Mathematics in Middle Schools

Mathematics in Middle SchoolsSource: Mathematics in School, Vol. 5, No. 3 (May, 1976), pp. 2-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211564 .

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Scholl This report was prepared for the Mathematical Association by members of the 5-13 Sub- committee of the Teaching Committee on the results of an enquiry initiated in 1973. The members of the sub-committee mainly concerned in collating the material and writing the report were Mrs. D. A. Carter, Mrs. J. M. Duffin, A. W. Riley (Chairman ofsub-committee), Miss R. K. Tobias. It is hoped that the report will stimulate discussion. Criticism or comment will be welcomed by The Editor Mathematics in School, Westhill College of Education, Weoley Park Road, Birmingham B29 6LL or Mr. A. W. Riley, Chairman of sub-committee, 27 Claremont Road, Wolverhampton, West Midlands, WV3 OEA.

Preamble The thirty years following the Education Act of 1945 have seen much lively thinking and experiment in the field of education of pupils aged between 11 and 18. A relatively recent development, which received official notice by the Department of Education and Science in 1965, has been the proposal to replace the traditional two-tier primary-secondary system with a three-tier system of first, middle and higher schools. In this the middle schools, with age ranges usually of 8 to 12 or 9 to 13, bridge the traditional break at 11+.

Development of the three-tier system has been rapid. It is understood that by January 1974 44 LEAs had schemes for some 1300 middle schools in being or plan- ned; in 1973 over 500 middle schools were at work, involving over 250,000 pupils and 10,000 teachers. The Mathematical Association therefore asked its Teaching Committee to examine possible implications for mathematics teaching in the new system. In June 1973 an enquiry was initiated in middle schools which were understood to have completed at least their first 4-years course. Several LEAs felt unable to allow an approach to their new schools on the not unreasonable grounds that the schools needed a longer period to settle down. Ulti- mately, complete replies to the Association's enquiry were received from only 38 schools, but these were from several areas and the information received was such as to justify publication.

Thanks are expressed to the Chief Education Officer and staffs of middle schools of the following Local Edu-

cation Authorities who co-operated in the enquiry: Eal- ing, Hull, Merton, Sheffield, Wallasey, Hertfordshire, Kent, Northumberland and Worcestershire.

PART I: RESULTS OF THE ENOUIRY

Introduction The aim of the enquiry was to gather information and initiate discussion on special opportunities or problems in teaching middle school mathematics, to make information available to schools generally and to consider what help appeared to be needed and how best it might be provided. Response on the latter point revealed a need so urgent that the present document has been framed in two parts. Part I is the report proper; Part II offers some suggestions for planning a mathematics scheme. Part II has been written by a small number of members of the Association's Five to Thirteen Sub- committee with the valuable help of study groups on middle school mathematics at the Association's Annual Conference in 1974 and 1975.

In the report, opinions on the middle school system are those of respondents to the enquiry, that is, of teachers in those schools. The attitude of the Mathemat- ical Association towards the new system is neutral; but it is suggested that the results of the enquiry may well merit the attention of all teachers of mathematics, particularly those with pupils aged between 8 and 14.

The practical initiation of any new system must be based on the availability and nature of existing build-

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TICS

ings and the money available for adaptation. In this respect it seems likely that Local Authorities have often linked the problem to the need for new secondary school places to meet the raising of the school leaving age to 16. From the point of view of numerical accommodation an attractive solution might be to make room for the extra 15-16 year olds by excluding the 11-12s. This consideration may have some bearing on the fact that, in 1973-74, among middle schools as a whole the 8-12 type prepond- erated in a ratio of roughly two to one.

On educational grounds justification of the three-tier scheme may be found in the increasing advocacy of post- ponement of entry to secondary education. There is considerable support for the view that the child of 11 is ordinarily not sufficiently mature mentally, tempera- mentally, or physically, to fit comfortably and profitably into a community whose age range may extend to 18. In the primary school the child has enjoyed experience of different aspects of a unified world, guided in the main by the same teacher in the same class group. The secondary school on the other hand accepts, usually from the outset, the analysis of the world of experience into separate subjects, each taught more or less formally by a specialist teacher. There is a strong consensus of opinion that only a small minority of children are ready for the formal secondary approach by the age of 11, and that the majority barely reach this stage by 13. Many middle school teachers speak of the benefit to the child of 11 to 13 of continuing to work in a primary school atmosphere with his own class teacher.

