MATHEMATICS�TO REASON NOT JUST TO DO

BY JOSEPH SPEARNortheastern University y Boston, Mass.

Some time ago at a meeting of mathematics teachers, theauthor suggested the use of reasoning methods rather than rulesand short cuts in the teaching of fractions in both arithmeticand algebra. The purpose was to try to eliminate mistakes dueto faulty memory in trying to remember short cuts and ruleswhich had no meaning. Work with equations was also discussedfrom a similar point of view. There seemed, then, to be someagreement that the methods suggested might be of value.

Recently at another meeting of the association, during theconsideration of problems presented to the high school algebrateacher arising from the different degrees of preparation amongthe students, the same topic of difficulty in fractions was againdiscussed and to many present the methods suggested earlierappeared as brand new ideas.The author is therefore moved to put his recommendations

in writing, not in the belief that there is anything new in them,but with the hope that teachers will at least give some of thesuggestions a trial. Much progress has been made in the lasttwo or three decades in methods of teaching history, geography,and reading. In arithmetic, however, many of the same oldtricks resorted to years ago are still being used, and the samecomplaint is continually heard that the students can’t even addfractions.Even though the method outlined here may not cure all evils,

it surely cannot turn out worse results than we see at present,and it may by chance make improvements in the student’slearning and his retention of learning. The method has beentried for eighteen years with college freshmen and has beenparticularly beneficial to those students who claim they havealways found mathematics hard to understand.

In order not to be too lengthy, the methods will merely beoutlined. The teacher can readily fill in between the lines wherenecessary.

First of all, in dealing with fractions, the only operation per-mitted, in order not to change the value of a fraction, is tomultiply it by unity or to divide it by unity. The unity may bedisguised in various forms in order to accomplish the desiredend, but multiplication or division by the factor of unity is theonly privilege permitted.

402

MATHEMATICS 403

The UNITY METHOD illustrated:

3 5 /3\/9\ /5\/4\ 27 20Ex; T^TXi^Xr)-^

27+20 47

36 ’"36’

Here each fraction is multiplied by a unity, the proper unitybeing chosen in order that the resulting fractions should eachhave the same common denominator. The so-called method ofcross-products, of multiplying the numerator 3 by the denomina-tor 9, and multiplying the numerator 5 by the denominator 4,should be entirely discarded. It may be a short cut in somecases, but leads to wrong thinking and no reasoning.

E.: ^-i.^ViWi-Vl).25-,!6 15 \6/\5/ \15/\27 30 30

25-8 1730 30

2 1 7 / 2 \ /20\ / 1 \ /45\ / 7 \ / 6 \Ex: T^-^-^X^-^^Xii)^^)^)

^

40 45 42 40+45-42 43’"Tso’^Tgo’Tgo"180=T8o’

The advantage of requiring that each unity be put in whereit is to be used as a multiplier is obvious. Indeed, for some stu-dents, where the instruction need be more forceful, it may bedesirable to have each unity written in with colored pencil. Thestudent here is using correct reasoning, and must realize thathe is not changing the values. No trick is resorted to. Eachfraction multiplied by unity does not change in value, althoughit does change in appearance. The method of finding the properunity to use in each case as a multiplier for each fraction doesnot present any difficult problem.Here is a similar example in algebra:

3x-2 7 5 3x-2 /4^/3\a2-62+3(6-a)~4(a+6) ~

(a-b^a+b) uA^7. 7 (-1}(–\(^ 5 (3}(^3(6-a)\-l/\47W5/ 4(a+6)\3/\a-6/

404 SCHOOL SCIENCE AND MATHEMATICS

(3^-2)(4)(3)+7(-l)(4)(a+6)-5(3)(a-6)������� etc.

(a-6)(o+6)(4)(3)

In the reduction of fractions, similar steps should be required.The reduction of a fraction to lowest terms by cancellationleads to vicious habits. In fact, it is suggested that the wordcancellation be used only in cases where a term subtracted fromitself gives zero. The word cancellation is now being used tocover a multitude of abuses. If reduction of fractions is handledonly by the unity method, the reasons involved in the workcan be seen and more easily remembered:

4 2 /2\ 2 2Ex: �==�(�) =�(i)=�

18 9 \ 2 / 9 9

x2-2x-3 {x-3){x+l) (x-3) {x+1)^-^-S’^+l)^-^)" {x-5) (x+1)

_(^-3) ^_^-3(x�S) x�5

Ex:

This treatment of fractions by the unity method should beginas soon as fractions are started in arithmetic, whether it be inthe 5th, 6th, or 7th grades. Perhaps it would be wise to post-pone the start of fractions until the upper grades, in order thatthe student may learn more easily and effectively than at pres-ent through the reasoning processes involved in applying theunity method. The method too often taught, and presented inmany grade school books, of multiplying the 3 by the 5, and

3 2 3X5+2X4the 2 by the 4 in �+�==������� does not teach the

45 4X5

student to reason, and in most cases becomes merely a matterof rote memory. The result of teaching such short-cut methodsis that the learning is not retained. Furthermore, the studentfrequently invents new short cuts and tricks, dubious systemsof arithmetic with which every teacher is only too familiar.

