recording mental calculations & standard written methods mathematics cpd course 04/05 nigel...
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Recording mental calculations & standard written methods
Mathematics CPD course 04/05
Nigel Davies
April 20, 2023 2
The approach to recordingChildren first need to become familiar with oral expressions of
mathematics by listening to an adult, then copying the adult and saying the expressions themselves.
The approach to recording is similar. In Nursery and Reception classes children will draw or make marks to represent their work.
Next, their teacher will show them how to write numerals. At first they can do this in the air, following an adult's directions. Or they may finger paint,
draw numerals in sand or trace a finger over the surface of numerals made from an assortment of textured materials.
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Recording in Reception
The next step is to model for the children: •ways of recording the steps in calculations so that they can keep
track of their mental work; •how to use mathematical notation (numerals, signs and symbols) to
communicate their methods and answers.
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Why write it?For communication to others
•to record answers •to describe or explain methods •to record findings •to present information or data for self or others to interpret or use
To clarify own thinking or ideas, or keep track of one's work •to help work out a calculation by recording interim steps •to help solve a problem by sketching a diagram •to help organise information so that it is easy to interpret •to help remember key facts or ideas
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Starting off“Three and one makes four”
The teacher demonstrates, using a physical model such as bricks, and a number line. The teacher says, then the children repeat, ‘Three and
one makes four’.
The teacher writes the number sentence 3 + 1 = 4 and speaks and points at each part – the children watch & listen
The teacher points at the number sentence & leads the children speaking in unison. They read the sentence together.
Later, the teacher modifies the language to ‘three add one equals four’
From the start, children get used to the number sentence as the written formulation of what is being done & said.
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Reception children’s recordings
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Encouraging early recordingEncourage recording in the context of 'play' by providing writing materials in
different areas of the classroom. Suggest where writing materials could be provided.
Suggest where numerals can occur in an early years classroom for children to look at and mimic.
Suggest some activities which would encourage children to 'write' numerals.
April 20, 2023 8
Encouraging early recordingEncourage recording in the context of 'play' by providing writing materials in
different areas of the classroom. Suggest where writing materials could be provided.
•In a Post Office area•On a writing table•In a painting corner
Suggest where numerals can occur in an early years classroom for children to look at and mimic.
Suggest some activities which would encourage children to 'write' numerals.
April 20, 2023 9
Encouraging early recordingEncourage recording in the context of 'play' by providing writing materials in
different areas of the classroom. Suggest where writing materials could be provided.
•In a Post Office area•On a writing table•In a painting corner
Suggest where numerals can occur in an early years classroom for children to look at and mimic.
•On a telephone•On 'kitchen' equipment •On a play clock
Suggest some activities which would encourage children to 'write' numerals.
April 20, 2023 10
Encouraging early recordingEncourage recording in the context of 'play' by providing writing materials in
different areas of the classroom. Suggest where writing materials could be provided.
•In a Post Office area•On a writing table•In a painting corner
Suggest where numerals can occur in an early years classroom for children to look at and mimic.
•On a telephone•On 'kitchen' equipment •On a play clock
Suggest some activities which would encourage children to 'write' numerals. •Make tickets for a raffle•Make price labels for the classroom shop•Make door numbers for a frieze of houses
April 20, 2023 11
Progression in recording signs & symbols
Year 1 teaching objective: •begin to use the +, - and = signs to record mental calculations in a number sentence
Year 2 teaching objectives:•use the +, - and = signs to record mental calculations in a number sentence•use the x, ÷ and = signs to record mental calculations in a number sentence•explain how a problem was solved orally and where appropriate in writing
Later, children will explain their methods of calculation, first orally, then using more efficient mathematical notation. This should be modelled by the teacher:
e.g. 15 + 7 = 15 + 5 + 2= 20 + 2= 22
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Early recordingChildren's recording in Reception and Year 1
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Recording in Year 2
April 20, 2023 14
“Recording” mental calculations
It is not usual to show working in mental calculations but, to develop mental methods and aid understanding, it helps to record mental steps.
