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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 task-specific 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 task-specific strategies they used were recorded on a trial-by-trial 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 computation-focused activities to strategy-learning 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., 8-9 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).
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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 strategy-differences 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 problem-solving 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 problem-solving 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 problem-solving strategies used by mathematically normal childrenhave 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 &Burlingham-Dubree,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).
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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 long-term 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 number-fact 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 & BurlinghamDubree, 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 strategy-use 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 task-specific 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 strategy-use 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
(1)Children with mathematical development in accordence with a developmental delay model.
(2)Children with mathematical development in accordance with a developmental difference model, that is, mathematically disabled children (MD-children).
About the sample
The MD children included:
(a)Children registered in the schools ordinary support services as in need of a special programme of mathematics teaching, and
(b)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 task-specific strategies they used were recorded on a trial-by-trial 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 pair-wise combinations of the integers 29. The (8) tie problems were excluded. (A tie problem is 2 + 2). The remaining problems, which consisted of 56 single-digit addition problems of the form "a + b", were divided in two equivalent halves (28 in each), one half for use at T-I and the other at T-II[2]. The two halves were counterbalanced. This means that all the 56 problems were pair-wise 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 trialbytrial basis. Several previous studies have shown that children can describe arithmetic problemsolving 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 followup 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 twoway 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 task-specific 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.