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vC`;a YSZQO{;qԽ/30szغIn13A5C3q,fّK7y0gѧykvve4#hlkp%VSt=pl4c;Ç1}ޙǤ==v);:yhvZfNױ#c;z}W;DV,`oS3a>p̓3lq@˴wL;d 흑Țs;wo1if#;9Q?[x\9+ws ;%vN?1-_I;(q;lJTgsu6{QLli.i87J\ߎ):SJH$ MΎ9Pe\iOwi3ǝvNaO!Z;yu~l'dQ 6cv\divdlGvQ`G;O=#;%g3\ui/2%ɠp;g=}YmǟoNwL.1OH;Fg"! ĩ)шNN64M}C0&isK;wnȫI! vniNع;bv8`;i;!ꫥs9\Q ; oZZly;ع@`;vݰ/I'ۡ;!Ne7Kbvh<A\+C!'!v;A` v;A!v; C!v;A` C!A` C!;A` Cv7QΧμ/DnvJ ']:3{`'})S'՜o@fR(\3zRr9s5(Ǻ`q;W Q9;F;;˛cGk;Untc2aMn&'4Iک'[;2eɎzRvv*NQq՚xTh65Z47;?54N5mΣNe^jS*FM1 xr` C!vA` C!A` C!;A` Cv;A!v; C!v;A` C!vA` C!;A` Cv;A` v;A!v; Ы3}IENDB`}( H/0DArialngsp9L70Wo 0DSimSungsp9L70Wo 0 DWingdingsp9L70Wo 0 `. @n?" dd@ @@`` `,K1 !"$ #%&',/*+()-.01/Xb$YBmU^7rb$o˒./r0AA@ʚ;ʚ;g4dddd0zppp@<4dddd 0L7<4!d!d. 0L0___PPT10 ppZ___PPT9<46?%O =v0Early intervention for mathematical difficulties Ann Dowker, University of Oxford30$Mathematical difficulties are common%%(Bynner and Parsons (1997) gave some Basic Skills Agency literacy and numeracy tests to a sample of 37-year-olds from the National Child Development Study cohort (which had included all individuals born in Britain in a single week in 1958). The numeracy tests included such tasks as working out change, calculating area, using charts and bus and train timetables, and working out percentages in practical contexts. According to the standards laid down by the Basic Skills Agency, nearly one-quarter of the cohort had 'very low' numeracy skills that would make everyday tasks difficult to complete successfully. This proportion was about four times as great as that classed as having very low literacy skills. XP542 Severe specific difficulties in arithmetic are found in about 6% of children ( Lewis, Hitch and Walker, 1994; Gross-Tsur, Manor and Shalev 1996; Bzufka, Hein and Neumarker , 2000). Both genders seem to be equally affected. Pu 5,) JIn order to study the nature of the arithmetical difficulties that children experience, and thus to understand the the best ways to intervene to help them, it is important to remember one crucial thing: arithmetic is not a single entity, but is made up of many components. These include knowledge of arithmetical facts; ability to carry out arithmetical procedures; understanding and using arithmetical principles such as commutativity and associativity; estimation; knowledge of mathematical knowledge; applying arithmetic to the solution of word problems and practical problems; etc.8KZbb.bFs C .* 5Experimental and educational findings with typically developing children, adults with brain damage, and children with mathematical difficulties have shown that it is possible for individuals to show marked discrepancies between almost any two possible components of arithmetic (Ginsburg, 1977; Dowker, 1998). 64`4 0/"Desirability of early intervention. It is desirable that interventions should take place at an early stage: in the early school years, or if possible the preschool years. This is not because of any 'critical period' or rigid timescale for learning. Age of starting formal education has little impact on the final outcome (TIMSS, 1996). People who, to varying degrees, lacked opportunity for or interest in learning arithmetic in school, may learn later as adults (Evans, 2000). $ZZ1-"The problem of mathematics anxiety##(But there is one important potential constraint on the timescale for learning arithmetic and other aspects of mathematics (apart, of course, from the practical constraints imposed by school curricula and the timing of public examinations). Many people develop anxiety about mathematics, which can be a distressing problem in itself, and also inhibits further progress in the subject (Fennema, 1989; Hembree, 1990; Ashcraft, Kirk and Hopko, 1998). ,Z2. This is rare in young children (Wigfield and Meece, 1988) and becomes much more common in adolescence. Intervening to improve arithmetical difficulties in young children may reduce the risk of later development of mathematics anxiety. In any case, interventions are easier and less painful if they take place before mathematics anxiety has set in . Therefore, while it is never too late to intervene to help people with their arithmetical difficulties, interventions may be particularly effective if they are early. BPt{, *' In view of the componential nature of arithmetic, understanding of the individual differences in specific components of early arithmetic should facilitate the teaching of number in the early years. In particular, if one could predict individual children's likely specific patterns of strengths and weaknesses, then it might be possible to target early interventions to focus on preventing or at least ameliorating the potential weaknesses ,Z+( b Some of