This is a preview of subscription content, log in via an institution to check access.
Bibliography
Abidjan, Conference on ‘Mathématique et Milieu en Afrique’ 1978-Report to appear (Conference included presentations and working groups devoted to linguistic problems).
Accra workshop, Languages and the Teaching of Science and Mathematics with Special Reference to Africa, London, Commonwealth Ass. for Science and Maths Edn., 1975. Report of seminar held from 27 Oct–1 Nov. 1975. Papers by Taiwo and Mmari. Bibliography (27).
Adelman, C. and Walker, R., Flander's System for the Analysis of Classroom Interaction-a Study of Failure in Research, CARE, University of East Anglia, 1973.
Aiken, L. R., ‘Verbal Factors in Mathematics Education: a Review of Research’, J. Res. Ed. 2, (1971), 304–313. A precursor-taking a more limited view-of Aiken (1972).
Aiken, L. R., Language Factors in Learning Mathematics, Mathematics Education Reports, ERIC, October 1972. A survey report with good bibliography (96). Considers factors affecting mathematical ability and experimental evidence, including readability tests, and the importance of verbalisation in conceptual learning. Relation between reading and mathematical development; the latter possibly similar to second language learning? Communication.
Allardice, B., ‘The Development of Written Representations for Some Mathematical Concepts’, J. Child. Math. Behaviour I(4) (1977), 135–160. Early encounters with symbolism.
Amidon, E. J. and Hunter, E., Improving Teching: Analysis Verbal Interaction in the Classroom, New York, Holt, Rinehart and Winston, 1966.
Arnold, H., ‘Why Children Talk: Language in the Primary Classroom’, Ed. for Teaching (1973).
A.T.M. (ed. Wheeler, D. H.), Notes on Mathematics in Primary Schools, London, Cambridge University Press, 1967. Contains many examples of classroom usage and many references, both implicit and explicit, to language.
A.T.M., Mathematical Reflections, London, Cambridge University Press, 1970. Collection of essays, some of which, e.g. Tahta, Caldwell, Thornton, directly concern language.
A.T.M. (ed. Wheeler, D. H.), Notes on Mathematics for Children, London, Cambridge University Press, 1977. Similar to A.T.M., 1967.
Ausubel, D.P., “Some Psychological and Educational Limitations of Learning by Discovery’, The Arithmetic Teacher 11 (1964), 290–302. Discussion of discovery methods which suggests that these may still be rote learning-of procedures rather than formulae. Verbalisation constitutes a part of abstraction so that verbal teaching is often adequate and quicker, when sequenced suitably.
Ausubel, D. P., ‘A Cognitive Structure View of Word and Concept Meaning’, in Readings in the Psychology of Cognition (Eds. Anderson, R. C. and Ausubel, D. P.), New York, Holt, Rinehart and Winston, 1965.
Balow, I. H., “Reading and Computation Ability as Determinants of Problem Solving’, The Arithmetic Teacher 11 (1964), 18–22. Survey of previous research and description of experimental procedure involving total ability ranges-unlike earlier work. Concludes that reading ability also is required for solving verbal problems.
Banwell, C., Saunders, K., and Tahta, D., Starting Points: for Teaching Mathematics in Middle and Secondary Schools, London, Oxford University Press, 1972. A collection of ideas for the presentation of mathematical concepts, together with some provocative remarks concerning mathematics education and teacher influence.
Barnes, D., Britton, J., Rosen, H., and the N.A.T.E., Language, the Learner and the School, Harmondsworth, Penguin, 1969, 1971 (revised ed.). Raises a wide variety of problems concerning language in education, especially relating to talking.
Barnes, D., ‘Language and Learning in the Classroom’, J. Curr. Studies 3(1) (1971(a)), 29–38.
Barnes, D., ‘Classroom Contexts for Language and Learning’, Ed. Review 23 (1971(b)), 235–247.
Barnes, D., From Communication to Curriculum. Harmondsworth, Penguin, 1976.
Bauersfeld, H., ‘Kommunikationsmuster im Mathematikunterricht’, IDM, Bielefeld, 1978. A discussion of the use of language in the classroom.
Bell, D., Hughes, E. R., and Rogers, J. Area, Weight and Volume: Monitoring and Encouraging Children's Conceptual Development, London, Nelson (for the Schools Council), 1975. Gives guidelines for teaching these concepts based on Piaget (including discussion of sign-signifier relationship), and especially the need for verbalisation-questioning in particular-at all stages, thus deducing from experience.
Bellack, A. A., Kliebard, H. M., Hyman, R. T., and Smith, F. L.Jr., The Language of the Classroom, New York, Teachers College Press, Teachers College Columbia University, 1966. Theoretical description of classroom discourse analysed as a game (Wittgenstein), with experimental analyses assessed in detail. Games consist of various moves which are fairly consistent in pattern of occurrence, and teacher controlled.
Bernstein, B., Class Codes and Control: 1 Theoretical Studies, 2 Applied Studies, London, Routledge and Kegan Paul, 1971, 1973. Collected essays of a leading exponent of socio-linguistics.
Braine, M. D. S., ‘Piaget on Reasoning: a Methodological Critique and Alternative Proposals', in Kessen, W. and Kuhlman, C. (eds.), Thought in the Young Child, Child Devt. Mono., 27(2) (1962) 41–61.
Britton, J., Language and Learning, Harmondsworth, Allen Lane, 1970.
Bruner, J. S., Towards a Theory of Instruction, Cambridge, Mass., Harvard Paperbacks, 1966. A collection of essays about learning, knowing, developing and teaching. References to language.
Bruner, J. S., Goodnow, J. J., and Austin, J. A., A Study of Thinking, New York, Wiley, 1956.
Bruner, J. S., Olver, R., and Greenfield, P., Studies in Cognitive Growth, New York, Wiley, 1966.
Brunner, R. B., ‘Reading Mathematical Exposition’, Ed. Research 18(3) (1976), 208–213. Attempts to provide an ‘operational definition of successful reading of mathematics’.
Cajori, F. A History of Mathematical Notations, La Salle, Open Court, 1928. An invaluable source for those who believe that history can provide guidance. Vol. II contains some interesting quotations (and references to writing) on language and symbolism by such authors as de Morgan, Babbage, Branford, Whitehead, etc. Also accounts of international attempts to standardise notation.
Call, R. J. and Wiggin, N. A., ‘Reading and Mathematics’, The Mathematics Teacher 59 (1966), 149–157. Experiments show that teaching reading of mathematics texts improves performance of verbal problems. Teachers need to be able to give specialised reading instruction.
