PROCEDURE IN MATHEMATICS 799

EDUCATIONAL PROCEDURE IN MATHEMATICSIS MISDIRECTEDBY J. DONALD WATSON

College of CommerceUniversity of Noire Dame, South Bend, Indiana

It is common knowledge that school curricula have everlagged behind the needs of a particular era. This is indeed truetoday. A recent study at the University of Notre Dame revealsan instance of this tardiness of the educational framework tobecome adjusted to the changed conditions.

In recent years an increasing number of university studentsthroughout the country have been majoring in business andthe closely associated social sciences. Properly, they should. Theproblems of today are in those fields. "A Century of Progress"is mere irony while invention and material improvement is ac-companied with economic and social disorders. But high schooltraining of students in mathematics is not directed to this end,for it is planned largely for the students who will enter engi-neering fields.The students majoring in business and social sciences need

a different type of mathematical preparation from that of theengineering and architectural students. They need commercialarithmetic and bookkeeping. They will be working with num-bers on tons, miles, etc., in statistical tabulations and withdollars and cents numbers taken from accounting records. Fun-damental operations, decimals, fractions, and percentage are thebasic tools. But these students must see how such tools are usedin an infinite variety of business and economic situations. Themathematical work of the Department of Commerce in Wash-ington or of a particular business enterprise is of this type. Theproblems require economic and social measurements, their com-pilation, and their interpretation. These students must knowhow to perform such calculations because (1) they should beable to compile date in these fields, (2) they should be able tointerpret their own or other published data on economic andsocial conditions, and (3) they should advance themselves abovethe routine level of figuring and into the stratosphere which de-mands judgment on larger problems. Advanced alegbra, geome-try, and trigonometry are not employed here.These facts show that the high school work in mathematics

of at least a fourth of the students entering colleges and uni-

800 SCHOOL SCIENCE AND MATHEMATICS

versifies is misdirected. Though it is admitted that algebra asabstract arithmetic is extremely important in its fundamentalphases to anyone, much of the high school algebra is directedtowards engineering. Geometry and trigonometry are ap-proaches to the surveying of land, erection of buildings, andcomplex problems of engineering. In the past decades the taskof surveying the American continent has been completed. Asthe country reaches its adulthood and as the need for houses,factories, and office buildings decreases, engineering and archi-tecture offer opportunities only to a restricted number of stu-dents.

In high school most students have three semesters of algebraand two semesters of geometry. If they are not going to college,they may possibly take commercial arithmetic and bookkeep-ing. Conversely, if high school students take commercial arith-metic and bookkeeping, they are barred from many collegesand universities because of the traditional insistence in collegerequirements for algebra and geometry.The study at the Universityof Notre Dame included 267 fresh-

men in the College of Commerce. These students were from 35states and the District of Columbia and from 197 different highschools widely scattered over the country. The number ofsemesters, on an average, that a student spent in his high schoolwork on various types of mathematics is revealed in the follow-ing table.

Algebra.................................2 71Geometry ...............................2 27Trigonometry............................0 31Bookkeeping.............................0 27Commercial arithmetic....................0 18

In December of 1934 an arithmetic test was given to 148 ofthose students who were going to take during the second semes-ter their first university course in mathematics of the type usedin business and the social sciences. The primary purpose of thetest was to classify the students in accordance to their degreeof preparedness. The test revealed that those who had had nocommercial arithmetic or bookkeeping in high school had anaverage score of 72.2 and those who had had one or severalsemesters had an average score of 82.2. Thus, those who hadtaken this type of mathematics in high school and who had notdevoted their time entirely to geometry and advanced algebrawere better prepared at the beginning of their university work

DIVISION BY ZERO 801

for the type of mathematics they planned to use during andafter their university years.The conclusion is twofold. First, colleges and universities

should broaden their entrance requirements relative to mathe-matics so as to recognize the need of business and social sciencestudents for a background of commercial arithmetic and book-keeping. Second, high schools should provide suitable coursesin commercial arithmetic and bookkeeping and allow studentsto plan their work with thought to their subsequent universitywork.

THE TREATMENT OF DIVISION BY ZERO

BY CECIL B. READTfie Municipal University of Wichila, Kansas

Judging by the reaction of students beginning college work,many teachers do not emphasize that division by zero is anoperation excluded in modern mathematics. Students readilygrant that co+co=2oo; co/co =1, etc. Infinity is just anothernumber; far less mysterious to the student than to an ablemathematician.

It would seem that any one could clearly understand thatwhen we say a/b = q, we define q as the number which, multi-plied by 6, will produce o. When 6=0, the definition is meaning-less, for if o==0, every number q, multiplied by zero gives zero;if fi^O, there is no number q which multiplied by zero will give ff.

Students often assume that division is always possible,neglecting to consider that the divisor may be zero. For ex-ample, rigorous treatment gives the value of (^�(^/(x�d)as x+a, unless x=a. When ^=o, the expression is meaningless,we assign the value x-}-a approached as x approaches a.

Division by zero cannot be "justified" by pretending that thedivision is really by an extremely small number, hence divisionby zero gives a quotient extremely large, co. Logically, this alsoimplies that 7�7 is not really zero, but some very small num-ber, for a symbol can not have two meanings.

Perhaps the greatest difficulty comes from an improperunderstanding of such expressions as tan 90° = oo, often foundin tables, handbooks, etc. Few students seem to realize thatthis is an abbreviation for the statement which describes thebehavior of tan x as x approaches 90°, and not a value for