In most areas the abandonment of the 11+ examination has already given the primary school child the opportunity of more relaxed development, and it will clearly be beneficial if such progress can continue without interruption until he is ready for the outlook and discipline of the secondary school. At 8 or 9 the child is beginning to see relations between his different obser- vations and experiences, developing in concrete terms patterns or structures which will form a basis for for- malisation in the secondary stage. Through extension of the "primary school" by a year or two, most children

may be expected to have secured much firmer foundations by the age of about 13. That the middle school offers such an opportunity is probably its strongest justification.

Taken as a whole, the three-tier system appears to offer a better opportunity than existed under the traditional system for each type of school to concentrate on its specialised task related to a fairly well defined stage of a child's development. We proceed now to consider the special task of the middle school with particular reference to mathematics.

Uncertain aim The aim of any school is most readily seen in its scheme of work. Response to the invitation to provide information on schemes varied from a list of topics on a single sheet to one or two compendious "teaching handbooks". Among more disquieting comments were "Previous to this year (i.e. during three years since the school started) there was no syllabus or scheme of work and no one in charge of maths".

and another "Little opportunity of co-ordinating in the past" - this from a school with 13 on staff of whom 6 teach mathematics, with no designated post. Possible minimal conditions for formulating a unified aim, though still hardly adequate, are implicit in the following rather better picture: "Head of math. advises on scheme content and on teaching, but individual teacher has considerable freedom to develop own methods. No round table discussions, but Head of math. consults individual teachers if problems arise".

It is not unlikely that this last quotation conceals the major handicap under which most middle schools are working; shortage of mathematics teachers makes almost impossible any withdrawal of the head of department from teaching to help his less experienced colleagues in the classroom. Furthermore, for guidance and co-ordination to be effective frequent and regular team meetings are essential.

Replies to the enquiry showed guidance as much more detailed for the lower than the upper classes. This may have reflected a school's outlook as "extended primary", or a tendency to use for the later years a standard intro- ductory secondary scheme such as SMP. Suggestions for work with the less able were rarely more specific than "More practical work".

It was not possible to find any certain signs of agree- ment on content and practice in mathematics teaching, but the general impression was of two years' primary plus two years' secondary work. Aims were uncertain, several schools apparently playing for safety by planning solely to meet the requirements of the higher school. It is probably fair to say that there was little evidence of consciousness of a middle school ethic.

Is there a "middle school ethic"? It has been a main criticism of the traditional system of primary-secondary schools that the break at 11+ resulted all too often in a quite disturbing experience for the pupil; the break was sudden and severe. In mathematics it could be shown diagrammatically (if exaggerated) as:

PRIMARY SECONDARY

CONCRETE DEDUCTIVE EXPERIMENTAL ABSTRACT

Adapting slightly a passage in a Mathematical Associa- tion report of nearly forty years ago (A Second Report on the Teaching of Geometry (1938) p.49) we would agree that "the human mind does not grow in an orderly fash- ion such as suggested in the diagram, nor is it always possible to be off with the old love before you are on with

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the new. The arrangement which we recommend is more properly represented like this:

FIRST MIDDLE HIGH or FURTHER or SECONDARY TERTIARY

CONCREE- EXPERIMENTAL DEDUCTIVE -ABSTRACT

The concrete approach will continue to some extent all through the course, while abstract consideration will begin gradually". We urge all middle school teachers to apply this scheme in thinking out an ethic for their type of school, and not in mathematics alone. We suggest that a main function of the middle school should be to offer an opportunity of a gentle beginning of the transition from concrete-experimental to generalisation and abstraction. In saying this we would emphasise that for the majority of children this transition will be barely under way by 13+. By the time they move to a higher school it is hoped that many children may have some acquaintance with "generalisation", in the sense that they can recognise that different situations may show similar patterns, or "work" according to the same rules. But any urge towards abstraction should be very care- fully controlled; many children, probably the majority, are unlikely to reach understanding of it, either in the higher school or in later life. The need for guidance The meagreness of the information offered in replies to the enquiry suggests that there is much uncertainty about what to teach and how to teach it. This impression is confirmed by the frequent requests for advice on syllabus and teaching method.

The present document is primarily a report on the results of an enquiry into current practice in middle schools. Since the enquiry did not produce any useful consensus of information in the all important field of syllabus and method, a report, strictly speaking, can offer nothing further; but we cannot leave it at that. Schools are looking for guidance and help. Some practical suggestions on planning a scheme are therefore set out separately in some detail in Part II. Although the content of Part II is necessarily coloured by the information - or lack of it - received, it is not a picture of what is going on in any one school. The suggestions are put forward as the result of discussion by a few experienced teachers; we hope they may offer a basis of discussion among middle school maths teachers, who in turn will offer their own comments, criticisms and suggestions. The problem of differing achievement standards Prominent among the problems besetting the schools is that of mixed ability teaching. It is commonly argued that to organise a school in streams based on ability levels tends to produce a community stratified socially as well as academically. Two alternative schemes are ordinarily suggested, mixed ability groups throughout, and subject sets; each has its problems.

