Maths in Primary School

Transcription

1 Maths in Primary Schoo Incuding Resuts of an INTO Survey Irish Nationa Teachers Organisation 35 Parne Square Dubin 1 Teephone: Fax: Emai: info@into.ie Web: Genera Secretary: John Carr Cumann Múinteoirí Éireann 35 Cearnóg Pharne Baie Atha Ciath 1 Guthán: Fax: Ríomhphost: info@into.ie Gréasán: Árd Rúnaí: John Carr

2 CONTENTS Foreword 1 CHAPTER 1 Mathematics in the Primary Schoo DISCUSSION DOCUMENT FOR THE INTO CONSULTATIVE CONFERENCE ON EDUCATION, KILKENNY CHAPTER 2 Resuts of INTO Survey on Mathematics, CHAPTER 3 Report from the Maths Discussion Groups at INTO Education Conference Bibiography 47 Appendix 1 SHARED MATHS / MATHS FOR FUN ACTIVITIES 49

3 Foreword It is generay accepted that mathematics education is an important dimension of a genera education. It is perceived as a core area of earning in probaby most education systems throughout the word. However, we may we question how we mathematics are taught in the primary schoo. Ireand has participated in a number of nationa and internationa assessments in the area of mathematics, the main findings of which are summarised in this report. Though Ireand performs reasonaby we in overa terms, there is no doubt that there is room for improvement, particuary in some aspects of the mathematics programme. The most recent internationa research, conducted by the OECD among fifteen year od pupis, indicates Ireand s performance in mathematics to be in the average range. Mathematica questions in rea-word contexts were set to assess not ony mathematica skis and knowedge but aso probem-soving strategies. The evidence from OECD countries shows that the greater the socia incusion in schoos the better the pupi performance in mathematics overa. Ireand s performance in the OECD PISA study (Programme for Internationa Student Assessment) raises issues which warrant further discussion in reation to the approach to teaching mathematics and the content of the curricuum at both primary and second eve. The primary maths curricuum of 1999 has incorporated a number of concerns that were identified by the INTO in its previous research on mathematics, and which were raised at the consutative conference on education in It is heartening to know that many of the recommendations the Education Committee made in 1990, particuary in reation to probem soving, overoading of content, estimation, menta and ora maths and use of mathematica games, have been incorporated into the revised curricuum. Initia responses from teachers to the revised curricuum are incuded in this report. Both the Department of Education and Science and the Nationa Counci for Curricuum and Assessment carried out evauations and reviews of the impementation of the mathematics curricuum in primary schoos during the schoo year The quaity of the mathematics curricuum was found to be good in the majority of schoos. However, a number of issues were identified which require addressing if the teaching of mathematics is to improve further. The INTO cas for additiona materias and resources and enhanced opportunities for professiona deveopment for teachers as key eements in supporting mathematics teaching in primary schoos. 1

4 Maths in the Primary Schoo I woud ike to acknowedge the work of the INTO Education Committee in preparing this report. It is a further contribution to education research carried out by teachers. The Committee carried out quantitative research, by issuing questionnaires to members seeking their views on the revised Maths curricuum and I woud ike to thank a the members who competed the questionnaire, which informs this report. I woud aso ike to take this opportunity to thank members of the Head Office team who prepared the report for pubication, under the genera direction of Deirbhie Nic Craith, Senior Officia. John Carr, MA (Ed) Genera Secretary February

5 1 Mathematics in the Primary Schoo Discussion Document circuated for the INTO Consutative Conference on Education, Kikenny, INTRODUCTION There is a genera acceptance of the importance of mathematics education. It is perceived as a core area of earning in probaby most educationa systems throughout the word. Indeed many internationa studies have sought to compare achievement in mathematics between various countries. Its incusion as a curricuar area in schoos refects its vaue in providing pupis with knowedge, skis and procedures which are necessary toos in understanding the physica environment and in exporing patterns and reationships. Mathematics, though enjoyabe and vaid in its own right, is aso reevant to earning in many other curricuar areas. The INTO, in its report on Mathematics in the Primary Schoo (1990 ) highighted a number of areas where changes were required in the primary curricuum. Many of the Education Committee s recommendations, particuary in reation to probem soving, overoading of content, estimation, menta and ora maths and the use of mathematica games have been incorporated into the revised mathematics programme in the Primary Curricuum (1999 ). The proposa to deveop a continuous curricuum for the compusory schoo years, as is common in most European countries, has not materiaised, as primary and post-primary curricua tend to be deveoped separatey, an issue which merits addressing by the Nationa Counci for Curricuum and Assessment (NCCA). The INTO argued strongy that the needs of pupis in compusory schooing shoud not be subordinated by assessment. It was aso 3