my current research involves investigating individual differences in components of 4-year-olds numerical abilities, and their relationship to each other. An ultimate aim is to examine their role in predicting strengths and weaknesses in components of arithmetical abilities at age 6 and beyond.2Z2)Interventions for primary school children**(Mathematical difficulties have not received the same attention over the years as, for example, literacy difficulties. However, research has been and is being carried out on programs for young primary school children which: involve individualized assessments are based on componential, non-unitary theories of mathematical development and mathematical difficulties; take into account individual children's strengths and weaknesses in specific components of arithmetic Z &Individualized intervention techniques''(f Individualized, component-based techniques of assessing and remediating mathematical difficulties have surprisingly early origins. They have been in existence at least since the 1920s (Buswell and John, 1927; Brownell, 1929; Greene and Buswell, 1930; Williams and Whitaker, 1937; Tilton, 1947). On the other hand, they have never been used very extensively.&gZG>=r,s -J.F. Weaver: one of the pioneers in this area..(Weaver (1954) put forward several important points that have since been strongly supported by the evidence: "Arithmetic competence is not a unitary thing but a composite of several types of quantitative ability: e.g. computational ability, problem-solving ability, etc."; "(T)hese abilities overlap to varying degrees, but most are sufficiently independent to warrant separate evaluations"; Children exhibit considerable variation in their profiles or patterns of ability in the various patterns of arithmetic instruction" .(pp. 300-301). (E)xcept for extreme cases of disability, which demand the aid of clinicians and special services, remedial teaching is basically good teaching, differentiated to meet specific instructional needs".,PA,/Why has such work had relatively little impact?00(xIf componential theories of arithmetical ability, and their applications to differentiated instruction and remediation in arithmetic have been advocated for at least 80 years, why have they had comparatively little impact on theory and practice? Practical problems. In under-resourced classrooms, it is difficult to provide individualized instruction. Limited communication of findings. Communications between teachers, researchers in education, researchers in psychology and policy-makers have been limited, as often have been communications between researchers within the same discipline in different countries and at different ByZX" Recent interventions: There have been a significant number of more recent individualized and small-group interventions with children with numeracy difficulties (e.g.. Askew, Bibby & Brown, 2001; Kaufmann, Pohl, Semenza & Delazer, 2003; Kroesbergen and Van Luit, 2003; Young-Loveridge, 2004). Many of these projects are still undergoing research, development and evaluation. Two of the most individualized programs, which place particular emphasis on componential theories of arithmetical development, will be discussed in some detail Zb (Two individualized intervention programs))( Two recent independently developed, individualized intervention programs which address numeracy in young children, and take componential approaches based on cognitive theories of arithmetic, are: The Mathematics Recovery program (Wright, Martland and Stafford, 2000; Wright, Martland, Stafford and Stanger, 2002); The Numeracy Recovery program (Dowker, 2001, 2003). Both are still undergoing further research and development.BPb\>{4Some important differences between the two programs.55( The Mathematics Recovery program is much more intensive than the Numeracy Recovery program. The Mathematics Recovery program places more emphasis on methods of counting and number representation, and the Numeracy Recovery program on estimation and derived fact strategy use. From a more theoretical point of view, the Mathematics Recovery program places greater emphasis on broad developmental stages, while the Numeracy Recovery program is treats mathematical development, to a greater extent, as involving potentially independent, separately developing skills and processes. $PCPD Despite these distinctive features, the two programs have other important common features besides being individualized and componential. Both programs are targeted at the often neglected early primary school age group (6 to 7 year olds). Both place a greater emphasis than most programs on collaboration between researchers and teachers.BTZ( Mathematics Recovery