Supporting children with gaps in their mathematical understanding

PrimaryNational Strategy

Supporting children with gaps intheir mathematical understandingWave 3 mathematics

Using the pack

Acknowledgements

Many people have contributed to this pack. We want to acknowledge theircontribution, thanking particularly the teachers, teaching assistants, mathematicsconsultants and other LEA staff who provided a great deal of useful feedback.

Disclaimer

The Department for Education and Skills wishes to make it clear that theDepartment and its agents accept no responsibility for the actual content of anymaterials suggested as information sources in this document, whether these are inthe form of printed publications or on a website.

In these materials icons, logos, software products and websites are used forcontextual and practical reasons. Their use should not be interpreted as anendorsement of particular companies or their products.

The websites referred to in these materials existed at the time of going to print.Users should check all website references carefully to see if they have changed andsubstitute other references where appropriate.

© Crown copyright 2005 Primary National Strategy I 2Wave 3 Mathematics I Using the pack DfES 1165-2005

Contents

Introduction 6

Three ‘waves’ 6

Introducing the Primary National Strategy Wave 3 mathematics materials 6

Primary National Strategy Wave 3 mathematics: purpose and rationale 7

The significance of effective Wave 3 provision for children with 7mathematical difficulties

– Messages from the research review

The design of the Primary National Strategy Wave 3 mathematics 8materials

– Guiding principles

– Mathematical themes

– Assessment overview

– Details of materials’ structure

Management guidance 13

Whole school 13

Leadership and management 13

Inclusion 14

– Circles of inclusion

School self-evaluation of Wave 3 provision 18

– Quantitative self-evaluation

– Qualitative self-evaluation

Classroom 19

Processes 20

– Flexibility to meet identified needs

– Using the tracking charts and teaching materials

Organisation 21

– Intervals between teaching sessions

– Resources and their management

Frequently asked questions 23

Continuing professional development 26

Outline for a staff meeting

Objectives

Key messages

Resources


Activity 1: Getting to know the Primary National Strategy Wave 3 27mathematics materials

Activity 2: Understanding the purpose of the tracking children’s 27learning charts

Activity 3: Focusing on the teaching activities 28

Activity 4: Assessment and planning process 28

Activity 5: Developing Wave 3 mathematics in our school 28

Appendices

Appendix 1:Tracking children’s learning chart – addition and subtraction 30

Appendix 2:Tracking children’s learning chart – multiplication and division 40

Appendix 3: Handouts for staff meeting 52

– Handout 1: Structure of Wave 3 mathematics materials 53

– Handout 2: Mathematical themes and assessment approaches 55in the Wave 3 materials

– Handout 3: Assessment and planning process 56

Appendix 4: Further references 57

Appendix 5: Resources 58

– Equipment

– Suggested Interactive Teaching Programs


Supporting children with gaps in theirmathematical understandingWave 3 mathematics

Using the pack

Aims

The aims of the Primary National Strategy Wave 3 mathematics pack are:

• to develop practice in supporting children’s mathematical development and self-confidence by providing a suggested model for Wave 3 mathematicsintervention;

• to support the identification of specific areas of mathematics that can preventchildren achieving expected levels of progress;

• to increase children’s rates of progress by using targeted approaches to tacklefundamental errors and misconceptions;

• to reduce the proportion of children achieving below level 3 in mathematics bythe end of Key Stage 2.


Introduction

Three ‘waves’

Provision for effective mathematics learning and teaching can be described in termsof three ‘waves’ of intervention.

Wave 1 The effective inclusion of all children in high quality learning andteaching of mathematics in the daily mathematics lesson.

Wave 2 Additional time-limited provision in the form of small-groupintervention to accelerate progress and enable children to workat age-related expectations.

Wave 3 Additional time-limited provision to enhance the progress ofidentified children where Waves 1 and 2 are not, on their own,having the desired effect. This will involve focused teachingactivities which tackle fundamental errors and misconceptionsthat are preventing progress.

Introducing the Primary National Strategy Wave 3mathematics materials

These materials have been developed during a Wave 3 mathematics pilot with 27LEAs. Feedback has influenced the revision of the teaching materials and theirpresentation in the pack.

The materials:

• are aimed mainly at Key Stage 2 children, having been designed with ageappropriate contexts and approaches;

• follow the principles for successful Wave 3 intervention that have been identifiedby research;

• aim to increase children’s rate of progress by providing focused teachingactivities which tackle fundamental errors and misconceptions;

• are applicable to any child who, for any reason, demonstrates fundamentalerrors and misconceptions;

• focus on the most commonly occurring types of mathematical difficulties withnumber and calculation;

• are not intended to be worked through from start to finish as a self-containedprogramme;

• provide a model that can be used by any adult working with a child who hasdemonstrated a need for a Wave 3 intervention.


The pack contains:

• Two sets of A4 booklets, one focusing on common errors/misconceptions inaddition and subtraction and the second on common errors/misconceptions inmultiplication and division.

In the booklets are teaching materials referenced by year group to the NationalNumeracy Strategy Framework for teaching mathematics Key objectives.

• An A4 book, Resources and index of games.

In this book are lists of mathematics equipment and everyday materialsreferenced in the A4 booklets, photocopiable resource sheets and an index ofgames contained in the A4 booklets.

• This A4 book, Using the pack. Within this book are:

– management guidance (whole-school and classroom);

– charts for tracking children’s learning in addition and subtraction, andmultiplication and division;

– a professional development session to introduce Wave 3 mathematicssupport, with particular reference to the use of the Primary National StrategyWave 3 mathematics materials.

• An interactive CD-ROM providing direct access from electronic versions of thetracking charts to the teaching materials in pdf and Word document formats.This enables the teaching materials to be easily adapted.

The Primary National Strategy Wave 3 mathematicsmaterials: purpose and rationale

This section considers messages from research that indicate the need for, andstructure of, effective Wave 3 provision.

The significance of effective Wave 3 provision for children withmathematical difficulties

Although in 2004 there was some improvement in the proportion of childrenachieving below level 3 in mathematics by the end of Key Stage 2, this proportionhas not changed significantly over the last four years. Research shows that targetedinterventions in mathematics can have a significant impact on children’sperformance and self-confidence.

A research review of what works for children with mathematical difficulties wascommissioned by DfES and published in 20041.

Messages from the research review:

The research review suggests that mathematical difficulties:

• are common, often quite specific, and show considerable individual variations;

• are equally common in boys and girls, in contrast to language and literacydifficulties which are more common in boys;


1 What works for children with mathematical difficulties? (DfES research report 554), available from DfES

publications (tel:0845 60 222 60) or can be downloaded from the website at www.dfes.gov.uk/research.

• can take several forms. The causes for such difficulties are varied and include,for example, individual characteristics, inadequate or inappropriate teaching,absence from school resulting in gaps in mathematics learning, lack ofpreschool home experience with mathematical activities and language;

and that:

• children with mathematical difficulties typically combine significant strengths withspecific weaknesses;

• some children have particular difficulties with the language of mathematics;

• difficulty in remembering number facts is a very common component ofarithmetical difficulties, often associated with dyslexia;

• some children can remember many number facts, but seem to lack strategies(including suitable counting strategies) for working out calculations when they donot know the answer; other children are the reverse of this;

• other common areas of difficulty include word-problem solving, representation ofplace value and the ability to solve multi-step arithmetic problems.

The research review endorses the following points:

• Children’s difficulties with calculation are highly susceptible to intervention. Theseinterventions can take place successfully at any time and can make an impact.

• Individualised work with children who are falling behind in number andcalculation can have a significant impact on their performance.

• The amount of time given to such individualised work does not, in many cases,need to be very large to be effective. Short but regular interventions ofindividualised work may bring a child to the point where they can profit muchbetter from the whole-class teaching that they receive.

• It is important to find out what specific strengths and weaknesses an individualchild has and to investigate particular misconceptions and incorrect strategies

• Interventions should ideally be targeted towards an individual child’s particulardifficulties. If they are so targeted, most children will not need very intensiveinterventions.

The design of the Primary National Strategy Wave 3 mathematicsmaterials

Guiding principles

The materials evolved as feedback was provided through pilot LEAs and inresponse to relevant research.

The guiding principles informing the design are:

• flexibility so that teachers can adapt them;

• sharing the purpose of each activity with the child to encourage reflection on,and ownership of, learning;

• highlighting and modelling key vocabulary throughout;

• teaching activities finishing with related activities for whole-class use, whereappropriate;

• use of a variety of images and models, aiming to include some the child may nothave met before;


• linking mathematics to familiar and relevant contexts;

• integrating and exemplifying mathematical problem solving;

• inclusion of games among teaching activities, possibly for sharing with parentsand carers.

Mathematical themes

The following are fundamental to the approach.

• Using and applying mathematics has been integrated. Often there are severalopportunities for problem-solving within one activity, but in each, one particularopportunity has been highlighted. Aspects such as the following are incorporated:

– encouraging children to discuss and explain in order to supportdevelopment of their mathematical reasoning;

– opportunities for children to make choices are woven into the activities, forexample selecting numbers and devising calculations;

– encouraging children’s own recording to communicate mathematicalthinking, focusing on efficiency;

– opportunities for evaluating the efficiency of methods of calculation.

• Development is emphasised and key vocabulary is listed in each activity. It isimportant for adults to use correct mathematical language. To facilitate this,examples are given in words, for example, ‘725 � 3’ is accompanied by whatthe adult could say to the child: ‘Seven hundred and twenty-five multiplied by 3’.

• There is a focus on progression in counting from the earliest stages through toYear 6 to support the development of secure counting skills.

• Throughout the materials, there is emphasis on the process of estimating first,then calculating and then checking. This is denoted by the following icon:

• Decimals are addressed within meaningful contexts, for example via displays ona calculator and as a part of measure.

• Structured equipment and everyday materials are used to model mathematicalconcepts, supporting children’s mathematical thinking and development ofmental imagery. Some links to ICT resources such as the Primary NationalStrategy Interactive Teaching Programs (ITPs) are included.

• A wide range of resources is used in the teaching sessions. Teachers’ selectionof these to suit the needs of their children is an important part of adapting thematerials.

Assessment overview

The materials reflect best practice in assessment for learning as a key tool for raisingachievement through:

• use of questions to elicit information about children’s understanding;

• sharing the purpose of the activity with the learners;

• encouraging children’s reflection on their learning and identification forthemselves of possible next steps.


Details of the materials’ structure

The materials focus on a selection of the key objectives in National NumeracyStrategy Framework for teaching mathematics, namely, addition and subtraction,and multiplication and division objectives. Research shows that children’s difficultieswith calculation are highly susceptible to intervention and that individualised workwith children who are falling behind in number and calculation has a significantinfluence on their performance.

In order to exemplify progression in calculation, Reception, Year 2, Year 4and Year 6 have been chosen as representative milestones. Under each yeargroup heading, associated knowledge and skills that contribute to understanding ofthe year group key objective are listed. (See first column of tracking chart.)

The whole primary age range is represented in the progression in the chart. The yeargroup labels provide a convenient link to the National Numeracy StrategyFramework for teaching mathematics progression in number and calculation. Asthey use the chart, teachers will need to ‘track back’ to find the error ormisconception appropriate for the child, irrespective of the year group to whichit is attributed in the progression.

Tracking charts

1 Key objective.

2 This column lists associated knowledge and skills that contribute tounderstanding of the key objective.

3 Common errors and misconceptions linked to specific knowledge and skills arelisted to support diagnosis of children’s difficulties.

4 Questions in this column can be used to help the teacher decide where thechild’s difficulties lie.

5 Examples of the types of teaching activity in the A4 booklets (see below).

6 This column provides ideas to develop when the child has improved theirunderstanding of the identified difficulty. The teacher can make use of theseideas to consolidate understanding and extend thinking.

7 Code referencing to an A4 teaching unit.

Six essential areas to support a child’s learning in calculation are the basis of thePrimary National Strategy Using models and images to support mathematicsteaching and learning in Years 1 to 3 (DfES 0508-2003 GCDI) and the focus on


Year 6 key objective Carry out column addition and subtraction of numbers involving decimals (NNS Framework for teaching mathematics, Supplement of Examples, Section 6,pages 49, 51)

Associated knowledge and skills

Apply knowledge of the number system to enable efficient counting of a large number of objects.

Add and subtract multiples of ten, a hundred and a thousand.1 Y6

Give an estimate by rounding, to determine whether the answer to a calculation is sensible.2 Y6

Tracking children’s learning through the NNS Framework for teaching mathematics (addition and subtraction)

Errors andmisconceptions

Has inefficient countingstrategies and/orinsecure understandingof the number system. 1 Y6 +/–

Rounding inaccurately,particularly whendecimals are involved,and having little senseof the size of thenumbers involved.2 Y6 +/–

Questions to identify errors andmisconceptions

Imagine you have a money box containing 2pand 1p coins. What do you think would be agood way to count these quickly to find out howmuch money there is?

What is 60 + 20? … 60 + 30? … 60 + 40?What changed when you found 60 + 40?

What is 40 + 40? … 400 + 400?Which answer is the larger?How is the calculation 40 + 400 + 4000 differentfrom the others?

What is 60 – 20? … 600 – 200? … 6000 – 2000?Explain how you worked these out.What is 6000 – 200? … 6000 – 20?

Is 26 nearer to 20 or 30?Is 271 nearer 270 or 280?Is 1.8 nearer to 1 or 2?Draw a sketch to illustrate your answer andexplain how you know.

Teaching to address the errorsand misconceptions

Practical opportunities to developefficient counting strategies for arange of objects, for example coins,cubes, conkers, collectable cards,stickers.

Count forwards and backwards intens, hundreds and thousands fromdifferent starting points, includingstarting numbers that are notmultiples of ten or a hundred. Usean empty number line to supportthis development.

Order multiples of a hundred and athousand.

Use number squares and/ornumber lines to consider the orderand comparative value of numbersto support rounding.

Next steps in moving towards thekey objective

Carry out simple calculations thatinvolve crossing the boundary fromhundreds to one thousand and viceversa, supported by an emptynumber line and extending this to avisualised image to develop mentalcalculation.

Consider pairs of items from acatalogue and ask child to estimatewhether a £10 (or £20, etc.) notewould be enough to buy both theitems?

15

6

43

2

7

these is reinforced in the Wave 3 mathematics pack.

These areas are:

• ordering numbers;

• counting on and back;

• partitioning and recombining;

• addition and subtraction facts within 20 (not just those that total 20);

• understanding of the four operations;

• problem-solving strategies.

A4 booklets – teaching units

The structure of each booklet is as follows:

• focus error/misconception;

• opening teaching activity addressing error/misconception;

• a number of Spotlights (short focused teaching activities from which to select);

• final Spotlight, which includes assessment opportunities, often encompassed ina game, key vocabulary checklist, and intended learning outcomes list.

Opening teaching activity Spotlight


© Crown copyright 2005 Primary National Strategy I 1DfES 1128-2005

Wave 3 m

athematics ad

ditio

n and sub

traction

Tracking b

ack to Year 4

1 Y4 �

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Has insecure understanding of the structure of the numbersystem, resulting in addition and subtraction errors anddifficulty with estimating

Opportunity for: developing mental images

Resources Key vocabulary

● Two sets of number cards 0–9 (Resource sheet 1)

● 100-square

● Place value (arrow) cards

● Long number line (or masking tape)

● Sticky notes

● Bundles of straws or other Base 10 equipment

● 1p coins for counting

Teaching activity Time 15–20 minutes

‘We are going to be working with numbers today, deciding which numbers are larger or smaller, and we

are going to order them on a long number line. This work will help you with estimating and with getting

more calculations correct.’

Lay out the two sets of number cards 0–9 on the table.

Can you make the numbers forty-three and thirty-four?

Support the child to make:

Which is the larger number, forty-three or thirty-four?

If the child knows which is larger, move on.

Display a long number line, say up to four hundred. (You could stick masking tape on the floor or wall

and write the numbers along it. You will need to use the number line throughout this set of activities.)

Ask the child to position the numbers on the line with sticky notes or paper.

If the child doesn’t know which is larger, you could count with the child up to 100 on a 100-square,

pointing out the thirties and the forties.

Then change the numbers to 40 and 30.

Count in tens with the child on the 100-square, establishing that if you had forty sweets, you

would have more than thirty sweets because forty is more than thirty.

Then you will need to make some more two-digit numbers and repeat the activity before you move on.

Note: If the child seems to have problems crossing boundaries, see 1 Y2 +/–. If the child seems

unsure of numbers, you might want to check that the child can count a large pile of coins so that

you can assess their counting skills (see also 1 YR +/–).

?

?

digit

larger/largest

smaller/smallest

more than

less than

units

ones/tens/hundreds/

thousands

column order

estimate

guess

nearer/nearest

before

after

rounding to the nearest

ten/hundred

4 3 3 4

Spotlight 2Has insecure understanding of the structure of the number system, resulting in addition and

subtraction errors and difficulty with estimating

Opportunity for: reasoning about numbers

Place value on the calculator Time 10–15 minutes

Resources

● Calculator each

● Large book or other screen

● Number line

Teaching activity

‘Today we are going to do some more work on place value and where numbers go on the number line.

We will particularly be thinking about any zeros that we see. So if you see a zero, tell me and I will

record the number for later.’

Can you see any numbers with zeros on the number line?

Follow what the child says, giving experience of reading numbers such as three hundred and six.

Prop up the large book so that you can enter numbers secretly.

Now I want you to key in three hundred and forty-two

into your calculator and I will do the same on mine.

Does yours look the same as mine?

Then you add a one-digit number, such as 3, and show the child your calculator.

What do you think I did to three hundred and forty-two to get to that number?

Can you make your number the same as mine in one move? (Meaning an operation key, a

number and the equals key.)

What did you do to make your number the same as mine?

Repeat until the child understands how to add and subtract single-digit numbers.

Record any numbers you use that have a zero in them.

Then subtract all the tens from the number secretly.

Show the child your screen.

What subtraction will take away the four? (Signal to the child that you have a number with a zero

in and record that for later use.)

?

?

?

If the child isn’t sure you will need to do some further adding and subtracting of one-digit numbers.

?

?

?

1 Y4 �

/�


Key vocabulary

digit

larger/largest

smaller/smallest

more than

less than

units

ones/tens/hundreds/

thousands

column

estimate

guess

nearer/nearest

order

before

after


ten/hundred

342

345

305

Error/misconceptionheading

Problem-solvingemphasis

Suggested time

Activity title(for Spotlights)

Key vocabulary

Resources

Teaching activity

Specific icons are used to improve access to the text:


Icons

Questions are incorporated for teachers to select from and addtheir own as appropriate.

Whole-class follow-on activity.

Symbol reminding of the necessity to estimate, calculate, then check.

This variation of the game is harder.

This variation of the game is easier.

12 x 2 = 24 Text within this symbol indicates an opportunity for recording.

Text within a shaded box indicates alternative approachesfor a child who is having difficulty with the activity.

Additional game at the end of some teaching units.

?

Management guidanceThis section outlines management issues connected with Wave 3 mathematicsintervention and provides some case studies to illustrate a range of practicedeveloped during the Primary National Strategy Wave 3 mathematics pilot.

Whole school

This section discusses issues connected with Wave 3 provision in the whole schoolcontext.