Cantieni, G. and Tremblay, R., ‘The Use of Concrete Mathematical Situations in Learning a Second Language: a Dual Learning Concept’, Tesol Quart. 7(3) (1973), 279–288. English/French-primary school mathematics interdisciplinary approach.
Capps, L. R., “Teaching Mathematical Concepts Using Language Arts Analogies’, The Arithmetic Teacher 17 (1970), 329–331. Lists mathematical and literal examples of concepts: commutativity, associativity, distributivity, positional value, expanded notation, etc.
Carnap, R., The Logical Syntax of Language, London, Kegan Paul, Trench, Trubner, 1937. A classic work on the philosophy of science leading to the conclusion ‘the logic of science is nothing other than the logical syntax of the language of science’.
Carpenter, E. and McLuhan, M. (eds.), Explorations in Communication, Boston, Beacon Press, 1960. A collection of short articles dealing with many aspects of communication-visual, verbal, tactile-with references to cultural, psychological and geometric factors affecting the processes involved.
carpenter, T. P. (ed.), ‘Notes from National Assessment: Word Problems’, The Arithmetic Teacher 23 (1976), 3, 389–393.
Carroll, J. B., Language and Thought, Englewood Cliffs, N.J., Prentice-Hall, 1964. Discusses psychological aspects of language as a means of communication and as a process of cognition and thinking. Summarises linguistic structure theories, language learning, and language disabilities.
Cashdan, A. and Grugeon, E. (eds.), Language in Education, London, Routledge and Kegan Paul, 1972. Varied essays not specifically mathematical, e.g., H. Rosen on “The Language of Textbooks’.
Chapman, L. R. (ed.), The Process of Learning Mathematics, Oxford, Pergamon, 1972. A collection of articles some of which, e.g., Skemp's, refer to language.
Chausard, M., ‘Mathématique et Langage’, Rev. Gén Enseignt. Déf audit 68(4) (1976), 215–35. An account of teaching mathematics to deaf children.
Chomsky, N., Language and Mind, New York, Harcourt, Brace and World, Inc., 1968. A key work of (possibly) the most influential contemporary linguist. It offers a philosophical survey of theories past and present, including Chomsky's own, with reference to the psychology of mind.
Clark, M., An Investigation into the New Zealand Forms I to IV Mathematics Syllabus, M.Sc. thesis, Victoria University of Wellington, 1976. Indicates extent of confusion caused by differences in the colloquial and mathematical use of words and the difficulty in assimilating the meaning of technical terms.
Clarkson, D. M., Children Talking Mathematics, M. Ed. thesis, University of Exeter, 1973. (See also ‘A Bit of Research’, Maths Teaching 65 (1973), 26–30.)
Clements, M. A. and Gough, J., ‘The Relative Contributions of School Experience and Cognitive Development to Mathematics Performance’, in Learning and Applying Mathematics (ed.)Williams, D., Victoria, Australia, Math. Assn. of Victoria, 1978. A rigorous critique of Collis's Piagetian-style research in which considerable attention is placed on the role of language: both the use of mathematical language by psychologists and the child's gradual acquisition of mathematical language.
Coard, B., How the West Indian Child is Educationally Sub-Normal in the British School System, London, New Beacon Books, 1971. The effects of linguistic differences.
Cole, M., et al. The Cultural Context of Learning and Thinking, New York, Basic Books, 1971. Includes a discussion of the cultural influences on classification, problem solving and logic.
Colmez, F., ‘Teaching Mathematics at Pre-Elementary and Elementary Level’, Survey paper prepared for 3rd ICME, Karlsruhe, 1976. Contains several interesting comments on language, including some on that ‘artificial language’ devised in the 1960s which embraces sentences such as ‘the set of all girls who do not wear dresses’.
Connolly, P. G., ‘The Language of Mathematical Operations’, Eng. Lang. J. 1 (1) (1970), 25–31.
Cormack, A., A Study of Language in Mathematics Text-Books, M. Phil Thesis, Chelsea College, 1978. Books for 11-year-olds.
Coulthard, R. M. et al., Discourse in the Classroom, CIIT, 1972.
Coulthard, R. M. and Sinclair, J. McH., Towards an Analysis of Discourse, London, Oxford University Press, 1975. Review of literature on analysis of classroom discourse, description of method of analysis, examples analysed. Draws analogies between linguistic, grammatical and discourse analysis.
Creber, P., Lost for Words, Marmondswoth, Penguin, 1972.
Crystal, D., Linguistics, Harmondsworth, Penguin, 1971. General introduction giving social and historical contexts for an outline of the major theories and controversies.
Dale, E. and Chall, J. L., ‘A Formula for Predicting Readability’, Ed. Res. Bull. (Jan. 1949), 11–20.
Davidson, J. E., ‘The Language Experience Approach to Story Problems’, The Arithmetic Teacher 25 (1977), 28.
Davis, R. B., Mathematics Teaching-With Special Reference to Epistemological Problems. Journal of Research and Development in Education, monog. 1 (1967). A discussion of the aims and methods of mathematics education by extensive (oblique) analogies, examples of particular teaching situations, and analysis of experimental techniques. Stimulates many questions-not specifically linguistic.
Davis, R. B., See also Journal of Children's Mathematical Behaviour.
Dienes, Z. P., The Six Stages in the Process of Learning Mathematics, Windsor, Berks., NFER 1973, (trans. P. L. Seaborne. Originally published as Les Six Etapes du Processus d'Apprentissage en Mathématiques) Brief description of the six stages into which Dienes classifies learning, followed by their exemplification in three specific cases. Stage 5 is concerned with the invention of language to describe a mathematical situation.
Earp, N. W., ‘Procedures for Teaching Reading in Mathematics’, The Arithmetic Teacher 17 (1970a), 575–579. Mentions some problems related to specifically mathematical reading, together with suggestions on methods of overcoming them. Verbal problem solving technique is given as a specific example.
Earp, N. W., ‘Observations on Teaching Reading in Mathematics’, J. of Reading 13 (1970b), 529–532.
Easton, J. B., ‘A Tudor Euclid’, Scripta Mathematica XXVII (1966), 339–355.
Ebeling, D. G., The Ability of Sixth Grade Students to Associate Mathematical Terms with Related Algorithms, Ed.D. thesis, Indiana, 1973.
