Mathematics and Inclusion Mathematics and Inclusion - PGCE

DfES 0606-2003Primary National StrategyContentsIntroduction 5Section 1: Observing three key groups 7Section 2: Planning inclusive teaching and learning in mathematics 19Section 3: Assessment approaches 34Appendix iITT examples of inclusion provision within course programmes 40Appendix iiVideo notes and suggested activities 44Appendix iiiDefinitions of inclusion 46Appendix ivResearch articles 49Appendix vBibliography 113Appendix viNational Numeracy Strategy inclusion resources 117Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 20033

Primary National Strategy DfES 0606-2003AcknowledgementsPatti Barber Institute of Education, University of LondonMarion Gatrell University of PlymouthCathy Hamilton Bath Spa University CollegeHelen Mulholland University of the West of EnglandJean Murray Brunel UniversityChristine O’Mahony Disability, Equality in EducationReg Wrathmell King Alfred’s College, WinchesterThe following publications, which appear in appendix iv, pages 49–112, are reproduced bypermission of the copyright holders.Page 50, Corbett, Jenny, Teaching approaches which support inclusive education: aconnective pedagogy, British Journal of Special Education, Volume 28, No. 2 (June 2001),pages 55–59. Richard Byers, ed. © Blackwell Publishing.Page 55, Eyre, D. and McClure, L., chapter 3, Mathematics, pp.64–89, in CurriculumProvision for the Gifted and Talented in the Primary School: English, Mathematics, Scienceand ICT. Published by NASEN/David Fulton Publishing Ltd, 2001. © D. Eyre, L. McClure.Page 81, Raiker, A., Spoken language mathematics, Cambridge Journal of SpecialEducation, Volume 32, No. 1, pp. 45–60. © Andrea Raiker.Page 110. Visser, John, Opening address at the ATM 2002 Easter conference. This materialwas subsequently published in the Easter Conference Supplement, Mathematics Teaching,No. 179, June 2002 (a journal of the Association of Teachers of Mathematics). © JohnVisser.Copnor Junior School, PortsmouthHoly Family RC Primary School, SouthamptonLittletown Primary, HonitonLympstone Church of England Primary School, LympstonePlymouth University, Faculty of Arts and EducationPortsmouth Primary SCITTPrimary Catholic Partnership SCITTStokehill First School, ExeterTidcombe Primary, Tiverton4© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyIntroductionPurpose of the materialsThese inclusion materials consist of a file and an accompanying video. They are intended toprovide a variety of resources, information and ideas to support tutors in:• helping trainees to consider the needs of all pupils when planning and assessingmathematics;• enabling trainees to make connections between recent and relevant research related toinclusion and the pupils’ learning experiences;• engaging trainees in debate and evaluation of their own practice with respect to inclusionissues in mathematics.The materials are designed to complement existing resources within course programmes.Key principlesThe definition of inclusion initiates much debate and discussion. Trainees may find it helpfulto consider the various definitions of inclusion and prioritise these in terms of their ownbeliefs and experiences. The activities in appendix iii on pages 46–48 may prove a usefulstarting point for this process.The materials reflect Wave 1 and Wave 2 of the National Literacy Strategy (NLS) and theNational Numeracy Strategy (NNS) model of three ‘Waves’ of support, as well as theinclusion references within the Professional Standards for Qualified Teacher Status (DfES,2002). Wave 1 is about what should be on offer for all pupils: the effective inclusion of allpupils in a high-quality daily mathematics lesson. Wave 2 concerns pupils who, supportedby the class teacher and other adults, can be expected to ‘catch up’ with their peers as aresult of this intervention. There is an additional Wave 3 level of support involving specifictargeted intervention for pupils identified as requiring SEN support.The information and ideas within the materials focus on specific and generic principles ofinclusion in relation to mathematics teaching and learning. The aspects and suggestedactivities within each section exemplify Corbett’s description of exceptionally skilful inclusionpractice where ‘There is a real effort made to involve the learners, to create situations inwhich they can meet with success and to build on their existing level of knowledge’(Corbett, 2001). The materials also highlight the principle that all pupils have strengths thatdeserve recognition.RationaleThe materials have been organised in sections to highlight some key principles, aid flexibilityof use and support existing practice. Use of labelled sections and space for additional notesenable tutors to select, modify and exemplify appropriately.Section 1 focuses on helping trainees to observe and analyse three different groups: pupilsrequiring additional support with mathematics; pupils learning English as an additionallanguage; and more able pupils. Although these three groups do not represent all groupswithin a school or course programme, it is likely trainees will encounter one or more of thesegroups within school placements.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 20035

Primary National Strategy DfES 0606-2003Sections 2 and 3 focus on ways in which planning and assessment can support inclusiveteaching and learning in mathematics.Appendices i to vi provide examples and exemplification of inclusion provision within currentInitial Teacher Training course programmes; suggested activities for the accompanyingvideo; definitions of inclusion; research articles; a bibliography and National NumeracyStrategy resources.Existing National Numeracy Strategy and QCA video sequences have been drawn togetherin the accompanying video Mathematics and Inclusion: materials for providers of InitialTeacher Training. This video highlights some key principles of inclusion within the teachingand learning of mathematics. In addition to these video sequences being referencedappropriately throughout the file, there is also a separate section of possible video activitiesand observation tasks. The video has a counter for ease of reference.6© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategySection 1 Observing three key groupsPupils have different learning styles. The grids on the following pages have been structuredto provide a basis for considering the learning needs of particular groups of pupils and theteaching strategies that support them. Analysis of this information may be used to informthe planning process. The grids focus on pupils who:• need additional support in mathematics;• are learning English as an additional language;• are more able at mathematics.Some questions within the grid for one particular group of pupils may also be appropriatefor use with other groups of pupils. An example of this may occur where a pupil requiringadditional support may have significant language issues and the language focus questionsfrom the grid for pupils learning English as an additional language may also be appropriate.Trainees may find it useful to identify particular questions and issues that they wish to raisewith the class teacher, if there is an opportunity, before and after the observation.Suggested use of gridsAdditional bullet points have been provided for further ideas to be added.• Use the information collected by trainees as a starting point for considering key planningissues, teaching strategies and assessment implications.• Adapt grid/s to provide a specific focus, e.g. on a pupil’s communication skills or the useof resources.• Ask trainees to add further observation focuses in the empty boxes.• Suggest trainees compile their own observation grid.• ..• ..Follow-up activities using information from the grids can be found on pages 17–18.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 20037

DfES 0606-2003 Primary National StrategyObserving pupils requiring additional support with mathematicsIt is useful to observe a pupil within a mathematical activity/in the daily mathematics lesson (DML) and if possible one in which additionaladults are working. It is helpful to look at relevant planning before the activity or lesson.Observation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLIs the pupil listening to the teacher or adult?Degree of attention?What strategies are used to gain andmaintain the attention of all pupils?Is the pupil listening to other pupils?Degree of attention?How are all pupils encouraged to listen toeach other?Is the pupil participating as required, e.g. byresponding to questions, chanting, watchingvisual aids or demonstrations, usingresources, etc.?How are all pupils encouraged to participate?Can the pupil engage in the mathematicalactivity without further guidance?How is the mathematical activity explained?How is the mathematical activity madeaccessible for all pupils?How are resources used or organised tosupport mathematical learning?How are expectations made explicit to thepupils?Is the pupil fully engaged in themathematical activity?If not, what is he/she doing instead?How are pupils supported or encouraged tostay on task?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 20038

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLWhen appropriate, is the pupil workingindependently of teachers, additional adultsand peers?How is independent working encouraged?When appropriate, is the pupil workingcooperatively or collaboratively with peers,e.g. working alongside peers/discussingpossible mathematical solutions/sharingresources and ideas?What strategies are used to promote andsupport cooperation or collaborationbetween pupils?When appropriate, does the pupil askhis/her own questions?Provide an example to illustrate this.When appropriate, are there opportunitiesfor pupils to ask questions?Is the pupil using everyday language and/ormathematical words/terms to discuss anddescribe his/her work?How is the use of mathematical words andterms promoted or developed?Does the pupil appear to understand andrecognise his/her own achievement(s)during the mathematical activity/DML?How are pupils’ mathematicalachievements celebrated?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 20039

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLIs there evidence of the pupil engaging withthe mathematics and making progress inmathematics, e.g. identifying patterns,making connections?What strategies are used to enable pupilsto engage with the mathematics?Does the pupil seek help? From whom?How are additional adults or peers used tosupport learning during the mathematicalactivity?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200310

