sense?”
Can the children write their own
sums?
Ask the children to calculate
7+3+5 and determine whether they
used number bonds in calculation.
Write it out as a two step sum
what numbers would write?
15
4+6=7+3
Inability to understand the role of
the = sign within a sum. i.e. it is
not simply the place where the
answer goes
16
7+3+5 = 10+5
Inability to build/see relationships
between numbers
Ability to see number bonds in
multi-step calculations
17
+3 is the
opposite of -3
Inability to see subtraction as the
inverse
Specific links between the 4 rules
should be taught at every
opportunity
18
21+2; 31+2;
41+2
Inability to see consistent pattern
in the unit column
Teacher writes down the sums
21+2; 31+2; 41+2 and asks the
children what 51+2 would be
Lack of extended tens and units
correlation, leading to counting on
Further work on mathematical
patterning
Teacher writes 8+20=28,
8+30=38, and then asks for the
answer to 8+40. The children
should explain why they know the
answer is 48.
Unable to recognise patterns to
help deduce addition facts with
numbers other than “tens
numbers”
Patterns relative to addition in tens
needs to be securely established
Write the sums 46+5, 46+15,
46+25 with the answers. Can the
children deduce the total of 46+35
19
20
8+20; 8+30
46+5, 46+15,
46+25
Understand the meaning of = not
necessarily being the end of the
sum
Write 3+2=5 then 5-2=4. Ask the
children why this is wrong.
Associated vocabulary: add, more, and, make, sum, total, altogether, score, double, one more, plus, near double, addition, equals,=
9
Mental Calculation Strategies: Progression from KS1 and into Key Stage 2
The following is a digest of the mental calculation strategies introduced by the National Numeracy Strategy. Whilst the major
calculation focus in KS2 is the development of the number line, these strategies are crucial in developing children’s mental capability
in calculation. The first question a child should ask before embarking on any calculation is; “Can I do this in my head?” These
strategies are central to developing a clear sense of number and a fluency in calculation that will enable children to tackle an
increasing number of mathematical problems mentally. To this end they supplement rather than supersede the school’s core work on
the number line. (See “School Calculation” document)
Whilst these may be taught discretely there is a single sentence tucked unobtrusively at the bottom of pages 46-49 of the NNS
document that simply states “Use and apply these skills in a variety of contexts, in mathematics and other subjects”. This is a vital
aspect of the child’s mathematical development. The strategies must be taught through a range of contexts so that children imbibe
them holistically into their learning and use them correctly in contexts beyond the constraints of the formal maths lesson.
NNS
Exemplar
Year 2
Year 3
Year 4
Year 5
HTU
TH HTU
Doubles
e.g. 8+9 = 8+8 +1
U
TU
Odd/Even
Relationship: 2 odds=even
U
TU
Add 9/99/999
e.g. 22+9 = 22+10-1
HTU
TH HTU
Add 11/101
e.g. 22+11=22+10+1
HTU
TH HTU
TU+TU
HTU+TU
HTU+HTU
Commutative Law
p34
e.g. 95+86=86+95
TU+TU
Associative Law
p34
e.g. (25+17)+18 = 42+18
TU+TU
HTU+TU
HTU+HTU
Derived facts-place
value
p38
e.g. 7+9=16 so
70+90=160
TU+TU
HTU+TU
Decimals
Counting on in steps
p40
e.g. 643+50 adding in tens
TU+TU
HTU+TU
Year 6
10
Partition
p40
e.g. 24+58=20+70 and
4+8
TU+TU
TU+TU
HTU+TU
Identify near doubles
p40
e.g. 36+35=35+35+1
TU+TU
HTU+HTU
Decimals
HTU
Add to a multiple of 10
p40
e.g. 74+58 = 74+60-2
TU+TU
TU+TU
HTU+TU
Decimals
Addition/Subtraction
p42
TU+TU
TU+TU
Add several numbers
p42
e.g. 56+14=70 so 7056+14
e.g.
70+71+75+77=70x4+13
U
TH,HTU
TU
TU
TH,HTU
Use Known Facts and place value to add or subtract a pair of numbers mentally (see NNS p44-47)
Add multiples of 10
p44
e.g. 570+250
Add multiples of 100
p44
e.g. 500+700+400
Add to multiple of 100
p44
e.g. 58+x=100
p44
e.g. 3200 + x = 4000
p46
p46
Add to multiple of
1000
Add single digit to
HTU
Add pair of 2 digits
TU
TU
HTU
HTU
HTU+HTU
H+H+H
TU
TU
HTU
TH,HTU
TH,HTU
Decimals
e.g. 6745+8
TU
TH,HTU
e.g. 0.05+0.3
TU
TU
Decimals
Decimals
2 places
11
Section 3
Understanding Subtraction: A School Based Progression
Stage
Sum
1
4-3=
Errors and Misconceptions
Understanding to Acquire
Assessment Task
Subtraction is the concept of
“taking away”
The concept of the fact that
subtraction involves some of the
original total being removed.
