PRIMARY. Mathematics. Guidelines for Teachers of Students with MILD. General Learning Disabilities

Transcription

1 PRIMARY Mathematics Guidelies for Teachers of Studets with MILD Geeral Learig Disabilities

2 Cotets Ratioale ad itroductio 3 School plaig 6 Classroom plaig 8 Approaches ad methodologies 12 Exemplars 52 Appedix 64

3 Ratioale ad itroductio Mathematics gives studets the laguage through which they ca iterpret, aalyse, describe, make predictios, ad solve problems i everyday life. It allows them to participate i a wide rage of mathematical experieces ad relatioships both i school ad i daily livig. Ratioale The aims ad objectives of the Primary School Curriculum, Mathematics are valid for all studets. However, ot all studets lear mathematics i a eve ad predictable maer. The abstract ad coceptual ature of mathematics poses particular challeges to studets with mild geeral learig disabilities. The teacher has a pivotal role i mediatig the objectives of the Primary School Curriculum, Mathematics for studets with mild geeral learig disabilities. To meet the eeds of studets with mild geeral learig disabilities, greater emphasis is placed o the social, rather tha the creative ad aesthetic value of mathematics without excludig those importat aspects. There will be particular emphasis o maagig moey, uderstadig timetables ad usig measures i everyday life situatios. The acquisitio of these skills must be prioritised i order to equip the studet to participate fully ad idepedetly i society. Itroductio A level of proficiecy i basic mathematics is eeded to cope idepedetly ad effectively with everyday livig icludig tellig the time, shoppig, readig timetables, cookig, measurig, ad so forth. These guidelies aim to provide teachers with a uderstadig of the particular barriers to learig mathematics that studets with mild geeral learig disabilities may ecouter, ad to provide some strategies that they ca employ i plaig mathematical experieces for their studets, whether they are i maistream classes or i special schools. As studets with mild geeral learig disabilities may be learig their mathematics i may differet settigs it is importat that the teacher iitially idetifies the poit at which the idividual studets are operatig. The sectio Before you start offers a quick checklist for teachers to idetify a startig poit withi the Primary School Curriculum, Mathematics for idividual studets, may of whom will be able to access this material. Additioal advice for teachig

4 mathematics to studets with mild geeral learig disabilities is provided here with the caveat that the age ad stage of developmet of the studet provides the basis for mathematics learig for these studets. The aims of the Primary School Curriculum, Mathematics are: to develop a positive attitude towards mathematics ad a appreciatio of both its practical ad its aesthetic aspects to develop problem-solvig abilities ad a facility for the applicatio of mathematics to everyday life to eable the child to use mathematical laguage effectively ad accurately to eable the child to acquire a uderstadig of mathematical cocepts ad processes to his/her appropriate level of developmet ad ability to eable the child to acquire proficiecy i fudametal mathematical skills ad i recallig basic umber facts. The abstract ad coceptual ature of mathematics poses particular challeges to studets with mild geeral learig disabilities. The Primary School Curriculum, Mathematics stresses the importace of active learig, thus providig opportuities for studets to maipulate, touch, ad see objects as they develop their uderstadig of mathematical cocepts. Learig withi a group i which studets are ecouraged to talk about ad explai how or why they did somethig will also support developmet of their ow thikig about mathematics. A itegrated approach to mathematics will help studets to uderstad the relevace of mathematics i their daily lives. It is importat that the teacher is fully aware of the difficulties, both persoal ad academic, ecoutered by studets with mild geeral learig disabilities. Persoal difficulties are very ofte uderpied by a poor self-image brought about by a log-term sese of failure. Failure may be oe outcome of low itellectual ability, ad ca lead to slow progress. Slow progress is further aggravated by poor memory. Laguage ad readig difficulties ca cofoud studets difficulties with mathematics. Poor academic progress cotributes to the poor self-image ad lack of motivatio, which i tur may lead to withdrawal or maladjustmet of studets. Low expectatios ca ihibit the studet s effort ad performace cotributig to a cycle of failure. Teachers of studets with mild geeral learig disabilities must strive to break that cycle by providig opportuities for studets to experiece success with mathematics. Particular issues i mathematics for studets with mild geeral learig disabilities May studets with mild geeral learig disabilities require a structured approach to mathematics. Opportuities to practice mathematics skills ad cocepts eable studets to cosolidate their learig. Direct teachig, usig explicit strategies, is essetial as some studets may acquire iappropriate or icorrect strategies from icidetal learig. While may studets lear by workig thigs out for themselves or observig how others work, whe kowledge or skills are beig used i a ew cotext it is importat to support studets by makig their learig explicit, sice trasfer of learig does ot always take place automatically. Studets with a mild geeral learig disability are ot always proficiet at usig the skills ad kowledge they have already acquired if the features of the ew task are sigificatly differet. They may have difficulty extractig the key features of a task ad igorig the less importat oes. For example, they may focus o the umbers i a problem but ot cosider what they are beig asked to do; hece they ofte just add all the umerals without cosiderig the purpose. They may also focus o icidetal iformatio ad fail to otice the saliet feature of the topic. For example, they may be able to cout by rote but fail to uderstad the use of umbers as labels for quatities (I am 7, I live i umber 7, she has 7 sweets).

5 Features of the Primary School Curriculum, Mathematics The followig features of the Primary School Curriculum are relevat to studets with mild geeral learig disabilities. Early mathematical activities are aimed at ecouragig more work i pre-mathematical activities to develop cocepts before commecig formal umber work. These activities are particularly beeficial for studets with mild geeral learig disabilities. Teachers ca frequetly revisit these activities i age-appropriate settigs. The itroductio of umber limits ecourages the cosolidatio of umber facts ad the developmet of the cocept of place value. This is reflected i the reductio i the use of complex calculatios ad i the presetatio of the same cocept i differet ways. The emphasis o accurate use of mathematical laguage ad uderstadig of symbols will cotribute to a greater uderstadig of mathematics for all studets. Studets with mild geeral learig disabilities will eed frequet opportuities to cosolidate their uderstadig of both symbols ad mathematical laguage. Calculators have bee itroduced at fourth class, but it is essetial that studets develop good estimatio skills from the earliest stages if they are to use them efficietly. This is particularly true of studets with mild geeral learig disabilities, who may cofuse the meaig of arithmetical sigs or have coceptual weakesses i some areas. The icreased emphasis o the use of maipulatives (cocrete materials) throughout the school, icludig the seior classes, meas that studets who may still eed the support of materials will ot be perceived as beig differet from their peers. The Primary School Curriculum also emphasises real-life problem-solvig, usig checkable aswers ad activities based i the studet s eviromet. Ecouragig studets to egage with ope-eded problems such as makig scarves for teddy, buildig a house usig a limited umber of bricks, or workig out how to sped a sum of moey o food for a party will help them to realise that there are may ways to solve a problem, ad that sometimes there is more tha oe right aswer. The itroductio of fu areas such as chace ad mathematical trails allows for differetiated approaches, which ca iclude all learers i excitig mathematical activities. O a trail some studets may be lookig for oe digit umbers, while others may be seekig up to three digit umbers or addig umbers together. For further ideas see Primary School Curriculum: Mathematics, Teacher Guidelies (Mathematical trails). The use of a broad rage of assessmet tools is essetial i the accurate idetificatio of studets stregths ad eeds. This is particularly true i mathematics where studets may eed to acquire certai skills or cocepts before proceedig to more complex learig. The emphasis o usig a variety of methods of recordig studets progress ecourages differetiatio of respose, which recogises differet learig styles. Some studets may be able to give a aswer verbally while others would beefit from producig a diagram or drawig. All studets should have the opportuity to preset their work i a variety of ways. I the measures strad aswers should be verifiable where possible to ecourage uderstadig ad the developmet of persoal bechmarks. This ca provide support to studets with mild geeral learig disabilities. Hadweighig ad pourig activities usig a variety of shapes of cotaier will assist i this area. If studets measure ad label objects such as bookshelves, desks, etc i the classroom, they ca build up visual bechmarks. For example, a studet ca stad beside the bookshelf ad see that he/she is taller tha a metre, or that the teacher is shorter tha the door, which is more tha two metres. Icreased emphasis o itegratio ad o likage i strads of the Primary School Curriculum, Mathematics meas that studets ca come to see mathematics as relevat ad coected to their ow lives. Frequet referece by the teacher to mathematical elemets i the course of learig is essetial, for example measurig the distace from the classroom to the office usig a trudle wheel i geography, usig time words i history (before, log ago, a year), measurig jumps or distaces ru i PE.

6 School plaig No school pla will be complete without takig due cogisace of the eeds of studets with mild geeral learig disabilities. With proper plaig for differetiatio to meet the eeds of studets with mild geeral learig disabilities, they ca achieve success ad participate fully i the curriculum at their ow level. Curriculum plaig While studets with mild geeral learig disabilities are expected to partake fully i the curriculum they will ot be expected to do the same tasks at the same level as the more able studets. For these studets, learig will cocetrate o the areas of the curriculum that provide them with essetial life skills, for example moey, time, ad measuremet. Studets with mild geeral learig disabilities will eed more opportuities to use cocrete materials ad egage i cocrete tasks rather tha workig from a textbook where their weakess i laguage would further aggravate their arithmetical difficulties. As their cocetratio spa is short, they beefit most from tasks which are short ad varied. Usig the same mathematical laguage ad methodology throughout the school beefits all studets, especially studets with mild geeral learig disabilities. This is especially true i such difficult areas as subtractio with reamig, ad i the teachig of fractios ad decimals. For example, i doig the algorithm 9-3, if oe teacher says ie take three ad aother says three from ie, this will cause cosiderable cofusio to the weaker studet whe she/he moves from oe teacher to aother or from class teacher to resource teacher. It follows, therefore, that it is importat to also keep parets iformed at regular itervals about what ad how studets are learig ad their geeral progress. As the studet with mild geeral learig disabilities requires the carig help of both paret ad teacher i order to make satisfactory progress, it is importat that parets ad teachers have opportuities to meet ad discuss the studet s progress. Regular commuicatio with the class teacher ad support teacher are also importat as each teacher is aware of how the studet is progressig i the classroom ad i learig support.

7 Good teachig practice eeds regular assessmet. Assessmet should be iformative ad eable the teacher to pla for further learig, ad report o the studet s progress. Teacher desiged tests ca support the studet by allowig for a certai amout of success. Whe failure occurs, the studet will beefit from a discussio of where she/he wet wrog ad reassurace that makig mistakes is part of the learig process. Studets with mild geeral learig disabilities should be provided with a test suited to their abilities ad stage of developmet. As studets with mild geeral learig disabilities may be at differet stages of developmet, differet teacherdesiged tests ad idividual learig profiles may be ecessary to best support their learig. Orgaisatioal plaig Orgaisatioal plaig ivolves cosideratio of resource requiremets, home-school policy, ad a policy o homework. As already oted, plaig will esure that the class teacher, support teacher, ad the paret are workig as a cohesive uit i supportig the studet with geeral learig disabilities. For example, parets could sped time with the studet usig mathematical games that ehace the learig of basic facts. Some parets may ot be familiar with a algorithm, for example decompositio, ad try ad teach the studet usig borrow payback, thus creatig cofusio. It is advised that teachers commuicate with parets, at the begiig of the school year as to the approaches ad methodologies used, so as to eable parets to support the studet s learig i the home cotext. Studets with mild geeral learig disabilities very ofte have poor short-term/log-term memory. Homework should be used as a opportuity for reiforcig topics covered recetly ad i the past ad should be appropriate to the child s level of progress ad achievemet. These studets eed costat reiforcemet.

8 Classroom plaig Classroom plaig for each school settig special school, special class, ad maistream class will differ accordig to the eeds of particular studets at ay give time. These studets progress at a slower rate tha maistream studets. Sice the Primary School Curriculum, Mathematics is sequetial ad depedet o kowledge gaied, they may experiece great difficulty i keepig abreast of their fellow studets, although they may be able to partake i the majority of mathematical activities i the class. If, for example, the teacher is teachig a ew cocept or ew algorithm he/she must be careful that the calculatios are simple, thus esurig that the lack of umber facts o the part of a studet with mild geeral learig disabilities does ot iterfere with his/her acquisitio of the cocept. For example whe usig additio with reamig, is a suitable example but is ot, sice i the secod case the studet will have great difficulty calculatig 8+9 ad will ot be able to cocetrate o the cocept ivolved. The situatio is starker whe it comes to multiplicatio. For example, 25 3 is easily maaged, whereas 49 8 is likely to brig about failure i the case of the studet with mild geeral learig disabilities. Classroom plaig for these studets will ivolve close co-operatio betwee the resource teacher ad the class teacher i order to esure that there is cotiuity ad reiforcemet i each other s work. While providig special suitable work for these studets the teacher should avail of ay opportuity to allow participatio i whole class work. Project work may provide such a opportuity. If, for example, the studets are makig a model of the school, the more able studets ca do the complex calculatios relatig to scale while studets with mild geeral learig disabilities ca be gaifully employed i measuremet. Mathematical kowledge ca be reiforced i other areas of the curriculum ad such opportuities should ot be missed whe dealig with the studet with mild geeral learig disabilities. For example, umber ad measure ca be reiforced whe doig ruig, jumpig, etc. i physical educatio.