Leadership and management

There will be a number of decisions to make in connection with Wave 3 provision tosuit the school’s organisation. Some examples are:

• strategic planning of the coordination of Wave 3 provision;

• responsibility for coordination of Wave 3 provision;

• clarification of roles of the headteacher, deputy head, mathematics coordinator,SENCO, teachers and teaching assistants;

• continuing professional development for staff;

• processes for targeting Wave 3 provision;

• timing of Wave 3 sessions;

• storage and access to Wave 3 materials and resources;

• organisation of children to allow involvement of other children when necessary;

• opportunities for involving parents and carers;

• communication about children’s progress and implications for schools’assessment procedures.

During the pilot, schools evolved a variety of solutions. Some of these are describedin the case studies.

In one pilot school, the teacher started the day ten minutes earlier and workedwith children just for ten minutes; sometimes working with one child, sometimesinvolving others.

Often in pilot schools, teaching assistants were given extra training: sometimesin a centrally organised LEA session, sometimes as part of a school-led activityfor a group of teaching assistants, sometimes by working with teachers sharingideas while looking at video clips from Using models and images to supportmathematics teaching and learning in Years 1 to 3.

One Year 3 class teacher in the pilot occasionally replaced the mental/oralstarter of her mathematics lessons with a whole-class session in the playgroundor hall. She used large number cards, etc. for activities based on the Wave 3mathematics pilot materials.


Inclusion

Understanding the importance of children retaining their entitlement to a dailymathematics lesson, schools in the pilot trialled a wide range of strategies formaking time for Wave 3 provision, for example:

• during registration;

• part of lunch time;

• during afternoon teaching sessions;

• during the mental/oral starter;

• during whole-class mathematics group time.

Principles that schools adhered to were flexibility and the intention that Wave 3provision should be for a short, focused period of time, rather than timetabled weekafter week for the same child.

The materials have been designed to enable teachers to continue Wave 3 activitieswithin a whole-class context. Suitable activities are indicated at the end of manySpotlights by the icon:

A pilot school replanned its staffing budget, increasing the hours worked bysome teaching assistants so they could work in partnership with class teacherson the Wave 3 intervention.

One of the headteachers in a pilot school said that she realised that she neededto be much more involved with what was going on in Wave 3 mathematics. Shesaid that there would need to be some changes to roles as well as the timetablechanges the school had already made to accommodate Wave 3 work. Sherealised that asking two teaching assistants to use the materials to support anNQT had overburdened the teaching assistants, although she was veryimpressed by what they had achieved.

The headteacher realised that the teacher had lost touch with the children withwhom the teaching assistants were working. The headteacher decided totimetable teacher and teaching assistant planning and review sessions duringwhole-school assemblies each week.

During the trialling of the materials, several schools gave extra training toteaching assistants so that they could work one-to-one with the child on theSpotlights the teacher selected. (The extra training was given within the school,and sessions with the LEA numeracy consultant were also organised forteaching assistants from several schools.)

Using teaching assistants in this way was often successful, but headteachers,SENCOs, mathematics coordinators and teachers identified the vital need forthe teacher to work with the child on the opening teaching activity and finalSpotlight in order to assess the progress the child had made and to decide onnext steps.


Games are included in many sets of teaching activities. Some pilot schools usedthese beyond the children targeted for Wave 3 intervention both in school and assupport for parents and carers to continue the focused teaching on particularerrors/misconceptions. Some of the teaching units contain an additional game,indicated by the icon:

An index of all the games in the teaching materials is provided in the Resources andindex of games book.

The materials are not linked to specific types of special educational need: they donot, for example, form a programme for dyslexic or dyscalculic children, or childrenwith dyspraxia, Down’s syndrome or cerebral palsy. They are intended to have abroad application to children with difficulties in cognition and learning, howeverthose difficulties may have arisen.

Circles of inclusion

The different forms of support which children may need with their mathematicallearning are represented on the following diagram.

These circles of inclusion provide a useful model for considering children’s needs inthe context of Wave 3 mathematics provision. The teaching materials are designedto take into account the three different aspects of inclusion as conveyed in thediagram, with the primary focus on the learning objectives and teaching stylescircles.

A teacher working with George, aged 9, observed his confusion with place valuewhen working with place value (arrow) cards. She did some diagnosticassessment with him using questions from the tracking chart. Then she selectedsome spotlights for him to work on with the teaching assistant. The teachingassistant found George ‘resistant’ to some of the activities but she found a placevalue game which he played enthusiastically with another child chosen by theteacher. When the teacher came to assess George’s progress at the end of twoweeks, he could work with the place value cards and said (smiling) ‘I really likemaths now.’ The teacher then decided to focus on George’s mental calculation

In one pilot LEA, the observation was made that children regularly taken onholiday in the first few weeks of September, often make errors relating to placevalue, because they miss that area of teaching each year. Some of thesechildren might be comparatively high achievers. A pilot school pointed this out toparents, but meanwhile implemented Wave 3 intervention to help the childrencatch up.


1 Learning objectives: setting suitable learning challenges enabled by the use ofthe tracking children’s learning charts to support the process of tracking back.

Wherever possible, these should be linked to the class topic and focus on theearlier stages in learning of that topic. Some teachers in the pilot chose Wave 3interventions to act as pre-learning for the targeted children before the topicsbecame part of whole-class learning.

2 Teaching styles: implicitly addressing the needs of visual, aural and kinestheticlearners in the presentation of the teaching activities. Additional scaffoldingopportunities are presented in the shaded boxes within the text.

3 Access: overcoming potential barriers to learning by using practical resourcesand common real-life contexts, but not attempting to cover comprehensively therange of access strategies which different children will need. The table belowsets out some additional access strategies for the more common types ofimpairment and special educational need for which teachers and teachingassistants may need to plan.


The circles of inclusion

Teachingstyles

Learningobjectives

Access

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3

Inclusion


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in to

p rig

ht-h

and

corn

er o

f num

eral

car

ds to

pre

vent

dire

ctio

nal c

onfu

sion

s•

Use

of s

quar

ed p

aper

for

layi

ng o

ut w

ritte

n ca

lcul

atio

ns•

Mat

hem

atic

al s

ymbo

ls p

rese

nted

in d

iffer

ent c

olou

rs –

for

exam

ple,

alw

ays

gree

n fo

r +

, blu

e fo

r x

– to

pre

vent

con

fusi

on b

etw

een

sym

bols

whe

re a

diffe

renc

e in

orie

ntat

ion

is a

ll th

at d

istin

guis

hes

one

from

ano

ther

• U

se o

f pre

-pre

pare

d fo

rmat

s fo

r ca

lcul

atio

ns, g

raph

s an

d ta

bles

• U

se o

f app

ropr

iate

sof

twar

e fo

r re

cord

ing

calc

ulat

ions

, gra

phs

and

tabl

es a

nd fo

r dr

awin

g/m

anip

ulat

ing

shap

es•

Layi

ng r

uler

s or

sca

les

alon

g co

ordi

nate

s w

hen

plot

ting

or re

adin

g th

em•

Teac

hing

the

child

to p

ut v

isua

l or

spat

ial i

nfor

mat

ion

into

ver

bal f

orm

, and

vic

e ve

rsa

• U

sing

non

-slip

mat

ting

and

stic

ky-t

ack

to a

ncho

r re

sour

ces

and

pape

r•

Rec

ordi

ng u

sing

num

eral

car

ds o

r ci

rclin

g nu

mer

als

on n

umbe

r lin

es a

nd g

rids,

rat

her

than

by

writ

ing

Lang

uage

and

com

mun

icat

ion

diffi

culti

es

• B

reak

ing

dow

n in

stru

ctio

ns a

nd e

xpla

natio

ns in

to ‘c

hunk

s’•

Reg

ular

che

cks

for

unde

rsta

ndin

g•

Vis

ual p

rom

pts

and

cues

and

the

oppo

rtun

ity to

man

ipul

ate

phys

ical

reso

urce

s w

hils

t men

tal c

alcu

latio

n qu

estio

ns a

re b

eing

ask

ed•

Jott

ings

to n

ote

inst

ruct

ions

and

eac

h of

the

step

s in

mul

ti-st

ep p

robl

ems

or m

enta

l cal

cula

tions

, to

help

ove

rcom

e pr

oble

ms

in w

orki

ng m

emor

y•

Voca

bula

ry c

hart

s an

d po

cket

glo

ssar

ies

with

mat

hem

atic

al w

ords

and

thei

r m

eani

ng; u

se o

f mne

mon

ics

to a

id re

call

• A

ll w

ords

rela

ted

to a

dditi

on p

rese

nted

in o

ne c

onsi

sten

t col

our.

Acc

ess

stra

teg

ies

for

mat

hem

atic

s


School self-evaluation of Wave 3 provision

Quantitative self-evaluation

Schools may find the following questions helpful in considering the effectiveness oftheir practice for low-attaining children.

• How does the percentage of our children who achieve below level 3 inmathematics at the end of Key Stage 2 or below level 2 at the end of Key Stage1 compare with the national averages and the averages for similar schools (FSMand prior attainment)?

• How do we evaluate our children’s progress from the beginning to the end of aperiod of Wave 3 intervention?

• How do we use data to identify those children who could benefit from Wave 3intervention?

• How does the progress that we achieve for children with low prior attainmentcompare with that achieved nationally/locally?

Qualitative self-evaluation

Schools can compare their own Wave 3 provision with a set of quality guidelinesderived from research and best practice.

‘We found the kinaesthetic experiences the best way to get through to them ...We used things they could touch … We did many activities in the hall on a bigscale so that they were using their bodies … They loved it, especially the games… I loved seeing the smiles on their faces.’

A teaching assistant who took responsibility for resources for Wave 3 mathematics interventions in the school

Comments/actions

Establishing priorities,analysing results andreviewing progress

• is informed by clear expectations andthe tracking of individual children’sprogress

• involves the diagnostic assessment ofchildren’s strengths and weaknesses

• incorporates regular review andassessment of progress as an intrinsicpart of the provision

Key activities Our Wave 3 provision inmathematics:

Continuing to improve thequality of learning andteaching

• is taught and overseen by personnel withappropriate skills and expertise to adaptand tailor teaching to the child’sidentified needs

• builds in assessment for learning as afundamental part of the activity

• ensures close connections betweenthe intervention and the teaching ofthe whole class


Leading intervention andmanaging and deployingresources to meet theneeds of all children

• is led by members of the school’sleadership team who are responsiblefor strategic planning

• is managed by an identified memberof staff who oversees the interventionon a day-to-day basis

• is part of a coherent whole-schoolapproach to the three waves ofintervention

• is based on as early an intervention aspossible

• ensures that a range of age-appropriate interventions are available

• ensures that over time the entitlementof all children to a broad andbalanced curriculum will bemaintained

• establishes regular monitoring andevaluation of the impact of Wave 3mathematics provision

Engaging andcommunicating withchildren and others

• ensures that children are involved inthe assessment of their own learningand progress

• develops children’s capacity to beindependent learners

• develops children’s self-confidenceand image of themselves assuccessful learners of mathematics

• ensures good communication andeffective partnerships between allinvolved in children’s learning,especially parents and carers

Identifying continuingprofessional developmentneeds

• uses an approach for which there isan infrastructure of support for bothteachers and teaching assistantswho are involved

• ensures that all staff understand thewhole-school approach to Wave 3mathematics provision and their rolewithin it

Classroom

This section outlines features of the Wave 3 mathematics materials and theirintended use in the classroom.

Key features of effective Wave 3 interventions as highlighted in the research reviewhave been incorporated in the design of the Wave 3 mathematics pack.

In particular these are:

• a focus on the most commonly occurring types of mathematical difficulties

The materials focus on number and calculation, tackling areas such asunderstanding the structure of number and operations between numbers.Problem-solving is integrated and exemplified in the materials, and opportunitiesare provided for children to develop mathematical vocabulary.

• an individualised approach based on the particular areas the child finds difficult

The materials reflect best practice in assessment for learning and includetracking children’s learning charts that support the identification of the particularknowledge, skills and understanding with which the child needs help.

• relatively small amounts of individualised intervention

The Wave 3 teaching activities provide brief, focused teaching sessions whichmake it possible for the child to benefit more fully from whole-class teaching.Where appropriate, the activities finish with related activities for whole-class usein order to reinforce individual learning and promote inclusive practice.

Processes

This section provides guidance to support effective use of Primary National StrategyWave 3 mathematics teaching materials.

Flexibility to meet identified needs

The materials are intended to be used flexibly but with the expectation thatdecisions about selecting from the materials will be based upon information from atracking children’s learning chart. Once a set of teaching activities related to aparticular error or misconception is identified, it is up to the teacher to decide whichactivities within the set are relevant for a particular child.

Further flexibility is available as teachers annotate and adapt activities. Wordversions of the teaching materials are provided on the CD-ROM to enable easyadaptation. As teachers become familiar with the model presented in thesematerials, they may well choose to use this framework to develop their ownteaching materials for misconceptions and errors not represented in the pack.

Using the tracking charts and teaching materials

Day-to-day assessment is the starting point. Assessment opportunities areembedded in all the teaching materials.

The following flow chart describes the process assumed in the design of thesePrimary National Strategy materials.


‘All activities were seen as being short, sharp and effective. The materials wereeasy to use by all involved – including parents who used some as ‘homework’activities.’

Extract from a report from a pilot school

‘All teachers felt that the children were enthusiastic about the activities becausethey were short and snappy and practical.’

Extract from a report from a pilot school

Organisation

Particular organisational issues that pilot schools tackled in a variety of ways were:

• intervals between teaching sessions;

• resources and their management.

Intervals between teaching sessions

To aid retention and recall it is intended that teaching sessions are organised with asmall interval of time between them. Pilot schools experimented with having anumber of days between Spotlights but found that if the interval was too long,previous understanding was not embedded. The following case studies illustrate thestrategies schools used to achieve effective timing.


Day-to-day assessmentAs part of day-to-dayteaching, errors and

misconceptions noted.

Final Spotlight – a learning check

Teaching session (often a game) designed to support the teacher in assessing progress. Key vocabulary and learning

outcomes provided to assist theprocess.

Tracking chartDiagnostic assessment –

particular error identified by tracking back from actual year group

Second column of tracking chart used to identify closest fit to identified error/misconception, irrespective of the child’s chronological year group. Questions in third column used to

confirm choice. Select appropriatebooklet.

SpotlightsFurther teaching sessions

Activities to reinforce understandingthrough a variety of activities with

supporting models and images andcontexts. Materials include

assessment prompts to encouragethe child’s reflection on learning.

Further Spotlights to be covered inlater teaching sessions to aid

retention and recall.

A4 booklet – first teachingactivity in the unit

Opening teaching sessionOpening session used to

further assess child's understanding and to support

choice of further activitiesfrom the set or to decide to

return to tracking chart.

Final column of tracking chart

Suggestions for teaching activities to support child’s developing understanding and progress

toward achieving key objective.


Resources and their management Each A4 book includes lists of resources for the teaching activities.

These draw on a wide range of resources which can be categorised as:

• mathematics resources available in most schools;

• readily accessible everyday resources;

• resource sheets provided as part of the pack.

In Appendix 5, page 58, there are lists of equipment suggested for the teachingmaterials in the pack.

The A4 booklets include reference to resource sheets provided as part of the pack in the book, Resources and index of games and on the CD-ROM.

Some Interactive Teaching Programs (ITPs) are referenced within the A4 books.Others will provide a very relevant and useful resource to support children’s learningduring Wave 3 sessions. A suggested list is included in Appendix 5, page 59.

ITPs can be downloaded from www.standards.dfes.gov.uk/primary for the latestversions, or from the CD-ROM.

The timetabling varied: some groups worked in early morning sessions or afterlunch. Other groups worked during the mathematics lesson: in the mental/oralstarter time in small groups or as individuals, or sometimes in plenary time.

Another pilot school found the suggested five minutes for the Spotlight sessionsdifficult to manage. They could not timetable the same children for intensivework quite as often as was suggested. They wanted to incorporate as muchrepetition as they could (to give the children a chance to remember importantinformation) so class teachers targeted specific children for early morningsessions to back up the Wave 3 work that the SENCO and teaching assistantswere also doing with those individuals and groups.

During the pilot study one school tried to stick as closely as they could to thesuggested pilot study timings but found that because of the demands of bothtimetabling and staffing, they just had to make the best ‘fit’ they could. At firsttheir pattern for Wave 3 interventions was to work with the child on Tuesday,Wednesday and Thursday. However, they found retention so poor that theychanged the whole school timetable (assemblies and so on) so that sessionscould be on Monday, Wednesday and Friday. They thought that this improvedretention and they were starting to send some activities home over theweekend, having recently begun to involve the parents of children involved.


Frequently asked questions

Should I only use the materials with one child at a time?

Although the materials are intended to be used with targeted children identified ashaving specific errors and misconceptions, the materials are flexible. In pilot schools,sometimes they were used with small groups where children had common needs.

A deputy head was in charge of Wave 3 mathematics interventions in herschool. She kept the resources in her room and involved the mathematicscoordinator and the SENCO in reviewing the materials and deciding whichchildren would benefit from extra help.

Two children from different classes were identified for one-to-one support, andthese children were withdrawn on average about twice a week, individually, afterthe whole-class mental mathematics start to the lesson. The child worked withthe deputy head for about 20 minutes of group time, and then returned to theclass for the plenary session. Sometimes, though, this proved confusing andpotentially disruptive, so the child did their own plenary with the deputy head.

At other times, in some of the classes, it was decided that the deputy headwould work with a group of children, using Wave 3 materials but keeping thechildren within the class lesson.

One teacher said she used some of the activities as a whole-class mental/oralstarter when she felt things weren’t going too well with a specific area. Anothersaid she felt some activities were useful in reaffirming or refining certain skills andconcepts.

In other year groups different children were highlighted at different times – if amisconception was identified, Wave 3 would be looked at to see if any activitieswould be of use or support. This meant that it was never regularly the samechildren – children were chosen as and when appropriate.

?

One pilot school bought a large plastic box to store resources and this was keptcentrally in school, beside the teaching materials. Resources were chosencarefully to promote a multi-sensory approach to learning; it was hoped thatusing colour, visual models, movement and sound would aid longer-termretention of concepts.

In one pilot school, the headteacher made it a spending priority to give eachteaching assistant their own pack of practical resources containing 100-squares,wipe-clean number lines, cubes, number cards, calculators, bead strings, dice,spinners, place value (arrow) cards, and so on. This funding decision followed onfrom a training session for teachers and teaching assistants when they watchedvideo clips from the CD-ROM in the Using models and images to supportmathematics teaching and learning in Years 1 to 3 pack.

In another school the parents and governors provided the money for every childin the school to have a basic pack of resources in a plastic wallet.

However do we fit in Wave 3 with everything else?

Some schools managed their Wave 3 intervention by using teaching assistant timepreviously deployed on non-specific in-class support for lower-attaining children inthe daily mathematics lesson. Other schools used a range of strategies to givespace for Wave 3 work, such as diverting some teaching assistant time from othersubject areas to focus more on mathematics. In one pilot school every class didtheir daily mathematics lesson at the same time so that children with similar errorsand misconceptions (from various year groups) could be grouped.

As the pilot progressed, many of the later teaching units included whole-class worklinked to the errors and misconceptions, often with a challenge and problem-solvingslant. This enabled teachers to weave Wave 3 work into the whole-class context.

I’m the mathematics coordinator; how can I get a whole-schoolperspective on Wave 3?

