Surfaces and Essences

by Douglas Hofstadter

  What about Division?

  For division, too, there is a widespread naïve analogy that dominates people’s thought, profoundly affecting how people conceive of the operation. Although one would tend to think that division is perfectly understood by anyone who has gone through school, this is an illusion.

  Here is a very simple “home experiment” that reveals the hidden presence of this naïve analogy, demonstrating how concrete and standard it is. There is no trap here, just a couple of straightforward challenges. The first challenge is a warm-up exercise: Invent a word problem involving division — that is, a problem whose solution requires just one operation of division. Hopefully, this will pose no problem for any reader. Here, for example, are a few division word-problems that were invented by university students:

  •4 friends agree to share 12 candies. How many candies will each one get?

  •90 acres of land is going to be divided into 6 equal parcels. What will the area of each parcel be?

  •A mother buys 20 apples for her 5 children. How many apples will each child get?

  •A theater has 120 seats arranged in 10 rows. How many seats are in each row?

  •A teacher buying food for a class picnic filled 4 grocery carts with a total of 20 watermelons. How many watermelons were in each cart?

  •It takes 12 yards of cloth to make 4 dresses. How many yards does it take to make one dress?

  Each of these problems involves a starting figure to be divided by something, and the result of the division is always smaller than that starting figure. Thus the first problem reduces 12 candies down to 3, the second problem reduces 90 acres down to 15, the third problem reduces 20 apples down to 4, and so forth.

  This observation is already significant, since it shows that when people are asked to invent division problems, they come up with situations where the key idea is “making smaller”. Just as the standard image of multiplication as repeated addition locks in the belief that multiplication necessarily involves growth, here it seems that there is a naïve analogy at work that locks in the image of division as involving shrinking. And this supposed fact about division jibes perfectly with the way the word “division” is used in everyday speech. When one speaks of dividing X up, one sees X as being broken into pieces, with each piece obviously being smaller than X itself. The 1988 edition of Webster’s New World Dictionary confirms this vision, defining division as follows: “a sharing or apportioning; distribution”. Moreover, division is often associated with the notion of weakening. For instance, the slogans “united we stand; divided we fall” and “divide and conquer” imply that if an entity is divided into pieces, it will be weaker than the original entity.

  Well, that was the first of our two small challenges. The second one is simply to invent one more division problem, subject to one extra constraint: the answer must be larger than the starting figure. Readers, to your marks!

  As you probably have noticed, this slight modification of the assignment changes everything. Observe, for instance, that none of the problems in the preceding list meets this constraint. We assume that most of our readers experienced a jump in difficulty between the first and second challenges. Whereas inventing a word problem involving division is a piece of cake for nearly everyone, inventing a division problem where the answer is bigger than the starting number is generally not easy at all. It requires a bit of mind-stretching, and for many people it simply is beyond their reach. After all, how can dividing something possibly result in something that’s larger?

  The reactions of typical university students to the second challenge are quite diverse. Some students are categorically negative: the challenge is simply impossible. For them, division is by definition incompatible with the idea of making larger. Therefore, instead of inventing a word problem meeting the requirement, they explain why the task makes no sense:

  •“Can’t be done. Division always makes things get smaller.”

  •“When you have a certain value at the outset and you divide it up, you necessarily have less at the end, so it’s not possible.”

  •“Division means sharing, and with equal-sized shares. So each person gets less than what there was at the start. Therefore, it’s impossible to invent a division problem where someone winds up with more than there was at the beginning.”

  •“Impossible, because dividing means cutting something up into pieces. To get more, you have to multiply, not divide!”

  •“No way, because whenever you divide something, you always reduce it!”

  Some other students acknowledge that division problems can indeed have the requested property, because from school they recall the fact that dividing by a number between 0 and 1 has this effect. However, they are convinced that this kind of formal mathematical operation doesn’t correspond to any situation in the real world, and so they assert that there can be no word problem that meets the requirement. At least they can’t think of any. Here are some comments along these lines:

  •“I could say ‘10/0.5’, which gives 20, but that’s just a calculation. You can’t make up a corresponding word problem, because in the real world you always divide by 2, 3, 4, and so on. That is, you always divide by numbers bigger than 1.”