Note that it is best to write each step as a separate equation underneath the previous one, not on the same line, to avoid incorrect use of the equals sign in statements like
19 + 19 = 19 + 10 = 29 + 9 = 38
April 20, 2023 15
Mental calculations & recordings
Children should be able to:
read number statements (equations) and record answers, e.g. 5 + ? = 9
record and explain mental steps using mathematical symbols,
e.g. 27 + 19 (adding 20 and subtracting 1):
27 + 20 = 47
47 – 1 = 46
use informal jottings as an aide-mémoire when working with larger numbers, e.g. on an empty number line.
-1
+20
27 4746
April 20, 2023 16
The challenges for teachersKnowing what to do after sound mental strategies are established.
Children may progress at different rates; some may grasp a standard method while others never do without considerable help.
Children may be able to carry out some standard methods successfully, e.g. for + and , but not – and ÷.
Children tend to forget a standard method if they have no understanding of what they are doing.
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Stages in teaching children to write equations
•Children are first encouraged to explain their thinking in words aloud.
•Children are shown how what they have said can be recorded using mathematical signs and symbols.
•Both the teacher and the children use written recording to support oral explanations of calculation methods.
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Number sentences
6 + 3 = 9
15 + = 20
(2 x 20) + (2 x 8) = 2 x 28
12 – 3 + 7 – 9 = 28 4
1/3
1.5 < < 2.5
What language would you use to describe these number sentences?
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The answer’s right – so what’s wrong?!
Example 1 7 + 5 ………………….= 12
Example 2 25 + 9 ……………………. = 34
Example 3 21 – 6 ………………….. = 15
Example 4 double 28 ……………………………… = 56
Example 5 24 x 8 ……………………….. = 192
Example 6 5 miles = 8 km
April 20, 2023 20
The answer’s right – so what’s wrong?!
Example 1 7 + 5 = 7 + 3 = 10 + 2 = 12
Example 2 25 + 9 = 25 + 10 = 35 – 1 = 34
Example 3 21 – 6 = 21 – 1 = 20 – 5 = 15
Example 4 double 28 = double 20 = 40 + double 8 = 56
Example 5 24 x 8 = 20 x 8 = 160 + 4 x 8 = 192
Example 6 5 miles = 8 km
April 20, 2023 21
Methods used by six Year 3 children to work out 19 + 19 a. 10 + 10 = 20
5 + 5 = 104 + 4 = 820 + 10 + 8 = 38
b. 19 + 10 = 2929 + 9 = 38
c. 20 + 20 = 4040 - 2 = 38
d. 19 + 10 + 10 = 3939 - 1 = 38
e. 18 + 20 = 38 f. 19 x 2 = 38
April 20, 2023 22
Drawings to aid mental calculation
Drawings can help children interpret a question and visualise what is going on.
For example: 3 groups of 4 children
half of 12
April 20, 2023 23
Re-ordering a calculationSometimes it is helpful to reorganise a calculation.
April 20, 2023 24
Empty number lines
April 20, 2023 25
More number lines
April 20, 2023 26
Children’s recordings in Years 2 & 3
April 20, 2023 27
More Year 2 & 3
April 20, 2023 28
Discussion points•What sorts of paper might be best for jottings?
•Does recording always need to be neat?
•What sorts of recording need to be kept? For what purpose?
•What do we need to show parents?
•Do the National Curriculum mathematics tests include any questions that require children to record their work in a particular way?
•What do we expect children to record in their number work at present? How should they do this?
•Do we need to modify or adapt our present practice?
April 20, 2023 29
Summary of Reception & Key Stage 1
In Reception and Key Stage 1, the focus is on mental calculation. Written calculations in columns are not introduced in the Framework until Key
Stage 2. Children record their number work to:
•communicate mental methods or answers; •remember key facts; •support calculation.
Children should be introduced to forms of recording appropriate to their stage of development:
•pictorial representations •visual representations including number lines •numbers and jottings •simple equations, setting out calculations or steps in a calculation using appropriate symbols.
April 20, 2023 30
Into Years 3 – 6
•Up to the end of Year 3, the emphasis is on children working mentally, with calculations recorded in horizontal number sentences, and with
some informal jottings for more challenging numbers.
•In Years 4 to 6, children are taught more formalised written methods of calculation, starting with expanded methods and working gradually
towards more compact standard methods by the end of Year 6.
