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Change in mathematics education since the late 1950's-ideas and realisation great britain

Mathematics in Schools, 1957–77 Aims and Realisations

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Notes and references

  1. The tripartite system's rise and fall, and the public examination system are described more fully in Griffiths, H.B. and Howson, A.G. Mathematics: Society and Curricula, Cambridge U.P., 1974.

  2. The Teaching of Mathematics in Primary Schools, Bell, 1956.

  3. The Jeffery Report: School Certificate Mathematics, Cambridge Local Examinations Syndicate, 1944. See also ‘Possible Changes in the Mathematical Syllabus for the School Certificate Examination’ Math. Gazette, 28 (1944), 125–143.

  4. ‘The Place of Mathematics in Secondary (Modern) Schools’ Math. Gazette, 30 (1946), 250–271.

  5. A report Mathematics in Secondary Modern Schools (Bell) was published in 1959. In an attempt to reach more teachers the Mathematical Association in 1971 established a second periodical, Mathematics in School (Longman).

  6. Fletcher T.J. (Ed.) Some Lessons in Mathematics, Cambridge U.P., 1964. A later volume, Notes on Mathematics in Primary Schools, Cambridge U.P. 1967, also merits particular attention.

  7. Thwaites, B. (Ed.) On Teaching Mathematics, Pergamon, 1961.

  8. Brief Descriptions of British mathematical projects are to be found in Mathematics Projects in British Secondary Schools, Bell for the Mathematical Association, 1968 (A revised guide: 1976). More detailed descriptions of selected projects can be found in Chapman, L.R. (Ed.) The Process of Learning Mathematics, Pergamon, 1972, and Watson, F.R. Developments in Mathematics Teaching, Open Books, 1976. (The latter is a short, very readable account of recent changes in mathematics teaching in Britain.)

  9. Weston, J.D. and Godwin, H.J. Some Exercises in Pure Mathematics, Cambridge U.P., 1968.

  10. Mathematics in Primary Schools (Curriculum Bulletin No. 1), HMSO, 1965.

  11. In Britain Dienes' more recent materials and apparatus are often associated with Bulmershe College of Higher Education, a centre through which his work has been propagated.

  12. See the chapter by A.G. Vosper on the Nuffield Foundation Mathematics Teaching Project in Chapman (Note 8).

  13. The paper (taken from the Director's Report for 1962–63) is reprinted in Thwaites, B. SMP: The first ten years, Cambridge U.P., 1972, and in Griffiths-Howson (Note 1).

  14. For a discussion of these changes and their implications for today see: HowsonA.G. ‘Charles Godfrey (1873–1924) and the reform of mathematical education’ Educ. Studies in Maths., 5 (1973), 157–180; Howson, A.G. ‘Milestone or Millstone?’, Math. Gazette, 57 (1973), 258–266.

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  15. Pace those who would produce the substitution x=2 sin2θ from up their sleeves!

  16. See SMP Books T and T4, Cambridge U.P., 1964 and 1965, and the preliminary drafts for these books.

  17. See, for example, the chapter on networks in SMP Book 4 (1968) which is now usually omitted and ignored (‘not on the examination syllabus’), but which includes such ideas as the ‘max flow-min cut’ result for transport networks.

  18. See, for example, Blakely, B.H., Data Processing, Cambridge U.P., 1973; corlett, P.N. and Tinsley, J.D., Practical Programming, Cambridge U.P., 1968; Some experimental ideas for teachers, Cambridge U.P., 1971; Teacher's companion: from 11 to 16, Cambridge U.P., 1974; and the Compack units, Cambridge U.P., 1977.

  19. Godfrey, C. and Siddons, A.W. Elementary Algebra, Cambridge U.P., 1912.

  20. The Teaching of Geometry in Schools, Bell, 1923.

  21. Regrettably, though, an unmotivated approach to matrices is to be found in some of the ‘traditional’ school texts which now include a chapter on matrices as a sign of ‘with-it’-ness. Thus for example, in Channon, J.B. et al. New General Mathematics (2), Longman, 1970, we find: “In Chapter 9 a vector was defined as a set or array of numbers. A matrix (plural matrices) is a set of numbers in the form of a rectangular arry .... A vector is in fact a special case or sub-set of a matrix.”

