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1: INTRODUCTION

Section Contents:

The Usefulness And Importance Of Maths

Numeracy matters. Poor numeracy skills are a major disadvantage in everyday life and in the job market. The National Child Development Study found that 25% of adults aged 37 had numeracy skills so low as to make it difficult to complete everyday tasks like shopping successfully (Bynner and Parsons 1997). Men with poor numeracy skills were more prone to unemployment, more likely to be in manual jobs, less likely to have had work related training and more likely to earn a low weekly wage. Restricted numeracy also had some impact on women, who were more likely to be in part time jobs: only 25% with poor numeracy held a full time job compared to 40% of those with adequate numeracy. Alarmingly, less than 20% of those with poor numeracy skills as adults were identified as poor at maths by their teachers. Furthermore, there was evidence that adequate literacy had little effect in cushioning the impact in adult life of poor numeracy.

However, it cannot be assumed that more teaching in school would solve the problem, since there is also concern about the relationship between the school mathematics curriculum (before and after the National Curriculum) and the mathematical demands of everyday life and employment. A survey by Raines (1988) indicated parents were frequently critical of the relevance of the school mathematics they had learned to later life. As Nunes, Schliemann and Carraher (1993) point out in their exploration of the relationship between school mathematics and street mathematics, children can carry out quite complex arithmetical calculations in relation to their life needs without any formal teaching, while subsequently proving unable to do equivalent problems in school, where the problem is not only decontextualised but a different and singular route to the solution is required. There is not only a lack of generalization of school mathematics to real life, there is a lack of generalization of the mathematics of real life to the school.

The Cockcroft Report (1982) was an important landmark for mathematics education in the UK and recognised and emphasised the importance of parental influence and the early age by which attitudes to mathematics are fixed. It defined the aims of mathematics teaching as developing powers of logical thought and equipping children with numerical skills. The research for the Cockcroft report suggested that after three years in secondary (high) school, children understood important mathematical concepts very little better than when they left primary (elementary) school, sometimes less well. More recently the National Curriculum assessment has suggested that a range as wide as seven years could be expected in the mathematical attainments of children at the beginning of their secondary education.

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Standards In Mathematics: Nationally & Internationally

Over the years considerable concern has been expressed about the performance of British children in mathematics at primary and secondary level when compared to the performance of children from other countries.

The report "Learning Mathematics & Science: The Second International Assessment of Educational Progress" (Foxman, 1992) involved 20 countries in examining the attainment of 9 and 13 year olds in maths and science (some countries only examined attainments at age 13). The results did not make reassuring reading. Pupils from England and Wales and the U.S.A. performed poorly in mathematics and science relative to pupils from other countries. Switzerland achieved the best results at age 13 in Europe, while pupils from Korea outperformed those from the other countries.

The Third International Mathematics and Science Study (e.g. Scottish Office 1996a) involved 40 participating countries in 1995. In the first two years of high school, Singapore, Korea, Japan and Hong Kong occupied the first four places. England, Scotland and the USA hovered around 20th place, below a number of Eastern European countries where children had experienced a year less in school. Scotland was characterised by relatively high numbers of hours devoted to mathematics teaching and high availability of remedial mathematics teaching, but low use of homework, high use of calculators, high use of worksheets and textbooks, and a greater tendency to blame poor results on low pupil ability.

A study of mathematics ability in adulthood in seven industrialised countries left Britain bottom of the list (Basic Skills Agency 1997). Japan was top. Women did worse than men. Subjects were aged 16 to 60 years, and the older subjects did better than the younger.

Caution is needed in the interpretation of these results. Such "league tables" often mask quite small absolute differences between countries, different outcomes at different ages and stages, and different distributions of scores between different countries (e.g. in 1991, able British mathematicians did as well as Koreans, but low attaining pupils did much worse). Nevertheless, there was little ground here for complacency.

The Scottish Office (1996b) Assessment of Achievement Programme (AAP) survey of maths in Scottish schools examined performance at Primary 4 (age 8), Primary 7 (age 11) and Secondary 2 (age 13) levels in mathematics, relating them to the Scottish 5-14 Curriculum (Scotland does not have the same mandatory "National Curriculum" as England and Wales and chronological year bands have different labels). The apparent drop in mathematics performance in several areas from 1991 to 1994 provoked widespread concern, as did the developing trend for mathematics performance to decline as children grew older. In Problem Solving some pupils often used a random approach rather than a systematic one. Consequently, more emphasis was subsequently placed on solving a wider range of problems in context with direct teaching of strategies coupled with more practical experience and group discussion.