9 Some aspects of the teachig of mathematics eed particular emphasis i the case of studets with mild geeral learig disability. These iclude equivalets, ad a example ca help studets to reiforce a correct iterpretatio of the symbol. mathematical laguage symbols + plus ad add = 7 materials worksheets ad textbooks usig ICT. Mathematical laguage Mathematics must be see as a laguage with its ow vocabulary of words, symbols ad tools that are used i particular circumstaces. May studets cofuse mathematical laguage with ordiary laguage. They say He s bigger tha me whe they mea older, or My table is loger tha his whe they mea wider. It is importat to teach mathematical laguage to the studets, ad to reiforce it o a daily basis. A list of mathematical words ca be kept either by the teacher or by the studets themselves (for example, taped ito a copybook), ad they ca tick them off as they lear to use them appropriately ad accurately, either orally or i writig. As already oted, keepig parets iformed of the words beig used ad the importace of practicig them frequetly will help the studet to make use of these words i real cotexts. Symbols Some studets may have difficulty with mathematical symbols. They will ofte call the plus sig add, ad others will use ad or plus. As already oted, cosistecy of approach is vital ad this is especially true if the class cotais studets who have come from differet schools or classes where they may have developed misuderstadigs i relatio to some of the symbols. This is particularly relevat if the studet is movig to a ew class or beig itegrated with aother class for mathematics. It is importat that parets ad all staff who are i cotact with the studet are aware of the termiology beig used. I the classroom, charts with the symbol, word Plus ad equals symbols are itroduced first, followed by the mius sig ad, subsequetly, the greater tha ad less tha sigs (see symbols chart i the teacher guidelies for mathematics i the Primary School Curriculum). Materials Age-appropriateess It is very importat that the materials used are age-appropriate. Brightly coloured couters ad iterlockig cubes may be appropriate for youger studets, but they ofte come to associate these with the ifat classes. Usig football cards or cois ca make a coutig activity more age-appropriate for older studets. Pretedig to be the ma o the turstile tottig up the takigs after a match ca make coutig more real to a older studet. Jobs that ivolve coutig ca be icorporated ito the mathematics class ad are usually good motivators, for example coutig out iformatio otes for distributio to other classes, havig a class sale of work for charity, costig ad buyig foodstuff for cookery classes, or examiig the cost of popular clothes such as jeas or favourite chocolate bars. Distributig milk, books, or pecils ca be very valuable activities that support the modellig of laguage such as Are there eough pecils for all your group? How may cartos do you usually eed for your group? Is there ayoe abset today? How may do you eed today? The teacher ca maipulate this actively by ot givig eough items to the studet ad askig how may more are eeded. Durig break times iformal discussio about luch items ca iclude such questios as, Who has the most sadwiches? or Whose carto of juice cotais the most milk?

10 Accessibility Aother importat feature of usig materials i the classroom is that they must be stored i ways that make them easily accessible to studets. Clearly labelled ad colour-coded boxes, a routie tidy-up sessio, ad procedures to maitai clea ad eat materials will ecourage studets to use the materials appropriately ad resposibly. Havig a mathematics table where the materials eeded for the lesso are easily accessible is importat ad ca be a role of resposibility for differet studets. It ca be chaged as topics chage, for example Our measures table, Sets of four, etc. Worksheets ad textbooks It is importat that worksheets ad textbooks are used carefully with studets who have a mild geeral learig disability. Some maistream textbooks move too quickly through the cotet areas ad therefore should be see as a complemet to other meas of istructio rather tha the oly source. With these studets it is particularly importat to maitai a balace betwee computatioal ad problem-solvig tasks ad betwee the use of cocrete ad symbolic represetatio. I usig worksheets ad textbooks, the followig questios should be cosidered: Is there too much text i the worksheet/textbook? (This could ihibit the ability of poor readers to egage with the mathematical cotet.) Is there a lot of clutter o the page? (Studets are ofte distracted by busy pages cotaiig too much uimportat iformatio or illustratio.) Are the spaces give for the aswer large eough for studets who may have poor motor cotrol or write their umbers very large? Are worksheets that are iteded for homework clear i their istructios so that parets kow what the studet is supposed to be doig? Do worksheets/textbooks offer a variety of task types, for example practical tasks, ope-eded ivestigatios, puzzles, games, project-based work, ad word problems? Do they preset material i differet ways for differet purposes ad learig styles, for example pictorially, diagrammatically, ad usig miimal laguage as appropriate? Is the prit size appropriate to the developmetal stage of the studet? Over time, do worksheets/textbooks offer a appropriate balace betwee the strads? Do they offer opportuities for collaborative or group work, for example active ivestigatio ad talk-about sectios, ad itegratio opportuities that focus o other areas of the curriculum? Do they reflect the iterests ad eviromet of the studet? Do they cosider the social aspects of mathematics, for example shoppig, measurig for a purpose, life-skills? Are computatioal activities carefully graded, offerig appropriate opportuities to cosolidate learig at each level? Makig sure that worksheets ad textbooks are ageappropriate is extremely importat, sice studets are usually very aware that a particular book is beig used by a youger brother/sister or by a more juior class withi the school, ad this ca seriously impact o their self-esteem. Usig ICT May software programmes ca be used at differet levels withi the oe group or class. Valuable teacher time ca be take up i establishig the correct startig poit for a particular studet. Colour codes ad symbolic represetatios taped to the frot of software boxes ca help, ad a card with clear istructios ca be give to the idividual studet. Oce studets have practiced this procedure, they will be able to locate ad load the software themselves, ad a older studet ca either help or supervise. It is helpful to keep symbols costat ad, where possible, iclude the studets i the choice of how iformatio ca be preseted symbolically. This activity is also part of the advice provided i the Primary School Curriculum: Mathematics, Teachers Guidelies. 10

11 Assessmet for learig Studets should have opportuities to experiece differet assessmet methods from a early age, ad play a active part i their ow assessmet. Early idetificatio of difficulties i mathematics is essetial if the studets are to achieve their full potetial. Assessmet should be see as a positive rather tha egative experiece that will help them i their future learig, ad clear, costructive feedback to studets is ecessary if they are to play a active part i their ow future learig. Good commuicatio is essetial betwee the teachers ad parets, other teachers ad professioals who come ito cotact with studets regardig each studet's progress, stregths ad future learig goals. A importat issue i assessmet is the evaluatio of a plaed test with regard to its aims ad its suitability for the studets for whom it is iteded. The laguage of the test is crucial. If the laguage is too difficult or wordy for studets who have a readig difficulty the test may ot accurately reflect their mathematical ability. Teachers should use a variety of assessmet tools that take accout of differet learig styles. Whe plaig a sequece of learig outcomes for a particular studet or group it is ecessary to keep i mid how progress will be assessed. Too ofte a pe ad paper test at the ed of a uit of work is used. However, it is possible to assess progress by observig, for example, how a studet hadles moey i the class shop, how accurately a studet ca cotiue a patter, or ca make a shape with his/her body. Whe assessig studet progress i the various areas of mathematics teachers should use a variety of assessmet tools, icludig teacher observatio, teacher-desiged tasks ad tests, portfolios, projects, work samples, ad criterio referece tests. See the assessmet sectio i the Itroductio to the Primary School Curriculum for further details o assessmet techiques. 11

12 Approaches ad methodologies The Primary School Curriculum recogises that studets lear withi a social cotext ad are active participats i the learig process. They aturally ivestigate ad rely o their ow methods of workig out mathematical problems ad therefore eed to experimet ad discuss i a supportive eviromet that eables them to build o ad develop their ideas. Studets with mild geeral learig disabilities eed to build up persoal support structures which they ca use whe faced with a ew situatio or problem. They eed to lear how to lear', how to verbalise what the problem is, ad how they ited to go about solvig it. I the classroom situatio, it may be good for the more able studet ad the less able studet alike if peer teachig is itroduced. The more able studet ca be gaifully employed i helpig his/her less able peer i reiforcig algorithms, learig basic facts, ad i providig reiforcemet of the cocepts already leared. Difficulties with umbers Studets with mild geeral learig disabilities fid it very difficult to memorise basic facts. Cosequetly, umber is a area that presets multiple difficulties. This has a debilitatig effect o their success rate as very ofte they may have a algorithm doe correctly but the aswer is wrog because of lack of kowledge of basic facts. I order to overcome this difficulty, the teacher should 12 choose figures that are easy to calculate whe teachig a ew cocept or algorithm allow studets to use a calculator or a oe hudred square whe calculatig more difficult basic facts if part of a algorithm is correct, mark the part that is correct ad discuss the error i the remaider part with the studet. Cocrete materials Whe teachig ew algorithms it is importat (where possible) to use cocrete materials, for example, i difficult areas like decompositio or fractio or decimal cocepts. Studets should have opportuities to use such materials util the cocept is well grouded. Whe mistakes occur after the cocrete materials are removed, they must be reitroduced to the studet durig the remediatio process.

13 Learig basic facts Learig basic facts is very ofte a log ad tedious task for the studet with mild geeral learig disabilities due to possible short-term ad log-term memory difficulties. However, these difficulties ca be alleviated by providig the studet with strategies for learig basic facts usig some of the large variety of table games available, for example domioes usig computer games to reiforce kowledge of basic facts. It is of crucial importace to match the work give with the studet s ability to do it, i order to esure that the studet will experiece success. Potetial areas of difficulty The potetial areas of difficulty are described i order to outlie the possible implicatios for learig ad suggest possible differetiated teachig strategies. Studets with mild geeral learig disabilities are more like their peers tha ualike. They have the same rage of iterests, the same eed for affirmatio ad success, ad exhibit a wide rage of learig styles. Not all studets with mild geeral learig disabilities will exhibit all of the potetial areas of difficulty, but it may be helpful for teachers to uderstad the implicatios that these may have for their teachig ad, cosequetly, for the studet s learig. Durig his/her early years a studet with mild geeral learig disabilities may have had restricted experieces i compariso to his/her more able peer. Restricted mobility, poor motor cotrol, or poor uderstadig may have resulted i fewer opportuities to use the laguage of mathematics for example coutig stairs, judgig distaces or heights, ad learig mathematical rhymes. Teachers should be aware that they may ot have leared these thigs icidetally. Some studets with mild geeral learig disabilities will have difficulties i problem-solvig due to iheret limitatios i their ability to abstract ad geeralise. Their ability to work idepedetly is costraied by poor attetio spa ad retetio. Hece, the learig process eeds to be broke dow ito short sequetial steps ad work doe eeds to be reiforced through over-learig ad repetitio. The difficulties experieced by these studets ca lead to poor self-esteem ad a fear of failure. These combied ca lead to a sese of helplessess that results i the studet costatly seekig help or refusig to proceed with eve the simplest of tasks. Studets may get trapped i the I ca t do maths sydrome. It is importat that such studets experiece success as ofte as possible ad participate i fu mathematical activities if they are to overcome this feelig. The settig of realistic learig targets by teacher ad studet may help i the achievemet of success ad the retur of cofidece. It is also importat for studets to realise that makig mistakes is a itegral part of the learig process, ad that they should ot be discouraged by their mistakes. While difficulties i umeracy ad literacy ofte overshadow the studet s learig experiece, it is importat to provide a wide variety of learig tasks i order to allow studets to show their skills ad develop cofidece i other areas of the curriculum. The use of the calculator, for example, ca eable the studet to sometimes bypass arithmetical difficulties ad work successfully o other mathematical topics. Studets eed costat ecouragemet. They eed to be ivolved i their ow learig ad have opportuities to discuss their difficulties. Studets have valuable isights ito their ow learig eeds ad this should ot be igored. Teachers should help studets towards a awareess of what their learig eeds are, i the cotext of what they ca do rather tha what they caot do. Effective teachig builds o studets stregths rather tha focusig o their weakesses. The tables that follow outlie characteristics that may be associated with a mild geeral learig disability, examie their implicatios for the learig of mathematics, ad suggest a rage of strategies that may assist the studet. 13

14 Sometimes it is the way i which the material is preseted to the studet that creates a barrier to learig. Usig a variety of approaches ad methodologies will facilitate differet learig styles i ay give class group. Before you start The followig lists are ot comprehesive but should serve as a guide i assessig where a studet is i relatio to mathematics. It is suggested that idividual teachers or groups of teachers develop their ow profilig system that will isure that istructio is tailored to meet their studets eeds ad levels of attaimet. Teachers ca use this iformatio to ask Is the studet s readig level ihibitig his/her ability to egage with the mathematical elemets? Whe a observatioal picture of the studets has bee created they ca be grouped accordigly or provided with idividual work. However, it is very importat to iclude large group or full class work occasioally, so that studets ca lear from seeig how other studets approach problems. All studets will beefit from a variety of teachig styles ad classroom activities. Studets with mild geeral learig disabilities will particularly beefit if the teacher is aware of their stregths ad weakesses before embarkig o a ew topic. The followig table outlies some observatios that the teacher may ote about the studet. Such iformal assessmet ca assist the teacher i selectig suitably differetiated methods for the class, ad may prove useful i providig feedback to parets ad studets. If so, ca the material be preseted orally, diagrammatically, pictorially? 14