  22. The British, have, however, rarely been able to equal the standard set by the ‘solution’ of (x−1)(x+3) = 0 to be found on p. 31 of Synopses for Secondary School Mathematics, OEEC, 1961.

  23. Here the terms ‘relational’ and ‘instrumental’ are used in the sense of Mellin-Olsen and Skemp. See, for example, SkempR.R., ‘Relational understanding and instrumental understanding’ Maths. Teaching, 77 (1976), 20–26.

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  24. SMP Supplementary Booklets 1–5, Cambridge U.P., 1977.

  25. The quotation is taken from the ‘General Introduction’ to the various Nuffield Guides which were published by Murray and Chambers (1967 onwards).

  26. Nuffield was, to a limited extent, to provide such material with its sets of open-ended problems for 11–13 year-olds: Problems (Green, Red and Purple sets), Murray/Chambers, 1969 onwards. Other such material has been supplied by the Leapfrogs group (Hutchinson, 1975 onwards). A key book describing this approach is Banwell, C. et al, Starting Points for Teaching Mathematics in Middle and Secondary Schools, Oxford U.P., 1972.

  27. A new project, ‘Nuffield Mathematics 5–11’, started in 1976 and intends to produce pupils' materials which will aim “to avoid an over-rigid structure but nonetheless to provide a framework both flexible and feasible.” See: AlbanyE., ‘Mathematics 5–11’ Maths. in School, 6 (3) (1977), 26–7.

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  28. Curriculum research and development in mathematics, Schools Council, 1973. (Similar leaflets describing Schools Council supported projects are issued periodically.)

  29. Gattegno brought Cuisenaire rods to the attention of British teachers.

  30. See, for example, Corston, G.B. ‘Spreading the new ideas’ in Howson, A.G. (Ed.) Children at School, Heinemann Ed., 1969; Cane, B. ‘Meeting Teacher's Needs’ in Watkins, R. (Ed.) In-service training: Structure and content, Ward Lock Ed., 1973; Cane, B. In-service Training, NFER, 1969; Adams, E. (Ed.) In-Service Education and Teachers' Centres, Pergamon, 1975.

  31. Most notably, Fletcher, H. et al, Mathematics for Schools, Addison Wesley, 1971-. Some would argue that this popular series has formalised what was intended to be informal. An alternative view, by one of the authors, can be found in Walker, R. ‘Mathematics for Schools: Six years later‘ Maths. in School 6 (3) (1977), 2–6.

  32. What's Going on in Primary Mathematics? Schools Council, 1974.

  33. Report of the Working Party on the Schools' Curriculum and Examinations (Lockwood Report), HMSO, 1964.

  34. This did mean, of course, that the Schools Council was left with some of the stonier areas of mathematical education to till. A fact that should be borne in mind when one comments on its failure significantly to influence school practice.

  35. The aims of the MMP are described in Mathematics for the Majority (Schools Council Working Paper 15), HMSO, 1967. Data concerning the allocation of teachers to the ‘lessable’ students can be found in Mathematics for the Majority Project: Evaluation in Case Study. Initial data, Schools Council (1970). (See also Watson (Note 8).)

  36. FloydP.J., ‘Mathematics for the Majority’ Dialogue (Schools Council Newsletter), 3 (1969), 4–5.

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  37. Published by Chatto and Windus (1970–74).

  38. Working Paper 14 (Note 35). (See also Griffiths-Howson (Note 1).)

  39. Published by Schofield and Sims (1974-). See also, for example, KanerP. ‘Mathematics for the Majority Continuation Project’, Maths. in School 1 (1) (1971), 18; Pass, N. ‘Using the Communications Pack’, Maths. in School 5 (1) (1976), 12–14. (Watson (Note 8) contains a useful account of the project's work.).

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  40. The quotations are taken from the project's newsletter (issues 1 and 3) and its draft package on indices.