Engle, P. L., The use of Vernacular Languages in Education, Center for Applied Linguistics, Arlington, Va., 1975. A survey paper: good bibliography (93). Not specifically mathematical. Also in Rev. Ed. Res. 45(2) (Spring, 1975), 283–325.
Exeter Congress, Developments in Mathematical Education (Howson, A. G., ed), London, Cambridge University Press, 1973. Proceedings of 2nd ICME Congress, Exeter, 1972, (which included a working group on language). Relevant articles include those by Leach, Philp and Thom.
Farnham, D. J., Children's Use of Language in Developing Mathematical Relationships, M. Ed. Dissertation, Exeter, 1975. An account of classroom-based investigatory work. Also survey of literature: extensive bibliography.
Fey, J. T., Patterns of Verbal Communication in Mathematics Classes, Ph.D. thesis, Columbia University, 1969.
Fielker, D. S., ‘Editorial’, Mathematics Teaching 80 (Sept. 1977), 2–3. Includes comments on language and workcards.
Flanders, N. A., Interaction Analysis in the Classroom, University of Minnesota, 1960.
Flanders, N. A., Analyzing Teaching Behaviour, Reading Mass., Addison Wesley, 1970.
Flössner, W., ‘Sprachbarriere-durch Mathematik’, Neue Unterrichtspraxis Hanover, 1974, 1, 13. Criticism of the language used in ‘new math.’ workbooks for primary schools-may lead to building up of a selective language barrier in the classroom.
Fordham, D. L., An Investigation of 3rd, 4th and 5th Graders' Knowledge of the Meaning of Selected Symbols Associated with Multiplication of Whole Numbers, Ed. D. thesis, Georgia, 1974.
Forseth, W. J., ‘Does the Study of Geometry Help Improve Reading Ability?’ The Mathematics Teacher 54 (1961), 12–13. Suggests that solving geometry problems is in some ways similar to reading, and describes experimental evidence to show that the study of geometry (as opposed to other subjects) improved reading ability.
Freudenthal, H., Mathematics as an Educational Task, Dordrecht, Holland, Reidel, 1973. A wide-ranging survey of mathematical education which includes many references to problems of language. An interesting critique on the use of language in Piagetian ‘tests’. See also Weeding and Sowing: Preface to a Science of Mathematical Education, Dordrecht, Holland, Reidel, 1977.
Fujiwara, S., ‘Problems in Mathematical Symbolism-The Third, 5. Ambiguous Words’. Reports of Mathematical Education, J.S.M.E. VIII (1964), 18–26.
Furth, H. G., Thinking without Language, New York, Free Press, 1966.
Furth, H. G., Piaget for Teachers, Englewood Cliffs, N.J., Prentice Hall, 1970.
Furth, H. G., Deafness and Learning, Belmont, Cal., Wadsworth, 1973. Piagetian ideas applied to the teaching of the deaf.
Gagné, R. M., ‘The Learning of Principles’, in Klausmeier, H. J. and Harris, C. W. Analyses of Concept Learning, New York, Academic Press, 1966.
Gallop, R. and Kirkman, D. F., ‘An Investigation into Relative Performances on a Bilingual Test Paper in Mechanical Mathematics’, Educational Research 15(1) (Nov. 1972), 63–71. Contradictory assessments of advantages of bilingualism in education led authors to assess simultaneous English/Welsh test papers. Bilingual papers preferable for bilinguals. Bibliography (29).
Garbe, D. G., Indians and Non-Indians of the Southwestern U.S.: Comparison of Concepts for Selected Mathematics Terms. Ph.D. thesis, Texas, 1973. Indians more likely than non-Indians to prefer verbal to symbolic representation for number word.
Gattegno, C., ‘Mathematics and the Deaf’, in For the Teaching of Mathematics (2), Reading, Educ. Explorers, 1963. Reflections on non-verbal teaching.
Gattegno, C., What We Owe Children, London, Routledge and Kegan Paul, 1971. Questions philosophical and psychological bases for education, and derives an approach-‘the subordination of teaching to learning’-from this. Asks many questions, especially concerning mathematics and reading.
Gay, J., ‘Mathematics, Language and Effective Teaching’, UNESCO, Paris, September 1974. The interactions of culture, language and mathematics teaching. A paper prepared for the Nairobi Conference.
Gay, J. and Cole, M., The New Mathematics and an Old Culture, New York, Holt, Rinehart and Winston, 1967. Cultural background to mathematics of the people (with comparisons with Western culture), linguistic structure's influence on thought/learning ability, suggestions for effective teaching.
Gay, J. and Cole, M., Mathematics and Logic in the Kpelle Language, Ibadan, University of Ibadan Press, 1971. Examples of how mathematical and logical thought appear in the Kpelle language.
Gentilhomme, Y., ‘De Saussure avait raison ou les mathématiques, pourquoi?’. Langues Modernes 65(3) (1971), 61–73. Comparison of the studies of linguistics and mathematics.
Giles, W. H., ‘Mathematics in Bilingualism: a Pragmatic Approach’, ISA Bulletin 55 (1969), 19–26.
Gilliland, J., Readability, London, University of London Press, 1972. General introduction describing recognised variables and tests used to measure them. Readable!
Gimpel, M., ‘Zu einigen Fragen der Häufigkeit des Auftretens mathematischer Begriffe in der Lehrbüchern für den Mathematikunterricht an der Oberschule’, Mathematik in der Schule 10 (1972) 2, 82–89. On the need for the frequent repetition of mathematical terms in textbooks.
Ginsberg, H., The Myth of the Deprived Child, New York, Prentice-Hall, 1972.
Goeppert, H. C. (ed.), Sprachverhalten im Unterricht, Munich, W. Fink, 1977.
Goutard, M., Mathematics and Children, Reading, England, Educational Explorers, 1964. Descriptions and examples of learning through experience.
Greenfield, P. M., ‘On Culture and Conservation’ in Bruner, J. S., Olver, R. R., and Greenfield, P. M., Studies in Cognitive Growth (q.v.).
Griffiths, H. B., ‘What is Mathematics Education?’, Int. J. Math. Educ. Sci. Technol. 6 (1975), 3–15. (See also ‘The Structure of Pure Mathematics’ in Wain, G. (ed.), Mathematical Education, Wokingham, Van Nostrand, Reinhold, 1978.)
Griffiths, H. B., Surfaces, London, Cambridge University Press, 1976.