15 / A geeric list for iformal teacher observatio of studets Pre-test usig a simple task i a area you are itroducig What is the state of readiess of the studet to do this task i terms of kowledge, skills, ad attitudes? What learig strategies are beig used by the studet? Ca the studet describe what he/she is doig ad how he/she is doig it? Does the studet uderstad why he/she is doig this activity? Is the studet s readig level ihibitig his/her ability to egage with the task? Is the studet s umeracy level ihibitig his/her ability to egage with the task? Does the studet have ay miscoceptios about the task? Idepedece How does the studet cope whe left aloe with a task? Does he/she give up easily, ask for help, begi the ext task, sit iactively, or become disruptive? Does the studet clearly uderstad what to do if he/she has a problem or is fiished work earlier tha others? Is the studet orgaised i his/her work? Does the studet check if he/she has the correct equipmet, for example pecil, pe, ruler, graph paper? Does he/she have other difficulties, such as hearig loss, poor visio, poor motor cotrol, or hyperactivity, which may eed to be cosidered? Ca the studet preset work i a way that ca be uderstood by others? Group work Does the studet appear to lear better i a group, aloe, or i pair-work? Ca he/she take turs ad liste to other studets resposes? Ca he/she preset work i a clear ad coheret maer o behalf of the group? 15 Istructios Does the studet liste to ad uderstad istructios? Ca the studet read istructios? Ca he/she follow istructios give by the teacher? Ca she/he follow more tha oe istructio effectively? Ca he/she make appropriate resposes to the istructios? Is the studet clear about routies for settig out materials ad clearig up after a activity? Choosig ad usig the right materials Does he/she choose appropriate support materials, for example umber lie or objects for coutig, additio, ad subtractio? Does he/she use materials appropriately, for example use a ruler correctly to measure, use a umber strip for coutig o, use blocks ad push away those already couted, use a umber square efficietly, use a calculator appropriately (i.e. make decisios about whe its use is ecessary)? Readiess for mathematics Ca the studet classify, compare, ad order objects ad pictures? Does he/she use labellig words efficietly whe questioed? ('Ca you fid a teddy for me? What ca you see? What is this?') Does he/she respod to differet aspects of a mathematical problem, for example fuctios, attributes, differeces, ad similarities? ('Ca you fid all the red objects for me, please? Ca you poit out the red circles?') Ca the studet uderstad egatives? ('Simo says do ot stad up. Fid all the blocks which are ot red.') Does he/she reaso, for example predict, project, idetify cause ad effect, ad suggest alteratives? ('Why? What if I?')

16 Addressig potetial areas of difficulty for studets with mild geeral learig disabilities s Potetial area of difficulty The studet may have a short-term memory difficulty. = Implicatios for learig Retetio of umber facts ca be a problem. + Possible strategies Ecourage the use of visual clues to aid memory. Use umber rhymes ad sogs. Provide the studet with strategies for rememberig facts such as doubles, ear doubles, etc. Practise estimatio skills so that a calculator ca be used efficietly. Work o makig umber operatios automatic through fu games such as table darts. s Potetial area of difficulty The studet may have a short attetio spa, lack of cocetratio, ad lack of applicatio. = Implicatios for learig The studet fids it difficult to stay o task. + Possible strategies Provide shorter tasks with clear rewards for stayig o task, such as computer use or game time. Keep periods of istructio short ad to the poit, ad recap frequetly. Provide short work sessios with achievable goals. Ecourage the studet to become aware of the difficulty ad to try to beat their target i stayig o task. Use teacher observatio efficietly ad ote achievemets, stregths, ad preferred learig styles for use i plaig future work. Use classroom maagemet which focuses o cotiget praise ad ecouragemet (i.e. rewardig the task behaviour ad ot just the aswer). This icludes acceptig good aswers that may ot be ecessarily correct, for example Mary foud a iterestig way of doig that problem; let s see how it works. The use of ope-eded problems that have differet possible aswers ca help to develop a positive attitude to problem-solvig. 16 s Potetial area of difficulty The studet may have difficulty i uderstadig mathematical cocepts/abstractios. = Implicatios for learig The studet fids mathematics particularly difficult ad has difficulty with coutig umbers, place value, ad with uderstadig what is happeig whe usig the four operatios. + Possible strategies Frequet practice of the cocepts to be leared should be varied by use of games, ICT, ad real-life problems relevat to the studet s experiece. Make the learig fu by usig fuy ames, silly scearios, or ulikely settigs, for example: 'Tommy wet to Mars for his holiday ad met five alies. They each had four arms. How may arms had the alies altogether? Draw the alies. What if they each had eight arms?'

17 s Potetial area of difficulty The studet may have difficulties with spatial awareess; he/she may ot have had opportuities to play with puzzles, blocks, etc. = Implicatios for learig The studet may have difficulty orgaisig materials, may display left/right cofusio i recordig, may ot recogise shapes if iverted, or costatly lose items i the room. + Possible strategies Plety of work with three dimesioal objects will be eeded, with particular attetio beig paid to the laguage of spatial awareess, for example up/dow, over/uder, shape words. Usig jigsaws, block puzzles, ad tagrams i a fu way ca help studets i this area. Work o awareess of ow persoal space, especially i PE, left ad right, distace from, etc. Maitai cosistet orgaisatio patters i the classroom. Use visual cues for locatio ad directio o charts or table top. Give oral cues related to the studet s ow positio, for example o your door side. Allow studets to remai i the same seatig positio for group work. s Potetial area of difficulty The studet may have difficulty applyig previously leared kowledge. + Possible strategies = Implicatios for learig The studet may fid it difficult to use a skill or cocept i aother settig such as measurig i geography or sciece. Draw the studet s attetio to what is happeig: This is just like the measurig we did last week. What did we use to measure our books? How did we place the ruler? Cosciously reiforce mathematical cocepts i other areas of the curriculum, for example sortig ad classifyig i sciece, space ad shape i art (pritig 2-D shapes, both radomly ad i sequeces/ patters). 17 s Potetial area of difficulty The studet may have difficulty with trasfer to real life. = Implicatios for learig The studet does ot use mathematics i real situatios. For example, he/she does ot use ay of the four operatios whe buyig goods i a shop, does ot see the eed to measure whe cookig, does ot recogise shapes i the eviromet. + Possible strategies Use real-life objects ad cois i play situatios. Discuss how pocket moey is spet. If possible, provide opportuities to hadle moey i a real shop or school shop. Make sure that parets are aware of the importace of coutig, hadlig moey, etc. at home, for example settig the table, dividig up food, sharig equally, weighig for cookig, or measurig whe doig DIY.

18 s Potetial area of difficulty The studet may have difficulty with visual sequecig. = Implicatios for learig The studet ca t copy from the board or from a book, or has difficulty with sequecig umbers, mirror writig, etc. + Possible strategies Start with tracig exercises, usig tracig paper. Teach studets how to chuk iformatio (for example copy oly oe part of the sum at a time), ad how to check that it is correct. Use umber rhymes ad sogs to reiforce sequeces. Use visual cues, for example r start here s Potetial area of difficulty The studet may experiece cofusio with sigs ad symbols. = Implicatios for learig The studet does ot read symbols ad asks questios like Is this a add sum? + Possible strategies Use charts ad discussio of everyday symbols, ad relate these to mathematical symbols. Ecourage studets to verbalise what they should do first, for example look for ad idetify the symbol, or apply the correct symbol if it is a writte problem. 18 s Potetial area of difficulty The studet may display poor vocabulary/other laguage difficulties. = Implicatios for learig The studet caot follow complex seteces or multiple meaigs, may process oly part of the istructio, for example whe told to put a red circle aroud all the big thigs he/she may process oly circle ad thigs. The studet fids it difficult to verbalise what he/she doig i mathematics, or to relate the vocabulary of mathematics to real-life situatios. + Possible strategies Idetify ad specifically target mathematical laguage, esurig that it is reiforced i differet settigs ad i other areas of the curriculum, for example locatio words (o, i, uder, etc.) i PE, drawig games that ivolve followig istructios cotaiig target words (draw a square i the middle of your page, put a blue triagle beside/uder/to the right of the square. Be clear i commuicatig to both studets ad parets the laguage that is beig covered each week, for example usig a ote i a copy or a wall chart to list the mathematics words of the week.

19 s Potetial area of difficulty The studet may experiece readig difficulties. (For writig difficulties, see the sectio o commuicatio ad laguage.) = Implicatios for mathematics Readig difficulties ca prevet studets from egagig with mathematics. They may be capable of completig the mathematical task but become frustrated ad cofused by prited words. + Possible strategies Provide alterative forms of problems usig visual presetatio of material. Ask the studet to pick out the parts of the problem that he/she ca read, ad to focus o what iformatio is relevat. There is ofte a lot of redudat iformatio i a writte problem. Avoid presetig the studet with pages of textbook problems by givig modified worksheets or verbally delivered istructios, for example Mary has six sweets ad she gives her brother four; how may has she left? Ecourage the studet to use drawigs to write dow the importat features, for example Mary has 'Take away four by crossig them out. How may are left?' s Potetial area of difficulty The studet has difficulty i followig istructios. = Implicatios for mathematics The studet becomes cofused whe faced with more tha oe istructio at a time. + Possible strategies Get the studet to repeat the istructio(s). Use short, clear istructios or pictorial cues, e.g. a picture of a copybook o the blackboard or o a card. Use cue sheets, for example: ' Take out a copy ad pecil (picture). What kid of problem is it? What do I eed to kow? What do I do ext?' Give clear guidace o how ad whe assistace will be give by the teacher/other studets durig the lesso. 19 s Potetial area of difficulty The studet may be overwhelmed by the learig process. = Implicatios for mathematics The studet becomes overwhelmed whe preseted with ew iformatio or skills, ad cosequetly caot lear. + Possible strategies Vary the materials give to a group some usig umber strips to five while others are usig them to te, where appropriate. Adapt the teachig style, for example use more discussio at the begiig ad at the ed of the lesso to help both teacher ad studet to uderstad how learig takes place. Itroduce variety i the resposes required. The same activity ca ofte be doe with a group or the class, but some studets ca be required to aswer orally, some by usig symbolic represetatio or some by usig a pictorial respose, for example a+ a a = aaa Vary the requiremets of the task. Oe group or idividual may oly have to do six of the calculatios whereas aother may have to do te or more. Set persoal targets for the studets so that they do ot feel that others are gettig less to do tha they are.

20 Selectig cotet Whe plaig for a class, group or idividual it is useful to have a checklist of elemets you wish to iclude. Some suggestios are give i the followig example. I order to maximise the cross-curricular elemets of mathematics two further plaig grids, with suggestios, are give i examples 2 ad 3. EXAMPLE 1 Does your mathematical programme cotai these elemets? activities that lik with other areas of mathematics (for example, usig moey to teach tes ad uits) direct teachig of mathematical Laguage activities that itegrate with other areas of the curriculum real-life problem-solvig opportuities variety i the materials to be used cosideratio of age-appropriateess direct teachig of mathematical strategies a balace betwee class, group, ad idividual teachig direct teachig of skills such as tur-takig or active listeig, ad good work habits such as havig materials ready, kowig what to do whe fiished, etc. methods of recordig progress such as the studet s ow log books, portfolios of completed work, idividual progress charts with clearly set idividual targets. 20 AIMS What do you wat to do? (strad/topic) What particularr area do you wat to focus o? (laguage, skill, cocept?) OBJECTIVES What learig outcomes do you expect for each studet or group, icludig both skills ad cocepts? (These should form the basis for assessmet ad iform decisios about what the ext target should be.) Plaig cross-curricular work i mathematics ORGANISATIONAL ISSUES Ca you elist the help of other teachers/parets/ assistats/studets? Is the ecessary equipmet easily available ad are the studets familiar with its use? Ca the topic be taught to the whole class, but with differetiated elemets as outlied i the sectio o differetiatio? ASSESSMENT How will you record the studets progress? How will you use this iformatio to pla your ext step? How will you reiforce the skills/cocepts i other areas?

