1
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
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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).
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 childrens 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 problems1
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 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
1The basic facts of arithmetic are the simple, closed number sentences we use when we compute. These
number sentences involve two one-digit addends if they are basic addition or subtraction facts, or two one-digit
factors if they are basic multipication or division facts. Examples of basic facts include the following: 6 +7 = 13,
12 - 8 = 4, 3  6 = 18, 27 : 9 = 3. Basic fact problem solving is to supply missing sums, addends, products, and
factors for these basic facts.
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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 childrens 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 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
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 childrens 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
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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 &
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 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.
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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 taskspecific 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 2-9. 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-II2.
2
Målingen ved T-I ble foretatt i løpet av de siste fire ukene av vårhalvåret og målingen ved T-II så nøyaktig
som mulig på samme dato to år senere.
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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
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
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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 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
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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 question-answer
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”.
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The more detailed procedure used by the individual pupil within the defined frame of reference for addition strategies
A1b, A1c and A1d may vary. Some children solve the problem 3 + 5 = ? with the help of A1b without first counting the
quantity 5 (because they know they have five fingers). Others count the quantity five first. When operating with larger
numbers, e.g. in the problem 3 + 8 = ?, the student usally counts the quantity 8 first.
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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 problem-solving 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 childrens consistent use of backup strategies might reflect both fact
retrieval problems and working memory problems (Garnett & Fleischner, 1983; Geary, 1990;
Geary, 1993; Geary, Bow-Thomas & 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 age-determined 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
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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 memory-retrieval 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 domain-specific 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 domain-specific 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
problem-solving 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 domain-specific
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 & Burlingham-Dubree, 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 childrens insufficient domainspecific strategy knowledge in itself limits the choices available to them. Accordingly, I
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suggest the existence of important individual differences in the wealth of domain-specific
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 domain-specific 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 domain-specific
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 &
Burlingham-Dubree, 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 childrens
performance showed a course of development with a large and constant discrepancy,
throughout the primary school period, between their level of performance for solving number-
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fact 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
childrens 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 individuals school instruction
had encouraged or discouraged such variability. If the above suggestion is valid, the
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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 ready-printed 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
ready-printed exercise books, concretes, or real-word 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 task-specific
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 knowledge-base of task-specific
strategies to perform a particular task or a particular problem-type, (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 strategy-use for solving
simple addition, subtraction and word problems applied in a long-term 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 childrens difficulties and the absence of an adequate domain-specific knowledge
base of task-specific strategies. However, to what degree early convergence (flatten) of the
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MD childrens 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 childrens domain-specific 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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17
Children with difficulties in mathematics (*)
18
Two main groups:
(1) Children with mathematical development in
accordance with the developmental delay model.
(2) 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.
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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 cross-sectional 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.
20
Aspects of strategy variability applied
(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.
About the sample
21
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. (*)
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 task-specific strategies in addition
22
(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
4
The more detailed procedure used by the individual pupil within the defined frame of reference for addition strategies
A1b, A1c and A1d may vary. Some children solve the problem 3 + 5 = ? with the help of A1b without first counting the
quantity 5 (because they know they have five fingers). Others count the quantity five first. When operating with larger
numbers, e.g. in the problem 3 + 8 = ?, the student usally counts the quantity 8 first.
23
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
question-answer 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
24
MD children:
(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
(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 strategylearning activities.
Main factors focused upon:
25
(1)
Strategy use, i.e., task-specific strategies
(2)
Accuracy and speed of processing
(3)
Comorbidity between mathematical and language based
difficulties