Renewing the Framework for teaching mathematics
Discussion paper
Use of calculators in teaching and learning mathematics
In mathematics, the calculator can be an effective teaching and learning tool
in the primary classroom. The use of a calculator is embedded within the
National Curriculum Key Stage 2 Programme of Study for mathematics.
Within Ma2 Number, Calculator methods, it states that children should be
taught to:
 Use a calculator for calculations involving several digits including
decimals.
 Use a calculator to solve number problems.
 Know how to enter and interpret money calculations and fractions.
 Know how to select the correct key sequence for calculations with
more than one operation.
It is therefore important that children are confident users of calculators. They
need to recognise that the calculator is a tool of which they are in control and
to understand how it can help them to develop their skills in mathematics. The
use of calculators is assessed in the end of Key Stage 2 statutory tests.
Evidence from the analysis of test scripts shows that children are unsure
about when and how they might use a calculator. They are not clear about
using calculator methods or recording their method.
In the primary classroom calculators can be used for different purposes by
teachers and children. These include:
 Teaching children how to use a calculator effectively to calculate, to
recognise how and when it is appropriate to do so and when other
methods are quicker or more reliable.
 Supporting the teaching of mathematics where the aim is to focus on
solving a problem rather than on the process of calculation.
 Providing a tool with which children can explore patterns in numbers
and identify properties and relationships.
 Consolidating children’s learning of number facts and calculation
strategies.
Children need to be taught the basic skills of entering numbers and
operations. However, to use a calculator effectively children require a secure
knowledge of number. They need to be able to interpret the values displayed
during this process of entering numbers and when they review their answers.
They need to decide if the answer displayed is sensible and if it needs any
adjustment to take account of rounding errors and to incorporate suppressed
zeros.
In the renewed Framework an emphasis in Key Stage 1 and the first two
years of Key Stage 2 there is an emphasis on helping children to secure their
knowledge of number facts and mental calculation strategies. They begin too
to develop written methods that they can apply more generally. The
introduction of the calculator in Year 4 would be an appropriate time when
children to begin to learn how they can use their knowledge of number facts to
solve problems with and without the aid of a calculator. They can begin to
compare their mental, written and calculator methods. As children learn how
to enter simple one-step calculations involving whole numbers, they can
explore the behaviour of the four operations and the properties of these
numbers. They start to recognise how their number knowledge can be applied
to calculations involving larger whole numbers.
Over the course of Years 5 and 6, children would learn how to use other
functions on the calculator and apply their skills and knowledge to decimal
numbers, fractions and negative numbers. They would solve multi-step
problems and use the calculator to generate sequences of numbers and
families of calculations. Children would recognise underlying properties and
principles that they can then apply when calculating mentally or on paper. The
aim is that by the end of Key Stage 2, children know how to transfer their
knowledge and understanding of numbers and the four operations to mental,
written and calculator methods of calculation. They will be able to explain and
record their methods in succinct and manageable ways.
The overall aim is that when children leave primary schools they:
 Have a secure knowledge of number facts and a good understanding
of the four operations.
 Are able to use this knowledge and understanding to carry out
calculations mentally and to apply general strategies when using
single-digit and two-digit numbers and particular strategies to special
cases involving larger numbers.
 Make use of diagrams and informal notes to help record steps and part
answers when using mental methods that generate more information
than they can keep in their heads.
 Have an efficient, reliable, compact written method of calculation for
each operation, the standard written method, which children can apply
with confidence when undertaking calculations that they cannot carry
out mentally.
 Use a calculator effectively, using their mental skills to monitor the
process, to check the steps involved and to decide if the numbers
displayed make sense.
Objectives
The objectives in the draft renewed Framework include specific references to
calculators in the Use and apply mathematics and Calculate efficiently
and accurately strands:
Use and apply mathematics
Calculate efficiently and accurately
Year 4
• Solve one- and two-step problems
involving numbers, money or
measures, including time; choose
and carry out appropriate
calculations, using calculator
methods where appropriate
Year 4
• Use a calculator to carry out oneand two-step calculations involving
all four operations; recognise
negative numbers in the display,
correct mistaken entries and
interpret the display correctly in the
context of money
Year 5
• Solve one and two-step problems
involving whole numbers and
decimals and all four operations,
choosing and using appropriate
methods, including calculator use
Year 5
• Use a calculator to solve problems,
including those involving decimals
or fractions, e.g. to find 34 of 150 g;
interpret the display correctly in the
context of measurement
Year 6
• Solve multi-step problems, and
problems involving fractions,
decimals and percentages,
choosing and using appropriate
and efficient methods at each
stage, including calculator use
Year 6
• Use a calculator to solve problems
involving multi-step calculations;
carry out calculations involving
time by converting hours and
minutes to minutes
Below are an expanded set of learning objectives for Years 4 to 6 to help a
school to review its policy and planned provision in the use of calculators in
the teaching and learning of mathematics. These are divided into the technical
knowledge and skills children need to use a calculator effectively, and the
interpretive skills and understanding they need to apply these skills to support
their learning. There is accompanying commentary on aspects of teaching
and learning.
A. Technical knowledge and skills needed to use a calculator