DfES 0606-2003 Primary National StrategyObserving pupils for whom English is an additional languageIt is useful to observe a pupil within a mathematical activity/in the daily mathematics lesson (DML) and if possible one in which additionaladults are working. It is helpful to look at relevant planning before the activity or lesson.Observation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLHow does the pupil communicate whencarrying out a mathematicalactivity/during the DML?Does he/she respond using resources,pictures, words, sentences, etc.?How does the teaching support pupils’communication skills when carrying out amathematical activity/in the DML?Does the pupil have opportunities toobserve, listen and talk aboutmathematics with other pupils?What opportunities are provided for pupilslearning English as an additional languageto observe, talk and listen duringmathematical activities?How does the pupil participate in themathematical activity, e.g. by using nonverbal/verbalresponses, carrying out orcopying actions, watching visual aids ordemonstrations, using resources,chanting, etc.?What strategies are used to enable pupilslearning English as an additional languageto participate in the mathematical activity,e.g. pre-tutoring?How does the pupil access themathematics in each part of theactivity/DML?What specific strategies are used to ensurepupils learning English as an additionallanguage can access the mathematics?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200311

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLDoes the pupil follow simple instructions?What strategies are used to help pupilslearning English as an additional languageto understand instructions?Is there evidence of the pupil engaging withthe mathematics and making progress, e.g.identifying patterns, using new vocabularyappropriately?What strategies are used to enable pupilsto engage with the mathematics?Does the pupil work individually/in pairs/asa member of a group?Does she/he have a preference?How are flexible groupings used to supportthe pupils’ learning?Does the pupil offer responses orsuggestions or ask questions?How is the pupil encouraged/supported inmaking a contribution?When appropriate, does the pupil workindependently on a mathematical activityand show initiative?How is independent learning developed?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200312

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLDoes the pupil use mathematicallanguage?Provide examples to illustrate this.How is the meaning of mathematical wordsor terms conveyed?What opportunities are provided forpractice and consolidation?Does the pupil appear to understand andrecognise his/her own achievement(s)during the mathematical activity/DML?How are pupils’ mathematicalachievements celebrated?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200313

DfES 0606-2003 Primary National StrategyObserving more able pupilsIt is useful to observe a pupil within a mathematical activity/in the daily mathematics lesson (DML) and if possible one in which additionaladults are working. It is helpful to look at relevant planning before the activity or lesson.Observation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLDoes the pupil grasp new material quickly?What strategies are used to take accountof this?Does the pupil appear systematic whenworking on mathematical problems?How is a systematic approach developedwithin the mathematical activity?Does the pupil quickly get to the ‘kernel’ of aproblem/activity, disregarding irrelevantinformation?How is this strategy supported, e.g. is thepupil encouraged to explain howjudgements are made about relevance?Is the pupil able to talk about mathematicalpatterns and relationships?Can they explain the pattern or relationship?Give an example.How are pupils encouraged to developtheir explanations of patterns andrelationships?Mathematics and InclusionMaterials for providers of Initial Teacher Training14

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupil© Crown copyright 2003© Crown copyright 2003Does the pupil use mathematicalsymbols confidently?Give an example.Notes Observation/issues linked to teachingand/or other adults working with pupilsat different times of the mathematicalactivity/in the DMLHow are pupils encouraged to usemathematical symbols to communicatetheir ideas or findings?Does the pupil approach amathematical problem or activity fromdifferent directions and persist infinding solutions?How are pupils encouraged to workflexibly?What strategies are used to developpersistence?Is the pupil able to explain or justifymethods and solutions?How are pupils encouraged to explain andjustify their strategies or solutions?Does the pupil use solutions toproblems as a starting point for newlines of enquiry?How are pupils encouraged to invent theirown problems?Mathematics and InclusionMaterials for providers of Initial Teacher Training15

DfES 0606-2003 Primary National StrategyObservation/issues linked to pupilNotes Observation/issues linked to teachingand/or other adults working with pupils atdifferent times of the mathematicalactivity/in the DMLIs the pupil able to argue and giveexplanations of his/her reasoning tosupport his/her findings?How are pupils helped to develop skills ofarguing and reasoning?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200316

DfES 0606-2003Primary National StrategyFollow-up activitiesThe following two activities may be useful in helping trainees to consider their ownperspectives and views concerning particular groups of pupils.Activity 1Ask trainees to discuss the following statements in small groups and choose threestatements they all agree with:Pupils who require additional support in mathematics:• provide an intellectual challenge for teachers which improves the teaching for all pupils;• need teachers to make learning objectives and expectations clear to all pupils;• can improve the teacher’s communication skills;• provide opportunities for other pupils to consolidate and deepen their understanding bypeer tutoring;• may have additional adult support that benefits the whole class;• can sometimes have additional resources that can be used by the whole class, e.g. avoice activator for the computer;• can develop the teacher’s skills in thinking laterally, e.g. realising that a learning objectiveneeds to be presented in different contexts;• can broaden the teacher’s understanding of the different learning styles in mathematics.Activity 2The following activity has been adapted from The Challenge of the Able Child, David George(1995).1. In pairs read through the following lists and check which statements are sometimes falseand which statements are sometimes true.2. Think of the most able child you know and circle which items apply to that particularindividual.Characteristics of able learners:• are more emotionally stable and mature than their peers;• prefer to work alone;• are model pupils;• are proud of their ability;• are organised and neat;• are well rounded;• are creative;• are good learners;• are very verbal;• have good handwriting;• have very supportive parents;• have a low tolerance of slower peers;Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200317

Primary National Strategy DfES 0606-2003• are perfectionists;• work harder than the average pupil.Identification procedures:• Able pupils always reveal their ability.• Able pupils are respected and looked up to by their peers.• Asian pupils are more likely to be able.• More girls tend to be able at mathematics than boys.• Able pupils tend to find socialising with peers difficult.Provision for able pupils:• Able pupils need constant challenges.• Teachers prefer to work with able pupils.• Acceleration can be harmful for able pupils.• Teachers of able pupils should be good at mathematics.• Able pupils learn things with only one presentation.• Able pupils should be grouped together.Further readingVisser’s article in appendix iv, pages 110–112, provides a useful overview of issues andimplications concerning the provision of effective inclusion within the teaching and learningof mathematics.18© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategySection 2 Planning inclusive teaching and learning inmathematicsThis section considers five aspects of planning that support inclusive teaching and learningin mathematics. These aspects are:• gathering initial information to inform planning;• identifying appropriate learning objectives;• planning appropriate mathematical activities;• teaching strategies that promote inclusive teaching and learning in mathematics;• organisational issues.For ease of reference these aspects are outlined within the three different groups: pupils requiring additional support with mathematics; pupils learning English as an additional language; more able pupils.The aspects outlined are not mutually exclusive for one particular group of pupils as theymay at times be appropriate for other groups. Tutors are invited to select or adapt theseaspects to support their course programme.Pupils requiring additional support with mathematicsGathering initial information to inform planningThe following points may be used to prompt discussion with trainees:• The class teacher and records of pupils’ progress (including targets) provide valuableinformation at the beginning of school experience.• Targets and Individual Education Plans for these pupils may address knowledgeof number and calculations, to ensure pupils have knowledge and understanding of basicfacts, skills, concepts and strategies. Such targets are clearly important, but there is awider and exciting world of mathematics out there that these pupils requiring additionalsupport are often unintentionally excluded from. These pupils also need access to the fullmathematics curriculum. Some pupils who struggle with number have strengths in shapeand space and problem-solving skills – refer to Koshy and Murray (2002) pages 106–121.• Practical use of resources may help pupils to demonstrate what they know and what isproving difficult.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200319

