Introduction

Mathematics is considered one of the 'basic subjects' in most countries. Methods of teaching, and the topics that are emphasized in teaching, vary over time and place. The ability to understand and deal with numbers has usually been treated as the most central topic, especially at the primary age. However, other topics such as shape and space, measurement, data interpretation, and applications of mathematics to real-world problems are also given importance, especially in the current British curriculum. Even within this topic of number, there has been a lot of debate as to which aspects deserve most emphasis. In recent times, both British and American educationists have recommended a reduced emphasis on written calculation at the early stages, and a greater emphasis on mental calculation, estimation and problem-solving (Cockcroft, 1981; Kilpatrick, Swafford, & Findell, 2001).

Key Research Questions

1. What is the relative emphasis on procedural and conceptual knowledge in mathematics instruction?
2. How does the use of concrete objects and sensory representations aid in the understanding of number?
3. What is the role of calculators, computers and other technology in mathematics education?
4. What is the need for individualized interventions when children are falling behind?

Recent Research Findings
 
       1.   What is the relative emphasis on procedural and conceptual knowledge in mathematics instruction?

There have been many debates in a number of countries over the years concerning whether mathematics instruction should place greater emphasis on the rote learning of procedures and facts or on understanding concepts. The former is often regarded as 'traditional' and the latter as 'modern'. In general, more recent educators and curricula have placed greater emphasis on conceptual knowledge. Piaget's theories were influential here, as they emphasized the importance of children acting as active explorers, rather than passive learners; and subsequent 'constructivist' theories have argued that the role of teachers is not so much to instruct directly as to provide children with a practical and social context in which they can make their own discoveries (e.g., Cobb, Yackell, & Wood, 1992).

On the other hand, it is simplistic to assume that developments in education have been a simple progression from emphasis on rote learning to emphasis on concepts. Baroody (2003) has pointed out that in early 20^th century America, there were strong conflicts between 'drill theory', emphasizing the rote learning of facts (e.g., Thorndike, 1921); 'meaning theory', emphasizing instruction in concepts (e.g., Brownell, 1938); and 'incidental learning theory', where children were expected to learn through exploration and satisfaction of their curiosity, rather than direct instruction (e.g., McLellan & Dewey, 1895.). Similar conflicts occurred in the UK (Brown, 2001; Cowan, 2003; McIntosh, 1977). As early as 1798, Edgeworth and Edgeworth argued for the greater importance of 'reasoning' over 'repeat(ing) by rote'.

Many educational techniques have been influenced by constructivism, and by earlier theories of the importance of reasoning. These range from techniques that encourage children to develop their own mental strategies, e.g., Cognitively Guided Instruction (Carpenter, Fennema, & Franke, 1993), to those that place greater emphasis on problem-solving in realistic contexts, e.g., Realistic Mathematics Education (Beishuizen, 1997; Freudenthal, 1973). In practice, there is considerable overlap between such methods.

Most educators agree that a focus on facts and learned procedures is insufficient, and that children must understand concepts if they are to develop a flexible 'adaptive expertise' rather than just 'routine expertise' which may be readily applied only to the contexts in which it was learned (Baroody, 2003; Hatano, 1982). It is also, however, generally accepted that rapid, automatic access to facts and procedures is often important, especially in reducing what might otherwise be a very heavy load on working memory (e.g., Geary, Liu, Chen, Saults, & Hoard, 1999).

       2.   How does the use of concrete objects and sensory representations aid in the understanding of number?

There is evidence that mathematical learning tends to be more effective if it occurs in several different contexts, involves different forms of representation, and if children are encouraged to relate these contexts and representations to each other (Ginsburg, 1977; Fuson, 1986, 1992). Otherwise, children may learn very efficiently in one context, or with one mode of representation, but not apply it to others. At one time, it was thought that if children are initially encouraged to use concrete objects to solve arithmetic problems, they will later be able to transfer their learning to other, more abstract problems. However, it has been found that children do not always make this connection, and even if they eventually learn how to do a particular type of problem in both concrete and numerical form, they may not connect the two. As a child informed Hart (1989), "Bricks is bricks and sums is sums". Moreover, children may not always understand the concrete materials themselves, or how these are related to number (Dowker, 2005; Hannell, 2005). Hannell (2005) quotes an adult reminiscing about his school experiences: "I never did really get what those little wooden blocks were all about. I said they were 'tens' and 'ones' because that was what the teacher said we had to call them; but it never, ever dawned on me that they could stand for anything real like ten kids, or ten dollars. They were just little bits of wood that we did things with" (p. 28).