The Mathematics Recovery program was designed in Australia by Wright and his colleagues (Wright et al, 2000, 2002). In this program, teachers provide intensive individualized intervention to low attaining 6 and 7 year olds. Children in the program undergo 30 minutes of individualized instruction per day over a period of 12 to 14 weeks.TZT < Children in the program improved very significantly on the topics that form the focus of the problem: often reaching age appropriate levels in these topics. The teachers who worked on the program found the experience very useful; felt that it helped them to gain a better understanding of children s mathematical development; and used ideas and techniques from the program in their subsequent classroom teaching.$ZZNumeracy RecoveryBThe Numeracy Recovery program (Dowker, 2001, 2003), piloted with 6 and 7 year olds (mostly Year 2) in some primary schools in Oxford, is funded by the Esmee Fairbairn Charitable Trust. The scheme involves working with children who have been identified by their teachers as having problems with arithmetic. 175 children (about 15% of the children in the relevant classes) have so far begun or undergone intervention Z, m These children are assessed on nine components of early numeracy, which are summarized and described below. /Components of numeracy addressed in the project00(The components that are the focus of the project: Counting procedures Counting principles Written symbolism for numbers The role of place value in number operations and arithmetic Word problem solving P Number fact retrieval Derived fact strategy use Arithmetical estimation Translation between arithmetical problems presented in concrete, verbal and numerical formats (e.g. representing the sum 3 + 2 = 5 by adding 3 counters to 2 counters, or by a word problem such as Sam had 3 sweets and his friend gave him 2 more, so now he has 5 ).$PVPW The components addressed here are not to be regarded as an all inclusive list of components of arithmetic, either from a mathematical or educational point of view. Rather, the components were selected because earlier research studies and discussions with teachers have indicated them to be important in early arithmetical development, and because research has shown them to vary considerably between individual children in the early school years.Z The children then receive weekly individual intervention (half an hour a week) in the particular components with which they have been found to have difficulty. The interventions are carried out by the classroom teachers, using techniques proposed by Dowker (2001). The teachers are released (each teacher for half a day weekly) for the intervention, by the employment of supply teachers for classroom teaching. Each child typically remains in the program for 30 weeks, though the time is sometimes shorter or longer, depending on teachers' assessments of the child's continuing need for intervention. New children join the project periodically.Z<:'Examples of derived fact strategy tasks(((<9 + 4 = 13 4 + 9 = 9 + 4 = 13 9 + 4 = 9 + 4 = 13 9 + 5 = =Z==; <9 + 4 = 13 6 + 7 = 9 + 4 = 13 8 + 4 = 9 + 4 = 13 7 + 5 = =Z=><Examples of translation tasksa) Translation from numerical to concrete: Children can be presented with written sums (e.g. "6 + 3 = 9", "8 - 2 = 6"), and will be invited to "show how to do this sum with the counters". (b) Translation from concrete to numerical: They can watch the teacher perform arithmetical operations with counters (e.g. adding 4 counters to 3 counters; subtracting 2 counters from 7 counters) and then write down the sum that the experimenter did. v+ZZ-ZZZ&& (The children in the project, together with some of their classmates and children of comparable levels of arithmetical ability from similar schools in the area from other schools, are given three standardized arithmetic tests: These are the British Abilities Scales Basic Number Skills subtest (1995 revision); The WOND Numerical Operations test, and the WISC Arithmetic subtest The first two place greatest emphasis on computation abilities and the latter on arithmetical reasoning. The children are retested at intervals of approximately six months. )Z) DThe initial scores on standardized tests, and retest scores after 6 months, of the first 146 children to take part in the project have now been analyzed. Not all of the data from 'control' children are yet available, but the first 75 'control' children to be retested showed no significant improvement in standard (i.e. age corrected) scores on any of the tests. Moreover, as the tests are standardized, it is possible to estimate the extent to which children are or are not improving relative to others of their age in the general population.