It is worth mentioning here that besides adding fractionsalong a horizontal line they should also be done in a verticalcolumn. Very often arithmetic books teach the addition offractions entirely in vertical columns, and then when the changecomes in algebra to adding fractions in a horizontal line thestudent thinks something new is happening.

MATHEMATICS 405

3 3 /5\ 15EX:+T4=+14(T)=+70^-4/14N

TW"4_ 4/14\ 56

"^^TW^TO3 3/7\ 21

-To--To\7/-7o15+56-21 50 5 /10\ 55^n-mWT^"70 70 7 \10/ 7 7

In complex fractions and in fractions with negative exponentsthis unity method not only tends to foster better habits, butalso results in quicker solutions than can be obtained by most

x+�y

other methods. In the example ��� the cross product and

short cut methods often lead to the following incorrect results:

xy+1 x+1 2xy+2either ��� i or �� ? or �:�� � By the unity method the

2 2y yy y

operation permitted is to multiply our fraction by one. Here

ythe unity takes on the appearance of � :

y

i ix-\-� x+�

Ex: y y / y\ xy+\

2~

2 \7/ 2y’

a c

T4"^In fractions of the type ���� too often the student is

m n

-rttaught to reduce the numerator to a single fraction, to reducethe denominator to a single fraction, to tumble the denominatorfraction up-side-down, and then multiply. This is neither goodmathematics nor good reasoning. Again it too often results innew inventions, as soon as the problem is changed into a little

406 SCHOOL SCIENCE AND MATHEMATICS

different form. The short cut of multiplying a by d and b by c,and m by d and ^ by b, in this particular problem may lead to

a c

T ~dthe correct answer. But in the example ���� the cross product

m n_i_

P q

ad-}~bcmethod too often leads to the incorrect result ����; the

mq-\-np

student discarding the denominators without realizing it. Thesteps here in the unity method are:

Ex:

Ex:

a ca c�+��(bd)+�(bd)b d/bd\ b d ’ /

ad+bcm n \bd/ m n md-\-nb�+- �W+�(bd)b d b d

a c

b d /bd?q\ adpq+bcpqm n \bdpq/ bdmq+bdnpp q

Reasons for introducing the following additional examplesare obvious:

Ex:3+^Va-x _3+^\/a-x^ 2\_6+\/a-x���4-^\l)~8

6 6x+5+� x+5+�

x x /^\ x^+Sx^+Gx6 S~ 6 8W/ ^-6^+8

x+5+6x-1/ x+5+6x~1 \/x2\ xs+5x2+6xEx:

1+6x-1+ 8x-2 \1+ 6x-1+ 8x-2/ \x2 / x2+ 6x+ 8

x{x+3)/x+2\ x{x+3) , . x(x+3)^-^^^V x+4:v/(^+4) \x+2/ x+4 x+4:

xy^+x^y xy^+x^y /x^y^ x^y+xy*Ex:

.t;-2+a;-ly~3+a;2 x~’^+x~ly~s+x2\x’^ys/ ys+x+x4ys

MATHEMATICS 407

ab-1 1 ab-1 1��+�b3x+c-2 ��+�b3x+c-2c2

_c 2 /2bcW\

ax^+ab+c-3~

ax^+ab+c-3 ^26AV2ac2x2+b4c3x3+2bcx2

~2abc3+2ab2c3x2+2bx2x(\-x)-^+x-^(\-x)

_gx" ��������������

x^l-x^-x-^l-x)-1

_

x^l-x^+x-^l-x) [-(l-^"] _^+(^^)2~

x(\-x^-x-^l-^LO- -x)x \~ x2-1

It is believed that the use of the unity method right from thestart will tend to eliminate many mistakes which persistentlycrop up in the work of altogether too large a number of thestudents in high school algebra and arithmetic classes, andwhich cause the classroom work to slow down so much that notsufficient ground can be covered by those desiring to go aheadfurther in the study, or what is even worse, cause teachers topass students into higher grades without proper preparation.If the student is never taught that the reduction of -^ is ob-tained by crossing out or cancelling a 2 from the numerator anddenominator, if the student is never taught that the reduction

xyof � is obtained by crossing out or cancelling the y^s from the

ay

numerator and denominator, then perhaps the student will

�^+3stop crossing out the 3^s in the fraction�� � If the student is

y+3

never taught to take things from the numerator to the denomi-nator, etc., perhaps he will not take the 2 down from the numer-

.yx+�

2 x+yator of ��� into the denominator and call it erroneously �� �

a 2a

^/4^^x^-x(4:-x2}-l!2{-x)A common mistake in the reduction of����������������

4:-x2

408 SCHOOL SCIENCE AND MATHEMATICS

\/4 � x^� x(� x)is to write it as -������� � The use of unity method

(4�^2)(4�^2)3/2has cured this evil in many, many cases.