Hence, it is important to present materials in a variety of contexts and a variety of sensory modalities, and encourage children to make links (Fuson, 1992). The use of a multisensory approach, and of materials that lend themselves well to such an approach, are currently recommended by numerous educators (Haseler, 2008; Henderson, Carne, & Brough, 2008) and advocated in the Williams Review (Williams, 2008). At the same time, it is pointed out that simply giving children the materials may not be sufficient; as Haseler (2008) states: "It is also important to remember that however 'good' we believe the equipment to be, it will only be of value to pupils if it is used by suitably trained staff who understand the rationale behind it" (p. 240).

       3.   What is the role of calculators, computers and other technology in mathematics education?

Much controversy has surrounded the question of whether allowing children to use calculators promotes an "unthinking" approach to mathematical problems. Some educators distrust calculators in general. Others consider that they are useful for older children and adults performing advanced calculations, but can interfere with the development of number sense if introduced too early. Dehaene (1997) expresses a contrary view: "I am convinced that by releasing children from the tedious and mechanical constraints of calculation, the calculator can help them concentrate on meaning. It allows them to sharpen their natural sense of approximation by offering them thousands of arithmetic examples" (p. 135).

Most studies of the effects of calculator use have shown few strong effects, in either direction, on arithmetical calculation or reasoning. A meta-analysis of over 80 studies (Hembree & Dessart, 1992) indicated that calculator use had little effect on the development of arithmetical computation skills. Problem-solving was better when pupils used calculators than when they did not. Experience with calculators had no effect, or a weak positive effect, on problem solving when calculators were not available.

      4.    What is the need for individualized interventions when children are falling behind?

Individualized interventions for children with mathematical difficulties have been advocated for at least since the 1920s (Buswell & John, 1926), but have had increasing emphasis in recent years (e.g., Dowker, 2005, 2007; Wright, Martland, & Stafford, 2005). For instance, in the UK, the government's Primary National Strategy has recently developed materials for mathematics emphasizing the individualized diagnosis of the errors and misconceptions shown by children with significant difficulties in mathematical learning (Gross, 2007). Williams (2008) has put forward a strong recommendation for early intervention for primary school children who are experiencing difficulties in mathematics (Recommendation 8 of his review). In particular, he recommends that children with serious difficulties in mathematics should receive intensive one-to-one intervention from a qualified teacher, though paired or small group work may be appropriate in some instances.

Dowker (2004, 2009) has set out some general principles for intervention for children with mathematical difficulties. Williams' (2008) report recommended that interventions should be individualized; but that in many cases they do not need to be very time-consuming or intensive to be effective. Interventions can take place at any time in a child's school career, but ideally should take place relatively early, both because mathematical difficulties can affect performance in other areas of the curriculum, and in order to reduce the risk of children developing negative attitudes and anxiety about mathematics. Interventions that focus on the specific components with which a particular child has difficulty are likely to be more effective than 'one size fits all' programmes. Therefore, intervention schemes should involve assessments of children's specific strengths and weaknesses within mathematics so that each individual child's weaknesses can be targeted effectively.

Conclusions

There has been a great deal of debate over the years as to how best to teach children mathematics. There is no firm consensus on any single method or approach; and indeed different approaches are likely to be appropriate to different children and different situations. Factual, procedural and conceptual knowledge are all important, and should not be seen as at 'war' with one another. Conceptual knowledge is often essential to the effective learning of facts and procedures. The use of concrete materials in teaching arithmetic is often very useful, as is the use of calculators, computers and other forms of technology; but no equipment is a panacea in itself. It is important to present materials in a variety of forms, and to encourage children to see the links between them. Interventions, especially those targeted to individual children's strengths and weaknesses, can be crucial in ameliorating difficulties in learning arithmetic.