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Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) A guide to the teaching of learnercentred mathematics B.K. Thekwane The author of this Guide, Mr B.K. Thekwane, has published this article with the purpose of inviting comments for improving the draft from classroom teachers, facilitators, advisory teachers and other stakeholders. Your comments as an expert in the field will be highly welcomed. 1. Preface It is clear that we live in an ever increasingly scientific and technological world. Hence, changing demands from society and the continual development in the field of technology have led to a radical change in the aims and objectives of school mathematics since independence in Namibia. As a result of this, the Ministry of Basic Education has adopted a new approach to teaching school mathematics, namely the learner-centred teaching approach. In the new approach, greater emphasis is placed on: ✴ ✴ ✴ ✴ understanding, communication, problem solving and autonomy the ability to reflect on own methods and thinking creating a positive self-image and attitude among all learners accepting the responsibility for own work In view of the complicated and challenging nature of the teaching, learning and application of Mathematics, the development of knowledge and skills should preferably take place within a flexible, safe and non-prescriptive environment. The teaching and learning environment should give learners the opportunity to give free expression to originality of thought and enable them to approach further studies and training purposefully and with dedication and self-confidence. 2. The purpose of the guide The purpose of this Guide is to provide Mathematics teachers with some ideas and guidelines as to how to teach Mathematics using a learner-centred approach. Teachers are encouraged to explore other approaches and should feel free to share them with other colleagues. This Guide will be updated from time to time as more ideas and experiences are obtained from classroom teachers, on how best to teach Mathematics in a learner- centred way in the Namibian set-up. Teachers are encouraged to implement these guidelines in an original and creative manner in their classrooms. Introduction Background Mathematics teachers experience a number of problems and challenges in the teaching and learning of Mathematics. They teach in the following environment: (i) The majority of teachers have very heavy classroom teaching loads. Virtually all teachers teach all periods every day. Moreover, teachers commonly have three to four different class preparations each day. Page 1 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) (ii) (iii) (iv) (v) A great percentage of teachers have extra-curricular responsibilities after normal school hours. The majority of teachers commonly have weak backgrounds in the subject matter of mathematics. Teachers commonly express a fear of or anxiety about mathematics. Teachers feel pressure from the constant focus of public attention on learners’ poor achievement. Teachers have to deal with learners’ problems, e.g. alcohol abuse, drug misuse, break-down of the traditional family life and the rise in single parenting. The above issues should be seen more as challenges then anything else and hence they need to be addressed. There is no doubt that the expectations for positive change in mathematics instruction and for improvement in learners’ achievement in mathematics are of top priority. The change in our schools is a process, and not an event. How successful we are implementing the new directions in the teaching and learning of mathematics depends on how we address the process of change. 3. Learner-centred education What is learner - centred education' The teaching of Mathematics using a learner-centred approach will be well understood once teachers first and foremost understand the pedagogical aims of the concept of learnercentred education in the Namibian context: (a) providing all learners with equal access to the knowledge and skills specified by the national curriculum (Goal: equality of educational conditions, resources, and treatment). (b) providing special support for particular groups of learners to rectify educational disadvantages (Goal: a commitment to equity and the removal of unjust or unfair discriminatory practices). (c) providing learning experiences which engage the learners’ powers of understanding, and in doing so, require them to go beyond the acquisition of mere surface knowledge (Goal: quality in education). (d) providing all learners with opportunities to shape the conditions which govern their access to knowledge, skills and understanding in classrooms and schools (Goal: education for, and not simply education about, democracy)[ (The Principles of Learner-Centred Education and Teachers’ construction of knowledge through Practice-Based Inquiry, p.2)]. It is in this broad context that learner-centred education should be understood and practised in Namibia. In learner-centred education, a learner is conceptualised as an active inquisitive human being, striving to acquire knowledge and skills to master his/her surrounding world. The learner brings to the school a wealth of knowledge and social experience gained from the family, the community and interaction with the