Assignment Report FOR INTERNAL USE RP/1979-80/1/4.4/01 NOT FOR GENERAL DISTRIBUTION Development of national infrastructure and strategies for science and technology education

ST.LUCIA, GRENADA, JAMAICA

MATHEMATICS TEACHING by Everard Barrett

The views expressed in this report are those of the author and not necessarily those of Unesco.

United Nations Educational, Scientific and Cultural Organization

Paris, March, 1981

Serial No. FMR/ED/STE/8I/IO8

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TABUS OF CONTENTS

PREFACE

GENERAL DESCRIPTION

THE EXISTING MATHEMATICS PROGRAMMES

A. i The St.Lucian Mathematics Programme

Curriculum Pedagogy Teacher Training External Examinations

B. The Grenadian Mathematics Programme

Curriculum Pedagogy Teacher Training External Examinations

C. The Jamaican Mathematics Programme

Curriculum Pedagogy Teacher Training External Examinations

EVALUATION

RECOMMENDATIONS

APPENDIX - Diary of the mission

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PREFACE

At the request of the Governments of Grenada, Jamaica and St. Lucia, the Director-General of Unesco arranged, under Unesco's Regular Programme for 1979-1980, for a consultant mission to be carried out from 14 to 29 July 1980 with the following terms of reference:

"To study and report on the existing mathematics programmes of the three countries and to recommend the steps which might be taken to improve these programmes."

The territories visited are three newly independent island-nations in the Caribbean.

Jamaica: an island of 4,400 square miles, has a population of 2.1 million people and became independent in 1962.

Grenada; an island of 11j5 square miles, has a population of approximately 100,000 people and became independent in 1974-.

St. Lucia; an island of 25^ square miles, has a population of 120,000 people and became independent in 1979»

All the countries are English-speaking and have an educational system based on the British Colonial pattern.

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GENERAL DESCRIPTION

1. This mission was undertaken 14-29 July 1980 at a time of year when schools were either closed or engrossed in the routines of closing for the summer.

2. I visited the following territories according to the following schedule:

July 14 - 18 St. Lucia 18 - 23 Grenada 23 - 29 Jamaica

3. During the tour I held meetings with representatives of the Ministries of Education, various representatives of high schools, primary schools, and other members of the educational system. In Jamaica I met the Unesco representative for the Caribbean, Mr. Hugh Cholmondely and the U.N. Resident Representative for Guyana and the Southern Caribbean, Mr. Trevor Gordon-Somers.

4. Unfortunately, the tour was made during the annual summer holidays when it was not possible to attend class sessions or perform analyses of student work books and texts. However, I had access to sufficient information to allow a fairly thorough assessment of the mathematics programmes in the three countries.

5. In all these countries, the school system is essentially operated by the governments and administered through the Ministry of Education.

6. Students typically enter the system at age six and study basic arithmetic with a syllabus geared to the Common Entrance Examination which permits entry to the High School System at age 11 or 12. At this point they are expected to be competent in:

(a) the computational arithmetic of whole numbers, fractions, decimals and per cents;

(b) basic geometry and geometric problems (perimeter and area of ̂simple plane figures);

(c) arithmetical problem solving.

7. Beyond primary, the highest priority in each territory has been preparation for the General Certificate of Education Ordinary Level Examination (G.C.E. n0" Level) developed and administered in Britain by universities such as London and Cambridge, to be taken by high school students throughout the British Commonwealth which embraces over 30 nations. Passes in five subjects at the ordinary level are regarded as proof of a satisfactory standard on graduation from high school. Under the British system of colonial education, students may attend high school for another two years and sit the G.C.E. advanced level.

8. Up to 1979 there were three syllabuses offered in Ordinary Level Mathematics: Syllabus A, Syllabus B and Syllabus C. Since 1979, Syllabus A has been dropped. A and B offer a traditional secondary school syllabus while C offers additional modern topics.

9. A school can opt for Syllabus B or Syllabus C. The choice of syllabus determines the areas of mathematical competence a student is expected to have by form 5*

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10. The best mathematics students can take a G.C.E. additional mathematics exam along with the B or C option. Relatively few students who pass Ordinary Level Mathematics go on to take Advanced Level Mathematics two years later in form 6B.

11. The G.C.E. Ordinary Level, a most important milestone in the life of every Caribbean child, has a great effect on the high school curriculum and on education in general. However, "The need for change in these examinations has long been recognized throughout the Caribbean and the Caribbean Examinations Council has been established to devise and administer, for the Caribbean as a whole, a more appropriate system of assessment to replace the G.C.E.".(Five-Year Education Plan, 1978-I983, Ministry of Education - Jamaica.)

12. The Caribbean Examinations Council's (CXC) mathematics examination is comparable to the G. CE "0"Level Syllabus C in both depth and breadth, and has been offered as an alternative to G.C.E. since 1978.Although most secondary schools still take "'0' Levels" there appears to be a strong commitment toward a complete phasing in of CXC.

l^. The education system in the region has been the subject of two interesting reports to which I had access and which helped greatly in my assessment of mathe­matics teaching, its operation and results. These reports are:

(i) "Report on Educational Appraisal Mission to Grenada and St.Lucia", September - October, 1979 by Shirley Gordon, World Hank Consultant.

(ii) "Change in Mathematics Education since the late 1950s - Ideas and Realization - West Indies", B.J. Wilson, Educational Studies in Mathematics 9, (1978) 355-379, Dordrecht, Reidel Publishing Co.

THE EXISTING MATHEMATICS PROGRAMMES

14. This report on the existing mathematics programmes in St.Lucia, Grenada and Jamaica will discuss the following:

I. Curriculum II. Pedagogy III. Teacher Training IV. External Examinations

A. The St.Lucian mathematics programme

I. Curriculum

15. On the primary level, St.Lucia has developed its own "St.Lucian Mathematics Project" under the direction of Dr. Desmond Broomes, a mathematician on the staff of the School of Education of the University of the West Indies. I was informed by Mr. Delmede, Chief Education Officer, that experimental materials have been written and tested by teachers over the past 10 or more years while new teachers have been trained to use the series which goes as far as Standard 4. It is largely based on standard British approaches and materials. The St.Lucian materials have been published (Caribbean Primary Mathematics) and are available to other West Indian countries.

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16. Dr. Robert Cluff, Unesco adviser in measurements and evaluation, informed me that there was no common curriculum in schools leading to the Common Entrance Examination as recently as 1978. During that year, he arranged a meeting of teachers to set curriculum objectives and from this a detailed outline for the Common Entrance Examination was developed. Each school received a copy.

17. Any reference to curriculum development in St,Lucia and Grenada is incomplete without special mention of the Caribbean Mathematics Project (CMP) during the 1970s which made a great impact on mathematics curriculum development in the Junior Secondary Schools of the Eastern Caribbean. The strategy of CMP was such that it involved the teachers themselves in the developmental process. "Thus the project attempted to take the weakest feature of the educational scene - the weakness of the teachers - and turn 1% into the strongest by proving to teachers that they had within themselves the power to develop skills of constructing diag­nostic tests, planning lessons, writing materials and organizing syllabuses which could respond to the rapidly changing circumstances of the world of the 1970s and beyond.

18. Developing the skills and confidence of the teachers was the primary aim of the project; the development of teaching materials was an important by-product. Essentially, curriculum development was used as a vehicle for a continuous pro­gramme of in-service training.

