**by A Ehlert · 2013 · Cited by 15 — ipn.uni-kiel.de/abt_erzw/ws12_k/pp_070113.pdf (accessed 4 March 2013). vermenigvuldigen en verdelen [Special remediating programme multiplication and.**

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Antje Ehlert & Annemarie FritzEvaluation of maths training programme for children with learning difficultiesAbstractIn the German school system, children are seen as educationally impaired when they are more than two grades behind in their performance in several areas of learning, and this has been the case for several years. A special problem is the fact that support measures are often effective only to a limited extent, or only for a short period. The study at hand focuses on the question of whether educationally impaired children with large deficits in mathematics can be supported successfully by means of a highly adaptive support measure (MARKO-T), and whether the effects of this support can be maintained over a certain period. For this, 32 educationally impaired third-graders with math deficits were supported individually with MARKO-T twice a week, over a period of ten weeks. As control group, 32 similarly impaired third-graders were paralleled according to the mathematical and cognitive achievements of the training group. Two further control groups, each with 32 unimpaired first-graders, were paralleled according to their mathematical and cognitive achievements, respectively. The results showed that the very poor mathematical performance of the educationally impaired children could be significantly improved with this support programme. Four months after the end of the training, significant support effects could still be established when compared to the educationally impaired control group. The comparison with the two control groups demonstrated that the developmental curve of the children with learning difficulties increased in a way that was comparable to that of the unimpaired first-graders.Keywords: educationally impaired children, evaluation of MARKO-training, training Antje Ehlert and Annemarie Fritz, Psychology Department, Faculty of Educational Sciences, University of Duisburg-Essen, visiting professor at Centre for Education Practice Research, Faculty of Education, University of Johannesburg Soweto Campus, e-mail address: antje. ehlert@uni-due.de Annemarie Fritz, e-mail address: fritz-stratmann@uni-due.deSouth African Journal of Childhood Education | 2013 3(1): 117-141 | ISSN: 2223-7674 |© UJ

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118IntroductionThe acquisition of mathematical competences is a complex process of learning that sets in long before formal schooling (e.g. Antell & Keating, 1983; Kobayashi, Hiraki, Mugitani & Hasegawa 2004; Starkey, Spelke & Gelman 1990; Wynn, 1992, Resnick, 1989, Gelman & Gallistel 1978, Fuson, 1992, Piaget & Szeminska, 1975). The relevance of this early acquisition of mathematical competences has been substantiated by numerous especially longitudinal studies. These studies verified empirically that there is a medium to high correlation between pre-school and primary school mathematical performance (Aunola, Leskinen, Lerkkanen & Nurmi 2004), and that the pre-school performance predicts the mathematical performance ability during primary school. Examples used to determine pre-school performance are, for instance, to what extent children are able to complete number rows. To determine this, three cards with digits between 0 and 20 are given (i.e. 4 Œ 5 Œ 6), on which one digit was missing: the first, the middle, or the last (Chard, Clarke, Baker, Otterstedt, Braun & Katz, 2005, also in Lembke & Foegen, 2009). In Jordan, Glutting & Ramineni (2010), the children counted to ten, counted stars in illustrations by pointing at the counted object, judged the counting skills of a finger puppet, determined precursory and successive numbers, and compared figures. For example, they were asked ‚Which is smaller: 8 or 6?™ and had to calculate number questions, such as: ‚Paul has 5 oranges. Maria takes 2 of his oranges away. How many oranges does Paul have now?™ Koponen, Aunola, Ahonen & Nurmi (2007) had the children identify the third and seventh object of a row, count a quantity of objects and match the counted number to a digit and had them draw quantities that contained two objects more, or one or two less than the presented quantity. Overall, the studies serve to ascertain that before school begins, children know more than merely the number-word-line and the counting out of quantities. They are capable of performing rather complex mathematical operations and already understand the relations between quantities and numbers.The question is whether this is really the case for all children entering school for the first time. Schipper (1998) investigated this question in critical re-analyses. On the one hand, he could confirm that many children entering school displayed high levels of competence. On the other hand, he detected a high variance in the children™s performances at school start. Even though many children entering school already possess significant arithmetic competences according to various studies, this is by no means the case for all children. The heterogeneity in performance amounted to up to three years. The reasons for this heterogeneity are manifold. The children™s chances to deal with quantities and numbers in pre-school (before they start school) are seen as essential. The child™s cognitive prerequisites are another factor (Geary, Brown & Samaranayake, 1991) as is the performance of his or her working memory (Logie, Gilhooly & Wynn et al ., 1994; Tronsky, 2005; Iuculano, Moro & Butterworth, 2011). In summary, it can be stated that children who start school with a high basic level of mathematical competence will maintain good performance (Aunola et al. , 2004). They may even go on to improve more than under-performing children.

