From transmitter of knowledge to mediator of learning:

How do I improve my practice as a mathematics educator as I encourage my learners to become independent learners and take responsibility for their own learning?

 

Gerhardus Adams, St Mary’s University College, Twickenham, and the Bulumko Secondary School, Khayelitsha

 

A paper presented at the SIG Action Research: Action Research - A Framework for Supporting Innovative Teaching Approaches for Diverse Student Audiences.

American Educational Research Association Annual Meeting

New York

Tuesday, 25 th March 2008.

Introduction

I am a mathematics educator at a working class school in a township on the outskirts of Cape Town. I am also studying for my masters degree through St Mary’s University College, Twickenham. I originally became a teacher because most people in my community, who had the opportunity, did so, or that it was the only course on offer with a bursary. During my service I had the opportunity to look deep into myself to find the deeper motivation for becoming an educator, that is, the significance of education in the renewal of society and my desire to teach mathematics. The writings of Lomax, McNiff, Schön, Whitehead, and others have broadened my horizons and made me acutely aware of my deeper commitment to the cause of education.

This paper is an account of the action enquiry I conducted in my school from July to September 2007. The focus of my enquiry was how I can improve my practice as a mathematics educator as I encourage my learners to become independent learners and take responsibility for their own learning. Throughout I have adopted the stance of a reflective practitioner (Schön, 1983) and aimed to find ways of developing a dialogical approach to my own professional education as I explain how I generate my own living educational theory of practice (Whitehead, 1989). I will describe how I have attempted to explore the idea of ‘mediation of learning’ in my classroom (Nieman and Monyani, 2006).

I find this idea alluring as I am guided by the words of Feuerstein (as cited in Nieman and Monyai, 2006: 9):

‘...the learning mediator serves only as an intermediary; a connecting and enriching link between separate elements: between the learner and their cultural heritage; between learners and their environment and between various aspects of the environment.’

Tibus (also cited in Nieman and Monyai, 2006: 13), says:

‘Suppose you are working on a problem in mathematics. If I were your mediator of learning, I would NOT be trying to help you to solve your problem. Instead, I would be working with you to help you understand your own approach to the problem. I would be helping you to understand how you are using your brain! Likewise, I would not be correcting or judging your response. Instead, I would be reflecting back to you the nature of your response so you could understand how it would appear to someone else.’

McNiff (1993: 4) simplifies and illuminate this idea further as she says

‘...I am compelled to believe that my work as a supporter is focused on the self-education of the [individual]. I am increasingly aware of my own role as a resource that may provide an appropriate environment in which people may grow, of my need to resist offering glib answers that are based on my own insights and experiences rather than encourage them to find their own answers. I am convinced of the need to encourage people to appreciate the power of the self, when that self engages in the process of her own development [and renewal]; of the power of the self to create her own understanding. This power allows us to apply our educational practices to the process of transforming our lives.’

I must point out at this stage that my research is still in progress and this is an interim report of progress so far.

Methodology

The purpose of all research is to generate new theory. Action research, which developed out of critical theory, is a form of research that enables practitioners to learn how they can improve their practice, individually and collectively, and make their claims that they have done so. It is a form of self-reflective enquiry undertaken by practitioners in social situations to improve the rationality and justice of their own social or educational practices (Carr and Kemmis 1986). The reason why I chose this approach is because I hold the same underpinning values as action research, and the principles of social justice, respect for others and high achievement, guide my actions as an educator and practitioner-researcher. Action research allows me to reflect and improve my practice in a situation that is real to me – my classroom. I think about practice as a setting not only for the application of knowledge but also for its generation (Schön, 1995). The question I am attempting to answer is what kinds of knowing are already embedded in competent practice. I therefore need to observe myself in practice, reflect on what I observe, describe it and reflect on my description. The reality of action research in education is that it goes beyond being merely a method of classroom inquiry, and embraces a philosophical statement about education. This enquiry will generate an account of my professional being; it is a reflection on my ontological values and practice.

