the kmp and ed. tech.: four years on

The KMP and Ed. Tech.: Four Years OnAuthor(s): Bertram BanksSource: Mathematics in School, Vol. 9, No. 2 (Mar., 1980), pp. 2-4Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213522 .

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TFhe ryaPsand ryaPsand

ed.tech.

Four yeats on

by Bertram Banks, Director of the Kent Mathematics Project

"The Kent Mathematics Project and Educational Technology" was the title of an article in the May 1975 issue of this journal. Much has happened since 1975 which should be interesting to many in the world of mathematical education.

The project is now being published by Ward Lock Educational and at this time (September, 1979), the first four levels of the mainstream material-bank are available to schools. The Schools Council contribution to KMP, known as the L project and consisting of material for very slow learners, is now in the hands of the publisher and its level 1 should be available well before Christmas 1979. Levels 2 and 3, which will complete this sub-project, are expected to be warehoused soon after Easter 1980, with mainstream levels 5, 6, 7 and 8 at approximate 6-monthly intervals thereafter. Incidentally, we call the Schools Council contribution the "L" project from "low level leavers", "limited language" or "Larcombe", the name of the material designer. We could hardly label the material from the already established F, M, S (fast, medium, slow) coding because it would have been S-minus, embarrassing to explain to an enquiring pupil. At least with "L" we could say "it stands for lovely".

After finalising the hierarchy of concept-building in the material-bank, we realised that very interesting implica- tions arose from the way ability was matched with chronological age throughout the scheme. For instance, we could see that a syllabus based on First year maths, Second year maths and so on, is doomed to produce failure for most of the pupils and is certainly unsuitable for the brighter children in any given group. Proper mathematics concept-building should be carefully structured and gaps in this structure are disastrous. Concept-building is also very personal"to each child and even to treat a class-group as a unit, let alone a year-group, ignores the fact that not only do children build concepts at different rates but that each child performs at a variable rate over periods of time. KMP therefore designed its scheme on a hierarchy of concept-building and organised personal courses within the structure.

We already had massive evidence of children being more comfortable because their concepts were developing logically (in their own personal world of learning maths they were always prepared carefully for each step) and work-rate and attitude to learning were significantly improved. Brighter pupils streaked ahead and all children were confident about their own stage of concept-development. Figure 1 shows how KMP levels relate to examination grades and Figure 2 shows how ability and chronological age match KMP levels and it must be understood that this Figure does not show a preconceived rationale but the result of about a 15-year evolution of a scheme in which the requirements of children have been given prior consideration. It is certainly not fair to design maths courses which guarantee failure for a percentage of pupils, and the new attitude from educational technology that if children fail it is the fault of the material (see 1975 article) sent KMP along its path of evolution, modifying material and system.

KMP Maths levels and examination grades

The approximate grade bands in Figure 1 are the result of seven years' experience of preparing candidates for 'O' level and CSE examinations. CSE grade 1 and 'O'-level grade C are

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roughly equivalent although syllabuses may be different. The examinations taken have been in modes 1, 2 and 3 and when correlations have been calculated between maths levels and exam results they have been significantly high whenever the examination has been properly designed and reliable. Level 9 is an overflow level for pupils who complete their KMP course well before the examination.



Fig. 1 5 4 3 2 1

i CSE grades i

KI(MP levels 1 2 3 4 5 6 7 8 9i

Primary andO '0' level grades Overflow early Secondary /

E D C B A

Ability and chronological ages KMP prefers not to classify children into ability categories because this encourages thinking in terms of groups of children which are then treated as units. Figure 2 therefore shows hypothetical individuals from examples of children who have worked on KMP. It does not show streamed groups.

The chronological ages are shown as numbers in the grid from 8 to 16 and are ages at the end of the academic year. Twelve (years) has been entered to show what level a typical child of the ability category would work in during the conventional First year of a Secondary school with transfer at 11+ years.

be working at a challenging level of concept-building which can be mastered and this should produce a contented child. KMP has overwhelming evidence to support this statement.

The development of KMP KMP is probably unique in the way its material and system have been developed. Concerned that KMP should, at the end of its development, be an effective learning tool for the teacher and pupil, we have allowed every aspect of the scheme and its management to evolve along a path of self-evaluation. Thus, taking the educational technology view that if something is not successful it is not the fault of the users but the designers, KMP was modified and repeatedly re-modified until it did work successfully. After about 15 years of evolution, one should not be surprised that KMP is a very effective tool. Classroom organisation, testing procedures, record-keeping, storage systems, flow diagrams and networks have all been evolved out of recognition of originals and the material has been subjected to the most rigorous of procedures for its improvement.

Fig. 2

KMP levels8 1 2 3 4 5 6 7 8 9 A YMPlevels Ability

(a) Exceptional 8 ....