However, recently concern has again been expressed by Her Majesty's Inspectorate about the low level of problem-solving skills among children in Scotland. A recent HMI Report (Standards and Quality in Primary Schools: Mathematics 1998-2001) (www.scotland.gov.uk/library3/education/saqm-00.asp) noted that problem-solving skills were still the number one priority for development and improvement in primary schools in Scotland.

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Attitudes To Maths

For children with learning difficulties, the hierarchical nature of mathematics as a subject of study had often meant 'going back to basics' - repeating what has already not been understood but with the effect of facing failure yet again - with the result that, as the Cockcroft Report observed:

'by the end of the primary years a child's attitude to mathematics is often becoming fixed ... and for many this meant being fixed as an attitude of rejection and antagonism'.

The survey by Raines (1988) showed that mathematics was an emotive topic for parents - even in middle class areas and schools with good home-school relations. Enjoyment of mathematics and self-concept were closely linked; very few parents who saw themselves as good at mathematics disliked doing it, whilst most of the parents who saw themselves as bad at mathematics hated it, especially at secondary school. The latter parents had strong memories of school experiences, often unfavourable. Many parents felt the aims of mathematics teaching in schools should include the development of confidence and enthusiasm as well as understanding. These findings led Raines to criticise "parental involvement in maths" schemes which were merely "shipping home the curriculum" as naively ignoring the affective and historical dimensions of parents' own reality.

Disliking mathematics can all too readily be construed as "normal". Adults seem prepared to say 'I'm no good at Maths" or "I never was any good at Maths at school' when they are very unlikely to be as ready to say 'I can't read'. Askew and Wiliam (1995) found that "pupils' self confidence and beliefs affect their success in mathematics", and argued that those who lacked confidence in their ability try to avoid challenges and show little persistence. The dangers of parents' modelling negative attitudes to mathematics which are then adopted by children and become a self-fulfilling prophecy is apparent.

Any attempt to involve parents in a more active role in mathematical education thus needs to make allowance for the influence of past experiences on their self image and confidence. The Problem-Solving with BP project is deliberately designed to deal with this, through inviting parents to a pre-project meeting initially without their children. This gives them the opportunity to play, learn and rehearse the games without embarrassment, provides both positive reinforcement and enthusiastic modelling and offers positive practice with non-threatening feedback. They also openly discuss misconceptions of their own everyday use of and effectiveness with mathematical knowledge, and consider how their own attitude to mathematics might affect their children's learning.

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Gender Differentials

Differences between girls and boys attainment in mathematics has been the focus of debate and concern for many years. However, the research evidence does not always substantiate popular assumptions.

For instance, Askew and Wiliam (1995) found that up to age 11 the achievement of girls and boys was comparable, while at age 15 boys scored higher than girls across mathematics topics, although the latter tendency was decreasing. In the TIMMS report mentioned above (Scottish Office 1996a) boys performed better than girls in 31 out of 38 countries in the early years of secondary school, but the difference was only statistically significant in six. As in previous studies, differences between countries were certainly greater than differences between genders (c.f. Hanna 1989).

The Assessment of Achievement Programme (Scottish Office 1996b) survey suggested little general difference in the performance of girls and boys on most tasks carried out in their survey. However, it was reported that girls outperformed boys in a "minority of tasks" at P4; while boys did better than girls at P7, again yielding some evidence of a relative gender decline over the years.

This echoed some earlier research from the USA, which had suggested that girls' interest and confidence in mathematics dropped in early adolescence before their actual performance in maths dropped (Fennema and Sherman 1978, Fennema and Meyer 1989). This led to the proposal that if their interest and confidence could be maintained, so could their performance. In the U.S.A. a number of programmes aiming to achieve this have been established (Campbell, 1995), such as `Eureka' and `Operation Smart'. The important common elements seem to be: many hands-on activities, enjoyment inbuilt, providing time for questions, relaxed activities with little or no emphasis on individual competition, and many opportunities to see that mathematics and science are as readily "girl" fields as boys'.