19. Wilson (p. ̂ 61) states, "The project had no specific intention to introduce new mathematical topics into schools". The course content is therefore similar to the curriculum traditionally developed in the British Colonial System, with its emphasis on rote learning of computation and problem solving. While this leads to the ability of the best students (a very small group) to solve fairly sophisti­cated problems at ages 10 to 12, it is based on mechanical repetition rather than on understanding of the problem solving or computational processes.

20. At an early stage of CMP, the Form One volume of a series of texts originating in Ghana, West Africa - the Joint Schools Project (JSP) - was introduced to pilot classes for use while its (CMPs) own materials were being developed. JSP was to have been used as the foundation from which local curriculum development efforts were to have sprung. Unfortunately, there was an eventual reversion to textbook dependence (many schools adopted JSP) as the time for external examinations approached.

21. Beyond Standard 4 and entry to the secondary school system, the Ministry of Education currently provides no mathematics curriculum (or syllabus) to its schools. Each school is essentially on its own and, in its own way, through the influences of CMP and JSP or a choice of standard British textbooks, tailors students' classroom experiences toward the external examinations in forms 5, 5 and 6. However, a secondary programme is now in the process of development with CXC (Caribbean Examinations Council) materials being used as a guide.

22. I will analyse the primary system by viewing it in the four specific areas outlined as a result of Dr. Cluff s meetings with the primary teachers:

(a) Number concepts; (b) Arithmetic skills; (c) Geometric problems; (d) Problem solving.

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23* Among the number concepts are the following:

1. Numeration; 2. Number sequences; 3. Comparison of fractions; 4-. Equivalence; 5. Factorization; 6. Multiples; 7. Exponential representation; 8. Conversion among fractions, decimals and percentages; 9. Conversion of improper fractions to mixed numbers;

10. Other traditional areas of primary concepts.

24. In the area of arithmetic skills the curriculum stresses computation almost exclusively based on rote memorization and on traditional rote teaching of the following skills: operations on whole numbers, fractions and decimals.

25. Computation of perimeters and areas of simple plane surfaces - parallelograms, triangles and circles are topics under "geometric problems".

26. The consensus among teachers was that because of the students' poor reading ability, the teaching of problem solving was extremely problematic. Generally, students are expected to develop competence in the use of computational skills for solving verbal problems which may require from one to several operations. Many of the problems are in the areas of consumer arithmetic and weights and measures.

27» On the secondary level, detailed syllabuses of the various G.C.E. Ordinary and Advanced levels options for form 5 have been developed. These syllabuses are the same options from which students in the same age cohort throughout the British Commonwealth must choose.

II. Pedagogy

28. In 1970, the region, including St.Lucia, undertook the Caribbean Mathematics Project (CMP). As stated before, CMP stressed teacher training, but in teaching methods it was the means for introducing new ideas.

29. The basic methods of group-class exposition with a blackboard, and verbal group repetition of examples continue to be used in the CMP system, but they have introduced locally produced classroom materials such as pebbles, bottle-tops and matchboxes, etc. In addition, teachers have been encouraged to have children work individually or in small groups - developments in a very positive direction which, given the proper tools, could provide a method for imparting greater com­petence in mathematics education.

30. Wilson states (p. 373) that the project discouraged teachers from utilizing a predetermined "syllabus of work and by concentrating its efforts on the produc­tion of self-contained teaching units, the project left able teachers free to develop their own approach to the curriculum".

31. Unfortunately, this system continues to be based on a high proportion of untrained teachers, and could lead to tremendous problems in those cases where teachers lack the necessary competence in syllabus preparation. The problem is further compounded by lack of a priority on changing the traditional curriculum with its emphasis on rote learning.

32. The timing of the mission made it impossible to determine if a random sampl­ing of teachers followed any suggested pedagogical strategies. However, informa­tion on teachers' pedagogical concerns was obtained upon my visit to the Anglican

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primary school in Castries. The principal, Mrs. King, stated that some of her teachers were unqualified and that every Friday they meet for lesson planning. She was concerned, however, that qualified new teachers were not willing to be guided by those with experience.

33» In the course of à meeting which Mrs. King arranged for the purpose of a discussion among 15 teachers and myself, they unanimously agreed that all children can learn mathematics well but they favoured separation of fast from slow learners with regular contact between them for mutually profitable exchange of experience.

34. At both visits with Mr. Delmede to the Anglican and San Souci schools, teachers expressed the need for a greater quantity and variety of concrete materials.

35» Mrs. Muriel Gill, director of Centre for Curriculum and Materials Development, has not yet launched an effort in the area of mathematics. At this time, the centre is developing materials for language arts. Mrs. Gill is critical of the spiral approach and thinks that the mathematics curriculum needs to be developed along "non-spiral" lines. Her realization that teachers were doing too much of the "talking and chalking" themselves (my quote) and that children were merely copying from the board, was a matter of grave concern to her, as well as the fact that teachers' enthusiasm and the level of student comprehension of concepts were generally low. She is convinced that "the teacher is the key".

H I . Teacher Training

36. While in Grenada, I had access to a report entitled "The Caribbean Mathematics Project: training the teacher as an agent of reform - by H. Martyn Cundy -Experiments and innovations in education, No. 32 - An International Bureau of Education Series - Unesco, Paris 1977"• I quote from the text: "Many teachers in the all-age primary schools had little or no pedagogical training; they had emerged from the upper classes at one end of the school to re-enter at the other end of the building as teachers of infants. The seven teachers, colleges in the islands, which now have an annual output of about 400 teachers, are working valiantly to remedy this situation, but the problem remains. In the specific field of mathematics in junior secondary schools, out of 60 teachers in 13 schools in 1975, 16 were untrained beyond '0' level mathematics (which probably most, though not all, would have passed); two had taken 'A' level mathematics and failed, three had passed 'A' level; 25 had been through a teachers' college; 4 had been to the University of the West Indies but had withdrawn without completing the course; three were university graduates, and the remaining seven were expatriate VSO or Peace Corps volunteers. Less than half, therefore, had had any professional training, and about one-third lacked the qualifications in content which would normally be expected of a secondary school teacher. In 1971 the situation would undoubtedly be much worse than this. In the junior secondary schools in particular most teaching of mathematics and science was in the hands of expatriate volunteers from the Peace Corps, the British Scheme for Voluntary Service Overseas, or the Canadian University Service Overseas".

37» The teaching force of the school system is summarized below:

Table I

Primary and all age Junior Secondary Senior Secondary

Total

Trained

331 62 42

435

Untrained

532 30 39

601

Total

863 92 81

1036

/. Source: Gordon, Shirley "Report on Educational Appraisal Mission

to Grenada and St. Lucia" World Bank 1979. Unpublished.

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38. As this table shows, of a total 1,036, only 4-35 or 42$ were trained. This reflects a tremendous need for teacher training. The Ministry of Education has conducted an in-service programme for up-grading mathematical competence of recruited teachers, most of whom were only graduates from the all-age primary system. Some of these teachers eventually go on to the two-year Teachers' College.

39. This programme has done very poorly over the past two years as measured by teachers' performance on the final examination. Gordon found (p. 30) that in 1978 only 18$ passed. In 1979, after Kj% were debarred from the final examina­tion, 34$ of those remaining then passed. As recently as October 1979, the in-service programme was suspended because of the poor results. An effective alternative for in-service teacher education is urgently necessary.

40. In the case of the Junior Secondary Schools the situation was slightly better as BT?° were trained, but in the case of the Senior Secondary Schools, only 52$ were trained.