environment. This knowledge and experience is a potential that can be utilised and drawn upon in teaching and learning. From the same perspective, the learner is seen as an individual with his/her own needs, pace of learning, experiences and abilities, and a leaner-centred education must take this into account. In the classroom, learning should clearly be a communicative and interactive process, drawing on a range of methods as appropriate for different groups of learners and the task at hand. These include group and pair work, learning by doing, self-and peer assessment, with emphasis on the supportive and managerial role of the teacher (The Pilot Curriculum Guide for Formal Senior Secondary Education, 1998, p.8). In the same vein, the Broad Curriculum for the Basic Education Teacher Diploma (1992), describes the learner-centred approach as follows: Learner-centred education presupposes that teachers have a holistic view of the learner valuing the learner’s life experience as a starting point for their studies. Teachers should be able to select content and methods on the Page 2 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) basis of the learners’ needs. They should use local and natural resources as an alternative or supplement to ready-made study materials, and thus develop their own and the learner’s creativity. A learner-centred approach demands a high degree of learner participation, contribution and production (p.3). From the above description one can discern that the new education system in Namibia views the learner as an active, curious being, eager to acquire knowledge. For Mathematics teaching and learning this means that learners are conceptualised as active mathematical thinkers who try to construct meaning and make sense for themselves of what they are doing on the basis of their personal experience. It also means that learners should develop their ways of thinking as their experience broadens, and they should always build on knowledge that they have already constructed. 4. Teacher-centred approach (TCA) versus learner-centred approach (LCA) The two approaches are outlined below so that the teacher can see and understand why there was a need for a paradigm shift. Table 1: Teacher-centred approach versus Learner-centred approach Teacher - centred • rote practice • rote memorisation of rules and formulas • single answers and single methods to find answers • use of drill worksheets • repetitive written practice • teaching by telling • teaching computation out of context • stressing memorisation • testing for grades only • being the dispenser of knowledge • • • use of cue words to determine operation to be used practising routine, one-step problems practising problems categorised by types Learner - centred • use of manipulative materials • co-operative group work • discussion of mathematics • questioning and making conjectures • justification of thinking • writing about mathematics • • • • • • • • • • Mathematics as Communication • • • Mathematics as Reasoning • doing fill-in-the blank worksheets answering questions that need only yes or no responses answering questions that need only numerical responses relying on authorities (teacher, answer key) • • • • problem-solving approach to instruction content integration use of calculators and computers being a facilitator of learning assessing learning as an integral part of instruction word problems with a variety of structures and solution everyday problems and applications problem-solving strategies open-ended problems and extended problem solving projects investigating and formulating questions from problem discussing problems reading mathematics writing mathematics listening to mathematical ideas Teaching Practices Mathematics as Problem Solving • • • drawing logical conclusions justifying answers and solution processes reasoning inductively and Page 3 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) Teacher - centred Mathematical Connections • • learning isolated topics developing skills out context Learner - centred deductively • connecting mathematics to other subjects and to the real world • connecting topics within mathematics • applying mathematics • having assessment be an integral part of teaching • focusing on a broad range of mathematical tasks and taking a holistic view of mathematics • developing problem situations that require applications of a number of mathematical ideas • using multiple assessment techniques, including written, oral and demonstration formats of Evaluation 6. having assessment by simply counting correct answers on tests for the sole purpose of assigning grades • focusing on a large number of specific and isolated skills • using exercises or word problems requiring only one or two skills • using only written tests The above can be summarised into five key major shifts in the environment of mathematics classrooms that are needed to move from the traditional teacher-centred practices to mathematics teaching for the empowerment of learners. The shifts are: • toward classrooms as mathematical communities - away from classroom as simply a collection of individuals; • toward logic and mathematical evidence - away from the teacher as the sole authority for right answers; • toward mathematical reasoning - away from merely memorising procedures; • toward conjecturing, inventing and problem solving - away from an emphasis on answer finding; and • toward connecting mathematics, its ideas, and its applications - away from treating mathematics as a body of isolated concepts and procedures. Table 1 pre-supposes that teachers as well as learners have certain roles to perform. 