21 EXAMPLE 2 Plaig cross-curricular work i mathematics This is ot a exhaustive list ad would eed to be adapted to suit the ages ad ability levels of the studets. The focus for the teacher should be o idetifyig the laguage to be reiforced ad to cosciously use it across subjects. It is very importat to draw the studets attetio to the fact that measurig i geography or sciece is just the same as i mathematics. SESE: Sciece Laguage of capacity: pour, fill, full, empty, liquid, solid; I thik My guess is Properties of differet materials lots of polystyree chips are lighter tha a few stoes; weighig activities. SESE: Geography Measurig the desk/room for mappig exercises, measurig ad recordig raifall ad temperature, examiig distaces from oe place to aother i plaig a jourey. 21 SPHE Cookery amouts of foodstuffs i packagig; value for moey; meaig of: 10% extra free, 25% off; amout of salt, sugar, etc. i processed foods; iterpretig packagig iformatio. STRAND: Measures PE Laguage of legth, weight, time durig activities (She ra the fastest; choose the heaviest ball; ca you ru farther tha that? Today we ll use the wide bech). Measurig jumps, ad distaces ru or walked usig a trudle wheel; timig races or settig time limits o games usig a stopwatch. (These activities ca be doe usig ostadard measures at first). Usig tally sheets to keep a record of laps ru or jumps take; estimatig how far a beabag or ball ca be throw.

22 EXAMPLE 3 Plaig topic work i mathematics Below is a example of how a area of mathematics (measures) ca be explored through topic work. The age-appropriateess of the activities must be cosidered at all times. Computer software packages ad books which support the vocabulary are easily available. Gloves ad scarves Comparig sizes of hads ad gloves; Usig appropriate laguage small, smaller tha, smallest, too big/small, just right, etc.; comparig legths of scarves; makig felt scarves for teddies ad sortig by legth, colour, etc. 22 Shoes Big ad small sizes of shoes; sizes of feet; relatioship betwee foot size ad height; data collectio o popular types of shoe.. TOPIC: Clothes STRAND: Measures Trousers Leg legths, waist sizes. Early mathematical activities Puttig clothes o i order logical sequecig. Coats ad jackets Heavy ad light; log ad short; sizes.

23 Strategies for teachig studets with mild geeral learig disabilities Geeral strategies The strategies outlied i this sectio use readily available cocrete materials, allow studets to experimet while learig ew cocepts, ad ecourage studets to discuss their ideas o workig out mathematical problems. I usig the strategies that follow, the followig approach should be take: (i) The teacher demostrates the algorithm o the board while usig the cocrete materials, iteractig with the studets ad posig questios as outlied i the idividual strategies. (ii) The studets apply the algorithm usig cocrete materials util the steps are iteralised ad the cocept is well grouded. (iii) The studets practice usig the algorithm without cocrete materials. (iv) A studet is guided to retur to the use of cocrete materials whe problems occur. The followig list of geeral strategies may prove helpful i the teachig ad learig of ew facts, cocepts, algorithms, ad skills. Draw out importat facts through classroom discussio. Try to pla for the cosistet use of laguage ad methodology throughout the school ad, if possible, i the home. Ecourage the studet to work at a slow pace ad i a very structured way. Avoid great leaps, takig very small, carefully thought out steps oe at a time. Provide opportuities for the studet to talk about a problem ad to say how he/she ca go about solvig it. Ecourage studets to practice ad apply the skill of estimatio as ofte as possible. Provide the studet with strategies for learig basic facts, ad use a variety of table games ad computer games to reiforce this kowledge. Use wall charts ad display boards to show the facts ad cocepts that are relevat to the curret classroom work. Provide regular opportuities for the studet to use cocrete materials util a ew cocept is well grouded. If a studet has difficulties with a ew cocept ad eeds remediatio, ecourage him/her to retur to usig cocrete materials. Choose figures that are easy to calculate whe teachig a ew cocept or algorithm. Use materials that ca be decostructed where appropriate, for example usig a budle of te lollipop sticks allows the studets to see the te sigle uits, ulike a te euro ote. Iitial use of lollipop sticks or Uifix cubes followed by the use of moey meas that the studet is movig from a cocrete to a more abstract use of materials. Use maageable materials where appropriate, for example moey is far more maageable tha lollipop sticks for larger umbers (hudreds), ad cheques may be appropriate for eve larger umbers (thousads). The studets should oly record the algorithm i the correct mathematical format whe they are very familiar with the algorithm ad eve the oly with easily maaged umbers. 23

24 Diagosig studet difficulties The diagosis of studet difficulties is oe of the most importat skills that a teacher ca develop. The table below aalyses some of the more commo studet errors ad suggests what may be the cause of the errors. It is icluded here to assist the teacher i the diagosis of errors so that a suitable remediatio strategy ca be udertake. Arithmetic operatio or cocept Coutig Writig umbers Example of error Twety-eight, twetyie, twety-te, twety-eleve, etc. Writig for twelve thousad five hudred Additio Additio Additio Additio Additio Subtractio Subtractio Subtractio Subtractio Further explaatio of error ad possible reaso for error Problems with matchig, coutig, ad place value Poor cocept of large umbers ad place value Not carryig Poor cocept of place value Carryig whe it does ot apply Poor cocept of place value Not makig te Poor cocept of place value Poor estimatio Addig aticlockwise Poor cocept of place value Carryig oe istead of two Poor cocept of place value Sayig 4 from 8 Forgettig to cross out Poor cocept of decompositio Crossig out whe there is o eed Poor cocept of decompositio Failure to uderstad what is happeig 24 Subtractio Poor cocept of decompositio

25 Arithmetic operatio or cocept Example of error Multiplicatio 34 x3 92 Multiplicatio 23 x3 79 Multiplicatio 25 x3 35 Multiplicatio 25 x3 615 Multiplicatio 29 x3 77 Multiplicatio 28 x3 102 Multiplicatio 28 x3 76 Multiplicatio 20 x3 63 Multiplicatio 302 x3 96 Divisio 2)58 24 Divisio 3)75 22 Divisio 2) Divisio 2) Decimals Decimals is bigger tha 3.26 Further explaatio of error ad possible reaso for error Not carryig Carryig whe ot applicable Addig the oe carried o to the two Does ot multiply by three Poor cocept of multiplicatio Puttig dow the fiftee, does ot carry Poor uderstadig of place value Carryig oe istead of two Problems with place value Carryig four istead of two Does ot uderstad place value Sayig three eights are sixtee Does ot kow tables/basic facts Sayig three zeros are three Problem with zero Failure to uderstad the fuctio of zero Not carryig oe Poor cocept of divisio Addig oe istead of te to the uits Problems with place value Igorig the oe Not recordig zero Iability to estimate Poor cocept of divisio Not recordig zero Iability to estimate Not puttig i the decimal poit Iability to estimate Uderstadig of decimals ot adequately developed Choosig based o the legth of the umber rather tha usig kowledge of place value Uderstadig of decimals ot adequately developed Measure: chagig uits 2.42km = 242m Cocept of measuremet ot adequately developed 25

26 The strategies that follow are desiged to assist studets who have difficulties such as those outlied i the table above icludig poor cocept of place value, poor uderstadig of the relevat algorithm, ad difficulty rememberig umber facts. Examples of strategies Each group of strategies covers a rage of differet levels o a topic from a particular strad uit. A list of ecessary materials ad resources is provided ad a overview of the type of methodology employed i the strategies is icluded. Specific class levels are ot stated as differet studets may beefit from these strategies at differet stages. Strad: Number STRATEGIES FOR TEACHING PLACE VALUE Materials ad resources: Lollipop sticks, elastic bads to group the lollipop sticks ito budles, a selectio of square pieces of cardboard measurig approximately 20cm 20cm for place value mats. Methodology: The strategies outlied below ivolve the use of cocrete materials ad teacher-studet dialogue. STRATEGY: Place value mats The teacher shows budles of lollipop sticks to the studets ad asks them to fid out how may there are i each budle. They discover that there are te i each. The they are asked to cout out te sigle lollipop sticks o their desks. The teacher asks them to make a te ad the poses the questio How may are left? The budle of te is placed o the tes mat. 26 Tes Uits 1 0 The followig teacher-studet dialogue draws out the cocept of place value. Teacher How may tes i the tes place? We write oe i the tes place. How may uits i the uits place? We write zero i the uits place. Now we all write the umber 10. Take the elastic off the te ad add oe more. How may have we ow? If I make a te how may will be left? Put them o their mats. Write the umber eleve. Studet Oe. Noe. Eleve. Oe. Eleve makes oe te ad oe uit. Tes Uits 1 1

27 The same approach ca be used to demostrate umbers up to ietee. For twety, the dialogue should lead to the fact that twety makes two tes with o uits left over, givig 20. All umbers up to 99 ca be demostrated i this way. STRATEGY: Pick up a umber Studet uderstadig of the cocept of umber ad place value ca be assessed by askig the studet to pick up a give umber of lollipop sticks. For example, if the studet is asked to pick up 21 lollipop sticks does the studet: pick up a budle of 10, aother budle of 10 ad 1 sigle stick. [Correct] cout out 21 sigle sticks. [Correct, but ow ask the studet to pick up 21 sticks i the quickest possible way] pick up a budle of 10, aother budle of 10, (sayig that is 20 ) ad a third budle of 10 (sayig that is 21 ). [Icorrect, ecourage the studet to cout the lollipop sticks to discover that they have 30, ot 21, ad to try agai?] Strad: Number STRATEGIES FOR TEACHING COUNTING Materials ad resources: Lollipop sticks, uifix cubes, moey, circular or square card, or paper for umber lie steps. Methodology: The strategies outlied below ivolve the use of cocrete materials, teacher-studet dialogue, ad discovery learig. STRATEGY: Walkig the umber lie Some studets will have difficulties recogisig umbers ad uderstadig the ordiality of umbers, for example 4 comes after 3 ad 5 comes after 4. Rather tha preset the umber lie as a straight lie o paper or o the board, it may be useful to preset it as a series of separate discs or squares as show i the diagram below: Iitially, large versios of the discs ca be placed o the floor by the studets. Havig to replace the discs i their correct order each day provides reiforcemet of the positios of the umbers for the studets. Studets ca ow be asked to walk, skip, or jump alog the umber lie. Teacher 'Stad o the umber 4. Now take oe step forward. Where are you ow?' Take aother step forward. Where are you ow? Studet 5 6 After cosiderable time has bee spet walkig the umber lie studets ca be asked hypothetical questios. Teacher 'If you are at 6 ad you take oe step forward, where are you ow?' Studet 7 This ca be reiforced with some simple writte exercises o addig 1.

28 This strategy ca be developed by askig the studet to take two steps forward leadig to simple writte exercises o addig 2. Studets the costruct their ow umber lie o their desks usig smaller discs provided by the teacher. For a studet experiecig difficulties, the umber lie ca be shorteed ad the exteded as the studet achieves proficiecy. I order to wea the studet off the umber lie some of the discs ca be iverted so that the umbers are o loger visible The umber of iverted discs ca be gradually icreased as the studet progresses util evetually the studet is able to perform the operatio without the aid of a umber lie. STRATEGY: Coutig o Before the umber lie is totally removed as a aid, the studets must be taught the skills of coutig o usig their figers. The teacher places two sets of objects o the table, for example a group of 8 objects ad a group of 3 objects. The studet is asked to hold up the umber of figers correspodig to the smaller set of objects (i this case 3 figers). As the teacher moves oe object from the smaller set ito the larger set, she/he presses the first of the three raised figers sayig, 8 ad aother 1 makes 9. While movig the secod object, she/he holds the secod figer sayig, 9 ad aother 1 makes 10. Fially, as the third object is moved, the third raised figer is held ad the teacher says, 10 ad aother 1 makes 11. The aim is that the studet will evetually be able to follow this procedure idepedetly. The reverse process applies whe teachig studets to subtract from umbers greater tha te. 28 STRATEGY: Twety-te To overcome the difficulty that some studets experiece with coutig beyod twety-ie, for example twetyie, twety-te, twety-eleve, etc., try teachig coutig, place value, ad writig umbers cocurretly. Egagig i activities such as makig the umbers with lollipop sticks, uifix cubes, or moey, or placig them o place value mats, ad learig to write the umbers will improve the studets uderstadig ad help to accelerate the learig process. STRATEGY: Which is bigger? 19 lollipops may appear to be more tha 21 lollipops especially whe the uits are spread out as i the diagram below It may be ecessary for studets to cout both sets of sticks i order to cofirm that 21 is bigger tha 19. It may help studets to make pairs of umbers such as 12 ad 21, 24 ad 42, 25 ad 52, etc. I this case physical appearace will more tha likely be eough to idicate which is the greater umber.

