Snorre A. Ostad:
Mathematical difficulties: Aspects of learner characteristics in developmental perspective
Lecture at Department of Experimental Psychology, Oxford University 22 May 2002
Abstract
Aim: The study, reported in this paper in general view, was designed to investigate the character and extent of differences between mathematically disabled children and their of mathematically normal peers (NM children) as reflected in the use of taskspecific strategies for solving elementary number fact problems and word problems when children move up through primary school, i.e., from grade 1 to grade 7. Particular concern was with the variability within the group of MD children, especially in light of the general literature showing substantial heterogeneity in the performance characteristics of the mathematically less able children. Method: The sample included 32 MD children in grade 1, 33 MD children in grade 3, 36 MD children in grade 5 and a corresponding number of MN children in each of the grades. The children were observed systematically over a period of two years, grade 1 children from the end of grade 1 to the end of grade 3, grade 3 children from the end of grade 3 to the end of grade 5, and grade 5 children from the end of grade 5 to the end of grade 7. The taskspecific strategies they used were recorded on a trialbytrial basis and classified as defined single variants of backup strategies and retrieval strategies, respectively. Result: The pattern of development showed the MD children as being characteristic of: (1) use of backup strategies only, (2) use of the most primary backup strategies, (3) small degree of variation in the use of strategy variants and, (4) limited degree of change in the use of strategies from year to year throughout the primary school. Early and striking convergence of the developmental curves follows a sequence that was fundamentally different (not only delayed) from that observed from that observed in normal achievers. The findings highlight the MD children’s need for mathematics instruction to shift from computationfocused activities to strategylearning activities.
Introduction and overview
The project involved the close cooperation of 12 primary schools, that is, all primary schools in two Norwegian urban municipalities. About 1000 children were included. The children were observed systematically over a period of two years, grade 1 children from the end of grade 1 to the end of grade 3, grade 3 children from the end of grade 3 to the end of grade 5, and grade 5 children from the end of grade 5 to the end of grade 7.
Several other investigators have determined that mathematical learning problems are relatively common (Badian, 1983; Kosc, 1974). More specifically, a more recent published study (Ostad, 1998) shows that the schools support services had picked out about 10% of the children in some primary schools as needing remedial programmes in mathematics when these children were in grade 2 (e.g., 89 year old children). Nevertheless, mathematical learning problems remain relatively neglected in the research literature (Geary, 1993). Only a few empirical studies of the cognitive mechanisms potentially contributing to mathematical learning problems have been conducted, even though much has been learned about the acquisition of basic mathematical concepts and procedures in mathematically normal children (e.g., Ashcraft, 1992; Fuson, 1982; Ginsburg, 1983; Siegler, 1990).
Success or failure in mathematics has often been defined by performance on standardized achievement tests. These tests, however, do not provide information about the mental processes that are likely contributing to the childrens achievement. One alternative approach is to compare groups of children, who vary in achievement levels, on tasks for which a developmental progression of skills are differentiated. Simple basic fact problems ^{1} and simple arithmetic word problems are examples of such tasks. The study I intend to described today followed this approach suggesting that comparisons of developmental differences of mathematically normal children (MN children) and mathematically disabled children (MD children) of varying age levels might provide useful information about factors potentially contributing to mathematical learning problems.
The first of these areas that was highlighted in this project was strategy use, the second accuracy and speed of processing (Ostad, 2000) the third area was comorbidity between mathematical and language based difficulties (Ostad, 1998b).
Actually, my plan for this presentation is to limit the discussion to the first area: Children’s strategy use, in particular, task specific strategydifferences between MD and MN children in a developmental perspective.