4.1. What a teacher at any level of teaching must know and be able to do to teach mathematics • • • • Setting goals and selecting or creating mathematical tasks to help learners achieve these goals; Stimulating and managing classroom discourse so that both the learners and the teacher are clear about what is being learned; Creating a classroom environment to support teaching and learning mathematics; and Analysing learner learning, the mathematical tasks, and the environment in which learners learn and teachers teach. 4.2. The teacher’s role in discourse The discourse of a classroom - the ways of representing, thinking, talking, agreeing and disagreeing - is central to what learners learn about mathematics as a domain of human inquiry with characteristic ways of knowing. Discourse is used to highlight the ways in which knowledge is constructed and exchanged in classrooms: Who talks' About what' In what ways' What do people write, what do they record and why' What questions are important' How do ideas change' Whose ideas and ways of thinking are accepted and whose are not' What makes an answer right or an idea true' Page 4 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) The discourse is shaped by the tasks in which learners engage and the nature of the learning environment; it also influences them. The mathematics teacher should orchestrate discourse by: • posing question and tasks that elicit, engage and challenge each learner’s thinking; • listening carefully to learners’ ideas; • asking learners to clarify and justify their ideas orally and in writing; • deciding what to pursue in depth from among the ideas that learners bring up during a discussion; • deciding when and how to attach mathematical notation and language to learners’ ideas; • deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a learner struggle with a difficulty; • monitoring learners’ participation in discussions and deciding when and how to encourage each learner to participate; and • using a variety of assessment methods to determine learners’ understanding of mathematics. Elaboration The teacher has a central role in orchestrating the oral and written discourse in ways that contribute to learners’ understanding of mathematics. Teachers should regularly follow learners’ statements with, “Why'” or by asking them to explain. Instead of doing virtually all the talking, modelling, and explaining themselves, teachers must encourage and expect learners to do so. Teachers must do more listening, and learners more reasoning. Knowledge of mathematics, of the curriculum and of learners should guide the teacher’s decisions about the path of the discourse. Beyond asking, clarifying or eliciting questions teachers should also, at times, provide information and lead learners. Decisions about when to let learners struggle to make sense of an idea or a problem without direct teacher input, when to ask leading questions, and when to tell learners something directly are crucial to orchestrating productive mathematical discourse in the classroom. Such decisions depend on the teachers’ understandings of mathematics and of their learners - on judgements about (the) things that learners can figure out on their own or collectively and those for which they will need input. Another aspect of the teacher’s role is to monitor and organise learners’ participation. Who is volunteering comments and who is not' How are learners responding to one another' What are different learners able to record on paper about their thinking' What are they able to put into words, in what kinds of contexts' Teachers must be committed to engaging every learner in contributing to the thinking of the class. They must think broadly about a variety of ways for learners to contribute to the class’ thinking - using means that are written, pictorial, and concrete as well as orally. 4.3. Learners’ role in discourse Mathematics teachers should promote classroom discourse in which learners: • • • • • • • listen to, respond to, and question the teacher and one another; use a variety of tools to reason, make connections, solve problems, and communicate; initiate problems and questions; make conjectures and present solutions; explore examples and counter-examples to investigate a conjecture; try to convince themselves and one another of the validity of particular representations, solutions, conjectures, and answers; rely on mathematical evidence and argument to determine validity. Page 5 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) The nature of classroom discourse is a major influence on what learners learn about mathematics. Learners should engage in making conjectures, proposing approaches and solutions to problems, and arguing about the validity of particular claims. Whether working in small or large groups, they should be the audience for one another’s comments, that is, they should speak to one another, aiming to convince or to question their peers. 