Whitehead (1989) says that education is a value-laden practical activity. We cannot distinguish a process as educational without making a value-judgement. Values are the human goals that we use to give our lives their particular form. They are embodied in our practice and their meaning can be communicated in the course of their emergence in practice. To understand our values we should observe and record our actions and the experience of the denial of our values. In using such records we can both experience ourselves as living contradictions and communicate our understanding of the value-laden practical activity of education. The significance of my learning and my theorising of practice is that I am learning to recognise, evaluate and live towards my educational values in practice.

By engaging in a reflective dialogue with my colleagues, participants, and myself, I will not only put my ‘living I’ at the centre of the development of a new educational theory of practice, but will also attempt to resolve the ‘living contradictions’ in my classroom. This methodology is appealing to me because it values the idea of dialogue as a method of enquiry and it improves the relationship between educational theory and professional development (McNiff, 1988). In particular I will follow the action research cycle as described by McNiff (1988) as follows:

An action-reflection cycle of planning, observing, reflecting, re-planning, and so on. These parts of the cycle can be subdivided into:

Planning – What is my concern? Why am I concerned?

Acting – What can I do about it? What will I do? How will I modify my practice in the light of my evaluations?

Observing – What kind of data will I gather to show the situation as it is? What kind of data will I collect to show the situation as it unfolds?

Reflecting – What can I do about it? How will I explain my educational influences in learning? How will I ensure that any conclusions I come to are reasonably fair and accurate? How will I evaluate the evidence-based account of my learning? How will I modify concerns, ideas and practice in the light of my evaluations?

This methodology allows a practitioner researcher to explore many different problems at the same time without losing sight of the main focus of the enquiry, because in reality one problem is often symptomatic of many other underlying problems (McNiff, 1988). It encourages me to account for my own professional development, offer explanations of how and why I have been prompted to change my practice and to demonstrate publicly that this change has led to improvement. This action-reflection cycle can be understood as a series of self-reflective questions that can also inform the writing of an account (McNiff and Whitehead, 2006) and I outline these as follows:

What is my concern?

I believe that this enquiry has afforded me the opportunity to make an effort to live my values in my practice. I believe, firstly, that learners can learn on their own and together. Secondly, I believe that an understanding and knowledge of mathematics is not only important for schooling but is essential to develop in learners a ‘critical awareness of how mathematical relationships are used in social, environmental, cultural and economic relations’ (Pretorius and Lemmer, 2004: 251). Thirdly, I believe that educators should not inevitably produce solutions to problems, but should first allow learners to think of possible solutions themselves. Educators should also get learners to understand that there is not necessarily only one solution to a problem, but that other possibilities and alternatives should also be explored (Nieman and Monyai, 2006).

In post-apartheid South Africa, educators are confronted with processes of change and renewal, the development of new curricula and terminology. Schools are subjected to periods of reform, intervention and new policies in the pursuit of school improvement and education in general. This drive for improved educational performance has resulted in a form of accountability that places tightly prescribed targets at the centre of systemic change and provides opportunities for educators to ask questions of the kind ‘What contribution can I make?’, ‘What is my concern?’ and ‘How do I improve my practice?’ (Whitehead and McNiff, 2006). In this new democratic dispensation, the role of educators has shifted from possessor and dispenser of knowledge to, amongst others, facilitator and mediator between learning outcomes and the learners’ desire and thirst for learning (Department of Education, 2003).

So, what is my concern?

I am concerned about my learners’ apparent inability to work cooperatively, to be independent learners and take responsibility for their own learning. I am concerned about my inability to move away from the notion that I perceive myself as the possessor and transmitter of knowledge and the learners as the recipients (Du Plessis et al., 2007). A relationship of dependency had begun to develop between the learners and me – they were relying mostly on me for their learning and understanding of mathematics (Davidoff and van der Berg, 1990). However, I am experiencing myself as a living contradiction (Whitehead, 1989) in that I am unable to manage my own internal dialogue as well as dialogue between the learners themselves and other sources of information.