Primary National Strategy DfES 0606-2003• Establishing an initial dialogue with a pupil before attempting to identify what pupils cando may help some pupils who associate teacher’s questions with previous experiencesof failure.• Asking pupils to explain an answer or strategy or to describe how they used theresources to achieve an answer will provide further useful information (refer to video clip4, Counter reading 11:43, for examples).• Allowing sufficient wait time for pupils to watch, listen and respond.• Pupils’ confidence and attitudes when undertaking mathematical activities.• Pupils’ verbal assessment of their capability in particular areas of mathematics.• Information recorded on the observation grids in section 1.Within part 2 of the National Literacy and Numeracy Strategy file Including all pupils in theliteracy hour and daily mathematics lesson (DfES 0465/2002) there is a section (Trackingchildren’s learning through the NNS Framework for teaching addition and subtraction) thatoutlines a suggested approach to identifying aspects of subtraction and addition that pupilsfind challenging. Strategies include tracking back to what a pupil can do, gathering oral andwritten evidence of where errors or misconceptions are occurring, framing questions toconfirm this and considering possible teaching approaches to progress learning.Frederickson and Cline (2002) also provide some helpful guidance (pages 351–356) onassessing pupils’ mathematical understanding and the role of assessment.Identifying appropriate learning objectivesIn planning appropriate learning objectives it may be helpful to review some of the followingpoints with trainees:• At times it may be appropriate to track back a learning objective to identify parallelactivities related to earlier learning objectives, so that pupils are working on the same areaof mathematics as their peers. At other times the mathematical activity can be made fullyaccessible for all pupils if the instructions and mathematical vocabulary are carefullyintroduced within the context of the activity.• Helpful pragmatic advice to support decisions concerning tracking back can be found onpage 13 of the National Literacy and Numeracy Strategy file: Including all children in theliteracy hour and daily mathematics lesson (DfES 0465/2002). An annotated version ofthe NNS medium-term plans can also be found in the booklet Tracking back through theNNS Framework on page 16 of this file. Each mathematical topic is referenced toexamples from the supplement of examples in the Framework and to the additional Plevels exemplification.• The NNS intervention programmes structure the learning objectives to help pupils who,with additional support, could work at age-related expectations within the Framework.Intervention programmes include Springboard 3, Springboard 4, Springboard 5 andSpringboard 6. More details and examples can be found on the NNS website(www.standards.dfes.gov.uk/numeracy).• Refer to research findings trainees may already be familiar with and evaluate suggestedareas for developing the mathematical knowledge and understanding of pupils requiringadditional support, e.g. moving on from 1 by 1 counting, learning key number facts,especially addition bonds to 10, building knowledge of first two-digit place value thenthree-digit place value (Thompson, 2001).20© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyPlanning appropriate mathematical activitiesTrainees may find it helpful to consider the following aspects when planning appropriatemathematical activities for pupils requiring additional support:• Pupils may have different preferred learning styles, e.g. some pupils may prefer activitiesrequiring manipulating and using resources, while other pupils may prefer activitiesinvolving discussion and recording. The NNS file Guidance to support pupils with specificneeds in the Daily Mathematics Lesson (DfEE 0545/2001) provides further guidance onthis and other aspects of planning and teaching mathematics for particular groups ofpupils.• Planning structured opportunities within the mathematical activity for pupils to discusstheir mathematics with peers and the teacher contributes to mathematical understandingand confidence. (Video clips 6 and 7, Counter readings 29:55 and 31:10, provide usefulexamples of this.)• Planning mathematical activities using ICT in a focused and structured way to developthe pupil’s understanding of mathematics. For example: to support the counting process (e.g. Counter); to ensure practice of relevant facts (e.g. Counter, multiplication grids in Monty); to encourage use of mental imagery (Monty); to generate number patterns (calculators); to practise addition skills in a motivating context (e.g. Toyshop).• Planning to use a range of recording opportunities within appropriate mathematicalactivities to ensure all pupils can contribute, e.g. spoken response, group response, ICT,charts, images, etc., will also benefit pupils.• Planning a degree of manageable challenge within the mathematical activity.For example, with the support of the constant function on the calculator or using Counter,explore multiplication facts beyond x10. Set challenges for finding the answer to the‘biggest’ sum. ‘Can anybody find out what 20 x 4 is? 100 x 4? 120 x 4? Can anyonecontinue the table beyond 120 x 4?’ Ask pupils questions such as ? x 4 = 144. The NNSpublication Mathematical Challenges for able pupils in Key Stages 1 and 2 (DfEE 0083/2000) offers a range of activities that can be readily adapted for this purpose.• Pattern-spotting activities and well-planned work discussing and exploring large numberscan help to intrigue pupils about mathematics. For example, ‘How many addition sumscan you make with the answer 20? How many subtraction sums can you make with theMathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200321

Primary National Strategy DfES 0606-2003same answer? Can you see any patterns in your answers?’ Trainees may also wish torefer to the DES publication Mathematics from 5–16 (1985) and Ernest (1999) in Casey,Ernest and Koshy. Ernest (1999) emphasises the importance of teaching mathematics todevelop enjoyment and positive attitudes: ‘…a positive attitude often leads to greatereffort and better attainment in mathematics…’.• Referring to research articles that raise pertinent issues concerning mathematicalactivities for pupils requiring additional support. Trickett and Sulke (1988) found that thesepupils are able to cope with activities involving elements of mathematical challenge,including a temporary state of confusion or puzzlement, but the dependency culturedeveloped in their classroom often limited such learning opportunities. Houssart (2002)also raises some interesting issues in her research into the relationship between themathematical activity and the pupils’ apparent refusal to carry out the activity. Issuescovered by Houssart (2002) include pupils’ preference for more challenging tasks, theneed for mathematical activities to be presented in a variety of ways and pupils’ preferredmethods of calculation.Teaching strategies that promote inclusive teaching and learning in mathematicsThere are a number of teaching strategies it is useful to review with trainees in terms of howthese strategies help pupils requiring additional support to make progress in theirmathematical learning. Examples of such strategies and the support they offer may include:• Selecting a range of resources to demonstrate and model the mathematics and exemplifyexplanations. Refer to video clips 6 and 7, Counter readings 29:55 and 31:10, forexamples. Refer also to the Corbett (2001) article in appendix iv, page 50, for furtherdetails of resource use.• Sharing clear learning intentions with pupils. Trainees may find the following sequencehelpful. Provide a clear instruction – what the pupils need to do – followed by a furtherinstruction about how the pupils should do this. Make clear the purpose of the activity,e.g. ‘We are learning to…’ and then agree the success criteria so pupils will know how torecognise that they have achieved the learning intention.• Using the NNS Interactive Teaching Programs (ITPs) to provide dynamic visual imagesthat model mathematical concepts and processes while promoting discussion andinteraction.• Considering when to split the mathematical learning into stages to ensure elements ofsuccess provide further motivation. Provide further scaffolding for this by using resources,peer and adult support. These strategies present the activity in a manageable way toprovide a balance of success and challenge. Instructions and explanations can also bebroken down into manageable chunks.• Ensuring expectations are communicated to pupils so that they understand what theyshould achieve within the activity. Pupils welcome a secure framework whereexpectations are explicit.• Offering opportunities for pupils to participate in paired discussion work duringmathematical activities. Video clip 4, Counter reading 11:43, provides useful examples ofthis strategy.• Presenting the same mathematical idea in a range of contexts, e.g. using place value inthe context of measures. A range of contexts supports concept development andreinforcement in mathematics.• Highlighting links to previous areas of mathematics to reinforce mathematics that has22© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategybeen covered and can be used within the activity. Pupils do not always readily make linkswithin and across areas of mathematics; making these explicit will help pupils use knownfacts and skills in a range of activities. It may be helpful to remind trainees of the researchof Askew et al (1997) on effective teachers making links across and within different areasof mathematics (connectivism). Examples of such links include emphasising + and – asinverses and the importance of making sense of calculations (making links betweendifferent representations of a mathematical calculation). A mathematical activity toillustrate this can be found in Askew (1998). The ‘think board’ activity encourages pupilsto record a calculation, tell a calculation story about the same calculation and to draw apicture of what it means.• Providing opportunities for pupils to make choices and suggest developments within theactivity. This strategy enables pupils to assume more autonomy within their learning.• Using prompting questions to direct and remind pupils of the knowledge and skills theycan apply to the activity, e.g. ‘What did you use last time when you were measuringobjects? Do you remember that you halved and halved again when you were dividingby…?’• Encouraging pupils to use a signal to indicate if they do not understand, e.g. thumbs upfor yes, sideways for not sure, down for no.• Selecting strategies to help pupils circumvent problems with rote recall of number facts.Strategies may include: encouraging the use of pattern; working from known facts to helpwith unknown facts; aids to recall basic number facts such as pupils’ own pocketnumber line, 100 square, number fact grid.• Demonstrating how to use approximation and checking when doing calculations and howto use jottings to note instructions for each of the steps in multi-step or mentalcalculations. These types of strategies help to overcome problems in working memory.• Using incorrect responses from pupils positively to highlight any part that is correct, inorder to prompt the same pupil to answer again or to consider the strategies used.Prompting pupils to explain incorrect answers often give clues to the correct answer andprovides a valuable teaching opportunity.Organisational issuesWhen reviewing organisational issues regarding pupils requiring additional support it may beuseful to consider:• access to the teacher and the resources;Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200323

