This article was downloaded by: [Florida Atlantic University]On: 10 November 2014, At: 10:55Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number:1072954 Registered office: Mortimer House, 37-41 Mortimer Street,London W1T 3JH, UK

Research in MathematicsEducationPublication details, including instructions forauthors and subscription information:http://www.tandfonline.com/loi/rrme20

SCAFFOLDING FOR SUCCESS:REFLECTIVE DISCOURSE ANDTHE EFFECTIVE TEACHINGOF MATHEMATICAL THINKINGSKILLSHoward Tanner a & Sonia Jones aa Education Department , University of Wales ,SwanseaPublished online: 14 Apr 2008.

To cite this article: Howard Tanner & Sonia Jones (2000) SCAFFOLDING FORSUCCESS: REFLECTIVE DISCOURSE AND THE EFFECTIVE TEACHING OF MATHEMATICALTHINKING SKILLS, Research in Mathematics Education, 2:1, 19-32, DOI:10.1080/14794800008520065

To link to this article: http://dx.doi.org/10.1080/14794800008520065

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of allthe information (the “Content”) contained in the publications on ourplatform. However, Taylor & Francis, our agents, and our licensorsmake no representations or warranties whatsoever as to the accuracy,completeness, or suitability for any purpose of the Content. Any opinionsand views expressed in this publication are the opinions and views of

the authors, and are not the views of or endorsed by Taylor & Francis.The accuracy of the Content should not be relied upon and should beindependently verified with primary sources of information. Taylor andFrancis shall not be liable for any losses, actions, claims, proceedings,demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, inrelation to or arising out of the use of the Content.

This article may be used for research, teaching, and private studypurposes. Any substantial or systematic reproduction, redistribution,reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of accessand use can be found at http://www.tandfonline.com/page/terms-and-conditions

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

SCAFFOLDING FOR SUCCESS: REFLECTIVE 2 DISCOURSE AND THE EFFECTIVE TEACHING OF MATHEMATICAL THINKING SKILLS

Howard Tanner and Sonia Jones

Education Department, University of Wales, Swansea

This paper describes a research study into the teaching of mathematical thinking skills. Nine classes of students (in total) who had followed a course emphasising metacognitive skills outperformed their control groups on assessments of those skills and were also more successful on measures of their mathematical development. However, participant observation data revealed that there were important variations in teaching style between teachers and the success of their classes varied considerably. Observational data was used to classzfi the teaching styles into four groups. The teaching styles of the two most successful groups, the 'dynamic scaffolders ' and the 'reflective scaffolders ', are analysed here.

LEARNING TO THINK MATHEMATICALLY

Learning to think mathematically is more than just learning to use mathematical techniques. Developing a facility with the tools of the trade is but one element in the development of mathematical thinking. From a constructivist viewpoint, the learner is a sense-maker and an active negotiator of meaning. Within this tradition, a clear distinction is often drawn between mathematical thinking and the knowledge base, strategies and techniques described as mathematics (Burton, 1984).

Learning to think mathematically means (a) developing a mathematical point of view - valuing the processes of mathematisation and abstraction and having the predilection to apply them, and (b) developing competence with the tools of the trade and using those tools in the service of the goal of understanding structure - mathematical sense making. (Schoenfeld, 1994, p. 60)

Mathematical thinkers have acquired a socially accepted way of seeing, representing and analysing their world, and an inclination to engage in the practices of mathematical communities.

The idea of mathematical sense making has its roots in constructivism; however, developing a mathematical point of view may be more akin to enculturation into a community. Two clusters of positions are identifiable which claim to explain learning in social contexts. According to the constructivist viewpoint "an individual makes sense of her surroundings, and tests hypotheses and sense making, by means of her actions, and through responses from others (and the environment)", whereas according to the sociocultural viewpoint "pupils learn to operate mathematically in contexts" and "a person's identity in a new context is constructed within that context" (Dawson, 1994, pp. 24-25).

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

This distinction can be surnrnarised as being the difference between individuals constructing their world by making sense of it, and being constructed by their world through participating in it. Taking a middle position between individualistic and collectivist perspectives, one might claim that "the teacher and students interactively constitute the culture of the classroom" through negotiation and communication (Bauersfeld, 1994). From this viewpoint, pupils should learn by participating in a classroom culture, which is characterised by subjective reconstruction of knowledge through negotiation of meaning in social interaction (Bauersfeld, 1988).

