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Collaborative Learning 
and 

Peer-tutoring 
in Mathematics 

Kathryn Joy Rowe 

A thesis presented in partial fulfilment of requirements for 
Master of Educational Studies (Mathematics) 

Massey University 
2002 



ii 

Abstract 

This study sought to promote learning by enhancing the level of higher order cognitive talk 

among collaborative groups engaged on mathematical tasks. An intervention, designed to 

utilise structures such as listening, multiple retelling, questioning, elaboration, and 

justification to promote high-level discourse, was trialled and refined using an action 

research classroom study. 

The collaborative skills training programme was based on Medcalfs peer-tutoring model 

(1997) and adapted to incorporate features of Lyman's Think-Pair-Share collaborative 

model (1992). The teacher' s role was seen as crucial to the development of collaborative 

group practices which establish the structures for high-level discourse. Collaborative group 

practices were reinforced in follow-up class discussions where the teacher facilitated student 

reflection on the mathematical strategies and the collaborative group strategies. It was also 

seen as important for the teacher to select appropriately levelled tasks which maintained the 

learner in his/her Zone of Proximal Development. 

Findings indicated that the structured intervention enhanced the level of higher order 

discourse between students and that it was an effective procedure to mediate learning. 

Several patterns of discourse were also identified that could provide useful indicators of 

higher level discourse to teachers during daily classroom observations. 



iii 

Acknowledgements 

I would like to extend heart-felt thanks to my supervisors Dr. Glenda Anthony and Brenda 

Bicknell at Massey University for their direction, critique and encouragement. Thank you 

too to the school staff and Board of Trustees for their continued support and 

encouragement both in my teaching practice and during the research study. 

Thank you to the Ministry of Education for providing a four week study award in which to 

complete the practical aspects of the research. I cannot commend the value of the award 

scheme enough for allowing teachers time to focus on their studies and the rewards this 

brings back to classroom practice. 

Thank you also to my family and friends who have put up with the stresses and strains of a 

life which is only work and study. Shortly a human being will return to your midst. 

A final thanks to all the students who have passed through my care, you continue to be my 

inspiration, and your success, my greatest joy. 

Kathryn Rowe 

November 2002 



iv 

Table of Contents 

Abstract 
Acknowledgements 
Table of Contents 
List of Figures 

(ii) 
(ii i) 
(iv) 
(vi) 
(vi) List of Tables 

Chapter 1: Introduction 

1. 1 Background 
1.2 Teaching Reforms and Collaborative Learning 
1.3 Definition of Terms 
1.4 Research Questions 
1.5 Overview 

Chapter 2: Literature Review 

2.1 Introduction 
2.2 Learning Theories and Discourse 

2.2.1 Individual Development of Thought 
2.2.2 Social Development of Thought 

2.3 Structuring Discourse in the Classroom 
2.3.1 Teachers and Students as a Mathematical Community 
2.3.2 Authority 
2.3.3 Transparency 

2.4 Collaborative Learning 
2.4.1 Cooperative Model 
2.4.2 Think-Pair-Share Model 

2.5 Peer-tutoring 
2.5.1 'Pause, Prompt, and Praise' Model 
2.5.2 'ASK to THINK-TEL WHY' Model 
2.5.3 Multiple Retelling Model 

2.6 Analysis of Classroom Discourse 
2.7 Literature rev iew summary 

Chapter 3: Methodology 

7 
10 
13 
13 
14 

16 
17 
17 
19 
21 
21 
23 
25 
27 
30 
32 
34 
36 
37 
38 
39 
42 

3.1 Introduction 43 
3.2 Action Research Design 43 
3.3 Setting 46 
3.4 Profile of the Sample Group 47 
3.5 Timeline 47 
3.6 Developing the Instructional Strategies 48 

3.6.1 Development of the Training Programme Through Cycles 1 and 2 51 
3.6.2 Cycle 2- The Main Study 56 

3. 7 Data Collection 60 
3.8 Data Analysis 63 
3.9 Ethical Considerations 65 



v 

Chapter 4: Results 
4.1 Introduction 
4.2 Transcript Data 
4.3 Academic Data . 
4.4 Identification of 'Oral Flags' 
4.5 Questionnaire Data 

4.5.1 Student Preferences for Group-work 
4.5.2 Authority 
4.5.3 Group Processes 
4.5.4 Mathematical Identity 

4 .6 Teachers' Anecdotal Observations 

6. Discussion and Conclusion 

5.1 Effects of Peer-tutoring 
5.2 Oral 'flags' which identify higher order cognitive thinking 
5.3 Students' Perceptions 
5.4 Teacher Perceptions 
5.5 Limitations of the Study 
5.6 Concluding Thoughts and Implications for Future Research 

References 

Appendices 

1: Ebbutt's (1985) table of broad classification of a range of insider activity currently occurring 

67 
69 
78 
80 
83 
83 
84 
85 
86 
86 

88 
91 
92 
93 
94 
95 

98 

in schools. 11 O 
2: Information sheet for students and caregivers. 111 
3: Student and caregiver consent form . 112 
4: Teacher consent form. 113 
5: Student response questionnaire. 114 
6: Fravillig , Murphy & Fuson's (1 999, p. 155) examples of instructional strategies employed 

to elicit, support and extend children's mathematical thinking. 11 5 
7: Example of coded transcript. 116 
8: Example of problem-solving task. 117 
9: NUMP stages and behaviour indicators for operational strategies for addition and 

subtraction and fractional knowledge. 118 



vi 

List of Figures: 

1: Thomas' model oftalk (1994). 41 
2: Kemmis' and McTaggart's action research planner (1981 ). 45 
3: Cycle 1 ofthis action research study. 49 
4: Cycle 2, the pilot study. 52 
5: Cycle 3, the main study. 56 
6: Classifications of collaborative group talk. 63 
7: Summary of the mean percentage oftalk for the sample group before and after the 

intervention. 67 
8: Mean percentages of talk before and after the intervention. 68 
9: Summary of the relationship between the mean percentages of talk. 68 
10: Mean percentage of task-related talk before and after the intervention according to 

grouping. 70 
11: Percentage of cognitive talk before and after the intervention. 74 
12: Mean percentage of cognitive talk before and after the intervention 

according to grouping. 75 
13: Mean percentage of higher order cognitive talk before and after the intervention 

according to grouping. 77 
14: Academic outcomes for fractional knowledge before and after the intervention. 79 

List of Tables: 

l : Daily Group Rotation During Mathematics. 48 
2: Percentages of Task -related and Non-task-related Talk Before and After the 

Intervention. 69 
3: Mean Percentage of Task-related Talk Before and After the Intervention According 

to Grouping. 7 0 
4: Percentage of Talk Contributed to the Group. 71 
5: Mean Deviation from Equal Talk Before and After the Intervention. 72 
6: Percentages of Task-related Talk Subcategorised as Cognitive and Social Talk 

Before and After the Intervention. 73 
7: Mean Percentage of Cognitive Talk Before and After the Intervention According 

to Grouping. 74 
8: Mean Percentage of Task-related Social Talk According to Grouping. 75 
9: Cognitive Talk Subcategorised as Higher and Lower Order Cognitive Talk Before 

and After the Intervention. 76 
10: Mean Percentage of Higher Order Cognitive Talk Before and After the Intervention 

According to Grouping. 77 
11: NUMP Diagnostic Interview Levels to Show Academic Outcomes Before and After 

the Intervention. 78 
12: Mean academic Outcomes Before and After the Intervention Using the NUMP 

Diagnostic Interview to the Nearest Level. 79 
13: Percentage of Class Who Preferred Working Alone, in a Group or Both Ways. 84 
14: Group Skills Level Indicated by Questionnaire Response. 85 



7 

Chapter 1. Introduction 

"Cognito, ergo sum" - Descartes 

1.1 Background 

The end of the twentieth century was characterised by the development of computer 

technology enabling vast quantities of information to be readily accessed by the world 

population. 

The information explosion will continue to expand ... Increasingly we will move 

away from defining educational success in terms of the quantity of information 

mastered. Instead, to a large extent, we will define educational success as the 

ability among students to generate, question, combine, categorise, re-

categorise, evaluate, and apply information. Secondary will be the content of the 

information; primary will be thinking skills- thus, the need for thinking skills 

structures in our classrooms. (Kagan, 1994, p. 111) 

As a response to the global ' information highway' and changing work-place structures, the 

Ministry of Education developed the New Zealand Curriculum Framework (Ministry of 

Education, 1993) that identified eight generic essential skills: physical skills, self­

management and competitive skills, communication skills, problem-solving skills, social and 

co-operative skills, information skills, numeracy skills, and work and study skills. 

Curriculum documents were revised during the 1990s to apply the New Zealand 

Curriculum Framework's essential skills to particular curriculum contexts. The first of these 

revised curriculum documents was Mathematics in the New Zealand Curriculum (Ministry 

of Education, 1992). This separated the mathematics curriculum into six key strands: 



8 

• 

• 

• 

• 

• 

• 

mathematical processes, 

number, 

measurement, 

geometry, 

algebra, and 

statistics . 

This core document and supporting documents such as Implementing Mathematical 

Processes (Ministry of Education, 1995) and Development Band Mathematics (Ministry of 

Education, 1996) were also influenced by the international mathematics community's 

emphasis on mathematical problem-solving which had arisen during the 1980s in the United 

Kingdom with the Cockcroft Report ( 1982) and the Agenda for Action Report in the 

United States ( 1980) and recognition of the importance of discourse and the value of 

collaborative learning practices in mathematics (National Council of the Teachers of 

Mathematics, 1989). For example, the Implementing Mathematical Processes document 

explicitly linked the need for 'new essential skills' with mathematical processes: 

The abUities- to think flexibly, be adaptable, be a creative problem solver, and 

be able to communicate and work co-operatively- have been highlighLed as 

essential skills for life in the world tomorrow. Mathematics teaching can play an 

important part in equipping our students with Lhese fundamental skills. This is one 

of the reasons why the mathematics curriculum emphasises the importance of 

developing the students' abilities to reason, think flexibly, communicate, solve 

problems and collaborate in a mathematical context. 