4.4. Learning environment/classroom environment The teacher of mathematics should create a learning environment that fosters the development of each learner’s mathematical power by: • encouraging learners to explore sound mathematics and grapple with significant ideas and problems; • helping learners to verbalise their mathematical ideas; • showing learners that many mathematical questions have more than one right answer; • teaching learners through experience the importance of careful reasoning and disciplined understanding; • building confidence in all learners that they can learn mathematics; • using the physical space and materials in ways that facilitate learners’ learning of mathematics; • providing a context that encourages the development of the mathematical skills and proficiency; • respecting and valuing learners’ ideas, ways of thinking, and consistently encouraging learners to: → work independently or collaboratively to make sense of mathematics and → take intellectual risks by raising questions. Learners are more likely to take risks in proposing their conjectures, strategies, and solutions in an environment in which a teacher respects learners’ ideas. Furthermore, teachers must teach learners to respect and be interested in one another’s ideas. Teachers should consistently expect learners to explain their ideas, to justify their solutions, and to persevere when they are stuck. Teachers must also help learners learn to expect and ask for justifications and explanations from one another. Teachers’ own explanations are not true simply because he or she “said so”. Learners also need to learn how to justify their own claims without becoming hostile or defensive. Learners’ learning of mathematics is enhanced in a learning environment that is built as a community of people collaborating to make sense of mathematical ideas. A key function of the teacher is to develop and nurture learners’ abilities to learn with and from others, to clarify definitions and terms to one another, consider one another’s ideas and solutions, and argue together about the validity of alternative approaches and answers. Classroom structures that can encourage and support this collaboration are varied; or learners may at time work independently, conferring with others as necessary; at other times learners may work in pairs or in small groups. Whole-class discussions are yet another profitable format. Take note that no single arrangement will work at all times. In order to ensure successful implementation of mathematics the role of mathematics teachers and learners and their classroom learning environment can further be seen in terms of the following principles, which should be stressed: 1. Learner autonomy during teaching and learning. The maintenance of learners’ autonomy is an absolute prerequisite for success. By autonomy is meant that learners: should never experience any obligation to use specific operations or strategies in solving a problem, or specific methods of computation; are allowed to choose methods according to their individual judgements; accept individual and collective responsibility to assess the sensibility of their answers; and ✴ ✴ ✴ Page 6 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) ✴ should have the freedom to develop and express their own thoughts and ideas in relation to the subject matter. The teacher has to ensure that the discussions among learners are effective: ✴ ✴ 3. learners should explain their own work in such a way that others may understand what they have done; and learners should listen to each other attentively so that differences and similarities between strategies and methods are recognised. 2. In the light of the above-mentioned, reflection is of the utmost importance, because real learning and progress take place during reflection on what was done initially. It is essential that learners should be led to think about: ✴ ✴ ✴ 4. what they have done; how others have done it; and why one way of doing it may be better than another way. Effective discussion and reflection form an integral part of interaction. Interaction could be best achievement through co-operative or group work. For instance, when different answers are discussed, the reaching of an agreement by learners, without the intervention of the teacher, is a powerful experience. Interaction leads to controlled learning, which is a natural process for any young learner. The learner-centred classroom is both well-organised and disciplined. Classroom activities should include well structured small group work as well as disciplined individual work. The teacher remains in full control of the class, and although interfering in a directive and prescriptive manner with learners’ mathematical thinking is refrained from, a welldefined and constructive programme for development should be provided. Enable every category of learner to gain access to the subject matter. Teaching interventions should protect the right of all learners to participate in learning activities. Teachers should be open and responsive to feedback from learners about the ways they enable or constrain their learning. Teachers should enable learners to respect and value each other point of view. Teachers should demonstrate a respect for standards of reasoning and evidence. 5. 6. 7. 8. 9. 10. 