1 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Identifying and overcoming barriers to
mathematics learning in Year 1
Introduction
Children who leave Key Stage 1 in primary schools with a good grasp of core mathematical
principles are in a strong position to build on this understanding and make good progress through
Key Stage 2. In turn, children who leave Key Stage 2 working at level 4 or above are likely to
obtain a good grade in GCSE mathematics, giving them a greater range of possible future life
choices. It is therefore important that schools ensure that as many children as possible make
good progress in mathematics in the early years of education.
Year 1 is an important year for children’s mathematical learning. In Year 1 children build on the
practical experiences they have enjoyed in the Early Years Foundation Stage curriculum, begin to
formalise their understanding of mathematics and start to develop the mathematical knowledge,
skills and strategies identified in the Programme of Study for National Curriculum in Key Stage 1.
Teachers report that, in Year 1, some children experience difficulties with key aspects of
mathematical learning and that these difficulties form a barrier to the children’s progress. The aim
of this study was to identify the common barriers that appear to inhibit children’s early learning
and understanding of mathematics. As number is such a major part of the early curriculum, this
study focused on clarifying those aspects of number that children commonly struggle to acquire,
yet which are vital in underpinning their emerging understanding and use of the number system,
place value and early calculation. The information in this project is based on observation of Year
1 and early Year 2 children’s working on a range of number-based activities and discussions with
their teachers. These observations were carried by a small team of consultants from schools
across seven local authorities.
This report summarises the key findings from the project. It outlines the common barriers to
learning identified for children in the project who were not making expected progress in
mathematics. It also includes some recommendations for actions and strategies that schools
might draw on where their assessment shows that pupils are experiencing some of these
difficulties. In this way, it is hoped that teachers and schools may be able to draw on the findings
of this study to help them consider and develop appropriate teaching strategies or intervention
approaches to support young children in overcoming key barriers to learning and so make good
progress in mathematics in Key Stage 1 and throughout their later education.
Structure of the report
Introduction.................................................................................................................................. 1
Summary of findings: common areas of difficulty ........................................................................ 2
Common areas of difficulty: related learning targets and teaching foci ....................................... 2
Related National Strategies resources ........................................................................................ 5
Project findings ............................................................................................................................ 5
Appendix A ................................................................................................................................ 10
Appendix B ................................................................................................................................ 23
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Summary of findings: common areas of difficulty
In this study, observations of children identified as making slower than expected progress in Year
1 as they worked on a range of number tasks revealed some common areas of difficulty. These
are summarised below. It is hoped that teachers and schools might find it useful to look at these
common areas of difficulty alongside their own assessment information for any Year 1 pupils
whose progress they are concerned about, and to identify any commonalities in learning that
might be inhibiting their progress. This should support schools in identifying, clarifying and
addressing the main barriers to progress in mathematics for these pupils.
Evidence from the study showed that children who made less than expected progress in
mathematics in Year 1, were hampered by a limited knowledge and understanding of number and
had acquired a narrow range of associated skills. In particular, children observed commonly
demonstrated:

an insecure grasp of the number sequence, leading to confusion when using ‘teen’ or ‘ty’
numbers and imprecision when counting backwards

limited understanding of how the counting and number skills they are developing could be
applied to practical situations and contexts

weak understanding of place value, for example not recognising that the value of each digit in
a 2-digit number is significantly different

reliance on basic counting strategies, for example always counting from one when combining
two sets of objects rather than counting on from the number that represented the size of one
of the sets

poor understanding of early subtraction concepts, such as finding ‘1 less than’, having to take
away 1 object and recount the set rather than using the counting sequence

difficulty interpreting and accurately recording simple subtraction number sentences,
confusing addition and subtraction notation and misplacing the numbers in the sentence

limited access to and use of mathematical vocabulary, and undeveloped language skills
needed to describe their actions, express their ideas or explain their thinking.
Common areas of difficulty: related learning targets and
teaching foci
In order to ensure that children overcome any identified barriers to learning and so make good
progress in mathematics, teachers and schools need to identify key learning targets that relate to
the children’s learning barriers and to design an appropriate range of teaching or intervention
activities to ensure that children attain these targets. To support teachers and schools in doing
this for any children who exhibit some of the common areas of difficulty identified above, a
number of learning targets have been identified (see below). These address the findings from the
project and identify the key next steps intended to help children to overcome these difficulties.
Each learning target is followed by a small set of suggested teaching foci that, where appropriate,
teachers might build into their planning and teaching or use to inform any intervention provision to
help pupils secure the learning target. These recommendations are drawn from a wide range of
evidence that draws from the study, observation of good practice in Key Stage 1 classrooms, and
wider experience of consultancy and support undertaken within primary schools.
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Learning target: Understand the difference between the ‘teen’ and the
‘ty’ numbers when counting
Teaching foci
Help children to:

pronounce the number names clearly, relate the teen numbers to numbers before 20 and
relate ‘ty’ to ‘groups of ten’, so that children are aware that for example ‘fourteen’ and ‘forty’
are different numbers

develop and use images for the ‘teen’ and ‘ty’ numbers. For example, discuss what it means
to be a ‘teenager’ and create posters or images for the ‘ty’ numbers that clearly show groups
of ten, for example using 10p pieces

use highlighting or colouring to clarify where the ‘teen’ numbers and the ‘ty’ numbers appear
on key models such as number tracks and hundred squares and to identify the key features of
the sequence of ‘teen’ numbers and of the sequence of ‘ty’ numbers.
Learning target: Count accurately and quickly, forward and backwards
and apply this counting to solving practical problems
Teaching foci
Help children to:

become confident in counting backwards as well as forwards

use actions, puppets and objects to illustrate counting rhymes and songs

use practical resources alongside counting activities to understand the link between the
spoken count and an increase or decrease in the number of objects

understand the importance of counting in everyday activities, for example through counting
the number of children in the class requiring school dinner each day and discussing why it is
important to send this number to the kitchen

develop understanding of the important roles played by numbers in the environment, for
example, through involving children in designing posters to give numerical information such as
the maximum number of children who should be in the role-play area at one time.
Learning target: Understand the value of each digit in 2-digit numbers
Teaching foci
Help children to:

group objects which number more than 10 into tens, exploring the relationship between the
number of tens and units and interpreting the values represented by the digits in the number

gain experience of counting out numbers in tens and ones using equipment such as linked
blocks, bundles of sticks or straws

discuss and rehearse how bundles or groups of 10 can help us count quickly

use models such as place value cards or bead strings to begin to understand how to partition
numbers into their tens and units parts.
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Learning target: Recognise how counting on and counting back can
provide efficient ways to add and subtract small amounts
Teaching foci
Help children to:

relate finding ‘one more’ than a given number to counting on one from that number in the
counting sequence and relate finding ‘one less’ to counting back one

read and record pairs of addition and subtraction number sentences linked to counting
activities to see the relationship between the symbols and the operations

understand how to find the total number of objects in two sets by counting on. For example,
children should experience adding the objects from one set into the other, one at a time,
counting on as they do so, and recognise that these numbers identify the total so far

understand how to find the number of objects left after some are taken away from a set by
counting back. For example, children should experience removing the objects from a set, one
at a time, counting back as they do so to give the amount of objects left at each stage

rehearse how to use fingers to keep track of how many objects they have counted on or back.
Learning target: Understand that subtraction can involve the processes
of taking away, counting back and finding how many more
Teaching foci
Help children to:

count back along a number track by physically jumping themselves or by moving an object
and counting the jumps

link jumping along a number track to counting back and to subtracting ones

begin to develop an understanding of subtraction as the inverse of addition, for example
through exploring the effect of adding then subtracting the same number

begin to recognise the relationship between subtracting and ‘finding the difference’ through
practical activities such as finding how many more cubes are in one tower than another

read and record subtraction number sentences alongside activities that involve subtraction

develop their understanding of the role of the subtraction and the equal signs and where each
number goes in the sentence.
Learning target: Extend their mathematics vocabulary and use
mathematical language to describe, explain and express their ideas
and methods
Teaching foci
Help children to:

respond to questions in full sentences, for example drawing on the words used in the
question, for example when responding to the question ‘What number is 1 more than 9?’,
children should be encourage to say ’10 is 1 more than 9’ rather than just say ‘ten’

repeat aloud words and sentences that they have heard so that they can then use to express
their ideas and methods
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
re-phrase answers so that they use accurate mathematical vocabulary within a complete
sentence

talk about the work they are engaged in and think aloud as they carry out some activity

develop the vocabulary and language needed to explain their method rather than just giving
their answer.
Related National Strategies resources
It is hoped that schools and Year 1 teachers will find the above information that has come out of
this project useful. Of course, schools need to draw on a wide range of information and resources
to support them in their ongoing assessment of children to help identify any difficulties in
mathematics that can inhibit their progress. These resources might also be useful in planning and
informing intervention provision to support children in overcoming these difficulties. The following
National Strategies resources contain information that is designed to support schools and
teachers in ensuring that as many children as possible make good progress in mathematics in
Key Stage 1.
National Strategies resources:

Numbers and patterns: laying foundations in mathematics (DCSF: 01011-2009FLR-EN)

Overcoming barriers in mathematics – helping children move from level 1 to level 2
(DCSF: 00021-2009)

Securing level 1 in mathematics (DCSF: 00041-2010BKT-EN)

Securing level 2 in mathematics (DCSF: 00687-2009BKT-EN)

Supporting children with gaps in their mathematical understanding (DCSF: 1168-2005G)

What I can do in mathematics level 1 (DCSF: 00952-2009DOC-EN-05)

What I can do in mathematics level 2 (DCSF: 00952-2009DOC-EN-01)
Project findings
Children who make slower than expected progress in mathematics
The key areas of difficulty identified in this report have been synthesized from the collated
findings from the project. In this section of the report, greater detail is given about how the
children making slower than expected progress in mathematics engaged in a range of number
activities. It may therefore provide teachers with greater insight into the understanding and skills
these children commonly demonstrated and the specific difficulties they had. These findings are
drawn from the observation notes made by the consultants involved in the project.
Understanding of the number sequence

Most children who were struggling with mathematics were able to count to 10 forwards and
backwards, but with less confidence. They were generally able to recognise the numbers on a
number line or track.

Children found it more difficult to start a count from a number other than 1 i.e. they knew the
count from 1 to 10, but found it difficult to start counting from a number such as 5.

Children confused the vocabulary of counting backwards and forwards and often counted in
the wrong direction.
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
Counting beyond 10 was problematic; they were less secure with the sequence of numbers
from ten to twenty. Children’s counting was hesitant and they became confused when
counting backwards particularly at the point where the pattern in the number names changes
in the teens, 20, 19, 18, 17, ... 13, 12, 11…

Children, including those making good progress, demonstrated confusion between the ‘ty’ and
the ‘teen’ numbers, for example when asked to count back from 20, counting: 20, 90, 80...
rather than 20, 19, 18...

When given number cards to 20, children were generally able to order them correctly and
could identify missing numbers in a sequence made from the number cards.
Understanding of how counting and number skills apply to practical
situations and contexts

When asked about why we use numbers, children often said that we needed numbers for
counting. Few children, however, could suggest real contexts in which such counting would be
useful. A few children explained that you needed numbers to find out how old someone was.