Many researchers have examined the issue of problem solving in mathematics, and considerable progress has been made during the 1980s and 1990s in describing the problemsolving process. The nature and influence of what affects problem solving has been described from many different perspectives. Among the most critical factors that have been shown to be associated with performance in mathematics includes the varying use of problemsolving strategies (Dowker, 1992, Dowker et al., 1996; Ostad, 1998a). A variety of findings, primarily based on chronometric data, have supported the suggestion that childrens strategy use vary with age and ability, but also that a single child will often use different strategies on different occasions. For instance, investigations concerned with development of problemsolving strategies used by mathematically normal children have shown an obvious progression, over time, from immature, inefficient counting strategies, through verbal counting, and finally to automatic fact retrieval from long term memory as children move through primary school . Thus, a normal development reflects an increase in the use of retrieval strategies, and a decrease in the use of backup strategies (e.g., Ashcraft, 1992; Carpenter & Moser, 1984; Geary, 1993; Siegler & Jenkins, 1989).
A growing body of research has provided useful information regarding the strategy characteristics of mathematically disabled children (e.g., Geary, 1993; Geary &BurlinghamDubree,1989; Goldman, Pellegrino & Mertz, 1988; Siegler, 1988).
As compared with that of their mathematically normal peers, these children are characterised by the use of developmentally immature problem solving strategies. That is, these children often use strategies more commonly employed by younger MN children (Geary, Widaman, Little & Cormier 1987; Goldman, Pellegrino & Mertz, l988).
However, the majority of studies that have been conducted on cognitive mechanisms potentially contributing to mathematical difficulties have shown several methodological limitations. Probably because of the time and effort needed to study such mechanisms in a longterm perspective, few studies of developmental differences between MD and MN children have been carried out. For instance, most often the reseachers on strategy use have focused on one single age level and on the youngest age groups in particular. Left unanswered, therefore, was whether the pattern of differences between MN children and MD children could be found throughout the elementary school years. Perhaps even more significantly, most often the starting point for the constitution of MD groups (samples) has been the childrens achievement on just one single mathematical test. It seems not enough consideration has been given to the fact that, for the youngest age groups, mathematical difficulties encountered during the above mentioned test may have a relatively short duration. Thus, it is possible that the researchers may have operated with heterogeneous samples, composed partly of children with temporary difficulties and partly of children with difficulties of a more permanent nature. Moreover, earlier studies of strategy use differences between MN and MD children have been dominated by chronometric procedures for the collection of data. Most commonly, the children have been instructed to solve simple addition numberfact problems using the strategy they themselves found most suitable for the case in hand, and also at the same time to respond as quickly as possible (e.g., Geary & Brown, 1991; Geary & Burlingham Dubree, 1989; Geary et al., 1987; Goldman, et al., l988; Svenson & Broquist, l975). Needless to say, investigations based on chronometric procedures have, in fact, provided valuable insight into the nature of arithmetical development. However, there are, I suggest, reasons for expecting that this emphasis on speed might influence the strategy use.
The present study was designed to address some of the above mentioned limitations: First, the study reports assessment of developmental differences between MD and MN children in a longitudinal perspective over an extended period of time: that is, the age range of 8 to 13 years and of the grades 1 to 7. This made it possible to obtain an overview of the development differences throughout the elementary school. Second, the children who were unsuccessful in mathematics for less than two years, were excluded from the group designated MD children in the report. Third, the samples of MD children were relatively large compared with the samples in earlier studies. Fourth, the strategyuse research data were recorded without focusing on the time the children spent in solving the problem. Fifth, the study of strategy use differences between MD and MN children was carrried out within a relatively broad frame of reference that included basic fact problems in addition, basic fact problems in subtraction, as well as arithmetic word problems.
Method  general
The present study was designed to determine potential deficits associated with the pattern of development that unfolds when children move up through primary school, as reflected in the use of taskspecific strategies.
The central theoretical viewpoint in the research includes aspects of strategy variability as a fundamental characteristic of mathematical cognition. In particular, four aspects of strategy variability were applied through an examination of (1) the use of backup strategies versus retrieval strategies, (2) the use of specific backup variants, (3) the number of different strategies used, and (4) the changes in strategyuse as children moved up through primary school, i.e., from grade 1 to grade 7.
As indicated above, the reseach design made it possible to observe aspects of the mathematical development in mathematically normal children and children with difficulties in mathematics. These observations identified two different groups of children with difficulties, that is