(Ministry of Education, 1995, p. 7) 

Mathematics in the New Zealand Curriculum required teachers to "design courses to 

provide their students with mathematical experiences which will enable the students to 



9 

achieve the broader aims and achievement objectives for the curriculum" (Ministry of 

Education, 1992, p. 18). This included developing courses to emphasise the essential skills 

of logic and reasoning, communicating ideas, problem-solving and collaborative group 

work. 

The processes of logic and reasorung and problem-solving were readily accepted by 

mathematics teachers as integral to mathematics learning (Holton, Anderson, & Thomas, 

1996). From the era of Platonic deduction and proofs mathematical thinking has emphasised 

logic and reasoning. Moreover, many of the established traits of effective thinkers-with 

cross-disciplinary characteristics of persistence, creativity, questioning, precision, 

metacognition, flexibility, listening, and the ability to restrain impulsiveness (Costa & 

O'Leary, 1989}-parallel those traits of good problem-solvers within Mathematics in the 

New Zealand Curriculum , namely flexibility, creativity, the ability to reflect, experiment, 

improve, and use divergent and convergent approaches. 

Parallel to the focus on processes the Mathematics in the New Zealand Curriculum 

encouraged New Zealand teachers to change their instructional practice of transmitting 

specific content in mathematics to interpreting and implementing mathematics through the 

essential skills base. This social constructivist approach suggested that classroom practices 

support students' development of mathematical understanding through social interaction 

involving conjecture, dialogue and critical examination. While teacher-to-learner 

transmission of information, facts and ideas retains a place in learning, it is only when we 

test, confirm, assimilate and accommodate ideas into our conceptual frameworks that 

knowledge takes on meaning (Hill, 1992). 



10 

The new expectation within the socio-constructivist approach was that individuals would 

develop better mathematical thinking through discussions with peers when they gave more 

coherent explanations, responded to questions and challenges, listened to and made sense of 

others' explanations, and asked for clarification of ideas. The use of such conceptually 

orientated explanations, involving alternative solution strategies, assists in building 

relationships between forms of representation strengthening the students' mathematical 

achievement (Fuchs, Fuchs, Karns, Hamlett, Dutka, & Katzaroff, 1996). 

Based on :findings from collaborative research, Cobb and Wood (1990) contend that when 

children engage in this type of mathematical 'talk', it can result in "learning opportunities 

that rarely arise in traditional instructional settings" (p. 34). Students learn to reason 

analytically and to communicate by explaining and justifying their mathematical ideas 

(Wood, 1999). 

1.2 Teaching Reforms and Collaborative Learning 

The effective use of mathematical talk to foster learning in collaborative groups required 

teachers' perceptions of their role to change whereby they became facilitators of learning 

(Von Glaserfeld, 1990, cited in Mayers & Britt, 1995) and in the 1990s teachers were seen 

to implement collaborative group work in the mathematics classroom in a range of ways 

(Thomas, 1994b) from putting students in groups and expecting them to work together, to 

a formal structured collaborative model with specific group and learning outcomes. Junior 

classrooms which used a Beginning School Mathematics (1993) model often had groups of 

children collectively engaged on activities that were designed to consolidate a core concept 

that had been introduced by the teacher. Examples of children using co-

operative/collaborative practices in later primary years included working in groups to 



11 

complete a one-off investigation or discussing a solution with a peer simply because of their 

physical proximity rather than because of collaborative engagement on a mutual task 

(Medcalf, 1995). 

Thomas' study (1994a) into discourse in the New Zealand junior mathematics classroom 

noted that though most talk was task-related, more was related to the social aspects than 

the cognitive aspects of the task. The challenge then was to consider and develop processes 

that would encourage ' learning enhancing' discourse. 

Thomas found that the social and cognitive nature of the task impacted significantly on the 

kind of talk which occurred and that problem-solving tasks were more likely to engage the 

children in explanations and abstract discussion compared to games or production tasks 

(making a model or collecting data). In order to reduce the social talk associated with 

organization within the groups most groups were reduced to pairs. Thomas also suggested 

that the ability to engage in collaborative abstract thought may be maturational and that 

"children in the first two years of school are not able to engage in substantial 'abstract' talk 

with their peers" (Thomas, 1994b, p. 327). 

While the intention of Mathematics in the New Zealand Curriculum (1992) was to support 

improved learning outcomes utilising the social-constructivist approach to learning 

including cooperative group contexts, two international studies in mathematics achievement 

appeared to provide evidence to the contrary. The duel studies-the Third International 

Mathematics and Science Study (1994) and the Third International Mathematics and 

Science Study - Repeat (1998), showed no significant improvements in mathematics 

achievement by New Zealand students. A recent Education Review Office (ERO) report 



12 

(2000) criticised the curriculum documents and queried "whether or not the structure and 

content of the curriculum documents provide a clear and practical guide for teachers, 

especially those who are not educated specialists in mathematics" (p. 104). The prevalence 

and effectiveness of teaching and learning in the New Zealand classroom by using small [co­

operative/ collaborative] groups was questioned when other countries which seldom used 

small groups achieved more highly: 

Inside the classroom, New Zealand primary school teachers typically use small 

group organization for teaching and learning mathematics. This assumes that 

students will learn better if they are taught in small groups for part of the time and 

then left to work individually or with groups not directly supervised by the teacher. 

(Education Review Office, 2000, p. 101) 

The Education Review Office report urged for a closer examination of small group teaching 

practices in New Zealand claiming that, " it is not self-evident, for instance, that the 

conventional New Zealand method is necessarily more child-centred, especially where it is 

the more or less exclusive arrangement" (p. 101 ). Specifically, ERO suggests we need 

further research information on group teaching "in terms of its contribution to developing 

mathematical understanding. This despite widespread claims as to its effectiveness, and the 

complexity of the process for teachers" (p. 101 ). Increasing research on teaching strategies 

and disseminating the findings of effective practice gives teachers better information on 

which to base their decisions. 

The thesis contributes to the body of research investigating group learning practices by 

investigating the use of specific discourse structures to enhance the level of higher order 

cognitive talk within the context of collaborative groups in the mathematics classroom. 



13 

1.3 Definition of Terms 

Definition of the terms associated with group learning practices are complex and often 

interpreted differently in different education settings intemationally--by teachers, 

researchers and policy makers. For the purposes of this research, the following definitisll$_ 

are provided as starting positions to guide the reader. A fuller discussion of their origin in 

relation to literature is provided in Chapter 2. 

• Collaborative learning is structured and organised small group work designed to 

actively engage students in ''the learning process through inquiry and discussion with 

their peers" (Davidson & Worsham, 1992). 

• Peer-tutoring is a form of co-operative learning in which students are trained to use 

specific instructional strategies to promote the learning of peers (Medcalf, 1992) 

1.4 Research Questions 

The aim of the study was to implement and investigate the effectiveness of a collaborative 

skills programme. The intervention programme was designed to increase students' 'learning 

enhancing' discourse, that is, to increase the amount of cognitive talk students engaged in 

while working in collaborative groups. Using an action research framework, information 

gathered from students' conversations as they worked in collaborative groups, anecdotal 

observations from their teachers, students' questionnaire responses and a diagnostic test 

formed the basis of subsequent modifications to the intervention. 

The following questions were investigated in the main study to evaluate the effectiveness of 

adaptations made to the structure of the programme and to examine resulting discourse 

patterns seen in the pilot study: 



14 

1. Does participation in the programme increase the frequency of 'higher order' cognitive 

interactions between children in the mathematics classroom? 

2. Are there 'oral flags' which identify higher order cognitive thinking? 

The collaborative skills programme was designed to be a practical classroom intervention 

which promoted peer scaffolding as an effective support for academic gains in mathematics. 

Effectiveness was addressed in relation to both teachers' and students' perceptions about 

the programme and through achievement gains in the Numeracy Development Project 

(Ministry of Education, 2002): 

3. How did the students think that participation in the programme affected their academic 

or perceived academic achievement? 

4. What changes did the teacher notice in individual/class interactions or attitudes when 

mathematical problem-solving? 

1.5 Overview 

Chapter 2 provides a theoretical framework for the role of discourse in learning and a 

rationale for the inclusion of specific techniques in the collaborative training programme. 

This is supported by a review of literature from America, Europe, Australia, and New 

Zealand, including discussion and evaluation of peer-tutoring and collaborative learning 

programmes. 

Chapter 3 describes the action research methodology. This includes a description of the 

pilot study and the results which led to modifications of the collaborative training 

programme for the main study. The data collection and analysis processes are described. 



15 

Chapter 4 presents and summarises the data collected for the results from the transcriptions, 

the teachers' anecdotal comments, the students' questionnaires, and the academic data 

recorded from the diagnostic interviews. 

In Chapter 5 the key findings related to the research questions are discussed along with the 

limitations of the study and concluding thoughts. Questions are raised for consideration in 

future research. 



16 

Chapter 2 Literature Review 

2.1 Introduction 

The literature review is divided into three sections. The first section considers the role of 

discourse within a social constructivist theory of learning whereby interactions are seen to 

promote learning by bringing about changes in the cognitive structures of those engaged in 

the interaction. The discussion of the socio-constructivist approach to teaching and 

learning, which underpins the current New Zealand mathematics curriculum, will include 

consideration of two theories of foundational relevance: Vygotsky's social learning theory 

(1978) and Piaget's developmental theory (1962). 

The second section reviews peer-directed programmes and research studies which structure 

students' interactions to promote their engagement in higher order discourse and thinking. 

The section focuses on peer-tutoring and cooperative/collaborative learning models, and 

highlights specific structures which enhance mathematical skills in reasoning and thinking, 

problem-solving, and communication-all of which are key process strands in New Zealand 

and overseas mathematics curricula documents. Though the role of discourse in the home 

influences thinking skills, for the purposes of this project the review will be confined to 

research which has occurred in the classroom. 

The third section investigates tools for the analysis of discourse in the classroom and 

provides a background and rationale for the inclusion and modification of the analysis tool 

used in this project. 



17 

2.2 Learning Theories and Discourse 

Within the socio-constructivist approach to learning, the process of learning is seen as the 

active construction of knowledge within a social community (Simon & Schiller, 1991, p. 

130). According to this approach learning is a by-product of interaction. Discussing new 

ideas with others transforms how we think about the ideas. Accordingly the process of 

accommodation: 

Allows us to organise and reconstruct the new material for ourselves and integrate 

it into our existing knowledge base, thereby allowing us to understand and 

remember it better... During such interaction with another, we clarify ideas, 

negotiate meaning, develop new skills, and construct new knowledge. 