You will need to link closely with senior management, and colleagues will need toidentify those children who are demonstrating misunderstandings and errors. Youmight need to help in the diagnostic assessment, using the tracking charts tosupport this.

How should we select the children for Wave 3 interventions?

It will be class teachers, in their day-to-day assessment during the dailymathematics lesson and in marking written work, who will note children’s errorsand misconceptions. They can use the questions in the third column of the trackingcharts to confirm their diagnosis.

I’m the SENCO, but my strengths are in literacy so what is my role inWave 3 mathematics?

You will have an overview of all the Wave 3 interventions in school (not justmathematics) and you will have the SEN Code of Practice in mind. You couldcoordinate staff who are involved, making sure that there is good communicationthroughout the school, and supporting class teachers in tracking children’s progressso that the impact of your Wave 3 provision can be evaluated.

?

In one pilot school the SENCO had the main responsibility for Wave 3mathematics because the mathematics coordinator was new. They plannedfrom the new academic year to work more closely together.

?

A pilot school mathematics coordinator said that in the new academic year sheintended to teach every class in the school and work very closely with theSENCO to judge where best to focus Wave 3 interventions.

?

To make the task easier, all year groups were asked to teach mathematics at thesame time. The small groups using Wave 3 materials were going to be taught for20 minutes, three times per week. It was agreed that the children would stay inclass during the initial teacher input and would then be withdrawn and returnedfor the plenary. However, it became clear that returning for the plenary disruptedthe remainder of the class, and did not make good use of the learning time forthe group themselves. It was decided that, for the limited time of the Wave 3intervention, the group would have their own plenary.

?



Will children be withdrawn from lessons or will Wave 3 interventionssometimes or always be kept within the class lesson?

Pilot schools said their practice varied considerably. In one school, when theteacher wanted to do the Wave 3 work, she stayed in the class working with thefocus group while the teaching assistant supervised the rest of the class as theyworked on the task the teacher had given them.

Schools where children were regularly withdrawn identified a problem with keepingthe teacher informed of those children’s progress; the children were also becomingout of touch with the whole-class work.

Pupils were taken out of the mental/oral starter approximately three times aweek. They worked in a separate room with the teaching assistant. They workedfrom the Spotlights, although the sessions invariably ran over the five minutessuggested in the pilot in order to ensure that the children gained useful support.

The Spotlights were chosen to link in with the week’s plan for the class. If thiswas not possible the teaching assistant selected a Spotlight from an area ofspecific concern or repeated something done previously. The programme wasnot used as a scheme or in a certain order, but rather as a resource to supportwhat was already happening in the main lessons.

?

We want to go and visit some schools where their Wave 3work seems to be going more smoothly than ours. Wethought we could send different teachers with a teachingassistant for a few visits then we will all compare notes andsee if we can find a better way to organise ourselves.

SENCO

We have been extremely pleased with the implementation of Wave 3mathematics at our school. It has boosted the confidence of all of thechildren involved and the positive effects have been witnessedthroughout the school. Many teachers have commented on thechange in children’s attitudes. A Year 2 teacher observed a child whowas previously petrified of mathematics lessons and is now muchmore cooperative and happier during lessons; she is very pleased withthe positive influence the group work has had.

Headteacher

Teaching assistants’ expertise andopinion has been valued and this hashad a positive effect on their motivation.

SENCO

The children have definitelybenefited, and their basic numeracyskills and confidence have improved.

Class teacher

Continuing professional developmentThis section contains suggestion for a staff meeting to introduce the PrimaryNational Strategy Wave 3 mathematics pack and strategies for Wave 3 mathematicsprovision.

Outline for a staff meeting

Objectives

• To provide some background to the development of the Primary NationalStrategy Wave 3 mathematics materials

• To develop practice in supporting children’s mathematical development and self-confidence by providing a suggested model for Wave 3 mathematicsintervention

• To familiarise staff with specific detail about using the Wave 3 mathematics pack

• To consider the identification of specific areas of mathematics that can preventchildren achieving expected levels of progress

Key messages

• It is important that the Wave 3 mathematics materials are seen as a suggestedmodel for providing Wave 3 intervention, which teachers should adapt to suit theneeds of their children.

• There are key features in the design of the materials to support good practice inworking with children with mathematical difficulties.

• Good day-to-day assessment is fundamental in supporting effective use ofWave 3 mathematics intervention.

Resources

• A complete Primary National Strategy Wave 3 mathematics pack Supportingchildren with gaps in their mathermatical understanding

• A copy of the addition and subtraction tracking children’s learning chart for eachmember of staff

• Copies of handouts 1, 2 and 3 from Appendix 3 for each member of staff

• Computer, etc. to demonstrate CD-ROM from pack

Introduction

1 Share the objectives for the session, setting these in the context of the three‘waves’ of inclusion.

Three ‘waves’

Provision for effective mathematics learning and teaching can be described interms of three ‘waves’ of intervention.

Wave 1 The effective inclusion of all children in high quality learningand teaching of mathematics in the daily mathematics lesson.


Wave 2 Additional time-limited provision in the form of small-groupintervention to accelerate progress and enable children towork at age-related expectations.

Wave 3 Additional time-limited provision to enhance the progress ofidentified children where Waves 1 and 2 are not, on theirown, having the desired effect. This will involve focusedteaching activities which tackle fundamental errors andmisconceptions that are preventing progress.

2 Briefly mention the development of the materials, their piloting and revision inresponse to feedback.

3 Refer to the research background that informed the design, mentioning that keyfeatures of effective Wave 3 interventions highlighted in the research review havebeen incorporated in the design of the Wave 3 mathematics pack.

In particular these are:

• a focus on the most commonly occurring types of mathematical difficulties

The materials focus on number and calculation, tackling areas such asunderstanding the structure of number and operations between numbers.Problem solving is integrated and exemplified in the materials, and opportunitiesare provided for children to develop mathematical vocabulary.

• an individualised approach based on the particular areas the child finds difficult

The materials reflect best practice in assessment for learning and includetracking children’s learning charts that support the identification of the particularknowledge, skills and understanding with which the child needs help.

• relatively small amounts of individualised intervention

The Wave 3 teaching activities provide brief, focused teaching sessions whichmake it possible for the child to benefit more fully from whole-class teaching.Where appropriate, the activities finish with related activities for whole-class usein order to reinforce individual learning and promote inclusive practice.

Activity 1: Getting to know the Primary National Strategy Wave 3mathematics materials

Introduce the components of the pack to illustrate what the pack has to offer. Brieflymention each of the items (see notes on page 7).

Activity 2: Understanding the purpose of the tracking children’s learning charts

1 Ask colleagues to scan Handout 1, Structure of Wave 3 mathematics materials,to begin to get an overview of the tracking chart. They should then scan theaddition and subtraction chart, working in pairs to identify a familiarerror/misconception from the second column.

2 Ask colleagues to look at the third column of the chart to find questions toconfirm their diagnosis.

How would they see themselves using these questions? One to one? Workingwith a small focus group including the target child?

Ask them to suggest further questions to confirm diagnosis of the chosenerror/misconception.


3 Explain that the fourth column contains an outline of work they could coverwhen they choose a booklet and the fifth column shows how teachers couldextend the work after the child has completed activities in the booklet.

4 Ask colleagues to focus on the code of their error/misconception, find therelated A4 booklet and read through the activities, identifying:

• helpful aspects for teaching and learning.

• any suggestions for improving the work in the booklet to suit the needs of their children.

Take feedback, emphasising the intention that teachers modify the materials asnecessary for their children.

Demonstrate the tracking back process using the CD-ROM interactive trackingchart.

Activity 3: Focusing on the teaching activities

As they look at the booklet they have selected and use Handout 2, Mathematicalthemes and assessment approaches in the Wave 3 materials, ask colleagues to findexamples of the mathematical themes listed on the handout.

Some questions to consider:

• What are the implications for your practice?

• What guidance would you need to give to your teaching assistant to supportthem in working with a child, or a couple of children, with Spotlight 1 in thebooklet?

At the end of Handout 2 is a list of the ways in which best practice in assessmentfor learning is reflected in the materials. Ask colleagues to look at this list and thebooklet they have been using to identify the ways in which assessment for learningis supported.

Activity 4: Assessment and planning process

Ask colleagues, in pairs, to use Handout 3, Assessment and planning process, todraw together the elements of the suggested model for using the materials tosupport children’s learning.

Ask for questions and suggestions about how the school can incorporate the use ofthis pack to support children’s learning.

Activity 5: Developing Wave 3 mathematics in our school

This part of the session will need to focus on the practicalities of Wave 3mathematics planning in the individual school.

Refer to material from this booklet to focus the discussion – See the section entitled‘Classroom’ page 17.

From this meeting, the leadership team will have gathered knowledge to inform theirfurther planning. Support will be found in the management guidance, starting onpage 12.

End by referring to some of the next steps identified by the pilot schools as theyplanned how to further improve Wave 3 provision for their children.



I think we might need to rethink ourassessment records so that there ismore space for the richness of theobservations we can do in Wave 3.

We’re going to do moreTA training.

My advice to anyone setting out to do Wave 3 wouldbe to keep asking yourself ‘Am I doing this the bestway?’ We kept changing what we did in the light ofcircumstances and we’re now on our third way ofdoing it … The results have been fantastic – moreconfidence and raised self-esteem and kids smilingand wanting to do maths.

Keep doing it! It’s great!

We want to involve theparents more.

Anything that gets the kids thisexcited about maths has to be worthputting lots of time and effort into. Wejust want to do more of it next year.

Appendix 1

Tracking children’s learning chart – addition andsubtraction

The tracking charts act as an initial diagnostic tool and a background framework forthe sets of teaching materials referenced to common misconceptions and errors.

Teaching unit codes are referenced against the errors and misconceptions listed inthe chart, for example 1 Y6 +/–



Year

6 k

ey o

bje

ctiv

eC

arry

out

col

umn

addi

tion

and

subt

ract

ion

of n

umbe

rs in

volv

ing

deci

mal

s (N

NS

Fra

mew

ork

for

teac

hing

mat

hem

atic

s, S

uppl

emen

t of

Exa

mpl

es, S

ectio

n 6,

page

s 49

, 51)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

App

ly k

now

ledg

e of

th

e nu

mbe

r sy

stem

to

ena

ble

effic

ient

co

untin

g of

a la

rge

num

ber

of o

bjec

ts.

Add

and

sub

trac

t m

ultip

les

of t

en, a

hu

ndre

d an

d a

thou

sand

.1

Y6

Giv

e an

est

imat

e by

ro

undi

ng, t

o de

term

ine

whe

ther

th

e an

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to

a ca

lcul

atio

n is

se

nsib

le.

2 Y

6

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

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tio

n)

Err

ors

and

mis

conc

epti

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Has

inef

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nt c

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stra

tegi

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nd/o

rin

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nder

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of t

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umbe

r sy

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. 1

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+/–

Rou

ndin

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accu

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ly,pa

rtic

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ly w

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deci

mal

s ar

e in

volv

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and

havi

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the

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the

num

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invo

lved

.2

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+/–

Que

stio

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fy e

rro

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isco

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ine

you

have

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con

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2pan

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ns. W

hat

do y

ou t

hink

wou

ld b

e a

good

way

to

coun

t th

ese

quic

kly

to fi

nd o

ut h

owm

uch

mon

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Wha

t is

60

+ 2

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60

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60

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hat

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n yo

u fo

und

60 +

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40

+ 4

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the

larg

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+ 4

00 +

400

0 di

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ntfro

m t

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Wha

t is

60 –

20?

… 6

00 –

200

? …

600

0 –

2000

?E

xpla

in h

ow y

ou w

orke

d th

ese

out.

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t is

600

0 –

200?

… 6

000

– 20

?

Is 2

6 ne

arer

to

20 o

r 30

?Is

271

nea

rer

270

or 2

80?

Is 1

.8 n

eare

r to

1 o

r 2?

Dra

w a

ske

tch

to il

lust

rate

you

r an

swer

and

expl

ain

how

you

kno

w.

Teac

hing

to

ad

dre

ss t

he e

rro

rsan

d m

isco

ncep

tio

ns

Pra

ctic

al o

ppor

tuni

ties

to d

evel

opef

ficie

nt c

ount

ing

stra

tegi

es fo

r a

rang

e of

obj

ects

, for

exa

mpl

e co

ins,

cube

s, c

onke

rs, c

olle

ctab

le c

ards

,st

icke

rs.

Cou

nt fo

rwar

ds a

nd b

ackw

ards

inte

ns, h

undr

eds

and

thou

sand

s fro

mdi

ffere

nt s

tart

ing

poin

ts, i

nclu

ding

star

ting

num

bers

tha

t ar

e no

tm

ultip

les

of t

en o

r a

hund

red.

Use

an e

mpt

y nu

mbe

r lin

e to

sup

port

this

dev

elop

men

t.

Ord

er m

ultip

les

of a

hun

dred

and

ath

ousa

nd.

Use

num

ber

squa

res

and/

ornu

mbe

r lin

es t

o co

nsid

er t

he o

rder

and

com

para

tive

valu

e of

num

bers

to s

uppo

rt r

ound

ing.

Nex

t st

eps

in m

ovi

ng t

ow

ard

s th

eke

y o

bje

ctiv

e

Car

ry o

ut s

impl

e ca

lcul

atio

ns t

hat

invo

lve

cros

sing

the

bou

ndar

y fro

mhu

ndre

ds t

o on

e th

ousa

nd a

nd v

ice

vers

a, s

uppo

rted

by

an e

mpt

ynu

mbe

r lin

e an

d ex

tend

ing

this

to

avi

sual

ised

imag

e to

dev

elop

men

tal

calc

ulat

ion.

Con

side

r pa

irs o

f ite

ms

from

aca

talo

gue

and

ask

child

to

estim

ate

whe

ther

a £

10 (o

r £2

0, e

tc.)

note

wou

ld b

e en

ough

to

buy

both

the

item

s?


Par

titio

n w

hole

nu

mbe

rs a

nd

deci

mal

num

bers

, an

d ad

d an

d su

btra

ct t

he

cons

titue

nt p

arts

an

d re

com

bine

to

com

plet

e th

e ca

lcul

atio

ns.

3 Y

6

Add

and

sub

trac

t a

pair

of n

umbe

rs

that

invo

lves

cr

ossi

ng

boun

darie

s,

reco

gnis

ing

whe

n to

adj

ust,

to

com

pens

ate

and

to

carr

y nu

mbe

rs

acro

ss t

he

boun

darie

s.

4a Y

64b

Y6

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Has

diff

icul

ty in

part

ition

ing

num

bers

with

zer

o pl

ace

hold

ers

and/

or n

umbe

rs le

ssth

an o

ne, f

or e

xam

ple

part

ition

ing

0.45

as

0.4

and

0.05

. 3

Y6

+/–

Has

diff

icul

ty in

choo

sing

sui

tabl

em

etho

ds fo

rca

lcul

atio

ns t

hat

cros

sbo

unda

ries:

add

ition

4a Y

6 +

/–H

as d

iffic

ulty

inch

oosi

ng s

uita

ble

met

hods

for

calc

ulat

ions

tha

t cr

oss

boun

darie

s: s

ubtr

actio

n4b

Y6

+/–

Wha

t is

the

val

ue o

f the

dig

it 1

in 3

010?

…

201

? …

6.1

?W

hat

do y

ou n

otic

e ab

out

the

valu

e of

the

dig

it 2

in 4

.2?

… 0

.2?

… 0

.25?

Writ

e th

e fo

llow

ing

stat

emen

ts:

507

is e

qual

to

50 +

774

03 is

equ

al t

o 70

00 +

40

+ 3

0.75

is e

qual

to

0.7

+ 0

.05

Whi

ch a

re t

rue?

How

do

you

know

?

Wha

t w

ould

you

add

to

37 t

o m

ake

the

near

est

mul

tiple

of 1

0?

Wha

t w

ould

you

add

to/

subt

ract

from

240

to

mak

e th

e ne

ares

t m

ultip

le o

f 100

?

Exp

lain

to

me,

usi

ng a

n em

pty

num

ber

line,

how

you

find

300

– 23

7.

Usi

ng a

n em

pty

num

ber

line,

exp

lain

how

you

wou

ld w

ork

out

5016

+ 3

700.

Wha

t is

217

– 6

? …

217

– 7

?W

hat

happ

ens

whe

n yo

u su

btra

ct 8

from

217

?E

xpla

in h

ow y

ou s

ubtr

act

18 fr

om 2

17.

Use

pla

ce v

alue

(arr

ow) c

ards

to

part

ition

and

rec

ombi

ne n

umbe

rs,

incl

udin

g nu

mbe

rs w

ith z

ero

plac

eho

lder

s an

d de

cim

als,

for

exam

ple

10.8

5 an

d 18

.05.

Con

tinue

to

prac

tise

coun

ting

upus

ing

an e

mpt

y nu

mbe

r lin

e an

dre

cord

the

ste

ps, f

ocus

ing

on t

hesi

gnifi

canc

e of

the

mul

tiple

s of

ten

and

hund

red

as m

ilest

ones

to

supp

ort

deve

lopi

ng u

nder

stan

ding

of c

ross

ing

boun

darie

s.

Use

pla

ce v

alue

(arr

ow) c

ards

to

part

ition

bef

ore

addi

ng a

ndsu

btra

ctin

g nu

mbe

rs, t

o illu

stra

teth

e si

gnifi

canc

e of

col

umns

to

supp

ort

calc

ulat

ion.

Pro

vide

or

ask

the

child

to

sele

ct a

calc

ulat

ion

usin

g nu

mbe

rs p

revi

ousl

ypa

rtiti

oned

and

ask

the

m t

o ex

plai

nho

w t

hey

can

tack

le it

sup

port

ed b

ypl

ace

valu

e (a

rrow

) car

ds t

ode

mon

stra

te. R

epea

t w

ith c

hild

choo

sing

furt

her

calc

ulat

ions

and

num

bers

.

Tran

sfer

cal

cula

tion

prac

tice

to a

vert

ical

num

ber

line,

tak

ing

the

oppo

rtun

ity t

o re

late

thi

s to

mea

sure

s.

Incr

ease

the

num

ber

of b

ound

arie

s to

be c

ross

ed, a

skin

g th

e ch

ild t

oco

mpa

re c

alcu

latio

n m

etho

dsco

nsid

erin

g th

eir

effic

ienc

y.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

to

war

ds

the

kno

wle

dg

e an

d m

isco

ncep

tio

nsm

isco

ncep

tio

nsan

d m

isco

ncep

tio

nske

y o

bje

ctiv

esk

ills


Year

4 k

ey o

bje

ctiv

eC

arry

out

col

umn

addi

tion

and

subt

ract

ion

of t

wo

inte

gers

less

tha

n 10

00 a

nd c

olum

n ad

ditio

n of

mor

e th

an t

wo

such

inte

gers

(NN

S F

ram

ewor

k fo

r te

achi

ngm

athe

mat

ics,

Sup

plem

ent

of E

xam

ples

, Sec

tion

6, p

ages

48,

50)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Kno

w t

he v

alue

of

each

dig

it in

a

thre

e-di

git

num

ber,

for

exam

ple

that

th

e 3

in 4

37 h

as a

va

lue

of 3

0.

Rec

ogni

se t

he

sign

ifica

nce

of e

ach

digi

t in

num

bers

up

to

1000

and

fin

d an

est

imat

e of

the

add

ition

or

sub

trac

tion

calc

ulat

ion.

