Division of Fractions Explained in PSSM 2000
(“PSSM” is the Principles and Standards for School Mathematics, published in 2000 by the National Council of Teachers of Mathematics (NCTM). It is their model for mathematics in the schools.)
page 219 of PSSM begins,
"The division of fractions has traditionally been quite vexing for students. Although "invert and multiply" has been a staple of conventional mathematics instruction and although it seems to be a simple way to remember how to divide fractions, students have for a long time had difficulty doing so.. . .” And the text continues,
“An alternative approach involves helping students understand the division of fractions by building on what they know about the division of whole numbers …”
by which they mean that A/B is construed as 'how many copies of B can be found in A' before a remainder stops the process. PSSM here illustrates this "procedure" with an example:
If 5 yards of ribbon are cut into pieces that are each 3/4 yard
long to make bows, how many bows can be made?
and a picture of the ribbon set against a yard-scale is given with the cutting points required to produce the 3/4 yard pieces:
Count 1 2 3 4 5 6 bows
Yardstick ------------1------------2------------3------------4-------------5 yards
The count, seen by eye, is 6 bows, with a remainder of 2/3 of a bow.
(How did they get that 2/3? Actually, they eyeballed it.)
This diagram is not a bad way to see one of the meanings of division by using a simple case, but – this is the only prescription given for the division of fractions in the middle school, nor does PSSM come back to it in the high school standards. This model is explicitly praised as exhibiting more “understanding” of the process of division than the “invert and multiply” rule, which in this case would produce the computation:
5 yards divided by 3/4 yards (per bow) is 5 / (3/4) = 5 x (4/3) = 20/3 = 62/3, the number of bows.
Now, the justification of the “invert and multiply” rule takes a bit of doing, for it is real mathematics, not fathomed in a day. Mere memorization of the procedure just given is indeed less illuminating than the ribbon diagram if it is the purpose of the division routine that is to be explained, but it is not the mere memorization of “invert and multiply” that is our purpose in explaining the origin and rationale for the rule, nor do we claim “invert and multiply” explains division of fractions all by itself.
The rule is in fact profound; the European pioneers of algebra didn’t even have it as recently as 500 years ago. Yet what can be developed in 500 years really can be learned by middle school children once the reasoning has been refined to modern standards, and in this case the refinement has enormous value that PSSM not only ignores but depreciates, considering it an imposition on children even to mention it. From the point of view of PSSM the question is, “Why go through all that when the picture is so simple, and gives the answer just as well? Any caveman could solve this problem by the diagram, without all the “invert and multiply” fol-de-rol that has plagued students (and their teachers) for generations.”
True; but suppose instead of 5 yards of ribbon there were 5040 yards, something not unheard of in factories that weave lengths of cloth for marketing? Yes, the diagram would work, but it would take a lot of counting, laying off repetitions of marks each 3/4 yard until reaching the end. Yes, one could think of some short cuts, since every 3 yards produces 4 bows. Well, suppose the length were 5040 yards and each bow were exactly .733 yards? Tolerances of this exactness are a commonplace in machine shops, where the ribbons are of steel, and need repeated cutting. The diagram would be impossible to use, while “invert and multiply” produces 5040/(733/1000) = 5,040,000/733, or a whiffle over 6875 bows (using long division, another skill carefully omitted in NCTM’s prescription book).
Yes, a calculator would give us this answer, provided someone knew how to enter the above calculation or its equivalent correctly into the machine, but neither the cave-man’s diagram nor the use of the machine is the real mathematics we wish to convey. The student of real mathematics who has learned “invert and multiply” along with its associated information about fractions, needed to prove the truth of the operation, has learned something real, a theorem or an algorithm, that gives him power over numbers, which neither the calculator nor the diagram can give him. It is not that we want future ribbon-cutters to avoid calculators; we all use them and so will they, when the time comes. We want them to understand more than this single prescription; they must know what is happening when any arithmetic combination of numbers is displayed in decimal form, or fractional form, and how to rearrange the symbols to produce a readable symbolic answer.
Experience has showed that children without such intimacy with actual numbers and calculations, performed on paper with their own fingers, become helpless even when provided with calculators, when computations have to be symbolized by them, as they do in any algebraic operation and in every part of the descriptive formulas of the sciences.
Every student who wants to understand chemistry or physics absolutely must understand immediately upon seeing it that for all real numbers (and not only integers) A, B, and C, A / (B/C) = AC/B. To have this information “in his bones” (to paraphrase C.P. Snow) he must first go through simpler examples, such as I have indicated above. But that’s only the beginning, for lessons concerning equivalent fractions are needed here, until the story gets to fractional numerators as well, something PSSM does not even consider via cave-drawings. Our student needs to know that (a/b)/(c/d) = (ad)/(bc), not to cut ribbons, but as an equivalence that will surely turn up in myriad forms in every science and craft. This understanding does not come from the ribbon diagram, but comes from a more profound understanding of the meaning of the fractional notation and how it behaves under arithmetic operations. It is this sophistication, which NCTM wishes to avoid because it deems it hard to teach, that distinguishes our use of mathematics from that of the cave-man; and not the fact that we have calculators to do our legwork while he did not.
There is, finally, an important use for “invert and multiply” that is not mentioned in the schools, with good reason perhaps, but which is vital to the understanding of many formulations in the sciences, and that is the matter of “dimensional analysis”. One cannot keep track of the dimensional consistency of a formula in physics or chemistry, or even in simple measurement with units of this or that sort, without it. Here is not the place to deliver a course in such analysis, but the example above, of the ribbon and the bows, will serve to illustrate the matter, and how in the present instance the arithmetic algorithm illustrates more than a numerical formulation.
We had 5 yards, and each bow to be cut from it was to be ¾ of a yard long. The computation would be symbolized as Y/B, where Y is the number of yards and B the length of a bow; what are the dimensions? Y is length (“l”), and B is length per bow (l/b). Why “l/b? Well, “b” is a new “unit” here, the unit by which “B” has been measured; this is as much as I can say without getting into a lengthy discussion. So the dimensional formula associated with Y/B is l/(l/b), which by the algebraic formulation of “invert and multiply” produces (bl)/l or b. We end up in units of b, i.e. the number of bows.
In 19th Century “practical” arithmetic books the units of a problem were given equal status with the numbers involved, and students in the very elementary grades, perhaps Grade 5 or younger, were required to present their answers orally, following a model given in the textbook. Even before fractions were introduced as such, a typical such problem (with integral quantities of sufficient simplicity to produce integers as answers) might be, “Farmer Brown has six cows, and sells them all for 90 dollars per cow. How much does he receive?” And the answer is “He sells six cows at 90 dollars per cow, so that six cows times 90 dollars per cow makes 540 dollars.”
The student here learns that (cows) X (dollars/cows) = dollars. This multiplication is not yet at the algebraic level of “invert and multiply”, but that would come in time, as in the ribbon example. The contempt in which “dime-store arithmetic” is held by today’s educators, blinded by the success of calculators in performing the purely arithmetic part of practical computations, is short-sighted, for the repetitious and tedious exercises of a hundred years ago did include some valuable conceptual lessons, lessons which cannot be taught directly. Nor is dimensional analysis part of pure mathematics at all, even though its concepts are integral to all the sciences to which mathematics has relevance.
Ralph A. Raimi
28 March 2005
revised 6 December 2005