The aim is that, for each of the four operations, as many children as possible can, by the age of 11, carry out a
standard written method.
April 20, 2023 31
Standard written methodsAdvantages
The methods are:
generalised they work with any numbers of a given type
reliable they are always give the right answer if done correctly
efficient they do not take too long to do
tried and tested people have confidence in them
known by all there are times when it helps if everyone uses the same method
Disadvantage
They are sometimes not easy to follow, particularly – and ÷.
April 20, 2023 32
Common standard written methods
5 1 42 8 52 3 2
4 1
crossed-out 5 replaced by 4 small 1 ‘borrowed’
Column subtraction by decomposition
2 3 04 9 67 2 6 1
Column addition, working from the right to the left
April 20, 2023 33
Common standard written methods
5 5
7 3 8 53small 3 ‘carried over’
Short division
4 8 6 X 29 7 2
2 8 X 4 71 1 2 0 1 9 61 3 1 6
Short and long multiplication
6 52 7 1 7 5 5 1 6 2 1 3 5 1 3 5
Long division
April 20, 2023 34
Weaknesses in written calculations
Children's errors are often due to misconceptions or misunderstood rules, not carelessness.
Some weaknesses in children's written calculations are:
•using a written method inappropriately, because they fail to recognise when a calculation is better done mentally, e.g. 4000 - 1997
•working inefficiently or too slowly, because they cannot recall the number facts they need, e.g. 7 + 8 = 15
•making errors or forgetting what to do, because they do not fully understand the written method they are attempting to use,
April 20, 2023 35
Common calculation erorsA. 99 B. 158
+ 101 + 184
1910 612
C. 945 D. 826
– 237 – 349
712 387
E. 434 F. 2000
– 276 – 108
258 902
6 1 1
2 1 1 1
April 20, 2023 36
Diagnosing misconceptions•It is important to diagnose each misconception, not simply re-teach
the method.
•To make the diagnosis, ask the child to explain how (s)he worked out the answer, and to work out in front of you a similar question,
talking aloud while doing it.
•Then deal with the misconception: for example, by careful discussion of place values, or showing an expanded
or alternative method.
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Children’s errors & misconceptions
•Simply correcting errors or re-teaching a method may help short term, but not necessarily in the long term. Children need to understand why a
particular method works - not just learn how to put marks on paper.
•The compactness of a standard written method can hide mathematical principles, e.g. children might use place value when working mentally, but be confused in written work if they do not understand how place
value links to jargon like 'carrying', or 'exchanging'.
April 20, 2023 38
Children’s errors & misconceptions
To reduce the frequency of errors and make them easier to deal with:
- begin by introducing expanded written methods;
- building on children's understanding of place value;
- refine methods gradually into a more compact and efficient layout.
April 20, 2023 39
Errors with decimals
8.2 – 4.55
8.2
- 4.55or
8.2
- 4.55
3.75
7 1
Typical errors :
April 20, 2023 40
Money Number Line8.2 – 4.55
8.208.005.004.55
+ 0.20+3+ 0.45
Think in terms of money change
£8.20 - £4.55
0.45 + 0.20 = 0.65
0.65 + 3 = 3.65
April 20, 2023 41
Stages in teaching written calculations
Up to the end of Year 3, the focus should be mental work, if necessary with jottings.
Once written methods are introduced, keep mental skills sharp by continuing to develop and apply them to appropriate examples.
Encourage children always to use mental methods as a first resort.
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Stages in teaching written calculations (contd.)
Show children how to set out written calculations vertically. First use expanded layouts that show clearly the place value of numbers, without adjustments of 'carrying', 'borrowing' or 'exchanging'. Introduce adjustments slowly and systematically.
Gradually refine the written record into a more compact standard method.
Extend to larger numbers and to decimals.
At any stage, children who persistently make errors should return to the stage they understood, until ready to move on.