Grossnickle, F. E. ‘Verbal Problem Solving’, The Arithmetic Teacher, 11 (1964), 12–17. Description of problem-solving procedure, based on assumption that three levels of maturity exist for this, together with specific examples.
Guilbaud, G. Th., ‘Les Langages d'Espaces’, in New Applications of Mathematics in Secondary Education, Luxemburg, 1975, 29–39. How terms and concepts of space language (geometry, topology) are used in every day life.
Gusset, J. C., Ghetto Children and Mathematics, 1971, (available from ERIC). The need to use the child's non-standard English in the mathematics lesson.
Hadamard, J., Psychology of Invention in the Mathematical Field, Princeton, Princeton University Press, 1949. ‘The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought’ Einstein.
Halliday, M. A. K., ‘Language and Experience’, Ed. Review 20(2) (1968), 95–106. Children's language problems.
Halliday, M. A. K., ‘Aspects of Sociolinguistic Research’, UNESCO, Paris, 1974. A survey paper prepared for the 1974 Nairobi Conference.
Halstead, M. H., Language Level, a Missing Concept in Information Theory, Tech Report 75, Purdue University, 1972.
Hanke, B. et al., ‘Soziale Interaktion im Unterricht’, Darstellung und Anwendung des Interaktionsanalyse — Systems von N. A. Flanders, Munich, Oldenbourg, 1974. Presents Flanders' system of social interaction in the classroom, and demonstrates its applicability to a practical example.
Hann, G., ‘Language, Logic and the Conditional’, Mathematics Teaching, 59 (1972), 18–20. Logical symbolism cannot be imposed on English language since motivation for connectives is often linguistic rather than logical. This leads to problems with communication, especially in giving exemplification to logical structures (through everyday examples) to aid teaching.
Heddens, J. W. and Smith, K. J., ‘The Readability of Elementary Mathematics Books’, The Arithmetic Teacher 11 (1964), 466–468. Readability formulae were applied to several textbooks. Found: (i) readability higher than assigned grade level; (ii) variation of readability level among books considered; (iii) variation within each textbook indicates some portions of texts should be comprehended by most students while other portions of same text are relatively difficult.
Hendrix, G., ‘Learning by Discovery’, The Mathematics Teacher 54 (1961), 290–299. Distinguishes three types of discovery method, which are each exemplified, and regards verbalisation as a means of communication not of conceptualisation. Conceptualisation proceeds by non-verbal awareness.
Henkin, L. A.,‘Linguistic Aspects of Mathematical Education’, in Lamon, W. E., Learning and the Nature of Mathematics (q.v.) On the introducation to young children of variables, quantifiers and other linguistic patterns.
Henry, J., Essays on Education, Harmondsworth, Penguin, 1971. A provocative discussion of sociological factors in education-expectations, aspiration, failure, motivation and the means by which these are communicated, cultivated, or destroyed, in several cultures but mainly American.
Hersee, J., ‘Notation and Language in School Mathematics’, Math. Gazette 61 (1977), 241–247. An attempt at standardisation of language and notation.
Hirabayashi, I., ‘Problems in Mathematical Symbolism-The Second, 4. Logical Words’. Reports of Mathematical Education, J.S.M.E. VII (1964), 43–52.
Hirabayashi, I., ‘Some Problems in Pedagogy of Mathematics II-Semiotic Aspects of Problems in Mathematics Education’. Bulletin of the Faculty of Education, Hiroshima University, Part 1, No. 22, 1973. ‘Children begin their learning of mathematics in the level of semantics and then pass into the level of syntax and later they often return again to the level of semantics.’
Hirabayashi, I. and Fujii, M., ‘Problems in Mathematical Education-The Fourth, 7. On Contraction of Mathematical Symbols’. Reports of Mathematical Education, J.S.M.E. X (1975), 1–14. Mathematics largely symbolisation, contraction of symbols, constitution of the new syntax of contracted symbols.
Hirabayashi, I. and Katayama, K. ‘Some Linguistic Aspects of Geometrical Diagrams’, Reports of Mathematical Education, J.S.M.E. XVII (1969), 1–14. Can geometrical diagrams be regarded as linguistic symbols, subject to rules of linguistics? Semantic, but not syntactical, rules found in geometry.
Hollands, R., ‘Language and Mathematics’, Mathematics for the Least Able, 11 to 14, Newsletter No. 11 (Feb. 1977), 4–5. The Scottish Centre for Mathematics, Science and Technical Education. On avoiding and alleviating reading problems.
Hunt, J. McV., Intelligence and Experience, New York, Ronald Press, 1961. A survey of several theories concerning human intelligence and thinking, including Piaget's work, and discussions of experimental evidence.
Iida, M., ‘Problems in Mathematical Symbolism-The Third, 6. On Easiness of Mathematical Symbolism’, Reports of Mathematical Education, J.S.M.E. VIII (1964), 27–32. Attempts to classify symbols according to factors such as ideographic content, degree of contraction, etc.
INRDP, Etude du rôle des moyens d'expression dans l'apprentissage mathématique. Various reports, Paris.
Irish, E. H., ‘Improving Problem Solving by Improving Verbal Generalisation’, The Arithmetic Teacher 11 (1964), 169–175. Description of experiments designed to test whether improved verbal generalisation aided verbal problem solving. The results show that it did.
Iwago, K., ‘Problems in Mathematical Symbolism-The Second, 3. Equivocal Words’, Reports of Mathematical Education, J.S.M.E. VII (1964), 35–42. “Equivocal word” if single “symbolism” used for different “substances”-shades of meaning. Collects and classifies equivocal words, with aim of deciding how to teach them.
Jansson, L. C., ‘Structural and Linguistic Variables that Contribute to Difficulty in the Judgement of Simple Verbal Deductive Arguments’, Educ. Studies Math. 5 (1974), 493–505. A statistically based study of deductive reasoning.
Jerman, M. E., ‘Problem Length as a Structural Variable in Verbal Arithmetic Problems’, Educ. Studies Math. 5 (1973), 109–123. Tests given to pupils in Grades 4,5,6,7,8. Difficulty of problem not solely influenced by numbers of words, but by their relation to other factors-possibly syntactic. Earlier work shows length variable (number of words) more important in upper grades.
Jerman, M. E. and Mirman, S., ‘Linguistic and Computational Variables in Problem Solving in Elementary Mathematics’, Educ. Studies Math. 5 (1974), 317–362. Linear regression model used to predict number of students who would solve verbal arithmetic problems correctly. Significant variables: number of long words, adverbs of time, mathematical terms, and length of longest sentence.
