FOOTNOTES

AmountPercentage of Correct FiguresInitial ScoreGainInitial ScoreGainInitially highest five individuals85617018next five "56516810next six "4622748next six "3885812next six "29245614

Achievement in arithmetic depends upon a number of different abilities. For example, accuracy in copying numbers depends upon eyesight, ability to perceive visual details, and short-term memory for these. Long column addition depends chiefly upon great strength of the addition combinations especially in higher decades, 'carrying,' and keeping one's place in the column. The solution of problems framed in words requires understanding of language, the analysis of the situation described into its elements, the selection of the right elements for use at each step and their use in the right relations.

Since the abilities which together constitute arithmetic ability are thus specialized, the individual who is the best of a thousand of his age or grade in respect to, say, adding integers, may occupy different stations, perhaps from 1st to 600th, in multiplying with integers, placing the decimal point in division with decimals, solving novel problems, copying figures, etc., etc. Such specialization is in part due to his having had, relatively to the others in the thousand, more or better training in certain of these abilities than in others, and to various circumstances of life which have caused him to have, relatively to the others in the thousand, greater interest in certain of these achievements than in others. The specialization is not wholly due thereto, however. Certain inborn characteristics of an individual predispose him to different degrees of superiority or inferiority to other men in different features of arithmetic.

We measure the extent to which ability of one sort goes with or fails to go with ability of some other sort by the coefficient of correlation between the two. If every individual keeps the same rank in the second ability—if the individual who is the best of the thousand in one is the best of the group in the other, and so on down the list—the correlation is 1.00. In proportion as the ranks of individuals vary in the two abilities the coefficient drops from 1.00, a coefficient of 0 meaning that the best individual in ability A is no more likely to be in first place in ability B than to be in any other rank.

The meanings of coefficients of correlation of .90, .70, .50, and 0 are shown by Tables 15, 16, 17 and 18.[26]

TABLE 15

Distribution of Arrays in Successive Tenths of the Group Whenr= .90

10TH9TH8TH7TH6TH5TH4TH3D2D1ST1st tenth.1.41.86.622.468.72d tenth.1.41.44.711.523.536.022.43d tenth.1.52.15.812.821.127.423.56.64th tenth.42.16.412.820.123.821.211.51.85th tenth.11.45.812.819.322.620.112.84.7.46th tenth.44.712.820.122.619.312.85.81.4.17th tenth1.811.521.223.820.112.86.42.1.48th tenth6.623.527.421.112.85.82.1.5.19th tenth22.436.023.511.54.71.4.4.110th tenth68.722.46.61.8.4.1

TABLE 16

Distribution of Arrays in Successive Tenths of the Group Whenr= .70

10TH9TH8TH7TH6TH5TH4TH3D2D1ST1st tenth.2.71.52.84.88.013.022.346.72d tenth.21.22.64.57.09.813.417.321.722.33d tenth.72.65.07.310.012.514.916.717.313.04th tenth1.54.57.39.812.013.714.814.913.48.05th tenth2.87.010.012.013.414.013.712.59.84.86th tenth4.89.812.513.714.013.412.010.07.02.87th tenth8.013.414.914.813.712.09.87.34.51.58th tenth13.017.316.714.912.510.07.35.02.6.79th tenth22.321.717.313.49.87.04.52.61.2.210th tenth46.722.313.08.04.82.81.5.7.2

TABLE 17

Distribution of Arrays of Successive Tenths of the Group Whenr= .50

10TH9TH8TH7TH6TH5TH4TH3D2D1ST1st tenth.82.03.24.66.28.110.513.918.031.82d tenth2.04.15.77.38.810.512.214.116.418.93d tenth3.25.77.48.910.011.212.313.314.113.94th tenth4.67.38.89.910.811.612.012.312.210.55th tenth6.28.810.010.811.311.511.611.210.58.16th tenth8.110.511.211.611.511.310.810.08.86.27th tenth10.512.212.312.011.610.89.98.87.54.68th tenth13.914.113.312.311.210.08.87.45.73.29th tenth18.916.414.112.210.58.87.35.74.12.010th tenth31.818.913.910.58.16.24.63.22.0.8

TABLE 18

Distribution of Arrays, in Successive Tenths of the Group Whenr= .0

10TH9TH8TH7TH6TH5TH4TH3D2D1ST1st tenth101010101010101010102d tenth101010101010101010103d tenth101010101010101010104th tenth101010101010101010105th tenth101010101010101010106th tenth101010101010101010107th tenth101010101010101010108th tenth101010101010101010109th tenth1010101010101010101010th tenth10101010101010101010

The significance of any coefficient of correlation depends upon the group of individuals for which it is determined. A correlation of .40 between computation and problem-solvingin eighth-grade pupils of 14 years would mean a much closer real relation than a correlation of .40 in all 14-year-olds, and a very, very much closer relation than a correlation of .40 for all children 8 to 15.

