876 



SCIENCE 



[N. S. Vol. XXXIX. No. 1015 



Our scheme of measuring mathematical 

 abilities resolves itself into two parts, as fol- 

 lows: 



1. A formula for " arraying " students in 

 order of ability, that is, for determining the 

 relative positions of the members of a class, so 

 as to establish the order of merit, or the rank 

 of each individual in the group. This formula 

 furnishes also preliminary estimates of abil- 

 ity. 



2. A revision of these preliminary estimates 

 so as to supplant them by an absolute stand- 

 ard. 



Part I 



Mathematical ability depends in part upon 

 knowledge of a subject and proficiency in 

 carrying on accurately the mechanical opera- 

 tions connected with it. This kind of ability 

 may be determined by the usual memory tests 

 conducted from day to day in the class-room, 

 and at longer intervals by examination. 



Mathematical ability is measured also by 

 the success in solving original exercises. 

 These tests are made in daily work, and also 

 in final examinations. 



The observation of instructors and the 

 teachings of the history of science suggest a 

 still further test of mathematical power, 

 namely, the diligence or tenacity displayed by 

 a pupil in pursuing his work. A pupil of only 

 average talents, but of great tenacity of pur- 

 pose, may achieve more in his life than a 

 bright pupil of limited powers of application. 

 A standard illustration is the case of Robert 

 Mayer, who as a pupil made only a moderate 

 record, but who, by his extraordinary tenacity 

 of purpose was led to the discovery of the law 

 of the conservation of energy. 



In Germany and Switzerland this feature is 

 being recognized in the records and reports of 

 scholarship. When I was a boy I received two 

 marks on every subject, one for Fleiss, or dili- 



is A. G. Steele, Pedagogical Seminary, Vol. 18, 

 1911, p. 523. 



18 W. L. Stevens, Popular Science Monthly, 

 Vol. 63, 1903, p. 312. 



17 D. Starch, Psychol. Bulletin, Vol. 10, 1913. 

 p. 74; Science, Vol. 38, 1913, p. 630. 



gence, the other for Fortgang, or progress. In 

 Germany this practise is in vogue to-day. 



According to our scheme, the mathematical 

 pupil is measured in three ways, as follows : 



1. By memory tests 



(o) In daily work Ma 



(&) In examination Mi 



2. By original exercises 



(o) In daily work Oa 



(6) In examinations ■ 06 



3. By diligence (tenacity) shown.... I) 



How these marks should be combined might 

 be a subject of legitimate debate. Following 

 custom, we use the weighted arithmetic mean, 

 as follows : 



^ ,, . ,, , Ma + rMi + sOa + tOi + uD 



Prelimmary Mark = = — ; ; ;— ;-; 



■' l-\-r + s + t-\-u 



where r, s, t, u are coefiicients determining rel- 

 ative weights. What weight should be given 

 to daily work, what to the examination? In 

 different schools the weights vary from daily 

 work f, final examinations i, to daily work i 

 and final examination i. A conservative esti- 

 mate would be to take s^l, r = i = M = J. 



Part II 

 After the relative place or rank of the 

 students in a class has been determined by 

 the process of Part I., we proceed to deter- 

 mine their marks on an absolute scale. We 

 shall assume that the pupils constitute a ran- 

 dom sample or " fair sample" of the student 

 body. What is the distribution of mental 

 ability, and of mathematical ability in par- 

 ticular? No one has been able to give a final 

 answer to this question. Francis Galton, Karl 

 Pearson and others have held that individuals 

 differ from each other in ability in such a 

 way as to conform with what is knovni as the 

 " normal frequency curve " or the " normal 

 curve " or the "Gaussian curve." Distances 

 along the horizontal line measure the stu- 

 dents' abilities. The corresponding ordi- 

 nates of this bell-shaped curve indicate the fre- 

 quency. In measuring physical characteris- 

 tics, it is easy to tell whether or not the 

 above curve represents the proper distribu- 

 tion. It is a singular fact that this curve has 

 been found to represent a general biological 