What is subtraction? Get the
children to use cubes to illustrate
their understanding
5-10=5
Confusion of the place of the
numbers within the context of the
sum
In principle subtraction involves
the smaller number being
subtracted from a larger total
Roll 2 die and make a subtraction
sum with them. get the children to
explain the rationale behind the
order of the numbers they chose.
3
5-1=3
Cannot identify number before a
given number
5-1=4
The order of the numbers needs to
Why is the answer 4?
be established.
How did you know?
4
8-0=0
Lack of understanding of the role
of zero.
The value of zero in a variety of
contexts (not to be confused with
zero as a place holder)
8-0=
Can they explain the answer?
10-4=6
When you have taken 4 away 6
remain.
The principle of subtraction using
10
Ask the question 10-4=6
Get the child to illustrate with
cubes
2
5
12
6
7
8
9
10-4=6
10-6=4 to put 4
back =10
10-9; 10-8
Number Bonds
10
10-7=3
11
“What is the
difference
between 7 and
10?”
If you start with 10 and 6 remain
then 4 must have been taken
Counting on principle to establish
the difference
Place 10 cubes on the table. Take
4 away without the children
knowing. With them being able to
see the 6 remaining cubes can
they calculate how many are in
your hand?
Conservation of number in
relation to addition and
subtraction be the inverse
Understanding that with 10 cubes
you can create different
subtraction sums all of which
total 10 in the number remaining
and the number removed
Give the children 10 cubes and
allow them to illustrate the
principle with the cubes
To build on number bonds of
other numbers
There is an pattern in the bonds to
10 i.e. 10-1; 10-2; 10-3 etc.
Allow the children to design their
own subtraction patterns. If this
fails…
Teacher writes 10-1; 10-2; 10-3
with answers. Can you write the
next sum in the sequence
Inability to see patterns in
calculation process
Whatever the number used the
family of number bonds will be
one more
If I have the number 8 how many
subtraction sums can I make in
that family?
Cannot see that 7+3 may help
them solve this equation
Ability to use “known fact” i.e.
Give the children 7+3=10. Then
7+3=10 to calculate (at speed) 10- ask them the answer to 10-7. How
7=3
do they know the answer is 3
Inability to understand the
Mathematical vocabulary of
“Difference” in the question
“What is the difference between 7
and 10?”
To understand the use of the word
“What is the difference between 7
“Difference in a mathematical
and 10?”
context”
13
Unable to translate
comprehension of “usual
subtraction” into new context.
Inability to use concept of inverse
operations
Unable to see that a subtraction
sum can contain more than two
numbers
Introduction to algebraic maths
such that concepts such as if cb=a then
a+b=c and b+a=c
Give the children the sum 10-=5
and ask if they can give the
answer
12
10-=5
13
10-2-3=
14
19-10=8
Lack of tens to units correlation
Establishing the introduction of
mathematical pattern
Write the sum 19-10=8 and ask
children to explain why it cannot
be correct.
15-7=12
Has no feel for the answer being
between less than 10
A concept of the relative value of
specific numbers
Give the children the sum 15-7
and on a number line get them to
guess where the answer might lie.
6-4=7-5
Inability to understand the role of
the = sign within a sum. i.e. it is
not simply the place where the
answer goes
Understand the meaning of = not
necessarily being the end of the
sum
Write the sum and ask “Does it
make sense?”
Can the children write their own
sums?
17
+3 is the opposite Inability to see subtraction as the
of -3
inverse
Specific links between the 4 rules
should be taught at every
opportunity
Write 10-7=3 then 3+8=10. Ask
the children why this is wrong.
18
19-2; 29-2; 39-2
etc.
Inability to see consistent pattern
in the unit column
Develop patterning skills using
tens
Write 19-2; 29-2; 39-2
What will 49-2 be?
19
59-20; 49-20;3920 etc.
Lack of extended tens and units
correlation, leading to counting
on
Further work on mathematical
patterning
Write 59-20; 49-20;39-20
What will 29-20 be?
15
16
Introducing subtraction with more Does it make sense? Can it be
than two numbers
done? Explain
14
20
21
46-5, 46-15, 4625
83-79=
Unable to recognise patterns to
Patterns relative to subtraction in
help deduce subtraction facts with
tens needs to be securely
numbers other than “tens
established
numbers”
The understanding that when
The child takes the 83 and starts
numbers are close together the
to subtract 7 tens
best strategy is to find the
difference by counting on.
Write the sums 46-5, 46-15, 4625 with the answers.
Can the children calculate 46-35?
Model 83-79 by taking away 7
tens and then 9 units.
Ask the children; Is there a
quicker way?