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119However, under-performing pupils tend to remain among the weakest performers, even at the end of fourth grade. Different longitudinal studies (for instance Stern, 1994) emphasises that later academic success depends largely on subject-specific prior knowledge. Thus, mathematically precursory competences have significant influence on academic performance (Helmke & Weinert, 1997). They are the prerequisites for understanding school mathematics.Problems in learning numeracy Dyscalculia and numeracy problemsChildren with numeracy problems or dyscalculia Œ we will discuss the difference later Œ usually start school already with a less accomplished level of prior knowledge. The problems do not to arise during school; they begin before. In addition, they increase if the lessons do not begin at the children™s respective level of knowledge. Below, we focus on the difficulties faced by weaker mathematicians, by limiting our attention to the field of arithmetic. First, we need to distinguish the different types of difficulties that they face and how these can be differentiated terminologically.Dyscalculia is diagnosed according to the classification criteria of the ICD-10 (International Classification of Diseases; Dilling, Mombour & Schmidt 2004), when a child™s performance in numeracy in a standardised arithmetic test ranks among the bottom 10% of the distribution. Secondly, intelligence should not be significantly below average (i.e., IQ > 70). Finally, a discrepancy between the expected numeracy skills, based on the performance in an intelligence test, and weak numeracy performance, should exist (Ehlert, Schroeders & Fritz, 2012). Another criterion states that the children™s problems ‚exist from the start™ and already affect the acquisition of the basic arithmetic operations. This definition applies to 4 Œ 6% of all children who are likely to have dyscalculia (Koumoula, Tsironi, Stamouli, Bardani, Siapati, Graham-Pavlou et al ., 2004; Von Aster, Schweiter & Weinhold, Zulauf, 2007). However, problems with numeracy occur far more frequently. Current results from the TIMSS (Bos, 2013) show e.g. that, in Germany, the mathematic performance of 19.3% of children in grade 4 does not meet the minimum requirements (data for other countries from TIMSS: Japan 6.8%; Australia 29.8%, Iran 66.8%). This means that almost 20% of German primary school children only possess a rudimentary knowledge of mathematics at the end of the fourth grade. What are the mathematical difficulties these children experience and what are the obstacles they do not seem to be able to negotiate by themselves? What all these children have in common is that they hang on to counting strategies and do not develop them further into effective strategies of decomposition and fact retrieval (Geary, 1990; Hanich et al., 2001). Without the necessary support, it is left to chance, whether they abandon finger counting, or verbal counting. When solving arithmetic problems and word problems, they lack the ability to choose between various problem-solving strategies (Hanich et al., 2001; Jordan & Montani, 1997; Barrouillet, Fayol & Lathulière,

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1201997; Ostad, 1997, 1998, 2000; Gross-Tsur, Manor & Shalev, 1996) and approaches for modelling the tasks. They commit many more errors, when they retrieve arithmetic facts from long-term memory (Hanich et al., 2001; Jordan & Montani, 1997; Ostad, 1997, 2000; Barrouillet et al., 1997; Fayol, Barrouillet & Marinthe, 1998; Geary, 1990). In children with dyscalculia, these problems are more comprehensive and persistent, but are not entirely different. Numeracy problems as a symptom of general learning problemsThe definition of dyscalculia, according to ICD-10, demands a discrepancy between IQ and mathematical performance and thus excludes children with a low IQ. By definition, children are also excluded from being diagnosed with ‚dyscalculia™ if, on top of a weakness in arithmetic, they exhibit further comprehensive learning difficulties . Within the German school system children are defined as ‚educationally impaired™, when they are more than two grades behind in their performance in several areas of learning. This definition applies when this has been the case for several years and the situation cannot be compensated for by support measures in the short term (Kanter, 1973). The deficits in performance should not be caused by insufficient learning opportunities or inappropriate schooling. The performance deficits should not be caused by insufficient learning opportunities or inappropriate schooling, though. A further criterion, often mentioned, is ‚impaired intelligence™ (IQ < 85). However, as an isolated factor, this is not enough to diagnose a learning difficulty (Klauer & Lauth, 1996). There is no comparable term to be found internationally for the German category Lernbehinderung (learning difficulty). The international parlance expressed in the ICD-10 (F 81.3 and F 81.9) ‚m ixed disorder of scholastic skills™ and ‚developmental disorder of scholastic skills,™ and as well as the term ‚Mild Mental Retardation™ do overlap, to a large extent, with the German term Lernbehinderung.Although a formal distinction is made, according to this definition, between children with weakness in arithmetic, empirical findings indicate that these children do not differ with regard to the appearance and extent of their difficulties in numeracy. So far, only a few studies exist for the area of mathematics. One such study compared the solutions of arithmetic tasks of arithmetically weak pupils, who did or did not fulfil the discrepancy criterion (Jimenez Gonzalez & Garcia Espinel, 1999; 2002). No significant differences could be found between the two groups of arithmetically weak children, neither in the frequency of solutions nor in the choice of solving strategies. A separate comparison of arithmetically weak children, who did, or did not fulfil the discrepancy criterion, could further demonstrate that both groups of children had the same mathematical concepts at their disposal. Thus, both groups showed a comparable grasp of mathematical concepts, independent of their IQs. This suggests that children of the same age, without an arithmetic weakness, are on a higher developmental level (Ehlert et al., 2012).The criteria, given for a comprehensive mathematical learning difficulty, describe comprehensive learning deficits of more than two years, which cannot be remedied by