The core part of enculturation into school mathematics "comes into effect on the meta-level and is 'learned' indirectly" (Cobb and Bauersfeld, 1995, p. 9). What is referred to as a "mathematical disposition" may be developed in this indirect manner through participation in "reflective classroom discourse" (Cobb, Boufi, McClain and Whitenack, 1997, p. 269). In a reflective discourse, teachers should manage the interplay of social norms and patterns of interaction to create opportunities for pupils to reason for themselves and to "engage in reflective thinking or reflective abstraction" (Wood, 1996, pp. 102- 103).

An issue which arises here is the extent to which the teacher can act as a genuinely neutral moderator of discussions amongst co-participants, acting as a director and guide of pupils' learning. There is an obvious power imbalance behvcen teachers and pupils in classrooms and teachers' comments carry great weight. What is significant is the manner in which power is expressed in action and the intentions underlying the action.

It is one thing for the teacher to cue students until they can act as though they had learned what the teacher had in mind all along and another for the teacher to express his or her authority in action by initiating and guiding the explicit negotiation of mathematical meanings. (Cobb, Wood, Yackel and Perlwitz, 1992, p. 486)

Scaffolding higher order thinking

Bruner (1985) uses the metaphor of scaffolding to describe the intervention of an adult or a more competent peer in the learning process to act "as a vicarious form of consciousness until such time as the learner is able to master his own action through his own consciousness and control" (pp. 24-25). The scaffolding metaphor is appealing in principle, but it was originally applied in one-to-one interactions in child psychology and is not easily translated into a practical classroom context (Maybin, Mercer and Stierer, 1992, p. 187). Moreover, scaffolding is an ill-defined construct in the literature, with one person's scaffolding being another's Socratic questioning. Indeed, the scaffolding metaphor has unfortunate associations in some respects:

Scaffolding on the building site is an unresponsive activity, the scaffolding erected and the building put up according to a predetermined plan. Scaffolding in the classroom must be much more dynamic, and ways must be found for the teacher and pupils to work jointly on activities. (Askew, Bliss and MacRae, 1995, p. 216)

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Scaffolding for Success

Bliss, Askew and MacRae (1996) claim that it is difficult to scaffold specialist knowledge and report that such scaffolding was generally absent from the teaching they observed. The dynamic scaffolding desired by Askew et al. may not be commonly achieved in classrooms.

It is possible to distinguish between two very different forms of interaction which might be described as scaffolding to support pupils' learning: funnelling and focusing (Bauersfeld, 1988; Wood, 1994). In funnelling it is the teacher, as the person with the expert knowledge, who selects the thinking strategies and controls the decision process to lead the discourse to a predetermined solution. The social processes of such a classroom obscure the mathematical structure of the problem which the pupil may construct only by choosing to reflect on regularities in the actions performed. If pupils do not appreciate these regularities and the reasons for them then "context- and problem-specific routines and skills" are likely to result (Bauersfeld, 1 988, p. 37). Mathematical logic and meaning are replaced by the social logic and meaning of the interaction.

In focusing the teacher's questions draw attention to critical features of the problem which might not yet be understood. The pupil is then expected to resolve perturbations which have thus been created (Wood, 1994, p. 160). Pupils as well as the teacher are expected to question assumptions and make conjectures, and the responsibility for identifying strategies and making decisions is devolved to the pupils. The use of such scaffolding then would lead to pupils who do not rely on the teacher to regulate and instigate thinking but are able instead to generalise particular learning experiences to more general thinking strategies (Jaworski, 1990, p. 98).

The achievement of higher levels of understanding and thoughtfulness in mathematics are claimed for such models of teaching (Prawat, 1991). Through "reflective discourse" teachers are able to "proactively support students' mathematical development" by guiding and if necessary initiating shifts in the discourse so that "what was previously done in action can become an explicit.topic of conversation" and thus "participation in this type of discourse constitutes conditions for the possibility of mathematical learning" (Cobb et al., 1997, pp. 264 - 269). The social character of the discourse may be arranged to lend social status to "the disposition to meaning construction activities" which is a "habit of thought" that can be learned (Resnick, 1988, p. 40).