6 Maths in the Primary Schoo argued that the mathematics curricuum shoud not be designed for pupis who woud need appied mathematics for future careers in engineering, accountancy or the sciences, but shoud instead focus on pupis needs in preparation for ife as aduts capabe of deaing with practica mathematics in rea-ife situations (INTO, 1990, p.64). In addition to its economic utiity and intrinsic socia vaue, mathematics education is an inteectua pursuit in its own right, a source of fascination, chaenge and enjoyment, (DES, 1999, p.3) and this is refected in the current mathematics curricuum in primary schoos. It is too eary to assess fuy the impact of the revised mathematics curricuum. Teachers were given an opportunity during the schoo year , to review their impementation of the mathematics curricuum, in addition to the Engish and Visua Arts curricuum. The NCCA, in order to assist schoos with this process, designed review tempates. A number of schoos were invited to return their tempates to the NCCA for incusion in a genera review and evauation of the curricuum. The NCCA has compied a report arising from the curricuum review process. The Inspectorate, as part of their ongoing work on schoo evauation, aso considered the impementation of the mathematics curricuum 1. The INTO carried out a survey of members to ascertain their views in reation to the current mathematics programme, and these findings are incuded in this report. Mathematics is recognised as one of the sciences and is used in everyday ife. Achievement in mathematics is reguary monitored by both teachers in cassrooms as an integra part of the teaching earning process and by the Department of Education and Science (DES), who on behaf of the State, gathers information on the genera performance of the educationa system. Ireand, therefore, has participated in many internationa assessments of mathematics earning, in addition to nationa assessments. A brief overview of Ireand s achievements are incuded in this report, highighting the areas in which Irish pupis achieve we and the areas where Irish pupis achieve poory by internationa comparison. Information from such assessments is usefu in guiding future curricuum deveopment in mathematics education and in informing the needs of teacher education, both initia teacher education and ongoing professiona deveopment. MATHEMATICS IN THE PRIMARY CURRICULUM The Primary Schoo Curricuum (1999 ) contains a number of revisions to the mathematics curricuum of The impementation process for the revised Primary Schoo Mathematics Curricuum began in September 2002 foowing an inservice 1 The Department s report An Evauation of Curricuum Impementation in Primary Schoos was pubished in May

7 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 programme provided by the Primary Curricuum Support Programme (PCSP). The aims of this curricuum are as foows: To deveop a positive attitude towards mathematics and an appreciation of both its practica and its aesthetic aspects. To deveop probem-soving abiities and a faciity for the appication of mathematics to everyday ife. To enabe the chid to use mathematica anguage effectivey and accuratey. To enabe the chid to acquire an understanding of mathematica concepts and processes to his/her appropriate eve of deveopment and abiity. To enabe the chid to acquire proficiency in fundamenta mathematica skis and in recaing basic number facts. (Primary Schoo Curricuum, 1999, p.12) Mathematics shoud be portrayed to chidren as being practica and reevant in their everyday ives. It shoud be integrated with other areas of the curricuum, such as socia, environmenta and scientific education (SESE), music and physica education. Whenever possibe it shoud be inked to the chidren s environment and their own experiences. Through the mathematics curricuum, chidren are being prepared to dea effectivey with the varied transactions of everyday ife and to make sense of the mass of information and data avaiabe through the media. It is essentia, therefore, that chidren see mathematics as reevant to their own ives. As stated in the teacher guideines for mathematics, the curricuum wi be a key factor in preparing chidren to meet the demands of the 21st century (p. 2). Chidren shoud enjoy mathematics and be catered for according to their abiity. They shoud ook forward to the chaenges of mathematics earning and be fuy equipped to face them with confidence and enthusiasm in order to experience the satisfaction of a job we done. There is a great emphasis on chidren being the instruments of their own earning. The constructivist approach is centra to the mathematics programme, where chidren must construct their own interna structures. They are encouraged to deveop their own mathematica strategies for soving probems by using their knowedge of one area to expore another and enhance their growth of reasoning. Access to concrete materias is considered necessary for pupis at a cass eves from infants through to sixth cass. Language aso pays an important roe. Chidren need to deveop the abiity to isten, question and discuss as we as to read and record. Discussion can be in pairs, groups or among the cass as a whoe. The teacher suppies mathematica anguage when necessary to enabe chidren to buid up an appropriate mathematica vocabuary. Concepts need to be adequatey deveoped oray before chidren record them in writing, using symbos and mathematica expressions. 5

8 Maths in the Primary Schoo STRUCTURE OF THE CURRICULUM The curricuum comprises five strands: Number. Agebra. Shape and space. Measures. Data. The strands form a network of reated and interdependent units which are further deveoped as strand units. An overview of the strands, and the main differences introduced to the curricuum in 1999 are outined beow. Number A ceiing has been paced on number work to aow for more extensive treatment of the mathematics programme as a whoe and to aow more time for concept deveopment. It paces ess emphasis than heretofore on ong, compex pen-and-paper cacuations and a greater emphasis on menta cacuations, estimation and probem-soving skis. Recording can be concrete, ora, pictoria or diagrammatic, or can incude mode-making. Work on fractions and decimas, in genera, wi pace more emphasis on understanding the reationships between them. Agebra Agebra has aways been part of the curricuum. It incudes patterns, sequences and statements such as 2 + _ = 5. It is, now, formay recognised at a eves. Positive and negative numbers are introduced in senior casses. It is expected that numbers used wi be kept sma so that the chidren can understand the concepts presented. Shape and space This strand expores spatia awareness and its appication to rea-ife situations. It is particuary suited to integration. Measures This strand has six strand units ength, area, weight, capacity, time and money. Probems shoud be mainy practica with the totas easiy verified by measuring. As with the previous curricuum chidren shoud be taught from an eary age to estimate. Chidren wi need to hande materias, investigate them and then to discuss and 6