April 20, 2023 43
Stages in addition
2. Vertical layout - expanded working, moving to adding the least significant digit first and extending to three-digit numbers. Partial sums are added mentally:
1. Mental method - partitioning, adding the tens first:
April 20, 2023 44
Expanded form for addition
296 + 357
296
357
200 + 90 + 6
300 + 50 + 7
500 + 140 + 13653
April 20, 2023 45
Addition (contd)3. Vertical layout, contracting the working to a compact, efficient form:
4. Bigger numbers and decimals.
April 20, 2023 46
Language used in compact method
Think carefully about the language you would use when explaining the expanded and then the contracted method to your class.
”Seven plus six equals thirteen, write down three and carry ten” (child writes carry digit underneath).
“Forty plus seventy equals one hundred and ten, plus the extra ten, which equals one hundred and twenty. Write down twenty and carry one hundred”
(child writes carry digit underneath). “Five hundred plus two hundred equals seven hundred, plus the
extra one hundred, which equals eight hundred.” The total is eight hundred and twenty-three.
April 20, 2023 47
Informal written methods of subtraction
Counting up (complementary addition)
Using negative numbers
April 20, 2023 48
Subtraction by decomposition
April 20, 2023 49
Expanded form for 803 - 526
- 500 20 6
- 500 20 6
800 00 3
⇨
700
⇨
200 70 7
- 500 20 6800 100 3
700 901
800 00 31
= 277
April 20, 2023 50
Written calculation : progression
•Before moving to the next stage, children should practise examples until they can do them accurately, with understanding. Encourage
checking - looking at answers to see if they make sense and sometimes using the inverse operation. Not all children in a class will be ready to move on at the same time - some may need to wait until
the next term or school year.
•If children persist in making errors, they should return to the previous stage until ready to move on.
•When revising or extending to harder numbers, it is a good idea to refer back to expanded methods. This helps to reinforce children's understanding and remind them that they have an alternative to fall
back on if having difficulties with a standard compact method.
April 20, 2023 51
Written calculation : progression
In existing classes, if children have been taught a standard method and have never met informal expanded methods, proceed as follows:
- Children who can use a standard method without making errors should continue to use it. (They might be introduced to other methods, but to explore for mathematical interest and understanding, not to adopt for
themselves).
- Children who make a significant number of errors should be introduced to informal expanded methods, since this may help them to understand
what they are doing and become more successful. Return to formal methods only when they can use an informal method accurately and
explain what they are doing.
April 20, 2023 52
Grading of difficulty in additionADDITION
Examples:
1. No 'carrying'
2. Extra digit in answer
3. Carrying U (units/ones) to T (tens)
4. Carrying T to H
April 20, 2023 53
Addition (contd)ADDITION Examples:
5. Carrying U to T and T to H
6. More than two numbers to be added
7. Different numbers of digits
April 20, 2023 54
Grading of difficulty in subtraction
SUBTRACTION Examples:
1. No adjustment
2. Adjustment T to U
3. Adjustment H to T
4. Adjustment H to T and T to U
5. Noughts
April 20, 2023 55
Common errors in multiplication & division
April 20, 2023 56
Progressing from mental multiplication to written
A useful way of recording intermediate steps when multiplying multi-digit numbers is the 'grid' or 'area' method.
Children need good knowledge of multiplication tables before they start on this method, otherwise their understanding of it is distracted by their
struggle to work out a multiplication fact.
They also need to be confident about multiplying by numbers like 300 or 30 by 10.
April 20, 2023 57
Grid method for 34 x 27
30 4
20
7
600 80
210 28
Answer = Answer = 600600 + + 8080 + + 210210 + + 2828
= 918
April 20, 2023 58
Ancient Egyptian x Method
The Ancient Egyptians had a different method of multiplication to ours.
For example, if they wanted to work out 64 x 23, they used this method …
April 20, 2023 59
Ancient Egyptian 64 x 23
64 x 23128 x 11256 x 5512 x 2
1024 x 1
Double 64 Halve 23 … ignore remainder
Double 128 Halve 11 … ignore remainder
Double 256 Halve 5 … ignore remainder
Double 512 Halve 2
Continue until you reach 1… then cross out any lines with an even number in this column
1472Add this column
April 20, 2023 60
Napier’s method : 274 x 63
1
2
4
2
2
4
2
1
0
6
1
2
2
6
7 4
3
6
1
1 2 6 27
Answer is circled numbers = 17262
April 20, 2023 61
Moving to vertical format
When children can use the grid method confidently:
•introduce them to a conventional vertical format (multiplying by the tens first is a more natural step on from the grid method);
•teach them to describe what they do by referring to the actual values of the digits in the columns, e.g. the first step in 38 x 7 is 'thirty
multiplied by seven';
•finally, help them to make the method more compact, by combining steps.