Children with mathematical development in accordence with a developmental delay model.

Children with mathematical development in accordance with a developmental difference model, that is, mathematically disabled children (MDchildren).
About the sample
The MD children included:

Children registered in the schools ordinary support services as in need of a special programme of mathematics teaching, and

among the 14% bottom group in mathematics achievement tests taken with a two year interval. Thus, in the present study, the definition of mathematically diabled children relate to the persistency of the mathematics difficulties and not to the way ”learning disabilities” is defined in reseach literature.
The sample included 32 MD children in grade 1, 33 MD children in grade 3, 36 MD children in grade 5 and a corresponding number of MN children in each of the grades. The taskspecific strategies they used were recorded on a trialbytrial basis and classified as defined single variants of backup strategies and retrieval strategies, respectively. (See Ostad, 1997b, 1998, 2000 for more details).
Method  Examples from addition Experimental Tasks
The addition stimuli were constructed from the 64 possible pairwise combinations of the integers 2 9. The (8) tie problems were excluded. (A tie problem is 2 + 2). The remaining problems, which consisted of 56 singledigit addition problems of the form "a + b", were divided in two equivalent halves (28 in each), one half for use at TI and the other at TII^{2}. The two halves were counterbalanced. This means that all the 56 problems were pairwise matched (e.g., 9 + 8 and 8 + 9). By means of drawing lots, each of the problems in the pairs became by chance a part of the one or the other of the two halves. The addition problems were vertically placed and presented at the centre of 21x10 cm "cards", one problem on each card.
Other apparatus: The following equipment was found on the table: paper, pencil, and 40 red and white Unifix rods (20 of each colour).
Procedure
To a large degree, the present study followed the procedures developed by Siegler (Siegler, 1987, 1988, 1990). The subject was seated at a table directly across from the experimenter, and was tested individually. The strategies used to solve the problem were recorded on a trial by trial basis. Several previous studies have shown that children can describe arithmetic problem solving strategies accurately if they are asked about them immediately after they have solved the problem (Siegler, 1987, 1988). The subjects were told that they were to solve the problems, which would be presented one at a time on cards in a random order, and they were encouraged to use whatever strategy made it easiest for them to obtain the answer (using fingers, rods, writing/painting on the cards, sounding out words, and so on). The only important thing was that the children should try hard as they could to arrive at the right answer. After each trial, the subjects were asked to describe how they had reached the answer. After solving the first problem the children were told: "We want to know how children of your age figure out the answers to these problems. Tell me, how did you figure out the answer to that problem?" The question, "How did you figure out the answer to that problem?" was repeated after each item unless the child volunteered the information before being asked, which he or she usually did after solving a few items. If the child's description was unclear, the experimenter would ask one or more follow up questions. For example, if the child simply said "I counted," the experimenter would ask, "What number did you begin counting at?" (See Siegler, 1988, p.844).
During the experimental session, the answer and the strategy used to solve each problem were recorded (in writing and on video tape) by the experimenter, and then classified. First, it was necessary to make a two way classification of strategies: backup versus retrieval strategies. A trial was classified as a retrieval trial when the children simply stated the answer after being presented with the item. If there was any visible or audible evidence of mediating computations such as counting in the arithmetic tasks, the trial was classified as a backup strategy trial.
Second, it was necessary to classify the backup strategies more specifically. Several studies indicate that individual children often use multiple strategies to solve a given problem. These include counting fingers, putting up fingers but answering without any apparent counting and counting aloud without any apparent external referent, and retrieval (Siegler & Shranger, 1984; Siegler, 1988). This list of alternative backup strategies is certainly not complete. My earlier studies indicate that children also use strategies such as counting concretes, painting appropriate dots or dot patterns that represent the numerals included in the problem, and so on (Ostad, 1991). Therefore, in the present study it was found necessary to develop a classification system which would take into account variations in the choice of strategies by children of varying ages and levels of achievement.
For this reason, it was necessary to undertake a preliminary observation of the children before starting to actually classify the strategies. A total of 30 pupils (15 from each class) were drawn from two randomly selected classes, one from the 2nd and one from the 6th grade. These pupils were asked to solve 28 addition problems (The same problems, and using the same procedure, as in the actual study). The strategies used by these pupils to solve the problems provided a basis for the system of classification to be used in the actual study, and described in the next section.
