The 16 ExamAuthor(s): Margaret BrownSource: Mathematics in School, Vol. 4, No. 2 (Mar., 1975), pp. 5-7Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211648 .

Accessed: 22/04/2014 12:34

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

The Mathematical Association is collaborating with JSTOR to digitize, preserve and extend access toMathematics in School.

http://www.jstor.org

This content downloaded from 143.167.204.106 on Tue, 22 Apr 2014 12:34:37 PMAll use subject to JSTOR Terms and Conditions

nby Margaret

nby nby Margaret Brown, Centre for Science Education, Chelsea College

In July 1970, the Governing Council of the Schools Council reached the conclusion "that there should be a single examination system at 16+". Since that date feasibility and development studies have been set up; the results of which are being discussed and assessed.

Some relevant questions are: Question 1. Is it likely that both O Level and CSE will be replaced by a single system of examining at 16+ in the near future? Answer. Very likely. The Schools Council, who are at present working with the examining boards in organizing operational trials of pilot schemes, are giving it top priority, and many teachers involved in the schemes seem to be enthusiastic. On time scale, the Schools Council requires representations from interested bodies to be submitted by mid-1975, and the final recommendation will hopefully be with the Secretary of State by late 1975 or early 1976. If all are in favour, the first full examination could be in 1979 or 1980. Question 2. Who will run the new examinations, what will they be called, and what will the grading system be? Answer. The trials in 1973 and 1974 have been organized by consortia of GCE and CSE examining boards. Those concerned with trials of mathematics examinations are:

London University Entrance and School Examinations Council (GCE) dnd South-East Regional Examinations Board (CSE); Cambridge University Local Examinations Syndicate (GCE) and East Anglian Examinations Board (CSE); Joint Matriculation Board (GCE), North-Western Secondary School Examinations Board and Associated Lancashire Schools Examining Board (CSE); Associated Examining Board (GCE) and Middlesex Regional Examining Board (CSE); Welsh Joint Education Committee (GCE & CSE).

In addition the SMP in cooperation with Oxford and Cambridge Schools Examinations Board (GCE) and Southern Regional Examinations Board (CSE) are investigating various schemes but these are non- operational and hence results do not count towards pupils' grades at present.

It is not known what type of board or consortium will run the eventual examinations, nor what it will be called (no prizes for GCSE!). There are likely to be about 6 grades, 3 corresponding to a GCE pass, and also an "ungraded" category at the lower end. Question 3. Who is taking part in the trials? Answer. Some 65,000-75,000 pupils altogether; about 10,000 in mathematics alone. Schools were invited by examining boards to participate, and in most schemes a sample representing all types of schools is being used.

(It is worth noting that in the case of JMB/ALSEB/ NWSSEB 140 schools volunteered for a total of 42 places.)

Pupils taking the pilot examinations will be awarded a CSE pass, and, for those gaining grade 1, a GCE pass also, on the result of these examinations alone. Question 4. Will it mean that all children in an area will follow the same syllabus up to 16+ and take the same examination? Answer. No, not necessarily. It is designed to be a "common system of examining" rather than a "common examination".

In fact, of the trial schemes, four have common papers across the ability range. In each case provision is made in the papers for pupils of varying ability; in the Camb/EA and Welsh (I) schemes both papers have a "difficulty gradient" so that only the brighter pupils are expected to be able to tackle the harder questions at the end of the papers, while in at least some of the JMB/ALSEB/NWSSEB and AEB/Mddx papers questions are clearly advertised to carry different weightings so that candidates may select appropriately. In all the above schemes there is a single syllabus for all abilities but pupils are not expected to cover topics to the same depth and of course weaker candidates may not complete it all.

In the London/SE scheme there is an optional third paper on an extended syllabus for brighter children, and in the Welsh (II) scheme there are 3 papers of which pupils are normally expected to attempt the first or last pair, although the possibility of trying all 3 remains open.