11. The principles outlined above provide the mathematics teachers with criteria for identifying pedagogical conditions that need to be changed to make their practices more consistent with the broad educational goals. 5. Some basic features of the learner-centred approach The objectives of the learner-centred approach to mathematics can be seen as directed at a strongly developed and versatile problem-solving ability. Learners, for example, can acquire number concept; the concepts of the basic operations as well as a variety of methods of computation by solving problems and by reflecting on and communicating about how they solved the problem. Page 7 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) Other important objectives are the development of a positive self-image, the ability to communicate about mathematics, the acceptance of responsibility for validating answers and a positive active attitude towards learning mathematics. The learner-centred approach to mathematics is further based on the following assumptions: ✴ young children have valuable mathematical ideas of their own and are able to develop new concepts and skills independently; knowledge cannot be effectively transmitted from one person to another, in the sense that learners simply learn which is being transmitted to them; nothing less and nothing more; and young children’s capacities and needs for real understanding of mathematical concepts and computational procedures are not limited. ✴ ✴ Initially, it is better for a learner to use relatively primitive strategies and methods which are thoroughly understood and which provide a sound basis for further development, than be required to use elegant methods imitatively and be entirely dependent on demonstration and prescription for further development. The ability to solve problems should include solving a specified range of problem types and the ability to solve non-routine problems. Well-developed concepts of the four basic operations and a variety of computational strategies, techniques and methods are essential for problem-solving. A basic principle is that learners should constantly experience the freedom to choose methods that they understand and are familiar with. Solving problems that make sense to the learners is the basic learning activity. This is done individually and sometimes co-operatively, while learners are expected to: ✴ explain their methods both verbally and in writing; ✴ assess the correctness and validity of their own methods; ✴ compare various methods of solving a problem; and ✴ identify errors. Social interaction between learners (effective communication) to a large extent determines the success of the learner-centred approach. Learners are able to develop effective computational and problem-solving skills by building on their informal knowledge, provided that: ✴ ✴ ✴ appropriate opportunities for development are created; they are given sensible problems to solve and that they are expected to solve these problems independently; and they are given extensive and well-structured opportunities to think about the own methods (e.g. by communicating in groups and by comparing their methods with the methods used by others). The following motivate some of the important points of departure of the learner-centred approach to learning mathematics: ✴ ✴ ✴ learners develop powerful ideas of their own, and can solve a variety of problems using their informal methods; learners do not necessarily learn what a teacher attempts to transmit to them directly; learners get a variety of perspectives about themselves, about learning and about mathematics from the way in which the classroom and their learning activities are organised. These perspectives can have profound (positive and negative) and persistent effects on their present and future learning; Page 8 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) ✴ learners have a substantial need that what they learn and do should make sense to them in terms of both logic and purpose, and have surprising capacities to make sense of mathematics. 6. Different types of knowledge The following types of knowledge as identified by Piaget that shed light on the different ways in which learners learn about numbers should be clearly reflected in the teaching programme. 1. Physical knowledge which can be defined as: ✴ the knowledge that the learner acquires physical objects, and ✴ properties of physical objects, e.g. an object’s colour or shape, or the way in which it behaves, like falling to earth if dropped. This type of knowledge forms the basis for the learner’s knowledge of number. Initially children learn about numbers by counting and by handling real objects. In the same way, the manipulation of three-and two-dimensional objects, and sand and water play, are essential for spatial orientation. 2. Social knowledge Social knowledge can only be learnt from others through interaction with people, e.g. by listening, watching, reading, asking questions, etc. Social knowledge, as mentioned below, should be conveyed to the learners: Rules of behaviour (at this school boys have to wear ties, but girls need not; you do not eat peas with your knife); 7. ways of communicating clearly (meanings of words, pronunciation, meanings of written symbols); and 8. Mathematical knowledge of this type includes: ✴ terminology, such as the number names; notation, such as the symbols for numbers and operations; and conventions, such as the order in which an arithmetical expression is evaluated, e.g. 20 - 7 x 2 is evaluated as 6 and not 26. 