# Language and mathematical education

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Mathematical Education
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## Bibliography

- Abidjan, Conference on ‘Mathématique et Milieu en Afrique’ 1978-Report to appear (Conference included presentations and working groups devoted to linguistic problems).Google Scholar
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*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).Google Scholar - Adelman, C. and Walker, R.,
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*J. Res. Ed.***2**, (1971), 304–313. A precursor-taking a more limited view-of Aiken (1972).Google Scholar - 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.Google Scholar - Allardice, B., ‘The Development of Written Representations for Some Mathematical Concepts’,
*J. Child. Math. Behaviour***I**(**4**) (1977), 135–160. Early encounters with symbolism.Google Scholar - Amidon, E. J. and Hunter, E.,
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*Ed. for Teaching*(1973).Google Scholar - 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.Google Scholar - A.T.M.,
*Mathematical Reflections*, London, Cambridge University Press, 1970. Collection of essays, some of which, e.g. Tahta, Caldwell, Thornton, directly concern language.Google Scholar - A.T.M. (ed. Wheeler, D. H.),
*Notes on Mathematics for Children*, London, Cambridge University Press, 1977. Similar to A.T.M., 1967.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Barnes, D., ‘Language and Learning in the Classroom’,
*J. Curr. Studies***3**(1) (1971(a)), 29–38.Google Scholar - Barnes, D., ‘Classroom Contexts for Language and Learning’,
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*From Communication to Curriculum*. Harmondsworth, Penguin, 1976.Google Scholar - Bauersfeld, H., ‘Kommunikationsmuster im Mathematikunterricht’, IDM, Bielefeld, 1978. A discussion of the use of language in the classroom.Google Scholar
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*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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Britton, J.,
*Language and Learning*, Harmondsworth, Allen Lane, 1970.Google Scholar - 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.Google Scholar - Bruner, J. S., Goodnow, J. J., and Austin, J. A.,
*A Study of Thinking*, New York, Wiley, 1956.Google Scholar - Bruner, J. S., Olver, R., and Greenfield, P.,
*Studies in Cognitive Growth*, New York, Wiley, 1966.Google Scholar - Brunner, R. B., ‘Reading Mathematical Exposition’,
*Ed. Research***18**(3) (1976), 208–213. Attempts to provide an ‘operational definition of successful reading of mathematics’.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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’.Google Scholar - 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.Google Scholar - carpenter, T. P. (ed.), ‘Notes from National Assessment: Word Problems’,
*The Arithmetic Teacher***23**(1976), 3, 389–393.Google Scholar - 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.Google Scholar - 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’.Google Scholar - 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.Google Scholar - Chausard, M., ‘Mathématique et Langage’,
*Rev. Gén Enseignt. Déf audit***68**(4) (1976), 215–35. An account of teaching mathematics to deaf children.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.)Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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’.Google Scholar
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*Eng. Lang. J.***1**(1) (1970), 25–31.Google Scholar - Cormack, A.,
*A Study of Language in Mathematics Text-Books*, M. Phil Thesis, Chelsea College, 1978. Books for 11-year-olds.Google Scholar - Coulthard, R. M.
*et al., Discourse in the Classroom*, CIIT, 1972.Google Scholar - 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.Google Scholar - Creber, P.,
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*Linguistics*, Harmondsworth, Penguin, 1971. General introduction giving social and historical contexts for an outline of the major theories and controversies.Google Scholar - Dale, E. and Chall, J. L., ‘A Formula for Predicting Readability’,
*Ed. Res. Bull.*(Jan. 1949), 11–20.Google Scholar - Davidson, J. E., ‘The Language Experience Approach to Story Problems’,
*The Arithmetic Teacher***25**(1977), 28.Google Scholar - 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.Google Scholar - Davis, R. B., See also
*Journal of Children's Mathematical Behaviour*.Google Scholar - 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.Google Scholar - 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.Google Scholar - Earp, N. W., ‘Observations on Teaching Reading in Mathematics’,
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*The Ability of Sixth Grade Students to Associate Mathematical Terms with Related Algorithms*, Ed.D. thesis, Indiana, 1973.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Fey, J. T.,
*Patterns of Verbal Communication in Mathematics Classes*, Ph.D. thesis, Columbia University, 1969.Google Scholar - Fielker, D. S., ‘Editorial’,
*Mathematics Teaching***80**(Sept. 1977), 2–3. Includes comments on language and workcards.Google Scholar - Flanders, N. A.,
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*Analyzing Teaching Behaviour*, Reading Mass., Addison Wesley, 1970.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Fujiwara, S., ‘Problems in Mathematical Symbolism-The Third, 5. Ambiguous Words’.
