BRYAN LANG AND P E T E R RUANE

G E O M E T R Y I N E N G L I S H S E C O N D A R Y S C H O O L S

ABSTRACT. In the past twenty years many countries have implemented new approaches to the teaching of geometry at school Ievet. In England, most schools seem to have changed towards an 'experimental science' approach to geometry, whilst a significant number appear to have continued the traditional geometry of Euclid. The new approach uses a great variety of techniques, and thereby loses coherence: it is based largely on prac- tical work and seems to have forsaken the deductive approach entirely.

The paper attempts to quantify the present situation in England and suggests a way of building upon the experimental science approach to provide a unified knowledge of spatial configurations through deductive methods.

1. I N T R O D U C T I O N

During the last hundred years, mathematical educators from a variety of

countries have expressed concern about the quality and quantity of geometry taught in the schools o f their native lands. In England, the Association for the Improvement of Geometrical Teaching (later to become the Mathematical

Association) was formed in 1871; fifty years have passed since the first publi- cation of Geometry and the Imagination by Hilbert and Cohn-Vossen (1952),

and nearly twenty since Coxeter's Introduction to Geometry (1961) appeared.

More recently, crusaders on behalf of geometry include Armitage (1973), Fletcher, T,I. (1971), Jeger (1964) and Meserve (1973). The efforts of these

and other geometers have led to the production of many time books on

geometry for teachers and other mathematicians but it is not clear how their influence has percolated through universities and colleges to the schools.

The main aims of this paper are to assess the status and nature o f geometry

as taught at present in English secondary schools and to suggest ways in which

the experiences o f traditional courses and the more progressive work could be combined to provide a more coherent and meaningful study. The paper

enquires into the amount of geometry being taught and analyses what type of geometry this is.

The difficulty in carrying out such a survey is that since there is no central

control over syllabuses for English schools, there is a wide variety of approaches to mathematics in general and to geometry in particular. Text books and syllabuses are not followed faithfully, but are interpreted according to the teacher's personal knowledge and interests. A recently published survey of aspects o f secondary education in England by Her Majesty's Inspectors o f Schools (1980) gives many interesting insights into present practice. The survey

Educational Studies in Mathematics 12 (1981) 123-132. 0013-1954/81/0121-0123 $01.00 Copyright �9 1981 D. Reidel Publishing Co., Dordrecht, Holland and Boston, U.S.A.

124 BRYAN LANG AND PETER RUANE

notes that more than 80% of pupils in its sample were following courses leading to O level or C.S.E. examinations, 1 and we believe that from the first

year of the five-year mathematics course, the work that the child does is largely influenced by the examination he or she will take at the end of the period. The universal instinct of teachers who prepare children for such examinations is to bias their teaching according to the trends set by examination papers of pre- vious years, and so we have assumed that the best indicator of the sort of geometry being taught in English schools is that which is tested in C.S.E or O level examination papers.

2. ANALYSIS OF EXAMINATION PAPERS

An analysis was made of mathematics papers set in the summer of 1978. This accounted for 19 different O level syllabuses from six examination boards, and 22 C.S.E syllabuses from eight examination boards.

Questions with any geometric content were isolated and classified as belong- ing to one of six categories as follows: (1) trigonometry, (2) mensuration, (3) vectors, (4) coordinates, (5) applied geometry, (6) pure geometry. Classification proved to be tentative in a number of cases, but difficulties were resolved by double marking. Marks were awarded by breaking down each question into component parts according to (1) to (6) above, using the published marking schemes given in some papers or relying upon our experience as setters and markers of this kind of examination. The following comments may help to

explain our thinking.

2.1. Trigonometry

Although this has strong geometric associations, a large portion of the marks may be awarded for analytic aspects such as manipulation of formulae and computation. It is, of course, possible to teach trigonometry in a highly arith- metic fashion with little dependence on geometric concepts such as similarity and enlargement.

2.2. Mensuration

Traditionally, this work is concerned with calculation of lengths, areas and volumes by use of geometric principles or formulae. It obviously overlaps with trigonometry and the comments of 2.1 apply.

G E O M E T R Y IN E N G L I S H S E C O N D A R Y SCHOOLS 125

2.3. Vectors

Most questions take the form of operations on ordered pairs of real numbers rather than using basis-free vectors to solve geometric problems.

2.4. Coordinates

As for vector questions, the major concern is with arithmetic or algebra manipulation rather than the use of coordinate methods to illustrate or derive

geometric theorems. Typically, a question might require the calculation of the gradient of the line joining two given points. Alternatively, a particular square

might be specified by the coordinates of its vertices and the pupil then asked

to study the effect of a particular transformation upon this square.

2.5. Applied Geometry

This category was reserved for pie charts, space-time graphs, etc., which

involve geometric ideas but which may not be taught in a geometric context.