The teacher of a mixed ability group must ensure that while the weakest pupil is given full opportunity to find interest and confidence in the mathematics within his limited powers, the most able must also have scope for his rapid progress. This makes heavy demands on the professional skill and the physical stamina of the teacher. The present study suggests that, with class sizes of to-day often 36 or more, mixed ability teaching in mathematics is not within the capabilities of the majority of teachers.

The other alternative, setting, involves simultaneous mathematics periods for different sets, unless balancing sets can be organised in some other subject. A sufficient number of mathematics teachers must be available, competent and confident to work with a minimum of guidance and oversight since the more experienced

mathematicians on the staff will be time-tabled with other mathematics sets at the same time. In present conditions this is often a major difficulty.

Very few replies to the questionnaire showed mixed ability teaching continued throughout the school. The general pattern was of mixed ability teaching for the first two years, sets for the final two years. It is interesting to note that this seems to retain a "break at 11+" inside the middle school. As a general principle it is suggested that, while the mixed ability teaching typical of most primary schools may well continue in the first year in middle school, the aim should be progression to more specialised teaching in sets.

Some suggestions for meeting the needs of pupils of differing abilities are included in Part II. The problem of the break between schools No one will question the importance of smooth continuous development in the work for each pupil. The opportunities of the middle school for smoothing the break at 11+ have been emphasised but, inevitably, there are now two breaks to be bridged instead of one, on entering the middle school and on leaving it for the higher school. For any scheme to be effective throughout a pupil's school career, close liaison between the teachers concerned, at all levels, is vital; its importance cannot be over-emphasised.

Unfortunately, the majority of replies show a very different picture, in which liaison between successive schools is often haphazard, if it exists at all. Between first and middle schools there was sometimes quite frankly no attempt at co-ordination; sample comments: "No co-ordination". "Little that I know of'. "Not tried - we have time to correct their mistakes".

Fortunately some schools show a more constructive outlook: "Staff visits in both directions; interchange schemes of work; subsequent discussions". "Special post for liaison with first school; most of staff familiar with content and method in first school and carry them forward in our first year".

The idea of a special post for inter-school liaison is recommended to those schools, apparently the majority at the date of the enquiry, which had not yet considered it.

There is evidence of more definite liaison between middle and high school, but emphasis is often on the latter's requirements: "Careful note of high school's requirements - incorporated in our syllabus."

SMP is often adopted in the upper middle school with a view to its continuance in high school, but a few schools appear more conscious of the possible development of middle school life in its own right: "Not affected by requirements of high school but we try to be aware of their work." "No syllabus demands by high school; they know what we are doing and are quite satisfied."

At both transfer points, first-middle and middle-high, undue reliance seems to be placed on the efficacy of agreed syllabus or common core and on record cards. Both are important in their place but neither is sufficient by itself; personal contact is needed. Comments from responding schools rarely mentioned liaison on teaching/learning methods, or on means of meeting the needs of mixed ability groups.

A general consideration affecting co-ordination with high schools is that the middle school is carrying pupils a year or two years further along the age range than did the old junior school, thereby reducing the time remain- ing before examinations at 16+. In this respect the popularity of SMP offers at least a safe basis for co- ordination; one school using SMP comments:

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"Most high schools do SMP so there is no problem; a term's coaching in trad is given to a top set of leavers known to be going to a trad high school."

Subsequent to the enquiry, a discussion on continuity produced a somewhat different though related question. Since practical considerations may rule out the simultaneous reorganisation in a three-tier system of all the schools in an area, a secondary/high school may continue to receive pupils from primary schools at 11+ and have a further intake from middle schools at 12+ or 13+. A teacher with experience of this situation expressed the opinion that at 13+ the ex-primary pupils, with two years of secondary school life behind them, were more mature in achievement and personality than the 13+ entrants from middle schools. If there is any sub- stance in this comment, its implications are wide. In particular, it could mean that the relative immaturity of 13+ children, detectable only in schools with mixed 11+/13+ entry, may be a concealed feature of the middle school system. Here certain dangers, once they are recognised, can be guarded against. One is that relaxa- tion of the traditional 11+ pressure and the extension into the middle school of primary school attitudes and techniques may lead to staff and children "taking things easy". That some middle schools are conscious of this danger is evident in comments such as "Advantage of having no pressure from exams; however perhaps this does have its bad effects also - some people tend to take too much advantage of it."