29 STRATEGY: Number balace Usig the umber balace, studets ca discover a rage of umber facts for themselves. For example, by placig a certai weight o umber 8 o the right-had side of the balace ad a similar weight o both of the umbers 5 ad 3 o the left-had side the two sides will balace This will also occur if the weights o the left-had side are placed o 7 ad 1, or 6 ad 2, or 4 ad 4, but it will ot happe with ay other combiatio of umbers. I this way studets ca discover the story of 8. A discovery may be more easily retaied by the studet tha a list of facts taught directly by the teacher. Strad: Number STRATEGIES FOR TEACHING ADDITION (NUMBER FACTS) Materials ad resources: hudred square, lollipop sticks. Methodology: The strategies outlied below are desiged to reduce to a miimum the umber of additio facts that must be memorised. If all of the strategies outlied below are successfully utilised very few facts remai to be committed to memory. The usefuless of the strategies will vary accordig to the ability of the studet, but mastery of eve a few of the strategies should be of beefit. A very similar approach ca be take to assist the studet i the memorisatio of subtractio facts. A studet who has great difficulty i memorisig the umber facts ca be provided with the basic facts i the form of a hudred square permaetly taped oto his/her desk. 29 STRATEGY: Start with the larger umber It is easier to start with the larger umber ad add o the smaller umber. For example, 8+4 is easier to work out by coutig o tha by attemptig 4+8. STRATEGY: Additio is commutative Plety of practice usig cocrete materials to show that additio is commutative (3+4 is the same as 4+3) will halve the umber of additio facts that a studet eeds to memorise. STRATEGY: Addig zero, oe, ad two Addig zero to a umber gives the umber itself. Addig oe to a umber gives the ext umber. Addig two to a umber gives the umber after the ext umber. STRATEGY: Addig o 10 Usig cocrete materials such as lollipop sticks studets ca see the patter that develops whe umbers are added to 10. For example: 10+1 = oe te + oe uit = = oe te + two uits = = oe te + three uits = 13

30 STRATEGY: Doubles The twelve double facts (1+1=2, 2+2=4, ad so o to 12+12=24) should be committed to memory at a early age. This should be possible if oe or two facts are take at a time. With the aid of flash cards ad oral repetitio, both idividually ad as a group, studets ca progress at their ow pace, movig oto a ew fact oce previous facts are well established. Oce these facts are memorised, facts such as 3+4 ca be treated as oe more tha 3+3, givig 3+4 = 7. Similarly, 5+7 ca be treated as two more tha 5+5, givig 5+7 = 12. The double facts ca also be used to fid oe less ad two less. STRATEGY: Nies ad eleves 9+6 is oe less tha 10+6 = 16, givig 9+6 = 15. It may also be useful to ote that whe addig o ie to a umber (called the added), where the aswer is i the tees, the added is the result of addig the tes digit to the uits digit. This patter ca be see i the example below. 9+6 = = = = = = 8 Similarly, whe addig eleve to a umber, it ca be treated as oe more tha addig te. Ad 11+4 is oe more tha 10+4 = 14, givig 11+4 = 15. Also addig eleve is very similar to addig oe, ad some cocrete work with lollipop sticks will demostrate the patter to the studets. STRATEGY: Addig twelve Addig twelve to a sigle digit is very similar to addig two. Cocrete work with lollipop sticks will demostrate the patter to the studets. For example, 12+9 ad ca be obtaied by addig oe less ad oe more respectively tha whe addig twelve to te. 30 STRATEGY: Use of games Reiforcemet of simple umber facts ca be achieved by playig simple card games such as Sap, Multo, ad Old Maid, ad by the appropriate use of suitable computer programmes. The itroductio of quizzes ca add a elemet of competitio to the learig of umber facts ad ca improve studet motivatio. Strad: Number STRATEGIES FOR TEACHING ADDITION Materials ad resources: lollipop sticks, magetic board. Methodology: May of the mistakes made by studets carryig out the additio algorithm are made due to poor uderstadig of the cocepts ivolved. The followig strategies suggest cocrete materials ad teacherstudet dialogue to explai ad practice the additio algorithm. STRATEGY: To carry or ot to carry A studet who isists o carryig whe it is ot appropriate ca be asked to make the umbers with lollipop sticks. Whe they ca see that there is ot eough to make a te, this ca be liked to the fact that we do ot carry. I additio, a list of additio sums ca be give to the studet with the istructio to circle the oes that ivolve carryig. The studet ca the cocetrate o whether or ot carryig is appropriate without actually havig to do the full calculatio.

31 STRATEGY: Magetic board While may studets ca cope with additio without reamig, may will experiece difficulty with additio with reamig uless it is doe with cocrete materials. I the followig example, the studet is asked to make two umbers ad combie them (with reamig) usig cocrete materials represetig tes ad uits displayed o a magetic board. The teacher-studet dialogue is described first. Teacher Studet Make twety-three. Make thirty-eight. (Figure 1). Add the uits. How may uits are there? Eleve (Figure 2). Have we eough to make a te? Yes. Make a te ad place it i the tes place. How may will be left? Oe. We write oe i the uits place. We write oe i the tes place (Figure 3). for the oe te we made. Now add the tes. How may tes are there? Six. We write six i the tes place. (Figure 4). How may altogether? Sixty-oe. 31 Figure 1 Figure 2 T U T U Figure 3 Figure 4 T U T U

32 Strad: Number Strategies for teachig subtractio Materials ad resources: lollipop sticks, magetic board, place value mats, preted euro otes ad cois to represet hudreds, tes, ad uits. Methodology: Subtractio with reamig ca be very problematic for the studet. I particular the laguage of subtractio ca vary from perso to perso. Plaig for cosistet laguage ad methodology i the school ad i the home ca greatly assist the learig process. The followig strategies suggest cocrete materials ad teacher-studet dialogue to explai ad practice the subtractio algorithm STRATEGY: Magetic board Subtractio with reamig ca cause difficulty for studets uless it is doe with cocrete materials. I the followig example, the studet is asked to make two umbers ad subtract them (with reamig) usig cocrete materials represetig tes ad uits displayed o a magetic board. The teacher-studet dialogue is first described for the algorithm. Teacher Studet Make fifty-oe. Three from oe we caot take. (Figure 1). Break up a te. How may tes ad uits do we have ow? We ow have four tes ad eleve uits (Figure 2). Take three from eleve. Three from eleve leaves eight. What does that leave? Take two from four. Two from four leaves two (Figure 3). What does that leave? How may do we have left altogether? Twety eight (Figure 4). 32 Figure 1 Figure 2 T U T U Figure 3 Figure 4 T U T U

33 STRATEGY: Subtractig larger umbers Usig moey istead of lollipop sticks ca make larger umbers easier to hadle. Moey is coveiet to use ad is familiar ad meaigful to the studet. Larger amouts of moey, for example thousads, ca be represeted by a cheque. The followig example ca be doe o a magetic board or o place value mats: Teacher Studet Make four hudred. Zero take five we caot do. (Figure 5). We eed to break up a te but there is o te there. We break up a hudred. How may tes ad uits are there ow? There are te tes ad o uits (Figure 6). We break up a te. How may tes ad uits are there ow? There are ie tes ad te uits (Figure 7). Te take five leaves? Five. Nie take four leaves? Five. Three take two leaves? Oe. How may do we have left altogether? Oe hudred ad fifty five (Figure 8). Figure 5 33 h T U Figure 6 h T U

34 Figure 7 h T U Figure 8 h T U 34

35 Strad: Number Strategies for teachig multiplicatio Materials ad resources: lollipop sticks, magetic board, place value mats, preted euro otes ad cois to represet hudreds, tes, ad uits. Methodology: Studets with mild geeral learig disabilities will beefit from the use of cocrete materials whe beig itroduced to multiplicatio. STRATEGY: Sigle digit by sigle digit Lollipop sticks ca be used to itroduce multiplicatio as repeated additio, as show below: = 4 x 3 = = 2 x 5 =10 Strategy: Two digits by oe digit Teacher Studet Make three twety fours. (Figure 1). What do three fours make? Twelve. How may tes is that? How may uits are left? Oe te. Two uits. Write two i the uits place ad put the te i the tes place. (Figure 2). Now, i the tes place what do three twos make? Ad the oe te we made gives? Six. Seve. So what do we have altogether? Sevety-two (Figure 3). 35 Figure 1 Figure 2 T U T U 24 x 3 24 x 13 2 Figure 3 T U 24 x 13 72

36 Strad: Number Strategies for teachig divisio Materials ad resources: lollipop sticks, magetic board, place value mats, preted euro otes ad cois to represet hudreds, tes, ad uits. Methodology: There are two cocepts ivolved i divisio. Oe is sharig. (For example, if 12 apples are shared betwee 3 studets how may apples will each studet get?) The other is repeated subtractio. (For example, how may 3s are i 12?) The strategies preseted below vary from iitial divisio activities to divisio with regroupig. Oce agai the emphasis is o the use of cocrete materials. Studets record the algorithm oly whe they have become very familiar with the steps ivolved. Strategy: Early divisio activities Before beig itroduced to the cocept of divisio, studets should have plety of practice coutig up i twos, threes, fours, ad so o usig lollipop sticks, beads, ad other cocrete materials. This skill ca be reiforced through writte exercises such as the oe below. Fill the missig umbers ito the boxes: 2, 4,, 8, 10, 5, 10,, 20,, 30 Practice through oral coutig ca provide further reiforcemet. It may take a log time to achieve proficiecy i the more difficult umbers, 6, 7, 8 ad 9, but studets may still lear short divisio through tasks based o the easier umbers. 36 Strategy: Divisio as groups Dialogues such as the oe show i the table below accompaied by suitable work with cocrete materials will follow the iitial divisio activities. Teacher Cout out six lollipop sticks. Put them ito groups of two. How may groups of two are there i six? How may twos i six? We write it like this: 2) 6 3 Cout out twelve lollipop sticks. This time we wat to fid out how may threes i twelve. So we put them ito groups of three. How may groups are there? Now may threes i twelve? We write it like this: 3) 12 4 Studet Three. Three. Four. Four.

37 Strategy: Divisio as sharig Whe movig o to dividig a two-digit umber by a sigle digit umber it may be better to use the cocept of sharig i order to get over the difficulty caused by place value, especially whe reamig is ivolved. The followig example shows the procedure that may be followed, ad the type of dialogue that may be ecouraged: Teacher We wat to share 75 betwee these three studets. Studet Make 75 with moey. (Figure 1) First we share the tes. Two tes each How may 10 ca we give to easch people (Figure 2) How may tes are left? Oe We write: 3) 75 2 How will we share out the last 10? Will we get a No way! scissors ad cut it ito three equal parts? Go to the bak ad chage the 10 ote for te oeeuro cois We had five uits ad we made te more uits. How Fiftee may uits do we have ow? (Figure 3) We write: 3) Now we share the uits. How may oe-euro cois ca we give to each pupil? We write: 3) How much moey does each pupil get altogether? Five each (Figure 4) Twety five euro each (Figure 5) 37 Figure 1 T U

38 Figure 2 Figure 3 T U Figure 4 38 Figure 5

39 Strad: Number Strategies for teachig fractios Materials ad resources: paper for paper-foldig activities, various shapes ad fractios of these shapes (circle, rectagle), lollipop sticks. Methodology: Fractios are part of a ew umber system ad cosequetly the learig of fractios is challegig for most studets. It is vital that studets use cocrete materials for the itroductio of these ew cocepts. The laguage of fractios must also be itroduced, iteralised, ad verbalised by the studets. The strategies below cocetrate o verbal exchage ad use of cocrete materials to itroduce various fractioal cocepts. Strategy: Itroducig the laguage of fractios The dialogue below takes place while studets have access to a umber of differet paper shapes or other suitable materials that ca be split ito halves. Teacher If you divide somethig ito two equal parts, what do you call each part? A half is writte like this: V How do you get a half of somethig? If you eat a half of a cake what part have you left? Put them ito groups of two. Studet A half. You divide it ito two equal parts. You still have a half left. 39 Strategy: Equivalece of fractios The same procedure as above is followed with thirds ad quarters, but whe a quarter is defied ad the studets are able to tell the teacher how to get a quarter or three quarters, the shapes represetig halves ad quarters are placed side-by-side ad the studets are asked what they ca discover from this. They should uderstad that a half is the same as two quarters. They ca ow be ecouraged to ope a discovery chart at the back of their copies. This chart ca be used to record iformatio discovered by the studets i the dialogues. This method of discoverig the equivalece of fractios is more meaigful to studets tha presetig them with a overcrowded fractio wall. Strategy: Fidig a fractio of a umber The laguage of fractios is a vital part of this cocept. The use of lollipop sticks allows the followig approach to be take: To fid a third of six lollipop sticks we divide the lollipop sticks ito three equal parts. This gives two sticks i each part. So oe third of six is two. I fidig a multiple fractio of a set it may be useful to first fid the uit fractio, ad the to stagger, physically, the multiple fractios as show i the diagram overleaf. The act of puttig the various uit fractios together ca help to cosolidate the cocept.

