Identifying and overcoming barriers to mathematics...

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

© Crown copyright 2011


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.



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


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


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.



Learning target: Recognise how counting on and counting back can provide efficient ways to add and subtract small amounts


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


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


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



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.



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 2-digit 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.



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.



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.




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.


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.



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

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.

Count reliably at least 20 objects, recognising that when re-arranged the number of objects stays the same

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

Extend the count up to and beyond 20 to 100 using similar activities to the previous column.

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?







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

Numbers and Patterns Phase 5 – learning focus

Instantly recognise, without counting, organised and random arrangements of small numbers of objects

Count the toys in this tray. I am going to cover the tray with this cloth.

How many toys are under the cloth? Can you tell me why?

I have removed the cloth do we need to count how many there are again?

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?

Compare and order numbers, using the related vocabulary

APP statement L1

Order numbers to 10

count back to zero

place 1–10 in ascending order

point to first, second etc. in a line

begin to count in two’s

Use the (=) sign

‘The numbers in this count are mixed up. Can you put them in order?’

4, 8, 2, 10, 1, 3, 0

Then:

18, 16, 17, 15, 13, 14, 12, 10, 11

Here is a row of four coloured counters. Which coloured counter is first, third etc?

(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







Check understanding

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

that there is no need to count when the number can be recognised without counting.

15 7 5 12

Here are some cards. Can you make me a number sentence using these cards?

4+1 8-1

7+1`

7-1

10-1 6

5

8 9

7 =

Count reliably up to 10 everyday objects (EOY)

APP statement L1

Count up to 10 objects e.g. estimate and check a number


Count reliably any arrangement of up to 10 objects

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.

I am going to put them in a circle like this. Count them for me again.

Now we will jumble them up can you count for me again?

Read and write numerals from 0 to 20, then beyond


Count large groups of objects by using efficient strategies

Count forwards and backwards within the number sequence 0 to 100

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.







1, 2, 3... what number comes next?

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.

Say the number that comes before and after a given number within the number sequence 0 to 100

Recognise, say and identify numerals 0 to 100

APP statement L1

Read, write numbers to 10 – perhaps with some reversal

Use knowledge of place value to position these numbers on a number track and number line

Look at this grid. Point to 16, 20 and twelve.

13 14

16

12

17 19

20

18

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


Estimate a number of objects that can be checked by counting

Count the number of red blocks in this tray.

Look at the blue blocks in this tray.

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.







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?


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

? 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

APP statement L1

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

Can you fold this piece of paper into two?

Can you cut along the fold so that you have two pieces?

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?







APP statement L1

Order numbers to 10

point to first, second etc. in a line

Check knowledge of: odd, even, every other

Similar activity and questions for say tubs with five marbles in each.

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)

How many are left?

What do you notice?

Check children’s knowledge and understanding that a half is one of two equal parts.



Knowing and using number facts

Initial assessment: expectations by the end of the Foundation Stage Expectations by the end of Year 1



Observe number relationships and patterns in the environment and use these to derive facts

How many toes do you have on this foot?

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?

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.

Derive and recall all pairs of numbers with a total of 10 and addition facts for totals to at least 5

Work out the corresponding subtraction facts

I have 4 counters how many more do I need to make a total of 10?

Can you tell me two numbers that add up to 10?

Can you tell me two more numbers that total 10?

How many do I need to add to 2 to make 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?

Find one more or one less than a number from 1 to 10

APP statement L1

Order numbers to 10 – say what number comes next, is one more/less

Can you put in order this set of cards? What number should come first? What number should come next? Which number comes last?

Are there any numbers which should come in between?

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


Count forwards and backwards in twos, fives and tens

See column 2 above for activities



Knowing and using number facts




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


Find the total by combining two groups where one group is screened (seen and then hidden) and counting on

Make two groups of objects up to a total of 10.

How many objects are in this set?

How many objects altogether?

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)

Recall the doubles of all numbers to at least 10

Begin to know some addition facts

e.g. doubles of numbers to double 5

How many fingers do you have on one hand? How many fingers do you have on two hands?

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?



Calculating




Begin to relate addition to combining two groups of objects and subtraction to ‘taking away’

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


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

Show me three fingers on your right hand. Show me two fingers on your right hand? How many fingers showing altogether?

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)

Relate addition to counting on

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


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

Can you pick up a handful of large buttons and put them on the table.

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?



Calculating




APP statement L1

Solve addition/subtraction problems involving up to 10 objects e.g. given a number work out ‘how many more to make.....’

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

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?

See also column 4 above – derive and recall numbers that make 10.

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?

Here is an egg box. It has four eggs in it.

How many more eggs do I need to fill the egg box?

If I take two blocks away from this pile, how many blocks will be left?

Also use sum, total, two more, two less, is the same as, gone etc

Understand subtraction as ‘take away’ and find a ‘difference’ by counting up

APP statement L1

Understand subtraction as ‘taking away’ objects from a set and finding how many are left


Understand subtraction as ‘take away’ and find a ‘difference’ by counting up

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?



Calculating




Remove a smaller number from a larger and find how many are left by counting back from the larger number

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

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?

Sam has five counters in one hand and six in the other.

How many counters does he have altogether

Count repeated groups of the same size

This row of four shelves has three books on each shelf.

How many books are there on the shelves?

See also section above count in two’s and fives

Solve practical problems that involve combining groups of 2, 5 or 10, or sharing into equal groups

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?



Calculating




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.



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)



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).



Focus/activity

Child’s response


(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



Focus/activity

Child’s response


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?



Here are some number cards can you put the cards in order starting at 1? (1- 5 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.



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.




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

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