^/^^-x^-x^-^^-x) r(4-^)1/2-! 4-^+.r2T^-v �

__________________________ _________ | ^^_________ .

4-X2 i(4-X2)^2] (4-.T2)3/2In the rationalizing of fractions the unity method frequently

1 /9

helps. Too often, since �==��? students also take a shortV2 2

1 \/2cut and write that �^ gives �� � By use of the unity method

�^/2 2

the danger of falling into this mistake is lessened.

Ex. J-^r^L2"^2y’2 21/2L23/2J 21 2

j- -JL r-22!3"!-22!3, -v/22-v/2 ~’2^3 L^J -

23^3 "^T" ’

Even if the best unity is not chosen, the chances of gettinga wrong result are lessened.

1 1 r2173"! 2l/3^21/3-| 22/3Ex: �:=�� �� ==�� �� =�� .

v/2 21/3L21/3J 22/3L21/3J 2

One word of caution should be given here. The student mustbe taught to be honest in his multiplication. That is, havingchosen a given unity, the multiplication must be honest, comewhat may. In trying to change students over from the oldshort-cut methods to this unity method, the author has oftenfound that at first, although the student chooses the correctunity, he then ignores it and writes the answer as he thinks itought to be judging from his old learning, without actuallydoing the multiplication. This is one of the difficulties thatalways arises when relearning is attempted. Probably if theunity method were taught from the start, this tendency wouldnever appear.When students become proficient in finding how to set up

the necessary unity, it is possible to learn quickly how, in onestep, to reduce into best form a fraction which is complex, con-taining negative exponents, having common factors in numer-

MATHEMATICS 409

ator and denominator, and requiring rationalization. However,as has been shown above, even though the most advantageousunity is not chosen, the work will not be wrong, but may re-quire an additional application of another unity. In the longrun, the unity method is really a good short method, and whatreally counts is that it is good common sense reasoning.One more suggestion is in order concerning equations. Just

as in fractions, the general use of the word cancellation shouldbe eliminated. Things are cancelled only when subtraction giveszero. In an equation, the requirement should always be tomultiply both sides by the same thing, or to add the samething to both sides. Transposing from one side to the otherleads to errors due to poor memory as to what it is that can betransposed. The use of cross multiplication to clear of fractionsis another vicious process which should never be taught. Given

X CLthe equation �=�; by multiplying both sides by 6, we get

^ o

(6)-^-=(6)-^- or 3x=2a.^ o

The 3 in the left member of the last equation is not the 3 fromthe right hand denominator of the first equation. It is the result

xof multiplying � by 6. In the solution of 5x = 7, to take the 5

from the left member and place it under the right member isnot good thinking. Both sides should be divided by 5, or multi-plied by ^ if one prefers.Examples need not be given of the ridiculous mistakes made

when the student thinks he is using the cross-product method,particularly when more than two terms appear in the equation.Nor need examples be given of the errors caused when studentsthink they are transposing, when they are really taking piecesof terms and factors and putting them upstairs, downstairs,and all over the place. To insist that both sides of the equationmust be multiplied by the same thing, helps the student to keepcontinuously in mind the fact that he is dealing with a balancedscale. When one member of the equation happens to be zero,this method will also incidentally finally teach that multiplica-tion by zero gives zero.

In a recent experiment, a large group of students were asked

410 SCHOOL SCIENCE AND MATHEMATICS

2to solve the equation �=5. The papers were immediately

x3

collected. The same students were then asked to solve�=7,x

but this time they had to indicate exactly what they did toboth sides of the equation. The results in the second problemwere almost entirely correct, while there were altogether toomany errors in the first set. What is still more interesting isthat a number of students who had the second problem correctwere surprised when they were shown that they had made anerror in the first problem.

If in an equation there appear fractions with negative ex-ponents, or fractions that are complex, these fractions shouldfirst be reduced by the application of the proper unity multi-plier. Then the solution of the equation should be undertakenby treating both sides alike, and if necessary the steps in thetreatment should be indicated, just as the unity multiplier isindicated in the treatment of fractions. Note -that the twoprocesses are entirely distinct. The unity method is not usedfor equations. The unity method is used in changing the ap-pearance of a fraction while its value is not altered. In an equa-tion, as long as it is kept balanced, the original value of amember on any one side of the equation need not be retained.Here, therefore, we can multiply both sides by the same thingor add the same thing to both sides.

In all that has been said here, the fundamental idea is tolead the student to understand the reasons for doing things.In fractions, the restriction of being permitted to multiply onlyby unity will help eliminate the ruination of many fractions.In dealing with equations, if the student learns to treat bothsides like the two sides of a balanced scale, he will save manyan equation from taking on meanings never intended for it.

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