Another danger is that the middle school may fail to take full account of the importance of introducing the pupil to secondary school attitudes and techniques such as, for upper sets, the ability to use a text book in inde- pendent study. Any advantage from the adoption of SMP or other standard scheme may be wasted unless close liaison on its use is maintained with the higher school.

In assessing reports on co-ordination between schools, allowance must be made for the quite severe handicaps affecting many schools through complicated flow patterns through first - middle - high schools. Cases of middle schools with 3 to 5 feeder schools are common, with extreme cases reporting 11, 15 and 20! To quote one Head: "Problem: seven feeder schools, seven high schools, some fully trad, some fully modern, some mixed; continuity problem almost intract- able."

The problem of breaks inside the school The importance of continuity of development within any school needs no emphasis. Replies indicate, however, that even here it cannot be taken for granted. The point has been touched upon in an earlier section (Uncertain Aim) and further relevant comment will be found in the section on staffing and in Part II.

The key problem - the mathematics staff Many middle schools have been and are being created either from former secondary modern schools or former primary schools, and for some years previous experience and staffing are bound to affect the approach to the new organisation. Staff may consist of (a) former primary teachers usually with little back-

ground mathematics, (b) former secondary teachers who may or may not have

specialised in mathematics, (c) new entrants to teaching. The initial balance or imbalance between (a) and (b) is likely to reflect the school's origin, and may influence its character and development.

In most of the sample schools the proportion of staff teaching some maths ranged from one-half to two- thirds. In about a quarter, practically all the staff were teaching some mathematics; most of these were the 8-12

type, suggesting a predominantly primary outlook. In order to avoid any suggestion that a mathematics

specialist should be a trained graduate, schools were asked to give their own definition of "specialist" as applicable in their school. Definitions ranged from "graduate" to "teacher whose time table is mainly maths" or "one with interest in and enthusiasm for maths". On the basis of the schools' own definitions the proportions of their maths staffs regarded as specialists ranged from 5% to 42%. It follows that, in all the schools, more than half the mathematics teachers made no claim to any specialist knowledge, and must therefore need a good deal of guidance and support. How can this be provided? The obvious answer is very difficult to implement in present conditions.

At a meeting of a study group in April 1974 it was stated that the supply of trained mathematics teachers was now sufficient to allow one to each middle school, but this scale of supply was not reflected in the replies to the enquiry. A further problem raised was the lack of continuity among staffs, the better qualified teachers often being tempted to move on to posts more rewarding both professionally and financially in higher schools.

The task of the head of department It is easy to stipulate, on paper, that there should be on every staff a responsible teacher to guide and co- ordinate the mathematics teaching. For this post as leader or head of department a certain degree of mathematical competence is clearly essential, though to specify any particular standard might discourage many who, leaders by force of circumstance, are conscienti- ously working towards a better command of their subject. In addition to mathematics itself, knowledge and experience of both primary and secondary teaching approaches and methods are desirable, as also some knowledge of the psychology of mathematics learning.

Whatever his professional equipment and experience, the leader should continue to develop his own back- ground knowledge of mathematics and mathematics teaching through study, consultation, and attendance at courses. He should satisfy himself on the place of mathematics in the integrated curriculum of his own school and in relation to associated first and higher schools. He must establish the school's mathematics scheme and teaching methods with the HIead of the school, with special attention to the time-tabling of new entrants to teaching and others needing help.

As head of department his main responsibility must be to plan the school's mathematics scheme, and to do his utmost to ensure that his colleagues can operate it confidently. Some detailed suggestions will be found in Part II.

Possible ways of meeting present difficulties Schools were invited to express opinions on the useful- ness of available help such as in-service courses, teachers' centres, or advice by experts. Only two such agencies earned generally favourable comment: teachers' centres, and mutual discussions between teachers in middle schools. While the attractions and advantages of such discussions are undeniable it is important that, having regard to the admitted dearth of mathematical strength in the schools, some qualified assessor should take part. The expert assessor should be careful to avoid assuming the mantle of omniscience; he should be at some pains to maintain the relationship of a colleague and not of a director or lecturer. In general, staffs do not want formal lectures or courses (apart from those under the wing of one or two famous national figures) nor do they welcome a deluge of advice - though much depends on the approach. Comments such as the following were too frequent to be ignored:

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"Enough advice available already to confuse any teacher", and "What we need is to be left to get on with the job".