If, with a little practice, children cannot use the compact efficient method without making errors, they should return to the expanded format.
April 20, 2023 62
Division
7 7281 4
287 = 4
7287 = 104
… could be addressed by partitioning :
700 7 = 100
April 20, 2023 63
Division using multiples of the divisor : 256 7
256
70
186
140
46
42
4
(10 x 7)
(20 x 7)
(6 x 7)
Answer: 36 remainder 4
April 20, 2023 64
DivisionWith division:
•first establish the process as repeated subtraction;
•introduce an expanded layout, subtracting simple multiples of the divisor repeatedly;
•help children to refine this process gradually, so that it ultimately becomes the conventional long division algorithm.
April 20, 2023 65
Stages in multiplication1. Mental method, using partitioning
2. Grid layout, expanded working
3. Extended to bigger numbers
April 20, 2023 66
Stages in multiplication (vertical format)
4. Vertical format, expanded working
5. Vertical layout, compact working
April 20, 2023 67
Division as sharingProblem: 20 cakes are to be divided between 4 people.How many cakes does each person get?
Practical method:Give 1 cake to each of the 4 people.Carry on like this until there are no cakes left.Then count how many cakes each person has.
April 20, 2023 68
Division as grouping (repeated subtraction)
Problem: 24 eggs are packed in boxes of 6.How many boxes are needed?
Practical method:Take 6 eggs and pack the first box.Continue until there are no eggs left.Then count how many boxes have been used.
April 20, 2023 69
Repeated subtraction
Consider 200 ÷ 6. The process is to subtract 6s from 200, until the remainder is less than 6. The answer is the number of 6s subtracted, plus the remainder.
200
- 6
194
- 6
188
……
14
- 6
8
- 6
……
20
- 6
2
April 20, 2023 70
Repeated subtraction : accelerating the processIt is quicker to subtract multiples of 6, such as 60. (Note: 10, 100 or 1000 times the divisor are easy multiples to find.)
10 lots of 6
10 lots of 6
10 lots of 6
3 lots of 6
200
- 60
140
- 60
80
- 60
20
- 18
2
April 20, 2023 71
Repeated subtraction : even quicker
•With good knowledge of multiplication tables and estimating skills, it is even quicker to choose an appropriate higher multiple to subtract, e.g. 180 (30 times 6).
30 lots of 6
3 lots of 6
Key teaching pointRather than introduce the most efficient method immediately, gradually refine the process, as children's estimating skills and speed at recalling
tables improve.
April 20, 2023 72
Grading of difficulty in multiplication
Multiplication Examples:
1. No 'carrying'
2. Extra digit
3. 'Carrying' but keeping in same decade
4. 'Carrying' and going into next decade
April 20, 2023 73
Grading of difficulty in multiplication
Multiplication Examples:
5. Noughts
6.Multiplying by multiples of 10
7. 'Long' multiplication
April 20, 2023 74
Grading of difficulty in division (single digit)
1. No remainder, no carrying
2. Remainder, no carrying
3. No remainder, carrying
4. Remainder, carrying
5. Placing of the quotient
6. Noughts in quotient
April 20, 2023 75
Grading of difficulty in division (double digit)
7. No remainder
8. Similar but remainder
9. Quotient not so apparent
10. Placing the quotient
April 20, 2023 76
Grading of difficulty in division (double digit)
11. No remainder
12. Remainder
13. Noughts in quotient
14. Divisors like 29, 39, 48, 45, 37, 24, 56
April 20, 2023 77
Questions to considerThinking of the children you work with, consider:
•Have we already in place a programme for systematic teaching of mental calculation, keeping mental skills sharp and ensuring that children always consider a
mental method as the first resort?
•To what extent do we already teach written calculation methods following the progression in the Framework?
•What changes should we consider and how should we phase them in?