9 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 compare their findings. The reasons for using standard measuring instruments shoud be expored in a practica way. Data Graphica representation and interpretation has aways been part of the curricuum, but data handing is, now, a separate strand. Interpreting and understanding visua representation is essentia, as the chid needs to be enabed to interpret data in an increasingy technoogica word. It is hoped that, where avaiabe, information technoogy wi be used by chidren in data-handing exercises. Chidren must understand how important it is to enter reevant data and ask cear questions if the information to be extracted from the database is to be of any use. The concept of chance is of great importance. It represents rea-ife mathematics and promotes thinking and discussion. Topics can be introduced through probems, practica experiments and simuations that hep to deveop intuitive foundations for future work and are fun for the chid. Linkage Strands can be taught in parae rather than one after the other. Chidren wi use their knowedge of one area to expore another. This faciitates the use of number throughout the mathematics curricuum. Cacuators Cacuators have been introduced to schoos for the first time from fourth to sixth casses. They hep the deveopment of probem-soving skis by aowing the chid to focus on the structure of a probem and possibe means of soution. They can be used to check estimates, to perform ong and compex computations and to provide exact resuts to difficut probems. However, the cacuator cannot be a substitute for practica activity with materias. Assessment Assessment shoud provide information that wi enabe the teacher to cater for individua differences in abiity, to assess previous earning, to address earning styes, and to resist pressure to push the chid to premature mechanica mastery of computationa facts and procedures. 7

10 Maths in the Primary Schoo ALTERNATIVE APPROACHES TO TEACHING MATHEMATICS In addition to traditiona approaches to the teaching of mathematics, a variety of aternative approaches are used by teachers to support the teaching and earning of mathematics a seection of which are outined beow. The Mathematics Recovery Programme The Mathematics Recovery Programme draws on research into the earning and teaching of eary mathematics undertaken by Professor Lesie Steffe and his coeagues at the University of Georgia in the US and by Professor Bob Wright who worked with him as a doctora student. Research has shown that there are aready vast differences in the mathematica knowedge of students when they begin schoo initiay (Aubrey, 1993 ; Young-Loveridge, 1989 ; Wright, 1991, 1994 ). This gap tends to remain and increase throughout their schooing. Strong negative attitudes to mathematics can deveop aong the way. The Mathematics Recovery Programme is directed towards eary intervention. ASSESSMENT Chidren are assessed by way of an Interview Schedue. This is an ora assessment seeking to find out what the chid knows, what strategies, if any, s/he is using and getting to the core of the concept probem. The assessment is concerned, primariy, with number, addressing the foowing topics faciity with number words and number word sequences; abiity to recognise, identify and write numeras; emerging strategies for adding and subtraction; knowedge of the tens and units aspect of the numeration system; emerging methods of notating in arithmetic; and abiity to ascribe number to spatia and tempora patterns. FOCUS AND KEY FEATURES OF PROGRAMME The focus of the programme is to identify ow-attaining students at an eary stage and, subsequenty, to provide a programme of intensive, individuaised teaching. The aim of the programme is to raise the student to a eve where s/he can return and earn successfuy in the mainstream cass. Instruction takes pace on a one-to-one basis and in conjunction with the cass teacher. 8

11 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 The Interview Schedue (ie, the assessment that is done) is used to deveop an individua teaching framework for each student invoved in the Mathematics Recovery Programme. The programme deas with eary number - counting on/back, counting in 2s, 3s and so on, counting in tens for pace vaue, visuaising and manipuating number. The instruction is probem-based and from a constructivist perspective. It is mainy ora extending the chid s current thinking. It shoud be chaenging but with a good possibiity of success, thus, increasing sef-esteem. Chidren are encouraged to refect on their own mathematica thinking. The programme invoves teaching cyces of weeks duration. Students are taught for 30 minutes daiy, four or five days a week. In Ireand, at present, earning support and specia needs resource teachers are using the programme. Those participating in the programme are finding it very successfu. There is great demand for the imited number of in-service paces avaiabe. Piot projects, supported by the Department of Education and Science, are panned for a number of schoos in Limerick. Regrouping for Mathematics Regrouping is a form of grouping where chidren remain in their norma teaching unit for most of the day and are assigned to groups, according to abiity, for certain subjects eg, mathematics. Assignment to groups is made on the basis of actua performance in the subject and there is mobiity between eves so that students can move from one to another according to their achievements. Instruction is adapted to the specific needs of the students. There are many advantages to grouping chidren according to abiity. For exampe, there is a greater chaenge to brighter students as they can progress faster and chidren with difficuties in earning mathematics wi gain more confidence and perform better if the cass work is aimed at their eve. In addition, teachers may find it easier and more productive working with groups of chidren of simiar abiity. There are various forms of grouping mixed-abiity grouping where chidren of different abiities are paced in the one cass and streaming, where chidren are paced in casses according to abiity. With regrouping chidren spend most of the day in a mixed-abiity setting and are regrouped for certain subjects such as mathematics. Therefore, the origina cass remains the primary reference group. Lyons (1999 ) in her research on regrouping sought to ascertain whether a regrouping system for mathematics at fifth and sixth cass eve contributed to the deveopment of a more positive attitude towards mathematics and an increased eve of confidence in approaching study of the subject. For the purpose of her research she chose an a-girs, argey midde-cass, convent schoo situated in an expanding urban area. There were three casses of each eve from second cass upwards. The 9

