Effective Teaching of Mathematics, The

Malcolm Simmons

• Publisher: Routledge
• Year: 2014
• ISBN:
  9781317901693
• Edition: 1
• Topic: Education
• Subtopic: Education General

First published in 1993. This series has been written for trainee teachers wishing to improve their teaching skills as well as for in-service teachers, especially those engaged in the supervision of trainees. The book provides an overview of mathematics teaching at secondary level and links established mathematics content to recent curriculum developments in mathematics teaching in England and Wales and in Scotland. The main purpose of the book is to instigate and complement good mathematics teaching practice in our classrooms.

CHAPTER 1

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Approaches to teaching and learning mathematics

During the last decade two reports have provided significant guidelines for practising mathematics teachers and for those training to become mathematics teachers to help bring about more effective teaching in their classrooms. These are Mathematics Counts: Report of the Committee of Inquiry into the Teaching of Mathematics in Schools (the Cockcroft Report: DES 1982) and Mathematics from 5 to 16: Curriculum Matters 3 (DES 1985). The former has had a significant impact on the deliberations for a national curriculum for mathematics in England and Wales, and has had some influence in the development of the mathematics curriculum in Scotland.

It is the purpose of this book not to judge to what extent these publications have been successful in bringing about change, but to attempt to offer the reader practical guidance towards becoming an effective mathematics teacher based upon the widely accepted principles set out in these publications. The book is aimed at those training to become mathematics teachers but it might also be of use to experienced teachers of other subjects wishing to teach mathematics.

Statutory orders detailing attainment targets and programmes of study now exist for the teaching of mathematics in England and Wales, and as yet non-statutory guidance of a similar nature exists in Scotland, but the inexperienced teacher is more likely to be concerned with the practicalities of ‘having a good lesson’ than with more distant but no less important questions concerning the teaching of mathematics in our schools. Because of these immediate considerations most relevance will be found in those parts of the reports dealing with ‘approaches to’ or ‘styles of’ teaching mathematics.

Mathematics from 5 to 16 states and discusses twelve principles influencing the approaches which might be adopted. These twelve principles enlarge upon the now famous paragraph 243 in the Cockcroft Report, which states the need for the following approaches to the teaching and learning of mathematics:

exposition by the teacher

discussion

appropriate practical work

consolidation and practice of fundamental skills

problem solving

investigative work.

It would be foolish and naïve to suggest that these principles generally or strictly adhered to would result in effective mathematics teaching, but they can be regarded as a good agenda for action. The quality of that action depends greatly upon the personal qualities of the teacher and the attitudes towards mathematics which can be engendered in pupils.

It is not suggested here, nor was it suggested in the Cockcroft Report, that mathematics teaching in any one lesson should fall only into one or two of these categories. However, it will be useful to consider briefly each one separately in order to appreciate the quite different demands each has on the teacher and pupils.

Exposition

Exposition is probably the aspect of teaching which the trainee teacher readily relates to as the ‘teaching act’, and in which he/she feels the most measureable gains are to be made as training proceeds; indeed, it is an aspect upon which many judgements are to fall. It is quite likely that this was the style followed by the student’s own teachers for much of the time and so is well within his/her experience.

This approach to classroom teaching, particularly if the whole class is being addressed, puts the teacher in the spotlight, usually centre stage, and the demands are not dissimilar to those of an actor with an expectant audience. Effectiveness in this situation requires many skills and attributes. The most important ones for the trainee teacher to concentrate on initially are:

to have, or at least show, confidence in the material being presented to communicate in a clear, unhurried, unambiguous manner.

If these do not come easily to the student then extra effort and guidance will be needed. Without these two basic requirements exposition will be ineffective or, worse, harmful to the progress of pupils. It is here that the similarity with the actor’s role ends; the pupil audience should take an active role in the proceedings, not a passive one.

Together with the need to present situations efficiently and to hold centre stage whilst doing so, the teacher also needs to stimulate pupils into thinking about and contributing to the topic under discussion. This requires skills which are more difficult to acquire, and during initial attempts the student will feel less secure, less sure of where things are going, less centre stage and more anxious about fulfilling objectives.

More will be said of this later. Done well, exposition can be a rich and rewarding classroom approach.

Successful exposition may take many different forms but the following are some of the qualities which should be present: it challenges and provokes the pupils to think; it is reactive to pupils’ needs and so it exploits questioning techniques and discussion; it is used at different points in the process of learning and so, for example, it may take the form of pulling together a variety of activities in which the pupils have been engaged; and it uses a variety of stimuli. (DES 1985: 38)

Discussion

It may appear from the previous quotation that bringing about discussion in the mathematics classroom is but a simple step away from a lesson in which the teacher has successfully achieved his objectives through exposition. The role which pupils will take relies heavily on the implications behind the word ‘exploits’. How does the teacher exploit questioning techniques and discussion? How is discussion generated? What demands are made on pupils? What skills are needed by the teacher?

These and other questions will be dealt with in greater detail in Chapter 5. The aim here is to give the reader some feeling for what lies behind the concept of good discussion in a mathematics classroom. Good discussion will not just happen, it has to be planned for, worked at and practised. It essentially requires a shift in the established roles of many teachers and their pupils. If the trainee teacher can begin to develop this as a possible classroom approach along with other approaches right from the start of training, much will be gained by not having to make such a deliberate change in approach later on, and a much more flexible attitude towards teaching and learning will be developed. Pupils too will gradually adapt and become more active and more responsible for their own learning. Clearly in order to have good discussion there must be something worthy of discussion, and the organizational factors for facilitating that discussion need some careful thought. Planning will inevitably be less detailed about where the lesson is to go, but this does not mean that there should be less planning; on the contrary, it is very likely that this approach will demand a far wider preparation than hitherto experienced.