41. Dr. Cluff reports statistically significant improvement in teacher performance over the last two years, based solely on teachers' own initiative.

42. Two important developments have led to this improvement:

1. As a result of the detailed curriculum outline whieh was developed by the teachers of St.Lucia under the guidance of Dr. Cluff, they (teachers) were able to see gaps in their knowledge of material children were to be taught.

2. Results on the previous year's examination were returned to every teacher individually. He or she could then privately compare his/her score in each sub-topic with the national mean and thus identify his weak "teaching-areas".

By seeing the gaps in their own knowledge, teachers were motivated to study on their own initiative. They were also, motivated to improve their teaching perfor­mance in those sub-topics for which they scored below the national mean.

43* It will be interesting to see if, in future years, the national mean rises in each sub-topic as teachers work harder to stay above it:

44. Prom discussions with an official of the Grenada Teachers' College, it is clear that their effort is totally concerned with getting students to pass the examinations which bring certification. There is almost no involvement in the schools.

45. A mathematics teacher on the secondary level acquires certified status by one of the following means:

(a) B.A. or B.Sc. in mathematics; (b) General B.Sc. in which mathematics was a major concentration; (c) Certification at "A" level plus graduation from Teachers' College

with mathematics as a specialty.

46. There is a high percentage of pre-trained personnel who were certified at "A" level mathematics and went directly into teaching.

47. Almost all education officials and teachers with whom I spoke reported wide­spread fear and dislike of mathematics among students and teachers alike.

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IT. External Examinations

48. The major thrust of the mathematics education effort is toward the external examinations: Common Entrance in standard 4, General Certificate of Education in form 5 and G.C.E. "A" level in form 6.

49. Dr. Cluff, whose meetings with teachers (in 1978) resulted in the develop­ment of curriculum outlines in all major subject areas of primary education, had developed examination items of mathematics in the areas of number concepts, arithmetical skills, geometric skills and problem solving. His measurements of the performance of St. Lucian children (in the standard 4 cohort) indicate that "in every category the mean St. Lucia level is approximately 30# below the expected mean level". (Quoting Dr. Cluff.)

50. The expected levels are the levels as determined by the textbooks and curri­culum guides for standard 4 which are quite close to international levels for the standard 4 age cohort.

51. As far as performance on Common Entrance Examination mathematics is concerned, records in the Overseas Examinations Office, Kingston, Jamaica, show the following results:

Table II

General Certificate of Education, 1978 Eastern Caribbean Area Table of Percentage Results

(School and Private Candidates)

0 Level

Mathematics Mathematics Mathematics

Syllabus A Syllabus B Syllabus C

Note: All three examinations academic level.

Number of candidates

526 525 820

are different, but of

% grade C and

the same

39.2 33.0 22.7

above

Source: Overseas Examinations Office, 2A Picadilly Road, Kingston, Jamaica, W.l.

52. Although this table shows results for the Eastern Caribbean area as a whole, it gives some indication of levels of performance in St. Lucia. Indeed, I was told there was no marked difference in the results between the territories. If we .contrast Caribbean results with those obtained in Britain by those students sitting the G.C.E. of the University of Cambridge during the same year, we note the following:

Table III

General Certificate of Education, I978 Ordinary Level - Home Centres

(The number of candidates examined and percentages gaining grade C and above are shown. Private candidates are not included.)

Number of candidates Percent passed

Mathematics Syllabus B Mathematics Syllabus C Mathematics (SMP) Additional Mathematics

9,886 13,628 9,584 844

59.9 67-6 62.1 70.4

Source: Overseas Examinations Office, 2A Picadilly Road, Kingston, Jamaica, W.l.

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53» The contrast in performance ás indicated by the two tables is startling as the British students performed better than twice as well as in St. Lucia. Further­more, based on my conversations with St.Lucian education officials, it seems that less than 40$ of students who enter form one go on to sit the G.C.E. "0" Level mathematics examination.

54. Dr. Cluff informed me that of 3,000+ students taking Common Entrance Examina­tions in I98O, only 1,200 (less than 40$) gained entry into secondary schools. Since less than 40$ of students who enter form one eventually take the G.C.E "0" Level mathematics examination, it follows that less than 16$ of all persons in the fifth form age cohort become candidates on the G.C.E. n0" Level mathematics examination. A glance at Table II shows that approximately 30$ of candidates pass either G.C.E. Syllabus A, B or C in mathematics. This means that less than 4.8$ of persons in the fifth form age cohort throughout St.Lucia pass G.C.E. "0" Level mathematics.

55» In the school year 1978-1979, a summary of the St.Lucian school system (Gordon p. 57) shows a total 1,328 students in Senior Secondary School and 123 students on "A" level. There is no indication of how many of the 123 students took "A" level mathematics. Since about 9$ of the number of students in senior secondary are on "A" level, and less than 4.8$ of persons in the fifth form age cohort pass "0" Level mathematics, we might approximate that less than .43$ of all persons at the "A" level age cohort in St.Lucia become candidates in "A" level mathematics.

56. Finally, if we assume the same rate of passing on "A" levels as on "0" level (30$, we estimate that less than .13$ of all persons at "A" level age cohort in St.Lucia pass "A" level mathematics.

57» Considering the tremendous emphasis by schools on the external examinations (especially G.C.E.), these approximations present a dismal picture.

58. The Ministry has recently adopted a proposal whereby examinations in form 3 of Junior Secondary and standard 7 of the All-Age Primary will permit the most successful students to enter the secondary school system in form 4. This bodes well for stimulating a well needed sense of purpose and commitment in previously uninspired teachers. It should also generate new impetus in programme organization and curriculum development, as well as bringing new hope and motivation to the hearts of many despairing St.Lucians.

B. The Grenadian mathematics programme

I. Curriculum

59* Unlike St.Lucia, Grenada has not developed its own primary mathematics project.

60. Mr. Victor Ashby, assistant to the Permanent Secretary in charge of tests, measurements and supervision of the mathematics programme informed me that all primary teachers received a publication issued by the Ministry of Education, Culture and Social Affairs entitled "Mathematics Curriculum for Primary Schools in Grenada". (Appendix 4) For the purpose of fulfilling curriculum objectives, teachers use a variety of commercially available materials among which are the St.Lucian Mathematics Project work books and teachers' guides.

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61. The introduction to the Grenadian Mathematics Curriculum Guide begins with a discussion of "New Mathematics in the Primary School". It stresses the impor­tance of students' exposure, before they reach the age of 11, to proper mathe­matical experiences and the encouragement of an appropriate response to those experiences, these factors being necessary for the development of the students' mathematical potential.

62. Some special features of the new (sometimes referred to as "modern") approach to mathematics appearing in the curriculum guide are:

1. "It provides opportunities for pupils to acquire ideas through their own investigations."

2. "It emphasizes the importance of pupils discovering mathematics rather than being told mathematics."

3. "It is mathematics taught in a practical way using concrete situations and materials."

4. "It permits greater understanding of mathematical concepts as well as greater appreciation of the structure, relationships and patterns of numbers and shape."

5. "It affords a better balance between computation and problem solving -the former being mainly a means toward the latter."

6. "It provides a uniform approach to the subject; (modern mathematics is not just about number and shapes. It occurs whenever the mind classi­fies and creates structures. The range of experience is tremendously increased, more of the pupils' experiences are mathematically relevant; it is easier for the teacher to create occasions for the pupils to use mathematics and find situations which release mathematical thinking)."