Higher levels of thought, the ability to argue from a hypothesis and to view reality as a reflection of theoretical possibilities represent, from a Piagetian viewpoint, the development of formal thought. Formal thought has been described as a systematic way of thinking: a generalised orientation towards problem solving with an improvement in the student's ability to organise and structure the elements of a problem (Sutherland, 1992). However, these key aspects of problem solving are metacognitive rather than conceptual in nature. It can be argued, therefore, that formal thought is underpinned by the development of metacognitive skills.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

Metacognition

Metacognition refers loosely to both the knowledge and the control which individuals have of their own cognitive systems. This dual nature includes both (a) the awareness that individuals have of their own knowledge, their strengths and weaknesses, their beliefs about themselves as learners and the nature of mathematics, and (b) their ability to regulate their own actions in the application of that knowledge (Flavell, 1976; Brown, 1987). The former aspect is passive in character and is characterised here as metacognitive knowledge or 'knowing what you know'. It refers to the knowledge and beliefs a person has about their own cognitive resources in a domain, how well they are likely to perform in that domain, the strategies and heuristics they might be able to use, and the nature of the domain itself (Flavell, 1987, pp. 22-23). Such subjective self-knowledge is likely to be an important influence on cognitive behaviour (Garofalo and Lester, 1985, p. 164) and may be shaped by participation in day-to-day classroom practices.

The second aspect of metacognition refers to the "active monitoring and consequent regulation and orchestration" of one's thinking (Flavell, 1976, p. 232) and is characterised here as metacognitive skill. The use of the term 'metacognitive skills' to distinguish metacognitive activities and processes from metacognitive knowledge is deliberate and seeks to identi@ those processes that might be improved by training.

THE MATHEMATICAL THINKING SKILLS PROJECT

This action research project aimed to improve pupils' performance in mathematics by developing their metacognitive skills. An earlier research study had identified the skills of planning, monitoring and evaluating as essential for successful practical problem solving and modelling, and had suggested classroom practices which would facilitate the development of these skills (see Tanner and Jones, 1994 for details). These practices included strategies to scaffold pupils' thinking and to encourage reflective discourse. The project aimed to evaluate the effect of a course based on such strategies in a quasi-experiment.

The teachers involved in the project agreed to try to adopt and develop experimental teaching approaches which aimed to develop metacognitive skills through social modelling and reflective discourse. One key strategy was referred to as Start-stop-go in which pupils were asked to read the problem, think in silence for a few minutes, and then discuss possible plans in small groups before the teacher led a brainstorming session which focused attention on key features. The intention was to constrain pupils to act as experts rather than novices by slowing down impulsive behaviour and encouraging the examination of several problem formulations. After the class had started work they were stopped at intervals for groups to report on progress; this was intended to encourage monitoring by pupils of their progress against their plans, and also against the progress of others.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Scafloolding for Success

It was intended that lessons would include plenaries in which groups would report their solutions to problems. Pupils would be required to question and discuss each others' findings and to evaluate approaches. Participation in peer assessment was intended to encourage the development of a reflective discourse and skills of self- assessment leading to metacognitive knowledge.

It was anticipated that the use of such teaching strategies would produce improved pupil performance in problem solving and modelling situations which were similar in character to those used on the course - "near transfer" (Shayer, 1992, p. 1 16). It was further hypothesised, however, that the development of metacognitive skills would lead to improved learning in mathematics through "far transfer" (ibid.) into the cognitive domain, i.e. the more usual content areas of mathematics which had not been specifically targeted by the project.

Methodology

The research has at its heart a quasi-experimental design involving pre-testing, post- testing and delayed-testing of control and experimental groups. 641 pupils in 12 pairs of classes were involved of whom 314 experienced intervention lessons and 327 followed their usual curriculum. However, it is not claimed that the experimental groups all received an identical experience or 'experimental treatment'. In fact, the novelty of the approaches employed and the materials used as context led to an expectation that the course experienced by pupils would vary considerably from class to class. In order to ensure that sufficient internal validity might be claimed, it was necessary to build into the research design ongoing observation and analysis of the way the course was being interpreted and experienced by teachers and pupils in the different social contexts which were developed in individual schools and classrooms.