#Z# The children in the intervention group have so far shown very significant improvements on standardized tests. (Average standard scores are 100 for the BAS Basic Number Skills subtest and the WOND Numerical Operations subtest, and 10 for the WISC Arithmetic subtest.) 63 Median standard scores on the BAS Basic Number Skills subtest were 96 initially and 100 after approximately six months. Median standard scores on the WOND Numerical Operations test were 91 initially and 94 after six months. Median standard scores on the WISC Arithmetic subtest were 7 initially, and 8 after six months (the means were 6.8 initially and 8.45 after six months). Wilcoxon tests showed that all these improvements were significant at the 0.01 levelP|M! One hundred and one of the 146 children have been retested over periods of at least a year, and have been maintaining their improvement. 74Further developments: Catch Up(Collaborators on this work: Graham Sigley Julie Lawes Wayne Holmes Peter Morris (research assistant) Alan Evans, University of Cardiff (evaluation) ,#`86*Catch Up is a not for profit charity which++(Sprovides a comprehensive training package, to support the management and delivery of the Catch Up Literacy Programme has developed the Catch Up Literacy Programme (an intervention for struggling readers) provides ongoing support, through the Catch Up Community, for those who deliver the Catch Up Literacy Programme to struggling readers 4RlR97 NCatch Up is currently undertaking an action research project, based closely on my research, in 7 local authorities with more than 200 children who struggle with numeracy. This additional research will then inform the development of Catch Up Numeracy , a comprehensive intervention programme, including accredited training and support, that local authorities will be able to implement effectively in real school contexts. ?? Training sessions are given to teachers and teaching assistants who will deliver Catch Up Numeracy. These involve three half-hour sessions. The assessments involve the same components as the pilot Numeracy Recovery scheme. They have undergone some refinement to make them applicable to a somewhat wider age-range (6 to 11). Also, they have been modified to make them manageable by teachers and teaching assistants following relatively brief training.ZB@ They involve formative assessments, followed by building a profile of the individual learner s needs and setting Catch Up Numeracy targets & l(@= A> :8&Children s progress>127 children were given a standardized maths test (Basic Number Screening Test) at the start and after 4 months. Control group A (20 participants; no intervention) made an average of 4.1 months progress. Control group B (25 participants; equal amount of time given to practicing school maths work): average of 5.4 months progress. Intervention group (82 participants): average of 9.2 months progress in 4 months. ZCA T-tests showed that neither control group made significantly more progress than the 4 months that would be expected. The children in the intervention group made very significantly more progress than the 4 months that would be expected (p<0.001). ANOVAs showed that the control groups did not differ significantly from one another, but that the intervention group differed significantly from both (p < 0.05).ZDB 4Still investigating progress of second batch of children, after 6 months. Control group A (9 participants; no intervention) made an average of 6.0 months progress (s.d. 8.49) Control group B (9 participants; equal amount of time given to practicing school maths work): average of 6.7 months progress (s.d. 9.1) Intervention group (37 participants): average of 12.73 months progress (s.d. 9.72) in 6 months. Z;9 >The National Numeracy Strategy (DfEE, 1999) incorporates some intervention techniques for children who are struggling with arithmetic. The main intervention is the 'Springboard' program, used with groups of children in Years 3 to 7 who have relatively mild arithmetical difficulties There are also the Primary National Mathematics Wave 3 materials: Supporting children with gaps in their mathematical knowledge $Z |FD Currently, the government and the charity Every Child Matters are beginning an Every Child Counts program, aimed at providing individualized intervention for all children with significant mathematical difficulties.GEImplications for theoryThus, the intervention was based originally on research that suggested that arithmetical ability is not a single entity but is made up of many components. This is proposed to be true in both typical and atypical development. 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