7. "Great emphasis is now placed on developing an understanding of the symbols and terms specific to mathematics."

8. "The objectives for teaching and learning mathematics are more clearly and specifically stated and are more closely related to the goals of education and the findings of the psychology of learning."

6^. The curriculum guide goes on to present "objectives for teaching and learning mathematics in the primary school".

"The primary pupil:

1. should be able to do basic computations with whole numbers, integers, fractions and decimals;

2. should understand and be able to analyse some geometrical aspects of his physical world;

3. should have a knowledge of basic mathematical processes, facts and concepts;

4. should acquire a vocabulary which will enable him to understand and communicate mathematical ideas;

5- should recognize the role of mathematics in the development of society;

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6. should appreciate the effectiveness and precision of mathematical thought;

7» should develop an ability to solve problems;

8. should cultivate an appreication and a liking for mathematics -its pattern and relationships;

9. should acquire study habits and mathematical skills which will motivate and facilitate his future work in the subject at the secondary level."

64. The introduction further discusses under "Teaching new mathematics in the primary schools":

(a) The nature of the child; (b) The nature of mathematics.

65 • Under "Learning new mathematics in the primary school" the introduction discusses:

(a) Opportunities which the learning of mathematics provides for the child.

. (b) Motivation and readiness factors.

66. Under "The Content of Primary School Mathematics" appears the following quotation:

"In the modern primary school a definite effort is made to emphasize the structure of mathematics, the relationships that integrate the work of dealing with the various number operations, and the development of power in quantitative thinking and in utilizing mathematical procedures in the affairs of daily life and in curriculum areas. Emphasis is placed on problem solving."

This section concludes with,"The content of the primary school mathematics pro­gramme is mainly built around the following areas:

1. Numberals and numeration systems.

2. Principles underlying number operations.

3. Relationships and generalizations.

4. Measurement.

5. Approximation.

6. Estimation.

7. Symbolism.

8. Proof.

9. Shape and space.

10. Statistics."

67« Syllabus details for each term of the school year (three terms in each) are provided in the remaining pages of the curriculum guide.

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68. The extent to which the ideas in "Mathematics Curriculum for Primary Schools in Grenada" are actually practiced in the classrooms of Grenada could have been determined had I been able to visit pupils and teachers at work in mathematics classes, unfortunately, the timing of the mission made this impossible.

69* Indications that the curriculum as outlined may not be fulfilled for the average learner are as follows:

(a) The "Year 3" syllabus has pupils "developing addition tables of components for numbers 2 to 20" and "addition of one and two place numbers with carrying". In two different meetings arranged by Mr. Ashby, each attended by teachers from various schools (approximately 10 teachers peij* meeting), it was agreed that addition facts (sum greater than 10 and less than 20) were introduced in Year 3» However, they also agreed that the average student learns these facts "by heart" in Year Five (standard 3), This would seem to create a problem in following the curriculum as out­lined since "Number Work" in Year 4 requires "multiplication and division of 2 and 3 digit numerals by single digit numerals with and without carrying" and "column addition with 1, 2 and 3 digit numerals with and without carrying".

(b) During the meetings with teachers from various schools, it was generally agreed that whereas the average learner was introduced to multiplication in standard one, the "tables" were learned by heart in standard 5 (Year 7)« A few teachers did exclaim that the average learner "never! " learns the multiplication tables by heart. In Term 2 of Year 6, however, the curriculum guide requires "mastery of multiplication tables 1-12".

( c ) The teachers generally agreed that addition of fractions with unlike denominators is mastered by the average learner in standard 6 (Year 8). However, "adding fractions with unrelated denominator" is introduced in standard 3 (Year 5)«

70. The time between introduction of a topic and its mastery seems very long indeed.

71. All references to CMP influences in St.Lucia apply to Grenada as well.

72. Beyond standard 5 (standard 5 in Grenada and standard 4 in St.Lucia belong to the same age cohort) and entry to the secondary school system, the Ministry of Education currently provides no mathematics curriculum (or syllabus) to its schools. As in St.Lucia, each school, in its own way, and through the influences of CMP, JSP, a choice of standard British text book and the syllabuses in the various G.C.E. "0" level options, tailors students' classroom experiences toward the external examinations in forms 3, 5 and 6. A secondary programme is now in the process of development with CXC materials being used as a guide.

II. Pedagogy

73. The first four paragraphs in the St. Lucia section on pedagogy apply equally to Grenada.

74. Almost every teacher and principal I spoke to expressed the need for more in-service teacher training. A system so hampered by large numbers of untrained teachers (see section on teacher training) needs a heavy infusion of in-service training.

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75» During my meetings with the two groups of teachers mentioned previously, I learned of the large gap of time between introduction to and mastery of the following :

(a) Addition facts - sum greater than 10 and less than 20.

(b) Multiplication facts up to "twelve times".

(c) Addition of fractions with unlike denominators.

Accordingly, for the purpose of offering solutions to the above problems and also to assess teachers' response to training, I presented some pedagogical strategies.

76. With respect to addition facts, the teachers said it would normally take two years for average pupils to learn them by heart. They estimated unanimously that this task could be accomplished in two to six weeks by means of my strategies.

77. With respect to multiplication facts, the teachers said it would normally take four years (if ever) for average pupils to learn them by heart. They estimated that this task could be accomplished in approximately four months by means of my strategies.

78. With respect to the addition of fractions with unlike denominators, the teachers said it would normally take three years for average pupils to attain mastery. They estimated that this could be accomplished in approximately six months by means of my strategies.

79. A member of one of the groups - a principal - said, "It would take more time to teach addition of fractions by your method, but once it is learned, it would not have to be taught over and over again". This and other comments regarding meaningfulness and usefulness of the strategies presented, indicated a positive attitude toward workshops on pedagogy.

80. In the Grenadian curriculum guide, some pedagogical strategies are offered:

(a) Pupils should learn through their own investigations.

(b) Pupils should discover mathematics.

(c) Pupils should be taught in a practical way by using concrete situa­tions and materials.

81. Due to lack of teacher training, the realization of these goals, with the possible exception of the third (the result of CMP influence), rarely exists in the island's schools. Teachers at St. George's Seventh Day Adventist and St. Dominic's Schools expressed concern about the shortage of concrete materials. Class sizes ranging from 36 to 52 were mentioned as a hindrance to individualized instruction by the St. Dominic's teachers. I have little doubt that such large class sizes are the rule rather than the exception.

III. Teacher training

82. Gordon (p. 11) found that 50.5^ of primary and 79«3^ of secondary teachers in Grenada were untrained in I978. Unfortunately, "There is not within the administra­tive system itself any drive for change or development". (Gordon, p. 12) In fact, teachers in St. George's Seventh Day Adventist School and the principal of St. Dominic's did confirm that there was no formal in-service training.

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83. The teachers of St. Dominic's reported that over one year ago, workshops in lesson plan development and demonstration lessons were conducted in their classrooms by students from the Teachers' College. They found the assistance to be very helpful. The student teachers were expected to return for observation of teachers' efforts with the lesson plans, but did not do so.

84. "The Teachers' College is accepting approximately 45 new students annually. The vast majority are in their mid-20' s and clearly well established in the non-intensive, verbal, collective responses of the elementary school tradition. Nor was there much evidence that the Teachers' College course was challenging these norms or offering effective alternatives in education." (Gordon p. 11).