Categorization of taskspecific strategies in addition
(A1) ADDITION STRATEGIES: BACKUP VARIANTS
(A1a) Count everything, and start again from the beginning. Example: 3 + 5 = ? The pupil counts concretes, e.g. fingers or Unifix rods. First "One, two, three" concretes and continues "one, two, three, four five" concretes. Then he/she starts from the beginning and counts all the elements in the two quantities "one, two, three, four, five, six, seven, eight". The concretes physically represent the integers in the problem.
(A1b) Count everything. Example: 3 + 5 = ? The pupil counts concretes, first "one, two, three" and continues by counting "four, five, six, seven, eight".
(A1c) Counting further. Example: 3 + 5 = ? The pupil uses concretes, but counts onwards from the first number. He/she counts "four, five, six, seven, eight".
(A1d) Minimum variant (minimum number of counting steps). Example: 3 + 5 = ? The pupils count concretes, but in this case count on from the numeral that represents the largest number, i.e. counting from the larger addend. In this example this means counting on from 5, i.e. "six, seven, eight".^{3}
(A1e) The drawing variant. Using a pencil, the pupil drawns lines, dots or suchlike on a piece of paper and afterwards counts these dots or lines to arrive at the answer with the help of A1a, A1b, A1c or A1d.
(A1f) "Touch points" on numerals.The pupil draws (or visualizes) dots in the numerals (an appropriate pattern or dots which represents the numeral). The addition takes place by the pupil using these dots to find the answer with the help of A1a, A1b, A1c, or A1d.
(A1g) Other counting variants.The pupil first draws a line of numbers and counts using that, or counts finger joints, or combines elements from two or more of the strategies described above.
(A1h) Verbal counting. The pupil counts audibly or moves his/her lips as if counting implicitly. The counting has no other directly observable external frame of reference.
(A2) ADDITION STRATEGIES: RETRIEVAL VARIANTS
(A2a) Knows the answer.The pupil recognizes the example and answers immediately.
(A3b) Derived fact variant I. The pupil knows the answer to the addition of various combinations and uses these answers as a basis for counting onwards. Example: 3 + 5 = ? The pupils knows that 3 + 3 = 6 and says, for example, "three plus three is sex .. plus two ... (counts on his/her fingers, for example, and says) seven, eight".
(A2c) Derived fact variant II. The pupil uses his knowledge of several questionanswer combinations which he/she uses as a basis for solving the problem without counting. If the problems is 8 + 6 = ?, the pupil proceeds from eight and says, e.g. "Eight plus two is ten. This means that there are four left. Ten plus four is forteen”.
CONCLUSIONS, DISCUSSION, AND RECOMMENDATIONS
In general, across times of measurement the MN children showed an increased reliance on retrieval strategies, and a decreased reliance on backup strategies. This change was consistent with earlier research assessing the strategy use development of basic arithmetic skills (e.g., Ashcraft, 1982; Geary et al., 1991; Goldman et al., 1988; Siegler & Jenkins, 1989). Nevertheless, the course of development clearly shows that backup strategies play a dominant role in the problemsolving process through the primary school stage. These findings provide substantial support for arguments advanced by Siegler (1988) that children make use of a mixture of strategies, usually combining counting with direct retrieval.
In contrast, the MD children characteristically used backup strategies almost exclusively throughout the same period. Earlier studies have already shown that MD children most frequently use reconstructive counting strategies and not retrieval strategies (Connor, 1983; Fleishner et al., 1982; Geary & Brown, 1991; Russell & Ginsburg, 1984). The present study documents that this applies to MD children year after year throughout the primary school stage. The MD childrens consistent use of backup strategies might reflect both fact retrieval problems and working memory problems (Garnett & Fleischner, 1983; Geary, 1990; Geary, 1993; Geary, BowThomas & Yao, 1992; Goldman et al., 1988). Since the data from this study showed that the typical MD children use backup strategies only during the whole phase of primary education, it would be a reasonable assumption that the exclusive use of backup strategies reflects a critical factor for normal development.
Among the MN children the characteristic course of development shows the use of new strategies, both backup and retrieval strategies. In a longitudinal perspective, a course of development was observed involving an agedetermined shift in strategy use, not only away from backup to retrieval, but also within the framework of the backup strategies themselves, that is, away from the most primary counting strategies, so that other backup strategies, especially verbal counting, were used more frequently (Carpenter & Moser, 1984). It was suggested that the corresponding data for the MD children would reflect a developmental delay model, establishing that the difference between the two ability groups would converge early in the elementary school years (Goldman et al., 1988, Geary, 1993). Unexpectedly therefore, the typical MD children were characterised not only by little use of retrieval strategies but also by much more frequent use of the most primary backup strategies throughout the whole primary school stage. These results seem to conflict with the arguments proposed by Geary (1993) that the development of the procedural and memoryretrieval skills of MD children are largely modular; that is, functionally distinct. Consistent with the developmental difference model (Goldman et al., 1988), the acquisition of strategy skills by MD children seemed to follow a sequence that is fundamentally different from that observed in normal achievers.