(King, 1997, p. 221) 

Therefore although "thinking is the personal process individuals use to create their own 

understanding" co-operative exchanges are seen as extremely beneficial because the 

resulting "shared visions and understandings enlarge the process spheres that individuals 

may explore, thus making enhancement of individual thought as boundless as the visions 

shared" (Davidson & Worsham, 1992, p. xx). 

The socio-constructivist theory of learning, and its promotion of the role of discourse, is 

underpinned by two major theorists' work: Piaget's cognitive developmental approach to 

learning (1967) and Vygotsky's social learning theory (1978). 

2.2.1 Individual Development of Thought 

Piaget (1967) believed that learning was acquired through the process of assimilation where 

the ideas or concepts, which he termed schemata, were developed in response to stimuli. 

Subsequent information or stimulus allowed for the development of a schema through a 



18 

process called accommodation. If new information compared favourably with existing 

schema it was accommodated into the schema, enlarging its cognitive reference. If the 

incoming stimulus compared unfavourably it was rejected, or a period of disequilibrium 

resulted while the individual searched for more information causing the reorganisation and 

reconceptualisation of the schema. Within this theory, discourse facilitates the exposure of 

one individual's schemata or concepts causing individuals to clarify, develop, expand and 

elaborate their thinking in defence of their particular schemata. 

Piaget's theory favours a classical psychological approach to the measurement of the child' s 

cognitive development. The child's independent activity and achievements are observed, 

passing through cognitive stages of maturity using stimuli that progress from concrete to 

semi-concrete to abstract. The first stage, is that of the sensorimotor period, 

(approximately 0-2 years) where the child responds from reflex action, then to the 

preoperational stage (2-7 years) with increasing imitation and experimentation, to the 

concrete operational stage (7-11 years), and then to the formal operational stage where the 

individual is able to visualise actions abstractly and perform them without physical 

experimentation. 

According to this cognitive theory a child's mathematical cognitive development progresses 

from intuition to formal operations, from exploration through play to structured games, to 

representation through pictures and symbols and then formal operations, from a social 

experience of number to the learned abstractions and formal practices of mathematics. As 

such, cognitive development is influenced by the child's physical and maturational ability to 

imitate. It begins as a 'tentative exploratory activity' (Dienes, 1959) initially determined 

largely by spatial experiences. With maturity the development of language takes on an 



19 

increasing role; "physical actions become internalised and generalised into concepts and 

relations to which may be attached, either words or mathematical symbols" (Dickson, 

Brown & Gibson, 1984, p. 12). 

For children, their schemata are qualitatively different to that of adults and commonly 

include partial understandings or misinterpretations. The development of their schemata is 

influenced by the linguistic and cognitive practices in their environment and are adjusted 

using an egocentric logic with subsequent experiences and social interactions. Thus in 

learning mathematics, individuals construct ideas, processes and unique understandings for 

themselves. 

2.2.2 Social Development of Thought 

In contrast to Piaget's cognitive developmental theory where the independent activities of 

the child are measured, Vygotsky (1978) emphasises the social aspects oflearning, claiming 

that each child has a higher level of cognitive development evidenced by his/her ability to 

imitate. In this way, Vygotsky focuses on the cognitive processes, which are still evolving, 

not those already mastered: "What the child can do in co-operation today he can do alone 

tomorrow" (Vygotsky, 1962, p. 104). 

Vygotsky's sociohistorical-cultural psychological theory of learning contends that the 

combination of speech and practical activity support the most significant intellectual 

development (Blanck, 1990). Vygotsky claims that because children's higher mental 

functions originate in the activities and social dialogues in which they participate, those 

children involved in collective discourse, verbal and non-verbal are able to develop their 

thinking more effectively compared to children who are alone. However, Vygotsky 



20 

qualifies the nature of social interactions claiming that a child can imitate only that which is 

within his/her developmental level or Zone of Proximal Development (ZPD}--the "distance 

between the actual development level as determined by the independent problem-solving 

and the level of potential development as determined through problem-solving under adult 

guidance or in collaboration with more capable peers" (Vygotsky, 1978, p. 86). 

Thus, within the education context students need to be provided with experiences within 

their ZPD, just above their level of independent functioning, so that cognitive processes are 

developed. In order to keep the students in their ZPD the task and the environment must be 

appropriately structured so that the demands on the students are appropriately challenging. 

The amount of intervention provided by the adult or more capable peer needs to be adjusted 

in response to the students' needs (Tharp & Gallimore, 1988, cited in Berk & Winster, 

1995). Effective scaffolding occurs when: 

The tutor or the aiding peer serves the learner as a vicarious form of 

consciousness until such a time as the learner is able to master his own 

action through his own consciousness and control. When the child 

achieves that conscious control over a new function or conceptual 

system then he is able to use it as a tool ... the tutor in effect performs 

the critical function of scaffolding. (Bruner, 1985, p.215) 

Discourse facilitates "the collaboration necessary between expert and novice to acquire the 

cognitive strategy" (Palinscar, 1986, p. 95). Without the use of scaffolding Gallimore 

claims that children will learn at a less than optimum rate (Gallimore, cited in Holton, 

Anderson & Thomas, 1996). 



21 

2.3 Structuring Discourse in the Classroom 

2.3.1 Teachers and Students as a Mathematical Community. 

The socio-constructivist approach to teaching and learning highlights the importance of 

providing social situations that allow children to construct and modify their mathematical 

knowledge through discourse (Yackel, Cobb, & Wood, 1991): 

To understand what they learn, they must enact for themselves verbs that permeate 

the mathematics curriculum: "examine ", "represent '', transform ", "solve ", 

"apply '', "prove ", "communicate ". This happens most readily when students work 

in groups, engage in discussions, make presentations, and in other ways take 

charge of their own learning. (National Research Council, 1989, p. 58, 59) 

Reform instruction advocates that teachers and students within the classroom setting form a 

mathematical community in which they construct ''their own knowledge individually and 

collectively, negotiating shared understandings and developing collaborative processes for 

validating ideas" (Neyland, 1994, p. 4). The teacher has a vital role as the expert who 

guides, models and interacts with students to provoke them to think mathematically, 

providing opportunities for students to verbalise their questions, explanations and strategies 

(Ministry of Education, 2002a). A teacher can be thought of as: 

A discourse guide and each classroom as a discourse village, a small 

language outpost from which roads lead to larger communities of educated 

discourse... Teachers are expected to help their students develop ways of talking, 

writing and thinking which will enable them to travel on wider intellectual 

journeys. (Mercer, 1995, p. 83) 

Peer discussions are seen as valuable not only if they lead to the correct answer but also 

because of the way they influence students' mathematical behaviour. The way students 



22 

think about and solve problems is 'influenced by discussions in which ideas are shared, 

challenged, and justi£ed' and is 'related positively to mathematics achievement' (Muth, 

1997, p. 72). In particular co-operative learning puts students "in situations where they 

learn that reading, writing, listening and speaking in a co-operative manner are all-important 

components of successful problem-solving" (Muth, 1997, p. 72). Co-operative learning has 

many other advantages for supporting effective learning practices: 

Students learn mathematical language from each other and are also able to 

learn, firsthand, the various problem-solving strategies that their peers use. 

Equally important, students learn that there are usually different ways to 

solve a problem. Finally, co-operative learning builds leadership, 

decision-making, and conflict management skills, and ideally, positive 

attitudes towards mathematics. (Muth, 1997, p. 72) 

Within the mathematics classroom the mam source of scaffolding is the teacher. The 

teacher' s scaffolding role is most commonly evidenced in classroom discourse when the 

teacher focuses, extends, clari£es and redirects the learner (Carnboume, 1988). Using an 

appropriate selection of tasks or problems the teacher builds on prior knowledge and 

maintains the student within the Zone of Proximal Development 

Another aspect of scaffolding by the teacher is the appropriate selection of tasks or 

problems to maintain the students within their ZPD (Diezmann, 1998). Stein, Grover, and 

Henningsen (1996) investigated the selection of mathematical tasks by teachers and 

concluded that appropriate task selection included having appropriate amounts of time to 

engage on the task. Inappropriate time limitations precipitated low-order recall. Sustained 

pressure for explanations and elaboration was important through successive multiple 

presentations of solutions and justi£cations to the wider group or class as this encouraged 



23 

the making of connections and hence higher order thinking. 

2.3.2 Authority 

Relationships and interactions within the classroom produce the construction of 

mathematical knowledge and identity, and are greatly influenced by the teacher's practices 

which establish social norms. Productive discourse practices in a classroom mathematical 

community not only support students' mathematical understandings but also give authority 

to their discourse, and value student authorship (Klein, 2002). The social goal in 

negotiating meaning is to resolve uncertainty and that in order to reach resolution students 

will appeal to the dominant authority. Prior experience, empirical evidence, a 

knowledgeable person or a text can represent this dominant authority (Clark & Helme, 

1997). Brown (1994, cited in Brown & Renshaw, 1999) describes the process of 'Collective 

Argumentation' in which "authority is attained through discourse practices that privilege 

socio-mathematical norms such as meaningfulness, communicability and testability" (p. 

114). 

In the absence of appropriate authorities students find it difficult to make metacognitive 

judgements about errors, anomalous results and the appropriateness of their problem 

solving strategies (Clarke & Helme, 1997). The teacher's role in the process oflearning to 

resolve uncertainty is important. If an autocratic teacher makes the majority of learning and 

organisational decisions in the classroom then students learn to rely on the teacher as their 

source of authority and to rescue them from uncertainty (Hill & Hill, 1990; Neyland, 1994). 

This can result in students developing poor decision making skills and exhibiting an 

unwillingness to attempt novel strategies or be open to other available authorities such as 

experience, empirical evidence, texts or knowledgeable group members (Davidson, 1985). 



24 

Teachers who consistently intervene when students are struggling take away their 

opportunities to make progress on their own (Stein, Grover, & Henningsen, 1996). 

Equally, a teacher who takes a laissez-faire approach and leaves the children to take 

responsibility for their own learning "does not provide for responsibility, cohesiveness, 

cooperative skills or academic achievement" (Hill & Hill, 1990, p. 18). In contrast, teachers 

who promote the model of positive interdependence required in collaborative learning and 

encourage an acceptance of peers' ideas which are presented with logic and evidence, 

support cognitive development, and the movement of children's beliefs from a position of 

dualism to ethical beliefs (Davidson, 1985). 