Two teachers from each of six secondary schools formed an action research network to develop and trial teaching strategies and materials, supported by members of the project team. In each school, two equivalent pairs of classes were identified to act as control and intervention groups. In every case the pairs of classes were either mixed ability groups or parallel sets. One pair was in Year 7 (1 1-12 years old) and one pair in Year 8 (12-13 years old). The control classes were taught by their normal mathematics teachers who had no direct involvement with the project. Although the matching of classes was on the recommendation of teachers rather than assessment by the researchers, there was no significant difference between the two groups in the pre-test. The later analysis of the data utilised a covariate approach to adjust for the slight inequalities that existed between groups at the beginning of the quasi-experiment.

Regular network meetings of teachers and the university researchers were held at which experiences were exchanged, strategies discussed and new activities devised and refined. The activities were structured into eight sections, each of which required different strategic skills and mathematical processes for their solution, for

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

example, systematic working, control of variables, explaining and proving. The intervention class teachers tried to teach activities from at least six different sections but selected for themselves which activities to attempt according to their own curriculum requirements. The intervention teaching occurred within a period of approximately twelve school weeks and took place during normal mathematics lessons. At least one lesson per fortnight (but not more than one per week) was to be devoted to the project activities. Thus the intervention classes spent less time than their control groups on the 'normal' content areas of mathematics.

Regular participant observations of lessons were undertaken by the university researchers to record the nature of the interventions made. To ensure reliability the initial lessons were observed jointly by the two university researchers and their observations compared. Lessons were sometimes audio or video taped and then jointly analysed. At the end of each of the lessons teachers were interviewed on tape about the aims of the lesson, the interventions used, and the extent to which the teachers felt the aims had been achieved.

Written test papers were designed to assess pupils' cognitive development based loosely on a neo-Piagetian structure (see Tanner, 1997 for details). Each test item was classified within one of four sequential stages of development, but account was also taken of anticipated memory requirements, and of the results of large scale studies such as the Concepts in Secondary Mathematics and Science Project and its sequels (Hart, 1981) and the Assessment and Performance Unit surveys (Foxman, 1985). Items emphasised comprehension rather than recall and so were set in simple problem solving contexts. The test items were structured into four sections representing the four content domains of the Mathemtics National Curriculum for England and Wales - Number, Algebra, Shape and Space, and Probability. The purpose of the test was not to determine a Piagetian level for pupils but to provide an assessment of their mathematical development.

The metacognitive skills of question posing, planning, evaluating and reflecting were assessed through a section in the written paper entitled "Planning and doing an experiment". Metacognitive self-knowledge was also assessed by asking students to predict the number of questions they would get correct before and after each section (referred to here as forecasting and postcasting). The overall test was reliable with Cronbach's alpha of 0.86.

The pupils were pre-tested before the intervention teaching began and post-tested at the end of the course in July. Delayed testing took place four months later after the school summer holidays when many pupils had moved on to different classes and teachers.

THE TEST RESULTS

The participant observations revealed that the extent to which the teachers were able to adopt the suggested approaches was variable. Three teachers and their classes were dropped from the final analysis for failing to follow the course or approaches to

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Scaffolding for Success

any appreciable extent (see Tanner, 1997 for details). The results that follow are for the remaining eight teachers with their nine classes (one intervention teacher taught two groups) and a total of 499 pupils.

Multivariate analysis of variance (MANOVA) was used to analyse the three levels of test (ie: pre, post and delayed) and two types of class (ie: control and intervention). A covariate approach (using pre-test scores as covariates) was used to add power to the analysis by adjusting for the small inequalities which existed between the pairs of classes at the start of the quasi-experiment. For simplicity, only the multivariate results are given here in Table 1 (but see Tanner and Jones, 1995 or Tanner, 1997 for further details).

Table 1: Multivariate tests of significance for the effect of type of class

Variable

Metacog Skill

Forecast

Postcast

Cognitive Dev

The active metacognitive skills of planning, monitoring and evaluating

Both control and intervention classes improved their scores on the active metacognitive skills over the period of the quasi-experiment, although the intervention classes improved more than the control classes in the post-test and this improvement was sustained in the delayed-test. The effect size was small (0.19), but significant at the 0.1% level for the written tests.