12 Maths in the Primary Schoo system of regrouping pupis of both fifth and sixth casses for mathematics had been in operation in the schoo for the previous five years. Teachers of the fifth and sixth casses moved from one maths group to another each term. Pupis were not tod the abiity eve to which they were assigned athough it became cear in the course of the study that they succeeded in working that out for themseves. The three mixed-abiity casses in both fifth and sixth were regrouped according to abiity TOP MIDDLE WEAK for the teaching of mathematics. For the purpose of the investigation pupis were chosen at random from each eve and both pupis and their parents were interviewed. RESULTS OF INVESTIGATION Amost a pupis were happy with their pacement and understood the importance of abiity as a criterion for the formation of the groups, and that they were being regrouped in an effort to assist them. The majority of the students understood that it was possibe to change groups. A minority in the weak-abiity group did not seem to be aware of this, possiby, because fexibiity ony existed in this particuar study between high-abiity and medium-abiity groups. According to Lyons, it is very important that those in the weak abiity group are aware of the possibiity of upward mobiity. The majority of pupis and parents from sixth cass were quite happy changing teachers for instruction during the year, whereas, on the whoe, they were unhappy with this arrangement in fifth cass. There were more chidren in sixth cass than in fifth cass who rated mathematics among their favourite subjects, eading to the concusion that the overa attitude towards the subject was becoming more positive since regrouping. Haf of fifth cass compared to 87.5% of sixth considered the work easier than the previous year. Most of sixth cass pupis commented that earning mathematics was more enjoyabe and over haf of fifth cass reported an improved attitude. These comments were generay supported by the parents of both casses. Pupis and parents, aike, from both fifth and sixth casses were agreed on the positive effects of regrouping on the nighty ritua of homework. This was most evident in the weaker abiity pupis. It was thought that the reativey new feeings of success for weaker abiity students woud, hopefuy, hep to break the cyce of repeated faiure and ead to an increase in confidence and improved attitude towards maths. The majority of pupis and parents from both fifth and sixth casses considered that their confidence in approaching mathematics had increased and no chid reported a reduction in confidence eves. Pupis in the midde and weaker groups were more sensitive to comments made about pacements. Those in fifth cass were more affected than those in sixth cass. It was perceived by the author that by the time pupis reached sixth cass, having been in the system for a year, they generay tended to accept the process as the norm and, 1 0

13 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 therefore, major discussion concerning it ceased. The majority of the parents and pupis of both fifth and sixth expressed a preference for continuing with the practice of regrouping. A regrouping system in any schoo woud need to be continuousy monitored and evauated. The focus of the system is on the deveopment of confidence and the growth of a more positive attitude towards the subject. The correct assignment of pupis to the different groups is essentia, and such assignments must be fexibe. There is a need aso to outine the phiosophy underying the system to both pupis and their parents when introduced initiay. According to Lyons a regrouping programme may offer pupis the support to deveop mathematica skis at their own pace, to experience success where they had previousy known ony faiure and to work at an appropriate individua eve. Parenta Invovement in Maths in Primary Schoo (A) SHARED MATHS / MATHS FOR FUN Shared Maths/Maths for Fun is a teaching strategy where parents are invoved in the cassroom. The chidren in the cass are divided into groups. One parent works with each group for a period of 40 minutes. The equipment is stored in crates, baskets or boxes. The groups move from one game/activity to another. A timer is set to buzz after 10 minutes signaing the end. Chidren tidy up before moving on to the next game/activity. To encourage ease of movement and a ow noise eve Merit Stickers may be given to groups who eave everything in its pace and move quiety from tabe to tabe. The group with the highest number of stickers (every group!) receive a Maths Bookmark at the end of the session, which asts six weeks. Suggested games and activities incude: Tangrams; Reationa Attribute Bocks; Pattern Bocks; Pentominoes; BINGO; Bank Baance; Snakes And Ladders; Dienes Bocks; My Computer and Time Games. See Appendix 1 for a description. (B) PAIRED MATHS Paired Maths is where parents are invoved in paying mathematicay-based games with their chidren at home. The parents of the chidren invoved attend a schoobased meeting. They are given instruction on how the system works. Each chid in the cass wi be given a game to take home for a period of one week. The whoe process asts six weeks. 1 1