There was an expected overlap with mensuration.

2.6. Pure Geometry

This category was reached by removing all the above types from those with

geometric connotations. The questions included in this category covered mainly transformation geometry or the traditional EucLidean approach.

The results of the analysis is summarised in Table I below. They show that the average total marks awarded to work on geometrical ideas (as defined above) was 52% for O level examinations and 43% for C.S.E. On both O level and C.S.E., 20% of the marks were awarded to questions on pure ge~)metry.

TABLE I O level (19 examinations from 6 boards; each consisted of two or three papers but we have scaled down the marks to a fraction of 100% for the complete examination)

Mean % mark Standard deviation Min. % Max. %

Trigonometry 17.2 Geometry 20.3 Mensuration 9.2 Vectors 4.6 Coordinates 0.2 Applied Geometry 0.4

Total 52

8.5 4.5 37 6.5 5.5 30 5.0 3.5 22 4.9 0 17 '

- 0 3

- 0 6 . 5

126 B R Y A N LANG AND P E T E R R U A N E

TABLE I (Contd.) C.S.E. (22 examinations from 8 boards)

Mean % mark Standard deviation Min. % Max. %

Trigonometry 5.4 3.9 0.5 10.5 Geometry 20.2 6.6 5.0 32.5 Mensuration 13.2 5.8 5.0 21.0 Vectors 2.8 3.4 0 8.5 Coordinates 0.6 - 0 5.0 Applied Geometry 1.1 - 0 6.0

Total 43

Before the advent o f modern mathematics in English schools, a typical O level

examination consisted o f three papers: (a) Ari thmetic, Trigonometry and

Mensuration, (b) Algebra, (c) Geometry. Hence it would seem that the marks,

52%, currently being awarded to work involving geometrical ideas is about the

same as it was formerly, but the marks awarded to pure geometry are about

13% lower for O level papers. I t is more difficult to assess changes in C.S.E.

papers because C.S.E. is a relatively new examination which coincided with the

advent o f modern mathematics in English schools.

3. F I N D I N G S

Our first conclusion is that there is no cause for concern about the total marks

allocated to geometrical questions as a whole, but we note that on O level

papers there is a substantial drop from 33% to 20% in the marks awarded to

pure geometry. The marking measures only the quant i ty of geometry being

examined and not its quality, so a breakdown of the pure geometry category

was undertaken.

Traditionally, geometry was examined in the context of deduction, some

questions giving specific measurements of lines or angles and some set in a

general framework comparable to the proofs of theorems in that no measure-

ments were involved. In either case, a body of knowledge in the form of

theorems was expected to be known and such results were applied to a new

configuration by means of a deductive argument to produce a new result or

theorem. On the other hand, some marks were available for single applications

o f one or more theorems with no chain of deduct ion necessary. For questions

involving deductive methods, it seems reasonable to suggest that two thirds o f

the geometry paper marks, that is 22% of the examination total , were available.

This can be compared with 1978 (Table II), when the more tradit ional examin-

at ion papers contain a similar total o f marks (21.6%) for this sort of work

G E O M E T R Y IN E N G L I S H S E C O N D A R Y S C H O O L S 127

whereas the m o d e m O level papers have only one third o f this amount (7.3%).

More significantly, deductive work on configurations with no specific measure-

ments given is almost non-existent (1.3%).

Thus our second conclusion is that while the amount of geometric work

examined at O level and C.S.E. is the same as it was in the 1950's, there was

in 1978 an almost total absence of deductive geometry.

TABLE II

Average For deduction with For generalized Total percentage marks specific measurements deductions

On 6 Traditional O Levels 8 13.6 21.6

On 3 Modern O Levels 6 1.3 7.3

On 5 Mixed O Levels 8.4 8 16.4

William Wynne Wilson (1977) writes of the School Mathematics Project

course for C.S.E. that:

Geometry throughout Books A to H is treated almost entirely as an experimental science, not a deductive one. The reader is told to draw, measure and to find things out and is asked questions starting

'What. . . ?' 'How many . . . ?' 'Can y o u . . . ?'

There is hardly ever any suggestion of giving a reason for anything or a question starting 'Why. . . ?' This is the case even at the end of the course where the last bit of geometry tackled is combinations of transformations and the method used is to see from plots on a diagram what the effect is of successive transformations on a particular square.

Wilson concludes that " . . . the pleasures exploited are those of the artist,

craftsman and physicist rather than those o f the mathematician."

Our analysis confirms the truth of these statements for modern O level and

C.S.E. examinations in general.