1 Y

4

Par

titio

n tw

o- a

nd

thre

e-di

git

num

bers

in

a n

umbe

r of

w

ays.

2

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Err

ors

and

mis

conc

epti

ons

Has

inse

cure

unde

rsta

ndin

g of

the

stru

ctur

e of

the

num

ber

syst

em, r

esul

ting

in a

dditi

on a

ndsu

btra

ctio

n er

rors

and

diffi

culty

with

estim

atin

g.1

Y4

+/–

Has

diff

icul

ty in

part

ition

ing,

for

exam

ple,

208

into

190

and

18 a

nd31

into

20

and

11.

2 Y

4 +

/–

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Is 4

3 sm

alle

r or

larg

er t

han

34?

Wha

t ab

out

343

and

334?

How

did

you

dec

ide?

Whi

ch is

the

larg

est/

smal

lest

num

ber:

216,

612

or

162?

How

do

you

know

?

Wha

t nu

mbe

r w

ould

you

add

to

437

tom

ake

477?

How

did

you

dec

ide?

Wha

t nu

mbe

r do

es 2

0 +

10

+ 1

repr

esen

t?Is

it t

he s

ame

as 2

0 +

11?

Wha

t sh

ould

I ad

d to

190

to

mak

e 20

8?W

hat

othe

r w

ays

can

you

part

ition

208

?

Teac

hing

to

ad

dre

ss t

he e

rro

rs a

ndm

isco

ncep

tio

ns

Use

a r

ange

of p

lace

val

ue m

odel

s an

d im

ages

suc

has

bun

dles

of s

traw

s, d

igit

card

s an

d st

ruct

ured

appa

ratu

s to

par

titio

n nu

mbe

rs a

nd r

ead

them

corr

ectly

.

Use

a n

umbe

r lin

e an

d pl

ace

valu

e (a

rrow

) car

ds to

iden

tify

posi

tion

of a

var

iety

of n

umbe

rs in

rela

tion

toth

eir

near

est t

en, n

eare

st h

undr

ed.

Mak

e tw

o-di

git

and

thre

e-di

git

num

bers

usi

ng d

igit

card

s. C

hang

e on

e di

git

at a

tim

e, fo

r ex

ampl

e 13

4 to

144,

est

ablis

hing

the

siz

e of

the

cha

nge,

in t

his

case

ten

mor

e. E

stab

lish

whe

ther

the

cha

nge

is s

igni

fican

t,fo

r ex

ampl

e th

at 1

54 is

clo

ser

to 2

00 t

han

100,

and

use

this

to

info

rm e

stim

ates

for

addi

tion

and

subt

ract

ion

calc

ulat

ions

.

Use

pla

ce v

alue

(arr

ow) c

ards

to

part

ition

and

com

bine

num

bers

with

zer

o as

pla

ce h

olde

r,di

scus

sing

the

pos

ition

of t

he z

ero

as a

pla

ce h

olde

r.

Use

num

ber

squa

res

or a

str

ing

of b

eads

to

mod

elth

e pa

rtiti

onin

g of

num

bers

in d

iffer

ent

way

s,re

cord

ing

the

asso

ciat

ed n

umbe

r st

atem

ents

, suc

has

43

= 4

0 +

3 =

30

+ 1

3 =

20

+ 2

3

Nex

t st

eps

in m

ovi

ng t

ow

ard

sth

e ke

y o

bje

ctiv

e

From

a s

et o

f dig

it ca

rds,

ask

the

child

to

sele

ct t

hree

, of

whi

ch t

wo

mus

t be

the

sam

e,fo

r ex

ampl

e 3,

3 a

nd 6

. Ask

the

child

to

mak

e an

d re

ad a

loud

diffe

rent

thr

ee-d

igit

num

bers

, for

exam

ple

thre

e hu

ndre

d an

dth

irty-

six.

Rep

eat

activ

ity.

Ask

the

chi

ld t

o es

timat

ean

swer

s fo

r so

me

give

nca

lcul

atio

ns in

volv

ing

two-

and

thre

e-di

git

num

bers

and

dis

cuss

with

chi

ld t

he e

stim

atio

nm

etho

ds u

sed.

Use

par

titio

ning

to

carr

y ou

t a

rang

e of

tw

o-di

git

calc

ulat

ions

,di

scus

sing

the

way

par

titio

ning

mak

es t

he c

alcu

latio

n m

ore

man

agea

ble.

Rep

eat

with

thr

ee-

digi

t nu

mbe

rs.


Kno

w w

hen

it is

m

ost

appr

opria

te t

o us

e co

lum

n ad

ditio

n or

sub

trac

tion

rath

er

than

car

ryin

g ou

t th

e ca

lcul

atio

n by

an

othe

r m

etho

d.3

Y4

Rec

ogni

se t

hat

the

oper

atio

n of

col

umn

addi

tion

can

appl

y to

mor

e th

an t

wo

num

bers

.4

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Doe

s no

t m

ake

sens

ible

deci

sion

s ab

out

whe

nto

use

cal

cula

tions

laid

out

in c

olum

ns.

3 Y

4 +

/–

Has

diff

icul

ty w

ithad

ding

thr

ee n

umbe

rsin

a c

olum

n, e

xcep

t by

addi

ng t

he fi

rst

two

and

then

the

last

one

.4

Y4

+/–

Wha

t is

700

– 1

? …

700

– 1

0? …

700

– 9

?W

hat

abou

t 30

+ 2

0? …

30

+ 2

1? …

30

+ 2

7?…

296

– 5

7? …

328

+ 1

87?

How

did

you

do

thes

e?

Ref

errin

g to

a w

ritte

n ad

ditio

n co

lum

n ca

lcul

atio

nin

volv

ing

thre

e nu

mbe

rs, a

sk t

he c

hild

to

talk

thro

ugh

thei

r st

eps

in c

alcu

latio

n.

Ask

the

chi

ld t

o ch

oose

pai

rs o

fnu

mbe

rs t

o ad

d to

geth

er,

high

light

ing

whe

re m

enta

l met

hods

are

mos

t ap

prop

riate

and

one

sw

here

writ

ten

met

hods

are

mor

eap

prop

riate

.

Use

pla

ce v

alue

(arr

ow) c

ards

to

dem

onst

rate

col

umn

addi

tion

ofm

ore

than

tw

o nu

mbe

rs a

nd fo

cus

on c

ompl

etin

g th

e ca

lcul

atio

n on

eco

lum

n at

a t

ime.

Ask

the

chi

ld t

o id

entif

y a

colle

ctio

n of

pairs

of n

umbe

rs t

hat

wou

ld d

efin

itely

be b

est

tack

led

by w

ritte

n ca

lcul

atio

nm

etho

ds a

nd e

xpla

in w

hy.

Ext

end

to a

ddin

g m

ore

than

thr

eenu

mbe

rs u

sing

a c

olum

n m

etho

d.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

to

war

ds

the

kno

wle

dg

e an

d m

isco

ncep

tio

nsm

isco

ncep

tio

nsan

d m

isco

ncep

tio

nske

y o

bje

ctiv

esk

ills


Year

2 k

ey o

bje

ctiv

eU

nder

stan

d th

at s

ubtr

actio

n is

the

inve

rse

of a

dditi

on; s

tate

the

sub

trac

tion

corr

espo

ndin

g to

a g

iven

add

ition

and

vic

e ve

rsa

(NN

S F

ram

ewor

k fo

r te

achi

ngm

athe

mat

ics,

Sup

plem

ent

of E

xam

ples

, Sec

tion

5, p

ages

25,

29,

35)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Cou

nt o

n an

d ba

ck

in o

nes

and

tens

.1

Y2

Iden

tify

pairs

of

num

bers

tha

t ad

d to

tw

enty

and

use

kn

own

num

ber

fact

s to

add

men

tally

.2

Y2

Find

a d

iffer

ence

by

coun

ting

up fr

om

the

smal

ler

to t

he

larg

er n

umbe

r.3

Y2

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Err

ors

and

mis

conc

epti

ons

Mak

es m

ista

kes

whe

nco

untin

g us

ing

teen

num

bers

and

/or

cros

sing

bou

ndar

ies.

1 Y

2 +

/–

Has

diff

icul

ty in

rem

embe

ring

num

ber

pairs

tot

allin

g be

twee

nte

n an

d tw

enty

,re

sulti

ng in

cal

cula

tion

erro

rs.

2 Y

2 +

/–

Cou

nts

up u

nrel

iabl

y;st

ill co

untin

g th

e sm

alle

rnu

mbe

r to

get

one

too

man

y in

the

ans

wer

.3

Y2

+/–

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Cho

ose

a te

en n

umbe

r an

d co

unt

back

war

dsfro

m y

our

num

ber

in o

nes.

Wha

t is

the

nex

tnu

mbe

r, th

e nu

mbe

r be

fore

?

Cou

nt fo

rwar

ds in

ten

s fro

m 7

. Wha

t nu

mbe

r is

ten

mor

e th

an 2

7? …

ten

less

tha

n 97

?

Wha

t is

3 +

5?

Wha

t is

13

+ 5

?H

ow d

id y

ou w

ork

that

out

?

Whi

ch p

air

of n

umbe

rs a

dds

to 1

8?A

re t

here

any

oth

er p

airs

?

How

man

y do

I ad

d on

to

get

from

thr

ee t

oei

ght?

I’ve

got

thre

e sh

erbe

t di

ps a

nd I

wan

t ei

ght.

How

man

y do

I ne

ed t

o bu

y?

Teac

hing

to

ad

dre

ss t

he e

rro

rsan

d m

isco

ncep

tio

ns

Use

a n

umbe

r lin

e an

d a

100-

squa

re t

o co

unt

forw

ards

and

back

war

ds in

one

s an

d te

ns.

Enc

oura

ge s

imila

r co

untin

gac

tiviti

es u

sing

a v

isua

lised

num

ber

line.

Use

rul

ers

and

num

ber

squa

res

tosu

ppor

t pr

actic

e w

ith r

ecal

l of

num

ber

pairs

bey

ond

ten

and

emph

asis

e pa

tter

ns in

num

ber

pairs

.

Use

a n

umbe

r lin

e to

dem

onst

rate

coun

ting

up b

y ju

mpi

ng in

one

s an

dhi

ghlig

htin

g th

e eq

uiva

lent

larg

erju

mp.

Thi

s w

ill illu

stra

te t

hat

the

star

ting

num

ber

posi

tion

is n

otin

clud

ed in

the

jum

p.

Nex

t st

eps

in m

ovi

ng t

ow

ard

s th

eke

y o

bje

ctiv

e

Use

kno

wn

fact

s ab

out

addi

ng a

ndsu

btra

ctin

g te

n to

wor

k ou

t ho

w t

oca

lcul

ate

usin

g ni

ne a

nd e

leve

n.

Ext

end

patt

erns

to

num

bers

bey

ond

20; u

se 4

+ 3

= 7

to

dedu

ce o

ther

num

ber

fact

s su

ch a

s 14

+ 3

= 4

+ 1

3 an

d so

on.

Ask

the

chi

ld t

o ch

oose

a n

ew ju

mp

size

, for

exa

mpl

e tw

os o

r fiv

es, t

o fin

dth

e di

ffere

nce

betw

een

two

give

nnu

mbe

rs. H

ow c

an y

ou c

heck

you

ran

swer

?


Rec

ogni

se

subt

ract

ion

as

taki

ng a

way

, fin

ding

th

e di

ffere

nce

and

com

plem

enta

ry

addi

tion.

4 Y

2

Rec

ogni

se, f

or

exam

ple,

tha

t su

btra

ctin

g 13

‘u

ndoe

s’ a

ddin

g 13

an

d vi

ce v

ersa

, and

th

at t

his

mea

ns t

hat

sinc

e 4

+ 1

3 =

17,

w

e ca

n st

ate

the

inve

rse

that

17

– 1

3 =

4.

5 Y

2

Dev

elop

and

re

cogn

ise

patt

erns

to

hel

p de

duce

ot

her

addi

tion

and

subt

ract

ion

fact

s.

6 Y

2

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Doe

s no

t re

late

find

ing

a di

ffere

nce

and

com

plem

enta

ry a

dditi

onto

the

ope

ratio

n of

subt

ract

ion.

4 Y

2 +

/–

Is in

secu

re in

mak

ing

links

bet

wee

n ad

ditio

nan

d su

btra

ctio

n an

d/or

reco

gnis

ing

inve

rses

.5

Y2

+/–

Doe

s no

t re

adily

use

num

ber

patt

erns

to

supp

ort

calc

ulat

ing,

for

exam

ple:

46 –

5 =

41,

so

46 –

15

= 3

1,46

– 2

5 =

21,

etc

.6

Y2

+/–

Wha

t is

the

diff

eren

ce b

etw

een

21 a

nd18

?H

ow d

id y

ou w

ork

it ou

t?W

hat

oper

atio

n di

d yo

u us

e?

Wha

t ot

her

way

s ca

n yo

u th

ink

of fo

rsu

btra

ctin

g 18

from

21?

Wha

t is

the

ans

wer

to

30 a

dd 2

0?If

30 a

dd 2

0 is

50,

wha

t is

50

subt

ract

20?

Wha

t is

17

subt

ract

8?

Writ

e a

num

ber

sent

ence

for

this

cal

cula

tion.

Use

the

thr

eenu

mbe

rs t

o w

rite

an a

dditi

on fa

ct.

Wha

t is

14

+ 5

? …

14 +

15?

…14

+25

?U

sing

thi

s in

form

atio

n, t

ell m

e w

hat

24 +

25

is.

Wha

t is

6 –

4?

…16

– 4

? …

26

– 4?

Now

wha

t do

you

thi

nk t

he a

nsw

er is

to

56 –

4?

How

did

you

wor

k th

at o

ut?

Illus

trat

e th

e co

nnec

tions

bet

wee

ndi

ffere

nt a

spec

ts o

f sub

trac

tion

usin

g a

rang

e of

mod

els

and

imag

es, s

uch

asnu

mbe

r lin

es, m

oney

, cou

nter

s or

Inte

ract

ive

Teac

hing

Pro

gram

mes

(ITP

s)su

ch a

s C

ount

ing

on a

nd b

ack,

Diff

eren

ces,

Num

ber

fact

s.

Use

dig

it ca

rds

and

num

ber

lines

to

mak

e an

d ch

eck

num

ber

stat

emen

tsth

at in

volv

e in

vers

e pa

irs s

uch

as

11 +

3 =

14,

14

– 3

= 1

1.

Use

num

ber

squa

res

to d

emon

stra

tean

d hi

ghlig

ht p

atte

rns

such

as

5 +

1 =

6, 1

5 +

1 =

16,

25

+ 1

= 2

632

– 2

= 3

0, 3

2 –

12 =

20

gett

ing

child

ren

to p

redi

ct a

nd c

heck

othe

r ca

ses.

Pro

vide

the

follo

win

g th

ree

ques

tions

:W

hat

is t

he d

iffer

ence

bet

wee

n 21

and

18?

21 –

18

= ?

18 +

= 2

1?W

hat

do y

ou n

otic

e ab

out

thes

e an

d th

eir

answ

ers?

Mak

e up

som

e of

you

r ow

n an

d ex

plai

nth

e pa

tter

ns y

ou a

re u

sing

.

Giv

e th

e ch

ild s

tate

men

ts t

o so

rt in

topa

irs in

volv

ing

inve

rses

, the

n in

to g

roup

sth

at m

ake

up a

set

of t

he fo

ur e

quiv

alen

tst

atem

ents

suc

h as

24

+ 3

= 2

7, 3

+ 2

4 =

27

27–

3 =

24,

27

– 24

=

3In

vite

the

chi

ld t

o m

ake

sets

of e

quiv

alen

tst

atem

ents

usi

ng n

umbe

rs t

hey

have

chos

en.

Ext

end

to in

clud

e pa

tter

ns w

here

the

calc

ulat

ions

cro

ss a

bou

ndar

y su

ch a

s14

+ 9

= 2

3, 1

4 +

19

= 3

356

– 9

= 4

7, 5

6 –

19 =

37.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

to

war

ds

the

kno

wle

dg

e an

d m

isco

ncep

tio

nsm

isco

ncep

tio

nsan

d m

isco

ncep

tio

nske

y o

bje

ctiv

esk

ills


Rec

epti

on

key

ob

ject

ive

Beg

in t

o re

late

add

ition

to

com

bini

ng t

wo

grou

ps o

f obj

ects

, and

sub

trac

tion

to ‘t

akin

g aw

ay’ (

NN

S F

ram

ewor

k fo

r te

achi

ng m

athe

mat

ics,

Sup

plem

ent

ofE

xam

ples

, Sec

tion

4, p

ages

14,

15,

16)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Cou

nt a

long

and

ba

ck o

n a

num

ber

trac

k to

and

from

a

give

n po

sitio

n.

Cou

nt o

bjec

ts s

et

out

in d

iffer

ent

arra

ngem

ents

; be

gin

to r

ecog

nise

sm

all n

umbe

rs

with

out

coun

ting

and

that

the

num

ber

of o

bjec

ts is

not

af

fect

ed b

y th

eir

posi

tion.

Cou

nt o

bjec

ts t

hat

are

out

of r

each

. 1

YR

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Err

ors

and

mis

conc

epti

ons

Can

onl

y be

gin

coun

ting

at o

ne;

inac

cura

tely

cou

nts

obje

cts

whe

nre

arra

nged

; has

no

cons

iste

nt r

ecog

nitio

nof

sm

all n

umbe

rs o

fob

ject

s; la

cks

syst

emat

ic a

ppro

ache

s.

1 Y

R +

/–

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Ask

the

chi

ld t

o ch

oose

a s

tart

ing

num

ber.

Wha

t ar

e th

e ne

xt t

wo

num

bers

?W

hat

num

ber

com

es a

fter

the

third

num

ber?

Wha

t co

mes

bef

ore

your

sta

rtin

g nu

mbe

r?

The

cont

ents

pan

el s

ays

ther

e ar

e tw

elve

. Let

’sch

eck.

Tip

the

m o

ut a

nd p

ut t

hem

bac

k. H

owm

any

are

ther

e no

w?

Can

you

tel

l me

how

man

y sw

eets

the

re a

rehe

re w

ithou

t co

untin

g th

em?

How

man

y sp

ots

are

in t

his

pict

ure?

Thro

w a

sm

all n

umbe

r of

obj

ects

ont

o th

eta

ble.

Can

you

cou

nt t

hem

with

out

touc

hing

the

m?

How

did

you

do

it?H

ow d

o yo

u kn

ow y

ou’re

cor

rect

?

Teac

hing

to

ad

dre

ss t

he e

rro

rs a

ndm

isco

ncep

tio

ns

Chi

ldre

n w

alk

forw

ards

and

bac

kwar

dsal

ong

a la

rge

num

ber

trac

k, a

long

anu

mbe

r lin

e, o

n a

snak

es a

nd la

dder

sbo

ard,

cou

ntin

g al

oud.

Sta

rt fr

om d

iffer

ent

posi

tions

; use

dig

it ca

rds

or d

ice

to s

elec

tst

art

and

to m

ove

one

forw

ards

, tw

oba

ckw

ards

, etc

.

Usi

ng s

ets

of m

ixed

obj

ects

to

coun

t an

dre

arra

nge,

ask

chi

ldre

n to

est

imat

e an

dch

eck

afte

r ea

ch r

earr

ange

men

t.

