There is much research that indicates that the school environment and teaching methods are important influences on the mathematical performance of children throughout the ability range. Appropriate teaching may prevent some mathematical ‘difficulties’ from ever becoming apparent; and many mathematical difficulties are undoubtedly mainly the result of limited or inappropriate teaching (or, worldwide, to a complete or near-complete lack of schooling). This report will not, however, deal with general educational influences, which have been extensively discussed elsewhere. Rather, it will focus on targeted interventions with children who demonstrate, or are at risk for, mathematical difficulties that are considerably greater than those typically experienced by others within the same educational system.
It is often important to distinguish between tools of access and tools of intervention. Tools of access are means of circumventing a difficulty which does not affect mathematics learning directly, but which may interfere with a child’s benefiting from standard forms of mathematics teaching or mathematical activities. Examples may include the provision of a sign language interpreter for a child with hearing impairment; oral presentation of material to a visually impaired or dyslexic pupil; or allowing a dyslexic or dyspraxic child to use a word processor instead of writing by hand. Tools of intervention involve remediating, or in some case preventing, difficulties with mathematical learning itself. The focus of this report is on tools of intervention, though the two may not always be sharply distinguishable. There have been far fewer intervention programmes for children with arithmetical difficulties than, for example, for children with literacy difficulties (Brooks, 2003). Nevertheless, there have been more such programme, and over a longer historical period, than is often realized.
Individual differences cannot be ignored or disregarded. Ignoring the existence of individual differences (whatever their sources) is not going to make them disappear. Individual differences are not solely created by the school environment, though they may be exaggerated by it. Unless expectations are reduced for all pupils, some pupils will struggle and become discouraged, if not 'math phobic', if their difficulties are not taken into account. Unless expectations are raised to the point of creating difficulty for a large number of pupils, some other pupils will find the tasks too easy and become bored. Pupils who are much better at some components of arithmetic than at others will not be helped to use their strengths to overcome their weaknesses. [None of this is to say that mathematical weaknesses are innate, fixed or immutable: merely to say that ignoring or disregarding them, or demanding that pupils perform at a higher level of ability, is not necessarily going to make them overcome their weaknesses.]
The needs of children with mathematical difficulties are being increasingly recognized in many other countries, including the United States (Ginsburg, 1977; Muller and Mercer, 1997; Woodward, 2004); Australia (Wright, Martland and Stafford, 2000; Wright, Martland, Stafford and Stanger, 2002; Van Krayenoord and Elkins, 2004); Germany (Bzufka, Hein and Neumarker, 2001); Belgium (Desoete, Roeyers and DeClercq, 2004), Spain (Casas and Garcia Castellar, 2004), Italy (Cornoldi and Lucangeli, 2004) and Japan (Tsuge, 2001; Woodward and Ono, 2004). The need to take mathematical difficulties into account, even while attempting to raise overall standards, has been discussed from an American perspective by Muller and Mercer (1997). They discuss the teaching of pupils with mathematical difficulties (specific or otherwise) in mainstream classrooms in the face of pressures to increase the rigour of mathematics education courses for all pupils, and to increase the mathematics requirements for high school graduation. The authors emphasize the importance of accommodating diversity, and adapting teaching to individual strengths and weaknesses, "Once the [individual's] strengths and weaknesses have been assessed, his or her mathematical needs should be explored, related to both short-term and long-term goals... When considering the diversity among all students with and without disabilities, it is unrealistic to assume that one curriculum or one set of standards will suit the math needs of everyone...Most individuals with disabilities are going to need accommodations or modifications in textbooks, assignments, teaching methods, tests and homework". Like a number of other researchers on intervention, the authors also emphasize the importance of teachers and researchers collaborating in devising and selecting effective methods of teaching pupils with and without disabilities.
Research strongly supports the view that children's arithmetical difficulties are highly susceptible to intervention. It is not the case that a large number of children are simply 'bad at maths', and that nothing can be done about it. For example, it is notable that Dowker’s (2001, 2003) study indicates strong improvements occurred in the WISC Arithmetic subtest: part of a test used predominantly to test ‘IQ’ rather than school achievement. Moreover, individualized work with children who are falling behind in arithmetic has a significant impact on their performance. The amount of time given to such individualized work does not, in many cases, need to be very large to be effective. Quite small amounts of individualized work may bring a child to the point where (s)he can profit much better from the general (‘Wave 1’) teaching that (s)he receives.
Further investigations are of course necessary show whether and to what extent the individualized interventions described here are more effective in improving children's arithmetic than other interventions which provide children with individual attention: e.g. interventions in literacy, or interventions in arithmetic which are conducted on a one-to-one basis but not targeted toward individual strengths and weaknesses. It is also desirable to investigate whether different approaches to such intervention (e.g. age when intervention starts; degree of intensiveness; the particular components emphasized) may be differentially appropriate to different groups of children.