Johnson, H. C., ‘The Effect of Instruction in Mathematical Vocabulary Upon Problem Solving in Arithmetic’, J. of Ed. Res. 38 (1944), 97–110.
Johnson, P., ‘Mathematics as Human Communication’ in Lamon, W. E., Learning and the nature of mathematics (q.v.).
Journal of Children's Mathematical Behaviour (Madison Project) (Appears somewhat infrequently: Vol. I, (1) (1971), (2) 1973, (3) 1975, (4) 1977). Many case studies of children's behaviour in the classroom. Considerable emphasis on the use of developing language and symbolism.
Kane, R. B., ‘The Readability of Mathematical English’, J. of Research in Science Teaching 5 (1968), 296–298. Discusses differences between mathematical English (ME) and ordinary English (OE), and points out the inappropriateness of tests designed for OE being applied to ME tests to assess readability.
Kane, R. B., ‘The Readability of Mathematics Textbooks Revisited’, Mathematics Teacher 63 (1970), 579–581. Discusses reasons why readability formulae are not a suitable measure of ME.
Kaplan, H. W. and E., ‘Development of Word Meaning Through Verbal Context: An Experimental Study’, Journ. Psych. 29 (1950), 251–257.
Keislar, E. R. and Stern, C., ‘Young Children's Use of Language in Inferential Behaviour’, J. of Research and Development in Education 3 (1969), 1, 15–29. (Reaction Paper by J. A. R. Wilson, 30–31). Experiments to test value of oral responding in young children.
Keitel, C. and Otte, M., Das Lehrbuchproblem als Gegenstand der Lehrerausbilding, EPAS 1, IDM Bielefeld, 1977. Questions relating to textbooks. Good bibliography (58-mainly in German).
Kennedy, M. L., ‘Young Children's Use of Written Symbolism to Solve Verbal Addition and Subtraction Problems’, J. Child. Math. Behaviour I(4) (1977), 122–134. How children operate with informal written symbolism. Operational competence in writing formal symbols not indicative of understanding.
Klare, G. R., The Measurement of Readability, Iowa State University Press, 1963. A survey of common formulae. Jumbo bibliography.
Krygowska, Z., ‘La texte mathématique dans l'enseignement’, Educ. Stud. Math. 2(3) (1969), 360–370. An analysis of (pedagogical) errors of presentation to be found in many 1960's ‘modern maths’ textbooks.
Kulm, G., ‘Language Level and Information Content Measures in Mathematical English’, and ‘Language Level Applied to the Information Content of Technical Prose’, Purdue University, Indiana, 1974. Suggests using Halstead's formulae for computer languages to assess information content and language level of technical prose. Investigations suggest measures are more objective than readability tests, and may also permit assessment of expected learning rate.
Kysilka, M. L., The Verbal Teaching Behaviors of Mathematics and Social Studies Teachers in Eighth and Eleventh Grades, Ph.D. thesis, University of Texas at Austin, 1970.
Labov, W., ‘The Logic of Non-Standard English’, in Language and Poverty (Williams, F., ed.), Chicago, Markham, 1970.
Lacey, P. A. and Weil, P. E., ‘Number, Reading, Language’, Language Arts 52(6) (1975), 776–82. The construction of exercises integrating mathematical concepts and reading development.
Lakatos, I., Proofs and Refutations, London, Cambridge University Press, 1976. The discourse of conjecture and refutation.
Lamanna, J. B., The Effect of Teacher Verbal Behaviour on Pupil Achievement in Problem Solving in Sixth Grade Mathematics, Ph.D. thesis, St. John's University, 1969.
Lamon, W. E. (ed.), Learning and the Nature of Mathematics, Chicago, S.R.A., 1972. A collection of essays on mathematical education. Some, e.g., those by Henkin and Williams, being particularly concerned with language.
Lawton, D., Social Class, Language and Education, London, Routledge and Kegan Paul, 1968. Review and reports of work done on socio-cultural factors affecting language learning and use in the context of education, psychological theories, and remedial language programmes.
Linville, W. J., The Effects of Syntax and Vocabulary Upon the Difficulty of Verbal Arithmetic Problems with Fourth Grade Students, Ph.D. thesis, Indiana University, 1970.
Lörcher, G. A., ‘Mathematik als Fremdsprache’, in Reflektierte Schulpraxis 10(1) (1974), 1–11. An analysis of books used in the lower secondary school.
Love, E. and Tahta, D., ‘Language Across the Curriculum: Mathematics’, Mathematics Teaching 79 (1977), 48–49. A series of ordered questions intended to stimulate further work.
Lovell, K., Intellectual Growth and Understanding Mathematics, Math. Ed. Report, ERIC, 1971.
Lunzer, E. A., Bell, A. E., and Shiu, C. M., The Acquisition of Some Mathematical Notions in Children of School Age, London, SSRC, 1976. Considers, inter alia, the transition from natural to mathematical language.
Luria, A. R. and Yudovich, F. J., Speech and the Development of Mental Processes in the Child, Harmondsworth, Penguin, 1971.
Lyda, W. J. and Duncan, F. M., ‘Quantitative Vocabulary and Problem Solving’, The Arithmetic Teacher 14 (1967), 289–291. Summary of factors isolated by previous research.
Mackay, C. E., ‘Language in Maths’, Mathematics for the Least Able, 11 to 14, Newsletter No. 11, 3–4, (Feb. 1977). The Scottish Centre for Mathematics Science and Technical Education. Restricted code of slow learners v. elaborate code of teachers.
Macnamara, J., ‘The Effects of Instruction in a Weaker Language’, J. of Social Issues 23(2) (1967), 120–134.
Maier, H., ‘Zum Problem der Sprache in Mathematikunterricht’, Beitrage zum Mathematikunterricht, Schroedel, 1975, 110–114. Language in primary mathematics.
Martin, N. et al., Understanding Children Talking, Harmondsworth, Penguin, 1976a. Listening to children in the classroom. Some specifically mathematical examples.
Martin, N. et al., Writing and Learning Across the Curriculum 11–16, London, Ward Lock Ed., 1976b.
Mathematics Teaching, Periodical of the Association of Teachers of Mathematics (A.T.M. (q.v.)). This quarterly publication contains many articles on the use of language-particularly that used in the classroom. See also the Association's more informal publications: ATM Supplement and Recognitions.