As the active metacognitive skills had been taught in practical mathematical modelling contexts, such near transfer might be considered unsurprising. Its achievement was non-trivial, however, as the pupils were required by the assessments to form their own problems within open situations, plan, identify and control variables, choose simple strategies, monitor their work, collect and organise their data, find relationships, evaluate and reflect on their results. These are the process skills of mathematics and are identified in the National Curriculum for England and Wales as worthy of learning in their own right.

Hotellings

.235

.O 13

.022

.021

Passive metacognitive knowledge or 'knowing what you know'

Hypoth

DF

2

2

2

2

F Value

43.67

2.30

3.78

3.89

The results here were not clear cut. Although the intervention classes improved in their forecasting more than the control classes in the post-tests and this improvement was sustained in delayed-testing, the effect of type of class was not significant at the 5% level. The postcasting of the intervention classes improved more than the control classes in the post-tests and this improvement was sustained in delayed-testing. This

Error

DF

37 1

363

341

3 69

Sig of

F

.OOO

.lo1

.024

.02 1

Effect

size

.I91

.O 13

.022

.02 1

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

was significant beyond the 5% level but was limited to an extremely small effect size of 0.02.

Mathematical cognitive development

This showed a similar pattern to the active metacognitive skills but a smaller effect. The intervention classes improved more than the control classes in the post-test and the advantage was largely sustained at the delayed-test but the effect size (0.02) was extremely small. The content of the cognitive section of the written paper was not taught directly by the activities and intervention class teachers avoided practising questions similar to those on the test. Given that the intervention classes had less teaching in the normal mathematics curriculum over the period of the quasi- experiment, it might have been expected that the control classes would generally have outperformed the intervention classes. This small overall effect is claimed, therefore, to be an example of mathematical thinking skills paying for themselves through far transfer.

THE FOUR TEACHING STYLES

Analysis of the qualitative data collected through participant observation led to the classification of the eight teachers into four characteristic groups according to the teaching styles employed. The teachers were classified solely on the basis of the observations made of their classroom behaviours. These were tuskers, rigid scaffoolders, dynamic scaffolders and reflective scaffolders (see Tanner, 1997 for detailed descriptions). There were two teachers and two classes in each group, apart from the reflective scaffoolders where one of the two teachers taught two classes.

The taskers focused on the demands of the task rather than the underlying aim of teaching metacognitive skills. Initially their teaching style was characterised by an extreme representation of constructivism as an autonomous, creative process in which teachers should not interfere. The pupils were left to sink or swim with the problem. In the later stages the teachers provided more structure especially for the planning stage, but the attention of the pupils was not drawn to the underpinning strategies and many pupils remained focused on the superficial objectives of the task rather-than the intended learning objectives - the metacognitive processes needed for its solution.

The rigid scaffoolders were far more directive in their approach to planning. Their emphasis was on demonstrating and sharing the teacher's own previously identified plan rather than helping pupils to develop their own plans. Organisational prompts - questions designed to help pupils to develop a framework to organise their thoughts - were used, but the scaffolding support provided by the questioning constrained pupils' thinking, funnelling them down a pre-determined path. The teacher's plan was then followed as if it were a recipe.

These two groups of teachers were the least successful. The taskers ' classes showed no advantage over their controls in any test. The rigid scaffoolders showed an

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

ScafSolding for Success

advantage in only the metacognitive delayed test with a very small effect size (0.09) significant at the 5% level. The remainder of this paper focuses on the other two groups of teachers.

The dynamic scaffolders

The dynamic scaffoolders made full use of the social structure of Start-stop-go to frame their pupils' behaviour and constrain them to act as experts rather than novices. This included the granting of significant autonomy to pupils, particularly in the early stages of planning. Their scaffolding was dynamic in character and was based on participation in a discourse in which differences in perspective were welcomed and encouraged. The most significant participant in the discourse was the teacher, who validated conjectures and used focusing questions to control its general direction ensuring that an acceptable whole class plan was generated.