I wish to encourage my learners to become independent and responsible, enjoy and appreciate the beauty of mathematics, excel in their learning and become active participants in each lesson. My research area is the teaching and learning of mathematics; my research question therefore becomes, ‘How do I improve my practice as a mathematics educator as I encourage my learners to become independent learners and take responsibility for their own learning?’

Why am I concerned?

I believe that learners can learn on their own and together; my professional commitments are grounded in my ontological and educational values of self- and cooperative learning, enquiry learning, respect for others and excellence in achievement. I believe, in the words of McNiff (1993: 4), that ‘educators must regard themselves as free thinkers, as creators of their own lives, in order to regard themselves as part of the educative process.’ During my tenure as an educator I have come to realize that I know ‘what I do and how I understand what I do’ (Schön 1995). Whitehead says that ‘my practice is not only a setting for the application of knowledge but for its generation’ (Whitehead, 1989 p.43). My practice of teaching and curriculum making are expressions of my moral and value-laden knowledge.

When I began my studies towards a teaching qualification I thought, as noted above, that I wanted to be a teacher because most people in my community, who had the opportunity, did. During my studies I had fond memories of my playing days as a child; when we played ‘school’ I was always the principal and the arithmetic teacher. During my service as a practicing educator I had an opportunity to look into myself to find the deeper motivation for becoming an educator, which is about the significance of education in the renewal of society, to act as an intermediary between the learners and the learning objectives and my desire to teach mathematics (Ryan and Cooper, 1984; Nieman and Monyai, 2006).

I have always believed that teaching is a profession of values and these are fundamental to understanding ourselves as teachers, how we relate to others and discharge our role competently and ethically. My major role as a mathematics teacher is to provide learners with opportunities to engage with real-life problems in different contexts and so consolidate and extend basic mathematical skills and communicate effectively. I am concerned about which methodologies and pedagogies of education are likely to influence my own learning, the learning of others and my teaching. I believe, in the words of Lomax (1994: 8), ‘that teaching is about influencing others to want to find out, showing them how to find out and knowing what is worth finding out.’ Evaluating my teaching is the process of making judgments about the quality and value of my teaching and reflecting on the extent to which learners achieve the pre-specified learning outcomes, the effectiveness of classroom management techniques and the way in which learners’ characteristics such as prior knowledge, gender and language influence their learning.

I will make judgments about the quality of my research in terms of how I transform my ontological values into my living standards of practice and into my living standards of judgment (Whitehead and McNiff, 2006).

How do I show the situation as it is and as it unfolds?

In my action enquiry I have employed various data collection methods including a reflective journal, audio-taped conversations, interviews and a questionnaire. I have asked the learners to complete a questionnaire, record observations in my journal and ask a critical friend to sit in on my lessons to make his own observations. Twenty-three learners responded to the questionnaire. In the questionnaire I asked the following questions: ‘Do you participate actively in the class?’ The results were as follows: 5 out of 23 said ‘all the time’, 15 said ‘some of the time’ and three said ‘very little’. Another question asked: ‘Are you allowed to solve problems on your own in class?’ The results were: 6 said ‘all the time’ 16 said ‘some of the time’ and two said ‘very little’.

When I was recording my observations in my journal during the first few weeks of my research project I found that most of the learners were waiting for an example before attempting to solve a problem and some were even waiting for me to write the solution on the chalkboard or made no attempt to find a solution. I asked the learners to give written responses to the question: ‘Suggestions on how I can improve lessons and activities in the class?’ The responses confirmed my observations as some wrote; ‘I think you must not do most of the work by yourself’’; ‘when you teach us something you must ask or let us do what you taught us on the chalkboard so that you can see our mistakes.’ (I have all the solutions!) In other responses some learners suggested that I do not give them enough opportunities to demonstrate their understanding or lack of understanding and that I am positioning myself as the transmitter and possessor of knowledge by saying: ‘You must not talk too much or too fast. When you teach us something, make sure that all of us understand before you move on.’ Finally it was suggested that I evaluate the learners’ work more often and not select books, but mark all the books every day.