Leder (1990) noted that in the mathematics classroom, boys interacted more frequently than girls with their teachers, both seeking and receiving more attention. The girls' lower frequency of interaction might be connected to their loss of interest and confidence and the subsequent falling off of their performance in mathematics, although the causal direction is difficult to establish.

Other international studies of gender differences in maths attainment have produced mixed results.

Many have found no gender differences (e.g. Stocking and Goldstein 1992 with high ability secondary students, Sedlacek 1990 at third grade level, Kohr et al. 1989 at fifth, eight and eleventh grade, and Tartre and Fennema 1995 at any grade 6 - 12 in a longitudinal study). Some of these have found race and socio-economic differences to have greater impact (e.g. Kohr et al. 1989, Sedlacek 1990). Some have found gender differences favouring males which were however small (e.g. the Hyde et al. 1990 meta-analysis of mathematical attitudes and affect and Manger's 1995 study of Norwegian third graders - although in the latter there were more marked gender differences at the extreme tails of the distribution.) Few studies have found differences reaching statistical significance, although Brown's (1991) study did so, finding second and fifth grade girls better than boys at reading while the boys were ahead in mathematics achievement. However, Brandon et al. (1987) found girls in Hawaii did better than boys in grades four, six, eight and ten.

Teachers and peers are of course not the only potential source of any gender stereotyping regarding mathematics. Children's beliefs about their own achievement seem related more to parental expectation about the child's achievement than to the parents' own level of achievement. There has been particular interest in maternal expectations of daughter's achievement in mathematics, but significant effects found only with more highly educated mothers and/or high achieving daughters (Jayaratne 1987, Dickens and Cornell 1990).

In summary, gender differences in mathematics achievement might be small in general and reducing, but they might nevertheless be educationally significant at later ages, at certain ability levels, in certain countries and cultural groups, and in certain mathematical topics. However, the transmission of negative gender stereotyping in relation to achievement expectations is clearly undesirable from any source, and enhancing parental encouragement in this regard would seem a useful component of any project.

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The Language Of Maths

Four essential components of children "doing mathematics" were suggested by Jones and Haylock (1985): concrete experience involving motor activity, language, pictures and symbols. As we have noted above, children and their parents and teachers talk and listen mathematics as well as visualise, read and write it, and discussion is crucial in developing understanding, especially for children with learning difficulties in this area (Daniels and Anghileri 1995).

As with other curricular areas (e.g. science), information transfer and processing is heavily dependent upon language. Mathematics has much specialist vocabulary, including that applied to abstract and complex concepts, as well as using some "everyday" vocabulary with more specific and restricted meanings. Durkin and Shire (1991) and Pimm (1991) suggested that words first encountered in a non-mathematical context (e.g. above, difference, figure, make, right, table, value) could cause particular difficulties for children owing to their ambiguity. The potential for confusion is enormous.

The linguistic aspect of learning mathematical concepts is thus of particular importance (Choat 1981). Concept formation is aided greatly by the ability to use the related language, whilst the learning of new concepts is closely associated with the acquisition of new words which are meaningful. Children might learn words without really understanding the associated concepts, while their understanding of some concepts might be underestimated because they do not use the "official" terminology. Jones & Haylock (1985) described activities in which children used language in the course of mathematical activities in ways that were meaningful to them. They emphasised the value of this with the slogan 'understanding means making connections' - between the concrete/motor, linguistic, pictorial and symbolic aspects of mathematics.

Correspondingly, too heavy a reliance on the medium of language in the process of teaching mathematics is likely to differentially disadvantage children whose language is not well developed.

In Problem-Solving with BP projects the need for the children to use and understand words and phrases in a mathematical context, through the discussion of joint and purposeful concrete activities, is strongly emphasised to both parents and peers (and through them, to teachers). One of the advantages of mathematical games is that they readily stimulate related discussion.

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Maths Games

At least at a superficial level, parents and teachers are likely to have some shared understanding of what "reading" is. This is much less likely with mathematics, however, as maths is a much larger and more varied area - certainly not a unitary skill. Asking parents to "do maths" with their children therefore holds greater possibilities for confusion and harm, and stronger support or scaffolding of the parent/child interaction is needed. The structured rules and materials of games provide this, while the choice of games avoids any danger of an autocratic "top-down" ethos likely to disempower the participants.