1 2 116 (b) Above average 9 12 16 (c) Average 9

. 12 16

(d) Below average 10 -12 16 (e) Slow 11- 12 16 (f) Very slow 12 1 6

The ability categories are difficult to name because "average" has many different meanings. The average Primary school child does not seem the same as the average Secondary pupil. An average Grammar school pupil is certainly different from an average Secondary Modern pupil.

Category (a) is undoubtedly exceptional although KMP has known better. This category means that a pupil has reached 'O'-level grade A standard and is tackling 'A'-level work by 16 years of age.

(b) Is for the pupil who obtains a grade A at 'O' level at the end of the fifth Secondary year.

(c) Is the kind of pupil who will manage at 16 years the old fashioned 'O'-level "scrape", that is, a nowadays 'O'-level grade C or CSE grade 1. This is not to be confused with the CSE average related to the normal distribution curve.

(d) Is much nearer the normal distribution curve "average". (e) Is a CSE grade 5 type. (f) Is the subject of the L project and will have language as

well as mathematics difficulties. See what I mean about categories being difficult to name?

"Average" is a useful term when we say that KMP level 1 material deals with concepts usually learnt by average 9 year- olds. Average Secondary children become complicated and CSE definitions using the normal distribution curve can be confusing, especially since Boards say that schools do not play the game because they will enter pupils below- the fourtieth percentile.

Now Figure 2, which is the result of many years of investi- gation, shows vividly the faults of, for instance, a First year Secondary school syllabus. We know that two causes of mis- behaviour, laziness, discontent and disaffection are failure and boredom. Yet in the First year of a Comprehensive school we can have an ability range for which concepts from KMP levels 1 to 5 are appropriate. When one considers that one KMP level is roughly equivalent to one year's concept development for an above average pupil, one can see the enormous range of mathematics concepts which should be in a syllabus for this year group. Inevitably, with a blanket syllabus, only a small percentage of pupils can be catered for properly, the vast majority either failing because the work is above them or be- coming bored because it is beneath. Ideally, each pupil should

Specifically, the material has been developed through two main procedures. Originally, a band of teachers from 10 experimental schools designed worksheets, booklets and tapes and through teachers' meetings the material was approved, printed up and made available to all the experimental schools. Later, the material was ,discussed, level by level, modified according to general criticism and re-issued to schools.

By 1972 it seemed that the teachers had written up all their favourite lessons and the source started to dry up. Two teachers released from schools for one day a week were helpful but it was clear that the massive material-bank would not be completed in a reasonable time, so two full-time material-writers were appointed to complete the job. From then on, a different procedure was followed in that material was designed, tried out with the material writer sitting next to a suitable pupil, and reactions noted. If necessary, the material was modified and tried on another pupil and when satisfied that it seemed to be working successfully, try-out copies were sent to six of the experimental schools with a request for reports from both teachers and pupils. Collation of these reports usually meant another modification and the material was then sent to the Kent printing department and issued to all KMP experimental schools. About half of the KMP mainstream material was developed through this procedure, and all the L material.

The development of all material culminated in a series of validation exercises. Conferences of experienced KMP teachers were asked to assess the success of each task using a criterion of 80/80, that is, in the opinion of each teacher, did a parti- cular task attain 80% success with 80% of the children? Of course, there were differences of opinion with some of the tasks so another parameter was introduced - ifa task satisfied the 80/80 criterion with 80% ofthe teachers, it was accepted into the scheme and would be published. Some tasks were rejected on the grounds that they were not really necessary and some which were important in concept development but did not satisfy the criterion were modified to bring them up to standard and tried out again.

A measure of the quality of most of the published material is therefore that it satisfies an 80/80180 criterion, that is, that 80% of the teachers have found it 80% successful with 80% of the pupils. Since the validation involved a large'number of teachers (70+ at one of the conferences), some having used

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KMP on their pupils for up to 12 years and involving hundreds of children, the validation exercises were a powerful ingredient in quality-control.

KMP and the teachers It was realised at the beginning that a scheme which gave the teacher overwhelming administrative work would not be acceptable. Fortunately, the style of material-design in KMP relies on pupils having answers easily available to them, so the teacher is immediately relieved of an immense load of marking. Tests, however, have to be marked by the teacher, who is also responsible for the selection of material for each personalised course. Test-marking and course-designing, however, only crop up when a pupil completes a module and this occurs, for most children, between 3 and 4 times a year. It turns out that the administrative work behind KMP takes no more time than for a conventional textbook and lesson scheme in which a teacher marks children's work properly and has to prepare the odd lesson.

What the teacher gets with KMP is a common-core course of material for each pupil in the class. This will provide long- term goals for the children and establish a foundation of concept- building from which the teacher can operate. Thus in any class of pupils the teacher can organise group sessions with any number of pupils, knowing exactly the level of concept-development and what can be learnt, and that the remainder of the children will get on with their work on their own modules. Such opportunities for all kinds of maths work often arising from the special interests of teachers and/or pupils are greatly appreciated by teachers who feel the need for such work. KMP says firmly that teachers should not give up teaching.