There is no doubt that the acceptability and effective- ness of help depends first of all on the attitude of the Head of the school and the leader of the maths team, and, on the other hand, of the adviser or assessor. Little can be done in a school whose Head's reply to the query on courses attended by staff was "Not known". But with a keen and interested Head, in-service courses can produce the needed stimulus and support, as witness the reply of a Head who "tries to have one member of staff on each course, especially prob- ationers; 11 out of 14 have now attended."

Some schools or groups of schools may be in a position to test the feasibility of this suggestion from one respon- dent: "For real effect all the members on staff should attend a course of at least two days duration, to be followed up within a definite time... this allows for trials of new material and resulting discussion in staff room".

The value of follow-up, trials, and collective staff discussion may underlie suggestions that a series of articles, each concentrating on one topic, would be welcome. It is proposed to produce such a series in Mathematics in School.

To sum up While the general picture seems on the surface to be one of difficulties, shortages, and weakness, underneath it all there is encouraging confidence in the special poten- tialities of the new type of school. There is the occasional teacher who doesn't like the new system, but he is heav- ily outnumbered by those who offer comments such as the following: tgives opportunity to develop a theme in which the child shows interest without having to watch the time because of exam demands". "The big advantage is that younger children's enthusiasm for the subject extends all the way to the top of the middle school. Freedom from exams enables a topic to be followed to a conclusion once interest is stimulated". "... advantage in having four-year flow at this stage where topics have been correlated in a flexible framework, child-based without external pressures, but in accordance with High School liaison". "Advantage of more specialist staff to advise non-specialists; opportunity for specialist teacher with a young form to stimulate early interest and sense of achievement".

In these comments there are points which may well provide food for thought by teachers in higher as well as middle schools, and the following final quotation seems to have implications for teachers in any type of school:

"Following secondary modern experience it is a joy to be in contact with the quick mind, and it is also useful to learn of the foundations of the subject with the less able".

PART II: SUGGESTIONS FOR PLANNING A SCHEME The place of mathematics in the Middle School curriculum Among those teaching mathematics in middle schools are many who make no claim to be mathematics teachers, much less mathematicians. Such teachers should not feel burdened with a sense of inadequacy; provided they can start with an interest in mathematics they can make a success of their teaching. Enthusiasm will grow as they go along, building up their own mathematical knowledge on the way. They will also come to appreciate increasingly the importance of bearing in mind continually certain basic ideas underlying the curriculum; these ideas may be stated as: (a) mathematics is worth studying for its own sake; it is

interesting and enjoyable; (b) mathematics is related to other school subjects and

to the world in general: it is useful; under this head- ing come topics of practical mathematics such as shopping sums or reading the plan of a house, but

there is much more to it; (c) mathematics is part of a unified purpose which runs

continuously through first to middle school and onwards to later schools and beyond. Mathematics is, ultimately, a single entity. If this sounds rather vague, it may be helpful to think of all-pervading principles which are applicable at all levels; the idea of pattern is an example.

The general scheme should be centred on these ideas. It is hardly necessary to add that at each stage it must be related to the interests, needs and abilities of the pupils. In this respect, replies to the enquiry only rarely offered specific suggestions for work with the less able. Main core concepts: role of the teacher in charge This section is primarily the concern of the teacher in charge of mathematics, but all other maths teachers should study it, too.

There are certain main core concepts or fundamental mathematical ideas which underlie and permeate the whole of the school's mathematics. Any piece of mathematics brings in some or all of them. Here are some of them:

Number and quantity: counting and measuring; Representation: symbols; diagrams and graphs,

spoken and written words, as well as numerals, letters, etc.;

Relation: difference, equality, ratio, etc.; Operation: such as the "four rules". Above and beyond these mathematical core concepts

are main concepts of still wider and more general nature, which enter into everything we think or say or do. Such all-embracing concepts are:

Language; Generalisation; Abstraction. If we think of our mathematics in terms of these general concepts we shall see how far-reaching are the links with what are usually considered other fields of interest. Conversely, any general idea may be examined to see if it has any mathematical content; for example, does the idea "democracy" contain any mathematical elements? Such considerations are not for the middle school pupil, but they may be illuminating for the teacher who is still not confident of his answer to the question "Why teach maths?"

The teacher in charge of mathematics should: FIRSTconsider the mathematical main core concepts

relevant to the middle school age range; THEN (1) make a list of the aspects of each which are

likely to be most prominent at different stages, and (2) make a list of possible teaching topics appropriate to each, bearing in mind the wide range of ability among his pupils.