40 of 12 Take 12 lollipop sticks ad divide them ito three equal parts. of 12 is 4 of 12 We cout the first third ad the secod third (Note the staggerig) of 12 is 8 W of 12 Take 12 lollipop sticks ad divide them ito four equal parts. W of 12 is 3 X of 12 We cout the first quarter ad the secod quarter ad the third quarter. (Note the staggerig) X of 12 is 9 Gettig a multiple fractio of a set directly, without havig first got a uit fractio of a set, is more difficult. It should ot be attempted util the above procedure is well established. Dialogue based o fidig multiple fractios of a cake could proceed like this: To get of a cake we first divide it ito three equal parts to make thirds. The we put two of the parts together to make two-thirds. To get W of a cake we first divide it ito four equal parts to make quarters. The we put three of the parts together to make three-quarters. 40 Strategy: Additio ad subtractio of fractios Oce agai, cocrete materials such as shapes o a magetic board are a useful aid to graspig the ecessary cocepts. However, additio ad subtractio of fractios should ot be attempted util the studets are very familiar with the equivalece of fractios. The eed for a commo ame or commo deomiator, whe addig ad subtractig fractios, ca be compared to askig, What is oe horse plus two dokeys? It is oe aimal plus two aimals, which is three aimals, aimals beig the ame that suits both?' Similarly, whe addig thirds ad quarters the ame that suits both is foud by coutig up i threes ad fours util a commo ame (twelve) is foud. The followig example shows how fractio shapes ca be used i the additio of simple fractios. The approach for more complex fractios ad for subtractio is similar. V W W W W W W W = 3 4 =

41 Strategy: Give a fractio, fid the whole Perhaps the most difficult fractio cocept to teach is whe the studet is give a multiple fractio of a set ad asked to fid the whole set, for example three-quarters of a umber is 30, fid the whole umber. Most studets fid this difficult because they associate a quarter with dividig by four. The followig approach which uses the uitary method may help: = 30c Three pecils cost 30c. 0 = 10c Oe pecil costs 10c = 40c Four pecils cost 40c. 0 W W W W W W W W W W W = 30 Three quarters is 30. = 30 Three quarters is 30. = 10 Oe quarter is 10. = 40 Four quarters or oe whole is 40. Fractios are a difficult cocept to teach, but with the help of cocrete materials, good teachig methodology, ad careful pacig, may studets with mild geeral learig disabilities ca experiece success. 41 Strad: Number Strategies for teachig decimals Materials ad resources: pecils, crayos, ad other coutable objects, lollipop sticks (uits ad teths), elastic bads to group the lollipop sticks ito budles, place value mats, measurig ad weighig istrumets, items of various weights, magetic board, Diees blocks, ad magetised Diees blocks. Methodology: The strategies outlied below ivolve the use of cocrete materials ad teacher/studet dialogue to reiforce the fudametal cocept that decimals ivolve bits, pieces, or fractios of whole umbers. Buildig o the already uderstood cocept of place value, the strategies also suggest likig with decimals i other areas of the curriculum, for example measure. Strategy: Itroducig decimals Uderstadig of decimals requires a solid grasp of the cocept of place value. Revisio of place value is a good startig poit for itroducig decimals. This ca lead o to group discussio o the size of various objects i the class, for example: This pecil is loger tha that pecil. It is twice as log. The small pecil is two times shorter tha the log pecil. The larger box of crayos holds more tha the small box. It holds three times as may crayos. The small box holds three times less tha the large box.

42 Whe the studets have iteralised this laguage the teacher makes oe hudred ad eleve with lollipop sticks, places them o the place value mats, ad writes the umber oe hudred ad eleve as show i the diagram. hudreds Tes Uits The followig teacher-studet dialogue draws out the cocept of decimals: Teacher Which is bigger, the te or the uit? (holdig up the te ad the uit) Which is bigger, the hudred or the te? (holdig up the hudred ad the te) So the te is te times bigger tha the uit ad the hudred is te times bigger tha the te. What comes ext? Studet The te is bigger. The te is te times bigger tha the uit. The hudred is bigger. The hudred is te times bigger tha the te because it takes te tes to make a hudred. Somethig that is te times bigger. Every time we move to the left it gets te times bigger. Oe thousad is te times bigger tha oe hudred. 42 The procedure is the reversed. Teacher Which is smaller, the hudred or the te? (holdig up the hudred ad the te) Which is smaller, the te or the uit? (holdig up the te ad the uit) So the te is te times smaller tha the hudred ad the uit is te times smaller tha the te. What comes ext? Studet The te is smaller. The te is te times smaller tha the hudred. The uit is smaller. The uit is te times smaller tha the te. Somethig that is te times smaller. Every time we move to the right it gets te times smaller. Before establishig what is te times smaller tha the uit some scaffoldig is required. Oe possible approach is to show a uit ad a half of a uit of a lollipop stick showig that the half is two times smaller tha the uit. The show a uit ad a quarter of a lollipop stick to demostrate that a quarter is four times smaller tha a uit. Fially establish that a teth is te times smaller tha a uit. Itroduce a teth as a ew umber i the family ad explai that it is give its ow placeholder. H T U TENTHS

43 Retur to the hudreds tes ad uits to reiforce. Te times smaller tha a hudred is te. Te times smaller tha te is a uit ad te times smaller tha a uit is oe teth. Teacher So the te is te times smaller tha the hudred ad the uit is te times smaller tha the te. What comes ext? Which is the odd oe out? Because it s the odd oe out, the teth is separated from the rest by a decimal poit. Aythig to the left of the decimal poit is made up of whole thigs. Aythig to the right of the decimal poit is made up of pieces, bits or fractios. Studet Te times smaller tha a uit is oe teth. The teth is the odd oe out because it is oly a piece or a fractio. All of the others are made up of whole lollipop sticks. H T U TENTHS This dialogue ad work with cocrete materials serves to establish the fuctio of the decimal poit as separatig whole thigs from pieces of thigs. 43 Strategy: Writig simple fractios as decimals The teacher draws the uits ad teths placeholders o the magetic board ad, usig lollipop sticks ad teths of lollipop sticks with magetic backig, asks a studet to pick up oe teth ad place it i the teths place. The the followig discussio takes place. U TENTHS Teacher How may uits do we have? We write zero i the uits place. How may teths do we have? We write oe i the teths place. What separates the whole thig from the pieces of thigs? We write the decimal poit betwee the uits ad the teths. Studet Noe. Oe. The decimal poit.

44 U TENTHS 0 1 The same laguage ad methodology is used for 2, 3,..., 9 ad so o Strategy: Writig simple decimals as fractios The studet is asked to pick up 0.1 lollipop sticks ad the followig diagram is show o the magetic board. U TENTHS 0 1 From the board, the studet ca see that 0.1 meas o uits ad oe teth. He/she picks up oe teth of a lollipop stick ad places it i the teths placeholder. Teacher-studet dialogue the takes place. Teacher I asked you to pick up 0.1 lollipop sticks. What did you pick up? So, 0.1 is oe teth. We write: = 10 Studet Oe teth of a lollipop stick. 44 U TENTHS 0 1 = 1 10 The same laguage ad methodology is used for 1.2, 1.3, 2.4, ad so o. Strategy: Usig decimals whe measurig legth After studets have leared about decimals ad about measure, further reiforcemet ca take place by likig these two cocepts. The followig exercise ivolves rewritig metres as cetimetres ad assumes uderstadig of simple relatioships betwee fractios of a metre ad cetimetres: 1 2 m = 50cm 1 m = 10cm m = 25cm 3 m = 30cm 10 The teacher draws a lie o the board measurig 1.1m ad explais that 1.1m meas oe full metre ad oe teth of a metre, ad is the same as 1,100cm. After further examples, studets are asked to draw 1.4m, 1.7m, 2.1m, ad so o.

45 Strategy: Usig decimals whe measurig weight This exercise assumes uderstadig of simple relatioships betwee fractios of a kilogram ad grams: 1 2 kg = 500g 1 4 kg = 250g 1 10 kg = 100g 3 10 kg = 300g Usig a selectio of 1kg bags ad smaller bags weighig 100g, the studets are asked to pick up 1.1kg. The work ca be displayed as follows: 1.1kg = 1kg kg U TENTHS sugar 1kg 100g 1 1 = 1,000g + 100g = 1,100g Whe rewritig grams as kilograms studets are asked to pick up 2300g i the quickest possible way ad to place the weights i the placeholders. The work ca be displayed as follows: 2,300g = 2kg +300g sugar 1kg U sugar 1kg TENTHS 100g 100g 100g = 2.3kg A similar approach ca be take with other uits of measure. Strategy: Operatios ivolvig decimals The magetic pieces used whe teachig fractios ca also be used as cocrete aids for the additio, subtractio, multiplicatio, ad divisio of decimals. Usig whole circles to represet uits ad teths of a circle to represet teths, the algorithm used follows that already outlied i the sectio o fractios. Strategy: Usig Diees blocks to explai hudredths ad thousadths Diees blocks ad magetised Diees blocks ca be used to make the cocepts of hudredths ad thousadths more cocrete. The Diees block measurig 10 cubes by 10 cubes by 10 cubes is used to represet oe uit. The 10 by 10 piece represets oe teth. The 1 by 10 piece represets oe hudredth ad a sigle cube represets oe thousadth. oe uit oe teth oe hudreth oe thousadth

46 The laguage ad methodology used for itroducig teths ca be used for itroducig hudredths ad thousadths. Evetually, studets preset their work as show i the followig diagrams. U Teths hudredths 1 = U Teths hudredths 2 3 = U Teths hudredths thousadths 1 2 = 1,

47 Strad: Measures Strategies for teachig legth ad weight Materials ad resources: lollipop sticks, place value mats, measurig ad weighig istrumets, strig, items of various weights. Methodology: The strategies preseted here focus o areas of difficulty with legth ad weight ad, i particular, o the use of decimals i measuremet. Strategy: Parts of a hudred ad a thousad Familiarity with fractios of a hudred ad a thousad will help the studet to deal with decimals i measure. A cocrete represetatio of fractios of a hudred could be provided usig lollipop sticks, ad appropriate dialogue will reiforce the cocepts. The first diagram shows oe hudred lollipop sticks. The secod diagram shows oe hudred lollipop sticks divided ito two equal parts. Each part is a half. A half of oe hudred is fifty. 47 Here the oe hudred is divided ito four equal parts. Each part is a quarter of a hudred. A quarter of a hudred is twety-five. Two quarters are fifty. Three-quarters are 75. Each part here is oe teth of a hudred, because the hudred is divided ito te equal parts. Oe teth of a hudred is te, two teths are twety, three teths are thirty, ad so o. Fially, the lollipop sticks ca be divided ito oe hudred equal parts. Each part is oe hudredth of a hudred. Oe hudredth of a hudred is oe. Two hudredths of a hudred is two, three hudredths of a hudred is three, ad so o. This work ca be reiforced with promietly displayed wall charts cotaiig facts that are particularly importat for the uderstadig of decimals i measuremet of 100 = of 100 = 1

48 A visual approach is also ecessary whe dealig with fractios of a thousad. A wall chart with oe thousad dots (or other small symbols or pictures) ca be displayed. 48 Alteratively, studets ca build up such a picture by attachig boxes of te dots to a display board util oe thousad is formed. This will help studets to have a metal image of oe thousad as a quatity. Relevat fractios of oe thousad are discussed as outlied already for oe hudred, ad charts are displayed to highlight appropriate facts of 1,000 = of 1,000 = ,000 of 1,000 = 1 Strategy: Decimal place value chart The followig chart ca be o permaet display i the classroom: TH H T U ,000 The studets use this to see that 0.1 meas oe teth, 2.3 meas two uits ad three teths, 0.01 meas oe hudredth, ad so o.

49 Strategy: Drawig lies Drawig lies of various legths ca reiforce the use of decimals i measure. The use of metres ad cetimetres illustrates parts of a hudred. Similarly, the use of cetimetres ad millimetres ca reiforce the cocept of teths. The followig simple example idicates the kid of approach that could be take whe drawig a lie that measures 1.1m. U m meas oe full metre ad oe teth of a metre. This is draw o the board ad the followig calculatio is discussed. 1.1m = 1 metre + V of a metre = 100cm + 10cm = 110cm This work cotiues so that studets uderstad that each example cotais a certai umber of whole metres ad a bit or a part of a metre. Strategy: How log is a piece of strig? Cuttig ad measurig pieces of strig ca provide a cocrete example of various decimal operatios. The table below outlies some examples: Operatio Workig with strig 2.4m + 1.8m Measure ad cut two pieces of strig with these legths. Put them ed to ed ad measure the whole legth. 2.4m - 1.8m Measure a piece of strig of legth 2.4 m. Measure 1.8m i from oe ed of it ad cut it off. Measure the piece that is left. 2.4 x 3 Cut three pieces of strig 2.4m log. Place them ed to ed ad measure the whole legth. 2.4m 2 Measure a piece of strig of legth 2.4m. Fold it so that the two eds meet ad cut the strig ito two equal pieces. Measure oe piece. 49 Strategy: A weight off your mid Parts of a thousad ca be demostrated cocretely with weight or capacity. Examiatio of the weights o the balace below ad appropriate dialogue lead to the fact that oe teth of a kilogram is oe hudred grams. sugar 1kg 100g 100g 100g 100g 100g 100g Give a decimal represetatio of a weight i kilograms, studets ca be asked to select the correct items, place them i the placeholders, ad write the result i grams as outlied already i the decimal strategies.