Children demonstrated limited awareness of the use of numbers in the environment. A few
children were able to suggest examples such as price tags showing you how much things cost
in a shop or house numbers to show you which house is which on the streets.

Most of the children were able to apply one-to-one correspondence to count a small set of
objects accurately, but those who were struggling with mathematics did not have effective
strategies for keeping track of the objects they had counted and lost track, particularly when
counting objects that were randomly arranged.

Children understood that the last number in the count described the number of objects in the
set but could not use this when adding two sets of objects.

Children had not acquired a secure grasp of the conservation of number and recounted a set
of objects when they were rearranged.

Children knew the sequence for counting in twos to 10, but they did not necessarily
understand its application as a counting strategy. For example, when asked to count socks
arranged into pairs, children counted each item individually and could not apply their
knowledge of the sequence of 2s to solve the problem of how many socks there were
altogether.
Understanding of place value

Children generally knew the sequence of numbers when counting in tens but, again,
demonstrated confusion between the ‘ty’ and ‘teen’ numbers. Several children counted: 10,
20, 30, 40, 50, 60, 70, 80, 90, 20.

When asked to read a range of larger numbers, children would confuse the ‘teen’ and ‘ty’
numbers e.g. confusing 13 and 30, 19 and 90. There were also examples where children
reversed the tens and units figures when identifying larger numbers e.g. 37 became 73.

Although many children were able to count with 2-digit numbers and could recognise and
write some 2-digit numbers, they were not aware of the value of each digit in the number or
that the value of each digit in a number is related to its position in the number.

Where children were starting to be able to draw on their understanding of place value in 2digit numbers, they were not always able to apply this understanding to ‘teen’ numbers. For
example, children who could count and represent accurately a 2-digit number using bunches
of ten sticks and single sticks still used 17 single sticks when asked to use these to make the
number 17.
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Using counting strategies to calculate

When adding two sets of objects each of which they had counted, children used a ‘count all’
strategy where they counted the total in the two combined sets, starting again at 1.

Only a few of the ‘on track’ children were able to use ‘counting on’ when adding small
numbers. Rarely did the children use a counting on from the bigger number strategy, with any
consistency. However, when prompted the children recognised that counting on from the
larger number made the calculation easier but many still lacked the confidence to use it.

Children were generally able to identify when to use addition to solve a problem, particularly
when it involved combining groups. When carrying out a recorded calculation involving
addition, children commonly resorted to a ‘count all’ strategy, often using their fingers to do so,
which led to mistakes being made as children lost track of the count.

Children were able to state the number that is ‘one more’ than a given number as they were
able to understand one more as the next number in the count or on the number track.
However, they were less confident when asked to find ‘one less’ and could not relate ‘one
less’ to the number before in the count or on a number track.

Children who were making good progress were starting to recall and use number facts to 10.
Children not making progress struggled to recall such facts and rarely used these to support
calculation, instead relying on basic counting in ones.
Understanding subtraction

Children understood the concept of taking away objects from a set. To find what was left,
children generally counted the remaining objects in ones. A few of the ‘on track’ children were
starting to use a counting back strategy when only 1 or 2 objects were taken away.

Very few of children making good progress were able to relate the terms ‘subtract’ or
‘subtraction’ to ‘take-away’. Hardly any of the children understood the word ‘fewer’ and none
of the children understood the term ‘difference’.

Most children struggled to solve problems involving subtraction unless the problem was
translated into taking away and counting objects left after those required had been removed.
Recording number sentences

Children in the study could carry out simple addition calculations and could record what they
had done using the addition sign and equals sign accurately in a number sentence. However,
even those who could deal with a ‘take away’ question struggled to record the subtraction
accurately using the minus sign in a number sentence.

When asked to record a number sentence to match a practical take away example, some
children were not able to recall or use the subtraction sign and used the addition symbol
instead.
Using mathematical vocabulary and language

There was considerable variation in the range of mathematical language used and understood
by the children in the study. Most children understood and could respond to and interpret
words that are used in everyday contexts such as ‘next’, ‘before’, ‘biggest’ and ‘altogether’.
They were more confident using mathematical words that related to addition such as ‘add’ and
‘more than’, than they were to words relating to subtraction, apart from ‘taking away’.

Children struggled to draw on accurate mathematical language to describe what they were
doing in mathematics. This restricted their ability to express their ideas or explain their
methods or reasoning.
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Overview of the approach taken in the project
The study was carried out to gather information on the following questions:

What difficulties do children commonly encounter when learning to count, record and
calculate?

Which key gaps in understanding do children who are identified as making slow progress in
mathematics in Year 1 commonly demonstrate?