85. Mr. McBarnett, Principal of Grenada Teachers' College, informed me that kQ$> of teachers working toward certification fear mathematics. (Mrs. Hagley, Principal of St. Peter's school, also informed me about some teachers' fear of mathematics.) He had great concern that there was no mathematics tutor at the college since 1974. However, Dr. Garth Baker, a Jamaican who teaches graduate mathematics at Harvard University, has been tutoring at Teachers' College this year. He reported that fear of mathematics was expressed to him by several students. Only 5 out of 30 passed his most recent test.

86. Dr. Baker informed me that in his attempts to teach students he was accused of "spoonfeeding" by the most competent members of his class.

87. Does this mean that if mathematics is learned easily there is something wrong with the pedagogy? Must the learners of mathematics necessarily experience difficulty? Do the student teachers think that most people are incapable of learn­ing mathematics? The charge of spoonfeeding leads me to doubt that the teachers see mathematics pedagogy as a flexible matter, the imaginative practice of which will bring mathematical competence to most learners. Teachers' apparent satisfac­tion with less than 50$ passing in their respective classes confirms my doubt of their pedagogical curiosity.

88. Grenadian teachers seemed no more capable of teaching for competence in problem ' solving than their counterparts in St.Lucia. Teachers in all four schools I visited (St. Peter's, St. Dominic's, St. George's Anglican Senior and St. George's Seventh Day Adventist), expressed concern over this area of the mathematics curri­culum.

89. All teachers in all four schools were confident that "on-the-spot" in-service workshops with classroom demonstrations would contribute greatly to their develop­ment. Speaking of his teachers, Mr. Arnold, Principal of St. Dominic's said, "They need a lot more training".

90. I received a copy of the Grenada Teachers' College "Mathematics Curriculum". In addition to general objectives, it outlined the following areas of mathematics in which primary teachers were expected to be competent: sets, numeration systems, number theory, algebraic expressions, graphs, business arithmetic, probability and statistics, measurements, planes and surfaces.

IV. External Examinations

91. As in St. Lucia, the major thrust in mathematics education is toward the Common Entrance Examination in standard 5, G.C.E. "0" level in form 5 and G. CE. "A" level in form 6.

92. I was told there was no marked difference in the results between the two territories in G.C.E. "0" and "A" levels. Consequently, the St. Lucia results will suffice to give some indication of performance on these examinations in Grenada.

/,.

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C. The Jamaican mathematics programme

I. Curriculum

9J5. By recommendation of the Jamaica Ministry of Education, all primary schools use the Ginn Series of workbooks and textbooks in grades 1-6 (Grade 6 is equivalent to standard 4 in St.Lucia and standard 5 in Grenada). This is an American series which offers teachers' manuals with workbooks in grades 1-2 and teachers' manuals with textbooks in grades 3-6. The Ministry has produced a "Mathematics Curriculum for Primary Schools" for the purpose of facilitating teachers' use of the Ginn Series: Teaching Mathematics We Need.

94. Under "Notes for Principals and Teachers" on page one appeared the following:

1. "The curriculum is arranged by grades, showing broadly in behavioral terms the scope of the material to be covered each term, and the corres­ponding reference pages in the Ginn Series: Teaching Mathematics We Need."

2. "The arrangement places the topics alphabetically for easy reference only and does not indicate the sequence in which they are to be taught."

j5. "It is recommended that the sequence developed in the text be followed."

95. Within the foreword of the booklet is the following statement: "In Jamaica, the Ministry of Education - Institute of Education Mathematics Project was started in September 1967 to meet the need to update the curriculum in our Primary Schools and lay the foundation for a modern mathematics curriculum in the Secondary Schools".

96. Topics by grade are as follows:

Grade 1 - Addition - subtraction, geometric ideas, inequalities, inverses, measurement, number(s), number line, numerals and numeration, properties, sets.

Grade 2 - Addition - subtraction, geometric ideas, inequalities, inverses, mathematical sentences, measurement, multiplication and division, number(s), number line, numerals and numeration, properties, sets.

Grade 3 - Addition — subtraction, geometric ideas, inequalities, inverses, mathematical sentences, measurement, multiplication - division, number(s), number line, numerals and numeration, properties, sets.

Grade 4 - Addition - subtraction, geometric ideas, graphical presentation, inequalities, inverses, mathematical sentences, measurement, multi­plication - division, number(s), number line, numerals and numeration, properties, sets.

Grade 5 - Addition - subtraction (whole numbers), geometric ideas, graphical presentation, inequalities, mathematical sentences, measurement, multiplication - division (whole numbers), number(s), numerals and numeration, problem solving, properties, rational numbers (frac­tions and decimals), sets.

Grade 6 - Geometric ideas, graphical presentation, mathematical sentences, measurement, number(s), numerals and numeration, problem solving, properties, ratio and per cent, rational numbers, sets.

97- The above is generally the international curriculum for the primary age cohort.

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98. The Ministry of Education sends to teachers a curriculum outline for grades 7-9 and advises that it be "considered as a proposed draft". Reference to pages within the series "A Natural Approach to Mathematics" appear through­out the curriculum outline. They co-ordinate curriculum topics to sections in the series. This is justified on the ground that teachers in the vast majority of schools offering grades 7-9 programmes have access to these texts. In fact, teachers are advised that "... one way in which the topics may be taught is to follow the order in which they are presented in 'A Natural Approach to Mathematics'n*

99» The following topics are offered in grade J: sets; number theory; relations, functions and graphs; geometric ideas; algebraic ideas; measurements; computa­tion; statistics; consumer arithmetic.

100. Topics offered in grade 8 are: sets, number theory, relations, functions and graphs; geometric ideas; transformation (motion) geometry; algebraic ideas; measurement; computation; probability and statistics; consumer arithmetic.

101. In grade 9 the following are offered: sets; number theory; relations, functions and graphs; geometry; reflection and line symmetry; translation; rotation; algebra; measurement; computation; probability and statistics; con­sumer arithmetic.

102. After each topic in the grade 7-9 curriculum guide there are brief comments which advise teachers concerning such matters as pupil motivation, rationale, articulation with previous work in the Ginn series, objectives, applications, related student activities, pedagogy and useful principles.

103. This curriculum guide needs more development and, toward that end, it urges teachers to complete certain evaluation forms and return them. When more fully developed, it should be very helpful to teachers.

104. The grades 10-11 mathematics syllabus is divided into three course levels: prefunctional, functional and continuing. The prefunctional course consists of seven units: whole numbers, fractions, decimals, percentages, simple equations, geometry, ratio and proportion. Students omit units which they have already mastered (as shown by scores on the placement test). The functional level course consists of 12 units: number patterns, elementary algebra, algebra, geometry, estimation, measurement I, measurement II, introduction to statistics, sampling and statistics, mapping and graphs, graphs, functions and graphs. The continuing level course con­sists of 11 units: enlargement and similarity, inequalities, estimation and measure­ment, linear relations, non-linear relations, calculating with logarithms, squares and square roots, symmetry, tangent, sine and cosine, relations and geometry.

105. The curriculum guide, grades 10-11, contains information concerning "strands" of topics which "flow" in a logical sequence. Beneath every listed topic is to be found an aim, content and material. An example follows:

Enlargement and similarity

Aim - skill in using enlargement to solve problems involving proportion.

Content - enlargement, similarity, proportion; arithmetical graphing and algebraic methods; ratio and percentage.