Unless the individuals concerned are very elaborately tested on several days, the correlations obtained are "attenuated" toward 0 by the "accidental" errors in the original measurements. This effect was not known until 1904; consequently the correlations in the earlier studies of arithmetic are all too low.

In general, the correlation between ability in any one important feature of computation and ability in any other important feature of computation is high. If we make enough tests to measure each individual exactly in:—

(A) Subtraction with integers and decimals,

(B) Multiplication with integers and decimals,

(D) Multiplication and division with common fractions, and

(E) Computing with percents,

we shall probably find the intercorrelations for a thousand 14-year-olds to be near .90. Addition of integers (F) will, however, correlate less closely with any of the above, being apparently dependent on simpler and more isolated abilities.

The correlation between problem-solving (G) and computation will be very much less, probably not over .60.

It should be noted that even when the correlation is as high as .90, there will be some individuals very high in one ability and very low in the other. Such disparities are to some extent, as Courtis ['13, pp. 67-75] and Cobb ['17] have argued, due to inborn characteristics of the individual in question which predispose him to very special sorts ofstrength and weakness. They are often due, however, to defects in his learning whereby he has acquired more ability than he needs in one line of work or has failed to acquire some needed ability which was well within his capacity.

In general, all correlations between an individual's divergence from the common type or average of his age for one arithmetical function, and his divergences from the average for any other arithmetical function, are positive. The correlation due to original capacity more than counterbalances the effects that robbing Peter to pay Paul may have.

Speed and accuracy are thus positively correlated. The individuals who do the most work in ten minutes will be above the average in a test of accuracy. The common notion that speed is opposed to accuracy is correct when it means that the same person will tend to make more errors if he works at too rapid a rate; but it is entirely wrong when it means that the kind of person who works more rapidly than the average person is likely to be less accurate than the average person.

Interest in arithmetic and ability at arithmetic are probably correlated positively in the sense that the pupil who has more interest than other pupils of his age tends in the long run to have more ability than they. They are certainly correlated in the sense that the pupil who 'likes' arithmetic better than geography or history tends to have relatively more ability in arithmetic, or, in other words, that the pupil who is more gifted at arithmetic than at drawing or English tends also to like it better than he likes these. These correlations are high.

It is correct then to think of mathematical ability as, in a sense, a unitary ability of which any one individual may have much or little, most individuals possessing a moderateamount of it. This is consistent, however, with the occasional appearance of individuals possessed of very great talents for this or that particular feature of mathematical ability and equally notable deficiencies in other features.

Finally it may be noted that ability in arithmetic, though occasionally found in men otherwise very stupid, is usually associated with superior intelligence in dealing with ideas and symbols of all sorts, and is one of the best early indications thereof.

FOOTNOTES[1]The following and later problems are taken from actual textbooks or courses of study or state examinations; to avoid invidious comparisons, they are not exact quotations, but are equivalents in principle and form, as stated in the preface.[2]The work of Mitchell has not been published, but the author has had the privilege of examining it.[3]The form of Test 6 quoted here is that given by Courtis ['11-'12, p. 20]. This differs a little from the other series of Test 6, shown on pages 43 and 44.[4]Eight or ten timesin all, not eight or ten times for each fact of the tables.[5]The facts concerning the present inaccuracy of school work in arithmetic will be found on pages 102 to 105.[6]McLellan and Ames,Public School Arithmetic[1900].[7]These concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work.[8]The very early learning of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, and perhaps a few more multiplications is not considered here. It is advisable. The treatment of 0 × 0, 0 × 1, 1 × 0, etc., is not considered here. It is probably best to defer the '× 0' bonds until after all the others are formed and are being used in short multiplication, and to form them in close connection with their use in short multiplication. The '0 ×' bonds may well be deferred until they are needed in 'long' multiplication, 0 × 0 coming last of all.[9]See page 76.[10]At the end of a volume or part, the count may be from as few as 5 or as many as 12 pages.[11]Certain paragraphs in this and the following chapter are taken from the author'sEducational Psychology, with slight modifications.[12]It should be noted that just as concretes give rise to abstractions, so these in turn give rise to still more abstract abstractions. Thus fourness, fiveness, twentyness, and the like give rise to 'integral-number-ness.' Similarly just as individuals are grouped into general classes, so classes are grouped into still more general classes. Half, quarter, sixth, and tenth are general notions, but 'one ...th' is more general; and 'fraction' is still more general.[13]They may, of course, also result in a fusion or an alternation of responses, but only rarely.[14]The more gifted children may be put to work using the principle after the first minute or two.[15]If desired this form may be used, with the appropriate difference in the form of the questions and statements.232300006966960[16]Courtis finds in the case of addition that "of all the individuals making mistakes at any given time in a class, at least one third, and usually two thirds, will be making mistakes in carrying or copying."[17]Facts concerning the conditions of learning in general will be found in the author'sEducational Psychology, Vol. 2, Chapter 8, or in theEducational Psychology, Briefer Course, Chapter 15.[18]See Thorndike ['00], King ['07], and Heck ['13].[19]A special type could be constructed that would use a large type body, say 14 point, with integers in 10 or 12 point and fractions much larger than now.[20]It will be still better if the 4 is replaced by an open-top 4.[21]For an account in English of their main findings see Howell ['14], pp. 149-251.[22]In hisHow We Think.[23]Compiled from data on p. 89 of Kruse ['18].[24]Siblings is used for children of the same parents.[25]Similar results have been obtained in the case of arithmetical and other abilities by Thorndike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and Thorndike ['13], Hahn and Thorndike ['14], and on a very large scale by Race in a study as yet unpublished.[26]Unless he has a thorough understanding of the underlying theory, the student should be very cautious in making inferences from coefficients of correlation.