The autonomy and responsibility of the pupils was limited by the teacher's desire to negotiate a plan to a pre-determined template. Their participation therefore could be compared to legitimate peripheral participation within an apprenticeship model of learning as their responsibility was limited to aspects of the task with the teacher taking overall responsibility for the design (see Lave and Wenger, 1991). The discourse focused on both procedural knowledge (strategies and processes) and conceptual knowledge. During the planning and monitoring sessions, articulation and objectification of explanation was encouraged; the explanation itself became the object of the discourse. This was the only form of evaluation or reflection used by the dynamic scaffolders, however, and may be characterised as "reflection in action" as opposed to "reflection on action" (Schon, 1990).

Table 2: Multivariate tests of significance for the effect of type of class for dynamic scaffolders

The dynamic scaffolders were very successful in accelerating the development of the active metacognitive skills of planning, monitoring and evaluating in the context of mathematical modelling - that is in near transfer - with a small to medium effect size (0.36), significant beyond the 0.1% level (see Table 2). However, they failed to achieve a significant advantage for their classes in either passive metacognitive self- knowledge or far transfer into the content areas of mathematics. It is conjectured that, although active metacognitive skills may be necessary in the learning of new

Sig of

F

.OOO

.226

.590

.453

Effect

size

0.36

0.04

0.02

0.02 i

Hypoth

DF

2

2

2

2

F Value

21.27

1.52

0.53

0.80

Variable

Metacog Slull

Forecast

Postcast

Cognitive Dev

Error

DF

77.

76

70

78

Hotellings

0.550

0.040

0.01 5

0.020

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

knowledge, they are not sufficient. Metacognitive self-knowledge may also be necessary for far transfer.

The reflective scaffolders

The reflective scaffolders also used the social structure of Start-stop-go to constrain their pupils to act as experts rather than novices, and dynamic scaffolding in a conjecturing atmosphere to lead the discourse in their classrooms. They granted their pupils more autonomy, however, encouraging several approaches to the problems rather than constraining the discourse to produce a class plan. Pupils thus had to evaluate their own plans in comparison with the other plans in the posing, planning and monitoring phases of the lessons.

The participation framework required pupils to draw on the help of the expert but to take a greater responsibility for an end product of their own design rather than as an apprentice taking limited responsibility for an element in the design of a master. The characteristic feature of the reflective scaffolders, however, was their focus on evaluation and reflection. During interim and final reporting back sessions, scientific argument was encouraged to make the explanation an object of the discourse. Peer and self-assessment was encouraged through group presentations of draft reports before redrafting for assessment. The teachers deliberately generated a reflective discourse after activities to encourage self-evaluation and reflection on process. Collective reflection is not the same as reflected abstraction, but it is conjectured that, during collective reflection, opportunities arise for pupils to reflect on and objectify their previous actions as they engage in reflective discourse (cf. Wheatley, 1991; Cobb et al., 1997).

Variable I Hotellings I F Value I Hypoth I Error I Sig of 1 Effect

Table 3: Multivariate tests for the effect of type of class for reflective scaffolders

Metacog Skill .

Forecast

Postcast

Cognitive Dev

The reflective scaffolders were very successful in accelerating the development of active metacognitive skills, achieving near transfer in practical modelling situations with a medium size of effect (0.4), significant beyond the 0.1% level of significance (see Table 3). They also succeeded in accelerating the development of passive metacognitive self-knowledge in forecasting and postcasting. The effect sizes were very small (0.07 and 0.14), but significant beyond the 5% and 1% levels respectively. They were the only group of classes to achieve this and it is conjectured that this was due to their emphasis on self-evaluation and reflection.

0.652

0.073

0.161

0.272

43.34

4.73

9.82

18.09

DF

2

2

2

2

DF

133

130

122

133

F

.OOO

.010

.OOO

.OOO

size

0.40

0.07

0.14

0.2 1

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Scaffolding for Success

The reflective scaffolders also succeeded in accelerating development in the content domains of mathematics measured by the cognitive test, again the only group of classes to achieve this far transfer. The size of effect was small (0.21), but was statistically significant beyond the 0.1% level and approximated to a year's development.