Material - chapters 1, 3 and 4- from Book 3 °f "Caribbean Maths" (JSP).

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106. Jamaica was not one of the group of territories to which CMP was originally introduced. I do not know of any CMP influence on curriculum development in this country. However, the JSP textbook series is one of the options being employed toward fulfilment of the grades 7-H syllabuses. Other options are British (mostly) and American in origin. The island participates in CXC developments and this organization seems destined to become the dominant influence in Jamaican curriculum development over the next 10 years.

II. Pedagogy

107» At the time of my visit to Jamaica, schools were closed for the summer and it was not possible to engage groups of teachers in pedagogical discussions. Prom lower level education officers I understood that because of the great thrust toward the Common Entrance Examination in grade 6, primary teachers tended to sacrifice developmental aspects of the Ginn series.

108. The Ginn series was introduced in 197^ under Ministry of Education sponsorship to facilitate a move away from traditional rote-mechanical practice. To that extent it can be helpful in Jamaica since it is highly developmental and contains much in the way of "explanations". But an infusion of textbooks, however excellent in exposition, can only realize effectiveness when teachers are prepared for their use by sufficient and intensive workshops.

109. The introduction to the draft assumes that students are entering grade 7 with a fair degree of mastery of the primary school mathematics curriculum for grades 1-6. This assumption may be somewhat optimistic in view of my conversation with the headmaster of one of the oldest and largest secondary schools in a fairly representative part of the city of Kingston. He claims at least 4C$ of students in forms one through 5 drop out before "O" level mathematics and the same per cent of form one students would answer incorrectly to 1/2 + l/j5. He was so concerned about the low level of computational skills displayed by students entering his school that he suggested heavy concentration on mathematics and English in form one (grade 7)-This educator seems to have much support for his postion since the Five-Year Educa­tion Plan contains the following statements:

"... there are an estimated 53/& of the children of age 11+, who at the end of primary education have not attained at acceptable standards in literary and numerary skills..."

"To counter the widespread learning deficiencies in critical learning skills at the secondary level, a major emphasis will be on mathematics and English."

110. Concern was expressed by both Mr. Radley Reid, mathematics teacher at Campion High School, and Miss Doreen Muir, education officer, for students' poor perfor­mance in the area of problem solving.

111. Based on this information it seems clear that pedagogical considerations in Jamaican primary schools are no more sophisticated than those of St.Lucia or Grenada.

112. Unfortunately, the large percentage of students (sometimes more than KOfo) who "drop out" before "0" level mathematics should also lead to serious concerns about mathematical pedagogy in the secondary schools of Jamaica. As we shall see in the last section, results of students (who do not drop out) on the G.C.E. "0" level mathematics examinations only serve to aggravate the concerns.

113. In analysing the problems of secondary education in Jamaica, the Five-Year Education Plan concludes that "much of the teaching at this stage of the system is still traditional and teacher oriented, and students do not develop either the spirit of enquiry-or creativity, or self-reliance...".

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III. Teacher Training

114. From a table (Staffing of Schools) in the Five-Year Plan, I found that yj% of primary teachers, 22$ of new secondary teachers, 18$ of secondary teachers, 26$ of comprehensive high teachers, 28$ of technical teachers and 55$ of indepen­dent school teachers were pre-trained.

115» The incidence of pre-trained teachers in Jamaica is less than in St.Lucia and Grenada. In spite of this much of the teaching is traditional and teacher oriented. I find it necessary to raise questions as to how effective are the techniques and methodologies employed by the trained teachers. Do the trained (and the untrained) teachers believe that the high rate of failure in and fear of mathematics is normal? Would more in-service training solve the problem? Are the pedagogical strategies being offered in the teacher training centres and the university of West Indies effective? Are teachers pedagogically curious? Do they see pedagogy as a flexible matter the imaginative practice of which permits virtually all children to learn mathematics?

IV". External Examinations

116. It was brought to my attention that Professor L.H.E. Reid of UWI School of Education develops items in arithmetic for the Common Entrance Examination and performs item analyses. Unfortunately, he was off the island on vacation during my visit. From his data I would have been able to make accurate assessments of students' competence in various arithmetical topics.

117* Records in the Overseas Examinations Office, Kingston, Jamaica show the following results:

Table IV

General Certificate of Education, Ordinary Level, June 1976 Table of percentage results (School and Private Candidates)

JAMAICA

Mathematics Syllabus A Mathematics Syllabus B Mathematics Syllabus C Additional Mathematics English Language

Number of candidates fo grade C and above

975 33-6 2,327 42.8 1,497 36.9 803 51.7

7,534

Table V

General Certificate of Education, Ordinary Level, June 1977 Table of percentage results (School and Private Candidates)

JAMAICA

Mathematics Syllabus A Mathematics Syllabus B Mathematics Syllabus C Additional Mathematics English Language

Number of candidates

568 3,368 1,630 811

8,013

% grade C and above

35.6 27.4 33-7 37.4

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118. The numbers of English Language students in 1976 and 1977 (7,534 and 8,013) respectively, are shown in the tables because, as stated by the director of the Overseas Examinations Office, each is within 3$ of the total numbers of students who sat one or more "0" level examinations. We will use one of these numbers later.

119. Comparison of Tables IV and V with Table II shows that Jamaican "0" level performance in mathematics is similar to that of the Eastern Caribbean. Table III shows that British students performed approximately twice as well as their Jamaican counterparts on the "0" level mathematics examination.

120. The table below shows G.C.E. Advanced Level performance in Jamaica, 1978:

Table VI

Advanced Level, 1978

JAMAICA

Mathematics A Mathematics B Pure Mathematics Applied Mathematics Statistics Further Mathematics

Number of candidates

313 140 27 18 3 1

Advanced level passes

51.1 50.7 11/27 4/18 1/3 0/1

121. Please note that the students who sat "An level mathematics in June, 1978 had taken rt0n levels in 1976.

122. In Table IV, the total number of students who took Mathematics Level A, B, and C is 4,799. The 803 students who took Additional Mathematics were the best among the 4,799 who sat for n 0 n level mathematics in 1976. Calculations based on Table IV show that in 1976, 1,876 students passed "0" level mathematics. We find that 39$ of students who took the "0" level mathematics examination in 1976 passed.

123. The number of students who sat for one or more "0" level examinations in 1976 was 7,534. If we subtract from this total the number of students who took "0n

level mathematics in 1976, we find that approximately 2,735 or 36$ of students dropped mathematics before taking the "0" level examination.

124. The total number of 5th form students in Jamaica during 1976 was 25,294 with an age range of 15-18. I obtained this information from the Jamaican Five Year Education Plan and the Ministry of Education. Since 4,799 students sat the examina­tion, we see that approximately 19$ of all 5th form students in Jamaica took ff0" level mathematics. The 1,876 students who passed are only 7% of all 5th form students in Jamaica.

125. The data below were drawn from Tables 5, 6, 7, 8 and 9 in the Five-Year Plan.

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Table VII

All-Age Primary New Secondary Schools Secondary High Schools Comprehensive High Schools Technical High Schools

The number of the 15-18 year age cohort attending school in '76/77

8,129 39,212 16,888 2,709 4,009

Total 70,9^7 |

126. As we can see, a total of 70,947 15-18 year olds attended school in 1976/77 and were spread across grades 7 through 13 (forms 1 through 6B), of the schools listed in the table. The 15-18 year age range is chosen for attention because the age of the G.C.E. n0 n Level student falls within it.