As much research has shown (e.g. Hughes 1983, Rogers & Miller 1984), if mathematical content can be contained in play form, motivation for learning will be so powerful that the question of 'relevance' will never arise for the child. Skemp (1989a,b) specifically devised games to provide shared experiences which gave rise to mathematical discussion among the children. The peer group interaction was seen as an important method of learning and much less threatening than being told what was wrong by the teacher. Skemp found his materials provided a focus for inservice training for teachers, which improved their own mathematical knowledge as well as providing insight into how children learn.

Games were adopted in Problem-Solving with BP to provide meaningful shared activities yielding practical experience of underlying mathematical concepts without the need for elaborate training. The emphasis was on activities which were intrinsically enjoyable, in order to create the opportunity for attitude change in children and parents alike. They enabled the matching of prediction against outcome, experience against expectation, in activities in which the children could learn from and compare ideas with their parents but in the context of an equality between partners forced by games of chance. Overlying all of this was the importance of the opportunity to explore the pragmatic use of the language of mathematics through the activities.

Having said all this, there are no universal panaceas, and the nature and quality of the games is all-important. Many "maths games" accompanying commercially published school-based maths schemes are utterly dreary and pedagogically inept. Those embarking on a Problem-Solving with BP project must choose the games with great discrimination and care.

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Parental Involvement In Maths

Various types of programmes to involve parents and other carers in the mathematical development of their children have become more common throughout the world, but perhaps particularly since the start of the 1990s in North America.

Reports offering a wealth of organisational detail are highlighted here. A series of six workshops for parents were described by Goldberg (1990), reviewing mathematical activities usable in the home with the aim of improving achievement in and attitudes toward maths for 8-12 year old children. A substantial guide for parents in Australia was developed by Costello et al. (1991), emphasising mathematics in everyday life and the importance of discussion and language. Home activities for children aged 5 to 13 years and their parents were detailed by Kanter, Dorfman and Hearn (1992) in a booklet published by the US Department of Education, with an emphasis on communication and developing positive attitudes.

Owens (1992) produced the delightfully titled "Parent-Helper Book for Those Who Want Arithmetic Made Touchable" - which in fact also covered geometry and other areas and included detailed activity guidelines. "Natural Math" was a programme offered to preschool and kindergarten Seminole and Chickasaw Head Start Native American and African American families (Sears and Medearis 1992, 1993), utilizing activities and games developed to relate to the home culture. The "Family Math" programme has proved very popular in the USA and Canada and has extended to parts of Australia (Onslow 1992).

Martin (1993) focused on parents in adult literacy classes, creating take-home parent and child activity kits designed for use with everyday materials, launched via parent workshops and including instruction sheets in Spanish. Planning and organisation considerations in establishing parent-teacher partnerships in mathematics were outlined by Neil (1994). "Beyond Facts and Flashcards" was the evocative title of a parent guide produced by Mokros (1996), designed to help parents uncover the mathematics in their daily lives through everyday experiences, suggesting many activities. Similar creations had come earlier from Valentine (1992), Leonard and Tracy (1993) and Sharp et al. (1995).

However, (with the exception of the long-standing Family Math programme), similar developments had begun somewhat earlier in the UK.

Following the surge in interest in parental involvement in reading at the beginning of the 1980s, a ripple effect became discernible in mathematics. By 1983 Jennings had published a report on parental involvement in maths with high school pupils, and a number of postgraduate research theses followed (e.g. Paskin 1986, Risk 1988).

Set in a deprived multi-cultural urban area of West Yorkshire, Alan Graham's "Sums for Mums" project, coupled with his book "Help Your Child With Maths" (Graham 1985), was another major development. The project targeted women and aimed to enhance their self-esteem as mathematicians with onward transmission to their daughters, this focus reflecting funding by the Equal Opportunities Commission. Although set in local schools, the workshop sessions were oriented to adult learners, and eschewed "conventional textbook maths". This was followed by PRISM (Parent Resource in Support of Maths). However, little substantive summative evaluation was reported. Nevertheless, Graham's (1985) book remains a major resource in the area.