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121short-term interventions. Thus, it must be assumed that difficulties in numeracy, which started at an early stage, have now become entrenched. Furthermore, longitudinal studies indicate that the differences in performance of children with, and those without, learning difficulties, become more pronounced in the course of development. This, in turn, implies that the performance gap between the two groups continues to increase. In order to support children with such pronounced learning deficits, we need a sophisticated way of diagnosing the learning difficulties that may accompany the learning process. This should provide detailed information regarding the progress, or a lack thereof, in learning. In addition, long-term support measures are needed which considers individual learning processes. Both measures will be discussed in more detail in the following section.Criteria for supportDiagnostics is of special significance in terms of the support required by children who experience difficulties with numeracy. In order to plan an intervention that is in line with the child™s state of development, a sophisticated analysis of the initial situation of mathematical learning is required. This should give a comprehensive description of the child™s current state of mathematical development, before the support starts.Nevertheless, in order to support a child successfully, it is not enough to limit diagnostics to the assessment of the initial learning situation and the determination of individual support requirements. A key factor for the success of the intervention is monitoring the support process to continuously adapt the training to the individual™s learning speed as well as the individual™s learning progress. Analyses of training studies show (Strathmann & Klauer, 2008; Strathmann, Klauer & Greisbach, 2010; Strathmann & Klauer, 2010) that a child™s development is not always marked by progress in learning. During the training, learning can also come to a halt, or there can even be a regression in learning. This phenomenon necessitates diagnostic monitoring that accompanies the learning process so that appropriate reactions to the learning process are possible. The development and implementation of diagnostically monitoring the learning process, however, is fraught with a number of problems (Klauer, 2011). If the learning process is to be represented in a theoretically correct manner, there ought to be different tasks in each test, yet, all of these have to be at the same level of difficulty, and have to assess the same latent trait. Only if this is the case, is it possible to guarantee that the learning curve is represented accurately, and that an improved performance is not due to re-testing, or because of variations in the difficulty of the tasks. Especially in mathematics, it is difficult to generate such test items as, for example, the number range limits the amount of possible tasks. One possible solution under discussion is the modelling of competence level models and the implementation of a probabilistic test theory (Klauer, 2011).The basic idea here is to break down a complex mathematical competence into partial competences, to put these in a hierarchical order, and then to develop tasks that require the respective partial competence for their solution. After that, one tests
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122psychometrically (e.g. with the Rasch model; Rasch, 1960) to determine whether the tasks, and the partial competences represented in them, really build upon one another hierarchically, as expected. Based on such an empirically validated competence level model, one can then determine, diagnostically, on which level of the competence scale the child currently operates, and which arithmetic concepts he or she has already developed. This provides a grid for the observation and evaluation of levels of competence.By means of Item Response Theory (IRT), the item difficulties and person abilities can be estimated on a common scale (see also Fritz et al in this journal). The probability of solving a task depends on the individual™s ability, and the score achieved (items solved correctly) can be used to locate a person with his/her ability at this scale. If the difficulty of the task is lower than the ability of the learner, it is likely that they will solve the task. Conversely, if the difficulty of the task is higher than the skill of the learner, it is rather unlikely that he or she will solve the task.If a test, such as this, that is based on a valid model is available, then the performance of each child can be located in the mathematical model, at any point in time, during the intervention. It is possible to determine whether the child shows an improvement in ability and progress in acquiring the next level of competence. Diagnosing the learning process does not then simply represent the learning curve, but also checks whether the child continues to progress in the developmental model, under the influence of the intervention.Aside from an accompanying diagnosis of the learning process, the training should be designed broadly to allow for a long-term support intervention mechanism that follows the development of the children, and does not merely focus on imparting a limited number of competences. Furthermore, the training should show sustainable