Put

sm

all n

umbe

rs o

f obj

ects

in fa

milia

r an

dun

fam

iliar

patt

erns

and

com

pare

with

know

n pa

tter

ns s

uch

as s

pots

on

dice

,di

spla

ys o

n w

all,

etc.

Cou

nt s

ound

s su

ch a

s dr

um b

eats

or

coin

sdr

oppi

ng in

to a

mon

ey b

ox; p

rovi

deco

unte

rs o

r pe

ncils

to r

ecor

d m

arks

as

they

coun

t; co

unt c

ount

ers

or th

eir

mar

ks o

n a

shee

t; co

mpa

re w

ays

to s

yste

mat

ical

lyco

unt p

artic

ular

arr

ange

men

ts, f

or e

xam

ple

win

dow

pan

es, s

quar

es o

n a

grid

, cha

irsar

ound

a ta

ble.

Nex

t st

eps

in m

ovi

ngto

war

ds

the

key

ob

ject

ive

Cov

er u

p se

lect

ed n

umbe

rs fo

rch

ildre

n to

iden

tify

as t

hey

coun

t; m

ove

forw

ard

and

back

war

ds fr

om d

iffer

ent

star

ting

poin

ts; c

ount

alo

ud a

ndsi

lent

ly t

o de

term

ine

posi

tion

afte

r a

mov

e.

Arr

ange

a k

now

n nu

mbe

r of

obje

cts

into

tw

o or

mor

e gr

oups

to e

stab

lish

that

the

tot

alre

mai

ns t

he s

ame;

cou

nt in

twos

; cou

nt o

bjec

ts a

rran

ged

inpa

irs; u

se r

ecog

nisa

ble

patt

erns

of s

mal

l num

bers

, suc

h as

3an

d 4,

to

intr

oduc

e co

untin

g in

thre

es a

nd fo

urs.

Est

imat

e th

e nu

mbe

r of

obj

ects

that

can

be

coun

ted

relia

bly;

chec

k by

cou

ntin

g, fi

rst

touc

hing

obje

cts

then

with

out

touc

hing

.


Find

one

mor

e an

d on

e le

ss t

han

a gi

ven

num

ber.

2 Y

R

Say

how

man

y th

ere

are

alto

geth

er

by c

ount

ing

all t

he

obje

cts

whe

n co

mbi

ning

gro

ups

for

addi

tion.

Sep

arat

e a

give

n nu

mbe

r of

obj

ects

in

to t

wo

or m

ore

grou

ps a

nd s

ay

how

man

y th

ere

are

in e

ach

grou

p.

3 Y

R

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Mis

unde

rsta

nds

mea

ning

of ‘

one

mor

e’an

d ‘o

ne le

ss’;

does

not

cons

iste

ntly

iden

tify

the

num

ber

befo

re o

r af

ter

a gi

ven

num

ber.

2 Y

R +

/–

Doe

s no

t re

late

the

com

bini

ng o

f gro

ups

of o

bjec

ts t

o ad

ditio

nan

d/or

doe

s no

tin

terp

ret

the

coun

ting

ofal

l of t

he o

bjec

ts a

s an

answ

er t

o th

e qu

estio

n‘H

ow m

any

are

ther

eal

toge

ther

?’3

YR

+/–

Her

e ar

e fo

ur c

ount

ers.

How

man

y w

ill yo

uha

ve if

I gi

ve y

ou o

ne m

ore?

Ther

e ar

e si

x sp

ots

show

ing

on m

y di

ce.

Imag

ine

ther

e is

one

less

spo

t. H

ow m

any

spot

s w

ould

the

re b

e?

Wha

t is

one

mor

e th

an s

even

? …

one

less

than

sev

en?

How

man

y sp

ots

are

ther

e on

thi

s bl

ue c

ard?

How

man

y sp

ots

are

ther

e on

thi

s re

d ca

rd?

How

man

y sp

ots

are

ther

e al

toge

ther

?

Ther

e ar

e th

ree

spoo

ns in

thi

s cu

p an

d tw

osp

oons

in t

his

cup.

How

man

y sp

oons

alto

geth

er?

How

do

you

know

?

List

en t

o th

ese

clap

s. H

ow m

any

wer

e th

ere?

Now

tel

l me

how

man

y ex

tra

clap

s I m

ake.

How

man

y cl

aps

is t

hat

alto

geth

er?

Ther

e ar

e se

ven

grap

es in

my

lunc

h bo

x. H

owm

any

red

grap

es a

nd g

reen

gra

pes

coul

dth

ere

be?

Use

the

se c

ubes

to

show

how

you

did

it.

The

child

cou

nts

alou

d fro

m a

giv

enst

artin

g nu

mbe

r, st

oppi

ng a

t pa

rtic

ular

num

bers

. Wha

t nu

mbe

r co

mes

nex

t?W

hat

num

ber

com

es b

efor

e?

Rel

ate

coun

ting

to a

set

of o

bjec

ts, a

ddon

e m

ore

and

ask

child

ren

to id

entif

yho

w m

any,

sim

ilarly

one

less

; arr

ange

obje

cts

alon

gsid

e a

num

ber

trac

k an

dke

ep t

akin

g on

e aw

ay t

o lin

k on

e le

ss t

oth

e nu

mbe

r be

fore

and

sim

ilarly

one

mor

e to

the

nex

t nu

mbe

r.

Use

con

tain

ers,

suc

h as

an

egg

box,

tha

tho

ld a

giv

en n

umbe

r of

obj

ects

. Giv

e th

ech

ild g

roup

s of

obj

ects

to

coun

t, fo

rex

ampl

e fo

ur o

bjec

ts a

nd t

wo

obje

cts,

and

plac

e th

e ob

ject

s in

the

con

tain

er fo

rch

ildre

n to

cou

nt a

ll th

e ob

ject

s. H

ide

the

cont

aine

r an

d as

k ho

w m

any

obje

cts

itho

lds.

Rec

ount

, che

ck a

nd c

onfir

m.

Rep

eat

with

oth

er n

umbe

r pa

irs.

Arr

ange

a s

mal

l num

ber

of b

iscu

its o

ntw

o pl

ates

. Ask

the

chi

ld t

o co

unt

the

bisc

uits

on

each

pla

te a

nd s

ay h

ow m

any

ther

e ar

e al

toge

ther

. Mov

e bi

scui

tsbe

twee

n th

e pl

ates

and

rep

eat

activ

ity.

Wha

t is

one

less

than

six

? …

one

mor

e th

an fo

ur?

Ask

me

anot

her

pair

of th

ese

ques

tions

with

the

sam

e an

swer

.

Cou

nt e

very

oth

er n

umbe

r;id

entif

y od

d an

d ev

en n

umbe

rs;

coun

t in

twos

; add

and

take

away

two

obje

cts

and

find

two

mor

e an

d tw

o le

ss th

an a

giv

ennu

mbe

r; bu

ild o

n pa

ttern

reco

gniti

on to

intro

duce

thre

em

ore,

thre

e le

ss, e

tc.

Arr

ange

a s

mal

l num

ber

ofbi

scui

ts o

n tw

o pl

ates

so

that

one

plat

e ha

s tw

o m

ore

bisc

uits

than

the

oth

er. A

sk t

he c

hild

to

say

how

man

y bi

scui

ts o

n ea

chpl

ate

and

how

man

y al

toge

ther

.R

epea

t w

ith d

iffer

ent

num

bers

of

bisc

uits

as

appr

opria

te. E

xten

dto

thr

ee p

late

s.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills


Say

how

man

y ar

e le

ft w

hen

som

e ob

ject

s ar

e ta

ken

away

, by

coun

ting

how

man

y ob

ject

s ar

e le

ft.

4 Y

R

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(ad

dit

ion

and

sub

trac

tio

n)

Is n

ot c

onfid

ent

abou

tw

hen

to s

top

coun

ting

whe

n ta

king

aw

ay(s

ubtr

actin

g) in

ans

wer

to t

he q

uest

ion

‘How

man

y ar

e le

ft?’

4 Y

R +

/–

Ther

e ar

e si

x in

thi

s bo

x. I

take

aw

ay t

wo

and

put

them

her

e. H

ow m

any

are

left?

Her

e ar

e ei

ght

cups

. Thr

ee c

ups

are

full

and

the

rest

are

em

pty.

How

man

y ar

e em

pty?

We

ate

four

of t

he s

ix c

akes

we

mad

e. H

owm

any

are

left?

How

did

you

wor

k th

is o

ut?

Use

a r

ange

of o

bjec

ts a

nd r

esou

rces

such

as

bead

s on

a s

trin

g. In

vite

the

child

to

take

one

aw

ay a

nd c

ount

the

rest

. How

man

y ar

e le

ft? R

epea

t th

eac

tivity

, inv

iting

the

chi

ld t

o ta

ke a

way

diffe

rent

num

bers

and

des

crib

e w

hat

has

happ

ened

.

Em

phas

ise

that

it is

the

num

ber

ofob

ject

s le

ft, n

ot t

he o

bjec

ts t

hat

we

take

away

, tha

t is

impo

rtan

t.

Con

tinue

to a

sk q

uest

ions

that

invo

lve

findi

ng h

ow m

any

are

left

afte

r ta

king

aw

ay o

r su

btra

ctin

g;as

k th

e ch

ild to

find

thei

r ow

nw

ay o

f rec

ordi

ng th

eir

answ

ers

and

to d

escr

ibe

thei

r so

lutio

nsus

ing

a ra

nge

of v

ocab

ular

y su

chas

:6

take

aw

ay 4

is 2

6 su

btra

ct 4

is 2

Sta

rt w

ith a

num

ber

and

take

anu

mbe

r aw

ay. H

ow m

any

are

left?

Ask

the

child

to s

tart

with

anot

her

num

ber

and

take

som

eaw

ay to

leav

e th

e sa

me

num

ber

as b

efor

e. R

ecor

d th

eca

lcul

atio

ns b

efor

e re

peat

ing

the

activ

ity.

Ask

the

child

to c

omm

ent o

nw

hat t

hey

notic

e an

d ex

plai

nho

w th

ey c

hose

thei

r nu

mbe

rs.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills

Appendix 2

Tracking children’s learning chart – multiplication anddivision

The tracking charts act as an initial diagnostic tool and a background framework forthe sets of teaching materials referenced to common misconceptions and errors.

Teaching unit codes are referenced against the errors and misconceptions listed inthe chart, for example 1 Y6 �/�



Year

6 k

ey o

bje

ctiv

eC

arry

out

long

mul

tiplic

atio

n of

a t

hree

-dig

it by

a t

wo-

digi

t in

tege

r an

d sh

ort

mul

tiplic

atio

n an

d di

visi

on o

f who

le n

umbe

rs (N

NS

Fra

mew

ork

for

teac

hing

mat

hem

atic

s, S

uppl

emen

t of

Exa

mpl

es, S

ectio

n 6,

pag

es 6

7 an

d 69

)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Kno

w b

y he

art

all

mul

tiplic

atio

n fa

cts

up t

o 10

�10

.

Der

ive

divi

sion

fact

s co

rres

pond

ing

to

mul

tiplic

atio

n fa

cts

up

to 1

0�

10.

Mul

tiply

and

div

ide

one-

, tw

o- a

nd th

ree-

digi

t nu

mbe

rs b

y te

n an

d on

e hu

ndre

d.

App

ly th

e as

soci

ativ

e la

w (b

ut n

ot b

y na

me)

to

mul

tiply

up

to t

hree

-dig

it nu

mbe

rs b

y m

ultip

les

of t

en a

nd h

undr

ed,

for

exam

ple:

147

�20

=

147

�2

�10

1 Y

6

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Err

ors

and

mis

conc

epti

ons

Q

uest

ions

to

iden

tify

err

ors

and

mis

conc

epti

ons

Te

achi

ng t

o a

dd

ress

the

err

ors

and

mis

conc

epti

ons

N

ext

step

s in

mo

ving

tow

ard

s th

e ke

yo

bje

ctiv

e

Mis

uses

hal

f-un

ders

tood

rul

esab

out

mul

tiply

ing

and

divi

ding

by

pow

ers

ofte

n an

d th

eas

soci

ativ

e la

w, f

orex

ampl

e:14

5�

30 =

145

000

1 Y

6 �

/÷

Can

you

tel

l me

a qu

ick

way

of m

ultip

lyin

g a

num

ber

by o

ne t

hous

and?

I hav

e th

irty-

seve

n on

my

calc

ulat

or. W

hat

sing

lem

ultip

licat

ion

shou

ld I

key

in t

o ch

ange

it t

o th

ree

thou

sand

sev

en h

undr

ed?

Wha

t is

3�

1? …

30�

10?

…30

0 �

100?

Tell

me

abou

t th

is p

atte

rn.

If I h

ad fo

ur t

hous

and

eigh

t hu

ndre

d on

my

calc

ulat

or, w

hat

sing

le d

ivis

ion

coul

d I k

ey in

to

chan

ge t

he d

ispl

ay t

o fo

rty-

eigh

t?

47�

10 =

470

Wha

t do

you

thi

nk 4

7�

20 e

qual

s?

Mul

tiplic

atio

n by

ten

, one

hun

dred

… u

sing

aca

lcul

ator

and

rec

ordi

ng n

umbe

rs b

efor

e an

daf

ter

mul

tiplic

atio

n in

HTU

col

umns

.

Rep

eat

for

divi

sion

.

Ask

the

chi

ld t

o de

scrib

e pa

tter

ns o

bser

ved

and

sugg

est

a ge

nera

lisat

ion.

Ask

the

chi

ld t

oillu

stra

te w

ith fu

rthe

r ex

ampl

es.

Dem

onst

rate

tha

t 42

�12

is t

he s

ame

as

42�

6�

2. W

hat

othe

r w

ays

can

we

mul

tiply

fort

y-tw

o by

tw

elve

?

Rep

eat

with

oth

er n

umbe

rs. U

se fa

ctor

‘tre

e’ t

osu

ppor

t th

is, f

or e

xam

ple:

4�

3

126

�2

3�

4

2�

6

Wha

t’s t

he b

est

way

you

can

thin

k of

to

find:

71 �

20?

202

�6?

82 �

300?

Ref

er t

o Ye

ar 4

cha

rt (1

Y4

and

2 Y

4) w

here

thi

s is

des

crib

ed in

con

nect

ion

with

a li

mite

d ra

nge

of m

ultip

licat

ion

and

divi

sion

fact

s. A

t Ye

ar 6

leve

l, th

enu

mbe

r ra

nge

will

exte

nd t

o in

clud

e m

ultip

licat

ion

fact

s up

to

10�

10 a

nd r

elat

ed d

ivis

ion

fact

s.


Pre

sent

the

re

mai

nder

in a

qu

otie

nt a

s a

who

le

num

ber

or fr

actio

n.2

Y6

Div

idin

g by

num

bers

sm

alle

r th

an o

ne,

for

exam

ple:

12 �

= 2

46

�=

18.

3 Y

6

Judg

e w

heth

er t

he

answ

er t

o a

mul

tiplic

atio

n or

di

visi

on c

alcu

latio

n is

rea

sona

ble.

4

Y6

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Has

diff

icul

ty, w

hen

appr

opria

te, i

nter

pret

ing

a re

mai

nder

as

afra

ctio

n, fo

r ex

ampl

e:16

�3

= 5

2 Y

6 �

/�

Inte

rpre

ts d

ivis

ion

assh

arin

g bu

t no

t as

grou

ping

(rep

eate

dsu

btra

ctio

n) s

o is

unab

le t

o in

terp

ret

aca

lcul

atio

n su

ch a

s 12

�.

3 Y

6 �

/�

Is n

ot c

onfid

ent

inm

akin

g re

ason

able

estim

ates

for

mul

tiplic

atio

n or

div

isio

nca

lcul

atio

ns.

4 Y

6 �

/÷

Ther

e ar

e tw

enty

-six

app

les

for

four

hors

es, s

o th

ey c

an h

ave

six

and

a ha

lfea

ch.

How

do

I kno

w t

his?

How

man

y ha

lf to

mat

oes

can

you

get

from

thr

ee w

hole

tom

atoe

s?

How

man

y qu

arte

rs o

f piz

za c

an y

ou g

etfro

m fo

ur p

izza

s?

Exp

lain

how

to

wor

k ou

t 5

�,

5�

.

600

�30

= 2

Can

thi

s ca

lcul

atio

n be

cor

rect

?H

ow d

o yo

u kn

ow?

Try

som

e m

ore,

suc

h as

:54

0 �

20 =

52

24 �

20 =

480

015

�=

30

24 �

= 2

2.

Wha

t is

a s

ensi

ble

answ

er fo

r ea

ch o

f the

follo

win

g di

visi

on c

alcu

latio

ns?

1. D

ivid

e tw

enty

-eig

ht e

ggs

by s

ix.

2. D

ivid

e tw

enty

-sev

en b

ars

of c

hoco

late

into

four

.

3. D

ivid

e £4

7 by

four

.

4. D

ivid

e si

xty-

two

child

ren

into

tea

ms

of t

en.

Mod

el w

ith p

ract

ical

con

text

s an

d eq

uipm

ent,

calc

ulat

ions

suc

h as

:

How

man

y ha

lves

in s

ix?

…qu

arte

rs in

eig

ht?

Dem

onst

rate

ass

ocia

ted

reco

rdin

g.

Use

a v

arie

ty o

f im

ages

and

con

text

s to

illust

rate

, for

exa

mpl

e, ju

mps

on

a nu

mbe

r lin

e,ch

ocol

ate

bars

, etc

.

Use

exa

mpl

es o

f cal

cula

tions

to

mod

el m

akin

ges

timat

es, f

or e

xam

ple:

We

know

thi

rty-

five

divi

ded

by s

even

equ

als

five.

Is t

hirt

y-th

ree

divi

ded

by s

even

mor

e or

less

tha

n th

is?

How

can

we

deci

de?

Rep

eat

usin

g ca

lcul

atio

ns s

uch

as:

44 �

1020

7 �

20

Ask

the

chi

ld t

o ju

stify

the

est

imat

es t

hey

sugg

est.

Writ

e a

divi

sion

que

stio

n fo

r yo

urpa

rtne

r w

hich

you

kno

w w

illha

ve a

frac

tion

as p

art o

f the

answ

er.

How

do

you

know

it w

ill ha

ve a

fract

ion

as p

art o

f the

ans

wer

?

Whe

n I d

ivid

e m

y m

yste

rynu

mbe

r by

hal

f the

ans

wer

iste

n.W

hat’s

my

mys

tery

num

ber?

Mak

e up

som

e m

ore

ques

tions

with

mys

tery

num

bers

.

If th

e qu

estio

n is

fifty

-fou

r di

vide

dby

sev

en, w

hat n

umbe

r fa

cts

coul

d yo

u us

e to

hel

p yo

ues

timat

e an

ans

wer

?

Expl

ain

how

eac

h of

the

fact

syo

u su

gges

t wou

ld b

e us

eful

tohe

lp y

ou m

ake

an e

stim

ate.

Wha

t abo

ut th

e qu

estio

n fiv

ehu

ndre

d an

d fo

ur d

ivid

ed b

y si

x?

Cho

ose

som

e m

ore

ques

tions

tous

e.

