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The Psychology of Mathematical Ability

Published online by Cambridge University Press:  03 November 2016

A. G. Hughes
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Extract

Both terms in the title of this paper are probably suspect. The mathematician, versed in a subject with hundreds of years of noble tradition, tends to be suspicious of a paper on psychology; the teacher of mathematics, immersed daily in psychological relationships with ordinary children, tends to be suspicious of a paper on mathematical ability. The discussion promises to be unsatisfactory, to the one because psychology lacks mathematical precision, to the other because mathematical ability seems to present no urgent psychological problems. The teacher-mathematician may then be pardoned if he reacts sceptically to our title with the petition, “Psychology, if there be a science of psychology, help me to cope with mathematical disability.” But these suspicions are not well founded. Not only is psychology, as we shall see, becoming increasingly mathematical, but also, if we interpret our terms as mathematicians should, disability is seen to be merely the negative aspect of ability. As such, it will not be excluded from our discussion.

Type
Research Article
Information
The Mathematical Gazette , Volume 14 , Issue 196 , October 1928 , pp. 205 - 215
Copyright
Copyright © Mathematical Association 1928

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Footnotes

*

A paper read before the Mathematical Association (London Branch) on February 25, 1928. It has been partly rewritten to meet some of the points raised in the subsequent discussion, and I have also had the benefit of some very helpful suggestions and criticisms from Professor Cyril Burt. In acknowledging my indebtedness to Professor Burt I ought to state that he does not necessarily subscribe to all the speculations I have advanced in this paper

References

page note 206 * J. B. S. Haldane, Possible Worlds, p. 139.

page note 206 † For a fuller defence at the “atomised” test, see P. B. Ballard, The New Examintr, Chapter VI.

page note 206 ‡ See, for example: P. B. Ballard, Group Tests: of Intelligence; Cyril Burt, Northumberland. Standardized Tests, General Intelligence.

page note 206 § A. L. Sogers, Experimental Tests of Mathematical Ability and their Prognostic Valus (1923).

page note 206 ‖ See, for example, Cyril Burt, Northumberland Standardized Test», Arithmetic.

page note 206 ¶ P. B. Ballard, The New Examiner, pp. 200-202.

page note 207 * In the explanation of the Two Factor Theory which follows, I am very deeply indebted to the expositions of Professor C. Spearman in his book, The Abilities of Man, and of Professor Cyril Burt in his pamphlet, The Measurement of Mental Capacities.

page note 208 * It may be roughly identified with what is popularly called “general intelligence.”

page note 208 † This criterion is due to Professor C. Spearman. He gives a mathematical proof of it in his recent book, The Abilities of Man, Appendix.

page note 210 * The existence of special common factors can also be shown by the method of partial correlation. As Spearman points out, the tetrad difference criterion can be easily derived from Yule’s formula for partial correlation. See Spearman, op. cit. pp. ii and iii, Appendix.

page note 210 † D. Collar, “A Statistical Survey of Arithmetical Ability,” Brit. Jnl. Psych, xi. 135-158 (1920).

page note 209 ‡ Brown and Burt had previously called attention to special arithmetical ability. See W. Brown, “An Objective Study of Mathematical Intelligence,” Biometrica, vii. (1910); “Some Experimental Results in the Correlation of Mental Abilities,” Brit. Jnl. Psych, iii. (1910); C. Burt, The Distribution and Relations of Educational Abilities, p. 58 (1917).

page note 209 § A. L. Rogers, op. cit.; W. S. Mack, “Investigation of Mathematical Ability in the Classroom,” Forum of Education, iv. 44-56 (1926); L. Fouracre, “Psychological Tests of Mathematical Ability,” Forum of Education, iv. 201-205 (1926). See also C. Spearman, op. cit. pp. 230-232.

page note 210 * Quoted from C. Spearman, op. cit. p. 232.

page note 210 † B. Branford, A Study of Mathematical Education, p. 123.

page note 210 ‡ A. N. Whitehead, An Introduction to Mathematics, p. 240.

page note 210 § See T. P. Nunn, “An Elementary School Syllabus In Mathematics,” Forum of Education, ii. (1924).

page note 210 ‖ Op. cü. p. 228.

page note 210 ¶ Fouracre, op. cit. p. 203.

page note 211 * The psychology of mathematical reasoning is therefore not discussed in this paper; we are immediately concerned with the special factors which differentiate mathematical ability from other forms of ability. Readers who wish to follow up the role of general intelligence in mathematical work will find a good introduction to a fairly extensive literature on the subject in The Psychology of Reasoning, chapters vii. viii. and ix. by E. Rignano.

page note 211 † The “collective common sense” of teachers seems to suggest that ability to do solid geometry may be distinct from ability to do plane geometry.

page note 211 ‡ V. Hazlitt, Ability (1927).

page note 211 § C. Burt, The Measurement of Mental Capacities, p. 41.

page note 212 * E. Jones, Psycho-Analysis, p. 648.

page note 212 † E. Eignano, op. cü. p. 159.

page note 212 ‡ This is perhaps not so true ol the advanced pure mathematician, for, as Santayana says, “It is a pleasant surprise to him and an added problem if he finds that the arts can use his calculations, or that the senses can verify them, much as if a composer found that the sailors could heave better when singing his songs.” G. Santayana, “Revolutions in Science,” The New Adelphi, March 1928.

page note 213 * Fehr, by a questionnaire sent out to 100 eminent mathematicians, found that in two-thirds of his cases mathematical ability was in the family. See W. Brown, Mind and Personality, chapter ix. “Mind and Mathematical Ability.”

page note 213 † On the subject of sex-differences in mathematical ability, see C. Burt, Mental and Scholastic Tests, pp. 295-302 and 403-406; A. E. Cameron, “A Comparative Study of the Mathematical Ability of Boys and Girls in Secondary Schools,” Brit. Jnl. Psych, xvi. 29-49 (1925).

page note 213 ‡ Aldous Huxley, Proper Studies, pp. 78 ff.

page note 214 * See M. McFarlane, “A Study of Practical Ability,” Brit. Jul. Psych. Monograph Supplement, viii. (1925); Cyril Burt, The Measurement of Mental Capacities, p. 41; C. Spearman, op. cit. p. 229.