  41. Published by Heinemann Educational (1975-). For accounts of the project's philosophy and aims the reader is referred to; for example, OrmellC.P. ‘Mathematics through the Imagination’ Dialogue 9 (1971), 10–11; Flemming, W. ‘Mathematics Applicable’, Maths. in School 6 (3) (1977), 14–16; Ormell, C.P. ‘Mathematics, Applicable versus Pure-and-Applied’, Int. J. Math. Educ. Sci. Technol. 3 (1972), 125–131.

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  42. The project devised an ‘Applicable Mathematics’ examination, administered by a GCE board, in which candidates, when defeated by a problem, may receive ‘hints’ which will anable them to work on. The number of ‘hints’ accepted can be taken into account by the examiner.

  43. Howson, A.G. and Eraut, M.R. Continuing Mathematics, Councils and Education Press, 1969.

  44. Published by Longman (1977-).

  45. Howson, A.G. ‘A critical evaluation of curriculum development in mathematical education’ New trends in mathematics teaching (vol. 4), UNESCO (to appear).

  46. SMP 7–13 is a course intended to cover the middle-years of schooling. See, for example, Rogerson, A. ‘SMP 7–13’ Maths. in School 2 (4), (1973), 7–8, and 4 (1) (1975), 4–6; Brighouse, A. ‘SMP 7–13’ Maths, in School 6 (4) 1977, 35–36. The first units, consisting mainly of workcards, were published by the Cambridge University Press in 1997.

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  47. See Secondary School Examinations other than the GCE, HMSO, 1960, the ‘Beloe Report’ which led to the establishment of the CSE.

  48. Examples of how this freedom has been exploited can be found in Griffiths-Howson (Note 1).

  49. SMP A-H series published by the Cambridge University Press, 1968-.

  50. SMP Workcards, Cambridge University Press, 1973.

  51. SMP Director's Report 1968–69.

  52. For a discussion of ‘mixed-ability teaching’ see Mixed Ability Teaching in Mathematics (Schools Council), Evans/Methuen 1977. (See also Morgan (Note 63).)

  53. Crawford, D.H. The Fife Mathematics Project, Oxford U.P., 1975.

  54. See, for example, BanksB. ‘The ‘Disaster Kit’’ Math. Gazette 40, (1971), 17–22; Banks, B. The Kent Mathematics Project Maths in School 4 (2) (1975), 2–4.

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  55. See, for example, GibbonsR. ‘An account of the Secondary Mathematics Individualized Learning Experiment’ Maths. in School 4 (5) (1975), 14–16; Langdon, N. ‘Smiling in the classroom’ Maths. in School 5 (5) (1976), 5–7. Other interesting local schemes include the City of Birmingham Structured Maths. Scheme, which has concentrated on providing basic support to poorly qualified teachers (see, for example, Graham, J.D. ‘Materials in a locally-based curriculum development centre’ Maths. in School 5 (5), 1976, 10–12), and the South Nottinghamshire Project which has provided material for mixed-ability classes in the 11–13 age range (to be published by Blackie).

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  56. Gibbons, R. (Note 55).

  57. Discussion of the contributions to curriculum development which can be made by ‘local’ projects can be found in Howson, A.G. ‘Teacher Involvement in Curriculum Development’ Schriftenreihe des IDM 6/1975, 267–288, and in the reference given in Note 45.

  58. This report was preceded by The supply and training of teachers of mathematics (Math. Assn./ATCDE) Bell, 1963.

  59. Combridge, J.T. writing in the Newsletter of the Mathematical Association.

  60. Teacher Education and Training (James Report), HMSO, 1972.

  61. The books, Modern Mathematics for Schools, were published by Blackie and Chambers (1965-).

  62. Robertson, A.P. ‘The future is a function of the past’ Proc. AAMT Biennial Conf. Perth, Australia, 1976. See also ‘Ten Years of Modern Mathematics for Schools’, Maths. in School 3 (6) (1974), 2–4. and Chapman (Note 8).