These observations by the learners show that they rely mostly on me for their learning and that I continue to take up most of the teaching and learning time. The data show that I talk too much in class, and that the learners are not given enough time to explore ideas, or express their feelings and thoughts. The broader aim of this data collection is to show that I need to move away from a transmission mode to a more interactive style of teaching.

What were my options for action? What did I do?

In this section I will describe how I have attempted to explore the idea of mediation of learning as I strive to position myself as a mediator between the learner and the learning material (Nieman and Monyai, 2003). I will report on the strategies I have drawn on in my classroom, what I did, interventions I introduced and how I did it. In my pursuit to become a mediator of learning and modify the way I do things, I have decided to transform the context in which we are operating by implementing the following measures in my classroom (Du Plessis et al., 2007; Harris and Muijs, 2005; Killen, 2000):

• I use brief written exercises in class to give learners time to think about what they have learnt, and to reflect on it. Learners wrote small tests on a weekly basis and I gave immediate feedback to learners. One learner remarked in our group discussions; ‘…I really feel that my mathematics is improving because my marks for the small tests continue to improve…’
• I ensure that learners experience themselves as successful by giving simpler tasks that gradually get more complex. I have asked learners to present their solutions on the chalkboard. Some learners suggested …’Sir, each group should get one chance per period to write their solutions on the chalk board, not only a few…’ I could sense that all of them wanted to experience this feeling of accomplishment.
• I negotiated my academic and behavioural expectations, in the form of ground rules developed by the learners themselves, and communicated these clearly to the learners by displaying them in the class. At the beginning of my enquiry I facilitated a process where the learners had to draw up a set of ten rules to guide our behaviour and actions in the class.
• I selected materials and activities that relate to their everyday life, drawn from newspapers and magazines and that lend themselves to group work. I decided to make the lessons more real by asking learners to collect data about the mass sum of grade 11 learners. They had to determine the body-mass index of ten learners, analyse the data again and work in their groups. This has led to lively discussion about obesity and diet, and the presenters of the groups had a field day with questions and comments and really had fun. Another learner remarked in the conversations afterwards:

‘...this was a very interesting lesson and we realise that mathematics is about solving and discussing real-life problems. I really enjoyed the tasks given to me today ... I felt free to express my own opinions and say what I wanted to say...’

• Managing the learning environment, I encouraged all learners to participate and provide a logical conclusion to all activities.

In addition to the above steps I also introduced measures to create a learning environment in which learners develop strong internal discipline and self-discipline (Cangelosi, 2003; Nieman and Monyai, 2006):

• To help learners to accept responsibility, I divided the learners into groups and each member of the group was given a specific responsibility and role. In order to get the learners to live out these roles I identified several roles that learners could play during any lesson. We decided to revolve these roles as we move from topic to topic and introduce new roles as we progress. I divided the forty learners into eight groups of five. When we were dealing with the topic of statistics: data handling we had the following roles in the class: Distributor of worksheets and collection of work that needed to be marked by myself, Time-keeper in group, Presenter of group solutions or attempted solution to problems and questions, Facilitator of discussions in the group to ensure that all group members participate in the discussions and Scriber that will write down the solutions of the group (Killen, 2000). All group members should be jointly accountable for completing the shared task with each member responsible for fulfilling their individual role. At first dividing the learners into groups took a lot of time, but as we progressed we became used to it.
• Help learners to set goals. At the beginning of the lesson I articulated and clearly explained the lesson outcome and how it could be assessed and asked the learners to set short term goals for that specific lesson.
• I directing learners into learning activities: When learners ask for answers to questions I do not give an answer but respond with a probing question. By giving directions I can minimize transition time, streamline communication procedures and reduce the amount of teacher-talk in the classroom by establishing signals or cues that nearly instantaneously communicate certain recurring expectations to learners. My observer (see below) said in a discussion afterwards ‘…your learners are really hungry for learning and I can sense an eagerness to learn and a sense of self-reliance.’
• I have prompted learners to write out responses to problems and questions in order to: (1) allow them to organize their thoughts in a way that enhances their understanding of the mathematics being studied; (2) this writing time serves as a silent period for all learners in which they can think about how to respond to questions; (3) to enable me to preview learners’ responses and decide which ones should be read out or written on the chalkboard, and finally to avoid the stammering and grasping for words that is typical of learners answering aloud in front of their peers. Some learners were reluctant to speak and were waiting for the correct answer.
• Monitor groups’ activities and provide guidance as needed without usurping learners’ responsibilities for designated tasks. Move from one group to another, cuing learners to be on-task without actually becoming a member of any one group. On many occasions I found myself spending too much time with one group or providing more assistance than needed.