Primary National Strategy DfES 0606-2003• careful planning of pupils’ seating;• having support in place for pupils who cannot ‘hold things in their head’, e.g. stickynotes, jottings, individual whiteboards;• giving time or support before responses are required, e.g. personal thinking time, partnertalk, more scaffolding until the pupil can answer correctly;• using buddying for paired work, e.g. placing a more settled pupil with a pupil who findsconcentration difficult;• providing a distraction-free area set up for pupils who need it to work in;• using alternatives for paper-and-pencil based tasks where appropriate;• using visual and tangible aids to summarise the mathematical learning;• using the strategies needed to enable all pupils to participate in the mental and oralstarter, e.g. opportunities to join in whole-class responses, planning of differentiatedquestions, appropriate use of ‘wait time’;• making well-planned use of any TA or additional teacher support available;• making expectations of the pupils’ achievement during the lesson/activity clear;• monitoring pupils’ achievement and providing positive feedback and encouragement;• encouraging pupils to reflect on and identify their learning – moving towards selfassessment;• giving mathematically specific praise and making it clear to pupils what they have learned;• planning for pupils to be engaged in the plenary, with support from the TA/supportteacher where appropriate;• considering the importance of communicating expectations of pupils’ progress toparents: supporting parents in their role in developing their children’s mathematicallearning in the broader context and within homework activities.Further pragmatic advice can be found in section 3 of the NLS file: Including all children inthe literacy hour and daily mathematics lesson (DfES 0465/2002).Pupils learning English as an additional languageGathering initial information to inform planningThe following points may be used to prompt discussion with trainees:• The class teacher and records of pupils’ progress (including targets) provide valuableinformation at the beginning of school experience.• Visual and other cues help pupils to make sense of the activity (refer to video clips 1 and2, Counter readings 00:28 and 03:20, for examples).• Resources can be used to demonstrate and model the activity and help pupils torespond (refer to video clip 7, Counter reading 31:10).• There is a need to distinguish between the different purposes of language within amathematical activity, particularly in terms of essential key vocabulary used within thecontext of a mathematical activity. Refer to Raiker’s article (2002) ‘Spoken Language andMathematics’ in appendix iv. This article also applies to other groups of pupils.• Allowing sufficient time for pupils to watch, listen and respond.Identifying appropriate learning objectivesIn planning appropriate learning objectives it may be helpful to review some of the following24© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategypoints with trainees:• Consultation and joint planning with the class teacher and EMA staff.• Appropriate learning objectives should be sufficiently challenging in terms of mathematicalcontent and also take account of accessibility for pupils who are speaking and listeningto English as an additional language. Attempts to simplify the language involved in themathematical activity may unintentionally result in simplifying the mathematics.• The Assessment toolkit to support pupils with English as an additional language (DfES0319/2002) is based on the key objectives for each strand of the Framework andprovides a useful starting point for helping to identify pupils’ mathematical skills andknowledge.Planning appropriate mathematical activitiesTrainees may find it helpful to consider the following aspects when planning appropriatemathematical activities for pupils who are learning English as an additional language:• It is easy to underestimate what pupils can do mathematically simply because they arenew learners of the English language.• The mathematical content has sufficient cognitive challenge.• The planned mathematical activities identify reinforcement activities or incrementalstepped challenges that can be selected appropriately as pupils undertake the activity.• Consider how pupils are able to access the mathematics successfully. Planning foraccess may include use of: resources; pre-tutoring to review necessary mathematicalskills; support of additional adults.• The use of precise, succinct instructional language illustrated by pictures/diagrams ifnecessary to ensure pupils can access and complete the activity.• Planning the development of mathematical vocabulary within the teacher’s demonstrationand modelling with opportunities for pupils to rehearse and explore the vocabulary withinthe follow-up activities. Video clip 2, Counter reading 03:20, illustrates these aspects.Teaching strategies that promote inclusive teaching and learning in mathematicsExamples of teaching strategies and the support they offer pupils learning English as anadditional language may include:• Actively encouraging pupils to join in when pupils are doing things in unison, e.g.counting, joining in actions, saying words. This will enable pupils to experience success ina secure setting.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200325

Primary National Strategy DfES 0606-2003• Building in thinking time to help pupils to process language and mathematics information.• Encouraging pupils to repeat the teacher’s question to check understanding and gainextra time. New skills and facts need more time to consider.• Using teaching strategies that encourage interaction and participation: Unison and repetition response – the teacher asks a question, repeats the question,judges thinking time and then gives a trigger for all pupils to answer together. Thisenables pupils to take risks in a secure context. Show me – using an individual whiteboard or A4 sheet folded into eight sections –every pupil is expected to display an answer – responses give immediate feedback tothe teacher to inform the next steps.‘The answer is 24. With a partner decide what is the question’ – this encourages arange of answers, creativity and paired work. This approach allows pupils to discusswhat they know, check ideas and test out answers with peers.• Using a variety of resources, e.g. number fans, whiteboards, etc. so pupils are providedwith visual cues to reinforce the mathematics.• Explaining the meaning of new words in context and rehearsing them several times –encouraging repetition in response to focused questions. This strategy helps pupils tomake sense of and use appropriate mathematical vocabulary.• Modelling answers to questions orally as well as in written form. This approach enablespupils to make links between spoken and written forms of communication.• Using pupils’ words and refining them to give more precise meaning. This helps pupils toextend their use of mathematical vocabulary.• Using gestures and drawings to support explanations. Pupils learning English as anadditional language draw on visual cues to support their developing understanding.Organisational issuesWhen reviewing organisational issues regarding pupils learning English as an additionallanguage it may be useful to select points from pages 21–22 and to also consider:• The use of pre-tutoring to support pupils’ participation, e.g. pre-tutoring of essentialmathematical vocabulary.• Grouping of pupils to support learning needs within the mathematical activity, e.g. theopportunity for paired discussion, so pupils can test out and exchange responses.• Placing a pupil with an adult or peer who shares the same first language (if possible) toprovide clarification of the mathematical activity and to support mathematical discussion.• Using a pupil who provides good mathematical models to work with a pupil learningEnglish as an additional language to provide exemplification and development of themathematics.• Placing a pupil learning English as an additional language with a pupil who provides goodlanguage models when the activity has a clear focus on vocabulary development andcommunicating mathematically.More able pupilsGathering initial information to inform planningThe following points may be used to prompt discussion with trainees.• Some more able pupils have stronger numerical ability than spatial ability and vice versa.• Some more able pupils may take short cuts and their route to a solution may not be26© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategyimmediately evident.• Some more able pupils prefer the routine of textbooks and work cards where goodmemorisation skills enable them to achieve a lot of correct answers, but they may find anactivity requiring logical thinking more challenging.• More able pupils have the capacity to learn and understand new ideas and ways ofworking quickly.• Pupils who appear to be generally good at mathematics may find use of mathematicallanguage and communication more challenging, as this involves concentration ondescription and explanation rather than just calculation.The attitudes and responses of able pupils may also be a useful aspect for trainees toconsider. For example, trainees may find it helpful to discuss the following examples in termsof how assessing these pupils’ capabilities may prove more challenging and how they wouldrespond:• Emma in Year 5 did not want to do a more challenging investigation into consecutiveproducts. She commented ‘The others will do a lot more (multiplication calculations) andget more right’.• Alice, a very able Year 1 pupil said ‘I want to use the beads too’ (a lower level activitywhich her friends had been set).The NNS materials: Supporting more able pupils in Year 5 (DfES 0449/2002) and Year 6More able pupils (DfES 0362/2003) outline further detailed information to inform planning.Identifying appropriate learning objectivesIn planning appropriate learning objectives it may be helpful to review some of the followingpoints with trainees:• Evaluate the different approaches that may be used in terms of accelerating the learningobjective, extending the learning objective and enriching the learning objective. A usefuldiscussion of these approaches can be found in Fielker (1997) and Eyre and McClure(2001).• The NNS Framework supplement of examples provides an overview of progression withinspecific learning objectives that trainees may find helpful when considering these differentapproaches.• Extension and enrichment will draw upon mathematical processes such as reasoning,Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200327