15
Mental Calculation Strategies: Subtraction
Progression into Key Stage 2
The following is a digest of the mental calculation strategies introduced by the National Numeracy Strategy. Whilst the major
calculation focus in KS2 is the development of the number line, these strategies are crucial in developing children’s mental capability
in calculation. The first question a child should ask before embarking on any calculation is; “Can I do this in my head?” These
strategies are central to developing a clear sense of number and a fluency in calculation that will enable children to tackle an
increasing number of mathematical problems mentally. To this end they supplement rather than supersede the school’s core work on
the number line. (See “School Calculation” document)
Whilst these may be taught discretely there is a single sentence tucked unobtrusively at the bottom of pages 46-49 of the NNS
document that simply states “Use and apply these skills in a variety of contexts, in mathematics and other subjects”. This is a vital
aspect of the child’s mathematical development. The strategies must be taught through a range of contexts so that children imbibe
them holistically into their learning and use them correctly in contexts beyond the constraints of the formal maths lesson.
NNS
Exemplar
Year 2
Year 3
Year 4
Halves
e.g. 16-8=8 as 8 is half of 16
TU
HTU
Odd/Even
Relationship: 2 odds=even
TU
HTU
TH HTU
Subtract 9/99/999
e.g. 22-9 = 22-10+1
TU/HTU
TH HTU
Subtract 11/101
e.g. 22-11=22-10-1
TU/HTU
TH HTU
Subtraction is noncommutative
p36
Understand 7-5 is not 5-7
Derived facts-place value
p38
160-90=70 so 1600-900=700
TU
Year 5
Year 6
TU-TU
TU-TU
HTU-TU
Decimals
16
Counting on in steps
p40
e.g. 387-50 count back in tens
TU-TU
HTU-TU
Partition
p40
e.g. 98-43=98-40-3
TU-TU
TU-TU
HTU-TU
Subtract 9 to multiples of
10
p40
e.g. 84-19=84-20+1
TU-TU
TU-TU
HTU-TU
Addition/Subtraction
p42
e.g. 83-25=58 so 83-58=25
TU-TU
TU-TU
Decimals
TH,HTU
Use Known Facts and place value to add or subtract a pair of numbers mentally (see NNS p44-47)
Subtract multiples of 10
p44
e.g. 130-50
Subtract multiples of 100
p44
Subtract a multiple of 10
or 100
Subtract single digit to
HTU
Find difference when
numbers lie either side of
a multiple of 1000
Subtract pair of 2 digits
crossing tens boundary
TU
TU
HTU
e.g. 1200-500
TU-TU
HTU-HTU
p44
e.g. 582-30 or 1263-400
TU-TU
HTU
p46
e.g. 900-7
HTU
TH,HTU
p46
e.g. 7003 -6988 Counting on
HTU
TH,HTU
TH,HTU
p46
e.g. 0.5-0.19
TU
Decimals
TH,HTU
HTU
Decimals 2
places
17
Section 4
Understanding Multiplication: A school Based Progression
Key Concepts
These frameworks are markedly different to those for addition and subtraction. There is a lot of incremental progress that can be made
in addition and subtraction, but in multiplication the stages of progression appear to hinge around a few key areas. The school sees two
key areas that hinder all mathematical development in relation to multiplication and both of them act as a gateway into a new arena of
understanding for children. Without these it would seem that children can make little progress and will keep needing to grapple with
the basics of calculating in multiplication.
The two key areas are as follows:
1. The children need to “see” the connection between multiplication and repeated addition. They need to “understand” that 3x4 is
3 groups of 4 and when these are totalled it comes to 12. They should of course be able to place this understanding into a real
life context e.g. All 3 friends had £4 each so they could afford the toy that cost £12.
2. The children need to “understand” the numerical rules that govern the multiplying and dividing of numbers by powers of 10
and the associated use of the “distributive law”. This provides children with the option of using partitioning to solve
calculations involving larger numbers.
Therefore, unlike addition and subtraction which can involve incremental steps which build on a child’s previous knowledge much of
our teaching of multiplication and division will be spent scaffolding a range of activities and learning experiences to address these two
key issues. This makes the assessment of progression easier in the sense that there is less to assess, but in another way harder because
the teacher needs to be assured that the child has a full and complete 360 degree understanding of the concept before moving on.
Prior Learning
The key area of prior learning required for the understanding of multiplication is that the children are comfortable and reasonably fluid
with a variety of addition calculations. There is a danger that we introduce multiplication too soon attempting to run it in tandem with
addition. The truth is that children need a secure grasp of addition, especially with regard to the concept of repeated addition, before
tackling the greater complexity of multiplication. I wonder in my own mind if we have a tendency to introduce the latter too early and
we should therefore not be frightened to delay this until we are sure the children have made good progress in their understanding of
addition. For instance the concept of 6+6+6+6 is probably harder than 44+6
18
Section 4
Understanding Multiplication: A school Based Progression
Sum
Stage 1
2+2+2+2 (with pictures)
Understanding to Acquire
The child cannot relate the repeated addition to the concept of multiplication, nor
can they hold the numbers in their head.