Perry (1970, cited in Davidson, 1985) describes students' beliefs about authority as 

cognitively developmental. At the initial level of development, Perry says students believe 

that all answers are right or wrong, and that only authority figures such as the teacher 

possess knowledge and answers. The next level of development shows students accepting 

some peers' viewpoints and becoming less dependent on texts or the teacher as the 

authority. In the later stages of cognitive development the students are able to value the 

opinions of others when these opinions are presented logically and with evidence. The 

students in this stage are also able to reason in a wider range of contexts and treat their 

teacher as a fellow learner rather than a person of authority. 

The teacher's pedagogy (beliefs and practices) influences the learner's sense of authority. 

Klein (2002) argues that the teacher can foster the learner's ability to apply constructed 

mathematical knowledge in new contexts by celebrating ''the student's presence and ways of 

making sense of mathematics and experience" (p. 391) and by giving them space " to 



25 

establish themselves in powerful ways in learning/teaching interactions as they construct 

mathematical ideas" (p. 392). 

Discourse moulds not only the content knowledge but also the learner' s mathematical 

identity and perception of his/her competence as students learn its linguistic code and how 

to relate to a subject (Blackredge & Hunt, 1985). The learner develops a sense of power 

through his/her legitimate use of mathematical ideas and practices, and hence a sense of 

internal authority. For the learner to develop this power they must be "capable of, 

respected and valued in, speaking/writing the accepted ' truths ' of a discourse, in enacting 

established ways 'of being ' and in going beyond these to forge something new" (Davies, 

1991 , cited in Klein, 2002, p. 391). Teachers consider the relationships within discourse; of 

what is spoken and who it is spoken by, the power of the speaker's social relationship or 

authority to develop socio norms within the mathematics classroom (Klein, 2002). 

2.3.3 Transparency 

One of the concerns for designing an intervention to enhance the level of higher order 

cognitive peer interactions is the degree to which the interaction amongst learners is 

structured. A balance needs to be achieved to enable the structure to promote both the 

exchange and challenging of ideas, and the freedom to pursue and develop individual 

thoughts (Cohen, 1994). The teacher has a major role in framing the language practices of 

the classroom as she acts as a translator for mathematical discourse "to help frame 

discussion, to pose questions, to suggest real life connections, to probe arguments, and to 

ask for evidence" (Adler, 1999, p. 51), and hence scaffold students' participation and 

provide access to school mathematics. 



26 

Adler (1999) warns that too much focus on the practice of mathematics discussion makes 

the discussion become the "main object of attention instead of a means to the mathematics" 

(p. 50). The discourse practices should be used as learning tools. 

Discourse as a social tool has a quality Lave and Wenger have labelled transparency (1991, 

cited in Adler, 1999). They describe transparency metaphorically as a window involving a 

complex interplay of visible and invisible characteristics: 

A window's invisibility is what makes it a window. It is an object through which 

the outside world becomes visible. However, set in a wall, the window is 

simultaneously highly visible. In other words, that one can see through it is 

precisely what makes it highly visible.... [Windows are like] 'mediating 

technologies ' in a practice ... [that] need to be visible so they can be noticed 

and used, and they need to be simultaneously invisible so that attention is focussed 

on the subject matter, the object of the attention in the practice. 

(Lave & Wenger, 1991, cited in Adler, 1999, p. 50) 

The discourse strategies for this project were developed with transparency in mind and 

aimed to enhance cognitive mathematical learning. The students were made aware of the 

significance of their talk to learning, and their teacher and the researcher modelled specific 

kinds of mathematical discourse to them. They were encouraged to model the same 

practices within collaborative groups and engage the strategies as a successful method of 

learning the mathematics. 



27 

2.4 Collaborative Learning 

The process of scaffolding can occur informally, or formally through the provision of 

structured co-operative and collaborative programmes that utilise discourse to enhance the 

teaching and learning of mathematics, enabling peers to construct robust knowledge 

structures as they clarify information and resolve discrepancies. Collaborative learning and 

peer-tutoring techniques provide scaffolding in the form of a peer who has authority 

because of their role (as tutor), their knowledge, or that they have the strategies to be able 

to utilise experience, evidence and texts in various combinations to provide authority. 

Most New Zealand classroom teachers utilise peer discussion as a fonn of scaffolding 

(Ministry of Education, 1997) as peers negotiate meanings and ideas while involved in 

shared tasks. However, the enhancement of achievement through peer scaffolding is more 

complex than assigning children to a group to work together to solve a problem (Peter­

Koop, 2002; Webb, 1989). Brown and Thomson (2000) make a finn distinction between 

what they call ' traditional group work' and learning together co-operatively to produce 

social or academic outcomes: 

Working round a table on individual tasks with the opportunity for discussion 

is not co-operative learning. Nor is having a team discussion, where some students 

can dominate or "hitch-hike". The core notion of co-operative learning is that 

when we co-operate, we work together to accomplish shared goals ... . While it helps 

to foster and develop interpersonal skills, co-operative learning is not a social 

skills programme. Rather it is an academic skills programme, a thinking skills 

approach that requires students to develop the necessary interpersonal and small 

group skills to enable them to work successfully together. 

(Brown & Thomson, 2000, p. 38) 



28 

Cooperatively styled programmes that have an active, focused approach to learning where 

both the teacher and the student are aware of the importance of scaffolded practice are 

more likely to lead to more effective learning (Brown & Thomson, 2000, p. 26). There is a 

wealth of methods currently employed in the classroom to promote collaborative learning. 

They build on the momentum of the cooperative programme development of the 1970s and 

group learning methods such as Constructive Controversy (Johnson & Johnson, 1987), 

Team Games (DeVries & Edwards, 1974), Group Investigation (Sharan & Sharan, 1976), 

Jigsaw Procedure (Aronson, Blaney, Stephan, Sikes & Snapp, 1978) and Student Teams 

Achievement (Slavin, 1989, 1996). Work has continued into the 1980s and 90s with 

Cohen's Complex Instruction approach (1986), Kagan's structural PIES approach (1994), 

Brubacher, Payne and Rickett's collaborative approach (1990) and Brown's Collective 

Argumentation (1994, cited in Brown & Renshaw, 1999). 

Johnson and Johnson define co-operative learning as "the instructional use of small groups 

so that students work together to maximise their own and other' s learning" (1994, p. 1). 

However, literature shows that others employ the term 'co-operative learning' in a loose 

fashion to include all group work. To avoid this ambiguity there has been a move by 

researchers in mathematics education to differentiate organised and structured group 

learning under the term 'collaborative learning' . 

Goos (2000) describes collaborative learning as those procedures "designed to engage 

students actively in the learning process through inquiry and discussion with their peers in 

small groups" (p. 39). It is a reciprocal process of mutuality where each other's reasoning 

and viewpoints are explored in order to construct a shared understanding of the task (Goos, 

2000). 



29 

Davidson and Worsham (1992, p. xiii) identify four common attributes for all collaborative 

learning with the premise that the group work is organised and structured ''to promote the 

participation and learning of all the group": 

1. There is student-to-student interaction in small groups. 

2. There is individual responsibility and accountability, or positive interdependence. 

3. There is structured cooperation. 

4. A learning task/activity is selected for group work. 

In the definitions of both Brown and Thomson (2000) and Johnson and Johnson (1994) the 

term 'co-operative learning' could be interchanged for 'collaborative learning' but to avoid 

confusion with other models of co-operative learning the term collaborative learning will 

be used in this study. While references will be made to co-operative learning research 

which conforms to the collaborative learning definition the study specifically differentiates 

collaborative learning, which involves the development of cognitive thinking through 

verbal negotiations of meaning with their peers, from that of involvement in cooperative 

group tasks. Cooperative group tasks are considered shared activities whereas collaborative 

learning is "a mutual task in which partners work together to produce something that 

neither could have produced alone" (Forman & Cazden, 1985, cited in Thomas, 1994, p. 

25). 

Working collaboratively in mathematics supports students' learning when they resolve 

contradictions that arise and when: 

... They attempt to make sense of a situation in terms of their current concepts 

and procedures, accounting for a surprise outcome (particularly when two 

alternative procedures lead to the same result), verbalising their mathematical 

I 

- I 



30 

thinking, explaining or justifj;ing a solution, resolving conflicting points of view, 

developing a framework that accommodates alternative solution methods, and 

formulating an explanation to clarifo another child's solution attempt. 

(Yackel, Cobb & Wood, 1991, p. 394) 

Proponents of collaborative leaming argue strongly that effective collaborative leaming is 

not automatic. Students do not osmotically obtain effective leaming strategies but rather 

they "gain them by exposure to capable models and from guided practice in their use" 

(Brown & Thomson, 2000, p. 33). This includes leaming how to deal with disagreements, 

as collaborative leaming is not about harmonising, as it often involves intellectual conflict 

(Hill & Hill, 1990). If students do not develop skills of positive interdependence whereby 

they work together to achieve the group's leaming goals through the synthesis of 

independent and collaborative contributions their leaming is likely to be no more successful 

than competitive or individualistic learning models (Kneip & Grossman, 1979; Qin, 

Johnson, & Johnson, 1995). 

2.4.1 Cooperative Model (Brown & Thomson, 2000; Johnson & Johnson, 1987) 

Brown and Thomson (2000) are two researchers in Special Education who have contributed 

strongly to New Zealand work on co-operative leaming and who differentiate it from 

traditional group work. They build on the extensive work of Roger Johnson and David 

Johnson who pioneered co-operative leaming as a teaching procedure (Johnson & Johnson, 

1994; Johnson, Johnson & Johnson-Holubec, 1993; Johnson, Johnson & Stanne, 2000: 

Johnson, Johnson & Zaidman, 1985; Yager, Johnson & Johnson, 1985). 

According to Brown and Thomson's model (2000), as students practise collaborative 

learning skills they move through four levels of functioning. These levels of functioning are 



31 

based on Johnson and Johnson's '4Fs': forming, functioning, formulating, and fermenting. 

'Forming' skills are basic skills required for groups to function such as moving and talking 

quietly, using eye contact and group members' names, and encouraging all group members 

to participate. 'Functioning' skills are those skills, which allow greater self-management 

within the group. Individual members maintain their given roles, all group members are 

included and encouraged, and the interactions are both courteous and positive. 