In the list of main core concepts given above it will be noticed that abstract terms, number and quantity, representation and so on are followed immediately by more tangible aspects or examples, such as counting or the use of diagrams. It is not suggested that the middle school pupil should be concerned explicitly with the basic abstractions, but it is hoped that his teacher will be willing to face them. Quantity, for example, becomes tangible, susceptible to study, only when it is measured, so that, at least for our age range, measuring may be considered the core concept, the main trunk of the struc- ture based on quantity. While measuring, in all its aspects - units to limits - will be a centre of attention throughout the middle school and thus for this stage a main core concept, it will later in the high school become an ancillary topic to the more sophisticated concept ratio. Planning round a concept: the maths involved in a teaching topic Measurement. The core concept measurement is taken

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as an example; it will figure in different ways at all levels in the middle school. To list its important aspects we think of the ways in which it may occur as the pupil's understanding develops.

Unit. For measurement to have any meaning, some unit must be chosen. This aspect of measurement is a general one, whatever kind of quantity is to be measured, length, weight, time and so on. The idea of unit is therefore an aspect of the general core concept of measurement. Including other aspects the list at this stage could appear as in the following table (which should be read from the bottom upwards, development being naturally thought of as a generally upward progression): ASPECTS OF MEASUREMENT

Approximation Averages Estimation Units

(A final aspect of measurement, limits, has been omitted as falling outside the general range of middle school work; but it should be included in the teacher's own consideration.)

Every time measurement arises some of these general aspects must be involved. The pupil will not study them as "aspects of the concept of measurement"; he will gradually become familiar with them as they arise in practical situations, i.e. as teaching/learning topics.

A teaching topic: Drawing a plan. Consideration of a typical, perhaps well-worn learning topic, drawing a plan of the school field, will show how the various aspects arise. Here the class teacher is directly concerned.

Estimation Pupils will discuss how long they think it is (estimation), and whether to

Unit measure it in centimetres, metres or kilometres.

The actual job of measuring will probably be done by pairs of pupils, and the collected results will be found to differ slightly among themselves: which is the right one? This brings in another

Average general aspect, average. It also offers an example of the important statisti-

Scatter cal idea of scatter of values. (What if nearly all the results agree roughly, but one is quite a long way out?)

Scale To do the actual drawing a scale must be chosen. This involves a core concept

Ratio ratio, different from measurement but connected with it.

Approximation Finding that it is not possible to draw a line the "exact" length needed, because a particular small fraction cannot be properly shown to scale, the idea of approximation arises. Approx- imation may have arisen in the origi- nal measuring, when a decision might have to be taken whether to record a length to the nearest metre or the nearest decimetre.

Examination of this simple familiar example shows that underlying each step there is a basic mathematical concept which will be found to apply to a wide range of situations not at first sight related to one another. It is these basic core concepts which give unity to mathema-

tics. The teacher should always dig down to discover the basic concepts underlying each topic the pupil is explor- ing, so that he may be ready to seize the opportunity when the pupil remarks "This seems to work the same way as..." The long-term aim is the recognition that mathematics as a whole displays complete unity. (One of the great advantages of "modern maths" is that it can show that this unity covers arithmetic, algebra and geometry, which for so long occupied separate compart- ments.)

The example just given can be repeated for all kinds of quantity as well as a wide variety of teaching/learning topics. In all of these will be found the same underlying core concepts. To show a complete picture it is suggested that a sheet be prepared for each main core concept, ruled in three columns. The general aspects of the concept may then be listed in the middle column, with specific mathematical topics on one side and teaching or practical topics on the other. The scheme for measurement would now be expanded to:

Table I Concept: Measurement

Kinds of quantity Aspects of core Teaching/learning to be measured concept topics

Temperature LANGUAGE Plan of school field Angle (Limits) Your birthday Volume Approximation in 1980? 1990? Area Averages How thick is a sheet Time Estimation of paper? Capacity Units Weighing flies and Weight elephants, Length etc.

A further example is given, for the main core concept position:

Table II Concept: Position

Math. topics Aspects of Teaching/learning involving position core concept topics

Polyhedra LANGUAGE The Globe: Shadows, Position in great circle,

projection space: 3D lat., long., Enlarging, Position on a thumb line

reducing surface: 2D Direction and Polygons Boundaries: bearing Graphs, 'between', Plans and maps;

grids, 'inside', etc. O.S. map lattices, Position on a Model making: co-ordinates line: 1D house, church, etc.