4. I N T E R P R E T A T I O N S AND R E C O M M E N D A T I O N S

The experimental sciences approach to geometry was in part a reaction against

the formality of traditional courses in secondary schools and the absence of

geometric experience in primary schools which was typical in England fifteen

years ago. It is now usual to find some experimental work in many primary

schools, but many text books now used in secondary schools repeat early work

in case some of their pupils have not seen it in primary school. If primary

128 BRYAN LANG AND PETER RUANE

schools were unanimous in their provision of geometric experiences, then the

early secondary years could be used to consolidate and extend such work and

lead towards the concepts needed to begin a deductive approach at the age of

fifteen. In Section 5 we outline a possible development.

Deductive argument is the central feature of mathematics and those who

have never encountered it can barely claim to have encountered mathematics. Geometry is a particularly good context for deductive proof since many results

are by no means intuitively obvious but rarely do they contradict intuition. For example, most children have no intuitive opinion about the relative lengths

of the sides of a right-angled triangle. Furthermore, the visual aspects of a geometric theorem can often suggest the structure that a deductive proof should take; this happens far less frequently in other branches of mathematics.

Hence, any result which is suspected from experimental work with specific cases may be worth the effort of justification and, further, the deductive method can be used to establish results o f yet more generality or unexpected- ness. The deductive approach, properly handled, provides a genuine method of

problem solving and can often put several seemingly unconnected results into context, thus simplifying the learning process by offering relational under- standing of the subject.

For all its mathematical and pedagogic shortcomings, the traditional school

course in the geometry of Euclid had an overall identity which we do not find in the geometry of modern mathematics examination papers and text books.

Topology, matrices, vectors, coordinates, transformations are all introduced but not developed in depth; pupils are likely to master none of these ideas and

to see no overall coherence in the subject. Isometries, enlargements and shears

are studied individually but ideas of invariance are never used to reveal the interesting properties of shape that arose in the traditional courses.

None of this is a plea for an axiomatic approach but it is a call to systematize

school geometry and display it as true mathematics. The systematization would consist of a 'local axiomatics' treatment of small sets of related theorems deduced from an agreed set of (not very basic) assumptions. This

formalization would provide a good context for discussion of central ideas such as invariance which are so important in later mathematics but which easily pass un-noticed in an experimental science approach.

5. A POSSIBLE DEVELOPMENT OF STAGE A AND STAGE B GEOMETRY 2

This section provides a very brief outline of work without reference to ways of implementation in the classroom. To help readers who wish to follow up the ideas, we name a few books which approach the work in a similar spirit to the

GEOMETRY IN ENGLISH SECONDARY SCHOOLS 129

one we have in mind; but we are not claiming either that these are the best, or that others which we have omitted have any defects. Many other activities could be included but we are merely attempting to convey the spirit of the work. No attempt has been made to put the topics in precise mathematical order (if one exists) and specific ages of children to whom they can be intro- duced is deliberately omitted.

Infant School (Age 5 to 7)

Recognition and naming of shapes and solids. Early concepts of length. Con- servation of volume. Fitting shapes together, colouring patterns. Symmetry by folding. Exploration of solids via boxes, cartogs, poleidoblocs. Use of tem- plates. Building with shapes. Use of geostrips, meccano, etc. Right angles, rectangles, squares. Tangrams. Measurement of lines.

Details are available in Fletcher, H. (1971), Fletcher and Ibbotson (1966) and Williams and Shuard (1970).

Junior School (Age 7 to 11)

Early practical work on area. Properties of squares, rectangles, triangles, circles by paper-folding, tessellation, fitting shapes into holes. Further use of geostrips, meccano, etc. Making solids from card, straws and pipe-cleaners. Angles, angle sum of a triangle. Work on length, up to scale drawing. Area as a count of units, perhaps approximate. Area of rectangles. Similar shapes by enlargement. Volume by counting cubes and displacement. Relating length, shape, area, volume to number (e.g., fractions, square numbers, triangle numbers, etc.). Height finding. ParaUel lines, angle sum of polygons. Rigidity of figures. Curve stitching. Use of protractors and compasses. Polyominoes. Introducing 7r via circle.

Details are in Fletcher, H. (1971), Fletcher and Ibbotson (1966), and Williams and Shuard (1970).

Secondary School (Age 11 to 13/14)

Development (not repetition) of the above work and a formalization of some of it. Areas of triangles and circles. Reflections, rotations and translations. Pythagoras' Theorem. Area/perimeter relationships. Similarity of polygons, particularly triangles, leading to first ideas of tangent, sine, etc.

Details are in SMP (1970), Scottish Mathematics Group (1971).

130 BRYAN LANG AND PETER RUANE

Secondary School (Age 14 upwards)

A Stage B approach to formalize the work which has been done earlier in a Stage A form, and to use deductive arguments to establish sets of related theorems. Results such as the fact that a triangle will tessellate, and the sym- metry properties of circles, which are intuitively evident from Stage A, are assumed true and provide the basis for the deductive element. Transformations such as isometrics and similarities are used to establish results, many of which are non-obvious. At this stage the principle is more important than choice of themes, but possible topics are: angles in a circle; angle properties of polygons; chords and tangents to a circle; Pythagoras' theorem; medians, altitudes and angle bisectors; theorems of Ceva and Menelaus; theorems of Pappus and Desargues. Each of these is set in a spatial context which is simple enough to allow the results to be easily appreciated, yet they are unexpected and significant.