Recognise the operations that the keys on the calculator
represent
Not all calculators have a key with the conventional division sign and they do
not all operate in the same way. Some have a key with an equals sign while
other models have an ENTER key. There is a difference too between the
subtraction key and the change sign key that assign a positive or negative
value to a number. Some calculators usefully display the full number
statement during the calculation, while others only display the number entered
or the answer. Spend time showing how the decimal point key is used. The
decimal point may already appear on the display at the end of the number.
After the key is pressed it appears to move with the number which can
confuse children. The +/- key changes the sign of the number displayed. It
toggles between + and -. This can be used to highlight the difference between
using + and - to represent the operations of addition and subtraction, and to
indicates whether the number is positive or negative. Children may miss the
sign that indicates a number is negative when this appears on the extreme left
of the display. Many calculators have a percentage key with the symbol %.
Using it is likely to cause confusion and it is better not to use it at all with most
primary children. Spend time pointing out to children the essential features of
the calculators available to them in the classroom.

Clear the display and memory before starting a calculation
It is good practice to remove any displayed numbers and to clear the
calculator’s memory if this is to be used to store new values. The CLEAR key
may be a combined CLEAR and CLEAR ENTRY key with C/CE or CE/C on it.
Clearing the memory can involve a CM (clear memory) key. Children are less
likely to make errors if they get into the habit of clearing the display and where
appropriate the memory, before starting a new calculation. Always ask
children to check if there are any ‘left-over’ numbers on the calculator before
they start using it.

Correct a wrong entry by using the CLEAR ENTRY key
Most children will clear the display and repeat the calculation if they think that
they made an error. In a more complex calculation it is quicker to clear the last
entry. Children should be taught how and when to use the CLEAR ENTRY or
CE key and when it is more appropriate to clear everything and just start
again. Get children into to the habit of using the CE key correctly rather than
starting again when they make an incorrect entry.

Store a value in the calculator’s memory and retrieve it during a
multi-step calculation
In Year 6 as children become more confident users of the calculator they can
be taught how to use the calculator’s memory. This involves storing and
retrieving numbers to assist them with multi-step calculations. The calculator
may have four keys associated with the memory: CM to clear the memory;
RM to retrieve the number stored in the memory; M+ to add the number
entered to the number in the memory; and M- to subtract the entered number
from the number in the memory. Make sure children understand the function
of each key and ensure that they do not inadvertently change the number in
the memory by using the wrong key.

Keep track of a calculation and record the method used
When using a calculator, Year 5 and 6 children should be taught how to
record their calculations, together with the answers they get, at each stage in
a multi-step calculation. They should be encouraged to check whether each
answer makes sense as they work through a problem. From time to time,
Year 6 children who are confident in using the memory should still be asked to
record the calculations involved. Children need to understand the difference
between recording their method and recording the steps they go through on
the calculator. Emphasise that recording the method is about recording the
number statements or the calculations involved.