14 Maths in the Primary Schoo ASSESSMENTS OF MATHEMATICAL ACHIEVEMENT Irish primary schoos have participated in a sma number of nationa and internationa assessments of mathematics achievement since the introduction of Curacam na Bunscoie (DES 1971 ). Ireand has aso participated in internationa assessments at second-eve. This section of the report highights the main findings of both nationa and internationa assessment in reation to mathematics, in addition to giving an overview of the main findings of a recent study on mathematics teaching and earning at junior cyce at second eve. The atter study raises some issues of reevance regarding the approach to mathematics teaching at primary eve. Nationa Assessments The Department of Education tested a nationa sampe of second and fourth casses in 1977 and sixth casses in Sixth casses were retested in 1984 and reports were issued by the DES in 1977, 1980 and The tests for second and fourth casses incuded items in seven content areas: Operations with Whoe Numbers, Whoe Number Structure, Measurement, Fractions and Decimas, Geometry, Graphs and Probems. At second cass eve, 85% achieved mastery in Operations with Whoe Numbers and 55% to 65% achieved mastery over six content areas. Items which caused most probems at second cass reated to subtraction and the commutative and distributive properties of addition. Girs sighty outperformed boys in a areas with a marked difference on a few items. At fourth cass, mastery eves were ower across the board with 75% mastery at Operations with Whoe Numbers and a 40% - 60% range in the other areas. Items of Unitary method, adding and subtracting decimas, interpreting graphs and timetabes, probems, reating fractions and decimas, symmetry, perimeter and ong division a achieved ess than 50% mastery. Again girs outperformed boys in most areas of the tests. The achievement tests for sixth casses covered 10 content areas in mathematics: Operations with Whoe Numbers, Whoe Number Structures, Fractiona Number Structure, Operations with Fractions, Decimas and Percentages, Metric Measure, Agebra, Geometry, Charts and Graphs and Probems. Geometry, Measurement, Probem Soving, Whoe Number Structure and Agebra were the areas of owest mastery and boys, on average, outperformed girs. In 1999, the Educationa Research Centre (ERC) tested nationa mathematica achievement in 4th casses in 5 areas: Number, Agebra, Shape and Space, Measures and Data. These correspond to the strand units in the Revised Primary Schoo Curricuum (NCCA 1999 ). Pupis performed best on Data (69% correct), Number (60%) and Agebra (58%) and poorest on Measures (54%) and Shape and Space 1 2

15 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 (46%). No significant gender differences arose but a much higher proportion of boys scored at or beow the 10th percentie. Internationa Assessments Ireand took part in a number of internationa surveys of mathematica achievement since the introduction of the 1971 Primary Schoo Curricuum. The Second Internationa Mathematics Study (SIMS) in targeted 13 year ods, some in sixth cass and some in first year, at post-primary eve. The Internationa Assessment of Educationa Progress (IAEP) conducted studies of 13 year ods in 1988 and both nine year ods and 13 year ods in The Third Internationa Maths and Science Study (TIMSS) took pace in 1995 and targeted third and fourth casses in primary schoos and first and second year pupis in post-primary schoos. IAEP I (1988) IAEP I (1988 ) carried out tests in five countries: Ireand, UK, USA, Korea, Spain and seven Canadian Provinces, in six content areas: Number and Operations; Agebra; Reation and Functions; Geometry; Measurement; Data Organisation and Interpretation and Logic and Probem Soving. Ireand, UK, Spain and French Canadians performed at the mean eve of a participants. Engish Canadians performed above the mean, whie Korea achieved we above the mean. The USA performed beow the mean. Ireand scored particuary bady in Data Organisation and Interpretation and reativey bady in Geometry and Measurement. The remaining three areas showed a much stronger performance. No significant gender differences were reported. However, a correation between a greater amount of time spent watching teevision and a ower maths achievement score was noted. IAEP II (1991) IAEP II (1991 ) tested 20 countries for 13 year ods and 14 countries for nine year ods. The internationa average for 13 year ods was 58% and Ireand scored 61%. At nine year od eve, the average score was 63% and Ireand s mean score was 60%. IAEP II covered five content areas: Number and Operations; Measurement; Geometry; Data Anaysis; Statistics and Probabiity; and Agebra and Functions. Ireand scored significanty beow the mean in both geometry and measurement and at about the mean in the other areas. Girs scored higher than boys in third and fourth casses but not significanty higher. TIMSS (1995) The TIMSS (1995 ) incuded 45 countries over two age groups nine year ods and 1 3

16 Maths in the Primary Schoo 13 year ods. The average score for the oder age group was 484 for first year students and 513 for second year students. The Irish pupis scored 500 for first year students and 527 for second year students. In third cass the internationa average was 470 and Irish pupis scored 476. In fourth cass Ireand scored 550 over a mean score of 529. The TIMSS (1995 ) tested six content areas: Whoe Numbers; Fractions and Proportionaity, Measure, Estimation and Number Sense, Data Representation, Anaysis and Probabiity, Geometry and Patterns, Reationships and Functions. Pupis in 4th cass scored at the mean in two areas: Measure, Estimation and Number Sense and Geometry. They scored significanty above the mean in the other four areas. No significant differences showed up between boys and girs. Fourth cass pupis reporting more than 100 books at home scored significanty higher than those reporting ess than 25 books and fourth cass pupis reporting a positive attitude to maths scored significanty higher than those who did not. Irish pupis in smaer casses scored higher than those in arger casses but not significant so. PISA 2000 The OECD Programme for Internationa Student Assessment (PISA) assesses the knowedge and skis achieved by students near the end of compusory schooing (ie, 15 year ods). In PISA 2000 Ireand scored just above the country average of 500 (502.9). Boys scored significanty higher than girs at this age which correated with that of most other countries tested. The number of books in the home was noted as an indicator of success in maths achievement. PISA 2003 PISA 2003 focused on mathematics and two reports were pubished Learning for Tomorrow s Word and Probem Soving for Tomorrow s Word. Mathematics was assessed in four domains, Shape and Space, Change and Reationships, Quantity and Uncertainty. Mathematica questions in rea-word contexts were set to assess not ony mathematica skis and knowedge but aso probem-soving strategies. Ireand finished within the OECD average range and 17th out of the 29 countries taking part. In the domains assessed, Ireand was significanty above the average in both Uncertainty and Change and Reationships; was on the average for Quantity, and beow average for Shape and Space. Maes outscored femaes in a four domains of mathematics assessed but the overa difference was not arge except in Shape and Space which was significant. These resuts seem at variance with Junior Certificate resuts in Ireand where femaes consistenty outperform mae students. The types of rea-word probems posed in the PISA 2003 survey may provide the answer to this variance in gender resuts. Pupi opinions 1 4

