Mathematics in a Comprehensive SchoolAuthor(s): Norman HarveySource: Mathematics in School, Vol. 3, No. 1 (Jan., 1974), pp. 7-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211134 .

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value for a probability which would be very difficult to calculate exactly.

After doing an experiment of this sort it is often a good idea to question how accuracy could be improved. The chief figure of importance here is the probability of a first division team beating a second division team being as low as 4. I suspect that an examination of figures for other years would yield a rather higher figure than this. There is, incidentally, no point in differentiating between home and away matches in this simulation, since the choice of whether a team plays home or away is random. It would not, however, be a difficult matter to extend the simulation in this direc- tion to make it more life-like.

International Soccer-A Comment In the May 1973 issue of Mathematics in School, Jas. A. Dunn looked at arrangements of points in the results of the home international football matches. He in fact set up a deterministic model by assuming that outcomes

of a home win, a draw or an away win were equally likely. In addition, and looking at all possible outcomes, he did not take into account the likelihood of the teams being ordered which makes the outcome 6-4-2-0 especially likely. To be fair Mr. Dunn did say "Column E is a bit doubtful". You may care to try to improve Mr. Dunn's results by working out a probability model. This would cope with the lesser likelihood of draws very easily, but it is more difficult to see how to cope with the ordering situation.

This final example illustrates nicely the difference between a deterministic model which has to make unreal assumptions but has fixed outcomes, and a probability model where the solutions vary according to random processes, but the assumptions can be made more lifelike. In the third article I propose to suggest some more random models which might be suitable for classroom investigation. If you are fed up with football by now, I promise that none of them will be drawn from this field. (See letter p. 36--Ed. )

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"The Ridings" is a Comprehensive School which when fully developed will have 1,500 pupils from 11-16+, covering the whole ability range, plus a Sixth Form of about 200 students. Organizing a department means that some plan must be adopted at the outset to avoid the "Alice in Wonderland" situation that "it just grew and grew". A plan in its turn implies either a strictly mechanistic (perhaps traditional?) approach or an attempt to think out a set of principles which can be applied to the department. The problem then becomes a linear one with the principles acting as restraints.

Since I used the word traditional above, I now hasten to add that such a method of creating an organization in no way implies the adoption of a modern syllabus. This is another matter and it is becoming a truism that modern topics can be taught in a formal way, just as traditional topics can be taught in an informal way.

Consider the following:

(1) High academic attainment is the requirement of a very few (2) Mathematics is a preparation for the future employ- ment of many (3) The amount of mathematics required for simply living is minimal--(a postcard has been suggested!) (4) Children, however unequal, should have equal

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opportunity (5) Rational thinking is advantageous to a civilized being (6) Purposive, mathematically viable, activity is of itself, better than no (directed) activity (7) The most able and the least able should be given special consideration (8) The traditional isolation of teachers is not con- ducive to co-operative work (9) Planned itineraries and clear aims are essential for such co-operation (10) Children have the right to know what is being done to and for them and the right to become involved in the decision-making of the department as it affects them directly.

Such a lengthy and diverse list of restraints might well appear to offer no restraint at all; on the other hand it might seem that very few teachers would agree with operational aspects of the above and, thus, the system might well break down. On the first count I would say that this article will reveal just how the restraints do affect decision-making; on the second, that teachers who agree with our methods tend to accept appointments here and that over 4a years not one teacher has left the department.

The solution now described is one of many within the set of possible solutions produced by the above restraints. It is not claimed to be the best or a best solution, but it works very well for us and for many reasons we are pleased with the outcome. (I avoid the dread word "results".) The decision to adopt a modern syllabus throughout the school, I deliberately push to the background as I believe this to be of no conse- quence with respect to the courses and opportunities offered to the children. I cannot forbear to mention, however, my firm belief that modern topics do help to arouse the interest of the children and reinforce any involvement brought about by the organization itself.

In the first two years, one teacher is given respon- sibility for recording and assessing the progress of the

children. In my opinion these years should begin with full mixed ability groups which gradually change into course groups, so that by the end of the second year, a course structure has been developed.

The work we do is planned rigorously so that a minimal target is set for each half-term of each year, twelve targets in the first two years. Each half-term unit culminates in a test on the minimal amount of work and these tests are standardized and repeated from year to year. Full statistics of cumulative test marks are kept for comparison from one year to another. This may seem very mechanical and un- imaginative but, in fact, we are able to give a great measure of independence to the children because of the formality of the work units. For instance, the children are given targets over a six-week period but how they get there is their own affair; they can, if they wish, go ahead at a faster rate than the suggested minimum.

Such checking as we do is restricted to grades for work done and standard of presentation. Of course, if we think a child is not working to his individual capacity, the grades suffer, and as they are given fort- nightly, this is of some importance to the child.

Marking is a communal activity and often takes place through several sets in the same period, although this is not necessarily so. Progress marks are simply an addition of ticks, such awards having been previously established by the staff.

The tests are set at the same time through sets, as far as is possible. They are taken very seriously by the child- ren, for whom we have a large measure of trust, in so far as passing information to other groups is concerned (is it in their interest?), and the tests are good measures of recall and comprehension. Generally speaking, these tests are multiple choice, unless this is not very sensible, as in the case with mappings, graph work, or some topological exercises.

At the end of the first year it has become apparent that some children are falling behind, unable to main-

1st Year 2nd Year 3rd Year 4th Year 5th Year

All children start at same point. The

1st year is diag- nostic. Minimum standards are set. SMP throughout. Tests each a2 year.

Continuation of Personal assessment. Major test at end of year. Pupils chosen for Courses 1 & 4.