50 Strad: Measures Strategies for teachig moey Materials ad resources: moey (cet ad euro cois ad otes), ufix cubes, other resources appropriate for shoppig role-playig. Methodology: The strategies outlied below cover practical additio ad subtractio experieces whe usig moey, uderstadig the value of cois ad otes, kowig what cois or otes are to be haded i whe payig for goods at various prices, ad kowig what chage to expect. Strategy: Pairs After coi recogitio has bee achieved, reiforcemet of the story of te ca precede simple shoppig activities. The umber balace strategy has already bee outlied. This card game further reiforces the pairs of umbers that combie to give te. Each pair of studets has eleve cards as show. J is the Joker J The cards are shuffled ad oe player gets six cards ad the other five. Both studets discard all matchig pairs, for example 8 + 2, The the player with the least umber of cards picks from the perso with the most. This cotiues util oe perso is left with the Joker. The other perso is the wier. After other suitable reiforcemet activities studets commit the story of te to memory. 50 Strategy: Shoppig with cet The table below shows a simple shoppig activity based o the story of te. Suitable items ca be amed i the item colum. Item Cost Chage out of 10c 8c 2c 5c 7c 6c 3c 2c 4c 1c 10c Studets ca also practice givig exact chage for amouts ot greater tha 10c. Some examples are give i the table below. Item Cost Exact cois 2c (2c) (1c + 1c) 4c (2c + 2c) (1c + 1c + 1c + 1c) 5c (5c) (2c + 2c + 1c) (1c + 1c + 1c + 1c + 1c)

51 A similar approach ca be take with the story of 20. Studets take part i a variety of activities usig cocrete materials such as ufix cubes util the pairs of umbers that combie to make twety are memorised. It is importat that the similarities betwee the story of te ad the story of twety are discussed explicitly. Simple shoppig activities with 20c follow. These exercises are repeated for 30c, 40c, 50c, 60c, 70c, 80c, ad 90c. Strategy: Shoppig with euro If real or play otes ad cois are ot available, te stacks of te ufix cubes ca be used to represet oe euro. Here the studets meet the same combiatio of umbers agai, for example: sped oe te ad there are ie tes (90) left. 1 matches with 9. Sped thirty (3 tes) ad there are seve tes left (70). 3 matches with 7. After reiforcemet of these facts, simple shoppig activities follow: Had i Cost Chage 11 10c 11 30c 11 60c 51 The tables below suggest how studets might deal with more complicated calculatios i stages. Had i Cost Chage 11 15c Sped 10c first 90c left Sped 5c ext 85c chage 11 64c Sped 60c first 40c left Sped 4c ext 36c chage Item Cost Had i Chage Sped 14 first 11 left Sped 60c ext 40c left Sped 2c ext 38c left Sped 12 first 18 left Sped 40c ext left Sped 8c ext left Where possible, studets should have access to euro cois ad otes whe egaged i shoppig activities. This will aid studets i their progressio to evetual memorisatio of the relevat umber facts, ad will help to situate their moey skills i the appropriate real-life cotext.

52 Exemplar 1: Mathematics Strad: Algebra Strad uit: Extedig patters Level: Juior ifats Materials/resources Learig targets Laguage Multi-liks, spools, buttos, cubes, blocks, couters, shells, care bears, beads, uts, toy cars, toy aimals, bottle tops, 2-D ad 3-D shapes, laces for threadig. Studets will exted their kowledge of patter accordig to colour, shape, ad size. They will - copy patter - cotiue/fiish patter - devise their ow patters. First, ext, after, eed more, repeat, agai, secod, last, comes after, loud, soft Methodology Direct teachig with the group ca be used to examie simple patters. Beads o a strig with a fixed begiig ca esure that studets are able to add to oe ed oly: blue, red, blue, red; What comes ext? > Studets copy the patter usig beads ad strig. Talk ad discussio will heighte studet awareess of the patter. Studets describe the patter. Repeat, usig a variety of materials, colours, shapes, ad sizes. > Studets copy a patter usig cocrete materials, followig a pictorial represetatio o card. Repeat usig variety of materials/variety of cards. > Studets cotiue/fiish a patter (colour, shape, ad size). > Studets create their ow patter (colour, shape, ad size). > Studets separate the colours ito bowls, takig turs. 52 Likage/itegratio Visual Arts Patter-makig with gummed paper shapes Spoge prits, for example had/foot prits usig differet coloured paits Examiig patters o sweets, for example liquorice allsorts Decoratig the witch s house (Hasel ad Gretel) with pattered sweets Makig Gray s patter bedspread (Little Red Ridig Hood) Makig Humpty Dumpty s wall Makig Mary, Mary Quite Cotrary s garde (bluebells ad cockle shells all i a row). Music Percussio istrumets: drum beat, the xylophoe, the triagle, the repeat (music patter will become more complex as studets improve). Voice: loud soud, soft soud; loud, soft; loud, soft Clappig activities Sog: Head, shoulders, kees ad toes.

53 Exemplar 1: Mathematics Developmet Maths trail: Lookig at patters (walls, wire, pavig slabs, railigs, flower beds). Differetiatio If studets are older, use other appropriate objects to form patters, for example football jerseys, euro cois, attribute blocks. The cotet of the exemplar may be spread over several lessos, depedig o the ability of the studets. Assessmet Has the studet difficulty orgaisig a patter (two colours)? Ca the studet exted a patter? Ca the studet create a patter? Ca the studet ame the primary colours? 53 Ca the studet recogise ad ame a circle, triagle, square, rectagle? Does the studet cocetrate well o the task? Is had domiace fully established?

54 Exemplar 2: 2: Mathematics Strads: Number, Measure, Shape ad space, Data Thematic approach: Birthday Level: Seior ifats Materials/resources Learig targets Laguage Paper plates (large ad small), paper cups, coloured cadles, balloos, pattered wrappig paper, lemoade, oe-had aalogue clock, caledar, large circle, triagle, square, rectagle, height chart, Uifix cubes, buildig blocks. Further cosolidatio of cocepts ad laguage itroduced i juior ifats o clock, 2-D shapes, circles, triagles, squares, umerals 1 to 5. Full/empty, large/small, more/less, heavy/light, youg/old, pour, tall, as tall as, short, shorter tha, yesterday, ow, February, sprig, dark, ight, adults, usually, eough. Laguage focus will deped o the ability of the studets. Methodology Use a direct teachig method, where appropriate, with teddy-bear s birthday as a startig poit. Which moth is it ow? Which seaso? What age is the teddy? five? six? What age will he be ext year? Will he be older or youger ext year? His birthday is today. What day is it today? How may people are comig to the party? Cout the boys. How may boys? Cout the girls. How may girls? How may cadles should we put o the cake? Allow oe studet to cout out the correct umber of cadles. What colours are the cadles? Poit to each cadle. What colour is it? How may balloos should we tie to the Wedy House door? How may balloos are red? How may are ot red? 54 Likage/itegratio What shape is the birthday cake (perhaps made from a circular or square biscuit ti)? What time does teddy usually go to bed at? After tea-time? Show the time o a clock. Music Sig Happy Birthday. Clap hads five times. Tap kees six times (ad repeat). Visual Arts Examie a patter o wrappig paper. Make wrappig paper. Draw a picture of the birthday party. Colour i a cake i the shape of a circle/square/rectagle/triagle. Make birthday cards. Make I am 5/6/7 badges. Usig Plasticie, make two sausages for each plate. Laguage See vocabulary above. SPHE Care should be take with matches/cadles. Feeligs: happiess (We are happy whe we get ice birthday presets). The studets are older this year tha last year. What ca the studets do ow that they could ot do last year? Skip? Ride a bike? Sig a ew sog?

55 Exemplar 2: Mathematics Developmet Studets ca set the table for the party. How may cups are eeded? How may plates? Reiforce the cocept of five. Studets build a tower of blocks, with Uifix cubes to represet ages 5/6/7. Is the 7 tower more tha the 5 tower? (Yes, a little more.) Is the 5 tower less tha the 7 tower? (Yes, a little less.) Differetiatio Use clear, precise istructios with activities. There may be a eed to repeat istructios several times. If the studets are older make the party more age-appropriate. For less able studets, use closed questios. (Is this glass full? Is this balloo red?). Spread the cotet over several lessos. Broade the cotet to iclude brow, purple, silver, ad gold for those studets who kow the primary colours. Narrow the cotet to oe colour for less able studets. Assessmet 55 Ca the studet distiguish betwee circle, square, rectagle, triagle? Ca the studet colour i shapes with reasoable accuracy? Ca the studet idetify the umeral 5 (6, 7)? Ca the studet build a tower of 5 (6, 7)? Ca the studet associate the symbol 5 (6, 7) with the umber group/tower? Has the studet coservatio of umber to 5 (6, 7)? Teacher observatio: Does the studet reverse umerals whe writig?

56 Exemplar 3: 3: Mathematics Strad: Measures Strad uit: Time Level: Seior ifats Materials/resources Learig targets Laguage Oe-had aalogue clock, two-had aalogue clock, various egg timers (1 miute, 3 miute), caledar, time/seaso pictures ad charts. Developig a uderstadig of the cocept of time. Tellig the time i oe-hour itervals. Developig a time vocabulary. After, before, morig, afteroo, eveig, ight, yesterday, today, tomorrow, breakfast-time, luchtime, tea-time, dier-time, bedtime, play-time, start, stop, slow, slower, slowest, fast, faster, fastest. Methodology > Fillig i a daily time chart: Today is. Tomorrow will be. Yesterday was. > Fillig i a birthday chart: referece to moths, seasos, chages i clothes with the seasos. > Advet caledar. > Referrig to pictorial represetatio of times of the day. > Makig a photographic record of studets ad some of their activities over time, for example platig seeds, plats i bloom, puttig up the tree before Christmas, Christmas presets. > Establishig what happes ext: the studets helpig the teacher to re-tell favourite stories, or predictig what will happe i stories the studets are ot familiar with. > Makig a cadle-clock with cadle ad pis (Pis fall as the cadle burs.) > Usig the oe-haded clock to teach the time i oe-hour itervals as soo as studets ca read the umerals to Likage/itegratio Mathematics: Number 3 o clock I will be 7 years old ext year. Music: Playig fast ad slow pieces. PE Ruig quickly, walkig slowly. Game: What time is it Mr. Wolf? SESE Platig bea seeds ad observig them over time. Evergrees, witer, Christmas, hiberatio, bulbs, sprig, St Patrick s Day, Easter, summer flowers, summer holidays, leaf fall, autum, Hallowee, harvest time. Laguage Nursery Rhymes: Hickory Dickory Dock, Twikle, Twikle, Little Star. Story-Time: Owl Babies (Mark Waddel). SPHE Feeligs: afraid of the dark.

57 Exemplar 3: Mathematics Differetiatio If studets are older use PE activities ivolvig a egg-timer or stop-watch. Ca you skip/hop/bouce the ball for two miutes? Assessmet Ca the studet tell a activity which takes a short/log time? Ca the studet sequece two/three picture cards? Ca the studet uderstad yesterday/today/tomorrow? Ca the studet uderstad before ad after? 57

58 Exemplar 4: 4: Mathematics Strad: Number Strad uit: Place value Level: First class Materials/resources Learig targets Laguage Abacus, Uifix cubes, couters, lollipop sticks, sets of coutig objects, beads, toys, shells, 100 squares. Studets will idetify groups of objects i sets of tes, ad i tes ad uits. Studets will assemble sets of tes ad uits usig cocrete materials, icludig moey. Oe less tha, oe more tha, equals, equal to, tes, uits, abacus, tes ad uits, more, before, after. Studets will record umber stories pictorially. Methodology Usig lollipop sticks or Uifix cubes, studets place te lollipop sticks i a row ad cout them. Now I have 10. Studets lear to place the budle of te o the tes mat sayig, I have oe te ad o uits. Mammy gave me aother 1. Now I have 11. Studets use the materials the record pictorially. Exted this skill to 99. Usig lollipop sticks/ Uifix cubes i a similar way, studets build up umbers to 99. Skip coutig by tes, usig 100 square. Establish a patter. Work with a abacus, for example 54 = 5 tes ad 4 uits. 58 Likage/itegratio Moey/shoppig activities. Algebra: substitute umerals o 100 square with a, b, c; what umber goes at a? b? c? Operatios: coutig i 5s, 10s. Costruct umber stories based o coutig i 10s. Lik to ordial umbers. My birthday is o the 3 rd. My birthday is o the 31 st. PE Skip coutig to ball boucig, skippig games. Music Body percussio i beats of te clappig, stampig, tappig. Sog: There were 10 i the bed Visual Arts Makig/modellig groups of te objects.

59 Exemplar 4: Mathematics Developmet Use umber dice i sakes ad ladders ad other board games. Use sakes ad ladders game for visual reiforcemet of 100 square. The first studet to get to 100 wis. Bak game: cet cois ad a pair of dice. The studets roll the dice, add the umbers obtaied, ad take that amout i cet. They bak amouts of te cet for a 10c coi. The wier is the oe who gais te 10c cois first. Differetiatio Studets who are kiaesthetic learers build towers, or write usig rice o the tray (or sadpaper/felt umerals). Trace, say, write. Less able studets may work oly to 50 or less iitially, depedig o ability. Assessmet Ca/does the studet 59 match a umeral to a set (0 99) uderstad 10 uits = oe te uderstad 46 = 4 tes ad 6 uits build towers i tes record a simple umber story pictorially uderstad oe more tha 20 is 21 uderstad oe less tha 20 is 19 tap out to 20 tap out to 30?