The following should be used to lead children through the “Repeated Addition as Multiplication” barrier
1. Pictures
Pictures provide a valuable visual illustration of “sets of” or “groups of”
2. Practical
Wherever possible there can be no better modelling of the concept of multiplication
than to use “real life” contexts. This allows children to see “real” multiplication in
“real” meaningful contexts – a principle supported by the Renewed framework
3. Array to illustrate groups
The use of arrays are probably the most powerful visual representation of
multiplication. At this juncture they should be limited to horizontal use only. To see
that 3x4 can also be 4x3 will probably only seek to confuse rather than aid
understanding.
4. Real life stories
The children should be able to verbalise their calculations in the form of meaningful
stories. They should fully comprehend that 4x2 is about 4 people having 2 legs each
and that this delivers a total of 8
5. Use of the number line (with
numbers and then as an empty
number line) to illustrate 2+2+2+2
This may be used as a visual strategy (not as a calculation strategy) but as a visual
tool to demonstrate 4 jumps of 2. It will also serve as an introduction to the number
line that the children will meet later. However the number line should not replace the
array which is a more powerful visual representation of multiplication.
19
The following should be used to consolidate children’s understanding of the “Multiplication process”
Stage 2
2+2+2+2 (no picture) is The child should be able to relate the principle of repeated addition and relate them clearly
4 groups of 2
to the multiplying concept of “groups of”
Stage 3
2x4=
Stage 4
Learning of tables
Stage 5
2x4 and 4x2
Stage 6
2x4 is not 4x2
Stage 7
6 x ▲ = 12
Children should be aware that symbols can stand for numbers and should thus be able to
complete calculations similar to 6 x ▲ = 12
Stage 8
Doubling relates to
halving
The children should begin to see the relationship between doubling and halving
Stage 9
Relation of the inverse
The children should develop the work on doubling and halving to understand that
multiplication is the inverse of division
Stage 10
Recognising Patterns
The children should increasingly be able to readily recognise patterns of multiples notably
those in the 2,10 and 5 times tables
At this point the child may be introduced to the formal recording of their calculation in the
standard horizontal format and the use of the x sign to represent “multiply by”
The relative “rote” learning of tables should run alongside the child’s work on
understanding the true principles of multiplication. The learning of tables should start at this
point in the child’s development.
The use of the array should be used to demonstrate the commutative law e.g. that 2 rows of
4 can also be represented as 4 columns of 2
The children should appreciate that whilst 2x4 yields the same total as 4x2 they should
understand the difference underlying the two calculations
Progression:
There should be a strategic progression through the times tables. This is related both to the ease with which particular table are learnt;
e.g. the 2x table and also their intrinsic usefulness relating to further developmental stages in Mathematical understanding; e.g. the 10x
table. The progression is as follows: 2,10,5,3,4,9,6,8,7
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Mental Calculation Strategies: Progression from KS1 and into Key Stage 2
The following is a digest of the mental calculation strategies introduced by the National Numeracy Strategy. Whilst the major
calculation focus in KS2 is the development of the number line, these strategies are crucial in developing children’s mental capability
in calculation. The first question a child should ask before embarking on any calculation is; “Can I do this in my head?” These
strategies are central to developing a clear sense of number and a fluency in calculation that will enable children to tackle an
increasing number of mathematical problems mentally. To this end they supplement rather than supersede the school’s core work on
the number line. (See “School Calculation” document)
Whilst these may be taught discretely there is a single sentence tucked unobtrusively at the bottom of pages 46-49 of the NNS
document that simply states “Use and apply these skills in a variety of contexts, in mathematics and other subjects”. This is a vital
aspect of the child’s mathematical development. The strategies must be taught through a range of contexts so that children imbibe
them holistically into their learning and use them correctly in contexts beyond the constraints of the formal maths lesson.
Mental Calculation Strategies: Multiplication
Progression into Key Stage 2
The following are Strategies that will be used in conjunction with the “Calculation Document” in the sense that the children will
have developed in their understanding to a point where they are ready to calculate using the school’s number line method
These strategies are not therefore linear in terms of their progression but should provide a useful checklist for staff looking to
provide children with a breadth of mental calculation strategies that will greatly enrich their understanding of multiplication
Multiplying by 1 leaves number
unchanged (Year 3)
The children need to appreciate that whilst most of their conceptual framework of
multiplication is that the number gets bigger, here the number remains the same
Multiplying by 0 = 0 (Year 3)
The children need to appreciate that whilst most of their conceptual framework of
multiplication is that the number gets bigger, here the number becomes zero
Doubling (Year 3)
1 x 25 = 25; 2 x 25 = 50; 4 x 25 = 100; 8 x 25 = 200 etc.
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Multiplication Relationships (Year 3)
Multiply by x1 x10 x100 (Year 3)
7 x 5 = 35 so… 35 ÷ 7 = 5 (Year 3)
Given numbers such as 2,5,10 the child can write four multiplication/division sums
The child can multiply by 1, 10 and 100 spotting the relationships and pattern in the
answers
The children should have an increasing understanding of the relationship between
multiplication and division
The children should understand and use the principles (but not the names) of the commutative, associative and distributive laws
as they apply to multiplication.