Earlier studies have shown that domainspecific knowledge, that is, substantial factual knowledge, is an important component in the effective strategy use (Ohlson & Rees, 1991; Pressley, Brokowski & Schneider, 1987, 1990). There is therefore reason to assume that the amount of domainspecific knowledge, that is, the amount of factual knowledge the child possesses about the various strategies and how and where to apply them, will be reflected in problemsolving through the range of variation in the strategies used.
Most frequent, when MN children were asked to solve basic fact problems and arithmetic word problems, they normally used several different strategy variants for this purpose. Thus, a course of development was observed which showed a gradual but marked increase in the number of strategy variants used as the children became older. This result might indicate that these children have at their disposal a rich amount of domainspecific strategy knowledge, that is, substantial knowledge of various strategies and their areas of application. Consequently, the study confirms the results of earlier studies (Ashcraft, 1982, 1990; Carpenter & Moser, 1982; Geary & BurlinghamDubree, 1989; Siegler & Shrager, 1984). The results further document that the number of different strategy variants used by the MN children increases as they move up through the primary school.
In the case of the MD child, however, the course of development showed far less frequent use of a large number of different strategies throughout the primary school. While a wealth of substantial strategy knowledge was typical of the MN children it seemed, on the other hand, that the MD children had a lack of strategies. As indicated above, the result of the present study gives argument to the suggestion that the MD childrens insufficient domainspecific strategy knowledge in itself limits the choices available to them. Accordingly, I suggest the existence of important individual differences in the wealth of domainspecific strategy knowledge. More precisely, I argue that the amount of factual knowledge the child possesses about the various strategies, and how and where to apply them, might be reflected in problem solving through the range of variation in the strategies used. If this suggestion is valid, there are reasons to assume that the quantity of domainspecific strategy knowledge could be a critical factor for normal development.
When the research on the individual child's strategy use was repeated two years after the first time of measurement, the MN children, who had already used several different strategy variants two years before, continued to change their strategy use in the direction of new strategy variants. This result could reflect an increase in the quantity of domainspecific strategy knowledge they possess, but could also relate to strategic flexibility indicating that children have the ability to "call forth" appropriate strategies by actively selecting and judging between the strategies at their disposal (Ashcraft, 1992; Siegler & Shrager, 1984; Geary & BurlinghamDubree, 1989; Siegler & Jenkins, 1989; Schneider, 1993). The MD children, on the other hand, did not change their strategy use to nearly the same degree. The typical MD child seemed to use the same strategy variant(s) again and again, year after year, right through the entire primary school, which implies that their pattern of development is characterised by strategic rigidity.
In other words, the study indicates that at an early stage, already in grade 1, the MD children seemed to adopt a characteristic pattern of development, featuring primary backup strategies, a minimum of strategies, and strategic rigidity. This pattern of development might provide substantial support for the suggestion that inefficient strategy use might be a consequence, in part, of perseverative use of primary backup strategies (e.g., Goldman et al., 1988). Several years ago, Gestalt psychologists labeled what may be a related phenomenon, that is, functional fixedness (Wertheimer, 1959). When a particular approach or procedure is practiced it can become fixed, making it difficult to think of the problem situation in another way.
According to the conceptual understanding hypothesis, strategies acquired in isolation from their conceptual basis tend to be error prone, and do not transfer easily to novel problems (Hiebert & Lefevre, 1986; Ohlsson & Rees, 1991). Thus, when the MD childrens performance showed a course of development with a large and constant discrepancy, throughout the primary school period, between their level of performance for solving numberfact problems and arithmetic word problems it could be argued that these children do not have a good conceptual understanding of arithmetic. That is, they approach simple arithmetic in a rather rigid, algorithmic manner, as they do counting.