Which aspects of early learning about number appear to act as barriers to progress in
mathematics in Year 1?
This study was carried out by a small group of consultants. Overall, the consultants spent time in
12 schools, across 7 different local authorities. The consultants selected schools that were known
to them and invited them to take part in the project. The schools involved were asked to identify a
small group of pupils whose progress in mathematics was below that expected and a small group
who were ‘on track’ for observation. These two groups of pupils were observed as part of the
study so that consultants were able to contrast the areas of mathematical understanding
demonstrated by the two groups in order to identify key gaps in understanding for pupils making
slow progress. The observations were discussed with the teachers to share findings and identify
any additional information about the pupils’ mathematical progress and attainment. This feedback
from the observations was intended to help the schools involved in supporting their ongoing
development of teaching and learning of mathematics in Key Stage 1. It was intended to inform
the work of their teachers, teaching assistants and Every Child Counts practitioners where they
were involved in the programme.
The project was based on observations of children working on a range of number-based
activities. Documentation and guidance was produced in order to support the consultants
engaged in carrying out these observations and to ensure quality and consistency of approach.
The document Expectations for Foundation Stage and Year 1 Children (see Appendix A), shows
the relationship between the Primary Framework for mathematics Year 1 objectives, the
Foundation Stage objectives and the learning focuses set out in phases 4, 5 and 6 in the
Numbers and Patterns resource. It also identifies some observation points and questions that can
help assess children’s understanding within these learning focuses.
Consultants were also provided with an Observation sheet (see Appendix B) giving a range of
suggested activities for the children to work on, and questions that they could draw on as they
interacted with children during the observation. These sheets contained space for consultants to
jot down key observation points that demonstrated children’s understanding or difficulties with
particular elements of counting and calculating. The two documents described above are included
in this report so that teachers may wish to draw on them to support their own observation of
children in Year 1 or early Year 2, and may incorporate such observation-based activities into
their own assessment practices.
Observations for the project were carried out early in the school year and included observations
of pupils who had recently entered Year 1 or had just entered Year 2. In this way, the study aimed
to assess the learning and understanding about number that children develop (and had
developed) at the start of and over Year 1. In total, consultants observed 64 children working. As
the children worked on the tasks, consultants noted the responses of the children and later
engaged them in a structured discussion about what they had been doing, in order to assess the
children’s understanding and application of number skills. They went on to analyse children’s
responses and assess the underlying understanding that was revealed for each child.
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Consultants then collated assessment points identifying common areas of understanding, gaps in
understanding, difficulties children had experienced and barriers to learning they had identified.
Key commonalities from the feedback of all the consultants involved in the project were identified
and form the findings presented in this report. This report is written so that teachers and schools
might reflect on commonalities between the findings from the project and the mathematical
development of their own Key Stage 1 pupils. In this way, teachers and schools may be able to
draw on these findings to help them consider and develop appropriate teaching strategies or
intervention approaches to support young children in overcoming key barriers to learning and so
make good progress in mathematics in Key Stage 1.
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Appendix A
Year 1 project: Identifying and overcoming barriers to mathematical learning in Year 1
Expectations for Foundation Stage and Year 1 children
The grid below shows the relationship between the Primary Framework for mathematics Y1 objectives, Foundation Stage objectives and the
learning focuses set out in phases 4, 5 and 6 in the Numbers and Patterns resource.
Following initial observation, as barriers to learning are encountered with an individual child, more thorough in depth work/questioning should
then take place by drawing on this document. Use the questions and activities set out below as a starting point for investigating further aspects
of mathematics that children are having more difficulty in understanding.
See also the associated recording document which practitioners can use to record/annotate the bits of mathematics the children find difficult
when they analyse the responses.
Notes on using these questions as starting points:

questions have been included in this document to help you to assess children’s understanding in some important areas of counting,
recording numbers and calculation. You will need to prioritise key areas to assess for each child/pair/group of children

the questions suggested provide starting points from which to engage children in discussion. You will need to adapt questions and add in
further questions in the light of children’s responses

in particular, throughout the assessment you should incorporate:
— prompting questions in order to encourage children to engage in discussion and to help to identify a secure base of understanding
— probing questions in order to assess children’s confidence and their depth of knowledge and understanding
— promoting questions that extend the ideas being discussed to determine children’s breadth of knowledge and understanding.
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Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Further questions/ activities to
test knowledge further
Primary Framework objectives
Further questions/activities to test
knowledge further
Say and use number names in order
in familiar contexts
Count one, two three.... counting
consistently through the unorthodox
teens – first to 10 then to 20. Look
out for teens numbers – not changing
the pattern
Count reliably at least 20 objects,
recognising that when re-arranged
the number of objects stays the
same
Extend the count up to and beyond
20 to 100 using similar activities to
the previous column.
Progress to beyond 20
Extend counting to include sounds
e.g. number of claps (including
clapping outside children’s vision),
count up to 10 objects that are out of
reach e.g. window panes
Explore children’s understanding/
recognition of numbers in the wider
sense. ‘How old are you? How old
will you be on your next birthday?
Last birthday? What number is the
house you live at?’
Look out for sticking points in the
count e.g. word omitted, words in the
wrong order, repeating a word
Count starting from a given number.
APP statement L1
Draw simple conclusions from their
work e.g. with support:

describe the different ways they
have sorted objects, what is the
same about objects in a set, how
sets differ

identify which set has most,
which object is biggest, smallest,
tallest etc.

explain numbers and
calculations, how many
altogether, how many used or
hidden, how many left, how
many each etc.
Estimate a number of objects that
can be checked by counting
Look out for changes in the decades.
Recognition that somethingty nine
signals a change e.g. twenty nine is
not followed by twenty ten.
‘Count up from one to as far as you
can, saying each number clearly.’
‘What number comes after 29? What
comes before 40?’
‘Can you count backwards from…
until you get to zero?’
Let’s count these red and blue
blocks. How many blocks did you
count? Now close your eyes and I
will rearrange them.
How many blocks are there now? Do
we need to count them all again?
Here are two towers of brick, how
many bricks in the first column? How
many in the second column? How
can you find out how many
altogether? Which tower is taller?
How do we know?
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Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/activities to test
knowledge further
Further questions/ activities to
test knowledge further
Say the number name that goes
before or after a given number ‘what
number comes next after six? What
number comes one before 6? Start
and stop counting at a given number
e.g. start at 2 and count to 16’.
Count on 3 numbers from 7
(encourage use of fingers)
Count back to include zero from a
given number. Start and stop at
given numbers. Count back 3 from
given numbers (fingers to help)
Know that numbers identify how
many objects are in a set
Count the toys in this tray. I am going
to cover the tray with this cloth.
Compare and order numbers, using
the related vocabulary
‘The numbers in this count are mixed
up. Can you put them in order?’
Numbers and Patterns Phase 5 –
learning focus
How many toys are under the cloth?
Can you tell me why?
APP statement L1
4, 8, 2, 10, 1, 3, 0
Order numbers to 10
Then:
Instantly recognise, without counting,
organised and random arrangements
of small numbers of objects
I have removed the cloth do we need
to count how many there are again?