Commutative (Year 4)
8 x 15 = 15 x 8
Associative (Year 4)
6 x 15 = 6 x (5 x 3) = (6 x 5) x 3 = 30 x 3 = 90
Distributive (Year 4)
18 x 5 = (10 + 8) x 5 = (10 x 5) + (8 x 5) = 50 + 40 = 90
Know all square numbers to 10 x 10
These should be known through rapid recall
Multiply by 5 (Year 4)
To multiply by 5 then multiply by 10 and halve the answer to get to 5
Multiply by 4 (Year 4)
Double and double again
Multiply by 20 (Year 4)
To multiply by 20 then multiply by 10 and double the answer
8 Times Table Facts (Year 4)
Work out the 8 times table facts by doubling the 4 times table
Quarters and Eighths
To find a quarter of a number then halve the number and halve again, halve again to
find an eighth
Multiply by 9 or 11
To multiply by 9 or 11, multiply by 10 and add or subtract the number
22
Multiply by 50 (Year 5)
To multiply by 50 then multiply by 100 and halve the answer to get to 50
Multiplying by 5 (Year 5)
To calculate 16 x 5 the children can halve the 16 and double the 5 i.e. 8 x 10
Halve an even number (Year 5)
16 x 51 This can be shortened to 8 x 51 =408 and 408 x 2 = 816
Use of brackets (Year 5)
The children should know that brackets determine the order of the operations and that
their contents are worked out first
Use Factors (Year 5)
15 x 6 This can become 15 x 3 = 45 and then 45 x 2 = 90
Multiply by 19 or 21 (Year 5)
To multiply by 9 or 11, multiply by 10 and add or subtract the number
Multiply by 15 (Year 6)
Multiply the number by 10, halve it and add the two resultant totals together (or
multiply by 30 and halve the total)
Multiply by 25 (Year 6)
To multiply by 25 times the number by 100 and divide by 4
Multiply by 24 (Year 6)
Work out 24 times table facts by doubling the 6 times table and doubling again
Multiply by 49 or 51 (Year 6)
To multiply by 49 or 51, multiply by 100, halve the total and add or subtract the number
Multiplication and fractions (Year 6)
Recognise that if 5 x 60 = 300 then 1/5 of 300 = 60 and 1/6 of 300 = 50
Decimal Fractions (Year 6)
Multiply a decimal fraction by a single digit e.g. 0.7 x 8
23
Section 5
Understanding Division: A School Based Progression
Key Concepts
As stated previously (multiplication) the frameworks for both division and multiplication are markedly different to those for addition
and subtraction. However division is the Achilles heel for all children, even into KS2 and beyond and this is not without good reason.
Whilst multiplication can be taught readily as “repeated addition”, this is not possible with division as there are two forms of division;
grouping and sharing. So whilst “repeated subtraction” can be taught cogently in terms of grouping this is not the case for sharing.
There is a ready recognition that children often access the first principles of division though the “sharing” concept as this is often
easier to demonstrate concretely. This is important but in terms of calculation strategies the number line operates using the “grouping”
principle. Therefore as children move out of Key Stage 1 they will be taught discretely the difference between “Sharing” and
“Grouping” and how these are calculated practically. They will then be introduced to the principle of grouping which they can readily
transfer as a calculation strategy through their use of the number line. The clarity of understanding they have with the two distinct
forms of calculation will assist greatly in their ability to have a full and complete understanding of the concept of division.
The school is firmly of the opinion that if children are to gain full cognitive understanding of the division principles, the concepts need
to be taught firmly in the context of “real life” problems. It is only in this arena that the children can fully grapple with and gain a full
understanding of the principles lying behind the numerical operations. The universal weakness children have in this area may simply
be down to the fact that whereas it is possible to cognitively understand the concepts of addition and subtraction abstractly, the
complexity of division requires a greater cognitive appreciation of what occurs within the calculation. If this is the case then the
subject is best explored practically allowing children to develop their “own understanding” of the principles involved.
As with its multiplying counterpart the children need to “understand” the numerical rules that govern the multiplying and dividing of
numbers by powers of 10 and the associated use of the “distributive law” before they can calculate effectively on the number line.
24
Section 5
Understanding Division: A School Based Progression
Sum
Stage 1
8-2-2-2-2=0 (with pictures)
Understanding to Acquire
The child cannot relate the repeated subtraction to the concept of division, nor can
they hold the numbers in their head. They also need to appreciate that reaching zero
determines that they have run out of objects to “Share” (or divide)
The following should be used to lead children through the “Repeated Subtraction as Division” barrier
1. Pictures
Pictures provide a valuable visual illustration of “sets of” or “groups of”
2. Practical
Wherever possible there can be no better modelling of the concept of multiplication
than to use “real life” contexts. This allows children to see “real” multiplication in
“real” meaningful contexts – a principle supported by the Renewed framework
3. Array to illustrate groups
The use of the array is a powerful visual representation of division especially for
grouping. To see that 12÷ 4=3 is a series of 3’s that move towards zero is key to a
child’s understanding of the grouping strategy
4. Real life stories
The children should be able to verbalise their calculations in the form of meaningful
stories. They should fully comprehend that 12÷ 4=3 is about 12 sweets being shared
between 4 people and that therefore each get 3 sweets each.