The next two levels of skills require more understanding of the task and the use of higher 

order thinking skills (Bloom, 1956). The third level of skills is 'formulating' and requires 

students to be able to apply and analyse ideas. Students engaged in a task ask for and listen 

to elaborations, justifications, and summaries from other group members. The fourth and 

final skill level is that of 'fermenting' when students are able to integrate ideas to form a 

concept or general principle. They are able to question, critique and evaluate peer ideas, 

and are able to develop and integrate other peer ideas into a new concept or application. 

This level aligns with synthesis and evaluation in Bloom's taxonomy (1956). At this level 

students are also able to handle controversy in a positive and constructive manner. 

Collaborative group work encourages vocalisation amongst peers and as the students 

develop the higher level thinking skills of formulating and fermenting they become more 

effective learners. 

Johnson, Johnson, Roy and Zaidman (1985) conducted research into the oral interactions of 

children (American Grade 4) and they found that a high proportion (90%) ohalk was task­

related. When they investigated the behaviours of high, medium and low achievers in the 



32 

groups they discovered that vocalisation had significantly more impact on achievement than 

listening. They noted that: 

Vocalizing task-related information orally striving to obtain more facts and 

information about the subject under discussion, giving explanations, providing 

rationales, and relating what is being learned to previously learned information, 

and disagreeing with other members' task-related conclusions were all significantly 

related to achievement. (Johnson, Johnson, Roy, & Zaidman, 1985, p. 318-19) 

They also found that medium achievers did not vocalize as much as high and low achievers 

in collaborative groups and concluded that medium achievers needed particular 

encouragement to elaborate in order to make more significant gains in achievement. 

From their results, Johnson et al. ( 1985) recommended that teachers encourage 

collaborative groups to elaborate upon material by providing a structure that increased the 

level of on-task vocalisation and introduced controversies so that students had to negotiate 

meanings from other students' task-related conclusions. 

2.4.3 Think-Pair-Share Model (Lyman, 1992) 

Lyman (1981, 1992) developed a cooperative learning strategy called 'Think- Pair- Share': 

A question or problem is posed then the students are given time to think individually. The 

students cannot talk or raise their hands but may write down or draw a diagram to represent 

their individual ideas. At a designated time, indicated by the teacher, the students share 

ideas. The sharing of ideas results in a collective understanding, generated when they 

challenge and elaborate upon each other's thinking. Ideas may be subsequently shared with 

a larger group or the class. Using a variation of the Think-Pair-Share model in 



33 

mathematical collaborative learning investigations, Neyland ( 1994) found that use of 

individual think time at the start of an investigation resulted in more students contributing to 

the mathematics subsequently developed by the group. Students did not run with the first 

idea and they began to value the alternative thinking styles of their classmates as well as 

utilising a wider range of problem solving strategies. 

The Think-Pair-Share model utilises the concept of 'wait-time' . Wait-time, a pause 

between the question and a student's response, provides an opportunity for a student to 

formulate a thoughtful reply, to recall relevant prior knowledge, or validate a possible 

solution. This skill can be developed by the students in the forming and functioning levels 

of collaborative group skills (Thomson & Brown, 2000). Various researchers and 

educationalists have investigated the use of wait-time to improve the quantity and quality of 

student verbal responses. Rowe (1974) and Tobin and Capie (1980) found that when 

teachers paused for three seconds or longer after posing their initial question, and again 

after the student's initial response, that the length of student responses increased, more 

frequent unsolicited contributions were made, explanations became more logical, students 

supported ideas with evidence, participation by less able class members increased, and the 

number of questions and speculative responses increased. The use of wait-time has also 

been shown to contribute to gains made by tutees in peer-tutoring reading programmes and 

written language programmes (Medcalf, 1999; Medcalf & Glynn, 1987). 



34 

2.5 Peer-tutoring 

The initial collaborative discourse model considered for this study was a peer-tutoring 

programme. Peer tutoring is an instructional programme where students teach other 

students. The idea of student as teacher and learner is not new and has been documented as 

far back as Roman times. More recently, the pupil-teacher of the nineteenth century formed 

a structured part of classrooms in the United Kingdom and the Colonies (Limbrick & 

McNaughton, 1985). Teachers for large class numbers or diverse age groups in small rural 

schools have formerly used cross-age peer tutoring in New Zealand as a coping mechanism. 

Since the mid 1980s peer tutoring has gained in popularity because of the increased use of 

co-operative learning models (Medcalf, 1995) and individualised instruction, and because of 

perceived cost-effectiveness benefits (Jenkins & Jenkins, 1988). 

Research has shown peer-tutoring programmes to produce positive academic gains in 

mathematics (Beirne-Smith, 1991 ; Fuchs, Fuchs, Hamlett, Phillips, Karns & Dutka, 1997; 

Greenwood, Sloane, & Baskin, 1974; Maheady, Sacca, & Harper, 1987); spelling (Garcia­

V asquez & Ehly, 1992; Mallette, Harper, Maheady, & Dempsey, 1991 ; Greenwood, 

Daiquiri & Carta, 1997); reading (Kreuger & Brown, 1999; Medcalf & Glynn, 1987; 

Robinson, Glynn, McNaughton & Quinn, 1979); written language (Huggard, 2002; 

Medcalf, 1994); and where students use English as a second language or are intellectually 

impaired (Bar-Eli & Raviv, 1982, cited in Garcia-Vazquez & Ehly, 1992). 

Positive social gains have also been documented when the peer tutoring has been used 

directly to effect behavioural changes (Schunk, 1987) and to raise self-esteem as a by­

product of increased academic success (Medcalf, 1995). 



35 

The term 'peer-tutoring' is used loosely and widely in literature to describe buddy systems, 

role-models, peer-testing, peer-monitoring and peer-teaching, but will be considered within 

this study as a programme in which students are trained to use specific instructional 

strategies to promote the leaming of peers. The strategies used by the child in the tutor role 

include promoting higher order cognitive thinking by asking appropriate questions, waiting 

so his/her partner has time to think before responding, listening attentively, and giving 

positive feed-back and encouragement (King, Staffieri, & Adelgais, 1998). The central role 

of peer tutoring is to provide structured prompting of cognitively orientated talk in order to 

promote active construction through the analysis and integration of ideas. However peers 

talking while working on tasks independently of their teachers is not enough to 

"substantially further their mathematical understanding" (Thomas, 1994, p. iii). Effective 

training of peer tutors is imperative. Without training "peer tutors are likely to be less than 

helpful in supporting the learning of others" (Medcalf, in press). 

Peer-tutoring in this study is not considered the same as peer-mentoring (the buddy­

system), or peer-testing, which involves peers carrying out ' low order' tasks together to 

capitalise on the influence children have on each other and the comfort they have in working 

together. Some of the literature regarding peer tutoring encompasses these mentoring and 

testing programmes, particularly American literature. Peer tutoring is distinct from peer 

mentoring in that tutors are trained and monitored to carry out their role. For the purpose 

of this study peer tutoring will exclude 'buddy-support ' or peer testing such as that used by 

Greenwood, Daquiri and Carta (1997). Though Greenwood et al.'s programme bears the 

name 'peer-tutoring' it is in fact a peer-testing programme which is used to provide 

immediate feedback to a partner rote-learning a set of spelling words and mathematics basic 

facts. It is a competitive progranune, with the tutee receiving a series of points (for 



36 

correctly spelt words) to contribute to the team total. This type of programme does not 

promote "greater use of higher level reasoning strategies and increased critical reasoning 

competencies" (Medcalf, 1995, p. 11) which is inherent in the training of tutors for 

collaborative style learning, peer-tutoring programmes. 

2.5.1 'Pause, Prompt and Praise' Model (Medcalf, 1992, 1994) 

Medcalf has developed a New Zealand based peer-tutoring programme for mathematics 

adapted from a research base associated with reading and written language. The 

mathematics peer-tutoring programme is "designed to help students identify errors in their 

thinking relating to their problem-solving efforts in maths, provide appropriate modelling 

and feedback and develop metacognitive skills" (Medcalf, p. 6, in press). In this programme 

peers are taught to perform four steps framed by the mnemonic WASP - Watch, Ask, 

Show, and Praise. The first step involves the tutor watching the tutee solve the problem. If 

the tutee has difficulties the tutor asks the tutee questions to redirect him/her to the next 

step or a previous error. If the tutee cannot proceed alone the tutor demonstrates (or 

shows) the next procedure in the problem or a similar example and then encourages the 

tutee to complete a similar problem independently. The last step is giving praise for 

completing the problem, or prompting the tutee as he/she works independently to complete 

the task with specific feedback on the issue the tutee was having difficulty with. 

Grant has trialled this approach in two pilot studies: one involving measurement with a class 

of nine and ten year olds, and the second (Grant, 2000, cited in Medcalf, in press) with 

twelve 9/10 year olds using mathematics games to develop computational skills. Both 

studies revealed increased self-esteem in tutors/tutees, increased on-task behaviour in all 

mathematics lessons, social benefits to the class and a higher average performance in the 



37 

post-unit test by the tutor group compared with the control group. In each study, the tutors 

were trained to respond within a particular mathematics unit (i.e., measurement and 

computational skills) and the tutors had specific mathematical knowledge beyond the level 

of the tutees. 

In contrast to Medcalfs programme, the tutor training in this study was developed so the 

instructional skills of the tutor could be applied generically to any strand of the mathematics 

curriculum. In this way the child in the role of tutor (or coach) did not have to be more 

mathematically knowledgeable about a specific task than the child in the role of tutee (see 

2.6.3). In the study this was achieved by introducing a collaborative component in which 

both students in a pair shared responsibility for the functional group skills and completion of 

the task. The programme and training is explained in detail in Chapter 3. 

2.5.2 'ASK to THINK-TEL WHY' Model (King, 1997) 

Sometimes peers, especially younger or less able students, are unable to verbalise their ideas 

so there can be no negotiation of meaning but by providing "rubrics, or key phrases, to 

explicitly encourage them to engage with each other's thinking" (Goos, 2000, p. 43) their 

ideas can be heard and evaluated. 