1 4

1 3

1 2

1 3

1 2

1 2

1 21 2

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills


Year

4 k

ey o

bje

ctiv

eU

se in

form

al p

enci

l and

pap

er m

etho

ds t

o su

ppor

t, re

cord

or

expl

ain

mul

tiplic

atio

ns a

nd d

ivis

ions

, dev

elop

and

ref

ine

writ

ten

met

hods

for

TU �

U,

TU �

U, a

nd fi

nd r

emai

nder

s af

ter

divi

sion

(NN

S F

ram

ewor

k fo

r te

achi

ng m

athe

mat

ics,

Sup

plem

ent

of E

xam

ples

, sec

tion

6, p

ages

56,

66,

68)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Kno

w m

ultip

licat

ion

fact

s fo

r 2,

3, 4

, 5

and

10 t

imes

tab

les.

1 Y

4

Kno

w t

he d

ivis

ion

fact

s w

hich

co

rres

pond

with

m

ultip

licat

ion

fact

s lis

ted

abov

e.

Und

erst

and

the

inve

rse

rela

tions

hip

conn

ectin

g m

ultip

licat

ion

and

divi

sion

.2

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Err

ors

and

mis

conc

epti

ons

Is n

ot c

onfid

ent

inre

callin

g m

ultip

licat

ion

fact

s.1

Y4

�/�

Is m

uddl

ed a

bout

the

corr

espo

nden

cebe

twee

n m

ultip

licat

ion

and

divi

sion

fact

s,re

cord

ing,

for

exam

ple:

3 �

5 =

15

so 5

�15

=

3

2 Y

4 �

/�

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Wha

t is

thr

ee m

ultip

lied

by fo

ur?

Dra

wm

e a

diag

ram

to

show

wha

t th

ism

eans

.

How

do

you

know

the

ans

wer

to

calc

ulat

ions

suc

h as

thr

ee m

ultip

lied

by z

ero

and

ten

mul

tiplie

d by

zer

o?

Wha

t is

four

mul

tiplie

d by

one

? H

owdo

you

kno

w?

Tell

me

the

mul

tiplic

atio

n fa

cts

this

four

by fi

ve a

rray

sho

ws

(sup

port

thi

squ

estio

n w

ith a

dia

gram

of t

he a

rray

).A

nd t

he d

ivis

ion

fact

s?

If yo

u kn

ow t

hree

mul

tiplie

d by

five

equa

ls fi

fteen

, wha

t nu

mbe

r se

nten

ces

usin

g di

visi

on d

o yo

u al

so k

now

?

Dra

w a

pic

ture

for

thre

e m

ultip

lied

bysi

x.W

hat

othe

r m

ultip

licat

ion

and

divi

sion

fact

s ca

n yo

u fin

d fro

m t

he p

ictu

re?

Teac

hing

to

ad

dre

ss t

he e

rro

rs a

ndm

isco

ncep

tio

ns

Pro

vide

mod

els

and

imag

es t

o illu

stra

te t

hem

ultip

licat

ion

fact

s, fo

r ex

ampl

e m

ultip

legr

oups

of b

eads

on

a be

ad s

trin

g, ju

mps

alon

g a

num

ber

line.

Use

rel

evan

t ro

ws

from

a m

ultip

licat

ion

squa

reas

a r

esou

rce

on w

hich

to

focu

s qu

estio

nsab

out

the

mul

tiplic

atio

n fa

cts

conn

ecte

d to

part

icul

ar n

umbe

rs, i

llust

rate

d w

ith a

gro

win

gar

ray

as a

con

nect

ed im

age,

for

exam

ple:

. . .

.. .

. .

. . .

.. .

. .

. . .

.. .

. .

. . .

.. .

. .

. . .

.. .

. .

Sho

w t

he c

hild

a t

hree

row

by

five

colu

mn

arra

y, m

ade

from

cou

nter

s la

id o

ut.

Dem

onst

rate

the

mul

tiplic

atio

n fa

ct t

hree

mul

tiplie

d by

five

equ

als

fifte

en fr

om t

hepa

tter

n:3

�5

= 1

5

How

cou

ld I

expl

ain

this

pat

tern

as

a di

visi

onfa

ct?

Sho

w m

e fro

m t

he c

ount

er p

atte

rn h

ow y

oukn

ow y

our

divi

sion

fact

?

Nex

t st

eps

in m

ovi

ng t

ow

ard

s th

eke

y o

bje

ctiv

e

Cov

er a

cris

s-cr

oss

patt

ern

with

are

ctan

gle

of c

ard

or p

aper

. Ask

the

child

to

imag

ine

the

cros

sove

rs(k

nots

, per

haps

). H

ow m

any

lines

acro

ss?

How

man

y cr

osso

vers

(kno

ts) i

n ea

ch?

Wha

t m

ultip

licat

ion

fact

will

tell

you

how

man

ycr

osso

vers

are

und

er t

he p

aper

?

Ask

the

chi

ld t

o m

ake

up h

idde

ncr

osso

ver

patt

erns

of t

heir

own

and

say

the

acco

mpa

nyin

g m

ultip

licat

ion

fact

s.

Wor

king

with

the

chi

ld, s

et o

ut t

hirt

y-fiv

e co

unte

rs in

a fi

ve b

y se

ven

arra

y.A

sk t

he c

hild

to

expl

ain

how

the

arr

ayca

n be

see

n to

rep

rese

nt t

wo

mul

tiplic

atio

n an

d tw

o di

visi

onca

lcul

atio

ns.

Ask

the

chi

ld t

o lis

t th

e ca

lcul

atio

ns.

48

1216


Und

erst

and

the

effe

ct o

f m

ultip

lyin

g w

hole

nu

mbe

rs le

ss t

han

a th

ousa

nd b

y te

n.

3 Y

4

Can

app

ly t

he

dist

ribut

ive

law

(but

not

by

nam

e) t

o m

ultip

lyin

g,

usin

g pa

rtiti

onin

g an

d re

com

bini

ng, f

or e

xam

ple:

14 �

3 =

(10

�3)

+

(4 �

3)=

30

+ 1

2 =

42,

so�

104

330

12

Kno

w h

ow t

o re

cord

TU

�U

mul

tiplic

atio

n ca

lcul

ated

by

a pa

rtiti

onin

g m

etho

d in

a g

rid fo

rmat

. 4

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Des

crib

es t

he o

pera

tion

of m

ultip

lyin

g by

ten

as

‘add

ing

a no

ught

’.3

Y4

�/�

Doe

s no

t ap

ply

part

ition

ing

and

reco

mbi

ning

whe

nm

ultip

lyin

g,fo

r ex

ampl

e:14

�3

is c

alcu

late

d as

(10

�3)

+ 4

= 3

4,or

�1

43

312

14 �

3 =

312

,co

nfus

ing

the

valu

e of

two-

digi

t nu

mbe

rs.

4 Y

4 �

/�

Wha

t is

the

ans

wer

to fo

rty-

six

mul

tiplie

dby

ten

? …

thre

ehu

ndre

d an

d fif

ty-o

nem

ultip

lied

by t

en?

How

do

you

know

?

Wha

t nu

mbe

r w

ould

you

part

ition

to

wor

kou

t tw

enty

-sev

enm

ultip

lied

by t

hree

?H

ow w

ould

you

reco

mbi

ne y

our

calc

ulat

ions

? C

an y

ou s

pot

the

mis

take

in t

his

calc

ulat

ion:

15

�7

= (1

0 �

7) +

5

= 7

5?

Mul

tiply

by

ten

usin

g B

ase

10 a

ppar

atus

on

aTh

HTU

boa

rd. F

or e

xam

ple,

put

fort

y-si

x on

the

boar

d (fo

ur t

ens,

six

one

s) a

nd la

bel w

ith d

igit

card

s. M

ultip

ly e

ach

piec

e by

ten

, mov

e th

e pi

eces

into

the

cor

rect

pos

ition

s to

bec

ome

four

hund

reds

, six

ten

s, z

ero

units

and

labe

l the

resu

lting

num

ber

with

dig

it ca

rds.

Rep

eat

for

othe

rtw

o-di

git

num

bers

.

Use

pla

ce v

alue

(arr

ow) c

ards

, mul

tiply

ing

the

num

ber

on e

ach

plac

e va

lue

card

by

ten.

Rep

lace

each

car

d by

the

mul

tiplie

d by

ten

vers

ion.

Brin

g th

eca

rds

toge

ther

to m

ake

the

com

plet

ed m

ultip

licat

ion.

Rol

l a d

ice

thre

e tim

es t

o ge

nera

te a

ran

ge o

fm

ultip

licat

ion

calc

ulat

ions

, for

exa

mpl

e 3,

5, 6

.

Rec

ord

36 �

5 to

dem

onst

rate

and

exp

lain

part

ition

ing

and

reco

mbi

ning

.

Ask

chi

ld t

o m

ake

anot

her

diffe

rent

mul

tiplic

atio

nca

lcul

atio

n to

dem

onst

rate

and

exp

lain

. Use

pla

ceva

lue

(arr

ow) c

ards

to

dem

onst

rate

par

titio

ning

of

two-

digi

t nu

mbe

rs p

rior

to m

ultip

licat

ion

and

agai

nto

dem

onst

rate

rec

ombi

ning

.

Ext

end

this

act

ivity

to

wor

k w

ith t

hree

-di

git

num

bers

mul

tiplie

d by

ten

, and

the

ntw

o-di

git

num

bers

mul

tiplie

d by

ten

the

nm

ultip

lied

by t

en a

gain

.

Enc

oura

ge c

hild

ren

to r

ecor

d th

eca

lcul

atio

ns a

nd s

ee t

he p

atte

rn a

sfo

llow

s: 46 �

10 =

46

046

0 �

10 =

460

046

00 �

10 =

460

00

Ext

end

to u

sing

par

tially

com

plet

edca

lcul

atio

ns fo

r ch

ild t

o co

mpl

ete,

for

exam

ple:

16 �

3 =

(10

�) +

(

�3)

= 3

0+

=

48

�10

318

Enc

oura

ge t

he c

hild

to

mak

e a

part

ially

com

plet

ed m

ultip

licat

ion

calc

ulat

ion

for

apa

rtne

r to

sol

ve.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

to

war

ds

the

kno

wle

dg

e an

d m

isco

ncep

tio

nser

rors

and

an

d m

isco

ncep

tio

nske

y o

bje

ctiv

esk

ills

mis

conc

epti

ons


Rec

ogni

se t

hat

the

com

mut

ativ

e la

w

hold

s fo

r m

ultip

licat

ion

but

not

for

divi

sion

. 5

Y4

Und

erst

and

the

idea

of r

emai

nder

, an

d w

hen

to r

ound

up

or

dow

n af

ter

divi

sion

. 6a

Y4

6b Y

4 6c

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Ass

umes

tha

t th

e co

mm

utat

ive

law

hol

ds fo

r di

visi

on a

lso,

for

exam

ple,

ass

umin

g th

at

15 �

3 =

5, s

o 3

� 1

5 =

5.

5 Y

4 �

/�

Writ

es a

rem

aind

er t

hat

is la

rger

than

the

div

isor

, for

exa

mpl

e,36

�7

= 4

rem

aind

er 8

.6a

Y4

�/�

Dis

card

s th

e re

mai

nder

;doe

s no

tun

ders

tand

its

sign

ifica

nce.

6b Y

4 �

/�

Doe

s no

t re

cogn

ise

whe

n a

rem

aind

er is

sig

nific

ant

in t

hede

cisi

on a

bout

whe

ther

to r

ound

up o

r do

wn.

6c Y

4 �

/�

How

do

you

wor

k ou

t yo

ur a

nsw

er t

oth

ree

divi

ded

by fi

fteen

? Ta

ke s

ome

cube

s, o

r us

e a

diag

ram

to

expl

ain

wha

t th

is c

alcu

latio

n m

eans

With

29p

to

spen

d, y

ou w

ant

to b

uyas

man

y sw

eets

at

3p e

ach

as y

ouca

n af

ford

? H

ow m

any

swee

ts c

anyo

u bu

y? S

how

you

r w

orki

ng o

ut o

n a

num

ber

line.

Whi

ch o

f the

se c

alcu

latio

ns is

corr

ect?

36 �

7 =

4 r

emai

nder

836

�7

= 3

rem

aind

er 1

536

�7

= 5

rem

aind

er 1

36 �

7 =

5.

Whi

ch d

o yo

u th

ink

is t

he b

est

one?

Why

?

Ther

e ar

e th

irty-

two

child

ren

and

each

tent

tak

es t

hree

chi

ldre

n. W

hat

is t

hele

ast

num

ber

of t

ents

the

y w

ill ne

ed?

Use

obj

ects

and

a n

umbe

r lin

e to

wor

kou

t ca

lcul

atio

ns s

uch

as 1

6�4,

rea

ding

this

as

divi

de s

ixte

en in

to g

roup

s of

four

.H

ow m

any

grou

ps a

re t

here

? Th

enco

mpa

re t

his

to 4

�16

rea

ding

thi

s as

divi

de fo

ur in

to g

roup

s of

six

teen

. How

man

y gr

oups

are

the

re?

Hig

hlig

ht t

hat

this

is n

ot t

he s

ame.

Use

a n

umbe

r lin

e to

hig

hlig

ht t

he id

ea o

fa

rem

aind

er. D

emon

stra

te r

epea

ted

subt

ract

ion

by ‘s

tepp

ing

back

’, fo

rex

ampl

e, 1

7 �

5 ca

n be

rep

rese

nted

by:

Pro

vide

opp

ortu

nitie

s to

con

side

r ho

wre

sults

of c

alcu

latio

ns m

ay b

e ro

unde

d up

or d

own

in o

rder

to

mak

e se

nse

of t

heco

ntex

t. Fo

r ex

ampl

e:

•H

ow m

any

tabl

es a

re n

eede

d to

sea

t a

clas

s of

thi

rty-

two

child

ren

if fiv

ech

ildre

n ca

n si

t at

a t

able

? Th

is w

ould

invo

lve

roun

ding

up

to s

eat

all t

hech

ildre

n.•

How

man

y £5

boo

ks c

an b

e bo

ught

with

£32

? Th

is w

ould

invo

lve

roun

ding

dow

n to

ens

ure

enou

gh m

oney

to

pay

for

the

purc

hase

.

Pro

vide

a r

ange

of d

ivis

ion

calc

ulat

ions

to

be s

orte

d in

toth

ose

with

ans

wer

s th

at a

rew

hole

num

bers

and

tho

sein

volv

ing

fract

ions

, for

exa

mpl

e,

10 �

5 =

5

� 1

0 =

16 �

8 =

8

� 1

6 =

Ext

end

to t

he c

onte

xt o

f mon

ey,

wei

ght

and

leng

th t

ode

mon

stra

te h

ow t

he r

emai

nder

can

be s

ubdi

vide

d in

to s

mal

ler

units

. For

exa

mpl

e, £

60 s

hare

deq

ually

bet

wee

n ei

ght.

The

rem

aind

er o

f £4

can

be d

ivid

edfu

rthe

r to

giv

e an

ans

wer

of

£7.5

0 ea

ch.

-5

-5

-

5-2

0

2

7

1

2

1

7

Ass

oci

ated

Err

ors

and

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

to

war

ds

kno

wle

dg

e m

isco

ncep

tio

nsm

isco

ncep

tio

nsan

d m

isco

ncep

tio

nsth

e ke

y o

bje

ctiv

esk

ills


Kno

w h

ow t

o re

cord

di

visi

on a

s re

peat

ed

subt

ract

ion,

with

ap

prop

riate

use

of

chun

king

. 7

Y4

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Con

tinue

s to

sub

trac

t tw

osw

hen

calc

ulat

ing

twen

ty d

ivid

edby

tw

o w

ithou

t us

ing

know

ledg

eth

at t

wo

mul

tiplie

d by

five

equ

als

ten.

7

Y4

�/�

Ask

the

chi

ld t

o lo

ok b

ack

at a

divi

sion

cal

cula

tion

they

hav

e ju

stco

mpl

eted

. Can

you

thi

nk o

f a q

uick

erw

ay t

o fin

d ou

t an

ans

wer

for

this

ques

tion?

Use

a b

ead

strin

g to

illu

stra

te:

•H

ow t

hirt

y di

vide

d by

tw

o ca

n be

calc

ulat

ed a

s th

irty

with

ten

sub

trac

ted

thre

e tim

es. E

very

ten

is fi

ve t

wos

.A

ltoge

ther

fifte

en t

wos

hav

e be

ensu

btra

cted

so

thirt

y di

vide

d by

tw

oeq

uals

fifte

en.

•H

ow t

o w

ork

out

seve

nty-

six

divi

ded

byfiv

e by

sta

rtin

g fro

m s

even

ty-s

ix b

eads

,re

mov

e te

n lo

ts o

f fiv

e, t

hat

is fi

fty, a

ndth

en fi

ve lo

ts o

f fiv

e, t

hat

is t

wen

ty-f

ive.

Alto

geth

er h

ow m

any

grou

ps o

f fiv

eha

ve b

een

slid

alo

ng t

he b

ead

strin

g?

76 -50

10 �

526 -2

55

�5

1so

76

�5

= 1

5 re

mai

nder

1

Ask

the

chi

ld t

o w

ork

out

fifty

-th

ree

divi

ded

by fo

ur u

sing

know

ledg

e of

the

four

tim

esta

ble,

and

to

reco

rd t

his

asre

peat

ed s

ubtr

actio

n us

ing

not

mor

e th

an t

hree

ste

ps.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills


Year

2 k

ey o

bje

ctiv

eU

nder

stan

d th

e op

erat

ion

of m

ultip

licat

ion

as r

epea

ted

addi

tion

or a

s de

scrib

ing

an a

rray

, and

beg

in to

und

erst

and

divi

sion

as

grou

ping

or

shar

ing;

kno

w b

y he

art

fact

s fo

r th

e 2

and

10 m

ultip

licat

ion

tabl

es; a

nd k

now

and

use

hal

ving

as

the

inve

rse

of d

oubl

ing.

(NN

S F

ram

ewor

k fo

r te

achi

ng m

athe

mat

ics,

Sup

plem

ent

of E

xam

ples

, se

ctio

n 5,

pag

es 4

7, 4

9, 5

3, 5

7)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Car

ry o

ut r

epea

ted

addi

tion,

rec

ogni

se

the

rela

tions

hip

betw

een

mul

tiplic

atio

n an

d re

peat

ed a

dditi

on,

and

use

the

asso

ciat

ed

voca

bula

ry o

f m

ultip

licat

ion.

1 Y

2

Mul

tiply

tw

o or

ten

by

a s

ingl

e-di

git

num

ber,

by c

ount

ing

up in

tw

os a

nd t

ens

from

zer

o.2

Y2

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Err

ors

and

mis

conc

epti

ons

Stil

l cou

nts

in o

nes

tofin

d ho

w m

any

ther

e ar

ein

a c

olle

ctio

n of

equ

algr

oups

; doe

s no

tun

ders

tand

voc

abul

ary,

for

exam

ple,

‘gro

ups

of’,

‘mul

tiplie

d by

’. 1

Y2

�/�

Doe

s no

t lin

k co

untin

gup

in e

qual

ste

ps t

o th

eop

erat

ion

ofm

ultip

licat

ion;

doe

s no

tus

e th

e vo

cabu

lary

asso

ciat

ed w

ithm

ultip

licat

ion.

2

Y2

�/�

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Sho

w a

n ar

ray

of s

ix r

ows

of t

wo.