17 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 amongst Irish students showed a more positive attitude to mathematics as a factor for improving education and ife skis than the OECD average. In reation to variation between both students and schoos, PISA found that Ireand had one of the owest ranges of student performance variation amongst OECD countries. Low achievers in Ireand performed stronger than many OECD countries but conversey high achievers did not perform as we as their counterparts. PISA aso reported that Ireand is one of the few countries with itte performance differences between schoos. This prompted the remark that parents can be confident of high and consistent performance standards across schoos in the entire education system. The variation of performance between schoos and between students, athough sma, was best expained by socio-economic factors. The initia recommendations of the PISA 2003 Report reate to the education system and schoos rather than pupis. The evidence from OECD countries shows that the greater the socia incusion in schoos the better the pupi performance in mathematics overa. The first recommendation, therefore, is to deveop socia incusion in schoos across the education system. Since there is itte variation in pupi performance between schoos, the second recommendation is that poicies aimed at improving the performance of ow achievers are ikey to be the most effective. PISA 2003 aso incuded a section on cross-curricuar probem soving which again paced Ireand within the OECD average simiar to that achieved in genera maths performance. A sma portion of PISA 2003 was aso given over to reading and science. First indicators were that Ireand was we above the OECD average in reading with ony three countries scoring higher than Ireand. In science, Ireand was again we above the OECD average but sti some way behind the highest rated countries. PISA 2006 wi be a major assessment of science in the OECD. The fu data for PISA 2003 remains to be competey interpreted and further reports wi be produced in due course. Summary Ireand has consistenty scored at about or just above the mean in the internationa studies in which it has participated. Items reating to number and data have been the most successfu, whie those reating to geometry, measurement and agebra have been ess so. Litte or no difference between the performance of boys and girs was detected except at 15 year od eve (PISA 2000 ). Pupis attitude to mathematics and socio-economic backgrounds seems to have some significance reating to mathematica achievement. 1 5

18 Maths in the Primary Schoo JUNIOR CYCLE MATHEMATICS IN POST PRIMARY SCHOOLS The Gender Equaity Committee of the Department of Education and Science initiated and funded a study reated to Junior Certificate Maths. The study is based on Exam Resuts ( ) and case studies of ten different second-eve schoos around the country. With the co-operation of teachers and students an intensive video study of twenty mathematics essons and six Engish essons invoving second year students was undertaken. The video studies were compemented by interviews with students, teachers and parents in order to examine the reationship between teaching practices and attitudes to earning. The study was designed to expore co-educationa and singe-sex schoos across different types of schoo background to gain understanding about pedagogica styes and priorities and their impact on the teaching and earning of mathematics. The impact of gender and socia cass on outcomes was a reated objective. The study outined two epistemoogica approaches to mathematics teaching. These contrasting perspectives are the absoutist and the reativist. The former is objective, consistent and knowedge based and favours a didactic approach to transmission. The atter is based on interaction between individuas, society and knowedge and is cuturay situated. This ends itsef more to probem based and constructivist earning. Second-eve teachers, who are preparing students for pubic examinations, are incined to favour didactic teaching where it is known to be rewarded with good examination resuts. In preparation for the main video study, an anaysis of Junior Certificate Mathematics examination resuts for was carried out. Students have a choice between three eves: Foundation, Ordinary and Higher. The uptake of each eve is approximatey in the ratio 1:3:2respectivey. Mathematics has a ow take-up at the Higher eve (36%) in comparison to Irish (40%) and Engish (61%). It is conjectured that this may reate to a static syabus that is perceived to be difficut. Gender differences vary widey with regard to type of schoo and socia background. Girs in genera tend to reject maths or under perform in maths more and this is more pronounced in disadvantaged schoos. However, it is shown that this cannot be expained by teaching or schoo-specific variabes aone. The wider socio-cutura context where gender identities are created and reinforced is the prevaiing basis for gender differences in both participation and performance. However, the number of girs taking Higher eve papers has risen significanty over the ast fifteen years. Where major differences exist in performance they are, increasingy, inked to socia cass background rather than gender. It woud appear that differences in take-up rates of different eves of mathematics between co-educationa and singe-sex schoos is 1 6