The 'ASK to THINK-TEL-WHY' programme was developed by King (1990, 1997) as an 

inquiry-based model of mutual peer-tutoring. Peer interactions were enhanced through the 

use of reciprocal peer-questioning strategies with generic stem questions. In King's model 

(1990) the question patterns created socio-cognitive conflict and forced the students to 

restructure their knowledge in order to respond to a particular pattern. They generated new 

examples and elaborated upon existing relationships between ideas in order to respond. 



38 

King found that students tended to interact at a "basic, concrete knowledge, retelling level" 

if left to their own devices, but by structuring the interaction with the provision of generic 

stem questions the students engaged "in the mutual exchange of ideas, explanations, 

justifications, speculations, inferences, hypotheses, conclusions, and other high level 

discourse known to promote peer-learning "(King, 1997, p. 222). 

Different kinds of questioning promote different levels of cognitive thinking. Asking, 

"What happened?" promotes lower order descriptive responses that recall facts. The 

information is restated within an integral context--reporting what happened during the 

concrete experience of the task. Asking "Why?" or "How?" type questions promote 'higher 

order' responses involving justifications, explanations and inferences. This encourages the 

integration of complex knowledge through making new connections between the task and 

external contexts (King, 1998). 

Similar models have been developed in Australia to assist peers, especially younger or less 

able students, to verbalise their ideas so there can be negotiation of meaning by providing 

"rubrics, or key phrases, to explicitly encourage them to engage with each other's thinking" 

(Goos, 2000, p. 43). These interactions engage students' higher order cognitive processes 

and assist them to build robust knowledge structures (King, 1997; King, Staffieri, & 

Adelgais, 1998). 

2.5.3 Multiple Retelling Model (Cesar, 2001) 

Another programme which focuses on the development of higher order cognition uses 

didactic peer interactions with the multiple retelling of solutions. This model was developed 

by Cesar (1998), who conducted a four-year project into social interactions of didactic pairs 



39 

and their mediating role lll knowledge apprehension, cognitive development and skills 

acquisition in 5th to 11th graders. The didactic pairs were initially suspicious of one 

another and disinclined to work cooperatively. Problems were overcome by the 

implementation of rules, or a didactic contract, where each individual had to be able to 

explain both problem-solving strategies in order that either one of them could present their 

work during the class discussions. In this way they explained their own problem-solving 

strategy and asked questions and actively listened so that they would also be able to explain 

their partner's strategy. From this study Cesar(l 998) argued that the interaction itself is 

enough to promote better performances. This contrasts with Vygotsky's belief that the 

social interactions of a more competent peer or adult best. 

2.6 Nature and Analysis of Classroom Discourse 

The complex interplay of content, context and relationships in the nature of discourse makes 

it difficult to select an adequate tool to measure and analyse classroom interactions. Three 

models were considered for this study: Thomas' model (1994); Johnson, Johnson, Roy, and 

Zaidman's model (1985); and King, Saffieri, and Adelgais' model (1998). The models and 

the implications suggested by the pilot research findings are discussed to provide a rationale 

for the intervention developed for this project. 

King, Staffieri, and Adelgais' model (1998), which measured the interactions in three peer­

tutoring conditions, was considered because it was developed to measure tutor responses in 

a pre-training and post-training situation. King et al. analysed the data using rates of 

interaction per minute and classified on-task interactions into three categories, those related 

to questioning, task statements, and supportive communication. This model required a 

question/statement or statement/statement interaction between tutor and tutee. After some_ 



40 

initial trial in the pilot study it was discarded because it was not able to be appropriately 

adapted to a multi-member group with equality of roles. 

Johnson et al. ' s model (1985) was considered because its development was supported by 

three decades of research by Johnson and Johnson in the area of co-operative learning. The 

particular model they used had been developed over three years as they observed co-

operative groups and categorised the groups' oral interactions. The five-factor 

classification system resulted from factor analysis and input from Roy and Lyons' 

observational model of oral rehearsal and cognitive processing (1982, cited in Johnson et 

al. , 1985). The five categorisations are: 

• 

• 

• 

• 

• 

interactions exchanging task related information; 

interactions elaborating information; 

interactions encouraging each other to learn; 

interactions disagreeing with each other's conclusions; and 

non-task comments and sharing feelings . 

This model, which was trialled during the pilot study, was extremely simple to administer 

with instances of talk able to be placed clearly in one of the five-factor categories. 

The third model considered was Thomas' (1994) as it had been developed and used in New 

Zealand mathematics classrooms specifically to investigate the interactions of children as 

they worked in small groups independently of the teacher. Thomas classified transcriptions 

of children's small group interactions, which had been videoed, using a computer 

programme to encode and identify patterns in verbal data. 

When Thomas (1994) conducted her study on the nature of talk in the New Zealand junior 



41 

mathematics classroom, teachers were surprised that 91% of the talk was task-related which 

is consistent with Johnson et al. 's findings (1985). Thomas also found that the usual pattern 

for non-task related talk was that it was "brief, intermittent and interspersed with task-

related talk" (p. 113). In most cases the children continued to work on the activity while 

they engaged in non-task related talk. However of the task-related talk, 53% was to do with 

social management aspects of the task and only 3 7% with cognitive aspects of the task. 

Thomas identified two further subcategories of cognitive talk (see Figure 1), that of 

reflection, when children considered their own thinking, and that of action talk, when they 

verbalised what they were doing, read aloud (reported on their writing), asked questions, 

gave help or commented to their peers about what they were doing with the expectation of 

a response. 

All Talk 
u u 

Non-task Task-related 

Social Cognitive 

Action Reflection 

Figure 1: Thomas' Model ofTalk (1994, p. 119). 

Thomas found that amongst young children there was very little reflective talk (1 %) unless 

the children were prompted by a teacher's question. This observation is also borne out by 

Higgins (1994) who found that the mathematical focus of an activity could be overrun by 

social and procedural problems in junior mathematics classrooms in New Zealand. Higgins 

also found that where teachers modelled mathematical language instead of everyday 



42 

language, and modelled discussion of an activity, that the children could apply these 

cooperative skills in later independent activities, by listening to explanations from other 

group members, asking questions of each other and explaining their thinking. Young­

Loveridge (1993) also found that younger children involved in junior mathematics 

classrooms required more monitoring, encouraging and modelling by teachers to remain on 

task and develop effective cognitive skills. 

In Thomas' study (1994), the majority of social talk was talk about the organization of 

materials for the task (29%) and about the organization of other children ( 16% ), and whose 

turn it was (20%). Social talk included a degree of social bargaining to gain acceptability 

and access to groups, articulation of the rules, disputes and comments about progress, such 

as who was winning a game/ who had the most/ who had completed their worksheet first. 

The amount of social talk increased as group numbers increased in size. 

2. 7 Literature Review Summary 

This literature review summarises the positive social and academic outcomes attributed to 

peer-tutoring and collaborative group-work. It emphasises the importance of the teacher ' s 

role to select 'rich' engaging tasks at an appropriate level, model discourse patterns, and 

utilise effective discourse strategies in collaborative programmes. 

The review highlights the need for collaborative group skills to be taught in order to move 

students from basic group forming and functioning levels to effectively and consistently 

implementing higher order cognitive thinking skills at fermenting and formulating levels. 



43 

Chapter 3 Methodology 

3.1 Introduction 

The aim of this action research study was to develop, implement, and refine a collaborative 

programme which has practical benefits to classroom teachers who wish to develop their 

students' higher order cognitive thinking in mathematics. The intervention was designed to 

support effective discourse structures of small groups working collaboratively on problem 

solving in the mathematics classroom. Scaffolding was provided by the teachers in the form 

of modelling and programme structure, and reinforced by competent peers during group 

work, in order to increase students' involvement in higher order cognitive interactions. The 

following questions formed the basis for the research as to the effectiveness of the 

intervention: 

1. Does participation in the programme increase the frequency of higher order cognitive 

interactions between children in the mathematics classroom? 

2. Are there 'oral flags ' which identify higher order cognitive thinking? 

3. How did the students think that participation in the programme affected their academic 

achievement? 

4. What changes did the teacher notice in individual/class interactions or attitudes when 

mathematical problem-solving? 

3.2 Action Research Design 

The research design is based on a classroom-based action research model involving three 

cycles (Kemmis & Taggart, 1981 ). This report focuses on Cycle 3 (Main Study) of the 

action research cycle, provides an overview of Cycle 1 and Cycle 2 (Pilot Study), and shows 

their relationship to the third cycle. 



44 

Lewin (1948) and Collier (1945, cited in Ebbutt, 1985) independently pioneered the 

concept of action research. The action research model "proceeds in a spiral of steps each of 

which is composed of a circle of planning, action and fact-finding about the result of the 

action." (Lewin, 1948, p. 205). Action research was initially used to deal with community 

issues in social planning but has subsequently been employed in the education arena. 

Kemmis and Taggart (1981) developed Lewin's spiral action research model in Australia as 

the Action Research Planner and their draft was furthered almost immediately by Elliott 

(1981) in Britain as the Action Research Framework for Self-Evaluation in Schools. Elliott 

( 1991) defines action research as "the study of a social situation with a view to improving 

the quality of action within it" (p. 61 ). 

Kemmis and Taggart (1981) summarised action research as a cycle (Figure 2), which 

involved planning, acting, observing, and evaluating: 

• 

• 

• 

• 

"to develop a plan of action to improve what is already happening; 

to act to implement the plan; 

to observe the effects of action in the context in which it occurs; and 

to reflect on those effects as a basis for further planning, subsequent action and so on, 

through a succession of cycles" (p. 7). 

This study design incorporates Kemmis and Taggart's spiral model within a social situation 

involving small groups of students engaged in collaborative mathematical problem solving. 

The action research model follows a cycle of planning, action, observation, and reflection, 

with adaptations made to the next level of the cycle after each phase. The main study 

focuses on the third cycle of the action research model. 



45 

Figure 2: Kemmis and McTaggart's Action Research Planner (1981) 

During the 1990s there was a rise in the amount of research undertaken within the context 

of the classroom; whereas before, the subject (s) undertook controlled, observed tasks and 

interviews outside the classroom. The action research model provided an ideal vehicle for 

increased teacher involvement in research projects (Ebbutt, 1985). 