All of this work can be done without using algebraic methods such as co-ordinates, vectors and matrices and in our view should be so treated. If it is

thought proper to include such topics in a secondary school mathematics syllabus, they would use results of geometry rather than provide them. A learner who has in a short time been faced with a wide variety of methods of dealing with geometry will have no judgement as to which is appropriate to solve a new problem and is likely to become confused and disheartened. The coherence of the subject will be less evident when a range of techniques is offered than when their number is limited to a few methods of wide applicability.

The nearest approach to such a development which we have found in print is in the First Edition of the texts of the Scottish Mathematics Group (1965). They take three results of tessellations of rectangles, congruent triangles and squares as axioms to develop the usual properties of these figures. They then use transformations to develop further properties of shape and the standard theorems of the traditional Euclid course; invariance is used where appropriate. It is interesting to note that the Second Edition has abandoned the approach to a great extent, perhaps because its planned start at 11 years of age was too early for pupils to have the necessary Stage A background.

6. CONCLUSIONS

The fact that geometry provides an opportunity to discover unexpected and significant results about the world around us makes it a particularly suitable topic to study at school level. Most countries in the Western world subscribe to

G E O M E T R Y IN E N G L I S H S E C O N D A R Y S C H O O L S 131

this belief, bu t they have varied considerably in their methods of obtaining

these results in a meaningful way, ranging from experimental science to an

axiomatic development , and from Euclid 's theorems to group and vector space

results as the goal. Servais and Varga (1971) include a number o f syllabuses

which show the differing approaches and, to quote UNESCO's N e w Trends in

Mathematics Teaching I V (1979, p. 39), "The question of geometry teaching

still remains open" . This article has argued for a goal of Euclid 's theorems

developed from a varied background of spatial experience via a ' local axio-

marie" deductive scheme using non-algebraic methods. We believe that such an

approach would provide a more coherent, understandable and worthwhile

s tudy than either the mixture o f many methods with very litt le deduction,

which exists at present in many English schools, or the axiomatic methods

favoured b y some other countries.

Chelmer lns t i tu te o f Higher Education

N O T E S

1 In England, General Certificate of Education Ordinary Level examinations (O level) are intended for the top 20% of the ability range at 16 years of age; Certificate of Secondary Education examinations (C.S.E.) are intended for the next 40% of ability at me same age. 2 Following a Report by the Mathematical Association (1923), it is the practice in England to refer to the experimental work in geometry as Stage A and the deductive part as Stage B, with a possible further systematization as Stage C.

R E F E R E N C E S

Armitage, J. V.: 1973, ~'lae place of geometry in a mathematical education', Mathematical Gazette LVII, 267-278.

Coxeter, H. S. M.: 1961, Introduction to Geometry, Wiley, New York. D.E.S.: 1980,Aspects o f Secondary Education in England, H.M.S.O., London, p. 111-163. Fleteher, D.E. and Ibbotson, J.: 1966, Geometry, 1, 2 and 3, Holmes McDougall,

Edinburgh. Fletcher, H.: 1971 onwards, Mathematics for Schools, Level I Books 1 to 7, Level II,

Books 1 to 10, Addison-Wesley, London. Fletcher, T.J.: 1971, ~ teaching of geometry: present problems and future aims',

Educational Studies in Mathematics, 3, 395-412. HUbert, D. and Cohn-Vossen, S.: 1952, Geometry and the Imagination, Chelsea. J,eger, M.: 1964, Transformation Geometly, George Alien and Unwin, London. Meserve, B.: 1973, 'Geometry as a gateway to mathematics', in Howson, A.G. (ed.),

Developments in Mathematical Education, Cambridge University Press, Cambridge, pp. 241-253.

Schools Mathematics Project: 1970, Books A to X,Cambridge University Press, Cambridge. Scottish Mathematics Group: 1965 (lst Edition), 1971 (2ndEdition),ModernMathematics

in Schools, Blackie, Glasgow.

132 B R Y A N L A N G A N D P E T E R R U A N E

Servais, W. and Varga, T.: 1971, Teaching School Mathematics, Penguin, London. pp. 181-216.

International Commission on Mathematics Instruction: 1979, New Trends in Mathematics Teaching, UNESCO, Paris, Vol. IV.

WiUiams, E.M. and Shuard, H.: 1970, Primary Mathematics Today, Longman, Essex. Wilson, W.W.: 1977, Geometry, in The Mathematics Curriculum series, Blackie, Glasgow,

p. 26.