Teachers certainly get to know their pupils far better than in class teaching. The KMP system sends each child to the teacher regularly. Children are more willing to admit difficulty in a confidential discussion than in front of a class, and from marking tests and talking about a child's interests, strengths and weak- nesses, the teacher builds up detailed knowledge of the pupil. A real bonus can be earned by the nurture of a teacher/pupil relationship based on close understanding of each individual and with the teacher in the role of guide, helper, and encourager.

KMP and gifted children Gifted children can cause problems in Primary schools and their treatment is often disastrous in mathematics. Even if a pupil is identified as gifted, there is seldom a teacher who can cope with specialised mathematical requirements and frequently the child's ability is squashed and his or her mathematical ability atrophied by boredom.

Throughout the history of KMP there have been instances of Primary school gifted, children happily forging ahead at their own speed on KMP material and entering Secondary schools as high as KMP level 6 (see Diagram 2). One boy, at age 10 years, working in level 4, won a scholarship to Eton, coming second out of 81 boys in the mathematics test. KMP has records of a family of gifted children who are one by one entering Grammar school and are being promoted a whole year.

Secondary schools have reported similar cases of KMP providing courses for gifted children. One High school boy was transferred at 13+ to a Technical school (parent choice), and immediately placed in the 'A' level group for mathematics. He was working in KMP levels 8 and 9. Another boy in a Comprehensive school worked through all KMP material available at the time (1972/73), and was tutored for 'A' level at which he obtained the highest grade at age 15+ years.

It is to be noted that not only does KMP provide material for gifted children but the classroom system allows advance ahead of others without embarrassment or undesirable social effects. In any given KMP group, there will be a range of levels being used by the pupils. KMP has no evidence that interest in another pupil's level is more than of trivial concern to any of the children.

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KMP and examinations From the beginning of KMP, we knew that examination work would have to be included in KMP objectives. Initially, benefits such as improved attitude, work-rate and behaviour were observed but these were almost standard in the reports on any work-card scheme, even if it were teacher-produced. What also went with the reports was disappointment that older Secondary pupils did not show such a degree of interest because, as we all know, "kids go off in the third year - puberty, you know, and all that, and anyway, you can't prepare for examinations on worksheet schemes". Wrong on all counts, reports KMP. We entered candidates for CSE and 'O' level in 1972, and results have improved ever since.

Modes 1, 2 and 3 examinations have all been tried. In a mode 1, the teacher compares the examination syllabus with KMP material and reserves so many periods a week for teaching topics not in KMP and, of course, examinology. A mode 2 with the London Universities 'O'-level Board has been running since 1977, and mode 3 exams have been taken by KMP schools since 1972. All pass grades for 'O' level and CSE have been obtained, sometimes with surprisingly good results compared with other subjects in the same school, and the theory that children build stronger mathematical concepts on personalised courses is certainly vindicated.

KMP style The educational technology behind KMP has ensured that its published packages are not only the products of learning theories but that the theories are sound. "Discovery", "induc- tive", "deductive", "investigational", "modular", "problem- solving" and so on learning theories all kick around the mathematical education world and often determine the style and content of maths schemes. Learning theories behind most textbooks seem to be conspicuous by their absence. KMP, however, has tried all identifiable theories and is a compound of all those that work, which seems more sensible than putting all one's eggs into one theory basket.

KMP cost The cost of KMP per pupil depends on the organisation of the school. One KMP Secondary school of 660 pupils uses only three mainstream workshops and one L workshop and the ratio of pupils per workshop is nearly 200 to 1. Work out the cost per pupil and it will come to about half the cost for textbooks covering the same mathematical topics suitable for the same range of ability. The total cost per pupil is also depressed by KMP only needing six of everything instead of 30 + for a class. Educational technology is cost-effective as well as being problem-solving in mathematical education.

KMP Enquiries Any enquiry about KMP can be addressed to the KMP office, West Kent Teachers' Centre, Deacon court, Culverden Park Road, Tunbridge Wells, Kent, for the attention of Mr B. Banks, Mrs A. Tourret (KMP Secondary Adviser) or Mr A. Larcombe (KMP Primary and L Project Adviser).

Postscript The first widespread publicity for the Kent Mathematics Project was for the L project and appeared in Schools Council literature as "Kent Mathematics Project: materials for very slow learning pupils". Such publicity reads as though KMP is a project only for very slow learners and it is hoped that this article corrects any misunderstandings about the nature of KMP. As one can see, the L project is only a contribution to KMP, generous, and for which we are grateful, because Secondary schools (and the later years of Middle Schools) can have all their pupils working on KMP. One admirable aspect has been the number of pupils in KMP schools who have gained confidence on L material and been transferred to mainstream KMP material.



the kmp and ed. tech.: four years on

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