Sets Pattern making Number line Assembly of squares

and cubes

Comparing this with the Measurement table it will be seen that the entries in the left-hand column are general mathematical descriptions while the right-hand column suggests jobs for the pupil. There is nothing hard and fast about this tabular arrangement, and schools will no doubt experiment with their own. But in any scheme, the teacher should keep clear in his mind the basic general concept, and the learning situations through which the pupil may be introduced to the concept.

The class teacher's contribution Separate tables may be made for each main core concept in the way just described. This will be a main responsibility of the teacher in charge, but it should not be left to him exclusively. The best way for the individual teacher to get a proper grasp of the scheme is to attempt his own analysis of each concept, comparing it afterwards with those of his colleagues.

Unless this main framework of the scheme is understood by the staff the work will risk being uncoordinated or aimless. The teacher in charge should make it his first duty, by round table discussion and personal guidance, to ensure that the general outlook is satisfactor-

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ily ventilated and understood before detailed planning is settled. The class teaching syllabus The complete set of main concept tables will form the school's mathematics scheme. They will be concerned first of all with the main content of the school's work, but should provide also for revision of first-school work and, at the other end of the range, introduction to high school work.

The class teacher will then select from the items in the third columns of the tables items suitable to the needs and abilities of his own pupils, working out in detail jobs or problems for them. He must bear in mind that: (a) every child should have some experience of every

concept; (b) the extent of the experience will vary with each

class and each child; (c) probably only a few children will cover all the

aspects and topics suggested for any particular concept.

Mixed ability teaching With the present emphasis on mixed ability teaching, the situation becomes increasingly complicated as pupils move up the school. Both class teacher and, for the school as a whole, the teacher in charge will need to be continually watchful to guard against two possible dangers: (a) lest topics may be omitted to such an extent that

some children are deprived of a balanced over-all diet;

(b) lest "popular" topics are repeated, unnecessarily and wastefully, as pupils move from class to class.

Detailed progress records have their part to play here. The point is also particularly important in connection with liaison with previous and later schools.

No single syllabus: the "learning spiral" From what has been said already it will be seen that, while the same basic considerations apply for all pupils at all levels, detailed treatment will vary widely. The syllabus can no longer be thought of as a list of consecu- tive topics, to be studied one by one, in the same order, by all pupils. But for each pupil the learning process necessarily follows a continuous line, if only chronologi- cally.

If for a particular pupil or group of pupils such a line

Figure 1

weight

length

time

angle

UN ITS

measure

Q se e

4 AfGU

93

capacity

volume

area

temperature

of advance is pencilled in on a concept analysis table, it will move among the topics and aspects in a complicated way. It will probably be helpful to think of it as a 3D spiral - a helix, if an irregular one - but to represent this on paper is difficult.

A learning spiral centred on the core concept Measurement Fig. 1 shows a framework for a learning spiral for Meas- urement. In this diagram the aspects of the core are shown in two concentric rings; language has the inner ring to itself because, whatever else we may be concerned with, language is always of basic importance.

Outside the core and its intrinsic aspects are shown the separate mathematical topics involving measurement (from column 1, Table I). In a 3D model these may be thought of as a cluster of vertical stems surrounding the main core. Fig. 1 is a plan view, or horizontal cross section, of the system.

Around the main core the learning spiral moves through the different learning topics (Column 3, Table I) using these to approach length, weight and the rest. But its visits to these are not once-for-all; it returns to examine each of them at different levels, from the infant school introduction "I'm heavier than you" through to the approximation appropriate in weighing flies or elephants. Fig. 2 shows an example of part of a possible learning spiral (to avoid undue complication the relevant elements appear more widely separated than they would be in practice).

Development along a spiral course Fig. 2 attempts to show in 2D how learning may advance along successive circuits of a spiral, that is, at different levels in successive terms or years. The pupil moving along his helical path may be thought of as climbing a circular ramp, examining and passing successive pillars at increasing heights as he moves around and upwards, the whole path supported by the core.

Figure 2

core concept rate

density

core concept ratio

specific gravity

weight

( <measurementt

volun+

1 2i 3 4

City

The dotted circles show successive periods of time

Alternatively we might think of a spiral thread, carrying a string of beads representing the activities and study topics experienced by the pupil; the reader is left to choose and thread his own beads.

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In the first circuit of the spiral the pupil will gain basic experience of measuring weight and capacity. In the next circuit, at a higher level, he may similarly study the rather more advanced idea, volume. At a still later level he will relate weight and volume in the idea of density (here the spiral draws together the three elements, but to show this in the diagram might be confus- ing). For mathematical completeness specific gravity is shown in a final circuit of the spiral, but this will be outside the range of all but a few middle school pupils.