count back to zero
18, 16, 17, 15, 13, 14, 12, 10, 11

place 1–10 in ascending order

point to first, second etc. in a line

Here is a row of four coloured
counters. Which coloured counter is
first, third etc?
begin to count in two’s
Show children a series of ‘flash’
cards asking how many objects are
on each card. How many… can you
see? How do you know? Do you
need to count to check?
Use the (=) sign
(extend up to 10)
Look at these number cards.
Which card shows the smallest
number?
Put the numbers in order from the
smallest to the largest
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Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/ activities to
test knowledge further
Check understanding
Further questions/activities to test
knowledge further
15
7
5

the purpose of counting is to tell
how many there are

the last number name is the
answer to ‘how many’

if two different counts give the
same answer something is
wrong
4+1
8-1
7+1`

that there is no need to count
when the number can be
recognised without counting.
7-1
10-1
6
5
8
9
Here are some cards. Can you make
me a number sentence using these
cards?
7
Count reliably up to 10 everyday
objects (EOY)
APP statement L1
Count up to 10 objects e.g. estimate
and check a number
12
I will start counting this group of
objects. Can you do this with me and
then continue?
Look at these toys in a line.
Count them for me.
Numbers and Patterns Phase 4 –
learning focus
I am going to put them in a circle like
this. Count them for me again.
Count reliably any arrangement of up
to 10 objects
Now we will jumble them up can you
count for me again?
Read and write numerals from 0 to
20, then beyond
Numbers and Patterns Phase 6 –
learning focus
Count large groups of objects by
using efficient strategies
=
Can you count these… carefully
How could you make sure you have
counted them correctly?
Ask children to count sets of objects
using a range of different formations
look for strategies children use.
Count forwards and backwards
within the number sequence 0 to 100
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14 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Further questions/ activities to
test knowledge further
Primary Framework objectives
Further questions/activities to test
knowledge further
1, 2, 3... what number comes next?
Say the number that comes before
and after a given number within the
number sequence 0 to 100
Look at this grid. Point to 16, 20 and
twelve.
5, 6, 7... what comes next?
Check understanding

whatever order a collection is
counted the number is the same
Look out for children strategies in
order to keep track of the count e.g.
moving each object across one by
one when counting.
Counting errors; counting the same
number twice, missing out a number
completely, error in the counting, not
giving a number name to an object
touched, counting the correct number
of objects but saying the wrong
number for the total.
13
14
16
APP statement L1
12
17
19
Read, write numbers to 10 – perhaps
with some reversal
20
18
Recognise, say and identify
numerals 0 to 100
Use knowledge of place value to
position these numbers on a number
track and number line
Which number is in the middle of the
grid?
Can you find a number bigger than
14? Smaller than 17?
Write the number 15 into the empty
box.
Estimate how many objects they can
see and check by counting
Count the number of red blocks in
this tray.
Numbers and Patterns Phase 5 –
learning focus
Look at the blue blocks in this tray.
Estimate a number of objects that
can be checked by counting
How many do you think are in this
tray?
Do you think there are more red or
blue blocks?
Say the number that is 1 more or
less that any given number, and 10
more or less for multiples of 10
APP statement L1
Order numbers to 10 – say what
number comes next, is one
more/less
What numbers are missing from this
number track?
How do you know?
What number is one more than 11?
One less than nine?
Etc.
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15 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/ activities to
test knowledge further
Count out the blue blocks so we can
decide if you were right.
Place a number of objects in a pot
(within the range 1 – 10)
How many objects do you think are
in the pot?
How can you check if you are
correct?
Numbers and Patterns Phase 5 –
learning focus
Compare sets of up to 20 objects
using language such as ‘more’ or
‘fewer’
Phase 4
Find one more or one less than a
number from 1 to 10
Further questions/activities to test
knowledge further
?
9
?
11
?
Look at this number line, can you tell
me which number is one more than
12, two more than 16, one less than
17, two less than 11.
Check for ‘teen’ confusion.
Ask similar questions without the
number line.
Count aloud in ones, two’s, fives or
tens
APP statement L1
Order numbers to 10

count back to zero

begin to count in two’s
Use language such as ‘more’ ‘less’ to
compare two numbers
Use ordinal numbers in different
contexts
Can you continue the count?
Stop when you get to 20: 2, 4, 6…
My bag has pairs of socks in it.
How many socks are in this one pair
of socks?
Two yes so if I get all the pairs of
socks from my bag, will you count
out the number of socks for me?
Can you do this by counting in twos?
How many are socks are there
altogether?
Similarly ask children to count pairs
of eggs in a box.
Use the vocabulary of halves and
quarters in context
Can you fold this piece of paper into
two?
APP statement L1
Can you cut along the fold so that
you have two pieces?
Begin to use the fraction one-half
e.g. halve shapes including folding
paper shapes, lengths of string

put water in a clear container so
that it is about ‘half-full’