This means that in all the schemes there would be the advantage to schools of having at least a basic common syllabus. It is interesting to note that none of the schemes can clearly be labelled either "traditional" or "modern" in the mathematical sense, although connoisseurs should not take too long to order them along the spectrum between the two extremes. The JMB/ALSEB/NWSSEB, the AEB/Mddx and the Lon- don/SE schemes each contain options in the syllabus to allow schools to bias it a little in their preferred direction. Question 5. Will there be any mode 3 facility available under the new regime, and will there be any course- work or project work included in the assessment? Answer. There will almost certainly be a mode 3 facility as the Schools Council have virtually guaranteed that

5

This content downloaded from 143.167.204.106 on Tue, 22 Apr 2014 12:34:37 PMAll use subject to JSTOR Terms and Conditions

none of the beneficial aspects of the present systems would be lost.

In principle there is no reason why coursework and project work should not be included under the mode 1 type examinations. (This may partly depend on the feasibility of organizing the necessary load of modera- tion.) However, none of the operational pilot schemes include project work, and only one, the London/SE, includes teacher assessment of coursework, allotting to this one third of the final marks based on a carefully structured system of assessments over the course of several terms.

The S.M.P. scheme is experimenting with assessment of both project work arising from a subject on the syllabus, and of topic work on Computer Appreciation and Programming, the History of Mathematics, Mathe- matical Design, or Mechanisms. The trials are taking place in a wide range of schools (including Eton!) over the current year and the results should prove interesting. Coursework assessment, on the basis of termly tests over the final two years, is also being investigated. Question 6. Presumably ordinary teachers have no say in the final decision? Answer. Not at all; the Schools Council have specifically invited the views of individuals and groups of teachers. Teachers who have not bothered to make their views known at the right time will have no right to complain if the final proposals are not to their liking. Anyone can write to the Schools Council individually, but clearly a representation coming from a group of teachers will carry much more weight.

The Mathematical Association organized a weekend seminar in September 1974, at which participants were able to hear and discuss full details of the various schemes with Dr. Gordon Barratt HMI and representa- tives of all but one of the schemes who kindly attended and were most helpful.

As a result of the Seminar a working party has been formed to propose a submission to the Schools Council for its meeting in April 1975. Although the timescale is short we would be very pleased to hear from members who have experience of the trials or who wish to offer views, in case there is an opportunity for further submission. (Please write to me at Bridges Place, London SW6 4HR. If you already take examinations of one of the boards involved in the trials, that Board should supply you with a syllabus and examination papers; otherwise a limited number of copies could be supplied although regrettably a small charge would have to be made to cover expenses.)

Question 8. What line will the Mathematical Association take in its submission? Answer. This depends on the views that the working party receive, as they are anxious to represent all members of the Association as far as possible. It may however be of interest to outline some of the general feelings expressed at the September Seminar:

(a) Support was given for a single externally moderated system of examination in mathematics at 16+. (b) None of the pilot schemes were judged to be in any serious way unsuitable, and there was praise for the work done in spite of a rushed timetable. (c) Although the idea of a 'common core' in the sylla- bus was generally welcomed, many were not convinced that it was possible in a common examination paper to both stimulate brighter pupils and prevent weaker ones from being deterred. The backwash effect of common papers might well produce inappropriate courses for these two groups since many teachers still tend to be guided principally by previous examination questions when planning their work. 6

(d) A 4-paper system was suggested in which pupils would take two consecutively numbered papers and where the norm would be papers 2 and 3. Less able pupils would take papers 1 and 2, paper 1 being essentially practical, while the very ablest would take paper 3 and the more testing paper 4. The aim is to avoid the difficulty in a 3-paper system where the cut-off between the first two and last two papers falls in the centre of the ability range rather than at both ends. If a pupil is entered for an unsuitable pair of papers the error is less serious for him, since, for example, it should be possible to get perhaps all but

Picks Theorem by Jas. A. Dunn, Education Centre, New University of Ulster, Coleraine

Pick's theorem is now well-known. It is written: A = P+I-1

where A represents the area of a polygon made on a nailboard with a rubber band; P represents the number of nails touched by the rubber band, that is the number of nails on the Perimeter; I represents the number of nails Inside the shape, untouched by the rubber band.