effects. This means that if knowledge or skills were actually developed during the training, these should still be evident after several months. A precursory overview of existing support or training programmes for the primary school age reveals that although there are some evaluated programmes such as, the MAthematics Strategy Training for Educational Remediation (MASTER) by Van Luit, Kaskens & Van the Krol, 1993 (see also Van Luit & Naglieri 1999, and Kroesbergen & Van Luit, 2002), or Tier 2 intervention by Bryant, Bryant, Gersten, Scammacca & Chavez (2008), these do not contain a diagnostic component to accompany the support process. Only the programme Tier 2 intervention is designed to adapt to the individual™s developmental process. Only one such programme is based on a mathematical stage model of early arithmetical learning and plans training over six stages of early arithmetic learning ( Mathematics Recovery Programme by Wright, Martland & Stafford, 2000). However, this programme has not been evaluated or validated.Below we present a support programme that takes into account all discussed aspects of a comprehensive intervention programme for children with early arithmetic problems.
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124Scope of applicationMARKO-T is a training programme that is accompanied by diagnostics for children aged between five and eight years old. It can be used for the targeted support of children with arithmetic weaknesses or retarded development during the transition from kindergarten to primary school, or in the first years of primary schooling or in schools for children with special-needs.Training structure The programme contains five training modules with a total of 57 training units. The training modules are aligned to the five levels of the model (Level I to V). Each of the five modules provides tasks that can be solved on the basis of the respective concept.In terms of content, each mathematical concept is acquired in small steps. Module I comprises 10 training units and contains comparisons of quantities, the one-to-one allocation, learning and consolidating the number-word line, as well as, counting and counting out smaller quantities.Module II is composed of 15 training units. This module focuses on the understanding that the number-word line follows a fixed order in which the numbers increase. With this knowledge, numbers can be compared regarding their position, and first arithmetic operations can be carried out.The requirements of Module III are also tied into 15 training units. The children work out that numbers are composed entities (units), that is, they represent the number of elements in a quantity. Each number stands for a specific cardinality; this cardinality increases successively in the number- word line. Numbers can therefore be compared according to their cardinality. As numbers are composed entities (units), they can be decomposed and recomposed. The number range is opened up to 20 and with the help of material that provides structures, decompositions and addition tasks, that use the power of 5 or the power of 10, can be practiced. The 10 training units of Module IV , which deals with the part-part-whole concept, help the children to grasp that numbers contain other numbers. Tasks such as ‚Give me 5, 3 of which should be red™ illustrate the determined connection between partial quantity Œ partial quantity Œ total quantity. Systematic number decompositions are practiced; decomposition and addition tasks are solved mentally.With the last Module V, the children acquire the relational aspect of numbers in 7 training units. The relational number concept makes differences and distances between numbers precisely definable. In the first task of this module, numbers are represented on the number line and differences between numbers are determined (e.g. ‚by 2 more/larger than™). Based on this understanding, comparing
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125tasks become solvable: ‚Marko and Rollo together have got 6 balls, M has got 2 more than R. How many has M got, how many has R got?™ The individual training units are each structured identically and consist of a greeting that is followed by a check of whether the constructs of the last training session are still known. Then the contents of the current training session is introduced and practiced. The session ends with a closing reflection and an overview on the next training unit. Structure and principlesThe tasks are embedded into problem contexts and are child-oriented and interactive. Dung beetle Marko is the child™s learning partner. Marko is continuously faced with problems that form the starting point for solving the tasks together. This way, the child repeatedly has to explain contents, approaches, or strategies, and has to focus and reflect on, or explain correct and false or ineffective procedures (which Marko then uses). By articulating their own, as well as, Marko™s strategies, the metacognition of children is stimulated and enhanced. The child is thus made aware of mathematical strategies that can be used for the further development of mathematical understanding. At the same time, the beetle can be employed to model problem-solving approaches. The performance requirements, within the training, are gradually raised to the child™s individual limits of performance. Successfully dealing with requirements, bordering on one™s own performance limits, also supports the development of success-oriented motivation. The fact that the child™s learning progress is verbalised at the end of each lesson also supports the experience of competence. With dung beetle Marko tasks can also be done competitively, which provides a ‚natural™ way of automatising strategies, since it is focused on speed. In this way, the mathematical knowledge and skills are consolidated and become readily available.Figure 1: Dung beetle Marko, the hand puppet in the training programme MARKO-T.