The IMPACT project, widely known in the UK, involves the class teacher sending home mathematical activities which parent and child carry out together (Merttens and Vass 1987, 1993). It is closely tied to traditional classroom activities which the child exports to the home, often involving whole classes completing the same "homework" simultaneously. This enables the activity to be closely articulated to current class teaching, but is a more "top-down" approach. Surprisingly, given its high profile, little evaluation evidence on Impact other than the anecdotal appears to be publicly available (Merttens & Vass 1993).

The Paired Maths project in Kirklees (West Yorkshire) involved mathematical games, structured in three levels to be appropriate for Key Stage 1, 2 and 3 pupils. A controlled study using pre- and post-testing of mathematical skills found marked advantages for early years (Key Stage 1) children who had participated in a parental involvement project of this kind. Positive subjective evaluation was reported from a year-wide project in a high school in a disadvantaged area. Further details will be found in Topping and Bamford (1998a,b).

Other projects akin to Problem-Solving with BP have sprung up in various parts of the country (Jennings, 1983; Woolgar, 1986; Perry & Simmons, 1987; Clive, 1989). One such in Scotland is the Play Along Maths programme described by Cheyne (1994, personal communication), a Home School Community Tutor. The programme targets families when children are just beginning school. Activities and games (including jigsaws and peg games, all cross-referenced to the prescribed 5-14 Curriculum) are sent home, coupled with "activity cards" bearing ideas and rules prepared by volunteer parents. Much emphasis is placed on language, under the rubric of "Chat-Along", and essential vocabulary amounting to 200 words is mapped out. Each game or activity is used 10-15 minutes per day for a week, then changed. Families keep diaries noting the game/activity, day, time and any comments - children adding smiley faces if they liked the game.

Neilan and Currie (1994) involved parents in a series of four workshops over a six week period, involving mathematical workbook tasks and problem solving activities which were continued at home and supplemented with number games. Pre and post norm-referenced number tests were applied to children in a mixed ability class of 18 five year olds randomly assigned to control and experimental groups. Uptake was 100% and attendance at workshops very good. The control group was offered involvement in the programme after post-test. Experimental and control parents were asked not to confer! In the event, the experimental and control groups were not equivalent on pre-test scores, those of the latter being lower. Experimental group children gained more on the test than a comparison group, but the difference did not reach statistical significance. Subjective feedback from parents was very positive.

In summary, a number of small and brief parental involvement in maths projects in the UK have nevertheless shown encouraging results, especially as three were controlled studies. Children involved ranged from very young primary school children to high school pupils, mostly but not exclusively those with mathematics difficulties. Gains have been demonstrated on various kinds of tests although not all gains reached statistical significance - this being elusive with small samples. Subjective feedback was ubiquitously positive. The time costs to parents of involvement was very various in different projects, and this has implications for the wider involvement of more parents.

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Peer Tutoring In Maths

Peer tutoring is characterised by specific role taking: at any point someone has the job of tutor or helper while the other (or the others) are in role as tutee(s). Quite specific procedures for interaction might also be outlined, in which the participants are likely to have training which is specific or generic or both. Both members of the pair should find some cognitive challenge in their joint activities and the tutor should be "learning by teaching". Gains for both helpers and helped are targeted; double added value.

Although there are a number of reports of peer tutoring in maths in the international literature, many in the USA have involved highly structured drill and practice routines in which the tutor merely apes the role of a traditional teacher, while those involving mathematical games have tended to be descriptive and omit substantive evaluation.

Topping and Bamford (1990) recorded the use of mathematical games in regular mathematics classes across the whole first year of a high school in a disadvantaged area, but evaluation was only through subjective feedback.

A more detailed report of a same-age peer tutoring project using games in a primary school in Scotland will be found in section 4 of this manual (distilled from Topping & Bamford, 1998b). This was thoroughly evaluated and found to be successful. This approach has been successfully replicated in Scotland on a cross-age tutoring basis by Topping, Campbell, Douglas and Smith (2001).

Also in Scotland, cross-age peer tutoring of mathematically 8-9 year olds by 12 year olds in a primary school using methods and materials drawn from the regular school mathematics curriculum was reported by Renwick (1995, personal communication).