Matsud, Y. et al., ‘On the Pattern of Reasoning in Daily Lives’. Reports of Mathematical Education, J.S.M.E. 31 (1977), 1–33. Discusses O'Brien's (etc.) work, and describes similar experiments in Japan-‘child's logic’ used more than ‘maths. logic’, but in some schools most children used ‘maths. logic’, ‘Child's logic’ not used exclusively, and patterns of thinking improved by instruction/training.
Miller, G. A. and Johnson-Laird, P. N., Language and Perception, London, Cambridge University Press, 1976. On concepts and language.
Mmari, G. R. V., ‘Tanzania's Experience in, and Efforts to Resolve, the Problem of Teaching Mathematics Through a Foreign Language’, in Accra Workshop Proceedings (q.v.). The problems of developing a workable language policy as the medium of instruction changes.
Morgan, J., Affective Consequences for the Learning and Teaching of Mathematics of an Individualised Learning Programme, University of Stirling, 1977. An unusual ‘evaluative’ study with valuable comments concerning, for example, classroom dialogue, and textual language and format.
Morris, R. W., ‘Linguistics Problems Encountered by Contemporary Curriculum Development Projects in Mathematics’, UNESCO, Paris, 1974. A wide-ranging comparative survey of problems arising in developing countries. Useful basic word list. (Preparatory paper for the Nairobi Conference.)
Morrison, A. and McIntyre, D., Schools and Socialization, Harmondsworth, Pelican Books, 1971.
Mugavero, A. C., ‘The Effect on Mathematical Achievement of Instruction in Four Language Skills’, Ed.D. thesis, Rochester, 1976. Specific language skills ((i) nontechnical vocabulary, (ii) technical vocabulary, (iii) identification of irrelevant words and phrases, (iv) ability to alter reading speed to adapt to material) might contribute to mathematical achievement.
Munroe, M. E., The Language of Mathematics, Ann Arbor, University of Michigan, 1963. An investigation of ‘Mathese’.
Nairobi Conference, Interactions Between Linguistics and Mathematical Education, UNESCO, Paris, 1974. Report of a symposium sponsored by UNESCO, CEDO and ICMI, Sept. 1–11, 1974. Particularly concerned with the problems of the developing countries in Anglophone Africa. Papers by Christiansen, Gay, Halliday, Morris, Strevens, etc. Bibliography (25).
Nesher, P. A., From Ordinary Language to Arithmetical Language in the Primary Grades. (What does it mean to teach ‘2+3=5?’), Ed.D. thesis, Harvard, 1972. Connections between the language of arithmetic, ordinary language, and the experience of the primary school child.
Nesher, P. and Teubal, E., ‘Verbal Cues as an Interfering Factor in Verbal Problem Solving’, Educ. Studies Math. 6 (1975), 41–51. The use of certain words which are always verbal cues results in an artificial mode of presentation of word-problems using specific and limited vocabulary-reflected in textbooks-connotations derived from problems, not experience with natural language. Key words found to influence choice of mathematical operation. Training should encourage revealing the underlying mathematical relations before giving a symbolic mathematical expression, and regardless of verbal formulation.
Nicholson, A. R., ‘Mathematics and Language’, Mathematics in School 6 No. 5, (Nov. 1977), 32–34. Investigation of CSE pupils' understanding of mathematical terms by contextual questions requiring words to be filled in. Designed (unlike earlier paper by Otterburn and Nicholson) to test just understanding, and not ability as such.
O'Brien, T. C., ‘Logical Thinking in Adolescents’, Educ. Studies Math. 4 (1972), 401–428. An investigation of the logical patterns used by children in Grades 9–12.
O'Brien, T. C., ‘Logical Thinking in College Students’, Educ. Studies Math. 5 (1973). 71–79. Tests given to College students who had attended a course in formal logic-‘child's logic’ used often and consistently. Context effects still evident, favouring causal items (contrapositive). Inverse easier than converse.
O'Brien, T. C., Shapiro, B. J., and Reali, N. C., ‘Logical Thinking-Language and Context’, Educ. Studies Math. 4 (1971), 201–219. Subjects (children in Grades 4–10) regard ‘If-then’ as meaning ‘If ... and only if ...’; ‘Child's Logic’-applied to ‘open’ schemes (inverse, converse)-partly due to underlying cognitive inability? Context may influence treatment of open schemes-probably more accurate on class inclusions than on causal items.
Otani, Y., The influence of the mother tongue on achievement in mathematics, 1975, (available from ERIC). Criticism of IEA study on international achievement for its failure to consider influence of mother tongue (Japanese).
Otterburn, M. K. and Nicholson, A. R., ‘The Language of (CSE) Mathematics’. Mathematics in School 5, No. 5, Nov. 1976, 18–20. Experimental report of tests applied to CSE pupils to evaluate their understanding of 36 words commonly used in maths. teaching and examining.
Ottmann, A., ‘Kann mit Mathematik Sprache erlernt werden? Bericht über einen Schulversuch in Zwei “Internationalen Grundschulklassen”’. Die Fachgruppe, Ludwigsburg, 1975, 1, 11–14. Reports of approaches to combined mathematical and language training for the children of foreign workers.
Paivio, A., ‘A Theoretical Analysis of the Role of Imagery in Learning and Memory’ in The Function and Nature of Imagery (Ed. Sheehan, P. W.), New York, Academic Press, 1972. A survey article which considers the role of verbal and imaginal codes.
Palzere. D. E., An Analysis of the Effects of Verbalization and Non-Verbalization in the Learning of Mathematics, Ph.D. thesis, University of Connecticut, 1968.
Peters, J., ‘Language and Mathematics Teaching in the Open University’, Teaching at a Distance 2, 31–34. Is traditional teaching language suitable for use in correspondence courses?
Philp, H., ‘Mathematical Education in Developing Countries-Some Problems of Teaching and Learning’ in Exeter Congress Proceedings (q.v.). The effects of language on the learning of mathematics in Papua New Guinea.
Piaget, J., The Language and Thought of the Child, London, Routledge and Kegan Paul, 1926. ‘Why does the child talk?’ ‘What are the functions of language?’ The first of a most influential series of studies of the child's mind.
Piaget, J., ‘How Children Form Mathematical Concepts’, Scientific American, Nov. 1953. Summary of Piaget's theory of development of mathematical concepts in children, with descriptions of the supporting experiments.
Piaget, J., The Construction of Reality in the Child, Basic Books, New York, 1954.
Piaget, J., Comments on Vygotsky's Critical Remarks Concerning ‘The Language and thought of the Child’ and ‘Judgement and Reasoning in the Child’, Cambridge, Mass., MIT Press, 1962.