A number of earlier studies have, on the basis of different criteria, characterised MD children as a heterogeneous group. These children show different levels of intelligence, different language skills etc.. Results from the present study indicated that approximately half of the MD children also had spelling difficulties. It would be a reasonable assumption that this observed heterogeneity within the group would be clearly reflected in the pattern of strategy development. Therefore, it may seem paradoxical that the present study documents surprisingly little variability in strategy use for solving both basic fact problems and arithmetic word problems. When the MD children (supposedly including children with different potentials for development) reach the end of grade 7, large deviations from the main pattern described above are a rare occurrence. Thus, the most striking feature of the pattern of strategy development, as is pictured in the results of this study, is the marked degree of similarity in the strategy use among the group of MD children as a whole. Perhaps even more significant, this pattern of development seems to have been almost permanently established early in primary school, probably at the end of first grade. The early and striking convergence (flatten) of the developmental curve is consistant with the developmental difference model (Goldman et al., 1988).
According to Ginsburg (1997), the most reasonable explanation for the MD childrens failure in mathematics is the conventional system of instruction. Several projects were initiated to address the need for better mathematics instruction. The importance of metacognition to mathematical problem solving is well acknowledged in the literature (Hiebert & Carpenter, 1992). Cognitive strategy instruction is a promising alternative to current approaches for teaching mathematics to students with learning difficulties (Montague, 1997).
Teachers should bear in mind the suggestion that children can be channelled into inappropriate development patterns, for which the teaching itself might be partly responsible. The consistency with which some MD children used material strategies could indicate that these strategies may very well have been restrictively taught. One could, for example, expect strategy variability to be influenced by the extent to which an individuals school instruction had encouraged or discouraged such variability. If the above suggestion is valid, the characteristic pattern of development of the MD children might have been created by excessive emphasis on teaching methods that invite the use of primary counting procedures. This seems particularly relevant when teaching the youngest age groups (as in the schools included in the study) is based to a large extent on readyprinted exercise books, often with concretes functioning as counting instruments; the main thing the pupil has to do is to count the concretes and write in the answers (Ostad, 1992). MD children require more than readyprinted exercise books, concretes, or realword practice in solving mathematical problems to become good problem solvers. By contrast, the results of the present study might suggest that remedy of mathematics difficulties should include instruction on taskspecific strategies involved in efficiently solving of arithmetic word problems and arithmetic basic fact problems. To address the needs of the MD children, there is evidence that the instructional methods generally need to change focus, early in the elementary school years, from how to learn more mathematics to how to learn mathematics by means of appropriate approaches, that is, providing MD children with instruction to help them become good strategy users and move beyond rote application of basic skills. The results of the present study suggest that good strategy users; (a) have available a knowledgebase of taskspecific strategies to perform a particular task or a particular problemtype, (b) are flexible in the use of particular strategies in specific situations, and (c) are actively engaged in monitoring the course of the solution and in evaluating of success.
By contrast, the pattern of development as reflected in the strategyuse for solving simple addition, subtraction and word problems applied in a longterm perspective throughout the elementary school years shows the typical MD child as being characteristic of: (1) use of backup strategies only, (2) use of the most primary backup strategies, (3) small degree of variation in the use of strategy variants and, (4) limited degree of change in the use of strategies from year to year throughout the primary school.
The basic question then becomes how do individual differences in strategy use relate to the acquisition of the performance on simple arithmetic word problems and simple basic fact problems. But do the MD children have sufficient knowledge of the different taskspecific strategies available to them? It could well be that their knowledge in this respect is limited to backup strategies alone. The results of the study indicate a possible relation between childrens difficulties and the absence of an adequate domainspecific knowledge base of taskspecific strategies. However, to what degree early convergence (flatten) of the MD childrens developmental curves can be counteracted by extended strategy instruction in the early age groups focused directly on the above noted points remains an open question.
Overall, the present study shows a need for a more comprehensive empirical study designed directly to reveal the relation between the childrens domainspecific strategy knowledge and their performance on arithmetic word problems and basic fact problems. It is up to future research to determine whether inadequate knowledge of the different taskspecific strategies may, in itself, be an obstacle to normal development. More valid classroom research is needed to find answers to these questions.
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Children with difficulties in mathematics (*)
Two main groups:

Children with mathematical development in accordance with the developmental delay model.

Children with mathematical development in accordance with the developmental difference model, that is, mathematically disabled children. (MD children)
(*) Children picked up by the school support service as needing a special programme in mathematics teaching.
Short overview
The schools included: 12 primary schools
Number of children included: about 1000
Grade levels included: Grade 1 – grade 7
Main research design:
Longitudinal combined with a crosssectional approach (*)
(*)The children were observed systematically over a period of two years, grade 1 children from the end of grade 1 to the end of grade 3, grade 3 children from the end of grade 3 to the end of grade 5, and grade 5 children from the end of grade 5 to the end of grade 7.
Aspects of strategy variability applied

the use of backup strategies versus retrieval strategies,

the use of specific backup variants,

the number of different strategies used, and

the changes in strategyuse as children moved up through primary school, i.e., from grade 1 to grade 7.
About the sample
The MD children included:

Children registered in the schools ordinary support services as in need of a special programme of mathematics teaching, and

among the 14% bottom group in mathematics achievement tests taken with a two year interval. (*)
The sample included 32 MD children in grade 1, 33 MD children in grade 3, 36 MD children in grade 5 and a corresponding number of mathematically normal children (MN children) in each of the grades.
(*) Thus, in the present study, the definition of mathematically diabled children relate to the persistency of the mathematics difficulties and not to the way ”learning disabilities” is defined in reseach literature.
Categorization of taskspecific strategies in addition
(A1) ADDITION STRATEGIES: BACKUP VARIANTS
(A1a) Count everything, and start again from the beginning. Example: 3 + 5 = ? The pupil counts concretes, e.g. fingers or Unifix rods. First "One, two, three" concretes and continues "one, two, three, four five" concretes. Then he/she starts from the beginning and counts all the elements in the two quantities "one, two, three, four, five, six, seven, eight". The concretes physically represent the integers in the problem.
(A1b) Count everything. Example: 3 + 5 = ? The pupil counts concretes, first "one, two, three" and continues by counting "four, five, six, seven, eight".
(A1c) Counting further. Example: 3 + 5 = ? The pupil uses concretes, but counts onwards from the first number. He/she counts "four, five, six, seven, eight".
(A1d) Minimum variant (minimum number of counting steps). Example: 3 + 5 = ? The pupils count concretes, but in this case count on from the numeral that represents the largest number, i.e. counting from the larger addend. In this example this means counting on from 5, i.e. "six, seven, eight".^{4}
(A1e) The drawing variant. Using a pencil, the pupil drawns lines, dots or suchlike on a piece of paper and afterwards counts these dots or lines to arrive at the answer with the help of A1a, A1b, A1c or A1d.
(A1f) "Touch points" on numerals.The pupil draws (or visualizes) dots in the numerals (an appropriate pattern or dots which represents the numeral). The addition takes place by the pupil using these dots to find the answer with the help of A1a, A1b, A1c, or A1d.
(A1g) Other counting variants.The pupil first draws a line of numbers and counts using that, or counts finger joints, or combines elements from two or more of the strategies described above.
(A1h) Verbal counting. The pupil counts audibly or moves his/her lips as if counting implicitly. The counting has no other directly observable external frame of reference.
(A2) ADDITION STRATEGIES: RETRIEVAL VARIANTS
(A2a) Knows the answer.The pupil recognizes the example and answers immediately.
(A3b) Derived fact variant I. The pupil knows the answer to the addition of various combinations and uses these answers as a basis for counting onwards. Example: 3 + 5 = ? The pupils knows that 3 + 3 = 6 and says, for example, "three plus three is sex .. plus two ... (counts on his/her fingers, for example, and says) seven, eight".
(A2c) Derived fact variant II. The pupil uses his knowledge of several questionanswer combinations which he/she uses as a basis for solving the problem without counting. If the problems is 8 + 6 = ?, the pupil proceeds from eight and says, e.g. "Eight plus two is ten. This means that there are four left. Ten plus four is forteen”.
Pattern of development
MD children:

use of backup strategies only,

use of the most primary backup strategies,

small degree of variation in the use of strategy variants

limited degree of change in the use of strategies from year to year throughout the primary school.
Early and striking convergence of the developmental curves follows a sequence that was fundamentally different (not only delayed) from that observed from that observed in normal achievers.
The findings highlight the MD children’s need for mathematics instruction to shift from computationfocused activities to strategylearning activities.
Main factors focused upon:

Strategy use, i.e., taskspecific strategies

Accuracy and speed of processing

Comorbidity between mathematical and language based difficulties
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