5. Use of the number line (with
numbers and then as an empty
number line) to illustrate
Unlike multiplication where the number line can be introduced as a visual tool
(rather than an effective calculation strategy in the early concept of understanding)
its use for division should be a little more judicious until the children have a deep
understanding of the difference between grouping and sharing.
Grouping and Sharing
The children should have a clear understanding of the difference between grouping
and sharing and be able to use them, calculating accurately, in their correct context.
25
Mental Calculation Strategies: Progression from KS1 and into Key Stage 2
The following is a digest of the mental calculation strategies introduced by the National Numeracy Strategy. Whilst the major
calculation focus in KS2 is the development of the number line, these strategies are crucial in developing children’s mental capability
in calculation. The first question a child should ask before embarking on any calculation is; “Can I do this in my head?” These
strategies are central to developing a clear sense of number and a fluency in calculation that will enable children to tackle an
increasing number of mathematical problems mentally. To this end they supplement rather than supersede the school’s core work on
the number line. (See “School Calculation” document)
Whilst these may be taught discretely there is a single sentence tucked unobtrusively at the bottom of pages 46-49 of the NNS
document that simply states “Use and apply these skills in a variety of contexts, in mathematics and other subjects”. This is a vital
aspect of the child’s mathematical development. The strategies must be taught through a range of contexts so that children imbibe
them holistically into their learning and use them correctly in contexts beyond the constraints of the formal maths lesson.
Mental Calculation Strategies: Division
Progression into Key Stage 2
The following are Strategies that will be used in conjunction with the “Calculation Document” in the sense that the children will
have developed in their understanding to a point where they are ready to calculate using the school’s number line method
These strategies are not therefore linear in terms of their progression but should provide a useful checklist for staff looking to
provide children with a breadth of mental calculation strategies that will greatly enrich their understanding of multiplication
Dividing by 1 leaves number
unchanged (Year 3)
The children need to appreciate that whilst most of their conceptual framework of
multiplication is that the number gets bigger, here the number remains the same
Dividing by 0 = 0 (Year 3)
The children need to appreciate that whilst most of their conceptual framework of
multiplication is that the number gets bigger, here the number becomes zero
Halving (Year 3)
Know and use halving as the inverse of doubling
Know doubles of multiples and the corresponding halves e.g. 36 ÷ 2, half of 130, 900 ÷ 2
Use known facts (Year 3)
Use doubling or halving, starting from known facts
(e.g. 8 x 4 is double 4 x 4).
26
Know halves of multiples 10 to 100
(Year 3)
Use known facts (Year 4)
7 x 5 = 35 so… 35 ÷ 7 = 5 (Year 3)
Fractions (Year 4)
Quarters (Year 4)
The children know for example; Half of 70
Use doubling or halving, starting from known facts.
For example: double/halve two-digit numbers by doubling/halving the tens first;
The children should have an increasing understanding of the relationship between
multiplication and division
Begin to relate fractions to division e.g. ½ , ¼ etc.
Find quarters by halving halves.
Use known facts (Year 4)
Use known number facts and place value to multiply and divide integers, including by 10
and then 100 (whole-number answers).
Sixths (Year 5)
Find sixths by halving thirds.
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Section 6
Assessment: Use of the Spreadsheet (Notes to teachers)
The assessment of the skills outlined in this document are held centrally on two spreadsheets; Foundational Understanding and Further
Understanding. The former will relate more to those children at KS1 whilst the latter (in general) will be for children in the lower
years of KS2. As ever the use of the document should be child centric rather than age related so in any year group there will be a wide
range of ability and this should be reflected in the spreadsheets.
Foundational Understanding
This aspect of the document relates to the understanding of basic principles and it is appropriate to use the progression outlined in the
stages as a skeleton scheme of work. With this in mind it is appropriate to use the assessment tasks that are outlined throughout the
document as a summative assessment to determine a child’s understanding of the principle taught. Whilst this may come in the form
of an assessment at the end of the lesson, it might be more cogent to make the assessment at a later date to ensure that the concept has
been securely retained. Indeed the procedure we have adopted in the past where we have taken children through large numbers of the
stages in one session may well be the way forward. The only caveat to this might be the warning that “You don’t make a pig fatter by
weighing it all the time”. To this end we should limit the element of testing as far as is possible, and the use of teaching assistants to
speed the process would appear to be a good way forward. The results should be recorded and the gaps that occur provide a
programme of personalised learning for each child.