halve an even number of objects
What do we call each piece?
If you keep one piece and I have the
other piece which of us has the
bigger piece? How can you check?
How many blocks/sweets (or
equivalent substitute to eggs) are in
the egg box?
Can you take half of them out?
How many did you take?
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16 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Counting and understanding number
Expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/ activities to
test knowledge further
How many are left?
APP statement L1
Order numbers to 10

point to first, second etc. in a line
Further questions/activities to test
knowledge further
Check knowledge of: odd, even,
every other
Similar activity and questions for say
tubs with five marbles in each.
What do you notice?
Check children’s knowledge and
understanding that a half is one of
two equal parts.
How many fingers do you have on
one hand, on two? Can you do this
by counting in fives?
How many fingers and toes do you
have? Count you find out by counting
in fives?
Count in tens to 100 forwards and
back, starting from any tens number.
Say the tens number that comes
before or after a given one
Here are three numbers e.g. 2, 5, 3
can you put them in order?
(Then within the range 1–10)
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17 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Knowing and using number facts
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Further questions/ activities to
test knowledge further
Primary Framework objectives
Further questions/activities to test
knowledge further
Observe number relationships and
patterns in the environment and use
these to derive facts
How many toes do you have on this
foot?
Derive and recall all pairs of numbers
with a total of 10 and addition facts
for totals to at least 5
I have 4 counters how many more do
I need to make a total of 10?
Cover up three toes. How many toes
can you see?
You have covered three toes and
you can see two toes. How many
toes do two toes and three toes
make altogether?
Work out the corresponding
subtraction facts
APP statement L1
Order numbers to 10 – say what
number comes next, is one
more/less
Can you tell me two more numbers
that total 10?
How many do I need to add to 2 to
make 10?
Now we have three blue cups on the
table; there are two red cups in this
bag. I’m not going to show them to
you yet but can you tell me how
many cups there are altogether.
Find one more or one less than a
number from 1 to 10
Can you tell me two numbers that
add up to 10?
I have 3 bricks in this box how many
more bricks do I need so that I have
a total of 10 in the box?
Here are some digit cards from 0 to
9. Can you put the numbers into
pairs so that each pair adds up to
10? Which number is left? What
would you need to make another
pair?
Can you put in order this set of
cards? What number should come
first? What number should come
next? Which number comes last?
Count on or back in ones, two’s fives
and tens and use this knowledge to
derive the multiples of 2, 5 and 10 to
the tenth multiple
Are there any numbers which should
come in between?
Numbers and Patterns Phase 6 –
learning focus
See column 2 above for activities
Count forwards and backwards in
twos, fives and tens
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18 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Knowing and using number facts
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/activities to test
knowledge further
How many fingers do you have on
one hand? How many fingers do you
have on two hands?
Further questions/ activities to
test knowledge further
I have 4 bricks and you have one
more. How many bricks do you
have? Can you count to check?
Here is a box that can hold 6 eggs.
You have five eggs, how many more
eggs do you need to fill the box?
What number is one more than... one
less than...... Using a number line
then with no apparatus
See also column 4 above
Select two groups of objects to make
a given total of objects
Make two groups of objects up to a
total of 10.
Recall the doubles of all numbers to
at least 10
Numbers and Patterns Phase 5 –
learning focus
How many objects are in this set?
Begin to know some addition facts
How many objects altogether?
e.g. doubles of numbers to double 5
Find the total by combining two
groups where one group is screened
(seen and then hidden) and counting
on
If you know there are 4 objects in this
set, can you count on to find out how
many objects in both sets?
(Change arrangement and number of
objects in the sets to extend)
You have 3 pennies and I have 3
pennies, how many do we have
altogether?
There are four wheels on this car,
how many wheels will there be on
two cars?
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19 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Calculating
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Further questions/ activities to
test knowledge further
Primary Framework objectives
Further questions/activities to test
knowledge further
Begin to relate addition to combining
two groups of objects and subtraction
to ‘taking away’
Show me three fingers on your right
hand. Show me two fingers on your
right hand? How many fingers
showing altogether?
Relate addition to counting on
Can you pick up a handful of large
buttons and put them on the table.
APP statement L1
Understand addition as finding the
total of two or more sets of objects
Understand subtraction as ‘taking
away’ objects from a set and finding
how many are left
Numbers and Patterns Phase 4 –
learning focus
Partition and recombine small groups
of up to 10 objects
Find the total number of objects in
two groups by counting all of them
Introduce the empty set (0)
Recognise that the number of objects
in a set does not change if they are
moved around
Remove objects from a small group
and count how many are left
Josh collects toy bears. He has six
and then is given 3 more for his
birthday. How many bears does he
have now?
Here is a shop with things to buy.
One object costs 2p one object costs
3p and one object costs 1p.
How many pennies do you need to
give the shop keeper?
Can you say how many pennies you
need to give the shopkeeper by
counting them all?
Can you say how many pennies you
need to give the shopkeeper by
counting on? (from biggest number)
APP statement L1
Add and subtract numbers of objects
to 10

Begin to add by counting on from
the number of objects in the first
set
Recognise that addition can be done
in any order
Numbers and Patterns Phase 6 –
learning focus
Relate addition to counting on and
recognise that addition can be done
in any order
Begin to find out how many have
been removed from a larger group of
objects by counting up from a
number
Use practical and informal written
methods to support the addition of a
one-digit or two-digit number
Count them to see how many you
picked up.
Put all the buttons into a pot. How
many buttons are in the pot?
Put another button in the pot. How
many buttons are in the pot now?
Look at this coat hanger. I’m going to
put 4 red pegs on the hanger first
then 3 blue pegs? How many pegs
altogether?
Write down the number sentence for
me.
Now I am going to turn the coat
hanger around.
The blue pegs now come first. How
many blue pegs? How many red
pegs?
How many altogether?
Write that down.
What do you notice?
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20 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Calculating
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/activities to test
knowledge further
APP statement L1
Can you put 5 red pegs on the coat
hanger and 4 blue pegs? How many
altogether? Now turn the coat hanger
around. How many blue pegs, how
many red pegs? How many
altogether? What do you notice?
Further questions/ activities to
test knowledge further
Solve addition/subtraction problems
involving up to 10 objects e.g. given
a number work out ‘how many more
to make.....’
In practical activities and discussion
begin to use the vocabulary involved
in adding and subtracting
My bag has four apples in it, let’s
count them.
If I put one apple back in my bag
how many apples are left on the
table?