Piaget, J., Six Psychological Studies, London, University of London Press, 1968. Outline of the mental development of the child, including the relationship between language and thought, (I, II, III).
Preston, M., ‘The Language of Early Mathematical Experience’, Mathematics in School 7(4) (1978), 31–32.
Pribnow, J. R., ‘Why Johnny Can't “Read” Word Problems’, School Science and Mathematics 69 (1969), 591–598. Exemplifies an approach to problem-solving which involves recognising general procedures.
Priesemann, G., ‘Die Fachsprachen des Unterrichts und das Problem der Verständigung’, Die höhere Schule, Düsseldorf 29 (1976), 3, 92–97. Specialised languages in school-language learning at all levels must remain in close relationship to human activities.
Proctor, V. H. and Wright, E. M., Systematic Observation of Verbal Interaction as a Method of Comparing Mathematics Lessons, ED 003 827, June 1961, (available from ERIC).
Reed, M. and Wainman, H., ‘Language Competence in Mathematics’, Int. J. Math. Educ. Sci. Technol. 9(1) (1978), 31–33. Asks whether particular problems exist in mathematics for those students who have to learn their discipline in a language which is not their mother tongue. One possible problem area is considered, the relationship between performance in mathematics and the understanding of mathematical text.
Ré éduc. orthophon ‘Mathématique et langage’, 13(85, 86) (1975), 385–480, 483–562. Two issues of the French periodical devoted to a report of a conference held to discuss ‘mathematics and language’. Contains eleven papers and accounts of discussions.
Reichenbach, H. and Walsch, W., ‘Einige, Probleme der Sprachschulung im Mathematikunterrich’, Zum logischen Denken im Mathematikunterricht, Berlin, VVW, 1975, 21–34. Interrelationship between language and logical thinking discussed and exemplified.
Renwick, E. M., Children Learning Mathematics, Ilfracombe, Stockwell, 1963. A collection of anecdotes, with interpretative comments, concerning children's mathematical misconceptions, and emotional reactions to mathematics learning.
Robinson, I., The New Grammarian's Funeral, London, Cambridge University Press, 1978. A critique of Chomsky's views of language.
Rogers, S. (ed.), Children and Language, London, Oxford University Press, 1975. A collection of papers concerning language in relation to society, thinking, meaning and the environment.
Rosen, C. and Rosen, H., The Language of Primary School Children, Harmondsworth, Penguin, 1973.
Rosen, H., Language and Class. A Critical Look at the Theories of Basil Bernstein, Bristol, Falling Wall Press, 1972.
Rotman, B., Mathematics as Language: An Essay in Semiotics, (typed notes) Bristol University, n.d. A resumé of the work of Saussure, Peirce and Jakobson and an attempt to apply their ideas to mathematics.
Sapir, E., Language, London, Rupert Hart-Davies, 1963.
Skemp, R. R., The Psychology of Learning Mathematics, Harmondsworth, Penguin, 1971. Outlines the theory of schematic learning, and describes aspects of learning mathematics-symbolisation, imagery, emotional factors-as they relate to this. Part B is concerned with presenting some fundamental mathematical ideas, based on theory of Part A.
Skemp, R. R., ‘Reflective Intelligence, and the Use of Symbols’, in Chapman, L. R. (q.v.).
Skemp, R. R., ‘Relational Mathematics and Instrumetal Mathematics-Some Further Thoughts’ University of Warwick, 1977.
Smedslund, J., ‘Development of Concrete Transitivity of Length in Children’, Child Dev., 34(2) (1963), 389–405.
Smedslund, J., ‘The Acquisition of Transitivity of Weight in Five to Seven-year old Children’, J. of Genetic Psych. 102 (1963b), 245–255.
Smith, E. B., Goodman, K. S., and Meredith, R., Language and Thinking in the Elementary School, New York, Holt, Rinehart and Winston, 1970.
Smith, F., ‘The Readability of Sixth Grade Word Problems’, School Science and Mathematics 71 (1971), 559–562. Dale-Chall formula applied to verbal problems aimed at 6th grade students.
Smith, K. J. and Heddens, J. W., ‘The Readability of Experimental Mathematics Materials’, The Arithmetic Teacher 11 (1964), 391–394. Brief outline of readability tests applied to various mathematics texts, finding that readability is generally inconsistent in a text and under-estimated by publishers/authors.
Steffe, L. P. (ed.), Research on Mathematical Thinking of Young Children, NCTM, 1975. General references to the development of linguistic and mathematical abilities. See, for example, the paper by Steffe and Carey.
Stenhouse, L., An Introduction to Curriculum Research and Development, London, Heinemann Educational, 1975.
Stern, C., ‘Acquisition of Problem-Solving Strategies in Young Children and its Relation to Verbalisation’, J. of Ed. Psych. 58 (1967), 245–252.
Stones, E., Readings in Educational Psychology, London, Methuen, 1970. A selection of papers and abstracts (usually edited and abridged). Relevant authors include Liublinskaya, Carroll, Sapir, Luria, Vigotsky, Piaget and Gagné. (Useful bibliographies.)
Strain, L. B., ‘Children's Literature: an aid in mathematics instruction’, The Arithmetic Teacher 16 (1969), 451–455. Advocates the use of children's literature not only for motivation of mathematics learning, but also as an aid to concept formation and development of linguistic skills-a stimulation to enquiry and discussion. Includes selection of examples.
Strevens, P., ‘What is Linguistics and How May it Help the Mathematics Teacher?’, UNESCO, Paris, 1974. An introductory paper prepared for the 1974 Nairobi Conference.
Stubbs, M., Language, Schools and Classrooms, London, Methuen, 1976. Discusses sociolinguistic aspects of classroom environment, with reference to some major theories and their shortcomings. Stresses complexity of the real situations, and advocates practical, well-recorded research.
Sutherland, J., ‘Aspects of Problem Solving’, Brit. J. of Ed. Psych. 11 (1941), 215–222 and 12 (1942), 35–46. An attempt to study the effect of vocabulary on efficiency in solving problems.
Sweet, A., ‘Children Need Talk’, Mathematics Teaching 58 (1972), 40–43. Teach mathematical concepts by verbalisation of concrete examples. Care in use of language-precision may avoid confusion. Recording badly is worse than no recording.