Further Understanding
a) The Number Line and the Iceberg Principle
These concepts run alongside the teaching of the number line as the school’s chosen formal written method. The child’s ability to
imbibe these further mental strategies is crucial as they are the underpinning of all mathematical understanding. To leave children with
a single strategy such as the number line, no matter how effective it is, narrows the mathematical curriculum and will by definition
remain relatively numerically impoverished. Whilst we would all be pleased for the child who could calculate 247+99 on a number
line, the fact that they cannot see that this sum could (and indeed should) be calculated mentally by adding 100 and subtracting 1,
should be a source of great concern to us. Where children have a deep understanding of the relationships between numbers and a
fluidity of understanding in how these operate then they will increasingly be able to revert to a mental calculation in the first instance
when presented with a given problem. Not only so but the number line is a creative strategy, where children have relative ownership
of the strategies they are able to use for any given calculation. Therefore the more strategies children have access to, and the deeper
their understanding the more effective and efficient their calculation processes will be. As the school’s own calculation document
states: The number line acts as a good bridging point between the “Mental” and the “Formal” methods of calculation. It is interesting
28
to note that although in Year 2 it is introduced as an introduction to “Pen and
Paper methods” in the KS3 strategy it comes under the heading of “Mental
Methods”. It is, in a very real sense, a mental method that is enhanced through the
visual use of the line. It is incumbent upon us therefore to use it as a method that is
supportive of the mental approach and a precursor to the formal written methods.
(Calculation Document p2) Therefore these additional strategies must not be seen as
a “bolt on” or an “optional extra” to the number line; they should be the core of
every child’s mathematical understanding and are the foundational constructs that
children should draw on to solve all mathematical problems. In this regard our
teaching of the Maths curriculum should be viewed as an iceberg in the sense that
whilst the most visible aspect may well be the number line the majority of our
teaching focuses on the understanding below the surface. The goal therefore is to
provide all children with a breadth of mental strategies that they can draw on and
use to solve a range of calculations effectively and efficiently.
b) The Principles of Assessment
In light of the above it would be inappropriate to assess these skills in the form of a
summative assessment. The key to understanding is not just to understand the
principle behind a given strategy but to use it appropriately. So, using our example
above of 247+99, whilst adding 100 and subtracting 1 is a very effective strategy for
this calculation the child who seeks to add 247+53 by adding 100 and subtracting 47
has not only missed the principle behind the original strategy but has missed another in the fact that 53+47 is a number bond of 100
and would be much more effective strategy in this instance. So whilst there will always be a place for short term assessments at the
end of each lesson to assess the child’s immediate understanding the assessments for the spreadsheet need to be different in nature.
These assessments should be similar to those applied in the Foundation stage, namely that they should be child initiated and set in an
appropriate context. As the school’s Foundation Assessment guidance states; this (assessment) process should be underpinned by the
observation of tasks within the curriculum as the “using of supplementary tests with Reception children is unreliable”. (Jan Dubiel:
Foundation Stage Profile, DFES). The same principle would hold in this context, whilst most children could be taught a strategy and
probably successfully apply this to a series of set calculations given in towards the end of that same lesson, it is a completely different
level of understanding to apply that principle to a calculation in an unrelated context at a later time. The latter should form the basis
for our assessment of true mathematical understanding.
29
c) Setting Expectations
It would be my wish that 80% (to use QCA’s numerical measure of “the majority of children”) of the children should have mastered
all the strategies found in the document by the end of Year 4. I am rapidly coming to the conclusion that in the upper years of KS2 the
children should all have a good understanding of all areas of Mathematics and therefore the vast majority of their work in numeracy
should be based on lessons that are rooted almost constantly in “Using and Applying” principles. There is very little in the Numeracy
strategy (bar concepts like ration and proportion, and the principles of fundamental algebra that can be drawn down form the KS3
curriculum) that is new to children. Much of the programmes of study simply build on the children’s earlier understanding as well as
encouraging the teacher to apply these concepts to ever challenging learning contexts.
d) Filling in the spreadsheet
The “Further Understanding” document is therefore completed slightly differently from its “Foundational Understanding” counterpart.
The assessments are filled in using a colour coded system (found on each page of the spreadsheet) which will record which National
Curriculum year the teacher believed the child fully understood and consistently used a given strategy. This will develop into a clear
audit of continuity and progression for each child as they move through KS2. Whilst there is a colour code for Year 5 and 6 it is
assumed that their use will be limited to those children who are either on the SEN register for Maths or have related issues with
numeracy.
30
Appendix 1: Mathematical Vocabulary
The use of correct Mathematical vocabulary is a key to children ability to communicate their findings to others. Mathematics is a
language and whilst much of its notation is abstract the communication of thoughts and ideas are best served through the use of correct
terminology. It is noticeable that children at the end of Key Stage 2 who have a rich grasp of mathematical vocabulary are more able
to succinctly and accurately communicate their work and conclusions to others.