  63. Modular Mathematics is published by Heinemann Educational (1975-) and the quotations are taken form the publisher's advertising material. Although the age-range ‘11–13’ is mentioned, the material was in fact prepared for use in Scottish secondary schools and so was directed at the ‘12–14’ age-range. Advertisements now claim that it is intended for 11–14 year-olds! See also, for example, SmithD.M. ‘Modular Mathematics’, Maths. in School 4 (2) (1975), 26–28. An interesting report of the project's work is Morgan, J. Affective consequences for the learning and teaching of mathematics of an individualised learning programme, Dept. of Education, Stirling University, 1977. Materials for the less able 14–16 year-olds have been produced in Scotland by the Mathematics Appreciation Project Group. They are published by Macmillan under the title: Mathematics for General Education.

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  64. See, for example, the differing views on the results of syllabus changes to be found in: ‘The mathematical needs of ‘A-level’ physics students’ Royal Soc./Inst. of Physics, Physics Edn., June 1973: ‘Report of the Working Party on Mathematics for Biologists’ Royal Soc./ Inst. of Biology, Int. J. Math. Ed Sc. Tech. 6 (1975), 123–135; ‘Mathematics and School Chemistry’ Royal Soc./Inst. of Chemistry, Educ. Sci.. Jan. 1974. ‘Mathematics for Social Science Students’ Economics Assn./Maths. Assn./Royal Economic Society. Math. Gazette, 57 (1973), 160–165.

  65. See, for example, HerseeJ., ‘MEI A level syllabuses’ Physics Ed. 7 (2) (1972), 80–81; MEI Supplement to Math. Gazette 42 (Dec. 1963).

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  66. See, for example, Mathematical needs of school leavers entering employment, I, II and III, IMA 1974–76; Mathematics at the School/Industry Interface, IMA, 1976; Report of the Working Party on School Mathematics in relation to Craft and Technician Apprenticeships in the Engineering Industry, Royal Soc./Council of Engineering Institutions, 1976; Basic Skills in Mathematics for Engineering Eng. Industry Training Board, 1977.

  67. Report of the Commissioners appointed to inquire into the State of Popular Education in England (Newcastle Commission) HMSO, 1861.

  68. The quotations are from Marjoram, D.T.E. ‘What the APU hopes to achieve’ in Towards standards a supplement to Education (15, 7, 77). See also, for example, BellA.W. ‘The APU and the 1978 Mathematics Survey’ Maths. Teaching 80 (1977), 24–27; Kyles, I. and Sumner, R. Tests of attainment in Mathematics in Schools N.F.E.R., 1975 and 1977.

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  69. The absence of such expensive exercises is not necessarily to be deplored (particularly, if one considers the outcomes of such projects elsewhere).

  70. See, for example, Armitage, J.V. ‘Cleansing the Augean stable’, Times Ed. Supp. 5.10.73; Boys, G.R.H. ‘A critical review’ Maths. in School 3 (2) (1974), 24–25; Boys, G.R.H. ‘The critical review’ Maths. in School 5 (3) (1976), 26–29. The books are to be published by Blackie.

  71. In this paper we have not listed all the Schools Council's contributions to curriculum development within the field of mathematics. Descriptions are available (see Note 28), but other projects to which the reader's attention is drawn are: Early Mathematical Experiences (mathematical ideas for nursery schoolchildren). See, for example, MatthewsJ. and MatthewsG. ‘Early Mathematical Experiences’ Maths. in School 5 (2) (1976), 22–23. The materials produced by the project will be published by Addison Wesley (1978). The development of science and mathematical concepts in children between the ages of 7–11. See, for example, Bell, D. et al. Area, weight and volume, Nelson, 1975. The Schools Council's Project on Statistical Education, although basically interdisciplinary, is also of considerable interest to mathematicians.

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  72. Studies based on the N and F proposals, Schools Council, 1977.

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Howson, A.G. Change in mathematics education since the late 1950's-ideas and realisation great britain. Educational Studies in Mathematics 9, 183–223 (1978). https://doi.org/10.1007/BF00221158

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