Use of other function keys
Children may have access to more sophisticated calculators that have
additional function keys such as square root or power keys and fraction
notation. Knowledge of how to use these keys is not a requirement of the Key
Stage 2 curriculum. However, it can provide children with the opportunities to
apply and extend their mathematics. For example, using square or power key
to generate the square numbers can take them well beyond manipulation of
three-digit numbers. Finding square roots of these square numbers
demonstrates the inverse operation to children. Finding the square roots of
intervening numbers such as the square root of 7 and then re-entering and
squaring the answer displayed shows children how calculators round their
displays. Use these keys to extend children’s mathematics but emphasise that
although they are extending their mathematical skills they will not be expected
to use these functions in the end-of-key-stage test.
B. Understanding and interpretation skills needed when using
a calculator to support learning in mathematics

Recognise the likely size of the answer and check answers
Children should be taught that calculation is a precise skill and to be good at it
requires a good understanding and knowledge of number. While children may
believe the calculator is a precise tool, remind them that the calculator only
responds to numbers the child enters and the keys they press. Children need
to remember that they may not be precise when it comes to entering values
into the calculator. This is often because the wrong digit is entered or the key
is not pressed hard enough. When using a calculator children should be
reminded that they should always check their results carefully. Teach children
a range of checking strategies, such as approximating, looking at the most
and least significant digits, checking the number of digits in the answer,
monitoring the position of the decimal point or carrying out the inverse
operation.

Recognise negative numbers in the display
Children are introduced to negative numbers in Year 4 and use them in
context. The calculator allows children to use both positive and negative
numbers in a calculation. However, negative numbers are often displayed on
the calculator as a result of some error in the calculation. Children need to
know when a number displayed is negative and not simply to ignore the sign.
They should go back and check if a negative value was expected and makes
sense. Year 6 children learn to find the difference between a positive and
negative integer and between two negative integers, again in a context that
gives meaning to the numbers involved. Using a calculator for these
calculations should be treated with caution as the manipulation of positive
and negative integers can easily be misinterpreted. For example finding the
difference between -3°C and +4°C may be misinterpreted as 1°C rather than
7°C if the signs are used as operations representing addition and subtraction.
The context too can change the sign in the answer. Asking ‘What is the
change in temperature from -3°C to 4°C?’ has an answer of 8°C or +8°C.
Asking ‘What is the change in temperature from 4°C to -3°C?’ leads to an
answer of -7°C. Children should recognise when a negative number is
displayed and check to see if this is a sensible answer.

Enter and interpret money and measurement calculations
Children need to understand when and why the decimal point can disappear
and can move about in the display. When £0.50 is entered, the number
displayed is likely to be 0.5 as trailing zeros are not shown in decimal
numbers. Multiplying 0.5 by 2 results in the number 1 being displayed and
the decimal point is no longer shown. Interpreting the results of a calculation
often causes difficulties, for example, 5.6 could mean £5.60, or 5 metres and
60 centimetres, or 5 kilograms and 600 grams, and so on. Children should be
taught why and when it is important to convert all measurements to the same
units before they carry out a calculation.

Carry out calculations involving time
Time does not lend itself easily to calculator use. Children will find it easier to
use a clock face or a time-line to do such calculations. These are less prone
to error than using a calculator. Finding the interval of time on a journey
starting at 08.38 and ending at 14.19 does not involve a decimal number
operation and children too often fall into the trap of thinking that it does. To
use a calculator they need to convert the times need to be convert the times
to minutes first then they can carry out the calculation. However, turning the
answer turned back to hours and minutes is less straightforward, so this
approach will require careful teaching.