127* Only three of the five categories of schools - secondary, comprehensive and technical - prepare students for "0" levels. This leaves 23,606 students spread across grades 7 through 13 (and belonging to the 15-18 age level cohort) in secon­dary, comprehensive and technical high schools. These students would either have taken "O" level examinations before 1977 or would be taking them within four years of 1976/77*

128. We noted before that about 36$ of students dropped mathematics before taking "0" levels in 1976. Assuming the same drop rate, 15,108 of the 23,606 students would either have taken mathematics at "0" levels before 1977 or would be taking it within four years of them. Assuming again the 39$ rate of passing (achieved by those who sat the examination in 1976), 5,892 would have passed by 1980.

129. In 1976, the total number of persons in the 15-18 year age cohort was 216,951. It follows that approximately 2.7$ of Jamaicans in the 15-18 year age cohort would have passed "0" level mathematics (4.8$ in St. Lucia) by I980.

130. Table VI shows that 500 of the students who passed "0" level mathematics in 1976 went on to take "A" level mathematics in I978. Of those,247 passed. The "A" level age range is 17-19 and the cohort totals 168,798. Approximately .15$ of the "A" level age cohort passed "A" level mathematics in 1980.

131. It is finally worth noting that of the 1,876 students who passed "0,r level mathematics in 1976, a mere 247 or 13$ went on to pass mathematics on "A" level.

132. When one considers the great significance of G.C.E. examinations in the lives of Jamaicans and the priority given to preparing for them, these numbers reflect a national crisis.

EVALUATION

133. The large number of untrained teachers makes it difficult for the system to operate with reasonable benefit to students.

134. The overwhelming bias toward the passing of the locally-set CEE in the primary system and the overseas-set G.C.E. in the secondary system has left little room for creativity and inhibited the developmental approach to teaching.

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135« Despite the laudable efforts of the CMP and the" St.Lucian Caribbean Primary Mathematics Project, there have not been sufficient basic changes in the curricu­lum and the system of rote memorization and learning continues.

136. Because of a severe reading comprehension problem, students suffer a serious disability in verbal problem solving.

iyj. The continuing shortage of teaching materials, particularly workbooks and textbooks, will hamper any serious further development of the educational system.

138. The Ministries of Education should provide common curriculum materials to all schools. This type of direction is particularly necessary where there are large numbers of untrained teachers and it provides a secure base upon vrhich to build a wonderful diversity. The CMP project might have had greater impact had this foundation been in place.

139* Blackboard presentation and verbal group repetition must urgently be replaced by more imaginative, classroom management procedures.

140. The teacher training programmes of the territories do not seem to deliver classroom effectiveness.

141. Effective alternatives for teacher training are urgently necessary. Dr. Cluff' s approach to stimulating a sense of responsibility for students' performance within teachers is part of the solution.

142. A well chosen textbook-workbook programme only brings potential for positive directions; but will not, by itself, produce solutions to mathematics education problems in the Caribbean.

143. The widespread fear of mathematics, in students as well as teachers, is a serious problem in all three territories.

144. The gap of time between introduction of topics and their mastery is far too wide. There are pedagogical strategies and programme organization procedures which effectively reduce these intervals to small fractions of their original dimensions.

145. Teachers' expectation of failure (a great threat to children) inflicts serious psychological damage upon learners wherever it exists. Caribbean teachers are no less guilty of this practice than teachers elsewhere.

146. Why teach "one digit" addition one year, "two digit" addition the next, "three digit" addition the next, and so on? The "spiral" approach, used universally within the three territories, encourages this type of curriculum organization. Mrs. Gill of the St.Lucia Curriculum Centre favours a non-spiral approach.

147. The Jamaican grades 7-9 curriculum materials contain some good ideas in their early stages. With continuous feedback and development they will become wonderful instruments for improvement of teacher organization and performance.

148. Some teachers expressed the need for revision of the St.Lucian Caribbean Mathematics Series.

149. Some problems of the St.Lucian Caribbean Primary Mathematics Project are:

(a) continual recruitment of untrained teachers; (b) workbooks are in short supply; (c) many students cannot afford to purchase a workbook.

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RECOMMENDATIONS

150. This set of recommendations begins with the following quotation from the Jamaica Five-Year Education Plan:

"During the years from birth to six, children begin to form basic attitudes toward themselves ... It is at this stage that all children should have a chance to acquire the basic skills and attitudes neces­sary for their ... intellectual ... development. "

151. Any serious approach to education anywhere in the world must reflect on the influence of formal education on the children during their most impressionable school years.

"Early in a child's mathematical development he/she encounters 'basic fact arithmetic1. This is the usual accumulation of addition, subtraction and multiplication facts.

These bits of information are traditionally committed to memory as a result of long, laborious efforts. This enormous struggle becomes one of children's earliest and most significant associations with the word 'mathematics'. It is a decidedly unpleasant association developed during the most impressionable years children spend in school.

In fact, the mental activity of pupils in the 'learning' of basic fact arithmethic bears almost no resemblance to the mental activity of mathe-maticiansi Why are children taken so far off the track toward mathematical insight upon their introduction to the subject? Very few ever get back 'on track'." (Personal Discovery Teacher's Guide Book I, published by The Barrett Learning Dynamics Corporation, I98O)

152. A contribution this approach makes to mathematics education is one which permits the mental activity of students, in their early school years and beyond, to resemble that of the mathematician. This can be realized in basic fact arith­metic by having children learn efficient cognitive processes by which facts such as 8+6, 13-7 or 8x7 can be calculated within 2 to 3 seconds by 6 or 7 year old children. How is this done? An example follows:

"... by means of a thinking strategy which utilizes a previously learned capacity for efficient mental doubling, virtually all seven-year-old children can respond within three seconds to the question, 'What is 8 times 7?' The thinking proceeds as follows:

If 2 times 7 equals 14, then 4 times 7 equals 28 and 8 times 7 equals 56.

This is an example of 'cognitive process learning' and the children are aware of a wonderful economy which, when articulated says, 'Anybody who knows two times knows four times and eight times'." (Personal Discovery, Book IV, published by The Barrett Learning Dynamics Corporation, I98O)

153* By such means, not only do children have the ability to think through the "facts" quickly without memorization,they also cannot "forget" them.

154. Useful information on the subject is contained in an article entitled Emphasizing Thinking Strategies in Basic Fact Instruction published in the Journal for Research in Mathematics Education 9: 214-227; May 1978, by Carol A. Thornton.

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155» It is critically important that children become impressed with their "cognitive process capabilities" while they are impressionable and that this type of psychological functioning be nurtured in all their future involvement with formal education. Accordingly, I recommend as follows:

1. Early introduction of learners to efficient cognitive processes for the learning of basic fact arithmetic.

2. Sustained involvement with cognitive processes for all learners through the learning of computational arithmetic.

(Recommendations 1 and 2 will greatly accelerate both the learning and retention of computational facts and processes. Much time .will be saved for more intensive work on conceptual development and problem solving before the Common Entrance Examination. This was a matter of grave concern among primary teachers in the three Caribbean territories.)

3. Rearrangement of the curriculum sequence into "clusters" of dynamically related material and, having accomplished this, permitting students to learn the contents of an entire cluster before moving on to the next.