19 Discussioon Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 not so much reated to their coeducationa status as it is to the socia cass composition of their schoo popuation and the tradition of the schoo. The schoos with the most disadvantaged students are the ones in which there is the highest take-up of Foundation and Ordinary eves. The most sociay seective schoos, fee-paying secondary, have the highest take-up rates at Higher eve. The anaysis aso suggests that teacher expectations and perspectives on students were infuenced by the students socia cass and background. The track, set, stream or band into which a student is paced infuences his/her experience of earning mathematics. Top tracks experience a more intense, work-focused and competitive earning environment than the ower tracks. Review of Case Studies There was a high eve of uniformity in how the essons were organised and presented. The didactic approach, teacher demonstration and student practice, was the preferred choice of teachers. Learning was most often equated with the memorisation of formuae and procedures. There was itte time devoted to probem-soving, to the practica appication of mathematics in the physica word or to aternative methods of soving probems. Teachers were more ikey to use ower-eve and dri questioning with the emphasis on giving the right answer. It was cear from the video that students were not encouraged to ask questions. Interaction was for the most part teacher-initiated with ony 4% student-initiated. In genera, boys received more teacher attention than girs did. The casses that were most gender-baanced were those in which the teachers were gender aware. There was a tendency for a sma number of boys to dominate casses in both coeducationa and in singe-sex boys schoos. Praise was imited, athough girs in a types of casses received somewhat more praise than boys did. Teachers Perspectives Teachers, generay, attributed students improvements to having an innate abiity and to being encouraged and supported by the teacher. They did not hod themseves responsibe for any observed deterioration in students performance. Students own attitudes, behaviour or ack of abiity were deemed to be the main reasons. A of the teachers adhered to the essentiaist view about mathematica abiity ie, that some students have a natura taent for the subject whie others do not. Six of the ten teachers caimed that students from poor backgrounds were disadvantaged in earning mathematics by their parents ack of knowedge of, and especiay interest in, education. Most of the teachers assigned homework in each esson and corrected it in the next. A number of teachers observed that certain students entered second eve schoos educationay disadvantaged in mathematics, raising questions as to why this 1 7

20 Maths in the Primary Schoo was the case, and why the probems had not been addressed at primary eve. Student Perspectives For pupis, mathematics, as a subject, seemed to be defined and interpreted in terms of the person who taught it. They rejected the essentiaist view of mathematics (innate abiity). Pupis were more ikey to state that what was required for success in schoo mathematics was having a good teacher and studying at home. They aso thought that good memorisation was important but not necessariy earning the text book by heart. They were quite positive about the vaue and importance of mathematics required for everyday ife, for empoyment and for further education purposes. They recognised it as an important subject that had both a short and ong term vaue. However, the study aso highighted that students were reuctant, even fearfu, to ask for hep in mathematics cass. In discussing their experience of earning mathematics throughout their primary and post-primary education, students said they, generay, preferred to ask their parents and/or other sibings for hep at home or, aternativey, to ask one of their friends or cassmates. Students spoke about finding it unnerving when questioned in cass and feeing under pressure to get it right. For students, teacher attitudes appear to have a big bearing on outcomes. Students reated their attitudes to mathematics in terms of their teacher. If the teacher was perceived as positive and supportive, they were abeed good. Teachers were criticised for being negative or going too fast. Parent Perspectives Parents from a types of socia backgrounds hed the essentiaist view on chidren s innate abiity for successfu earning in mathematics. Whie very few parents activey avoided co-educationa schoos there were those who activey chose singe-sex schoos as these were reported to be strong academicay and/or were prestigious because of their sociay seective intake. Parenta expectations were a significant infuence on student/teacher attitudes and performance. Possibe Impications for Primary Schoo MathsTeaching As evidenced from the above study, didactic teaching is the norm at second eve. However, whie constructivist teaching is the basis of the revised primary curricuum (1999 ) and was aso impicit in the 1971 curricuum, there is sti a ot of didactic teaching at primary eve. If primary teaching in the future evoves to a constructivist approach as is envisaged, the students shoud be transferring to second eve with greater knowedge and more positive attitudes to mathematics. However, they may 1 8

21 Discussion Document Circuated for the INTO Consutative Conference on Education, Kikenny 2004 we be ess suitaby equipped to cope with a didactic approach, if such practices remain at second eve. Students are often streamed at entry to second-eve. Students who eave primary with ow attainment eves, irrespective of abiity, wi be unabe to enter the higher streams at second eve. This raises the question of the quaity of mathematics education at primary eve. Disadvantaged schoos and disadvantaged students are being discriminated against both sociay and educationay. In second eve schoos where streaming takes pace, there is no rea mechanism for the disadvantaged pupi to change streams. This paces a burden on primary eve to bring chidren to a sufficienty high eve to give them the option to enter the highest stream on entry to second eve. Gender differences are sma in comparison to socia differences. Parenta vaues and student and teacher attitudes have a major effect on mathematics achievements. Attitudes to mathematics and resuting teaching styes and practices at primary eve need to be taiored to deveop support and positiveness towards mathematics and to reduce pupi anxiety. Parents aso need to be informed about how the system works as we as encouraged to deveop positive attitudes to mathematics and education in genera. Student attitudes appear to be very dependent on their concept of what a good teacher is, in that teaching styes that incorporate positive reinforcement, supportive and steady progress through topics and an emphasis on anxiety-free environments need to be encouraged for maximum effect. Teachers aso need to be aware of gender differences between boys and girs and aso between dominant and non-dominant boys in making casses more incusive for a students. 1 9