60 Exemplar 5: 5: Mathematics Strad: Measures Strad uit: Moey/time Thematic approach: The jumble sale Level: Third ad fourth class Materials/resources Learig targets Laguage Moey (usig real cois). Items for jumble sale. The studets will uderstad - simple additio/subtractio, ad moey calculatios ivolvig what chage will I get? - roudig off 99c to 1 ad 1.99 to 2. How much?, How may?, amout, value, altogether, more, most, sell, pay, bought, euro, cet. Methodology Preparatio talk ad discussio: > Studets each have five cois i their purses/wallets. How may differet total amouts ca they list? > The studets make lucky dips for the jumble sale, usig a combiatio of small items, for example pecils 15c, erasers 10c, sweets 20c, stickers 15c. How much will each lucky dip cost? What chage will they give/get from 1? > I buy a book for 50c ad a puzzle for 25c; how much do I sped? How much chage do I get from 1? Estimate first. Will it be more or less tha 1? > I bought a toy for 99c ad a watch for How much did I sped? Ecourage studets to adjust these amouts ad to chage their aswer accordigly. Why are so may items i shops priced this way? Discuss. 60 Likage/itegratio Mathematics: Time Date of jumble sale, time of jumble sale. Visual Arts Makig jumble sale ivitatios. Makig for sale posters.

61 Exemplar 5: Mathematics Developmet Use of calculator i moey calculatios. Examie i a practical way how 25c + 75c = 50c + 50c. Differetiatio Some studets will have difficulty uderstadig that 2.6 euro = Sellers may wish to use their calculators whe they are totallig a strig of amouts. They will eed to uderstad 46c must be etered as 0.46 o the calculator. Assessmet Ca the studet add simple amouts uderstad chage 61 use a calculator i simple moey calculatios uderstad roudig up, for example 1.99 to 2?

62 Exemplar 6: 6: Mathematics Strad: Number Strad uit: Percetages Level: Fifth ad sixth class Materials/resources Learig targets Laguage Percetage boards, Diee s blocks, squared paper, calculators, catalogues (clothes, travel, music), ewspapers. Studets will uderstad bechmark percetages: 10%, 25%, 50%. Cet, euro, America cet, cetury, cetimetre, cetipede, per cet, percetage, loss. Methodology > Talk ad discussio. > Percetages i daily life: orage (100% pure); beas (99% fat free). > Revisio of previous kowledge usig percetage boards. > Moey (50% of 1 = 50c, 25% of 1 = 25c). > Based o catalogues, studets ca discuss, estimate ad calculate discouts of 50% durig a sale. This ca be followed by discouts of 25%, 10%. > Usig a 100 square, studets fill i percetage amouts: 50%, 25%, 10%. 62 Likage/itegratio Mathematics Fractios, decimal fractios. Geography Huma eviromets. Visual Arts Studets look for discouted prices i ewspapers ad make their ow sales posters advertisig discouts i a clothes/music/travel shop.

63 Exemplar 6: Mathematics Developmet Establish key poits of referece: 8 is close to 8 which is 1 or 50% If I got 40 of my spelligs correct i a test, what percetage would that be? 80 Multiple-choice questios: If I got 10 marks for history what percetage would that be: 10%, 25%, 50%? 20 A computer game costs 120. If I get 25% off, how much do I have to pay? Games: percetage domioes. Differetiatio Use of calculators to check aswers. Estimatio (50% of 415 is roughly about 207, discuss.) Covert percetages to fractios (50% = 50 = 1) Covert fractios to percetages: multiply by 100 (1 100 = 33 1%) The studet reports o some of the above as if explaiig it to someoe youger. Assessmet Does the studet uderstad the cocept of per cet use some estimatio/metal computatio strategies covert a fractio to a percetage (ad vice versa) usig a calculator uderstad the use of per cet i huma eviromets covert a fractio to a percetage (ad vice versa) without usig a calculator?

64 Appedix Overview of cotet Mathematics Strads Early mathematical activities Strad uits classifyig matchig comparig orderig Number coutig comparig ad orderig place value operatios - additio - subtractio - multiplicatio - divisio fractios decimals 64 Patter ad sequece umber patters ad sequeces umber seteces Shape ad space spatial awareess 2-D ad 3-D shapes symmetry lies ad agles Measures legth weight capacity time moey Data recogisig ad iterpretig data chace

65 Checklist for assessig studet s skills developmet Ca the studet Early mathematical activities 4 Classify > atted to ad participate i the matchig of 3-D shapes > classify objects o the basis of oe attribute (colour, shape, size, ad texture) > classify objects o the basis of two attributes > select from a assortmet of objects, oe that serves the same fuctio as a give object > classify socially related objects > select from a assortmet of objects, oe similar to a give object Match > match pairs of idetical objects i oe-to-oe correspodece 65 > match sets of idetical objects i oe-to-oe correspodece > match equivalet sets of cocrete objects i oe-to-oe correspodece > match o-equivalet sets of cocrete objects i oe-to-oe correspodece Compare > compare objects accordig to legth, width, height, size, ad weight Order > order objects accordig to legth, size, weight, or height?

66 Ca the studet Algebra - Patter ad sequece 4 > atted to repeated souds ad actios > respod to repeated souds, patters, ad movemets > imitate repeated souds, patters, ad movemets > iitiate ad create his/her ow repetitive souds ad movemets > atted to the sequecig of two or three familiar routies > idetify some of the patters i daily routies > match sets of idetical objects i oe-to-oe correspodece > correctly sequece two or three evets > demostrate uderstadig of first, ext, last 66 > follow the correct sequece i carryig out activities > correctly sequece pictures that depict familiar activities > use familiar 2-D ad 3-D objects to copy/cotiue patters (i colour, shape, or size) > observe ad talk about patters > copy, cotiue, ad exted patters?

67 Ca the studet Number 4 > demostrate uderstadig of oe as opposed to a lot > demostrate uderstadig of some ad more > demostrate uderstadig of oe-to-oe correspodece > rote cout to a give umber > show a awareess of umerals > show awareess of umber i the eviromet > show awareess of umber i stories > show awareess that umber ames come i a fixed order > liste to, respod to, ad participate i umber rhymes, stories, ad games > talk with uderstadig about umbers of persoal sigificace > show uderstadig that umbers are used for coutig > trace/copy umerals, without a cosistet error such as writig particular umbers backwards or upside dow 67 > order umerals > respod to the laguage of ordial umber: first, last > cout accurately the umber of objects i a set, coutig each object oly oce (otig maximum umber couted accurately) > match symbols to sets > idetify a set which has more/less objects > estimate the umber of objects i a set > idetify the empty set ad recogise zero > combie sets of objects > partitio sets of objects > use the symbols + ad = to costruct simple word seteces ivolvig additio > use the symbols ad = to costruct simple word seteces ivolvig subtractio > use metal strategies, for example coutig o > recogise umber patters, for example coutig i twos, fives, tes > cout i twos, fives, tes > cout i twos, fives, tes, startig at ay umber > show uderstadig of the mathematical sigs eeded for a task > idetify ad record place value > solve simple oe-step oral problems?

68 Ca the studet Space ad spatial awareess 4 > show a awareess of the positio of his/her ow body ad body parts i space > demostrate movemet i differet parts of the body ad how the body ca move i space > atted to ad respod to the laguage of movemet ad positioig > follow istructios related to movemet ad positioig > use the laguage of movemet ad positioig > observe ad describe people ad objects i differet positios i space (o, uder, far away, beside, etc.) > give ad follow simple directios > follow istructios to positio themselves i relatio to others > recogise right ad left i real situatios Shape 4 > atted to ad participate i the idetificatio of 3-D shapes 68 > atted to ad participate i the matchig of 3-D shapes > combie 3-D shapes to make other shapes > sort 3-D shapes (regular ad irregular) > atted to ad participate i the matchig of 2-D shapes > atted to ad participate i the sortig of 2-D shapes > combie 2-D shapes to make other shapes or pictures > sort 2-D shapes > trace ad copy 2-D shapes > ame 2-D shapes (list oes kow)?

69 Ca the studet Measures - legth 4 > show a awareess of the cocept of legth through the use of appropriate vocabulary (log, short, loger tha, shorter tha, etc.) > idetify which of two objects is log/short > compare ad order two objects accordig to legth or height > estimate ad measure legth usig o-stadard uits Measures - weight 4 > show a awareess of the cocept of weight through the use of appropriate vocabulary (heavy, light, heavier tha, lighter tha, etc.) > idetify which of two objects is heavy/light 69 > compare ad order two objects accordig to weight > estimate ad measure weight usig o-stadard uits Measures - capacity 4 > show a awareess of the cocept of capacity through the use of appropriate vocabulary (full, empty, early full, early empty, etc.) > idetify which of two objects is full/empty > compare ad order two cotaiers accordig to capacity > estimate ad measure capacity usig o-stadard uits?

70 Ca the studet Measures - moey 4 > demostrate uderstadig that moey is ecessary to pay for goods > sort ad match cois > demostrate uderstadig that some cois are worth more tha others > recogise some cois ad otes > use the vocabulary of moey (buy, sell, how much?, cois) > calculate simple bills > use moey for social purposes with or without help Measures - time 4 70 > show a uderstadig of time as related to self > show a awareess of specific times at school (break, music time) > show awareess of day ad ight > show awareess of the daily patters of familiar evets > sequece pictures of daily evets > recogise the preset time, for example, today > idetify thigs that happeed i the recet past > show uderstadig that thigs will happe i the future > use the laguage of time to discuss evets > ame the days of the week?

71 Ca the studet Data - collectig ad processig 4 > sort by puttig similar objects together > make a set of objects with a give property > sort ad classify sets of objects by oe criterio > compare two objects, idetifyig similarities ad differeces > sort ad classify (with help) sets of objects by up to two criteria Data - represetig ad iterpretig 4 > represet data usig studets ad objects > represet data usig pictures 71 > iterpret data which is represeted by real objects or pictures?

72 Overview of skills developmet The skills that the studet should develop are evidet throughout the Primary School Curriculum, Mathematics. Opportuities should be provided for the studet to acquire the kowledge, cocepts, ad skills required for everyday livig ad for use i other subject areas: applyig ad problem-solvig commuicatig ad expressig itegratig ad coectig reasoig implemetig uderstadig ad recallig. The overview of skills developmet which follows is take from the Teacher Guidelies, Mathematics. The maistream class levels have bee retaied to show the developmetal sequece. However, studets with mild geeral learig disabilities may eed to sped loger at differet stages or to have the trasferability of the skills explicitly idetified for them. 72

73 Skills developmet Class levels i the maistream curriculum Ifat classes Problem-solvig Commuicatig ad expressig Itegratig ad coectig Reasoig Implemetig Recallig select appropriate materials ad processes for mathematical tasks select ad apply appropriate strategies to complete tasks or solve problems liste to ad discuss other studets descriptios/ explaatios discuss/record usig diagrams, pictures ad symbols uderstad the mathematical ideas behid procedures make guesses ad carry out experimets to test them recogise ad create mathematical patters ad relatioships execute procedures efficietly recall ad use termiology ad facts recogise solutios to problems 73 Class levels i the maistream curriculum First ad secod classes Problem-solvig Commuicatig ad expressig Itegratig ad coectig Reasoig Implemetig Recallig select appropriate materials ad processes for mathematical tasks/applicatios apply cocepts ad processes i a variety of cotexts liste to ad discuss other studets descriptios/ explaatios discuss/record usig diagrams, pictures ad symbols uderstad the mathematical ideas behid procedures make guesses ad carry out experimets to test them recogise ad create mathematical patters ad relatioships execute procedures efficietly recall ad use termiology ad facts

74 Class levels i the maistream curriculum Third ad fourth classes Problem-solvig Commuicatig ad expressig Itegratig ad coectig Reasoig Implemetig Recallig aalyse problems ad pla a approach to solvig them select ad use a variety of strategies to complete tasks/ projects or solve problems evaluate solutios to problems discuss ad explai the processes used/results of mathematical activities/ projects/ problems discuss ad record processes ad results usig a variety of methods discuss problems preseted verbally or diagramatically/ carry out aalyses coect iformally acquired mathematical ideas/ processes with formal mathematical ideas/ processes uderstad the coectios betwee mathematical procedures ad cocepts represet mathematical ideas ad processes i differet modes: verbal, pictorial, diagrammatic, symbolic make iformal deductios explore ad create mathematical patters ad relatioships reaso systematically i a mathematical cotext execute stadard procedures efficietly with a variety of tools recall ad use termiology, facts ad defiitios 74 Class levels i the maistream curriculum Fifth ad sixth classes Problem-solvig Commuicatig ad expressig Itegratig ad coectig Reasoig Implemetig Recallig reflect upo ad evaluate solutios to problems discuss ad explai processes used ad results i a orgaised way search for, ivestigate ad create mathematical patters ad relatioships discuss problems/carry out aalyses