(This clustering facilitates cognitive process functioning on the part of the learner since he/she acquires insight into the internal dynamics. When teachers become involved with "cognitive process teaching" they learn mathe­matics themselves. Teacher training potential is greatly enhanced since the clustering permits teachers' acquisition of the mental dynamics and, consequently, of their own mental model of how children should be functioning.)

k. On-the-spot classroom demonstrations of the above methodologies followed (on the same day) by workshops for discussion of the demonstration lesson, enhancing teachers' understanding of cognitive process learning and lesson planning.

(There might be ten demonstration lessons per school year per pilot class. Recommendation k will serve as the much needed alternative to traditional teacher training in the three territories. Classroom demonstration lessons on cognitive process teaching by an effective practitioner can serve the following purposes:

(a) dispel the fear of mathematics when observers consistently see virtually all children learning material previously regarded as difficult;

(b) reduce the gap of time between the introduction of topics and their mastery;

(c) Transform teachers' expectations of failure to enthusiastic anticipation of success.)

5. Development of teachers' awareness of the importance of careful and precise classroom articulation and dialogue for communicating their own mental models of how students may think about mathematics.

(Language is a most important medium of communication. Where teachers have previously acquired a mental model of how children should be thinking, it is subsequently important that they become sensitive to means for communicating it.)

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6. Observation of the demonstration lessons and workshops by a cadre of teachers who would, in turn, conduct demonstrations of the techniques and methodologies in schools throughout the three territories.

(This will serve the purpose of developing territorial self-reliance in teacher-training, curriculum development and overall organization of the • educational effort.)

7. Part of the language difficulty students have in problem solving is because the problems are outside the students' everyday experiences. Problems drawn from situations which exist in the society should be selected in such a way that appropriate mathematics can be developed and utilized in seeking the solutions. To permit development of competence in problem solving, we must make it possible for children to struggle with large numbers of problems. In time, they will accumulate useful problem-solving procedures and processes.

(We learn to solve problems by solving problemsl Traditional schedules do not permit time for a sufficiently intensive involvement with problem solving. Cognitive process involvement accelerates learning and provides much time for problem solving. )

8. Teachers' guides and children's workbooks appropriate for the task of teaching arithmetical computation through cognitive process involvement must be made available.

(Existing programmes do not have to be abandoned. The materials suggested here can be used as a supplement to those already in use. A statement in the Jamaican Five-Year Plan for education reads as follows:

"Thus the strategy for providing quality education is an integrated one, involving the curricular programmes and areas which support their development and implementation.")

9. In territories where an ambitious textbook-workbook programme exists (such as Jamaica with its Ginn series), the teamed cadre mentioned in recommendation 6 can be very effective in facilitating implementation and articulation of programmes.

(One of the long-term objectives in the Jamaican Five-Year Plan is to "provide a well qualified cadre of teacher trainers ... for all stages and types of education including a systematic re-training and upgrading programme".)

10. In view of the fact that the timing of the visits made it impossible to observe teachers and students at work on their daily classroom activities, I recommend a second mission to accomplish the following:

(a) a study of the mathematics programme in process of implementation at a random sample of schools;

(b) investigate the feasibility of establishing a pilot project in mathematics education.

(The pedagogical ideas suggested above were of great interest to education officials and teachers with whom I spoke. In fact, conversations with the Chief Education Officers in both St.Lucia and Grenada lead me to believe they might welcome projects in their countries whereby these pedagogical possibili­ties may become apparent.

Listed below are three programmes recommended by the consultant which might be made available in all territories as examples which involve learners in mental processes.

26 -

The Comprehensive School Mathematics Programme

This project is an attempt to individualize mathematics curricula for grades K-12. The basis of this project is the utilization of a variety of activity packages created to facilitate independent student work. The project concerns itself with a disciplined approach to mathematics educa­tion. Various topics are included in the general curriculum such as whole number operations, integers, number theory, probability and statistics and functions. In addition to books, films and games, audio-visual tapes are used to reinforce activities relative to addition, subtraction, multiplica­tion, division and problem solving.

The Greater Cleveland Mathematics Programme of the Educational Research Council of America

This programme is a comprehensive, as well as a sequential curriculum for K-12. Inherent in E.R.C.A. are three basic functions:

to approach computational skills through understanding of structure;

to interpret new situations;

to develop problem solving strategies by perceiving pattern and structure in mathematical situations.

Throughout the programme, an intentional approach to the logic and structure of mathematics is utilized while stressing guided discourse techniques. Available are teachers guides and pupil textbooks for grade K-6.

Barrett Educational Science Techniques

The thrust of Teacher Guides I, H , and III is the presentation of efficient cognitive strategies, starting from kindergarten, for the mastery of all arithmetical computations. They explain techniques which permit, for example,

almost all 6-year old children to know all addition facts without memorization in a few days;

almost all 6 year old children to know all subtraction facts without memoriza­tion in a few days;

almost all 7 year old children to know all multiplication facts without memorization within six weeks.

The guides also provide suggested dialogue to facilitate communication of mental strategies. Among its pages are a data sheet outlining highly signifi­cant results on various standardized tests in the United States. Guides and textbooks are available for supplementary use with existing mathematics pro­grammes. To facilitate implementation of the programme, consultants are avail­able for classroom demonstrations and teachers' workshops.

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APPENDIX

DIARY OF THE MISSION

A. 14 July - 18 July St. Lucia

15 July Interviews at the:

Ministry of Education - Permanent Secretary, Mr. Thomas. - Chief Education Officer, Mr. Delmede, - Unesco Representative in measure­ments and evaluation, Dr. Cluff.

San Souci School - Discussions with Principal and / teachers.

16 July Anglican School - Discussions with Principal and teachers.

Teacher's College - Interview, Principal, Mrs. Thomas. Curriculum Develop­ment Centre - Interview, Mrs. Gill, Director.

17 July Interview, Dr.Cluff Further discussion with Mr. Delmede. Summary of findings with Mr. Thomas.

B. 18 July - 23 July Grenada

19 July (Saturday) Planning Session with Mr. Ashby, Education Officer for measurement and evaluation.

20 July (Sunday) Discussion with Mr. Ashby.

21 July Ministry of Education - Carol Davis, Education Officer. St. George's Anglican Senior School. St. George's Seventh Day Adventist School. St. Dominic's Primary. St. Peter's Primary.

22 July First meeting with approximately 10 teachers from various schools. Teacher's College - Second meeting with approximately 10 teachers from various schools. Interviewed Mr. McBarnett, Principal of Teacher's College and Dr. Garth Baker, Mathematics Tutor. Technical School - Interviewed Mr. K. Roberts, Principal. St. James Hotel - Discussions with Dr. Garth Baker and Mr. Victor Ashby.

C. 23 July - 4 August Jamaica

24 July Ministry of Education - Discussions with Ms. Doreen Muir, Education Officer for Mathematics.

25 July Ministry of Education - Discussions with Mrs. Grace Eweka, Director of Curriculum Division.

- Discussion with Mr. Ross Murray, Chief Education Officer.

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July Excelsior Education Centre -

Overseas Education Office -

Discussions with Mrs. Fay Jenkinson, Headmistress, Pre-primary and Primary division. Data on G.C.E. results in mathematics.

July Excelsior Education Centre -

Overseas Education Office -

Introduced to the complex and its facilities by Hon. A. Wesley Powell, Director. Data on G.C.E. results in mathematics.

July Ministry of Education At my local residence

Data Interviewed Mr. Radley Reid, High School mathematics teacher.

July Ministry of Education - Discussions with Mrs. Eweka, Ms Muir, Ms. Campbell and Ms. McKenzie.