Carry out calculations with more than one step
Children should be taught how to use the calculator to carry out increasingly
complex calculations, such as finding three eighths of £96 or sharing equally
the sum of four quantities among three recipients. To use a calculator
effectively, children need to know that the order of operations in a calculation
is important. Some calculators have brackets that children can use to help
sequence the order of the operations. If brackets are not available children
need to know how to enter a calculation such as: £138.45 - £8.24 x 6 where
the order of calculation is not left to right. They should be taught how to use
the calculator to test their observations and the conjectures they make from
calculations they can do in their heads. For example, after working out
mentally calculations such as (30+ 40) ÷ 5 and (30 ÷ 5) + (40 ÷ 5) and
discovering they give the same answer they can use the calculator test
equivalents involving decimal numbers. Children should be taught how to
select the correct sequence of operations in calculations involving more than
one step.

Recognise and interpret rounding errors
Calculators generally work with more numbers than are displayed. The
displayed values are usually rounded before display rather than simply
truncated at the point of display. This can sometimes lead to build up and
magnification of errors, but this will not affect most calculations children
undertake. For example, the answer to a calculation might be displayed as
0.999999. This may represent an answer of 1 as some rounding has been
carried out by the calculator’s operating system during the course of the
calculations. There are cases where some interpretation of the number
displayed need to carried out by children. For example, sharing a sum of
money may result in three or more decimal places being displayed. Children
should be taught to decide whether an answer that is displayed as a decimal
makes sense in the context of the calculation or the problem.

Use the division operation to enter a fraction
Children should be taught to use the division operation to enter fractions and
to recognise the decimal equivalent displayed. For example, when ¾ is
entered as 3, /, 4, the number displayed is 0. 75, its decimal equivalent. The
calculator is a useful tool for children to establish that all tenths have
equivalent decimal numbers with one decimal place, while all hundredths
have equivalent decimal numbers with one or two decimal places. Children
should be taught to recognise decimal representations of familiar fractions
and be able to convert from one representation to the other.

Recognise recurring decimals
Using the calculator children will discover that some fractions entered will fill
the display and often exhibit repeating patterns in the decimal digits they can
see. They should recognise which of the fractions with which they are familiar
with display this characteristic and recur, and explore the patterns displayed
by other fractions. They should be taught to recognise that not all the digits in
the decimal representation may recur, for example, entering 1, /, 6 leads to
the number 0.166666 being displayed. Multiplying decimal representations of
fractions shows how displays can change significantly and teaches children
to look at the whole number on the screen. For example, multiplying the
decimal representation of one-sixth by 2 then by 3 has a significant effect.

Decide when a calculator is an appropriate tool to use
Children are usually given calculators when they are set a task so they know
that on that occasion they are allowed to use them to solve the problem.
Children need to recognise that any tool is designed to be used for a purpose
and the calculator is no exception. They should be given opportunities to
decide when a calculator might be helpful and when mental or written
methods are more effective and efficient. Showing children that they have the
knowledge and skills that enable them to calculate in less time than it takes
them to key the numbers and operations into a calculator is a useful ploy.
Children need to be taught that they can decide for themselves when to use a
calculator if it supports their learning and is not a replacement for more
efficient methods and is not counter to the objectives of the learning.
Spending time after an activity, discussing how the calculator supported the
activity and what the children learned as a result is worthwhile.
Overview of calculator skills in Years 4, 5 and 6
By the end of Year 4, children should know how to:

Clear the display before starting a calculation
Children are less likely to make errors if they develop the habit of clearing
the display as a matter of course before starting a new calculation.

Correct mistaken entries by using the CLEAR ENTRY key
Children tend to trust the calculator which never goes wrong without
thinking about the possibility that they have made the mistake when
entering numbers. Most children will clear the display and repeat the
calculation if they think that they have made an error. They need to learn
how and when to use the CE key.

Carry out one- and two-step calculations involving all four
operations
Most children have little difficulty with entering a one-step calculation to
work out for example £4.55 × 17. However, when they are solving word
problems, children are not always clear which values and what operations
to use. They may misinterpret the question and enter the wrong calculation
using the wrong values. It is good practice to get them to write down the
calculations involved so they can check they have identified and used the
right operations and numbers.

Interpret the display correctly, particularly in the context of money
Children need to be taught how to interpret the displayed numbers
particularly large numbers as there no gaps to help them read the answer
correctly. Decimal numbers can cause confusion when there is only one
decimal place and the value has to be put into a context such as money.