22 Maths in the Primary Schoo 2 0

23 2 Resuts of INTO Survey on Mathematics (2004) INTRODUCTION The Education Committee of INTO conducted a nationa survey on the teaching of mathematics in the primary schoo in March A tota of 505 (51%) questionnaires were returned and processed. Where responses were made by a or most participants, resuts are given in percentages. Questions that required opinions or a variety of responses are reported by giving the actua number of teacher repies. It was decided to incude some opinions that were expressed by reativey few respondents since they provide additiona information and iustrate practice in some instances. Schoo Detais Surveys were returned from 505 teachers of whom 20% taught infant casses, 15% taught junior casses, 20% were in midde casses and 24% were in senior casses. The remaining 20% taught in muticasses. Respondents reported that 23% were in one to four teacher schoos, 44% were in five to 16 teacher schoos and 33% were in arger schoos. Describing their ocation, 32% of teachers said they were in rura schoos, whie 35% were in sma towns and 29% were in city schoos. Three-quarters of teachers repying taught in co-educationa schoos, 13% taught in boys schoos and 10% taught in girs schoos. One-quarter of respondents were working in schoos designated disadvantaged and 6% were teaching through the medium of Irish. Teachers reported a variety of cass sizes. One-tenth of teachers had casses of up to 15 pupis, one-fifth had pupis, one-quarter had pupis, one-third had pupis and one-eighth taught casses of over 30 pupis. Learning Support Regarding the provision of Learning Support in Mathematics (LSM), 58% of teachers 2 1

24 Maths in the Primary Schoo reported that their schoos provided LSM, whie 39% teachers reported that they did not. However, reativey itte time out of tota earning support time is aocated to maths by most of those schoos providing LSM. In tota 184 (36%) teachers reported chidren in their casses receiving LSM, as foows: Tabe 1: Numbers of Pupis per Cass receiving LSM/Number of Teachers with Pupis receiving LSM Number of Pupis in Cass receiving Learning Support in Maths Number of teachers who reported pupis in their cass receiving earning support in Maths 1 pupi 34 teachers 2 pupis 42 teachers 3 pupis 35 teachers 4 pupis 29 teachers 5 pupis 21 teachers 6 pupis 13 teachers More than 6 pupis 10 teachers When respondents were asked how many shoud be receiving LSM, they reported as foows: 48% fet up to 4 pupis needed LSM, 31% reported five to eight pupis in need and 6% fet even more pupis required assistance. A tota of 218 (43%) teachers reported that chidren received support in maths from the specia needs resource teacher. Eighty six teachers reported one such pupi, 63 teachers had two pupis, 31 had three pupis, 20 had four pupis and 17 had more than four pupis receiving such hep. Detais of Respondents Four-fifths of respondents were femae and one-fifth of respondents were mae, which corresponds with the genera representation of mae and femae teachers in the profession. Ony 16% of respondents had mathematics as a degree subject whie 82% reported that they had not. Regarding the number of years spent teaching, 31% had ess than 10 years teaching experience, 23% had years experience, 29% had years experience and 18% had more than 30 years experience as a teacher. Professiona Deveopment /Inservice Apart from PCSP inservice seminars in Mathematics Teaching, 30% had attended 2 2

25 Resuts of INTO Survey on Mathematics, 2004 INTO Maths courses and 22% had attended Education Centre Maths courses. A sma number had attended other Maths courses, whie 25% reported not having attended any other inservice courses. Three-fifths of teachers reported that their schoo had avaied of the cuiditheoireacht service for Mathematics, whie one-third had not. TEACHING AND LEARNING MATHEMATICS Mathematica Activities Teachers were asked how often they used six different mathematica activities. The resuts are outined in Tabe 2 beow. There was ibera use of estimation, menta maths, memorisation and probem soving, especiay in the oder casses. However, ess than one-fifth made much use of aternative agorithms or computerised procedures, and this was mosty in the senior casses. Tabe 2: How often teachers used six different mathematica activities Methodoogies Very Often Often Sometimes Never Aternative 4% 16% 27.6% 33.0% agorithms Computerised 4% 12% 41.8% 38.8% procedures Estimation 25% 48% 21.0% 0.8% Memorisation of 33% 37% 20.0% 7.0% facts/formuae Menta maths 48% 40% 10.6% 1.4% Probem soving 35% 51% 12.2% 1.4% Teaching Methodoogies Teachers indicated how often they used eight particuar teaching methodoogies in mathematics. The resuts are outined in Tabe 3 beow. The environment was reguary used by ess than one-fifth of respondents but its most frequent use was in infant casses. Active and coaborative earning, tak and discussion, ora computation and inkage within the maths programme were very frequenty used by teachers. Integration and investigation were aso used frequenty. Integration was practised twice as often in infant casses as in more senior casses. As expected, eary earning activities 2 3