Express Group Course 1

Continuation of common course SMP Course 2/3

CSE Mode 3 SMP Practical

Calculating Course 4

O-level SMP

O-level course SMP Course 2

CSE Mode 1 Course 3

CSE Mode 3

1st Year Post 'O'

O-level Nov. Start Post 'O'

O-level

CSE Mode 1

CSE Mode 3

Fig. 1

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tain the target, exposed to failure, or what you will. For these children we have arranged reinforcement courses which keep them broadly within the main stream but moving at a slower pace. As it happens, three such children have just taken O-levels in the Lower Sixth, at a slower pace, you see! In no case is a child removed from the main direction of our curricu- lum. A remedial group of some 20 children, outside the department, nevertheless have basic training in our course work.

This situation is maintained during terms 4 and 5 of the 2-year period. Following our dictum that the less able need special consideration, we are not committed to full mixed ability in Year 2, but believe in with- drawing some 25/30 children from 250 for special attention. I appreciate the arguments for and against such treatment, but feel that the sight of the very able shooting ahead confidently and incomprehensibly within a group is inhibiting to those who have great difficulty. Mixed ability for the remainder is acceptable.

After term 5, we give a common setting test on the first two years' work and as a result of this and after consulting our statistics we select a group of very able children for special treatment. These children take O-level after four years and then enjoy a leisurely 3-year amble through A-level, pausing here and there, we hope, for numerical refreshment. By the end of Year 2, three broad groups have emerged; matters are discussed with them and sometimes with parents; and letters are sent home explaining the course structure of the department. We summarize it in Fig. 1.

The Mode 3 course begins in the 3rd Year. It is a 3-year course, aimed at producing Grade 3, 4, or 5 CSE. The course has three elements in equal parts.

(1) Modern syllabus, based on the main course. (2) Calculating methods, i.e. machines, slide rules, logarithms. (3) Practical work such as geometric constructions, models, statistical work, number patterns and the like.

During Year 5 each child selects a special topic and spends a term working on it. These topics and the practical work are given a percentage mark and an assessment.

Parts 1 and 2 are examinable at the end of the course, in the normal way. Each of the three sections carries about equal weight at moderation. After the Mode 3 mock examination, children who perform very badly are told that we cannot see fit to enter them but that they may, if they so wish, pay their own entry. Handing over the fee can have a salutory effect on performance. It is pleasant to be proved wrong.

I emphasize--we do not categorize children as failures. They know that the Mode 3 course is difficult, for we have made it so, but it does suit them because it is varied and each element reinforces the others. It holds their interest, obeys our restraints and produces results which mostly satisfy us. It does something else unfortunately; because the children are encouraged to communicate with us we become aware of the depression which accompanies a continuum of rejec- tion. We try to fight this but we do not succeed very well. However, since there are, each year, some 90 children in the Mode 3 Course, it is not in any sense a sink for the unwanted. It is also very well equipped.

For the express group the future is assured, I leave them forging ahead exponentially!

In the central area there are now some 120 children categorized as Course 2/3, which will be either O-level or Mode 1 CSE.

Once again targets and standardized tests are set (this applies equally to the Mode 3 Course).

Within courses we now aim at mixed ability groups and this persists until the beginning of Year 4. Last year we operated a voluntary group which would take O-level in November of the 5th Year. These children might, we thought, be self-motivated. Some of the volunteers seemed highly unsuitable for this course and we advised them accordingly. If, as a result of this advice, we failed to agree, then we tended to let them have their way. We find that children are usually thoughtful and realistic, they like taking part in this sort of decision making and great interest is aroused. The list this year is a long one! The last group is very keen and relationships are good. Since they appeared to choose themselves, other groups are not adversely affected.

During Year 4, children are offered O-level or CSE Mode 1 Courses and a high degree of discussion and involvement is encouraged. It is really dependent on their individual interests, set positions are not revealed, but in all cases we try to suit the child. The Mode 1 alternative course is very general and offers something for everyone. You may say that we are examination orientated. Well, in a way this is so, but what is the alternative? Examinations are with us; the alternative to a course structure can be the simple classification of a child as success or failure; examination or non- examination. We are dealing in the main with the sons and daughters of ordinary people and things being what they are, a piece of paper gives them a start in the world. I add, without comment, that if, overnight, the examination system were replaced by a system of con- tinuous assessment, this would suit us very well.

I must now add a short description of the courses designed for the Sixth Forms and post O-level groups. These range from CSE improving, through O-level, Additional Level, Mode 3 Computer Studies and several distinct A-level Courses. For these we have designed a Sixth Form mixed ability structure of topic boxes through which students will programme themselves to their desired end. A Sixth Form tutor will have some 15 students of mixed ability and a comprehensive system of testing and progress checking has been organized. It is the aim of the department to provide courses at all levels so that no student may feel rejected by the system. The amount of organization and cross- pollination is enormous but the end result is a logical development of the years in the main school.

I conclude with a brief note on the lecture system which we have developed from Year 3 onwards.

It soon became apparent that individual study can be inbuilt and non-reinforcing. Children forge ahead and comprehension can suffer. There is a lack of recall under any system; we attempt to do something about it with note folders and mixed set lectures. Anything from 60 to 90 children are installed in the lecture room for an hour long session of note taking and prob- lem solving. When three staff are involved in the system this works very well--three lectures per half-term unit, one lecture per teacher. On these occasions we employ the reactionary (?) procedure of a set homework for all ninety with parity of treatment and of presentation. The children thus build up a comprehensive set of notes for their particular course and it is hoped that the one lecture per half-term per teacher will be of high quality.

I have done no more than sketch the organization of this department; some of the procedures and details have not been mentioned at all. The advantages, we all agree, are the enormous involvement of the children, the accuracy of measurement, the confidence of all staff, and the general air of knowing what we are about, within our accepted restraints.

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