When the stage is reached when different elements are connected, such as weight-volume and length-time, it will be seen that links with other main core concepts emerge. The spiral used as an example will not have been followed to the exclusion of other mathematical work (spirals), and it is quite likely that the core concept rate may have been met earlier in problems about quantity-cost or length-time.

Main core mathematics arising from pupil's work It will be useful now to look at the scheme from the other end - the classroom point of view, starting with the pupil's activity and examining this for basic mathematical content. Reading and making diagrams and plans, maps and models, usually provides plenty of interest in the job itself, and by its recurrence at different levels offers expanding mathematical experience and understanding. Here is a list of main core concepts which will be found to be involved repeatedly, with some of the mathematical topics based upon them:

1.Comparison and correspondence: one-to-one correspondence, as scale length to full-size length; matching edges of net for solid model.

2. Measurement: units, approximation, limits all involved continually: (a) enlarging and reducing (the central concern in

all plans and models); property of similarity; (b) lengths; (c) angles, bearings; (d) co-ordinates, grids, etc. (e) patterns in 2D and 3D; (f) nets of solids; (g) area; (h) weight.

3. Position: 2 (a), (b), (c), (d), (e) and (f) above.

4. Ratio: 2 (a), (e), (g) and (h) above.

5. Number: throughout, especially fractions.

Relation to main core concepts in other subjects Some of the topics listed above will arise in other sub-

MAPS SCALE and RATIO Plan of classroom .............1 1 Enlarging and reducing or playground Simple surveying ...............2 2 Plotting on grids and

lattices Scale of a map ..................3 3 Co-ordinates,

ordered pairs Making and reading ..........4/ 4 Similar figures,

shadows simple maps Scales and grids 5 5 Representative atlas and O.S. fraction Shapes on different ..........6/ 6 Graph of multiplier maps: 21", 1/50,000 Relief, contours .................7 \7 Equivalent fractions Vertical sections ................8 8 Graphs of constant slope slope

Fig. 3 Some links between maps and maths

jects as well as in mathematics; patterns belong also to Art, maps to Geography. Core concepts of art and geography, and of other subjects, should be examined to identify their links with mathematics. The class teaching organisation of the middle school offers the opportunity to refer to these links as they crop up in either subject.

Links between concepts and topics in different subjects may be represented in a diagram like Fig. 3 which shows links between maps and the mathematical concepts of scale and ratio. This diagram however shows items piecemeal or disconnected, unlike the continuous flow of a spiral.

A note on language The special position of language in relation to any mathematical core concept was mentioned in a previous section. The mathematics teacher, as such, may think of language as an ancillary topic, contributing to precision of mathematical statement; the relation may equally be thought of in the converse sense, the precision of mathematics clarifying verbal language. Language, verbal or mathematical, will develop best when the child uses it, at first orally, then refining it in discussion with classmates and teacher, producing finally a statement in the form of a complete sentence. This applies with equal force whether we are teaching English or mathematics. In mathematics the pupil's statement may be a symbolic demonstration, at first usually pictor- ial. The teacher should keep in mind always the importance of developing and practising spoken, written, and symbolic forms of mathematical language and ability to change from one form to another.

In discussion of any topic, mathematical or other, at the child's level there is often a temptation to speak colloquially; this may be useful within limits, but it should be controlled against excess. The habit of precise statements in both mathematical and verbal language should be practised and inculcated on every possible occasion.

The school "math syllabus" The complete set of core analysis sheets compiled as suggested in Tables I and II will form the mathematics scheme for the school. In the details of column 1 and, particularly column 3 it should be regarded as fluid from year to year and from one teacher to another. Several copies of the core analyses should be prepared for use by teachers taking parallel groups; the entries for column 3 may be left for each teacher to fill in for his own class. Copies of each detailed sheet should be available for immediate colleagues and, particularly, for the teacher receiving the pupils in the next year, in the same or a higher school.

Progress records of individual pupils would be framed and read with reference to these detailed sheets.

To sum up The pattern of framework suggested in this paper for planning the scheme is not the only possible one but, whatever pattern is followed, the same underlying principles apply.

A series of teaching topics will be worked out in forth- coming articles in Mathematics in School, beginning with one on Area. No attempt will be made to present these in the form here suggested, but it will be found that each can be analysed, and all of them co-ordinated, under the basic pattern of core concept, mathematical topic, teaching topic, and spiralling line of progression.

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Mathematics in Middle Schools