DISCUSSION AND CONCLUSION

The teachers characterised as dynamic scaffolders employed a model of cognitive apprenticeship that included authentic tasks, student autonomy and dynamic scaffolding. Although they were very effective in teaching for near transfer, they failed to achieve far transfer. The importance of pupils' individual construction was subordinated to the dynamics of the apprenticeship model, whereas the reflective scaffolders encouraged both cognitive apprenticeship and individual construction. The dynamics of Start-stop-go were internalised through participation in social processes. The learning of this procedural knowledge was achieved through an apprenticeship model, organised and controlled by the teacher. The pupils learned to internalise the processes of scientific argument and argue with themselves through participating in a scientific discourse led by an expert using dynamic scaffolding. They also learned that mathematics made sense and that they could make their own sense of what occurred by making their own tentative conjectures and constructions and linking them with prior schemata.

In the classes taught by the reflective scaffolders, reflective abstraction and the objectification of explanation were encouraged through participation in reflective discourse. Collective reflection provided both a social model and an opportunity for reflected abstraction by the individual pupil. It is suggested here that the processes of problem solving, once objectified through individual reflection, became knowledge rather than mere information and were thus capable of being used elsewhere.

The reflective discourse focused on the processes of problem solving, abstraction and generalisation with the intention of understanding mathematical structure. Participation in reflective discourse may have encouraged the development of a mathematical disposition or point of view (Schoenfeld, 1994; Cobb et al., 1997) which might be expected to be transferable.

Planning, monitoring and evaluating are classified here as active metacognitive skills. These skills are generic rather than specific and it appears that they can be transferred to similar modelling contexts. The findings of this research study suggest that participation in reflective discourse can encourage objectification and the development of metacognitive self-knowledge. It is further conjectured that teaching approaches which support the development of active metacognitive skills in combination with passive metacognitive knowledge enhance not just the application of previously known mathematics to new contexts but also enhance the learning of new mathematics..

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

REFERENCES

Askew, M., Bliss, J. and MacRae, S.: 1995, 'Scaffolding in mathematics science and technology.' In P. Murphy, M. Selinger, J. Bourne, and M. Briggs (eds.), Subject Learning in the Primary Curriculum (pp. 209-2 17). London: RoutledgeIOUP.

Bliss, J., Askew, M. and MacRae, S.: 1996, 'Effective teaching and learning: scaffolding revisited.' Oxford Review of Education, 22(1), 3 7-61.

Bauersfeld, H.: 1988, 'Interaction, construction and knowledge: alternative perspectives for mathematics education.' In D. Grouws, T. Cooney, and D. Jones (eds.), Effective Mathematics Teaching (pp. 27-46). NCTM, Reston, VA: Lawrence Erlbaum.

Bauersfeld, H.: 1994, 'Theoretical perspectives on interaction in the mathematics classroom.' In R. Biehler, R. W. Scholz, R. Straesser, and R. Winkelmann (eds.), The Didactics of Mathematics as a ScientiJic Discipline (pp. 133- 146). Dordrecht: Kluwer.

Brown, A. L.: 1987, 'Metacognition, executive control, self regulation and other more mysterious mechanisms.' In F. E. Weinert and R. H. Kluwe (eds.), Metacognition, Motivation and Understanding (pp. 65-116). New Jersey: Lawrence Erlbaum.

Bruner, J. S.: 1985, 'Vygotsky: a historical and conceptual perspective.' In J. V. Wertsch (ed.), Culture, Communication and Cognition: Vygotskian Perspectives (pp. 2 1-34). Cambridge: Cambridge University Press.

Burton, L: 1984, 'Mathematical thinking: the struggle for meaning.' Journal for Research in Mathematics Education, 15(1), 35-49.

Cobb, P., Wood, T., Yackel, E. and Perlwitz, M.: 1992, 'A follow-up assessment of a second-grade problem centred mathematics project.' Educational Studies in Mathematics, 23(5), 483-504.

Cobb, P., and Bauersfeld, H.: 1995, The Emergence of Mathematical Meaning: Interaction in Classroom Cultures (pp. 1 - 16). Hillsdale, NJ: Lawrence Erlbaum.

Cobb, P., Boufi, A., McClain, K. and Whitenack, J.: 1997, 'Reflective discourse and collective reflection.' Journal for Research in Mathematics Education, 28(3), 258- 277.

Dawson, S.: 1994, 'Conference news.' Philosophy of Mathematics Education Newsletter, 7, 24-25.

Flavell, J. H.: 1976, 'Metacognitive aspects of problem solving.' In L. B. Resnick (ed.), The Nature of Intelligence (pp. 23 1-235). Hillsdale, NJ: Lawrence Erlbaum.