How

man

y se

ts o

f tw

o ca

n yo

u se

e?H

ow m

any

are

ther

e al

toge

ther

?R

epea

t w

ith s

ix r

ows

of t

en.

How

can

you

qui

ckly

wor

k ou

t tw

om

ultip

lied

by s

even

?

Wha

t st

eps

coul

d yo

u co

unt

up in

to

help

you

?

How

man

y st

eps

do w

e ne

ed?

Teac

hing

to

ad

dre

ss t

he e

rro

rs a

ndm

isco

ncep

tio

ns

Rev

eal a

n ar

ray

of t

wos

, one

row

at

a tim

e.E

ncou

rage

the

chi

ld t

o im

agin

e th

e ne

xt r

owan

d to

say

how

man

y th

ey w

ill se

e al

toge

ther

whe

n th

at r

ow is

rev

eale

d. R

epea

t w

ith p

airs

of c

ount

ers

set

out

alon

g a

num

ber

line.

Enc

oura

ge t

he c

hild

to

step

ove

r th

e pa

irsw

ith t

heir

finge

r as

the

y co

unt

up in

tw

os.

How

man

y tw

os?

Wha

t ca

lcul

atio

n ha

ve w

eju

st d

one?

Usi

ng s

even

pap

er s

trip

s ea

ch s

how

ing

two

dots

, mak

e an

arr

ay s

even

row

s by

tw

oco

lum

ns. W

ith t

he c

hild

, arr

ange

the

str

ips

into

a ‘n

umbe

r lin

e’ o

f sev

en p

airs

of d

ots.

Cou

nt u

p in

tw

os t

o fo

urte

en. C

ount

the

ste

psan

d re

cord

2 �

7 =

14.

Rea

rran

ge t

he s

trip

sba

ck in

to t

he s

even

by

two

arra

y an

dem

phas

ise

the

link

to t

wo

mul

tiplie

d by

sev

eneq

uals

four

teen

. Add

and

rem

ove

strip

s an

dre

peat

.

Nex

t st

eps

in m

ovi

ng t

ow

ard

s th

eke

y o

bje

ctiv

e

Enc

oura

ge c

ount

ing

alou

d in

tw

osan

d te

ns u

sing

imag

es a

nd r

esou

rces

such

as

num

ber

lines

, cou

nter

s an

dar

rays

.

Mak

e th

e co

nnec

tion

betw

een

the

mul

tiplic

atio

n fa

cts

writ

ten

as a

tab

le:

2 �

1 =

22

�2

= 4

, etc

.

Dis

play

the

num

ber

stat

emen

t 10

�6

= ?

Ask

the

chi

ld t

o us

e pa

per

strip

s w

ithte

n do

ts o

n ea

ch, t

o fo

rm a

n ar

ray

that

rep

rese

nts

ten

mul

tiplie

d by

six

.

How

man

y do

ts a

re t

here

?

Wha

t if

you

wer

e to

arr

ange

the

str

ips

in a

num

ber

line?

How

man

y do

tsw

ould

the

re b

e? H

ow d

o yo

u kn

ow?

Ask

the

chi

ld t

o ch

oose

a d

iffer

ent

arra

y.


Inte

rpre

t an

arr

ay a

s a

repe

ated

add

ition

an

d as

a

mul

tiplic

atio

n, a

nd

reco

gnis

e ho

w t

he

arra

y ca

n be

de

scrib

ed a

s, fo

r ex

ampl

e,

3 +

3 +

3 +

3, 3

�4,

or

as

4 +

4 +

4, 4

�3.

3 Y

2

Rec

ogni

se t

hat

doub

ling

and

mul

tiply

ing

by t

wo

are

the

sam

e, a

nd

use

know

n m

ultip

licat

ion

fact

s an

d pa

rtiti

onin

g to

do

uble

num

bers

to

fifte

en.

4a Y

24b

Y2

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Doe

s no

t fo

cus

on‘ro

ws

of’ o

r ‘c

olum

nsof

’, bu

t on

ly s

ees

anar

ray

as a

col

lect

ion

ofon

es.

3 Y

2 �

/�

Has

diff

icul

ty r

elat

ing

mul

tiply

ing

by t

wo

tokn

own

fact

s ab

out

doub

les;

rec

ords

doub

le fo

ur a

s 4

+ 4

.4a

Y2

�/�

Doe

s no

t us

epa

rtiti

onin

g to

find

doub

le t

wel

ve o

r do

uble

thirt

y-fiv

e.4b

Y2

�/�

Dis

play

six

row

s of

tw

o. H

ow m

any

row

s ar

e th

ere?

How

man

y do

ts a

reth

ere

in e

ach?

How

can

you

des

crib

eth

e ar

ray?

Turn

the

arr

ay t

hrou

gh 9

0°. H

ow h

asth

e ar

ray

chan

ged?

Can

you

des

crib

eth

e ar

ray

to m

e no

w?

Wha

t’s t

he s

ame

and

wha

t’s d

iffer

ent?

Wha

t is

six

mul

tiplie

d by

tw

o? H

owdi

d yo

u w

ork

it ou

t? W

hat

is d

oubl

esi

x? W

hat

did

you

notic

e?

Wha

t is

the

ans

wer

to

doub

le t

en?

…do

uble

four

?H

ow c

an w

e us

e th

ese

answ

ers

tofin

d do

uble

four

teen

?

How

can

we

wor

k ou

t do

uble

thi

rty

five?

Dis

play

the

firs

t ro

w o

f a fo

ur-b

y-fiv

ear

ray.

Her

e ar

e fiv

e do

ts. T

here

are

four

row

s, w

hich

are

exa

ctly

the

sam

e.M

ake

the

arra

y w

ith t

hese

cou

nter

s. T

ell

me

abou

t yo

ur a

rray

. Wha

t nu

mbe

rsco

uld

you

add

toge

ther

to

find

the

tota

l? W

hat

num

bers

cou

ld y

oum

ultip

ly?

Use

a b

ead

strin

g to

mak

e a

set

of fi

vebe

ads.

Ask

wha

t do

uble

thi

s se

t of

bead

s w

ould

be.

Mak

e an

othe

r se

t of

five

bead

s. U

se t

he v

ocab

ular

y do

uble

five

bead

s, fi

ve b

eads

tw

ice,

five

bea

dsm

ultip

lied

by t

wo.

Agr

ee t

hat

ther

e ar

ete

n be

ads

in e

ach

case

and

rec

ord:

D

oubl

e 5

= 1

0 an

d 5

�2

= 1

0.

Use

the

bea

ds t

o do

uble

ten

.

Mak

e a

set

of fi

fteen

bea

ds. S

epar

ate

into

ten

and

five

and

dou

ble

the

ten

and

five.

Com

bine

to

mak

e th

irty.

Rec

ord:

D

oubl

e 15

= D

oubl

e 10

+ D

oubl

e 5

= 2

0 +

10

= 3

0.

Rep

eat

for

othe

r tw

o-di

git

num

bers

.

Rev

eal t

he fi

rst

row

and

firs

t co

lum

n of

afo

ur-b

y-fiv

e ar

ray

of d

ots.

Des

crib

e th

is a

rray

to

me.

Writ

e a

num

ber

sent

ence

to

mat

ch t

he a

rray

.

Mak

e up

a n

umbe

r sen

tenc

e th

at m

atch

esan

othe

r ar

ray

and

draw

the

arr

ay.

Illus

trat

e th

e fo

llow

ing

with

Bas

e ap

para

tus

and

ask

the

child

to

cont

inue

the

sepa

tter

ns, u

sing

par

titio

ning

as

appr

opria

te.

2 +

2 =

43

+ 3

= 6

4 +

4 =

8, e

tc.

10 +

10

= 2

011

+ 1

1 =

22

12 +

12

= 2

413

+ 1

3 =

26,

etc

.

20 +

20

= 4

025

+ 2

5 =

50

30 +

30

= 6

035

+ 3

5 =

70,

etc

.

Use

app

ropr

iate

voc

abul

ary

as y

ou r

ecor

dth

ese

patt

erns

as

mul

tiplic

atio

ns, f

orex

ampl

e:3

�2

= 6

13 �

2 =

26

25 �

2 =

50

• •

•

•

•• • •

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills


Rec

ogni

ses

that

w

hen

findi

ng t

he

doub

le o

f a n

umbe

r, ha

lf th

e an

swer

is

the

orig

inal

num

ber.

Use

s th

e in

vers

e of

do

uble

to fi

nd h

alve

s of

sm

all e

ven

num

bers

to

ten,

us

ing

know

n fa

cts.

5

Y2

Sha

re a

giv

en

num

ber

of o

bjec

ts

out

equa

lly,

reco

gnis

e th

e re

latio

nshi

p be

twee

n sh

arin

g eq

ually

and

di

visi

on a

nd u

se t

he

voca

bula

ry o

f div

isio

n to

des

crib

e th

e pr

oces

s, fo

r ex

ampl

e ‘d

ivid

e by

’, ‘s

hare

eq

ually

’.6

Y2

Beg

in t

o un

ders

tand

di

visi

on a

s re

peat

ed

subt

ract

ion

or

grou

ping

.7

Y2

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Doe

s no

t us

ekn

owle

dge

of d

oubl

esto

find

hal

f of a

num

ber;

for

exam

ple,

con

tinue

sto

find

hal

f by

shar

ing,

usin

g a

‘one

for

you’

appr

oach

and

can

not

appl

y kn

owle

dge

ofdo

uble

s.5

Y2

�/�

Is n

ot s

yste

mat

ic w

hen

shar

ing

into

equ

algr

oups

, usi

ng a

‘one

for

you’

app

roac

h; d

oes

not

use

the

lang

uage

of

divi

sion

to

desc

ribe

the

proc

ess.

6 Y

2 �

/�

Doe

s no

t un

ders

tand

that

‘set

s of

’ or

‘gro

ups

of’ n

eed

to b

esu

btra

cted

to

solv

e th

epr

oble

m.

7 Y

2 �

/�

Wha

t do

you

thi

nk h

alf o

f ten

is?

How

do y

ou k

now

?

Can

you

thi

nk o

f a n

umbe

r th

at’s

eas

yto

hal

ve?

Why

do

you

thin

k it’

s ea

sy t

oha

lve

your

num

ber?

Her

e ar

e tw

elve

cou

nter

s, s

hare

the

mou

t eq

ually

into

the

se t

hree

box

es. H

owm

any

coun

ters

are

the

re in

eac

h bo

x?H

ere

are

six

coin

s, c

an y

ou d

ivid

e th

embe

twee

n th

ree

purs

es?

How

man

y w

illbe

in e

ach

purs

e?

Alto

geth

er w

e ha

d ei

ghte

en c

ount

ers

and

ther

e ar

e si

x in

eac

h bo

x, c

an y

oude

scrib

e w

hat

we

have

don

e?

Ther

e ar

e fo

urte

en c

hild

ren

and

the

child

ren

are

aske

d to

wor

k in

pai

rs(tw

os).

How

man

y pa

irs a

re t

here

?

Tell

the

child

you

hav

e th

ree

coun

ters

inyo

ur le

ft ha

nd a

nd t

hree

in y

our

right

hand

. How

man

y co

unte

rs h

ave

Ial

toge

ther

? G

ive

the

child

eig

htco

unte

rs. Y

ou h

ave

doub

le t

he c

ount

ers

I hav

e in

my

hand

, how

man

y ha

ve I

got?

S

how

four

num

bers

. Wha

t is

dou

ble

four

? W

hat

is h

alf e

ight

? et

c.

Sha

re a

set

of p

ictu

re c

ards

equ

ally

betw

een

a nu

mbe

r of

pla

yers

. Des

crib

eth

e pr

oces

s us

ing

the

corr

ect

lang

uage

of d

ivis

ion,

for

exam

ple

‘twen

ty c

ards

divi

ded

betw

een

four

pla

yers

giv

es e

ach

play

er fi

ve c

ards

’.

Sho

w t

he c

hild

tw

elve

cub

es. A

sk t

hem

to t

ake

two

cube

s an

d m

ake

a to

wer

.R

epea

t to

mak

e an

othe

r to

wer

and

agai

n to

mak

e a

third

tow

er. M

odel

the

lang

uage

at

each

sta

ge, f

or e

xam

ple

‘four

cub

es m

akes

tw

o to

wer

s’. T

hen

ask

the

child

how

man

y to

wer

s of

tw

oth

ey c

an m

ake

usin

g tw

elve

cub

es,

show

ing

the

arra

y of

tw

o ro

ws

by s

ixco

lum

ns s

o th

at t

he t

hey

can

rela

te t

hepr

oble

m t

o a

visu

al im

age.

I thi

nk o

f a n

umbe

r an

d do

uble

it a

ndm

y an

swer

is e

ight

. Wha

t nu

mbe

r w

as I

thin

king

of?

At

the

cam

p th

ere

are

six

tent

s. T

hech

ildre

n w

onde

r ho

w m

any

will

besl

eepi

ng in

eac

h te

nt. T

hey

know

tha

tth

ere

are

eigh

teen

chi

ldre

n in

the

gro

up,

and

ther

e ha

s to

be

the

sam

e nu

mbe

r in

each

ten

t. H

ow m

any

slee

p in

eac

hte

nt?

Invi

te t

he c

hild

to

use

twel

ve c

ubes

to

mak

e to

wer

s of

thr

ee a

nd a

sk t

hem

to

expl

ain

how

man

y to

wer

s ca

n be

mad

e.R

epea

t w

ith t

he t

ower

s of

four

and

six

.

Ask

the

chi

ld t

o ch

ose

an e

ven

num

ber

of c

ubes

and

pre

dict

how

man

y to

wer

sof

tw

o th

ey w

ould

be

able

to

mak

e. G

etth

em t

o ch

eck

thei

r an

swer

aga

inst

the

irpr

edic

tion

and

refle

ct o

n w

hat

they

notic

e.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills


Rec

epti

on

key

ob

ject

ive

Use

dev

elop

ing

mat

hem

atic

al id

eas

and

met

hods

to

solv

e pr

actic

al p

robl

ems

(NN

S F

ram

ewor

k fo

r te

achi

ng m

athe

mat

ics,

Sup

plem

ent

of E

xam

ples

,se

ctio

n 4,

pag

e 20

)

Ass

oci

ated

kn

ow

led

ge

and

sk

ills

Cou

nt in

tw

os a

nd

talk

abo

ut h

ow

man

y pa

irs.

1 Y

R

Say

how

man

y al

toge

ther

in a

do

uble

; rec

ogni

se

that

find

ing

a do

uble

m

eans

form

ing

a pa

ir or

add

ing

a nu

mbe

r to

itse

lf.

2 Y

R

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Err

ors

and

mis

conc

epti

ons

Con

fuse

s nu

mbe

rsw

hen

coun

ting

in t

wos

;ha

s di

fficu

ltyun

ders

tand

ing

a pa

irco

nsis

ts o

f tw

o ob

ject

s.1

YR

�/�

Has

diff

icul

ty w

ithid

entif

ying

dou

bles

and

addi

ng a

sm

all n

umbe

rto

itse

lf, fo

r ex

ampl

e 2

+ 2

, to

mak

e tw

ice

asm

any.

2 Y

R �

/�

Que

stio

ns t

o id

enti

fy e

rro

rs a

ndm

isco

ncep

tio

ns

Can

you

mat

ch t

his

sock

/glo

ve t

o m

ake

a pa

ir?

Can

you

find

som

e m

ore

pairs

? H

ow

man

y pa

irs h

ave

you

mad

e?

With

in r

ole

play

, eng

age

with

chi

ld a

ndas

k qu

estio

ns, f

or e

xam

ple:

If th

ere

are

two

whe

els

on t

his

scoo

ter,

how

man

y w

heel

s on

tw

o sc

oote

rs?

How

man

y gl

oves

mak

e a

pair?

Teac

hing

to

ad

dre

ss t

he e

rro

rs a

ndm

isco

ncep

tio

ns

Find

mat

chin

g pa

irs fr

om a

col

lect

ion

ofpi

ctur

es o

r ob

ject

s, in

trod

ucin

g th

evo

cabu

lary

with

in t

he c

onte

xt o

f the

activ

ity a

nd e

ncou

ragi

ng t

he c

hild

to

use

the

corr

ect

voca

bula

ry.

Sho

w o

ne a

nim

al/o

bjec

t an

d ad

d a

seco

nd o

ne (d

oubl

e it)

. Str

ess

voca

bula

ry a

nd e

ncou

rage

chi

ld t

oex

plai

n w

hat

you

are

doin

g.

Nex

t st

eps

in m

ovi

ng t

ow

ard

s th

eke

y o

bje

ctiv

e

Pla

y sn

ap/p

elm

anis

m a

nd c

ount

in t

wos

to s

ee h

ow m

any

pairs

of c

ards

you

have

mad

e.

How

man

y pa

irs o

f eye

s on

the

se t

hree

face

s? If

you

dra

w fo

ur fa

ces

how

man

ypa

irs o

f eye

s w

ould

you

nee

d? H

owm

any

eyes

? W

hat

if yo

u dr

aw…

?

Ext

end

to n

on-m

atch

ing

pairs

, for

exam

ple

pairs

of c

hild

ren.

Ext

end

to a

ran

ge o

f dou

bles

con

text

s,fo

r ex

ampl

e:

Put

four

cow

s in

the

fiel

d, n

ow p

utdo

uble

tha

t nu

mbe

r in

the

nex

t fie

ld.

How

man

y al

toge

ther

? W

hat

is d

oubl

efo

ur?

Wha

t if

you

put

five

cow

s in

the

field

thi

s tim

e an

d do

uble

tha

t nu

mbe

r in

the

next

fiel

d…?


Sol

ve s

impl

e pr

oble

ms

(whe

re

ther

e ar

e no

re

mai

nder

s) t

hat

invo

lve

coun

ting

the

num

bers

of o

bjec

ts

by o

rgan

isin

g th

em

into

gro

ups

of e

qual

si

ze, i

nclu

ding

mor

e th

an t

wo

grou

ps;

talk

abo

ut t

he

num

ber

of g

roup

s an

d ob

ject

s, a

nd

reco

rd t

heir

answ

ers

in t

heir

own

way

.3

YR

Sha

re o

bjec

ts o

ut

fairl

y in

to t

wo

or

mor

e gr

oups

, tal

k ab

out

whe

ther

the

re

will

be a

ny le

ft ov

er.

4 Y

R

Cou

nt in

ten

s fo

rwar

ds a

nd

back

war

ds fr

om a

te

ns n

umbe

r (i.

e.

mul

tiple

of t

en),

and

iden

tify

the

tens

nu

mbe

r be

fore

and

af

ter

a gi

ven

tens

nu

mbe

r.5

YR

Sol

ve p

robl

ems

that

invo

lve

findi

ng

halv

es.

6 Y

R

Trac

king

chi

ldre

n’s

lear

ning

thr

oug

h th

e N

NS

Fra

mew

ork

for

teac

hin

g m

ath

emat

ics

(mul

tip

licat

ion

and

div

isio

n)

Mak

es u

nequ

al g

roup

san

d is

una

ble

toco

mpa

re t

he g

roup

s.3

YR

�/�

Whe

n sh

arin

g, c

anso

met

imes

mak

e eq

ual

grou

ps, b

ut h

as n

ost

rate

gies

to

deal

with

any

left

over

. 4

YR

�/�

Has

diff

icul

ty w

ithco

untin

g re

liabl

y in

ten

sfro

m a

mul

tiple

of t

en.