Recognise negative numbers and use the sign-change key
Children may miss the minus sign that indicates a negative number. This
usually appears on the extreme left of the display. It will appear if a
subtraction calculation has been entered in the wrong order, for example.
By the end of Year 5, children should know how to:

Estimate the likely size of the answer and check answers
appropriately
This is an important skill – errors in making entries often lead to answers
that are nonsense, particularly when decimals or fractions are involved.
Using some checking strategies, such as rounding and making an
estimate, or carrying out the inverse operation will help children to avoid
such errors.

Carry out measurement calculations and interpret the answer
Entering decimal numbers to carry out calculations involving
measurements can cause difficulty. It is not always clear if the decimal
point has been entered until other entries are made and the numbers start
moving along the display. For example, 5.6 could mean 5 metres and 60
centimetres, or 5 kilograms and 600 grams, and so on. In addition, children
need to be taught to change all measurements to the same units before
they do a calculation. This is best done manually before the values are
entered.

Solve problems involving fractions
To find ¾ of 260g children need to be taught that this is represented by the
calculation 260 x 3 ÷ 4, or recognise that the decimal equivalent of ¾ is 0.
75 and use the calculation 0.75 x 260. These approaches support
children’s understanding of how a fraction is used as an operator and the
equivalent representations of number as a fractions or a decimal. The use
of fraction keys that are available in some calculators is likely to cause
confusion and should be avoided.
By the end of Year 6, children should know how to:

Solve problems involving multi-step calculations
Children need to be familiar with the order of operations so that they choose
the correct sequence of operations in calculations involving more than one
step. They also need to practise jotting down parts of a calculation as they go
along. Calculations such as: 8 × (37 + 58), 43% of £285, or 3⁄8 of 980 km are
all multi-step and need to be taught and practised. At Key Stage 2, there
should be no particular need to use a calculator to work out percentages, but
if necessary the percentage can be represented as a fraction or decimal. So
43% of £285 can be calculated as 285 × 0.43, or as 285 × 43 ÷ 100. Basic
calculators usually have a memory. While there is no requirement for children
to use the memory at Key Stage 2, children in Year 6 will probably enjoy
learning to use it.

Recognise rounding errors
Although rounding errors are rare on modern calculators, they can still
occur. Children need to know when answers are likely to have been
rounded. For example, using the calculator to explore decimal
representations of fractions will show that 1/9 is represented by 0.1111111
but when this number is multiplied by 9 the answer displayed is 0.9999999
not 1.

Recognise recurring decimals
Children should be familiar with decimals representations such as 1/3 and
0.3333333. They also need to recognise that not all digits may recur in a
decimal representation of a fraction, as in 1/6 where the decimal equivalent
is 0.1666666 or 1/7 which is 0.1428571 and the six digits 142857 recur.

Carry out calculations involving time
Calculations involving time can cause particular difficulty in Key Stage 2. The
best approach to calculating time intervals is to use a time line, not a
calculator. If a calculator is used to find the time that elapses from 6:40 to
10:05, then the hours and minutes must be turned into minutes and the
difference found. However, turning the difference of 205 minutes back into
hours by dividing by 60, will display 3.4166666 hours and the decimal part
must be multiplied again by 60. The value 3.4166666 hours actually
represents 3 hours 25 minutes. For most children it is best to avoid using a
calculator for time calculations in Key Stage 2 and use a time-line instead.
Summary
By the end of Key Stage 2 children should know:
• How to clear the display, correct entry errors, and what to do if the
estimated answer and calculator answer do not agree.
• The order in which to use the keys for calculations involving more than one
step.
• How to enter numbers and interpret the display when the numbers
represent money, metric measurements or fractions.
• When and how to use function keys such as the square root key and the
sign-change key.
• How to select from the display the number of figures appropriate to the
context of a calculation.
• That it is better to choose written or mental methods for calculations
involving percentages or mixed units of time (e.g. hours and minutes).