3. Logic-mathematical knowledge refers to that type of knowledge that learners construct themselves by thinking beyond the knowledge that was obtained from handling physical objects and from listening to others. ➔ ➔ ➔ The learner who notices the underlying pattern of the number names (forty-one follows on forty) has constructed logic-mathematical knowledge which goes beyond the social knowledge of the number names. The most important aspect of logicalmathematical knowledge that the young learner has to construct regarding numbers is numeracy. Learners learn mathematics well only when they construct their own mathematical understandings. To understand what they learn, they must enact for themselves verbs that permeate the mathematical curriculum: “examine, represent, transform, solve, apply, prove, communicate and etc.” This happens most readily when learners work in groups, engage in discussion, make presentations, and in other ways take charge of their own learning. One would, for example, expect teachers to stimulate learners to ask questions like the following: ➲ ➲ ➲ ➲ ➲ ➲ What do others think about what Kanu said' Do you agree' Disagree' Does any one have the same answer but a different way to explain it' Would you ask the rest of the class that question' Do you understand what they are saying' Can you convince the rest of us that makes sense' Page 9 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) Helping learners to rely more on themselves to determine whether something is mathematically correct: ➲ ➲ ➲ ➲ ➲ ➲ Why do you think that' Why is that true' How did you reach that conclusion' Does that make sense' Can you make a model to show that' How would you prove that' Helping learners learn to conjecture, invent, and solve problems: ➲ What would happen if…' What, if, not' ➲ Do you see a pattern' ➲ What are some possibilities here' ➲ Can you predict the next one' ➲ How did you think about the problem' ➲ What decision do you think she/he should make' ➲ What is alike and what is different about your method of solution and hers/his. Helping learners to connect mathematics, its ideas and its applications: ➲ ➲ ➲ ➲ ➲ How does this relate to...' What ideas that we have learned before were useful in solving this problem' Have we ever solved a problem like this before' What use of mathematics did you find in the newspaper yesterday morning' Can you give me an example' The above-mentioned questions will ensure the implementation of the paradigm shift from traditional teacher-centred approaches to learner-centred approaches. These elements of good learning are not realised overnight. They take much practice and training. In conclusion, teachers should realise that greater success can be achieved in the teaching and learning of mathematics, if teachers can actualise the following hints: • • • • • • encourage learners to try concepts and allow time for them to learn from their mistakes without embarrassment; keep a positive attitude and be excited; continue to search to expand your teaching skills; encourage better learners to tutor others; let learners know that you are proud of their efforts; and allow learners to succeed. The information above is of the utmost importance. It contains the essential information for a successful implementation of the learner-centred approach. All teachers, especially teachers at the Lower Primary phase level, should therefore internalise its contents and integrate it in their teaching programme, which should be evaluated regularly according to the principles discussed. Page 10 of 11 Reform Forum: Journal for Educational Reform in Namibia, Volume 14 (May 2001) References Bryant, P., & Nunes, T., (1997). Learning and Teaching Mathematics: An International Perspective. Psychology Press Ltd Publishers, UK. Chaka, M., (1997). Learner-Centred Education in Namibia: A case study, M.Ed. Thesis, University of Alberta, Canada. Dahlstrom, L. & Zeichner, K., (1999). Democratic Teacher Education Reform in Africa: The Case of Namibia. Westview Press, USA. Elliot, J., (1999). The principles of Learner-Centred Education and teachers’ construction of knowledge through Practice-Based Inquiry. Centre for Applied Research in Education, University of Anglia, England. McCombs, B.L., (1997). Learner-Centred Psychological Principles: A framework for school redesign and reform. Ministry of Basic Education and Culture, (1992). The Broad Curriculum for Basic Education Teacher Diploma, Windhoek, Namibia. Ministry of Basic Education and Culture, (1998). Pilot Curriculum Guide for Formal Senior Secondary Education, Windhoek, Namibia. The National Council of Teachers of Mathematics, (1996). Professional standards for teaching Mathematics. INC,USA. Sibuku, C.M., (1996). Beginning Teachers’ perceptions of a Learner-Centred Approach to teaching in Namibia. M ED Thesis, University of Alberta. Shinyemba, D.N (1999). Learner-Centred Education; Development of Teachers’ Concepts and Practice of Teaching in the Context of Namibian School Reform. M. Phil. Thesis, Oxford Brookes University. Schmidt, N., (1996). Improving learning and teaching from a learner-centred approach. Learning Resource Centre, University of Guelph. Page 11 of 11