*Reports of Mathematical Education*, J.S.M.E.**VIII**(1964), 18–26.Google Scholar - Furth, H. G.,
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*Deafness and Learning*, Belmont, Cal., Wadsworth, 1973. Piagetian ideas applied to the teaching of the deaf.Google Scholar - Gagné, R. M., ‘The Learning of Principles’, in Klausmeier, H. J. and Harris, C. W.
*Analyses of Concept Learning*, New York, Academic Press, 1966.Google Scholar - 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).Google Scholar - 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.Google Scholar - Gattegno, C., ‘Mathematics and the Deaf’, in
*For the Teaching of Mathematics*(**2**), Reading, Educ. Explorers, 1963. Reflections on non-verbal teaching.Google Scholar - Gattegno, C.,
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*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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Giles, W. H., ‘Mathematics in Bilingualism: a Pragmatic Approach’,
*ISA Bulletin***55**(1969), 19–26.Google Scholar - Gilliland, J.,
*Readability*, London, University of London Press, 1972. General introduction describing recognised variables and tests used to measure them. Readable!Google Scholar - 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.Google Scholar - Ginsberg, H.,
*The Myth of the Deprived Child*, New York, Prentice-Hall, 1972.Google Scholar - Goeppert, H. C. (ed.),
*Sprachverhalten im Unterricht*, Munich, W. Fink, 1977.Google Scholar - Goutard, M.,
*Mathematics and Children*, Reading, England, Educational Explorers, 1964. Descriptions and examples of learning through experience.Google Scholar - Greenfield, P. M., ‘On Culture and Conservation’ in Bruner, J. S., Olver, R. R., and Greenfield, P. M.,
*Studies in Cognitive Growth*(q.v.).Google Scholar - 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.)Google Scholar - Griffiths, H. B.,
*Surfaces*, London, Cambridge University Press, 1976.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Halliday, M. A. K., ‘Language and Experience’,
*Ed. Review***20**(2) (1968), 95–106. Children's language problems.Google Scholar - Halliday, M. A. K., ‘Aspects of Sociolinguistic Research’, UNESCO, Paris, 1974. A survey paper prepared for the 1974 Nairobi Conference.Google Scholar
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*Language Level, a Missing Concept in Information Theory*, Tech Report 75, Purdue University, 1972.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Hersee, J., ‘Notation and Language in School Mathematics’,
*Math. Gazette***61**(1977), 241–247. An attempt at standardisation of language and notation.Google Scholar - Hirabayashi, I., ‘Problems in Mathematical Symbolism-The Second, 4. Logical Words’.
*Reports of Mathematical Education*, J.S.M.E.**VII**(1964), 43–52.Google Scholar - 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.’Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - INRDP,
*Etude du rôle des moyens d'expression dans l'apprentissage mathématique*. Various reports, Paris.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - 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.Google Scholar - Johnson, H. C., ‘The Effect of Instruction in Mathematical Vocabulary Upon Problem Solving in Arithmetic’,
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*Learning and the nature of mathematics*(q.v.).Google Scholar *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.Google Scholar- 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.Google Scholar - 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.Google Scholar - Kaplan, H. W. and E., ‘Development of Word Meaning Through Verbal Context: An Experimental Study’,
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*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.Google Scholar - 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).Google Scholar - 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.Google Scholar - Klare, G. R.,
*The Measurement of Readability*, Iowa State University Press, 1963. A survey of common formulae. Jumbo bibliography.Google Scholar - 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.Google Scholar - 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.Google Scholar
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*The Verbal Teaching Behaviors of Mathematics and Social Studies Teachers in Eighth and Eleventh Grades*, Ph.D. thesis, University of Texas at Austin, 1970.Google Scholar - Labov, W., ‘The Logic of Non-Standard English’, in
*Language and Poverty*(Williams, F., ed.), Chicago, Markham, 1970.Google Scholar - 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.Google Scholar - Lakatos, I.,
*Proofs and Refutations*, London, Cambridge University Press, 1976. The discourse of conjecture and refutation.Google Scholar - 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.Google Scholar - Lamon, W. E. (ed.),
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