Taiwo, C. O., ‘Primary School Mathematics in an African Society’, London, Commonwealth Secretariat, 1968. General description of Yoruba culture and education in Nigeria, with passing mention of language problems. A paper prepared for the Commonwealth Conference on Mathematics in Schools, Trinidad, 1968. See also ‘Accra’.
Takai, K., ‘Relations between Language and the Teaching of Numbers’. Paper presented at the meeting of Canadian and Japanese Mathematics Educators, University of Alberta, August 1975. Describes Japanese number system(s), and teaching of multiplication.
Thom, R., ‘Modern Mathematics: Does it Exist?’ in Exeter Congress Proceedings (q.v.). Valuable ideas on language, grammar and meaning within mathematics.
Thom, R., ‘“Modern” Mathematics: An Educational and Philosophic Error’, Amer. Scientist 59 (6) (Dec., 1971), 695–699. On the hazards of mixing distinct semantic fields.
Thomas, G. R., ‘Precision, Accuracy, and Other Ambiguities’, The Mathematics Teacher 58 (1965), 509–511. Ambiguity of meaning implied by mathematician's or scientist's uses of words ‘precision’, ‘accuracy’, to mean ‘uncertainty’, ‘no uncertainty’.
Thornton, E. C. B., ‘The Power of Words’, Mathematics Teaching 38 (1967), 6–7. Confusion arises in manipulating arithmetic symbols through use of misleading, rote phrases, (e.g., “cancelling”, “turning upside down”, “taking out the brackets”, etc.) which are not sufficiently informal to be distinguished from formal operations. A mathematical explanation would be preferable. Other phrases: “into” and “turning it into”, “taking ... as”, etc.
Thornton, E. C. B., ‘Any must go’, in Mathematical Reflections, (see under A.T.M.). Discusses the uses made of logical connectives and quatifiers, with some exemplary proofs, and elucidates the logically inconsistent use of the word ‘any’ within such proofs.
Toda, K. et al., ‘Problems in Mathematical Symbolism-The First, 1. Mathematics and Symbolism, 2. Synonyms’, Reports of Mathematical Education, J.S.M.E. VI (1963), 25–36. Relationship between concepts and symbolism.
Trevino, B. A. G., An Analysis of the Effectiveness of a Bilingual Program in the Teaching of Mathematics in the Primary Grades, Ph.D. thesis, University of Texas, 1968.
Trueblood, C. R., ‘Promoting Problem-Solving Skills Through Non-Verbal Problems’, The Arithmetic Teacher 16 (1969), 7–9. Suggests that illustrated problems, where pupils discuss a verbal form, permit development of solving skills and verbalisation skills and hence may overcome reading difficulties. Greater interest if such problems are created by pupils. Examples are given.
UNESCO-UNICEF (Asian Region), Thé Development of Science and Mathematics Concepts in Children, UNESCO, Bangkok, 1972. Report of a conference held in Bangkok. Although only indirectly concerned with language, the problem is almost everywhere implicit.
Vollrath, H.-J., ‘The Understanding of Similarity and Shape in Classifying Tasks’, Educ. Studies Math. 8 (1977), 211–224. Description of experimental procedure of classifying tests which illustrate that colloquial and mathematical usage of a word do not coincide. Emphasis on acquiring vocabulary was a possible source of further confusion.
Vygotsky, L. S., Thought and Language, New York: MIT Press and John Wiley, 1962. A psychological study of the development of language in children: contrasts with Piagetian ideas.
Wallace, J. G., Concept Growth and the Education of the Child. A Survey of Research on Conceptualisation. Slough, NFER, 1965. To what extent is the learning of concepts facilitated by the use of verbal labels?
Wallace, J. G., An Enquiry into the Development of Concepts of Number in Young Children Involving a Comparison of Verbal and Non-Verbal Methods of Assessment and Acceleration, Ph.D. thesis, University of Bristol, 1967.
Weigand, R. B., ‘Readability of Mathematics Textbooks’, J. of Reading, 11 (1967), 201–4.
Weiner, S. L., ‘On the Development of More and Less’, J. of Exp. Ch. Psych. 17(2) (1974), 271–287. Language and mathematical development amongst pre-school children.
Whitehead, A. N., An Introduction to Mathematics, London, Oxford University Press, 1911.
Whorf, B. L., Language, Thought and Reality, New York, MIT Press and Wiley, 1956. Collected writings which cover ideas of theoretical linguistic analysis, language in psychology of mind, and specific examples from South American languages which are analysed for structure and meaning of English.
Williams, J. D., ‘Some Peculiarities of Calculative Thinking and Their Consequences’, in Lamon, W. E. (ed.), Learning and the Nature of Mathematics (q.v.).
Wilkinson, A. M., The Foundations of Language, London, Oxford University Press, 1971. The interaction of language and social learning.
Wilkinson, A. M., Language and Education, London, Oxford University Press, 1975.
Wilmon, B., ‘Reading in the Content Area: A “New Math” Terminology List for the Primary Grades’, Elementary English 48 (1971), 463–471. Suggests that specialised vocabulary is important for mathematics readability, and reviews reports of compilations of word lists.
Wittgenstein, L. J. J., Philosophical Investigations, Oxford, Basil Blackwell, 1958. Meaning, knowing, understanding. “Philosophy is a battle against the bewitchment of our intelligence by means of language”. (See also: Hardwick, C. S., Language Learning in Wittgenstein's Later Philosophy, The Hague, Mouton, 1971.)
Wode, D., ‘Mathematikunterricht und Spracherziehung’, Der Mathematikunterricht, 22 (1976), 1, 44–64. Need for more precise use of language in primary schools. One of five papers in a special issue devoted to ‘logic and language’.
Wrigley, J., ‘The Factorial Nature of Ability in Elementary Mathematics’, B. J. of Ed. Psych. 28, 1958, 61–78. The relationship between verbal and mathematical abilities. (Wrigley's findings have met with opposition-see Aiken, 1972.)
Young, A., Outline of a Project for Introducing Bilingual Education into Basic Schools in Jamaica, Kingston, Jamaica, 1977.
Zaslavsky, C., Africa Counts: Number and Pattern in African Culture, Boston, Mass., Prindle, Weber and Schmidt, 1973. Historical and cultural descriptions of mathematical concepts in African countries, with studies of Nigerian and East African concepts in particular.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Austin, J.L., Howson, A.G. Language and mathematical education. Educ Stud Math 10, 161–197 (1979). https://doi.org/10.1007/BF00230986
Issue Date:
DOI: https://doi.org/10.1007/BF00230986