So, did these interventions help or contribute to my learning, the learning of the learners and possibly the learning of the observer in my class?

How do I ensure that any conclusions I come to are reasonably fair and accurate?

I wrote to one of my colleagues in our school to invite him to be my critical observer and to some of my colleagues on the MA programme, asking them to validate my claim to knowledge. The school principal has supported my enquiry from the beginning and has given permission to use my findings to encourage other educators to get involved in a similar exercise. In communication between myself and my critical observer my commitment and drive to generate my own living theory of practice was clearly illustrated. I looked for validation of my claim that I had done so in the form of the communications between myself and my observer and from my validation group. My critical observer commented that I was ‘achieving success in my objective to improve the quality of teaching and learning in your class, by observing how learners take responsibility for their own learning.’ Critical feedback from my validation group comprising my colleagues and mentor from the MA in Education programme indicates the same.

How do I modify my ideas and practices in the light of my evaluation?

The fact that learners attempted to solve the problems on their own and learn from their mistakes has increased their competence and performance in mathematics. The group work and pairing was a successful strategy and the work was rewarding although weaker learners were intimidated and tended to disappear, an issue that I need to address in the next stage of my action enquiry. I believe that through my actions and interventions I am making progress as I attempt to practice as a mediator of learning. The learners have begun to show that they are capable of taking responsibility for their own learning and realise that they can learn independently. Independent and mediated learning are capacities that cannot be perfected overnight. The small steps we have taken so far form the building blocks for the next action reflection cycle.

References

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Carr, W. and Kemmis, S. (1986) Becoming Critical. London, Falmer.

Davidoff, S. and van den Berg, O. (1990) Changing your teaching: The challenge of the classroom. Cape Town, Centaur.

Department of Education (2003) Policy Handbook for Educators in South Africa, ELRC. Pretoria, Universal Print Group.

Du Plessis, P, Conley, L and Du Plessis, E. (2007) Teaching and Learning in South African Schools. Pretoria, Van Schaik.

Harris, A. & Muijs, D. (2005) Improving School Through Teacher Leadership. Maidenhead, Open University Press.

Killen, R. (2000) Teaching Strategies for Outcomes-based Education. Cape Town, Juta.

Lomax, P. (1994) ‘The Narrative of an Educational Journey or Crossing the Track.’ Inaugural address as Professor of Educational Research in the school of Teacher Training at Kingston University. Kingston, Kingston University.

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Ryan, K. and Cooper, J. (1984) Those Who Can, Teach. Boston, Houghton Mifflin

Pretorius, F. And Lemmer, E. (2004) Teaching in South African Schools: The teacher as leader, administrator and manager. Johannesburg, Macmillan.

Schön, D. (1983) The Reflective Practitioner: How Professionals Think in Action. New York, Basic Books.

Schön, D. (1995) ‘Knowing in action: The New Scholarship Requires a New Epistemology’, Change, November-December pp. 27-34.

Whitehead, J. (1989) ‘Creating a living educational theory from questions of the kind, “How do I improve my practice?”’ Cambridge Journal of Education, 19(1): pp. 41-52.

Whitehead, J. and McNiff, J. (2006) Action Research: Living Theory, London, Sage.

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