One of the most inhibiting factors against the development of discussion in the secondary classroom has been the presence of public examinations and to a lesser extent the internal examination. As examination time looms nearer, lessons become more dedicated to the preparing of pupils for the type of examination questions they are likely to get, rather than taking a more extensive format in which pupils are active, develop ideas by doing, refine ideas by discussing and even arguing, and reflect on findings. The new examination syllabuses and curricular guidelines now give encouragement to follow a wider approach. The following two assessment objectives are included in the National Criteria for mathematics:

Any scheme of assessment will test the ability of candidates to:

3.16 respond orally to questions about mathematics, discuss mathematical ideas and carry out mental calculations;

3.17 carry out practical and investigational work, and undertake extended pieces of work. (SEC 1986)

In describing how ‘contexts for learning’, ‘kinds of learning’ and ‘aspects of mathematics’ interrelate, the Report of the Review and Development Group on Mathematics for Scotland, Mathematics 5–14 (SED 1990) stresses the importance of identifying a context, developing positive attitudes and awareness, and using problem solving to learn concepts, facts and skills. Appropriate discussion between teacher and pupils and amongst pupils themselves can play a major part in bringing about these desirable outcomes.

Appropriate practical work

It is unfortunate that much practical work in mathematics is confined to the least able groups. This practice reinforces attitudes that mathematics, ‘respectable mathematics’, is highly abstract and that resorting to simple models and apparatus detracts from the true spirit of mathematics. It is through appropriate practical work, the emphasis being on the word ‘appropriate’, that ideas about what doing mathematics really is can begin to change. The sensitive use of practical work can be a great unifying factor within a classroom, providing a means to seek common goals, to share ideas and experiences and to delight in corporate success.

Practical work in the mathematics classroom need not always be directly related to the needs of employers or to the direct or more obvious needs of individuals seeking employment. There will always be a place for such approaches but not to the exclusion of the experiential, often very simple approaches designed to develop mathematical thinking within the individual which leads to a clearer understanding of the concepts involved. It is through the use of such practical work that the inexperienced teacher is likely to become more aware of the importance of the processes experienced by the pupils, and to realize that mathematics teaching is not concerned merely with the identification and acquisition of techniques to be remembered and demonstrated at a predetermined future date.

Appropriate practical work also provides the teacher with an excellent opportunity to learn more about the children, to learn names and faces, to find out what level of thinking each child is capable of, to diagnose weaknesses without testing and to provide help without taking away initiative. It provides space and time to think!

Consolidation and practice of fundamental skills

The student teacher is likely to find considerable pressure brought to bear on this aspect of teaching – a pressure which has for many years dominated all too easily the pattern of lessons. This is highlighted by the often heard criticisms of new mathematics schemes that allegedly do not provide sufficient practice of fundamental skills. Trainee teachers will of course need to discover the importance of such skills and where necessary diagnose weaknesses and teach explicitly towards them. The teacher should however be developing ideas about creating situations where the need to have certain skills becomes apparent to the learner. Motivating pupils in this way is extremely important if the teacher is to avoid the drudgery of mere practice lessons. This naturally leads into the introduction of problem solving.

Problem solving and investigative work

Mathematical problem solving should be an accepted integral part of any mathematics programme. By ‘accepted’ I mean accepted on the part of the pupils. On the one hand it should not be regarded by them or the teacher as an add-on activity requiring little attention, and on the other hand pupils should not become over-anxious and worried at the prospect of not being able to cope with a problem situation. Given sufficient planning and preparation, this aspect of mathematics learning can be the most rewarding and enjoyable for all concerned.

Situations where pupils can be motivated – and motivated for the right reasons (i.e. task motivated) – should never be far from view when lessons are planned or evaluated. In fact motivation could always be included in a lesson plan as an unwritten but assumed aim of any lesson. Matching problems to pupil abilities and past experiences is a major factor affecting motivation. If we assume for the moment that a good match can be made and that other factors are present which produce the kind of classroom where thinking can take place, then motivation for many will be directed into action, and largely negative non-cognitive affects such as anxiety, pressure and non-perseverance which naturally arise during the problem-solving process will have less impact.

Interest in problem solving has widened from eminent researchers such as Polya (1945) and Wickelgren (1974), who advocated that the right kind of experiences could improve problem-solving skills, to educationalists and teachers who are now faced with developing these activities in the classroom. The nature of problem solving in the classroom is continuing to change as teachers and pupils become more familiar with modern technology. A growing familiarity is a prerequisite for skilful use, but it is unlikely to be sufficient for those ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Editor’s Preface
• Author’s preface
• Acknowledgements
• 1 Approaches to teaching and learning mathematics
• 2 Some early considerations when planning lessons
• 3 Effective approaches to the teaching of mathematics
• 4 Enriching the process of learning
• 5 Communication, language and mathematics
• 6 Investigative mathematics through problem solving
• 7 Mathematics and turtle geometry
• Bibliography
• Index