Flavell, J. H.: 1987. 'Speculations about the nature and development of metacognition.' In F. E. Weinert and R. H. Kluwe (eds.), Metacognition, Motivation and Understanding (pp. 2 1-30). New Jersey: Lawrence Erlbaum.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Scaffolding for Success

Foxman, D.: 1985, Mathematical Development: Review of the First Phase of Monitoring: Report on the Series of Annual Surveys of the Mathematical Performance of 11 and 15 Year Olds Held from 1978-1982 Inclusive. London: HMSO.

Garofalo, J. and Lester, F.: 1985, 'Metacognition, cognitive monitoring and mathematical performance.' Journal for Research in Mathematics Education, 16(3), 163-176.

Hart, K. M.: 198 1, Children 's Understanding of Mathematics 11 - 16. London: John Murray.

Jaworski, B.: 1990, 'Scaffolding: a crutch or a support for pupils' sense-making in learning mathematics?' In G. Booker, P. Cobb and T. N. de Mendicuti (eds.) Proceedings of the 14th Conference of the International Group for the Psychology of Mathematics Education Vol. 3 (pp. 91-98). Oaxaca, Mexico. Program Committee of the 14th PME Conference.

Lave, J., and Wenger, E.: 199 1, Situated Learning: Legitimate Peripheral Participation. Cambridge: Cambridge University Press.

Maybin, J., Mercer, N. and Stierer, B.: 1992, "'Scaffolding" learning in the classroom.' In K. Norman (ed.), Thinking Voices (pp. 186-195). London: Hodder and Stoughton.

Prawat, R.: 199 1, 'The value of ideas: the immersion approach to the development of thinking.' Educational Researcher, 20(2), 3-10.

Resnick, L. B.: 1988, 'Treating mathematics as an ill structured discipline.' In R. I. Charles and E. A. Silver (eds.), The Teaching and Assessing of Mathematical Problem Solving (pp. 32-60). Reston, VA: National Council of Teachers of Mathematics.

Schoenfeld, A. H. (Ed.): 1994, Mathematical Thinking and Problem Solving. Hillsdale, NJ: Lawrence Erlbaum.

Schon, D.: 1990, Educating the Reflective Practitioner. Oxford: Jossey-Bass Publishers.

Shayer, M.: 1992, 'Problems and issues in intervention studies.' In A. Demetriou, M. Shayer and A. Efklides (eds.), Neo-Piagetian Theories of Cognitive Development: Implications and Applications for Education (pp. 107- 12 1). London: Routledge.

Sutherland, P.: 1992, Cognitive Development Today. London: Paul Chapman Publishers.

Tanner, H. F. R., and Jones, S. A.: 1994, 'Using peer and self assessment to develop modelling skills with students aged 11 to 16: a socio-constructive view.' Educational Studies in Mathematics, 2 7(4), 4 1 3 -43 1.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14

Research in Mathematics Education Volume 2

Tanner, H. F. R., and Jones, S. A.: 1995, 'Teaching mathematical thinking skills to accelerate cognitive development.' In L. Meira and D. Carraher (eds.) Proceedings of the 19th Conference of the International Group for the Psychology of Mathematics Education Vol. 3 (pp. 12 1 - 128). Recife, Brazil. Universidade Federal de Pernambuco.

Tanner, H. F. R.: 1997, Using and Applying Mathematics: Developing Mathematical Thinking Through Practical Problem Solving and Modelling. Unpublished Ph.D. thesis. Swansea: University of Wales Swansea.

Wheatley, G. N.: 199 1, 'Constructivist perspectives on science and mathematics learning.' Science Education, 75(1), 9-2 1.

Wood, T.: 1994, 'Patterns of interaction and the culture of mathematics classrooms.' In S. Lennan (ed.), Cultural Perspectives on the Mathematics Classroom (pp. 149- 168). Dordrecht, Netherlands: Kluwer Academic Publishers.

Wood, T.: 1996, 'Events in learning mathematics: insights from research in classrooms. ' Educational Studies in Mathematics, 30, 85- 105.

Dow

nloa

ded

by [

Flor

ida

Atla

ntic

Uni

vers

ity]

at 1

0:55

10

Nov

embe

r 20

14