5 Y

R �

/�

Whe

n ha

lvin

g, m

akes

two

uneq

ual g

roup

s or

split

s a

sing

le o

bjec

tun

equa

lly.

6 Y

R �

/�

Can

you

put

the

sam

e nu

mbe

r of

spo

tson

eac

h w

ing

of t

he la

dybi

rd?

Can

you

sha

re o

ut t

hese

bis

cuits

fairl

yso

the

re a

re t

he s

ame

num

ber

ofbi

scui

ts o

n ea

ch p

late

?

Can

you

sha

re t

hese

sw

eets

equ

ally

betw

een

you

and

a fri

end?

Will

ther

e be

any

left

over

?

Can

you

sha

re t

hese

mar

ker

pens

betw

een

the

grou

p? W

ill th

ere

be a

nyle

ft ov

er?

Wha

t w

ill yo

u do

with

any

left

over

one

s?

Can

you

say

all

the

num

bers

mar

ked

onth

is n

umbe

r lin

e? F

or e

xam

ple,

ten

,tw

enty

…

Can

you

put

the

car

ds in

the

cor

rect

orde

r so

we

can

coun

t in

ten

s to

one

hund

red?

How

man

y sh

eep

in t

he fi

eld?

Can

you

put

half

of t

hem

in t

he p

en?

How

man

yar

e in

the

pen

? H

ow m

any

are

in t

hefie

ld?

Pra

ctic

al a

ctiv

ities

set

in c

onte

xt; f

orex

ampl

e pl

antin

g bu

lbs,

giv

ing

out

stic

kers

, sha

ring

bisc

uits

.

Rel

ate

to s

torie

s su

ch a

s Th

e D

oorb

ell

Ran

g, b

y P

at H

utch

ins

(Mul

berr

y B

ooks

,19

86, I

SB

N 0

688

0 92

34 9

)

Pra

ctic

al s

harin

g ac

tiviti

es r

elat

ed t

odi

ffere

nt c

onte

xts;

for

exam

ple

shar

ing

clas

sroo

m r

esou

rces

, mak

ing

grou

ps fo

rP

E.

Pro

vide

vis

ual r

esou

rces

suc

h as

a 1

00-

squa

re, a

ten

s nu

mbe

r tr

ack,

larg

enu

mer

al c

ards

from

10

to 1

00 t

o ho

ld u

pto

dem

onst

rate

cou

ntin

gfo

rwar

ds/b

ackw

ards

in t

ens

from

diffe

rent

sta

rtin

g nu

mbe

rs. P

rovi

de c

lues

to id

entif

y pa

rtic

ular

num

bers

, for

exam

ple:

Can

you

sta

nd o

n th

e te

ns n

umbe

r th

atco

mes

just

afte

r 60

?

Pra

ctic

al a

ctiv

ities

with

diff

eren

t nu

mbe

rsan

d ty

pes

of o

bjec

ts t

o ex

perie

nce

halv

ing.

Incl

ude

activ

ities

invo

lvin

gm

easu

res;

for

exam

ple

capa

city

, mas

s,m

oney

.

Ext

end

to c

onsi

derin

g gr

oups

and

obje

cts.

How

man

y pl

ates

for

thre

edo

gs?

How

man

y do

g bi

scui

ts d

o w

ene

ed s

o th

at t

hey

can

have

thr

ee e

ach?

Can

we

put

som

e m

arks

/num

bers

on

each

pla

te t

o he

lp u

s re

mem

ber?

Wha

tif

we

had

twel

ve b

iscu

its –

how

man

y fo

rea

ch d

og?

Ext

end

by li

nkin

g to

mak

ing

sim

ple

pred

ictio

ns. C

an y

ou s

hare

the

se s

even

cube

s eq

ually

into

tw

o gr

oups

? W

hyno

t? C

an y

ou c

hoos

e a

num

ber

ofcu

bes

that

will

shar

e eq

ually

with

non

ele

ft ov

er?

Ext

end

to m

enta

l im

ager

y us

ing

a 10

0-sq

uare

, for

exa

mpl

e ‘L

ook

care

fully

at

10. C

lose

you

r ey

es (o

r lo

ok a

t th

ece

iling)

. Can

you

stil

l see

the

10?

Now

look

at

the

10 a

nd t

he 2

0. C

lose

you

rey

es (o

r lo

ok a

t th

e ce

iling)

. Can

you

see

the

10 a

nd 2

0? W

hat

num

ber

com

esne

xt?

etc.

Ext

end

to a

ctiv

ities

tha

t in

volv

e th

e ch

ildin

find

ing

cont

exts

for

halv

ing.

Ass

oci

ated

Err

ors

and

Que

stio

ns t

o id

enti

fy e

rro

rs a

nd

Teac

hing

to

ad

dre

ss t

he e

rro

rsN

ext

step

s in

mo

ving

kn

ow

led

ge

and

mis

conc

epti

ons

mis

conc

epti

ons

and

mis

conc

epti

ons

tow

ard

s th

e ke

y o

bje

ctiv

esk

ills

Appendix 3: Handouts for staff meeting

Handout 1: Structure of Wave 3 mathematics materials

Handout 2: Mathematical themes and assessment approaches in the Wave 3materials

Handout 3: Assessment and planning process



Handout 1

Structure of Wave 3 mathematics materialsIn order to exemplify progression in calculation, Reception, Year 2, Year 4 and Year6 have been chosen as representative milestones. Under each year group heading,associated knowledge and skills that contribute to understanding of the year groupkey objective are listed. (See first column of tracking chart.)

The whole primary age range is represented in the progression in the chart. The yeargroup labels provide a convenient link to the National Numeracy StrategyFramework for teaching mathematics progression in number and calculation. Asthey use the chart, teachers will need to ‘track back’ to find the error ormisconception appropriate for the child, irrespective of the year group to whichit is attributed in the progression.

Tracking charts

1 Key objective.

2 This column lists associated knowledge and skills that contribute tounderstanding of the key objective.

3 Common errors and misconceptions linked to specific knowledge and skills arelisted to support diagnosis of children’s difficulties.

4 Questions in this column can be used to help the teacher decide where thechild’s difficulties lie.

5 Examples of the types of teaching activity in the A4 booklets (see below).

6 This column provides ideas to develop when the child has improved theirunderstanding of the identified difficulty. The teacher can make use of theseideas to consolidate understanding and extend thinking.

7 Code referencing to an A4 teaching unit.

A4 booklets – teaching units

The structure of each booklet is as follows:

• focus error/misconception;

• opening teaching activity addressing error/misconception;

• a number of Spotlights (short focused teaching activities from which to select);

• final Spotlight, which includes assessment opportunities, often encompassed ina game, key vocabulary checklist, and intended learning outcomes list;

Year 6 key objective Carry out column addition and subtraction of numbers involving decimals (NNS Framework for teaching mathematics, Supplement of Examples, Section 6,pages 49, 51)

Associated knowledge and skills

Apply knowledge of the number system to enable efficient counting of a large number of objects.

Add and subtract multiples of ten, a hundred and a thousand.1 Y6

Give an estimate by rounding, to determine whether the answer to a calculation is sensible.2 Y6

Tracking children’s learning through the NNS Framework for teaching mathematics (addition and subtraction)

Errors andmisconceptions

Has inefficient countingstrategies and/orinsecure understandingof the number system. 1 Y6 +/–

Rounding inaccurately,particularly whendecimals are involved,and having little senseof the size of thenumbers involved.2 Y6 +/–

Questions to identify errors andmisconceptions

Imagine you have a money box containing 2pand 1p coins. What do you think would be agood way to count these quickly to find out howmuch money there is?

What is 60 + 20? … 60 + 30? … 60 + 40?What changed when you found 60 + 40?

What is 40 + 40? … 400 + 400?Which answer is the larger?How is the calculation 40 + 400 + 4000 differentfrom the others?

What is 60 – 20? … 600 – 200? … 6000 – 2000?Explain how you worked these out.What is 6000 – 200? … 6000 – 20?

Is 26 nearer to 20 or 30?Is 271 nearer 270 or 280?Is 1.8 nearer to 1 or 2?Draw a sketch to illustrate your answer andexplain how you know.

Teaching to address the errorsand misconceptions

Practical opportunities to developefficient counting strategies for arange of objects, for example coins,cubes, conkers, collectable cards,stickers.

Count forwards and backwards intens, hundreds and thousands fromdifferent starting points, includingstarting numbers that are notmultiples of ten or a hundred. Usean empty number line to supportthis development.

Order multiples of a hundred and athousand.

Use number squares and/ornumber lines to consider the orderand comparative value of numbersto support rounding.

Next steps in moving towards thekey objective

Carry out simple calculations thatinvolve crossing the boundary fromhundreds to one thousand and viceversa, supported by an emptynumber line and extending this to avisualised image to develop mentalcalculation.

Consider pairs of items from acatalogue and ask child to estimatewhether a £10 (or £20, etc.) notewould be enough to buy both theitems?

15

6

43

2

7

Opening teaching activity Spotlight

Specific icons are used to improve access to the text:


Icons

Questions are incorporated for teachers to select from and addtheir own as appropriate.

Whole-class follow-on activity.

Symbol reminding of the necessity to estimate, calculate, then check.

This variation of the game is harder.

This variation of the game is easier.

12 x 2 = 24 Text within this symbol indicates an opportunity for recording.

Text within a shaded box indicates alternative approachesfor a child who is having difficulty with the activity.

Additional game at the end of some teaching units.

?


Wave 3 m

athematics ad

ditio

n and sub

traction

Tracking b

ack to Year 4

1 Y4 �

/�

Has insecure understanding of the structure of the numbersystem, resulting in addition and subtraction errors anddifficulty with estimating

Opportunity for: developing mental images

Resources Key vocabulary

● Two sets of number cards 0–9 (Resource sheet 1)

● 100-square

● Place value (arrow) cards

● Long number line (or masking tape)

● Sticky notes

● Bundles of straws or other Base 10 equipment

● 1p coins for counting

Teaching activity Time 15–20 minutes

‘We are going to be working with numbers today, deciding which numbers are larger or smaller, and we

are going to order them on a long number line. This work will help you with estimating and with getting

more calculations correct.’

Lay out the two sets of number cards 0–9 on the table.

Can you make the numbers forty-three and thirty-four?

Support the child to make:

Which is the larger number, forty-three or thirty-four?

If the child knows which is larger, move on.

Display a long number line, say up to four hundred. (You could stick masking tape on the floor or wall

and write the numbers along it. You will need to use the number line throughout this set of activities.)

Ask the child to position the numbers on the line with sticky notes or paper.

If the child doesn’t know which is larger, you could count with the child up to 100 on a 100-square,

pointing out the thirties and the forties.

Then change the numbers to 40 and 30.

Count in tens with the child on the 100-square, establishing that if you had forty sweets, you

would have more than thirty sweets because forty is more than thirty.

Then you will need to make some more two-digit numbers and repeat the activity before you move on.

Note: If the child seems to have problems crossing boundaries, see 1 Y2 +/–. If the child seems

unsure of numbers, you might want to check that the child can count a large pile of coins so that

you can assess their counting skills (see also 1 YR +/–).

?

?

digit

larger/largest

smaller/smallest

more than

less than

units

ones/tens/hundreds/

thousands

column order

estimate

guess

nearer/nearest

before

after


ten/hundred

4 3 3 4

Spotlight 2Has insecure understanding of the structure of the number system, resulting in addition and

subtraction errors and difficulty with estimating

Opportunity for: reasoning about numbers

Place value on the calculator Time 10–15 minutes

Resources

● Calculator each

● Large book or other screen

● Number line

Teaching activity

‘Today we are going to do some more work on place value and where numbers go on the number line.

We will particularly be thinking about any zeros that we see. So if you see a zero, tell me and I will

record the number for later.’

Can you see any numbers with zeros on the number line?

Follow what the child says, giving experience of reading numbers such as three hundred and six.

Prop up the large book so that you can enter numbers secretly.

Now I want you to key in three hundred and forty-two

into your calculator and I will do the same on mine.

Does yours look the same as mine?

Then you add a one-digit number, such as 3, and show the child your calculator.

What do you think I did to three hundred and forty-two to get to that number?

Can you make your number the same as mine in one move? (Meaning an operation key, a

number and the equals key.)

What did you do to make your number the same as mine?

Repeat until the child understands how to add and subtract single-digit numbers.

Record any numbers you use that have a zero in them.

Then subtract all the tens from the number secretly.

Show the child your screen.

What subtraction will take away the four? (Signal to the child that you have a number with a zero

in and record that for later use.)

?

?

?

If the child isn’t sure you will need to do some further adding and subtracting of one-digit numbers.

?

?

?

1 Y4 �

/�


Key vocabulary

digit

larger/largest

smaller/smallest

more than

less than

units

ones/tens/hundreds/

thousands

column

estimate

guess

nearer/nearest

order

before

after


ten/hundred

342

345

305

Error/misconceptionheading

Problem-solvingemphasis

Suggested time

Activity title(for Spotlights)

Key vocabulary

Resources

Teaching activity

Handout 2

Mathematical themes and assessment approaches in the Wave 3materials The following are fundamental to the approach in these Primary National Strategymaterials.

• Using and applying mathematics has been integrated. Often there are severalopportunities for problem solving within one activity, but in each, one particularopportunity has been highlighted. Aspects such as the following areincorporated:

– encouraging children to discuss and explain in order to supportdevelopment of their mathematical reasoning;

– opportunities for children to make choices are woven into the activities, forexample selecting numbers and devising calculations;

– encouraging children’s own recording to communicate mathematicalthinking, focusing on efficiency;

– opportunities for evaluating the efficiency of methods of calculation.

• Development is emphasised and key vocabulary is listed in each activity. It isimportant for adults to use correct mathematical language. To facilitate this,examples are given in words, for example, 725 � 3 is accompanied by what theadult could say to the child: ‘Seven hundred and twenty-five multiplied by 3’.

• There is a focus on progression in counting from the earliest stages through toYear 6 to support the development of secure counting skills.

• Throughout the materials, there is emphasis on the process of estimating first,then calculating and then checking. This is denoted by the following icon:

• Decimals are addressed within meaningful contexts, for example via displays ona calculator and as a part of measure.

• Structured equipment and everyday materials are used to model mathematicalconcepts, supporting children’s mathematical thinking and development ofmental imagery. Some links to ICT resources such as the Primary NationalStrategy Interactive Teaching Programs (ITPs) are included.

• A wide range of resources is used in the teaching sessions. Teachers’ selectionof these to suit the needs of their children is an important part of adapting thematerials.

The materials reflect best practice in assessment for learning as a key tool for raisingachievement through:

• use of questions to elicit information about children’s understanding;

• sharing the purpose of the activity with the learners;

• encouraging children’s reflection on their learning and identification forthemselves of possible next steps.


Handout 3

Assessment and planning processDay-to-day assessment is the starting point. Assessment opportunities areembedded in all the teaching materials.

The following flow chart describes the process assumed in the design of thesePrimary National Strategy materials.


Day-to-day assessmentAs part of day-to-dayteaching, errors and

misconceptions noted.

Final Spotlight – a learning check

Teaching session (often a game) designed to support the teacher in assessing progress. Key vocabulary and learning

outcomes provided to assist theprocess.

Tracking chartDiagnostic assessment –

particular error identified by tracking back from actual year group

Second column of tracking chart used to identify closest fit to identified error/misconception, irrespective of the child’s chronological year group. Questions in third column used to

confirm choice. Select appropriatebooklet.

SpotlightsFurther teaching sessions

Activities to reinforce understandingthrough a variety of activities with

supporting models and images andcontexts. Materials include

assessment prompts to encouragethe child’s reflection on learning.

Further Spotlights to be covered inlater teaching sessions to aid

retention and recall.

A4 booklet – first teachingactivity in the unit

Opening teaching sessionOpening session used to

further assess child's understanding and to support

choice of further activitiesfrom the set or to decide to

return to tracking chart.

Final column of tracking chart

Suggestions for teaching activities to support child’s developing understanding and progress

toward achieving key objective.

Appendix 4: Further referencesWhat works for children with mathematical difficulties? (DfES research report 554).This is available from DfES publications. (tel: 0845 60 222 60) or can bedownloaded from the website at www.dfes.gov.uk/research.

Using models and images to support mathematics teaching and learning in Years 1 to 3 (DfES 0508-2003 GCDI)

Including all children in the literacy hour and daily mathematics lesson(DfES 0465/2002)

Guidance to support pupils with specific needs in the daily mathematics lesson(DfES 0545/2001)

Framework for teaching mathematics from Reception to Year 6

Primary National Strategy website: www.standards.dfes.gov.uk/primary



Appendix 5: ResourcesEquipmentThe lists below show all the mathematical equipment and everyday materialsrequired in the teaching activities.

There is an assumption that regular classroom stationery, such as paper (includingsquared paper), felt tip pens, pencils, labels, sticky notes and whiteboards with drywipe pens will be available.

Mathematical equipment Everyday materials

Cubes Small counting items, cars, animals, etc

Counters Food items, such as apples, tangerines, biscuits

Beads Hoops, balls, beanbags

Money, including notes Small and large bricks

Number lines to 30/100 Cloth

Wipe-clean number line Jars, boxes, pots, jugs, beakers

Floor or playground number line Tins and tin tray

Number track Draught pieces

Bead string Game board, such as Ludo

Counting stick Magnet and paper clips

Small 100-square String, ribbon

Wipe-clean 100-square Paper plates

Large 100-square with removable Cups, plates spoonsnumbers

Dominoes Tiddlywinks game

Large foam number tiles ‘Eggs’ (ping pong balls) and egg boxes

Dice Coat hanger and pegs

Calculator Cloth bags to hold cards

Place value (arrow) cards Purses

Wipe-clean place value board Paper bags

Abacus Small character toys, for example astronauts

Bundles of straws Fish shapes

Base 10 equipment Framed picture

Cuisenaire or other rods Pairs of socks/gloves

One/two-minute timer Lolly sticks

Plastic tocker timers Marbles

Sand timers Sand tray

Stopwatch Mirror

Large digital clock Finger puppets

Teaching clock (analogue) Construction bricks with pairs of prongs

Metre rulers Items for rewards

Mathematical equipment cont.

Measuring tape

Weights

Balance scales

2-D shapes

3-D shapes

Straight-sided clear containers/measuring cylinders

Plastic or other fraction pieces

Sorting circles

Suggested Interactive Teaching Programs (ITPs)Interactive Teaching Programs for teaching calculations (downloadable from the CD-ROM, and the latest versions from the Primary National Strategy websitewww.standards.dfes.gov.uk/primary).

The list can be added to as new programs become available.

Addition, subtraction and place value Multiplication and division

Difference Number gridUses a number line to find difference Multiples shown on a 100-square

Number line GroupingAddition and difference on a number line Division on a number line

Place value Multiplication gridHundreds, tens and units on place Grid method of multiplyingvalue cards

Counting on and back Division grid Simulates a 100-bead string Long division

Number facts Remainders after divisionAddition and subtraction number Gives remainder as a number or a sentences fraction

Ordering numbers Multiplication factsRelating counters to a number line Multiplication as repeated addition

Twenty cards Moving digitsSequences Multiplying and dividing by 10 and 100

Number dialsMultiples


Documents

Supporting children with gaps in their mathematical understanding