Primary National Strategy DfES 0606-2003generalising, conjecturing and proving. Trainees may not realise that the calculationstrategy of adding 10 and subtracting 1 in order to add 9, is just as much ageneralisation as a rule about the nth term of a repeating pattern.Planning appropriate mathematical activitiesTrainees may find it helpful to consider the following aspects when planning appropriatemathematical activities for more able pupils:• Some more able pupils may prefer mathematical activities with clearly defined parametersregarding methods and answers rather than mathematical activities that require the use ofmathematical reasoning, making decisions and communicating approaches and findings.• Activities should also offer the opportunity to work at mathematics at a deeper level. Eyreand McClure (2001) provide a number of detailed examples of mathematical activitiesbased on working at a deeper level and within a broader range (see appendix iv page 55).• The NNS booklet Year 6 More able children course handbook provides some helpfulexamples of more able pupils’ responses to mathematical activities as well as guidanceon developing teaching strategies to support their learning. Video clips 3 and 4, Counterreadings 07:55 and 11:43, provide exemplification of appropriate activities for Key Stages1 and 2.• Appropriately challenging activities for more able pupils may involve more sophisticatedmathematical reasoning and frequent aspects of proof. Video clip 5, Counter reading23:08, and clip 6, Counter reading 29:55, offer useful examples of able pupils usingmathematical reasoning and developing ideas of mathematical proof.Teaching strategies that promote inclusive teaching and learning in mathematicsTeaching strategies to support more able pupils may at times be appropriate for othergroups of pupils but it may be useful to consider how these strategies specifically supportmore able pupils:• More able pupils can be asked to communicate succinctly to peers and involve peers inthe evaluation of the approaches used. Encouraging more able pupils to explain to peershow to tackle a particularly challenging aspect of mathematics enables them to extendtheir own understanding as well as helping peers.• Consider strategies to develop a mathematical activity by focusing on the questions andanswers posed to the pupil. Such strategies may include: changing part of the question to encourage pupils to consider further examples orcounter examples; removing some of the question, e.g. changing 9367 – 2649 to 9 67 – 2649encourages pupils to consider possibilities; giving the answer with the question, e.g. ‘What right-angled triangles can you find witha hypotenuse of 17cm?’ This provides opportunities for pupils to demonstrate andapply mathematical knowledge; posing questions based on: is it always, sometimes or never true? e.g. ‘Is it always,sometimes or never true that if a is less than b then a squared is less than b squared?’This strategy encourages pupils to reason, justify and communicate mathematically.• Asking pupils to consider what is the same and what is different, e.g. ‘What is the sameand what is different about regular and symmetric polygons?’ This strategy providesopportunities for pupils to use mathematical reasoning and explain their mathematicalunderstanding.28© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategy• Giving a definition of... e.g. ‘Give a definition of a rectangle which only mentions itsdiagonals.’ This encourages pupils to apply and communicate their mathematicalknowledge.• Asking able pupils to devise questions for each other (they also have to provide theanswer sheet). This encourages pupils to apply and demonstrate their mathematicalunderstanding.• Video clip 3, Counter reading 07:55, of young children working mathematically providesuseful examples of the types of questions that support and extend mathematical thinking.Trainees may find it useful to develop examples of setting more challenging mathematicalactivities for more able young children within the classroom context, such as the role-playarea. Other examples could be providing calculators as a resource for these children touse to explore large numbers, and playing games without fixed rules to explore a rangeof strategies.• The type of questions teachers ask has a profound effect on the type of learning thattakes place. If the focus of questioning is to extend and deepen mathematical thinkingthen it may be appropriate to consider the use of promoting questions to: set pupils challenges so they can apply their ideas and reasoning skills and deepentheir mathematical understanding; encourage pupils to enquire by setting up and testing hypotheses of their own; stimulate discussion about efficiency and the merits of alternative strategies; foster pupils’ ability to think, to review their approaches and solutions, and identifyother, more efficient, strategies they might use.Examples of promoting questions include: ‘Do you think that the number of columns in the grid makes a difference?’ ‘Have you thought of all the possibilities? How can you be sure?’The NNS booklet Year 6 More able children course handbook (session 2) discussesteaching strategies that offer opportunities for more able pupils to work on the same aspectof mathematics as the rest of the class.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200329

Primary National Strategy DfES 0606-2003Organisational issuesWhen reviewing organisational issues regarding more able pupils it may be useful toconsider:• The impact of some methods of organisation on the more able pupils’ experience oflearning mathematics, e.g. if the more able pupil is always offered different opportunitiesfrom their peers their experiences become exclusive and learning is isolated.• If the more able pupil is perceived as the ‘expert’ then intellectual risk-taking may provedifficult. In the same way peer group pressure may suggest that it is not desirable to beable at mathematics.Flexible organisation may help more able pupils who are not particularly competent in anarea of mathematics to work easily alongside peers. A further perspective is offered by Eyreand McClure (2001) in appendix iv page 55 .When considering specific organisational issues such as grouping of pupils it may be helpfulto consider current research in this area. Boaler and Wiliam (2001) present a useful overviewof research studies and acknowledge that there is a strong belief that ability grouping (or‘setting’) is necessary for effective teaching and that ability grouping raises levels ofachievement. Whitburn’s (2001) research indicates that ability grouping, of itself, does notraise standards and in some cases can lower them. The NNS Framework pages 18–20highlights the need for flexibility in grouping pupils and the fact that setting does notnecessarily close the attainment gap across the year group over time.A number of the aspects covered in this section that are significant to inclusive practice inthe teaching and learning of mathematics are represented within the following inclusiondiagrams:• diagram for more able pupils (page 31);• diagram for pupils who are learning English as an additional language (page 32);• diagram for pupils requiring additional support (page 33).These diagrams provide a useful overview and help trainees to identify the elements of theirpractice that already take account of these aspects, and the elements of their practice theyare still developing.30© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyThe circles of inclusionMore able pupilsLearning objectivesSetting suitable learningchallenges.Tracking forward.Providing opportunitiesfor enrichment.Teaching stylesResponding to pupils’diverse needs.Pace, developing reasoning,generalising and explaining.InclusionAccessOvercomingpotential barriers to learning.Using a variety of recording methods, wherethe barrier to learning is the speed of writtenrecording compared to thought processes.Providing opportunities for pupils to worktogether to overcome isolation issue.Encouraging use of resources as apositive aspect of workingmathematically.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200331

Primary National Strategy DfES 0606-2003The circles of inclusionPupils learning English as an additional languageLearning objectivesSetting suitablelearning challenges.Tracking back.InclusionTeaching stylesResponding to pupils’ diverseneeds.Adopting a multi-sensoryapproach using visual cues,mime, gesture to support oraldelivery.Using clear language ofinstruction.Encouraging participation byallowing sufficient thinkingtime and opportunitiesfor paireddiscussion.AccessOvercomingpotential barriers to learning.Using culturally appropriate texts and images.Strategies to overcome linguistic barriers –using home language, pre-teachingvocabulary.Using teaching assistant and peersupport.32© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyThe circles of inclusionPupils requiring additional supportLearning objectivesSetting suitablelearning challenges.Tracking back.Providing opportunitiesfor enrichment.Teaching stylesVisual, aural andkinaesthetic learning.Differentiated questioning.Shorter tasks.Scaffolding to makeopen-ended tasks moremanageable.InclusionAccessOvercomingpotential barriers to learning.Resources – use of visual prompts and cuesand the opportunity to manipulate physicalresources when responding to mathematicalquestions.Opportunity to use a variety of recordingmethods.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200333

Primary National Strategy DfES 0606-2003Section 3 Assessment approachesAssessment approaches that support the inclusive learning for all pupils will reflect a rangeof learning styles to ensure that all pupils can demonstrate their understanding andattainment through appropriate means. These approaches are based on the characteristicsof effective assessment for learning, with particular reference to the characteristic thathighlights a commitment that every pupil can improve.Trainees may find it helpful to consider the characteristics of effective day-to-dayassessment:• using questions and sharing comments with pupils;• making observations of pupils during teaching and while they work;• holding discussions with pupils;• analysing work, reporting to pupils and guiding their improvements;• testing pupils informally in a variety of ways;• engaging pupils in the assessment process.The following assessment approaches may be useful in helping trainees to focus onparticular groups of pupils to find out levels of pupils’ understanding and to use thisinformation to promote effective mathematics teaching and learning. The approaches areneither exhaustive nor exclusive, for example, observation may inform questioning thatmay lead to discussion. Trainees could be encouraged to identify further assessmentapproaches and then to consider how the information will be used to inform mathematicsplanning and teaching.OpportunitiesIt is useful to ask trainees to consider the types of assessment opportunities that will enableall pupils to demonstrate what they can do, what they know and what they understand.Pupils may need to be reminded of the links between various mathematical areas to helpthem use existing knowledge more readily. These planned assessment opportunities willenable pupils to recall mathematical knowledge; to use and apply the mathematics theyhave learned; and support the development of mathematical thinking. Such opportunitiesmay take place within the main part of the mathematics lesson or within the plenary.If the plenary is used as an assessment opportunity it is important to ensure that pupils areclear about the learning goals within the main activity, in terms of what they are expected toachieve, so that they can make a full contribution to the plenary. An example of this may beto remind pupils during the main activity that they need to be considering a challengingquestion based on what they are learning, to ask the class during plenary. When selectingthe pupils’ challenging questions the teacher can make explicit the types of questions thatallow pupils to use and apply the mathematics from the main activity. Further suggestedexamples of using the plenary as an assessment opportunity for pupils to use and applymathematics can be found in a number of the NNS Unit Plans, e.g. Year 4 Unit 4 Reasoningabout shape, days 3 and 4 (www.standards.dfes.gov.uk/numeracy).The opportunities should also take account of different communication styles that pupils willuse, e.g. pictorial responses, practical application, and the use of resources including ICT.Eyre and McClure (2001) provide some useful examples of these different styles in relation toyounger and older more able pupils. Trainees may also find it useful to refer to the chart34© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategyoutlining alternatives to written recording that can be found at the back of the NationalLiteracy and Numeracy Strategy file: Including all pupils in the literacy hour and dailymathematics lesson (DfES 0465/2002).Effective questioningSome questions are more effective than others at providing trainees with assessmentopportunities and information. It is useful to remind trainees that changing the way aquestion is phrased can influence:• the mathematical thinking a pupil needs to use;• the language demands made on the pupil;• the opportunity for pupils to respond fully and demonstrate their mathematicalunderstanding.To help trainees select questions appropriately for assessment purposes, you may wish toreview the types of questions related to mathematics teaching and learning on page 4 of theNNS Mathematical Vocabulary booklet (DfEE, 0313/2000) as well as considering thefollowing types of questions: Prompting questions help pupils access the mathematics they are being taught andencourage them to answer questions that may not be familiar. Such questions can alsohelp pupils to tackle a problem they are unsure of starting. For example, ‘If you know32–12=20, can you now calculate 32–13? What about 31–12?’ Probing questions help pupils develop their mathematics, recognise when they havemade a mistake and become more confident in communicating and reasoning. Youmay find it useful to encourage trainees to review and modify the examples of probingquestions related to the key objectives in section 2 of NNS Using assess and reviewlessons (DfES 0632/2001). Promoting questions build on and deepen pupils’ understanding of the mathematicsthey have been taught. They are designed to help pupils to draw out a pattern or tohighlight the structure they have been using but may not have noticed. For example‘When you divide by 10 the number gets smaller, when you divide by 1 it stays thesame. What happens if you divide by 0.1?’Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200335