choose which of given pairs of
numbers add to a given total

solve measuring problems such
as ‘how many balance with...’

solve problems involving 1p or
£1 coins
Understand subtraction as ‘take
away’ and find a ‘difference’ by
counting up
APP statement L1
Here is an egg box. It has four eggs
in it.
Understand subtraction as ‘taking
away’ objects from a set and finding
how many are left
How many more eggs do I need to fill
the egg box?
Numbers and Patterns Phase 6 –
learning focus
If I take two blocks away from this
pile, how many blocks will be left?
Understand subtraction as ‘take
away’ and find a ‘difference’ by
counting up
Also use sum, total, two more, two
less, is the same as, gone etc
See also column 4 above – derive
and recall numbers that make 10.
Use counters on a number track to
help you with these questions.
Bilal has seven computer games.
Anya has two fewer. How many
computer games does Anya have?
There are 11 birds on a roof, six fly
away. How many are left?
Here is the number 3 on the number
line and here is the number 8. What
is the difference between the two
numbers?
How many numbers in between?
If Bobby is 6 years old and his sister
is four years old what is the
difference in their ages?
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21 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Calculating
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/activities to test
knowledge further
Remove a smaller number from a
larger and find how many are left by
counting back from the larger
number
Write a number sentence for each of
these problems. Billy buys a box of
12 eggs. He cooks 4 of them. How
many are still in the box?
Use practical and informal written
methods to support the subtraction of
a one digit or two digit number and a
multiple of ten from a two digit
number
Sam has five counters in one hand
and six in the other.
Count repeated groups of the same
size
Further questions/ activities to
test knowledge further
This row of four shelves has three
books on each shelf.
How many books are there on the
shelves?
How many counters does he have
altogether
Solve practical problems that involve
combining groups of 2, 5 or 10, or
sharing into equal groups
See also section above count in
two’s and fives
Share objects into equal groups and
count how many in each group
Can you share these pencils out into
these three jars?
Is there the same number of pencils
in each jar?
How many are there in each jar?
Use the vocabulary related to
addition and subtraction and symbols
to describe and record addition and
subtraction number sentences
APP statement L1
Record their work e.g. record their
work with objects, pictures or
diagrams
Begin to use the symbols ‘+’ and ‘=’
to record additions
There are 5 caterpillars on a leaf and
a bird comes along and eats 2 of
them.
How many caterpillars are left on the
leaf?
Draw a picture to show how you
solved the problem.
Can you write a number sentence to
match it?
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22 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Calculating
Initial assessment: expectations by the end of the Foundation Stage
Expectations by the end of Year 1
Foundation stage objectives
Primary Framework objectives
Further questions/ activities to
test knowledge further
Further questions/activities to test
knowledge further
Can you write a number sentence for
this problem?
There are eight pennies in this purse.
I spend 5p. How much money will be
left?
I want to save 10p.
How much more money do I need?
Range of vocabulary to incorporate into questions and/or check children’s understanding
number, count, pattern, forward, backward, next, before, between, about, more/more than, less/less than, most ,least, sequence, order, count on, count
back, add together, total, take away, how many/altogether? How many left?
first, second, third, fourth, fifth...
the teens numbers; the ty’s numbers; zero, number names to 100.
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23 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Appendix B
Year 1 project: Identifying and overcoming barriers to mathematics learning in Year 1
Observation sheet
Use this observation sheet as your starting point for assessing children’s understanding of number
Notes on using these questions as starting points:

questions have been included in this document to help you to assess children’s understanding in some important areas of counting,
recording numbers and calculation. You will need to prioritise key areas to assess for each child/pair of children

the questions suggested provide starting points from which to engage children in discussion. You will need to adapt questions and add in
further questions in the light of children’s responses

in particular, throughout the assessment you should incorporate:
— prompting questions in order to encourage children to engage in discussion and to help to identify a secure base of understanding
— probing questions in order to assess children’s confidence and their depth of knowledge and understanding
— promoting questions that extend the ideas being discussed to determine children’s breadth of knowledge and understanding.

School:

Child:
(Include gender, age including months)
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24 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Focus/activity
Child’s response
(Please record specific difficulties the child experiences)
Do you know this song? Can you sing it with me?
One, two, three, four, five once I caught a fish alive,
Six, seven, eight, nine, ten, then I put it back again.
Why did you let it go?
Because it bit my finger so
Which finger did it bite?
This little finger on my right
What do we call those words that we have been singing? (e.g. one, two,
three....)
When and how do you use numbers? a) in school b) out of school?
Can you count up to 10 with me? Can you count backwards from 10 with me.
Can you do this on your own?
Extend the count to 20 where possible to test out individual children’s
knowledge of numbers beyond 10 (extension questions to 20 are set out in
brackets for each question below)
I’m going to start counting can you carry on when I stop?
I’m going to count. When I stop can you tell me which number comes next
1, 2, 3......4, 5, 6.....? (Then 9, 10, 11.......? 15, 16......? )
NB Try counting in twos with the odd and even numbers: 1, 3, 5, 7… and 2,
4, 6 …. (Then 11, 13, 15........)
Can you point to the number 1 on this number line, can you point to the
number 7? (Can you point to number 11, 13 etc.)
Now point to 3 and count on to 8 pointing to each number in turn then to 10).
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25 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Focus/activity
Child’s response
(Please record specific difficulties the child experiences)
(Point to 14 and count on to 16 then to 20)
Can you point to 6 (then 10) and count back to 1?
(Can you point to 17 and count back to 10)
Point to 4. Which number comes next? Which number comes before?
Point to 5. Which number is one more than 5? Which number is one less
than 5?
(Point to 12 which number comes next? Which number comes before?
Point to 15. Which number is one more than? Which number is one less
than?)
Here are some counters/beads. Count them out for me (4 say)? Add one
more to the pile, how many do we have now? Can you add two more etc?
(Extend – can you count out 12 counters/beads? Add one more how many
do we have now?)
Here are some counters/beads. Count them out for me (6 say)? How many
do we have if we take one away? How many do we have if we subtract two
from six?
(Extend – can you count out 15? How many do we have if we take one
away? How many do we have if we subtract one?)
Here are some counters/beads. Give me 7 counters/beads? How many will I
have if you give me one less? How many will I have if you give me 2 less?
(Give me 17 counters/beads? How many will I have if you give me one less?
How many will I have if you give me 2 less?)
Can you tell me what this number is? Choose number cards in the range
0–5, then 5–10. (Extend using number cards 10–20)
Can you count out that number of counters/beads.?
There are three counters/beads in this box. How do we know there are
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26 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Focus/activity
Child’s response
(Please record specific difficulties the child experiences)
three, how can we check?
Now I’m going to hide them. How many counters/beads are hidden?
Can you count how many counters/beads are in this box? (5)
Now we will hide them. How many counters/beads are hidden?
(Extend activity with 13 counters/beads in the box)
Can you count how many blocks are in this line?
Now I’m going to mix them up (re-arrange them). Can you tell me how many
blocks there are now? How do you know there are (7) blocks? Do you need
to count them again?
Can you mix up the blocks?
How many blocks do you think there are now? How do you know?
(Extend activity using 14 blocks)
Can you count out how many counters/beads are in this box? (Within
number range 1– 5 then to 10.)
Can you add one more?
How many will we have then?
If you take one away how many will you have then?
(Extend the activity selecting a number in the 10–15 range then 15–20)
How many red counters/beads are in this group?
Look at the blue group
How many blue counters/beads do you think there are?
Do you think there are more (less) blue .........?
Can you count out and check?
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27 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Here are some number cards can you put the cards in order starting at 1? (15 then to 10 then to 20)
Which number comes first? Which number comes last?
Here are 3 number cards (2, 4, 5 then 12, 15, 17) etc. Can you put these
cards in order?
Which number/s are missing?
Here are five coloured counters in a line. Which colour comes second in the
line, third etc?
Can you make a line with the counters? What colour counter have you put
first (third etc) in your line? (Alternatively could use coat hanger and pegs)
Here are some pairs of socks? How many socks are there altogether?
How many socks are in a pair?
Can you count in two’s and tell me how many pairs of socks we have?
Here are some pencils (3) for the children on this table. How many pencils
are there?
Are there enough pencils for all four children to have one each?
How many more pencils do we need?
(Here are (7) pencils. Are there enough for all four children to have two
each? How many more pencils do we need?)
We will count 5 pennies put them in this purse and close it. How many
pennies do we have in the purse? Now I’m going to give you 3 more pennies.
How many pennies do we have altogether?
How can you check?
(Extend to 12 pennies and 3 more)
Show me 3 fingers on your right hand. Show me two fingers on your left
hand.
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28 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
How many fingers showing altogether?
(I’m showing you ten fingers. Can you show me four more?
How many fingers are we holding up altogether?)
Here are 6 counters/beads can you take away 2. How many do you have
left? (Here are 15 counters can you take away 2. How many do you have
left?)
Here are 5 pennies. You spend 3 pennies on a toy. How much do you have
left? (Here are 16 pennies you spend four pennies how many do you have
left?)
Count 6 counters/beads into an open box. Take some out - 4. How many are
still in the box? (Count 14, take out 4, how many are still in the box?)
Give 4 children in the group 3 pennies each then pose the question. There
are 4 children on our table each child has 3 pennies how many pennies do
we have altogether? Can you count to find out? (Extend to 5 pennies each)
Here are 3 pencil pots, can you share these 9 pencils into the three pots?
How many are there in each pot? Does each pot have the same number? Do
you have any left? (Extend to 3 pencil pots and 15 pencils)
Incorporate the following vocabulary into your questioning whenever possible
number, count, pattern, forward, backward, next, before, between, about,
more/more than, less/less than, most ,least, how many more to make,
sequence, order, count on, count back,
add together, total, take away, how many/altogether? How many left? difference between
first, second, third, fourth, fifth....
the teens numbers; the ty’s numbers; zero, number names to 100.
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29 of 29 Identifying and overcoming barriers to mathematics learning in Year 1
Resources
Set of 1–20 digit cards
Number lines 1–10 and 1–20100 square
Empty boxes, Coloured counters
Sets of coloured beads, bricks, buttons, small toys etc. for counting out groups of objects
Plastic coat hanger with sets of coloured pegs
Pencils and pencil pots
Purse and up to 20 x 1p coins
Pairs of socks or gloves
4 small cars
paper for folding and/or recording
© Crown copyright 2011