It is important to compare the different programmes using similar forms of assessment. At present, as pointed out by Kroesbergen and Van Luit (2003) and by Rohrbeck et al (2003), it is difficult to compare programmes, because most researchers and project managers have worked in relative isolation, unaware of each other’s programmes. Most programmes have involved different methods of sampling and different forms of assessment, rendering it difficult or impossible to make valid comparisons. It would also be desirable to investigate the potential for similar types of intervention in areas of mathematics other than numeracy: e.g. geometry and measurement.
It would also be desirable to investigate who are the best people to carry out individual and small-group interventions: classroom teachers (perhaps being relieved of classroom duties for short periods for the purpose); learning support assistants; special needs teachers; or in certain cases parents? Future goals should include further development and investigation of individualized and smallgroup interventions with children outside the age range (schoolchildren over 7) which has so far received most attention from this point of view . Action research by teachers could play a significant role in such studies. In particular, given the already-demonstrated importance of preschool interventions with at-risk children, it would be desirable to have more investigations of methods of assessing preschool children’s early mathematical abilities; of predicting different forms of mathematical difficulty; and of targeting early interventions to have maximum impact in preventing such difficulties. Among school age children, it would be desirable to focus more on interventions for young children: those under 7. This younger age group seems particularly important from this point of view, as the first two years of school are when much of the foundation is being laid for later mathematical learning; and identifying and intervening with difficulties at this stage has the potential prevent children from developing inappropriate arithmetical strategies which may handicap them in later work, and from developing negative attitudes toward arithmetic.
At the other end of the scale, another goal of research should be to investigate the role of targeted interventions for adults with mathematical difficulties. Most intervention programmes have been with children or adolescents. Since numeracy difficulties have lifelong implications, it is important that more work be carried out on diagnosis and intervention for such difficulties in adults. Numeracy is increasingly included in 'basic skills' programmes for adults; though most such programmes do not differentiate between adults who have not learned such skills due to lack of educational opportunity, and those who have specific difficulties in arithmetic. Greater communication and collaboration between teachers, researchers and policy-makers is vital. This was indeed pointed out by Piaget (1971), but has only rarely been put into practice.
It is clear that many children have difficulties with some or most aspects of arithmetic. Arithmetical thinking involves such a wide variety of components; so that there are many forms and causes of arithmetical difficulty, and many degrees of severity. Only a minority of children have dyscalculia as defined as severe specific difficulties with most or all aspects of arithmetic. However, a very significant proportion of the population have difficulties with certain aspects of arithmetic, which are sufficient to cause them at least some
practical and educational problems, especially without intervention. It is important to recognize those individuals who do have major difficulties with very basic number concepts: e.g. perceiving and distinguishing small quantities (Butterworth, 1999, 2002), which are likely to result in severe general mathematical difficulties. Most children with mathematical difficulties do not have such extreme problems, and should be seen more as part of a continuum of mathematical performance. This implies that there is also a continuum of intervention needs. This may range from:
No active intervention needs, but pupil may benefit from teachers and others being aware of his or her specific strengths and weaknesses;
through:
Need for flexible adaptations of programme of activities within whole-class setting ('Wave One' intervention').
Need for small group provision ('Wave Two' intervention)
Need for individualized ('Wave Three') provision of an infrequent and/or non-intensive nature.
Need for intensive individualized ('Wave Three') provision
to
Need for totally individualized programme and/or special educational setting.
Children may require different degrees and types of intervention at different times in their school career, or for different aspects of the mathematics curriculum. Relatively small amounts of individual intervention may make it possible for a child to benefit far more fully from whole-class teaching. There are some children who have an greater than average chance of developing arithmetical difficulties, and should be targeted for assessments. These include children with any degree of general learning difficulties, and those with dyslexia and/or oral language difficulties. Also, arithmetical difficulties sometimes run in families, so that children whose parents or siblings have trouble with arithmetic are at increased risk of having difficulties with arithmetic themselves. However, not all children with these risk factors will have difficulty with arithmetic; and some children with none of these risk factors have specific difficulties in arithmetic.
Children who have difficulty with mathematics usually (but not always) have particular difficulty in remembering number facts. This is probably especially true of children who are also dyslexic or have other language difficulties. Often, such children can learn to compensate for their difficulties by using strategies based on reasoning; and eventually by using a calculator effectively. However, some children have more serious difficulties and are restricted to using cumbersome strategies that are based on counting. Interventions can also help these children, but may need to be longer and/or more intensive. No two children with arithmetical difficulties are the same. It is important to find out what specific strengths and weaknesses an individual child has; and to investigate particular misconceptions and incorrect strategies that they may have. Interventions should ideally be targeted toward an individual child's particular difficulties. If they are so targeted, then most children may not need very intensive interventions.
Mathematics assessment is useful when it enables teachers to identify the particular problems that children are experiencing, and to profile their strengths and weaknesses. Forms of assessment that compare children in a linear fashion and label some children simply as ‘bad at maths’ are likely to be counter-productive. Peer tuition and computerized teaching play a useful role in mathematics interventions, but cannot substitute for interaction with a teacher.
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