Primary National Strategy DfES 0606-2003Trainees will find section 2: Using effective questioning techniques, in the bookletAssessment for learning – Using assessment to raise achievement in mathematics (QCA,2003) helpful in terms of examples of effective questions, and the use of questions to findout pupils’ specific misconceptions. Refer also to pages 28–29 for examples of questioningthat promotes mathematical thinking.It may be useful to discuss with trainees that some pupils will not respond to verbalquestioning for a variety of reasons, e.g. pupils who are learning English as an additionallanguage or pupils requiring additional support whose language skills are weak may use anon-verbal framework for thinking. These pupils may be able to demonstrate understandingwhen a calculation is shown to them using objects, a computer screen calculation,pictograms, etc.ObservationPlanning for assessment may involve using systematic observations as a basis fordiscussion between the class teacher and the trainee. It may be useful to select aconsidered observation and/or a confirmatory observation of particular pupils duringteaching and while they work. A considered observation will have been planned to takeaccount of earlier assessment information in order to provide further information aboutachievement. A confirmatory observation will be part of ongoing assessment practices thatfocus on pupils’ attainment and progress. Trainees may find it helpful to consult NNS Usingassess and review lessons (DfES 0632/2001) to support their planning of systematicobservations:It may be helpful to highlight and discuss with trainees the potential issues arising fromsystematic observations.• Pupil produces illegible written response due to poor print skills, physical impairment, lackof confidence which may suggest the pupil does not understand the mathematics.• Pupil has different methods of demonstrating understanding that may not be recognised,e.g. sorts according to a different criteria. It may be useful to consider if the preferredcriteria is demonstrating understanding.• Pupil has to interpret and use a range of mathematical terms to describe an operation,e.g. add, plus, more.• Pupil may find it difficult to find appropriate words to explain a strategy or an answeralthough the mathematics is secure.• Pupil may have insufficient thinking time to respond to questions.• Pupil may have difficulty holding a question or problem in his or her head while workingout a response.• Pupil may be unable to recall number facts and tables to apply them.Trainees could also use the information gathered from the observation grids in section 1pages 8–16 to consider the pupil’s approach, motivation and attitudes to mathematics aswell as his/her understanding of the mathematics. El-Naggar (1996), pages 18–21,highlights significant issues concerning the observation of pupils with specific learningdifficulties in mathematics that are also helpful for pupils learning English as an additionallanguage and more able pupils.36© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyDiscussionTrainees may use a focused discussion to develop an assessment dialogue with pupils. Forexample, trainees may hold a brief impromptu discussion to follow up an impromptuobservation or any surprises concerning a pupil’s behaviour or responses to themathematics that is being taught.Trainees may also hold illustrative discussions to assess pupils’ understanding, to diagnosethe reasons for any misunderstanding or misconceptions and to resolve difficulties. Thesetypes of discussion may involve using resources, further examples and/or diagrams. NNSLearning from mistakes, misunderstandings and misconceptions in mathematics (DfES0527/2002) offers a workbook and a range of resources including video clips that traineeswould find useful to support such discussions.Trainees may also use informal discussions to follow up earlier assessment and diagnoses,to discuss progress, targets and any peer- or self-assessments that have been made.Planning discussions within a number conferencing context offers a structured approach toassessment discussions and helps trainees to think about:• how to make precise assessments of pupils’ knowledge and understanding fromrelevant data;• how teachers’ questions and interactions with pupils affect and influence theassessment process.One approach used by an institution involves trainees defining an appropriate part of a keyobjective from NNS for mental mathematics (e.g. using near doubles) and devising a seriesof questions which they feel would assess this. The trainee then undertakes a ten-minuteconference with one pupil using a tape recorder. The trainee transcribes the data and bringsit back to the institution for analysis using an analytical framework. This framework requirestrainees to use their data precisely in considering such questions as:• What does this data show that the pupil knows, understands and can do?• What kinds of questions are you asking? How is the pupil responding to your questions?• Overall, how effective is the number conference, as you designed it, in enabling the pupilto discuss his or her knowledge and understanding?Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200337

Primary National Strategy DfES 0606-2003The BEAM materials (Denvir and Bibby 2001) also use interview techniques to obtaindetailed assessments of pupils’ knowledge and progression in numeracy, based on Denvir’sresearch at King’s College.Analysing pupils’ verbal/written/non-verbal responsesListening to pupils’ responses and analysing them will provide further information aboutlevels of understanding and difficulties pupils may be experiencing. Analysis may also involveassessing pupils’ achievement against the learning objectives in order to monitor pupils’progress. This type of analysis will also inform the feedback trainees provide to help pupilsrecognise the required next steps and how to take them.Trainees may find it helpful to refer to pages 10–12 in the booklet Assessment for learning –Using assessment to raise achievement in mathematics (QCA, 2003). Exemplars of pupils’responses to questions are provided, with the assessment notes triggered by the differentresponses given by pupils. The Ian Sugarman video (clip 1) in the NNS Guide for yourprofessional development book 4 provides an appropriate example for trainees to analyse.Analysing pupils’ written responses needs to take account of whether this is undertakenwith or without the pupils present. If analysis is undertaken without the pupils present,consideration should be given to how well the pupils have understood the task and what thepupils appear to know and do not know.Analysis of written work with the pupils present will provide the opportunity to clarify thepupils’ response, discuss any errors or misconceptions and consider improvements andnext steps.TestingInformal testing of pupils may be carried out using a variety of strategies. Trainees may wishto introduce brief review tests that draw upon what has been taught previously, to assessaspects of mathematics that may need revision or reteaching quickly. Information from thesereviews can be used to guide the rest of the lesson or plan future teaching. Anotherstrategy may involve conducting short and sharp informal recall tests with planned orspontaneous questions to provide immediate assessment information about the pupils’ability to recall mathematical knowledge and facts. NNS Learning from mistakes,misunderstandings and misconceptions in mathematics (DfES 0527/2002) offers a range offormative assessment tasks with accompanying notes that would provide a useful startingpoint for trainees.FeedbackEffective feedback should prompt pupils to think more deeply and to engage moreproductively in improving their mathematical work. Feedback needs to be focused andbalanced in terms of making clear how well the pupil has done, what still needsimprovement and how to begin that improvement. This requires the trainee to have a goodunderstanding of the mathematics within the teaching objective and the intended learningoutcomes. Developing constructive comments requires a great deal of effort, and in manycases, it is time consuming. Black, Harrison, Lee, Marshall and Wiliam (2002) report that asteachers tried to create useful comments many realised that they needed to reassess thenature of the activities they had set. In some cases, the activity revealed the pupils’understandings and misunderstandings, but other activities focused mainly on conveyinginformation.38© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyIt may be useful to broaden the scope of feedback in terms of suggesting trainees considerthe feedback pupils receive when undertaking mathematical activities using ICT. Higgins(2001) presents a number of case studies focusing on the complexity of feedback offered bymathematical computer programs and the need for the teacher to enhance the quality ofthis feedback.EngagingEngaging pupils in the assessment process is a key principle of assessment for learning.Supported pairs or small groups of pupils reviewing each other’s work, with guidance or achecklist outlining what to look for, provides a useful bridge to the process of selfassessment.Black, Harrison, Lee, Marshall and Wiliam (2002) suggest that ‘self-assessmentwill only happen if teachers help pupils, particularly the low-attainers, to develop the skill.This takes time and practice.’A particular idea teachers found effective is for pupils to use ‘traffic light’ icons, labelling theirwork green, amber or red according to whether they think they have good, partial or littleunderstanding. This helps the pupil to communicate in a simple way that could then bedeveloped in terms of justifying their judgements in a peer group, so linking peer- and selfassessment.Further strategies can also be found in Black, Harrison, Lee, Marshall andWiliam (2002).Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200339