ADDITION AND SUBTRACTION
MULTIPLICATION AND DIVISION
add, addition, more, plus, increase
sum, total, altogether
score
double, near double
how many more to make…?
subtract, subtraction, take (away), minus, decrease
leave, how many are left/left over?
difference between
half, halve
how many more/fewer is… than…?
how much more/less is…?
equals, sign, is the same as
tens boundary, hundreds boundary
units boundary, tenths boundary
inverse
lots of, groups of
times, multiply, multiplication, multiplied by
multiple of, product
once, twice, three times… ten times…
times as (big, long, wide… and so on)
repeated addition
array, row, column
double, halve
share, share equally
one each, two each, three each…
group in pairs, threes… tens
equal groups of
divide, division, divided by, divided into
remainder
factor, quotient, divisible by
inverse
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Appendix 2: Instant Recall of Times Tables and Number Bonds
Known Facts are the tools of Creative Mathematics
The use of “Known Facts” (as the Numeracy Strategy calls them) is a crucial mathematical building block for children. To be aware of
the number bonds to 10 and the times tables up to 10x10 allows children to make good progress in all areas of Mathematics. Whilst
the child may have learnt them by rote and may lack a full appreciation of their understanding; these “known facts” are the gateway to
learning effectively in the area of creative Mathematics. When undertaking higher ordered mathematical tasks children should be
solely focused on “choosing the appropriate operation and method of calculation” (NNS p82-3), where instant recall is lacking they
often struggle to fulfil the task because they are engaged in two parallel tasks; namely selecting the correct calculation method as well
as developing strategies to solve that same calculation. In the sense that the former is a higher order skill the children who lack secure
knowledge of “known facts” struggle to fulfil such tasks.
Learning without Understanding
Whilst it seems a little incongruous to learn these facts without a full understanding of their mathematical meaning, the rote aspect of
the learning must be seen as a process not as an end goal. The key is to put these mathematical tools into the hands of children so they
can explore the underlying understanding as they use them in a range of contexts.
Known and Derived Facts
The rote learning should be restricted the number bonds to 10 (addition only) and the times table up to 10x10 (multiplication only).
The goal of seeking “instant recall” in facts beyond these has the potential to stunt true mathematical understanding. Research shows
that “those who can make links between known and deduced facts number progress, because the known and the derived support each
other.” (M. Askew “They’re counting on us” June 2008) In this sense…“these two aspects of mental mathematics – known facts and
derived facts – are complimentary. Higher attaining pupils are able to use known facts to figure out other number facts.” (Gray EM
1991 – from M. Askew 2008). The reality is that if children develop an over-reliance on rote learning they tend to resort to a default
mechanism for calculation and this is often “counting on”. As Askew points out; “This is one of the reasons that the national
Numeracy strategy initially advocated delaying the introduction of standard algorithms. The danger is that children can appear to be
confident in working with large numbers but they are actually only dealing with single digits and using counting on procedures”.
On the basis of this understanding the school would seek to adopt the following:
1. All rote learning should be limited to the number bonds to 10 and times table up to 10x10. Children should not “know” what
12x12 is they should be encouraged to use cogent strategies to derive the answer from known facts. Eventually these “derived”
32
facts may well become “known” facts but this needs to come out of a correct philosophical understanding that seeks to take
children deeper in their understanding of how numbers relate to one another in a range of calculations. The truth is that as a
“child’s range of known number facts expands, the range of strategies available for deriving new facts expands as well” (M.
Askew June 2008) However this richness of mathematical understanding does not occur if the process is bypassed through rote
learning.
2. The “instant recall” does exactly what it says on the tin. When gauging whether children have “instant recall” teachers should
work on the basis that a child should be able to answer times tables questions at a rate of one every 2 seconds in a test situation.
Any speed less than this would tend to imply that the children are “calculating” the answers and they should be pushed towards an
instant recall of these facts.
3. The school should be very careful in to bridge the gap between known and derived facts for children, gradually weaning them
away from “counting on” around the end of KS1 and into the early years of KS2. Children should be actively encouraged to see
the connection between known facts and derived facts and not treat each calculation as if coming to a blank canvas. The truth is
many children see their times tables as the end goal not as the tools that enable them to solve more complex calculations. So
children should be shown that to know 3x3=9 also provides a solution to 30x3, and 30x30. It is this network of mathematical
connections that will truly foster a rich mathematical understanding in children.
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Appendix 3: Examples of Misunderstanding
1. Two of the school’s most able mathematicians were working in a more able group of 4 looking at the concept of Algebra. They
had calculated 33 – 8t = 15 with ease and were recalibrating the equation 6x – 27 = 0. They got to x = 27 ÷ 6 + 0 and debated
whether they needed the zero. They concluded that they did not, they were then asked whether they would if the operation was
“multiply by 0”. They suggested that 4.5 x 0 was 45. They checked it on a calculator, finding the answer was zero they assumed
the calculator was broken and got a new one. Strangely the same answer appeared. It took some time before the concept hit home
that 4 x 0 is “4 lots of nothing”
2. A year 4 above average child was calculating 236 x 12 – a concept comfortably into Year 5. Upon finding she needed to calculate
236 x 2 she told me the answer was 4 … 6 … and then 12 which made 4612. I pointed out to her that the 12 was in the wrong
column and asked her what the 1 stood for “Units” she informed me, and the 2; “that is units as well” she said. So they are both
units I suggested “Oh yes she said”
34