However, the idea of teacher as researcher alarmed some members of the academic research 

community who believed a teacher' s research would not be academically rigorous (Carnine 

& Gersten, 2000). This argument was given support by those who did not differentiate 

' normal' teaching practices from research, and who believed action research was an integral 

part of 'good teaching' and that it was inherent in the daily cycle of planning, assessment 

and evaluation in the classroom (Sharples, 1983, cited in Kelly, 1985). Others believed that 

by allowing teachers to maintain a dual role as researcher and teacher, 'coal-face' questions 

about learning and teaching would be more likely to be pursued in the microcosm of the 

classroom and that this in turn would provide impetus for further global research (Cochran­

Smith, & Lytle, 1993). 



46 

Ebbutt (1985) worked with Elliott using the educational action research model and created 

a broad classification system to differentiate daily teaching practices from those of 'teacher-

researcher' (see Appendix 1). The role of teacher-researcher in the second and third cycles 

of this study falls within Ebbutt ' s ' classic ' action research model due to the degree of 

systemization, the testing of hypotheses beyond one classroom, and the collaboration 

undertaken with other teachers working in the same field of interest. The principles of 

planning, action, observation, and reflection are used for the systematic collection of 

information, and to reflect upon and plan change while maintaining flexibility and 

responsiveness to the context (Poskitt, 1994). 

3.3 The Setting 

The research was undertaken at a three classroom rural primary school (Roll approximately 

60, decile 41 
). Each of the three classrooms was multilevel: The junior classroom included 

all Year 1, 2 children; the middle classroom Year 3, 4, 5 children; and the senior classroom 

Year 6, 7, 8 children. 

The pilot study took place in the middle classroom in 200 I (see summary in 3 .6.1 ). The 

main study was undertaken over four weeks in Term 2, 2002, in the senior classroom. 

3.4 Profile of the Sample Group 

A composite class of 25 Year 6, 7 and 8 children were involved in the collaborative training 

programme. The class had two teachers (teaching principal and release teacher) who took 

1 Funding in New Zealand schools is currently allocated per capita according to a socio-economic decile 
rating from 1 to 10, with decile l being the lowest and receiving the greatest funding per capita. A school 
receives a decile number according to the average of the socio-economic location of the children's 
residential addresses. 



47 

joint responsibility for the implementation of the mathematics programme. Twenty children 

volunteered to be part of the research study but data analysis was completed for only 15 

Year 6 and 8 children who were present for the entire training programme. The sample 

group consisted of four girls and two boys from Year 6, and three girls and six boys from 

Year 8. Nine identified themselves as Pakeha/European, three as Maori, two as 

Pakeha/Maori, and one as Maori/Pacific Islander. All the Year 6 children had been involved 

in the pilot study the year before. None of the Year 8 children had been involved in the 

pilot study. 

The class of twenty-five children from which the sample group was taken was concurrently 

involved in the 2002 Ministry of Education Numeracy Project. The numeracy teaching, 

during the period of the study, involved addition and subtraction of whole numbers and 

fractions. The classroom teachers used the draft Diagnostic Interview for the Numeracy 

Project (Ministry of Education, 2002c) to arrange sub-groupings and assess academic gains. 

3.5 Tirneline 

Previous Year Pilot Study with researcher' s class. 

Term 1 Literature Review. 

Term 2 Week 1 • Classroom teacher administers Numeracy Project Diagnostic 
Interview. 

• Classroom teacher organises learning groups. 

• Researcher audiotapes samples of each learning group's pre-training 
discourse while engaged in problem solving. 

Term 2 Week 2-5 Collaborative training programme commences as an integrated 
component of the students' mathematics class. 



48 

Term2 Week 6 Post-training audiotaping of each learning group's discourse while 
involved in collaborative problem solving. 

Term 3 Week 5 Post-unit Numeracy Project Diagnostic Interview. 

Table 1: Daily group rotation during mathematics. 

Activity during Mathematics Class First 20 Second 20 Third 20 
Minutes Minutes Minutes 

Researcher - collaborative training in Group 1 Group 3 Group 2 
library. 

Independent- Problem-solving in Group 2 Group 1 Group 3 
groups in classroom. 

Teacher- instruction in classroom Group 3 Group 2 Group 1 

3.6 Developing the Instructional Strategies 

3.6.1 Development of the Training Programme through Cycles 1 and 2 

The first cycle of the action research model (Figure 3) was used when the researcher 

worked (as a collaborative teacher) with a Resource Teacher for Learning and Behaviour 

(RTLB) on a reading intervention plan for a student in 2000. The intervention plan was 

based on training pupil tutors using Medcalf and Glynn's (1987) 'Pause, Prompt and Praise' 

Model. In this model, tutors are specifically taught to praise tutees for self-corrections, 

correctly reading sentences, and the implementation of reading strategies. They are trained 

to pause to give the tutee time to think and to prompt if necessary with a series of reading 

strategies. These tutor-tutee interactions are designed to promote the high order cognitive 

skills of explanations, justifications, speculations, hypotheses, inferences, conclusions, and 

questioning. 



49 

Reflect 
Classroom Teacher and RTLB 
discuss the Peer-tutoring 
Programme and its high level 
of success in the development 
of the tutors' self-esteem, 
reading skill s and reading 
achievement 

The Classroom Teacher and 
the RTI ,B raise the question as 
to whether the principles of the 
programme can be generalised 
to other curr iculum areas such 
as writ ing and mathematics. 

Observe 
RTUJ makes Running Records 
of the tutors' reading behaviour 
and levels, before and after the 
programme. 

Revised Plan 
RTLB plans 
adaptation for 
written English and 
Classroom teacher 
plans adaptation for 
mathematics. 

Plan 
Use "Pause, Prompt and 
Praise" method of peer­
tutoring as an intervention 
for a child with a low 
reading level. Aim is to 
develop self-esteem, 
reading skills and readin g 
achievement for th is child. 

Act 
RTLB implements a peer­
tutoring programme with 
four children as tutors. 
Classroom Teacher is 
instructed in th e principles 
of the programme and 
schedules daily tutor/tutee 
meetings. 
Children act as tutors for 
four members of junior 
class. 

Figure 3: Cycle 1 of this Action Research Study: The classroom teacher becomes 
involved in research at Ebbutt's Self-Evaluation Action Research Mode. (Refor to 
Appendix I). 

The action phase of Cycle 1 invo lved the RTLB implementing the peer-tutoring programme 

with fo ur children. The classroom teacher was instructed in the principles of the 

programme and scheduled daily tutor/tutee meetings in which the fo ur students tutored 

junior class members in reading. The observation phase involved the RTLB recording the 

tutors' reading levels and behaviours using the Running Record system (Clay, 1985) before 

and after the peer-tutor training. The classroom teacher and the RTLB reflected on the 

success of the programme as an intervention and discussed whether the principles of the 

programme could be generalised and implemented as an intervention in other curriculum 

areas. 



50 

In the second rotation of the action research cycle the classroom teacher, now in the role of 

researcher, developed an intervention based on Medcalf s peer-tutoring procedure (WASP) 

for mathematics (Medcalf, in press). Medcalf had adapted this model from the 'Pause, 

Prompt and Praise' for reading (Glynn & Medcalf, 1987). The following steps outline this 

procedure: 

W=Watch 

Watch as your friend shows you how they work out the problem. 

A= Ask 

Ask questions to check out your friend ' s understanding 

E.g. "How did you get that answer?", 

"What do you think the next step is?", 

"Can you see where you went wrong?". 

Sometimes asking the right question will be enough for the tutor to realise their 

error and be able to self-correct or complete the problem independently. If so 

the tutor may skip the next step and proceed to prompt and praise. 

S=Show 

Show your friend how to complete the problem. This will often involve 

modelling a particular step in the process that a tutee is experiencing difficulty 

with and/or completing incorrectly. The tutor can assist the tutee to complete a 

similar problem with their help and encourage the student to then attempt another 

example independently. 

P=Prompt and Praise 

Prompt your friend to complete a similar problem on their own and praise their 

efforts. The tutor prompts the tutee to attempt a similar problem independently. 

Before doing this the tutee is asked to tell the tutor what they will do to solve 



51 

the problem and complete the task. The tutor watches and provides specific 

positive feedback with special emphasis on the step or issue that the tutee was 

having difficulty with. 

This pilot study, Cycle 2, aimed to test Medcalf's findings of social and academic benefits 

through replication, but also to evaluate and refine the type of questioning format in the 

' Ask ' phase. Questions were specifically designed to increase higher order thinking and 

scaffolding. The pilot study, undertaken within the teacher-researcher's own class of 

(twenty-four) Year 4 and 5 children, involved making anecdotal observations and audio 

recordings of classroom peer interactions during the collaborative group work in 

mathematics. The Year 4/5 class was concurrently involved with a pilot study for written 

language with the RTLB (Huggard, 2002). This reinforced the establishment of a 

classroom community in which students learned cooperatively and questioned each other's 

thinking. 



52 

Reflect 
Think-Pair-Share model best 
promoted mathematical 
learning with peer support by 
requiring individual 
accountability, idea 
construction through 
negotiation of meaning and 
justification by presenting 
ideas to partners and the 
wider group. 
Thomas' model has NZ 
research base and easy to use 
initial categories of on task/ 
olI task, and social/cognitive 
interactions. Need to redefine 
subcategories of cognitive 
reflection and cognitive 
action. 

Observe 
Transcribe and analyse 
interactions of children 
engaged in mathematical 
problem solving in small 
groups. 
Make anecdotal observations 
of interactions that appear to 
promote mathematical 
learning amongst peers. 

Plan 
Review literature for: 
1. Examples of peer-tutoring, 
cooperative/ collaborative learning 
programmes. 
2. Analysis tools for discourse in the 
mathematics classroom. 
3. The use of cognitive thinking 
skills to promote learning. 

Use these literature ideas to develop 
and trial a pilot intervention which 
promotes peer learning in the 
mathematics classroom. 

Act 
1. Use WASP and Think-Pair-Share 
learning models. 

2. Trial ways to quantify qualitative 
data in the analysis of classroom 
interactions: Thomas' Model 
(1994); Johnson et al. 's Model 
(1985); and King et al's Model 
(1998). 