Primary National Strategy DfES 0606-2003Appendix iITT examples of inclusion provision within courseprogrammesThe following pages contain examples from mathematics tutors of how they cover inclusionwithin course programmes.Example 1Here are examples of some of the activities I do with trainees that consider inclusion issues.This is not a comprehensive list. I try to embed inclusion into most mathematics sessions.Counting in SwahiliI introduce the session by counting to 10 in Swahili several times then I ask the traineeswhat is going on. We then identify which numbers are easy to remember, which ones arehard. Then I ask the trainees to join in with me and count forwards and backwards.Analysis• The possible difficulties that bilingual children have in learning a counting system that iscompletely different from their own (language structure, vowel and consonant sounds).• The vocabulary and the sounds of words.• The resources and support that can help, e.g. pictures, numerals, songs, rhymes, etc.• The benefits of class chanting.• The irregularity of our number words.• Children with hearing difficulties not being able to distinguish between all of the sounds.This has links with Early Years as well. We also list the difficulties children with dyslexia have,e.g. remembering the order, formation of numerals, etc.Multicultural issuesThe trainees are introduced to games from other countries such as Oware. They are askedto analyse the structure of the game and the mathematics involved. We also look wider andtalk about the mathematics involved, e.g. geometry (symmetry) from African cloth, weaving,etc. and try to recreate the patterns.There is also the series called ‘Mathematics from Many Cultures’ and the Oxfammathematics opportunities cards.The development of the number systemPlace valueI get the trainees to decode five scripts (e.g. Arabic, Bengali, Chinese, etc). They are givensix numbers in the five scripts not in order and they have to match the six numbers for eachscript by looking at the ‘numerals’. We then analyse how each script is made and thesimilarities and differences between the scripts and our number system. This helps thetrainees appreciate some of the reasons why children might make mistakes with how theywrite numerals or how they partition numbers.40© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyMental imageryWhen the trainees cover mental imagery for geometry we consider the differentrepresentations pupils have and consider why pupils use these representations and howthey help with the mathematics. As there are a number of trainees with dyslexia, they cantalk from their own experience. We try out some activities and talk about the images used.Some of the difficulties identified include: not being able to follow all of the instructions;manipulating things in space; left and right orientation, etc.Example 2One session which works very well focuses on use of the NNS folder Teaching the dailymathematics lesson for children with specific learning difficulties, which covers dyslexia,autistic spectrum, etc. We divide the trainees into groups and give each group one of thebooklets from the NNS folder and a copy of an NNS unit plan lesson. I usually pick one thatis about written calculation strategies and where the mental/oral starter has no link to themain activity.The trainees then prepare a short presentation on their particular ‘special need’ and thensay which parts of the lesson would present particular difficulties, what provision they wouldneed to make by way of alternative resources, and how the lesson would have to beadapted.Example 3Here are some of the aspects of inclusion from our mathematics course programme.• When we consider ability and ability grouping, the trainees are divided into two sets, Forand Against. The Fors read a summary of Ofsted’s report on ability grouping and theAgainsts have a TES report summarising Anita Straker’s comments on setting by ability.Each set prepares three key points for a debate and we use this as the basis fordiscussing their own experience as pupils and as trainee teachers in school.• We look at the three principles of inclusion and the NNS and discuss implications fordifferentiation and needs of particular groups.• We consider the NNS point on meeting individual needs and closing the gap of lowattainment – NNS Guide for your professional development book 1 (i.e. making links withdifferentiation but maintaining high expectations).• We ask trainees to have a go at tracking back a particular mathematics topic from theyear group they currently work with, including looking at P levels.• We use the Ofsted video (doubles lesson) to identify how a wide range of ability can becatered for – trainees have to identify, for example, the easiest and most challengingquestions asked in the mental and oral starter. In the main part of the lesson, againdifferentiated tasks are discussed.• We ask trainees to rank order statements from the NNS folder including all children in theliteracy hour and daily mathematics lesson – ‘dyslexia friendly classroom’ giving reasonsfor choices, e.g. ‘your top three’.• We recognise the need to plan beyond the ‘normal’ range of differentiation, e.g. for thegifted and talented. Trainees brainstorm characteristics of more able pupils inmathematics and then compare them with those identified elsewhere, e.g. NNS bookletMathematical challenges for able pupils in Key Stages 1 and 2 and the QCA website ongifted and talented pupils. We discuss acceleration versus enrichment by looking atMathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200341

Primary National Strategy DfES 0606-2003sample medium-term plans and the NNS booklet Mathematical challenges for able pupilsin Key Stages 1 and 2.• For pupils learning English as an additional language – mathematical vocabulary – weconsider lists of mathematical vocabulary that cause confusion, e.g. words used ineveryday life. We come up with strategies to address these, e.g. display mathematicalvocabulary, develop/use mathematics class dictionaries.• We look at written mathematical texts using Clements in Dickson, Brown and Gibson(1984). We use examples for reading, comprehension and transformation difficulties(taken from Clements' analysis) including National Curriculum test questions.• We identify activities for promoting talk/discussion, e.g. ‘feely bag’ with mathematicalshapes; ‘peek-a-boo’ – slowly revealing a digit/mathematical shape; back-to-backinstructions to form a partner’s shape with five interlocking cubes; sorting or classifyingtriangles by side or angle. Relate to Straker (1993).• Final year trainees are given a mock interview question requiring them to presentstrategies for teaching mathematics to pupils who are learning English as an additionallanguage. Their responses are compared with Hounslow’s website suggestions (websitereference, appendix v, page 114). We also use the NNS EAL video to exemplify.• We have also used ATM's ‘Talking maths talking languages’ giving trainees the problemswritten in other languages and getting them to think about what strategies they use todecipher meaning.Example 4Here are some examples of how we integrate and provide discrete coverage of inclusionaspects in our mathematics programmes:• School experience task – observe approaches to teaching and planning mathematics inthe school with a particular focus on children learning English as an additional language.This is followed up with further guidance during the taught sessions.• Approaches to assessment include how this assessment can be used to inform theplanning and provision of a differentiated mathematics curriculum with a particular focuson children with English as an additional language.• We also develop the use of assessment to inform the planning of a differentiatedprogramme of mathematics suitable for all children, especially those with specialeducational needs. Trainees are asked to bring the assessments (observations, samplesof work across the range of needs) gained during the school experience weeks and anexample of their planning.• We consider ways of linking mathematics across the curriculum, including theopportunities for multicultural mathematical activities.• The trainees visit special schools for a week during the cross-phase visits in February andMarch. They are briefed for this in both the preceding maths sessions and theProfessional Studies course. At this time there are also two sessions in the ProfessionalStudies course that focus on a) the current debate concerning inclusion, and b) the SENCode of Practice and recent legislation relating to pupils with special educational needs.• The mathematics course is tied in closely with other courses. EAL provision is developedwithin the literacy courses and in Professional Studies courses, e.g. the professionalstudies session on a diagnostic and formative assessment session links closely withmathematics session 5 on diagnostic and formative assessment, and both have anemphasis on the provision for pupils who are learning English as an additional language.42© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyExample 5The Year 3 trainees address issues of Inclusion in a Professional Studies module insemester 1. This is built on in semester 2, in Core Studies, in a school-based moduleentitled ‘Extending the core: English, mathematics and science’.The trainees have two key lectures and three taught sessions of two hours in English,mathematics and science before going into school to teach small groups of children for foursessions, thus enabling immediate juxtaposition of theory and practice. The trainees areallocated to schools, with between seven and fourteen trainees working in each of theseven schools. They work in twos with small groups of children, having negotiated andplanned the lessons in either English, mathematics or science with the school-based tutor,thus ensuring a quality learning experience for both the children and the trainees. Thisapproach fosters team building and collaborative approaches to teaching and learning.Trainees often work in groups at the university but traditional school experience can leavetrainees feeling isolated and under constant scrutiny. Concerns about discipline and controlare not central to the experience. What is central is trainees’ analysis of how children learnmathematics in an inclusive classroom and the strategies needed to ensure inclusivity.Trainees are encouraged to see the school-based component as problem solving, involvingreflective thinking and decision making, which develops confidence.Traditionally, the experience in school has placed an emphasis on rigorous assessment,whereas here trainees’ ability to reflect and learn is critical and therefore they areencouraged to be creative and take risks. The trainees have also benefited from planning,teaching and evaluating in twos, taking turns being a ‘critical friend’. Following from thisexperience, there are two more taught-based sessions at the university in each of the threesubjects, prior to the trainees’ PowerPoint presentations of their particular ‘case studies’.Yet again, the school-based tutors play a significant role and are involved in jointassessment with the university-based tutors. Such school involvement presentsopportunities for developing more shared understanding and ownership of ITE among allstakeholders. Trainee motivation and learning is extended; children with particular needsreceive a very high level of attention in a small group; and partner schools are presentedwith an opportunity to experience new ideas.Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200343