[1]The following and later problems are taken from actual textbooks or courses of study or state examinations; to avoid invidious comparisons, they are not exact quotations, but are equivalents in principle and form, as stated in the preface.

[2]The work of Mitchell has not been published, but the author has had the privilege of examining it.

[3]The form of Test 6 quoted here is that given by Courtis ['11-'12, p. 20]. This differs a little from the other series of Test 6, shown on pages 43 and 44.

[4]Eight or ten timesin all, not eight or ten times for each fact of the tables.

[5]The facts concerning the present inaccuracy of school work in arithmetic will be found on pages 102 to 105.

[6]McLellan and Ames,Public School Arithmetic[1900].

[7]These concern allowances for two errors occurring in the same example and for the same wrong answer being obtained in both original work and check work.

[8]The very early learning of 2 × 2, 2 × 3, 3 × 2, 2 × 4, 4 × 2, 3 × 3, and perhaps a few more multiplications is not considered here. It is advisable. The treatment of 0 × 0, 0 × 1, 1 × 0, etc., is not considered here. It is probably best to defer the '× 0' bonds until after all the others are formed and are being used in short multiplication, and to form them in close connection with their use in short multiplication. The '0 ×' bonds may well be deferred until they are needed in 'long' multiplication, 0 × 0 coming last of all.

[9]See page 76.

[10]At the end of a volume or part, the count may be from as few as 5 or as many as 12 pages.

[11]Certain paragraphs in this and the following chapter are taken from the author'sEducational Psychology, with slight modifications.

[12]It should be noted that just as concretes give rise to abstractions, so these in turn give rise to still more abstract abstractions. Thus fourness, fiveness, twentyness, and the like give rise to 'integral-number-ness.' Similarly just as individuals are grouped into general classes, so classes are grouped into still more general classes. Half, quarter, sixth, and tenth are general notions, but 'one ...th' is more general; and 'fraction' is still more general.

[13]They may, of course, also result in a fusion or an alternation of responses, but only rarely.

[14]The more gifted children may be put to work using the principle after the first minute or two.

[15]If desired this form may be used, with the appropriate difference in the form of the questions and statements.232300006966960

[15]If desired this form may be used, with the appropriate difference in the form of the questions and statements.

[16]Courtis finds in the case of addition that "of all the individuals making mistakes at any given time in a class, at least one third, and usually two thirds, will be making mistakes in carrying or copying."

[17]Facts concerning the conditions of learning in general will be found in the author'sEducational Psychology, Vol. 2, Chapter 8, or in theEducational Psychology, Briefer Course, Chapter 15.

[18]See Thorndike ['00], King ['07], and Heck ['13].

[19]A special type could be constructed that would use a large type body, say 14 point, with integers in 10 or 12 point and fractions much larger than now.

[20]It will be still better if the 4 is replaced by an open-top 4.

[21]For an account in English of their main findings see Howell ['14], pp. 149-251.

[22]In hisHow We Think.

[23]Compiled from data on p. 89 of Kruse ['18].

[24]Siblings is used for children of the same parents.

[25]Similar results have been obtained in the case of arithmetical and other abilities by Thorndike ['08, '10, '15, '16], Whitley ['11], Starch ['11], Wells ['12], Kirby ['13], Donovan and Thorndike ['13], Hahn and Thorndike ['14], and on a very large scale by Race in a study as yet unpublished.

[26]Unless he has a thorough understanding of the underlying theory, the student should be very cautious in making inferences from coefficients of correlation.

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