WHY SOLVE for X?' EWING C. SCOTT Briar College, Sweet Briar, Virginia
Sweet
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HE average student in college freshman chemistry courses is arithmetically infantile. The so-called "mathematical difficulties" in such courses are almost entirely matters of seventh- and eighth-grade arithmetic. College freshmen can do problems of this grade by an automatic process if the labels on the quantities are those of everyday life, but fail completely if they are unfamiliar. With many students, it is not necessary to go as high as the seventh grade to find this state of affairs. Say, "There are 12 eggs in a dozen, how many dozen in 42 eggs?'' The student answers, "Three and a half." But say, "There are 12 g. of carbon in a gram atom of it, how many gram atoms of carbon in 42 grams!" and you find that the student doesn't know whether to divide 12 by 42 or to multiply. I have even said to a student, "If one mole is two equivalents, how many moles is one equivalent?'' and more than once been answered, "Two." For the benefit of a dean who seemed to be taking freshman complaints of mathematical diiculties in Chemistry 1 a t their face value, I once devoted the opening session of the course to a little examination in grade school arithmetic. The hardest problem ran, "If a car can go 80 miles on a dollar's worth of 18# gasoline, how far could i t go on a dollar's worth of 166 gasoline?" S i t y per cent. of the class got below sixty per cent. of the. q u i z It is true that this was in a college for women, but I have been assured by a wellknown professor a t Yale that the situation is no different there. The one process involving arithmetical reasoning which any considerable number of students can employ in solving problems is what our fathers called, "the rule of three," ratio and proportion. There seems something particularly appealing in the orderly design of dots. On the other hand an even larger number than use the dots, use the equivalent form in which the ratios are expressed as fractions, with the sign of equality in the middle. Perhaps there it is the delightfully mechanical process of cross-multiplying that appeals. However that may be, the fact remains that students use this process in season and out, even using it, and frequently incorrectly, in doing things that should be solved by inspection. They are like the engineering student who was so wedded to his pocket slide-rule that when someone asked him suddenly, "What's two times two?" he whipped the slide-rule out, and after a rapid adjustment, read, "Three point nine nine!" Personally, I think that ratio and ~ronortionwas
put into the hands of students by the Devil. My ohjections to the process are twofold. F i s t , a ratio and proportion can be set up in no less than sixteen different ways, half of which are correct and the other half wrong, and it is difficult to tell by inspection whether a given set-up is right or wrong. In the second place, it is necessary to solve for X, and each bit of algebraic juggling done by a student offers one more opportunity for error, and an opportunity all too rarely foregone. This is particularly true owing to the ingenious way in which students manage regularly to get their X into the denominator on the right-hand side. To illustrate my first point, look a t the possibilities:
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V,=r T* T
T:T::V:V T2:Vs::V:T
z-7
T* - T iK -
T:T,::V,:V V * : T s : : T: V
T2- V -
v
V,-T
vz
T
To make the case easier I have chosen a case of direct proportion, but remember that gas law problems have an equal number of inverse proportions. Furthermore, students rarely label their figures. I wonder how many of you have by this time seen that all of the eight differentequations on the left are correct, and those on the right wrong. Of the statements in the center, the upper eight are correct, the lower eight wrong. Faced with a statement from a student which may he in any one of this multiplicity of forms, even the instructor finds it difficult, as I have said, to tell by inspection whether it is one of the correct forms or one of the incorrect ones. Usually he must repeat in his mind the reasonine Drocess involved in the oridnal --writing. The studin; definitely cannot tell anything * Presented before the Division of Chemical Education at the by inspection, and if he endeavors to check his work by , taro. ninety-third meeting of the A. C. s., chapel H ~ I I~~~~h lina, April 13, 1937. repeating the reasoning process, he inevitably does just 342
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exactly that, repeats it right or wrong, by the hypnotic power of the written word. My second objection to this process hardly needs more than to be stated. You all know from experience how indiscriminately students get correct or incorrect results in solving such set-ups for X, regardless of whether the set-up itself is correct or not. In any case, any unnecessary mathematical process gives an unnecessary chance for error. The process of solving an equation for X is completely unnecessary in doing ordinary chemical problems, because the thing can he set up already solved for X, and with even less effort than is required to set up a proportion with the X in some more or less inaccessible position. When a problem is set up solved for X there is only one correct way of writing it. Not only that, but it is easy to detect errors by inspection. Consider the following example: If 10 ml. of 6 N HzS04is diluted with water and added to an excess of Zn, what will be the volume of the gas liberated, if measured over water a t 18"C., with the barometer a t 745 mm.?
X ml. & 273 18"A. 745-15 mm.
672 ml. H a 273"A. 76Omm.
X = 672 ml. H1X
+
291 "A.
--
760 mm.
X --273"A. 730 mm.
The student should reason as follows in setting this up, "I want ml. of HZfor my answer, so I must start out with ml. of Hz. A t a higher temperature the volume will be larger, so in multiplying by the ratio of the temperature I put the larger one on top. The volume will also increase when the pressure is decreased, so the ratio of the pressures also goes large side up. Checking, the units of temperature and of pressure cancel out, leaving the fractions pure numbers, so my answer has the dimensions of ml. of Ha which was what I wanted." Students brought up on proportion hate to abandon it for the above method, hut when persuaded to do so they make far fewer errors. The main objection to this method is from students who wish to substitute values mechanically in an equation. They don't like a Z n + 2H+ + SO4-- = Zn++ SO4-- f HZ method which requires a thoughtful consideration and 10 ml. = 0.01 1. of 6 N acid = 0.06 equivalent. 0.06 understanding of the nature of gases, much prefening equivalent = 0.03 mole of HzSO* = 0.03 mole of HZ to parrot a formula. Personally, I consider this ob= 0.06 X 22,400 ml. of Hz S.T.P. jection a very valid recommendation of the method.
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