Mathematically able tutors were paired with able tutees, and less able tutors with less able tutees. The tutors visited the tutees' class weekly and worked with gender balanced groups of three tutees, but each group of tutees was assigned three tutors, thus engaging in three half hour sessions per week. Video process data was gathered but no outcome evaluation reported.

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Tutoring The Maths Curriculum - Duolog Maths

Mathematical games have many advantages for scaffolding the interaction between children learning mathematics and their non-professional tutors, but they also have some limitations. Children might become highly motivated to engage with mathematical games, but not regard the gains as connected with the "proper maths" which constitutes the school curriculum. Gains in confidence and competence in mathematics must generalise to other mathematical problem-solving, both within and outside school, and this might not happen automatically or spontaneously, but need to be engineered.

Just as Paired Reading is a set of generalised tutoring behaviours for reading which can be applied to any book (Topping, 2001), there was a need for a set of generalised tutoring behaviours which could be applied to any mathematical problem. Mathematics is a very wide ranging area, and this proved considerably more difficult for mathematics than for reading, not least as there was less assurance that the tutor would necessarily be competent to solve the problem themselves. However, after a study of successful tutoring behaviours by professional teachers, a set of (much simpler and complementary) tutoring behaviours suitable for use by non-professionals was designed. This system was termed "Duolog Maths". A duologue is a dialogue between two people, and the title was selected as the preferred name by a large group of teachers in the USA - hence the American spelling. Section 5 of this manual gives more detail about the Duolog Maths procedure.

Evaluation of the Duolog procedure to date has been with parents rather than peer tutors. This was a controlled study in Glasgow with P4 to P7 pupils who were weaker mathematicians. Pre-post curriculum-based mathematics tests showed the experimental group made significantly greater gains than the control group (Topping, Kearney, McGee & Pugh, 2001) - see section 7 of this manual.

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Problem-Solving with BP

The Problem-Solving with BP project seeks to blend selected elements of all the important factors and methods outlined above together to create an especially powerful package - novel in its combination more than in its elements, but robust and durable as well as effective - lending itself to widespread replication in many different contexts with minimal resource requirements. Problem-Solving with BP has a strong emphasis on discussion and language and on concrete hands-on activity, with a de-emphasis on written recording.

It can include games which can be played in different ways and at different levels of abstraction and complexity, offering choice from a pool of activities structured to ensure experience of a breadth of mathematical experiences, avoiding a narrow preoccupation with number. These blend co-operation and competition, operating through cognitive conflict between equal (or equalised) partners and also through apprenticeship with a more capable partner.

Problem-Solving with BP also includes a set of generic maths tutoring behaviours or skills (Duolog Maths) which can be applied to any curriculum materials, tasks or problems posed by the teacher or selected as particularly interesting by the pupil.

Or the project can include both games and regular curricular materials (simultaneously or sequentially).

Similarly, the project can operate through peer tutoring, or through parental involvement, or through a combination of both (usually sequentially rather than simultaneously).

The options for operating Problem-Solving with BP are outlined in the figure below:

Peers Parents
Games
Curriculum (Duolog Maths)

The particularly distinctive features of the organisational system of Problem-Solving with BP include:

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Structure of This Manual

After this Introduction, a section on selecting Mathematical Games follows. As such games may be used more for parental involvement than peer tutoring, a section on Organising Parental Involvement comes next. However, games may also be used in peer tutoring, and a section on Peer Tutoring Using Games follows. The generic set of skills or behaviours for tutoring in relation to any curriculum content, tasks, materials or problems known as Duolog Maths is described in the next section. This is likely to be mainly used in peer tutoring, and a section on Organising Peer Tutoring comes next. The main text culminates in a brief section discussing the evaluation of Problem-Solving with BP projects. The References and Bibliography lead to a section of Reproducible masters, and a final section of reproducible Overhead Projection Masters.

Remember that this manual should be read in conjunction with: Topping, K. J. & Bamford, J. (1998). The Paired Maths handbook: Parental involvement and peer tutoring in mathematics. London: Fulton; Bristol PA: Taylor & Francis (ISBN 1-85346-497-X) and the Problem-Solving web site:
http://www.dundee.ac.uk/eswce/research/projects/problem-solving

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