It might be interesting to begin by assuming that this relation is known and investigating its implications. I am not suggesting that it be suddenly pulled out of a hat. Whatever one's general view about doing that, this is much too nice a problem for it; and it has, at an elementary level, the very pleasant and unusual quality of surprise. Its potential as an opportunity for in- vestigation and problem-solving processes is well-known and documented. But after it has been established, with whatever rigour seems appropriate, then the direction of attack might be changed.

The relation contains the variables (A,P,I) and the first problem is to decide which one to fiddle with. It's possible to start with any one of the three and produce similar arguments. Starting with P, a bit of reflection suggests that, if there's to be a shape at all at least three nails are needed on the perimeter, also P must be a whole number, so:

(i) Pe(3,4,5, }... Similarly:

(ii) IE {0,1,2, ...} Taking the smallest value of each, i.e. P = 3, I= 0, and substituting in the relation gives A = , and a little bit of thought suggests:

(iii) Ae 1(2,,3,4,5, . . . With this in mind, returning to the relation

A = !P+I-1 it is suddenly trivially obvious that when P is even, A is integral.

Since this work is being done on a nailboard, the shapes that can be made are more interesting than the formulae and numbers associated with them, although the manipulation of the formula has suggested a problem, "What shapes are possible on a nailboard when A is known?" (1) Begin with A = ,

then P+21 = 3 and the only possible solution of this is P = 3, 3, = 0.

.So (A,P,I) = (,3,0)

This content downloaded from 143.167.204.106 on Tue, 22 Apr 2014 12:34:37 PMAll use subject to JSTOR Terms and Conditions

the highest grade by a good performance in papers 2 and 3. (e) Support was given for a "mixed-mode" examination in which all pupils were required to sit a general short-answer paper 1 but where the second paper could be replaced by some acceptable form of internal "mode 3 type" assessment if teachers favoured this. (f) Short-answer questions in paper 1 were preferred to multiple-choice or true-false questions, and this was supported by the evidence gathered from SMP trials. (g) More work should be done to evaluate teacher assessment of both coursework and project work.

Regret was expressed that so few of the schemes were attempting to do this, and further experiment was encouraged. (h) Project work should not be seen as simply for the less able. (It was reported that in some areas mode 3 syllabuses were getting a poor response from employers for this reason.) Bright pupils at present doing O-Level might have at least as much to gain since working in this manner could develop, and assess, important qualities which the straight written paper may not allow to be recognized. m

and the hunt for polygons

This shape must be a triangle on a unit base and there is an infinite set of them.

Some examples are shown in diagram 1. (2) A = 1, then P+2I = 4 and the only possible solution

of this is P=4, I = O So (A,P,I) = (1,4,0).

For this I found two kinds of triangle and a parallelogram.

See diagram 2. Have I missed any? (3) A = 2, then P+21 = 5 and this has two solutions

(P,I) = (3,1) or (5,0) So (A,P,I) = (-,3,1) or (2,5,0)

What shapes are possible? How does it go on? I had just about reached this stage when Mathematics

in School (July 1974) arrived with the article "Polygon Hunt". The sets of polygons on a nine-pin board were wide-open for a Pick-style analysis, so I wrote the values of P, I, and A for each triangle shown on Page 17. I was surprised to find only one of them with area 2, as I could visualize two (see diagram 1). This led

Diagram 1

Diagram 2

me to a mistake. Triangles 5 and 7 are congruent, and so 7 can be replaced with the missing one. (See also The Mathematical Gazette, No. 390, Dec. 1970, p. 359).

There are still some interesting points for discussion. Why can the hexagons alone include cases where P = 9? In fact the range of values of P for each shape can be discussed. Would Pick's theorem assist in the creation of those polygons, and would it provide conviction as to when they have all been found?

Editor's Footnote The triangle (number 5) in Professor Ewbank's article was incorrectly drawn by the artist. It should have been

Diagram 3

5

The error was also discovered by second year secondary pupils at the Sacred Heart Convent, Lewisham, who discovered three extra polygons:

Diagram 4

69 70 71

Professor Ewbank has received a further three polygons making a grand total, so far, of 74.

Diagram 5

72 73 74

7

This content downloaded from 143.167.204.106 on Tue, 22 Apr 2014 12:34:37 PMAll use subject to JSTOR Terms and Conditions