Primary National Strategy DfES 0606-2003Appendix iiVideo notes and suggested activitiesSuggested focuses and questionsAdditional bullet points have been provided for further ideas to be added1. Supporting pupils learning English as an additional language (Counterreading: 00:28)YR bilingual teacher Sufiah demonstrating use of home language duringmathematical activities. Role of nursery nurse highlighted.• How does the teacher enable pupils to participate?• How is diversity valued?•2. Partnership teaching to support pupils learning English as an additionallanguage (Counter reading: 03:20)Y5 Partnership teaching – Emma class teacher and John EMTAGcoordinator. Clip covers discussion re: joint planning, partnership teaching.• Mathematical vocabulary development – alternatives?• Effectiveness of teacher intervention – effective examples, suggestions for development?• Quality of inclusion – teacher and pupil perspective – alternatives?•3. Young children communicating mathematically (Counter reading: 07:55)Snippet to illustrate young child explaining infinity.• Identify the questioning strategies used by the interviewer•Snippet of Noah’s calculation strategy using derived facts.• How efficient is Noah’s calculation strategy?•Snippet of Tom’s facility with number and number operations.• What discovery does Tom make while baking?• How does Tom interpret division?•4. Inclusive teaching and learning in the daily mathematics lesson(Counter reading: 11:43)Y5 Children calculating.Focus on the inclusive teaching strategies such as:• ethos that teaches pupils to think for themselves – this thinking is valuable;• teacher’s use of strategies for building confidence, maintaining self-esteem (starts with44© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National Strategywhat pupils can do, uses positive feedback, handles wrong answers sensitively, praisesdifferent methods, returns to the pupil who made a mistake to give her a chance ofsuccess);• questioning strategies – use of open questions to extend mathematical thinking, targetsindividuals to ensure participation;• differentiation within main activity – to support less confident/less experienced;• encouraging peer-evaluation of answers and explanations of strategies;• assessment – keeping a mental record of pupils’ answers for later discussion, and totackle pupils’ errors.•5. Working with Mathematically Gifted and Talented Children (QCA)(Counter reading: 23:08)Y1/2 – Snippet of Mary, followed by Mazher and Laureen.• How would you plan the next steps for these children?• What teaching strategies would you use to challenge their thinking?•Y5/6 – Stephen/Phillip.• Analyse the levels of questioning used by the teacher.• Consider Stephen’s explanations – what do they suggest about his ability to generalise?•6. Explaining mathematical thinking (Counter reading: 29:55)Y3 Division lesson – Jasper’s demonstration and explanations.• Consider Jasper’s use of language to explain and justify his method and answer.• What would be the next steps for Jasper?•7. Developing inclusive teaching of mathematics (Counter reading: 31:10)Teacher’s explanation of strategies to ensure all pupils can access themathematics and how she deals with incorrect answers.• Strategy of children discussing and working together, choosing who they wishto help them.• How does the teacher support kinaesthetic learners?• How does the teacher help pupils to recognise that they are able to think and workthings out ?• What models and images help all pupils, especially those who need additional support,to access the mathematics?• Identify three strategies the teacher uses to help all pupils develop confidence in theirmathematical ability.•Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200345

Primary National Strategy DfES 0606-2003Appendix iiiDefinitions of inclusion: 9-diamond activityIt is useful to carry out this activityin a small group. Cut out the ninediamonds. Discuss and agree anorder of preference – startingwith the most preferred definitionat the top, followed by twoothers and then threedefinitions that arereasonable. Place theleast appropriatedefinitions at thebottom of thediamond.dInclusion is an importantcomponent of a quality educationbInclusion isconcerned with thelearning and participationof all trainees vulnerable toexclusionary pressures, not onlythose with impairments orthose who arecategorised as‘having specialeducationalneeds’gInclusion treasures diversityand builds communityaInclusion is aboutequal opportunitiesfor all pupils withparticular attention to‘different groups’, i.e. boys andgirls, minority ethnic and faithgroups, EAL, SEN, giftedand talented, ‘lookedafter’ children, sickchildren,children atrisk…iInclusionis about aphilosophy ofacceptance andabout providing aframework within which allchildren (regardless of theprovenance of their difficulty atschool) can be valued equally,treated with respect andprovided with equalopportunities atschoolcInclusion is an equality ofopportunity issue: this in turnrelates to quality of accessfor all children to thestatutory curriculumhInclusion is abouta child’s right tobelong to their localmainstream school, to bevalued for who they are and beprovided with all thesupport they need tothrive in themainstreamschoolfInclusion ineducation involves theprocesses of increasingparticipation of trainees in,and reducing exclusion from,the cultures, curricula andcommunities of localschoolseInclusion is an active process46© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyDefinition sourcesa. Ofsted Evaluating Educational Inclusion – guidance for inspectors and schools (2001)b. Centre for Studies on Inclusive Education (CSIE) Index Process and the SchoolDevelopment Planning Cycle (DfEE sent the Index to all schools in March/April 2000)c. Ollerton, M (2001) – Inclusion and entitlement, equality of opportunity and quality ofcurriculum provision. Support for Learning vol 16 no 1. NASENd. Disability Equality in Education (DEE) (2002) Inclusion in Schools Course Booke. Corbett, J (2001) Teaching approaches which support inclusive education: aconnective pedagogy. British Journal of Special Education vol 28 no 1f. Centre for Studies on Inclusive Education (CSIE) Index Process and the SchoolDevelopment Planning Cycle (DfEE sent the Index to all schools in March/April 2000)g. source unknownh. Disability Equality in Education (DEE) (2002) Inclusion in Schools Course Booki. Thomas, G (1997) Inclusive schools for an inclusive society. British Journal of SpecialEducation vol 24 no 3Mathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200347

Primary National Strategy DfES 0606-2003Rank these statements in order 1 to 5, wherestatement 1 most closely reflects what inclusionmeans to youdisabled andnon-disabledchildren beingable to learnand playtogetherchanges inmainstreamschools so thatthey can meeta wider rangeof needsall childrengoing tomainstreamschoolsschools notbeing allowedto excludebadly behavedor violentchildrenall childrenable to go totheir localschool48© Crown copyright 2003Mathematics and InclusionMaterials for providers of Initial Teacher Training

DfES 0606-2003Primary National StrategyAppendix ivResearch articlesCorbett, J (2001) Teaching approaches which support inclusive education: a connectivepedagogy. British Journal of Special Education vol 28 no 2 June 2001 pp 55–59Eyre, D and McClure, L (2001) Curriculum Provision for the Gifted and Talented in thePrimary School, English, Mathematics, Science and ICT – chapter 3 Mathematics pp 64–89NASEN/FultonRaiker, A (De Montfort University, Bedford) (2002) Spoken language and mathematics.Cambridge Journal of Education vol 32 no 1 2002 pp 45–60 Carfax PublishingVisser, J (2002) Mathematics, Inclusion and Pupils. Mathematics Teaching no179, EasterConference SupplementMathematics and InclusionMaterials for providers of Initial Teacher Training© Crown copyright 200349

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Copies of this document may be available from:DfES PublicationsTel: 0845 60 222 60Fax: 0845 60 333 60Textphone: 0845 60 555 60e-mail: dfes@prolog.uk.comRef: DfES 0605-2003 G© Crown copyright 2003Produced by theDepartment for Education and Skillswww.dfes.gov.ukIf this is not available in hard copy it can bedownloaded from:www.standards.dfes.gov.ukThe content of this publication may be reproducedfree of charge by Schools and Local EducationAuthorities provided that the material isacknowledged as Crown copyright, the publicationtitle is specified, it is reproduced accurately and notused in a misleading context. Anyone else wishingto reuse part or all of the content of this publicationshould apply to HMSO for a core licence.The permission to reproduce Crown copyrightprotected material does not extend to anymaterial in this publication which is identifiedas being the copyright of a third party.Applications to reproduce the material from thispublication should be addressed to:HMSO, The Licensing Division, St Clements House2–16 Colegate, Norwich NR3 1BQFax: 01603 723000e-mail: hmsolicensing@cabinet-office.x.gsi.gov.ukAcorn/St Ives 11/2003