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126DiagnosticsAfter having analysed the individual™s learning abilities, a training programme aims to design learning conditions that facilitate individualised learning progress. As was explained in detail above, a sophisticated description of the initial learning situation, as well as, a diagnostic assessment of learning process is needed. In MARKO-T, the diagnostics of the initial level of competence is done by means of a diagnostic instrument called the MARKO-D. A series of MARKO tests have been designed, namely; the MARKO-D0 (Ricken, Fritz & Balzer, 2013) for pre-school age and the MARKO-D1 (Fritz, Ehlert, Ricken & Balzer, in prep.) for the first grade. The MARKO-D instrument allows a child™s performance to be assessed according to social norms or according to criterion norms:Assessment by social norm: The child™s performance can be assessed based on the percentile ranking scale and T-value scale. The child™s performance is compared by reference to the peer group.Assessment by criterion norm : regarding the IRT approach, the child™s performance can additionally be assessed qualitatively with reference to the level of competence. As it cannot necessarily be assumed that a child will solve all tasks of a level of competence without error, a 75% criterion was introduced. This criterion states that at least 75% of all items of a competence level have to be solved successfully, before the level is seen as being understood. Regarding levels of competence, a child™s performance profile can look like this: Level 1: 100% Œ Level II: 90% Œ Level III: 60% Œ Level IV: 30% Œ Level V: 0%. This profile means that the child™s performance is allocated to Level III. The concepts of Level I and II are understood and the child is about to develop an understanding of cardinality (concept of Level III). He/she can occasionally solve part-part-whole tasks from Level IV, but the acquisition of Levels IV and V are future steps in his/her development.Since MARKO-D and MARKO-T are based on a common model of mathematical development, the diagnostics of the MARKO-D can be used to deduce the individual support needs as well as the learning goals for the MARKO-T programme. The individual entry level for a child into the training programme can thus be determined by means, of not only a descriptive, but also a prescriptive diagnostics. The support programme begins with consolidating the concept of the level the child is currently on (Level III in the example above). After that, the support focuses on facilitating the acquisition of the concept needed for the next developmental level.At the beginning of each training unit, the contents of the last session are checked in the sense of a status diagnostics (see Figure 2). If the child can solve the tasks without difficulty, the training continues. If significant problems occur, the previous training unit is repeated. In addition, the problematic requirements are trained once more and thus the conceptual contents are revised in detail. Conversely, a rapid individual developmental speed can be adjusted. At certain points of the training, the possibility exists to skip training units in case of a very high success rate.
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127Figure 2: Diagnostics in the support programme MARKO-T. In order to adapt the training to the individual learning speed and learning progress of the child, his or her learning process is additionally checked by ‚diagnostic windows™ during the practicing phase (Leutner, 1992, 1993, 2004). Only when the specific goals of each training unit are achieved, work on new tasks, with a higher requirement structure, will begin. This is to ensure that conceptual understanding is built systematically and that mathematic competence is acquired successively. Tasks at the next level or of a more complex nature will only be presented and worked on when the necessary requirements, in terms of mathematical understanding, have been acquired and secured.On the following page is an overview of the structure of a training unit (Figure 3).
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