Revised Plan 
Refinement of pilot programme to promote ·'wait-time", specific feedback , 
multiple presentations of solutions with explanations and justifications. 
Reduce off-task and social interactions by careful selection of rich tasks by 
teacher and reinforcement of cooperative group skills with teacher modelling. 
Define collaborative learning cf. cooperative learning. 
Redefine cognitive subcategories in terms of Bloom' s taxonomy, as lower or 
higher order cognitive thinking. 
Identify more verbal flags that indicate cognitive thinking. 

Figure 4: Cycle 2 the Pilot Study: The Teacher as Researcher in Ebbutt's Classic 
Action Research Mode. (Refer to Appendix 1 ). 

The data from the second cycle was quantified and analysed using three models of discourse 

analysis: Thomas (1994), Johnson, Johnson, Roy, and Zaidman (1985), and King, Staffieri, 

and Adelgais (1998). 

Johnson et al.'s model (1985) enabled children's talk to be clearly classified into either one 

category or another. Though it was the easiest of the three to administer for collaborative 



53 

groups, it did not give the researcher enough information about the type of information 

exchanged. 

Thomas' model (1994) was both hierarchical and linear which suited discussion that 

modelled a hierarchical model of cognitive thinking such as Bloom's taxonomy (Bloom, 

1956). The model also defined the concept of a ' turn' as the utterance made by one child 

before another child speaks. A tum could be a single word, a phrase, or one or more 

sentences (Sharan & Sachar, 1988, cited in Thomas, 1994) and this was used to define 

instances of data. Though the Thomas' model distinguished the classification of cognitive 

talk into subcategories of action and reflection a child' s tum could be enmeshed with 

elements of both. 

King et al. ' s model ( 1998) required a question/ statement or statement/statement interaction 

between tutor and tutee and was found to be inappropriate for analysing collaborative group 

discourse. 

These considerations alerted the researcher to the need to clearly match the method used to 

analyse the children's interactions with either the collaborative group model or peer­

tutoring, dependent upon the research questions to be considered in the main study. For 

this reason, Lyman's Think-Pair-Share method of cooperative learning (1992) was trialled 

as an alternative to Medcalf's WASP technique (in press). The Think-Pair-Share method 

promoted mathematical learning with peer support by requiring individual accountability, 

idea construction through negotiation of meaning, and the development of higher order 

cognitive thinking when students elaborated and justified their ideas and their solutions to 

partners and the wider group. 



54 

The pilot study, conducted over twenty weeks for eighty hours, provided insights into 

factors which supported effective mathematical learning by children in collaborative groups. 

These factors included: 

1. Careful task selection at an appropriate level to reduce social/procedural task-related 

talk and increase cognitive task-related talk; and to maintain students in their zone 

of proximal development (Vygotsky, 1978). 

2. Modelling and the provision of feedback by the teacher (and other groups) to 

develop and encourage collaborative strategies which reflect positive 

interdependence including: 

• 

• 

• 

• 

• 

listening to explanations from other group members; 

asking questions to clarify own/other's thinking; 

providing evidence to justify conclusions/solutions ("What's your evidence?"); 

praising and encouraging with positive specific feedback; and 

giving and receiving help . 

3. Development of mathematical strategies by teaching modelling of mathematical 

language. 

4. Provision of individual thinking time to allow for the development of multiple strategies 

as a base for negotiating meaning and solutions. 

5. Group size effects, particularly with reference to maturational age. Pairs usually resulted 

in each partner sharing the talk equally (50% of all talk). Groups of three or four could be 

dominated by a particular member. 



55 

In preparation for Cycle 3, which was to comprise the main study, the researcher used the 

information gained in the second cycle to: 

• Select the Think-Pair-Share technique as the basis for collaborative learning in small 

groups. 

• Select rich, appropriately levelled tasks to reduce task-related social interactions. 

• Reinforce and model cooperative group skills by the teacher to reduce off-task social 

interactions. 

• Select Thomas' model to analyse interactions but with the modification that the 

subcategories of cognitive interactions were replaced with lower cognitive interactions 

and higher cognitive interactions. The lower cognitive interactions were defined as 

those involving recall and recognition (knowledge) and translating symbols and words 

(comprehension). The higher cognitive interactions were defined as those, which 

involved elements of application, analysis, synthesis and evaluation based on Bloom's 

taxonomy ( 1956). 

• Define and use the term 'collaborative· instead of ' cooperative'. 

• Be aware of phenomena such as children using 'verbal flags ' (e.g., 'so'), to identify the 

linking of ideas and hence higher cognitive thinking. 

The next cycle of the action research study was to investigate the use of collaborative 

training to promote increased higher-level cognitive interactions between children. 



56 

3.6.2 The Main Study - Cycle Three 

The main study, Cycle 3 (see Figure 5), involved the initiation of the collaborative/peer-

tutoring training programme based on Lyman's Think-Pair-Share model (1994). The aim of 

the intervention was to develop the interpersonal skills necessary for effective collaborative 

work and to develop the higher order thinking skills of the students. 

Reflect 
Verbal nags were identified, the 
link (conjunctions), the run-up 
(rehearsal), wail-time (demand 
for individual cognitive thinking 
time). Arc there more? 
Was task selection to keep 
children motivated and in ZPD 
the primary reason the number 
of cognitive interactions 
increased? 
Did the number of higher 
cognitive interactions increase 
significantly? 
What limitations faced the 
programme? 
Did the children have enough 
forming and fun ction ing skills to 
reach fermenting and 
formulating levels? 

Observe 
Interactions of children recorded 
as transcripts before and after the 
training programme. 
Transcripts analysed. 
Response questionnaire 
completed by children aficr the 
programme. 
Verbal response and anecdotal 
comments given by classroom 
teachers. 

Figure 5: Cycle 3 The Main Study. 

Revised Plan 
Future resca rch 

Act 
25 children work with 
researcher in 3 
collaborative groups to 
develop interpersonal 
group skill s and thinking 
skills for collaborative 
learning. 

Programme rein forced by 
teacher modelling and 
small group work in the 
classroom. 

Plan 
Use Lyman·s Think-Pair­
Share method as an 
intervention for a 
mathematics class. 
Aim is to develop 
interpersonal ski lls 
necessary for effective 
collaborative work and 
higher order thinkin g 
ski lls. 

The collaborative-peer training was developed so that the students would advance through 

Johnson and Johnson's '4Fs' group skills: forming, functioning, formulating, and 

fermenting. In the initial group discussion between the researcher and students, basic self-



57 

management skills were discussed at the group 'forming' level. These included principles of 

courtesy such as orientating the body to the speaker/listener and one person speaking at a 

time, as well as principles of encouragement such as no 'put downs', including all group 

members in discussion, and praising attempts or ideas. 

Four key strategies were discussed in subsequent sessions to develop the students' higher­

level group skills of functioning, formulating and fermenting. These strategies included: 

• wait and give individual's time to think for his/herself; 

• be specific with feedback; 

• give help when asked in the form of a specific strategy, idea or question rather than an 

answer; and 

• support agreement or disagreement with evidence. 

The process involved two key ideas for the teacher: modelling and maintaining the ' 4Fs' in 

class discussions and group work, and selecting rich tasks which maintained the students in 

their ZPD. 

Initially during the training, the students worked in pairs using Medcalf's (1992) peer­

tutoring model. One member of the pair adopted the authority role of tutor or coach and 

was responsible for the use of the instructional prompts while his/her partner solved a 

mathematical task. When the tutee had solved his/her problem (or developed a solution to 

the best of his/her ability with the aid of the tutor) the pair swapped roles and the other 

partner completed a different problem with the same mathematics concept. The tutor/tutee 

role was alternated so that individuals would develop a greater awareness of the effects of 

their feedback. Pairs were encouraged to reflect and comment on the implementation of the 



58 

collaborative group skills and their own learning. 

The positive interdependence of Lyman's Think-Pair-Share model (1992) of collaborative 

learning was incorporated into the next stage of training. Individuals worked on problems 

by themselves for a specified time and then they shared their solution and the evidence to 

support their conclusion with a partner. If a problem had a finite solution the partners 

compared answers and evidence. An identical answer meant they were either right or both 

wrong, and two different answers meant they needed to review their evidence to convince 

their partner their solution was correct. When the partners reached an agreed solution they 

presented the solution and evidence to another pair. Multiple solution paths were viewed 

and compared. Any answer could be challenged as long as the challenge was supported by 

reasoned argument. 

Open-ended questions with more than one solution were approached in a similar manner. 

The students had individual thinking time and then presented their solution to a partner. If a 

partner was not convinced of the alternate solution he/she again challenged the partner for 

more corroborative evidence and the partner was required to justify the solution with more 

information. Alternative solutions and evidence were presented to the entire group. The 

children were encouraged to compare their solution and reasoning to those of other 

presentations. In this way each different solution was considered, and flawed solutions 

were challenged and discarded. Variations of this model occurred when a pair worked 

together and presented their solution to another pair and then the wider group; or 

individuals worked on the same problem and presented their solution and reasoning to a 

partner and then combined their solution and evidence or selected what they considered the 

strongest solution and most convincing evidence to present to the wider group. 



59 

The following extract is a recording of a cooperative group work training session, which 

shows the Think- Pair- Share pattern used by two boys during problem solving and the 

reinforcement of the model by the teacher. 

Teacher: [Speaking to whole class] I want you to work on problem four on page ten. 2 

You 'fl have about ten minutes to work out your own answer. You can have 
more time if you need it. I'll check in about ten minutes. Remember this is 
your own time for ten minutes. You can write stuff down, draw pictures but 
don 't talk to your partner. You 'll have a chance to compare ideas after. Have 
you found page four Jeffery? That 's right ... the question at the bottom. [The 
teacher reads the problem aloud both to focus the group and because several 
children have reading difficulties]. Jan needs exactly 5 litres of water. She has 
only these two containers. How can Jan use her containers to measure exactly 5 

litres ofwater? ... Okay? ... gofor it. [Teacher withdraws to her desk and observes 
the class. After four minutes she notices an early finisher and comments aloud.] If 
you have an answer to the problem you might like to write it down or draw a 

picture so it 's easy for your partner to understand what you've done. [Ten minutes 
passes]. Does anyone want more time? ... Okay Cherie, why doesn 't your partner 
check her solution until you 're ready. The rest of you can show your ideas to your 
partner. Remember if you see something you like or realise there 's something you 
want to change you can change it. I want to hear people backing up their ideas 
with evidence. [Teacher moves about the room interacting with groups then focuses