  •“Yes, it’s possible — for instance, ‘5/0.2’ — but I can’t think of any actual situation that this formula would describe.”

  •“Any time you divide by a quantity less than 1 you get a larger answer, but I can’t think of any real situation where it works like that.”

  •“When you divide something by one-half, you get more, sure — but the thing is, it’s not possible to divide anything by one-half!”

  Then there are some students who invent various problems that seem to them to work, but they cheat in one way or another, because the problems they give don’t match the assignment. For example:

  •“Rachel has 20 bottles of wine. She sells half of them at 8 dollars apiece. How much money does she get?”

  •“Eric had 8 marbles. In a game, he won half again as many. How many marbles did he wind up with?”

  Despite all these protests, it is perfectly possible to devise a division word-problem whose answer is larger than the starting number. Some people find good examples:

  •“How many half-pound hamburgers can I make with 4 pounds of meat?”

  •“If I have 3 days to prepare for an exam, and it takes me 1/5 a day to read a book, how many books can I read before my exam?”

  •“I have 10 dollars, and a chocolate mint costs a quarter (of a dollar). How many chocolate mints can I buy?”

  •“How many scarves can I make out of a 3-yard roll of cloth if each scarf requires 3/8 of a yard?”

  It turns out, however, that to come up with a problem such as these last four is quite hard. Among 100 undergraduate students, roughly 25 came up with a problem of this type, while the other 75 couldn’t do so, and were split into roughly equal-sized groups associated with the three types of failures quoted above. And so we see that an arithmetical operation that in theory should have been completely mastered in elementary school still gives a great deal of trouble to adults, even university students. Could it be that division problems lie so far back in their past that they’ve forgotten what they once knew about division? Well, no, because the same challenge was set to 250 seventh-graders, all of whom had been studying division for the previous three years, and so for them this kind of challenge was very fresh in their minds (indeed, they had studied problems involving divisors smaller than 1 for at least one full year), and yet it turned out that over three-fourths of them said that it’s impossible to invent a situation where division gives a larger answer, and of the 250, only one single student invented a word problem that correctly met the challenge.

  Why is it So Hard to Dream up Such Problems?

  It’s a common belief that when situations are concrete, people think more clearly, but this challenge shows that concreteness is no guarantee of clear thinking. The kinds of problems invented by univer
sity students in both parts of our little test featured essentially the same kinds of everyday items (cakes, candies, glasses of water, books, scarves, and so forth), and they were set in the same kinds of environments (kitchens, schools, trips, shopping, and so forth). What, then, is the nature of the conceptual gulf between solving the first challenge, which virtually everyone was able to do, and solving the second challenge, which so few people could do?

  The explanation is that the two challenges belong to two quite different categories of problems. They do not rest on the same naïve analogy. To be specific, the problems dreamt up in response to the first challenge, which didn’t ask for a larger answer than the initial value, were all problems involving sharing. The examples we quoted above were selected in order to give readers some variety, but in truth, two-thirds of the problems invented were extremely routine, always involving sharing the same kinds of things — relatively uniform everyday objects — among the same kinds of recipients — children, siblings, or friends. From the examples cited, it’s obvious that the division word-problems that people spontaneously come up with nearly always involve the concept of sharing, and more specifically, the splitting-up of a certain quantity into a number of precisely equal shares. The most typical case involves countable items (candies, apples, marbles) shared among people, and the word “sharing” often shows up explicitly in the problem’s statement. Nonetheless, there are more abstract kinds of sharing that show up in a few of the word problems suggested.

  In such cases, one has to imagine a more abstract manner of sharing than merely distributing a given set of objects to a given set of people. It might still involve the distribution of entities, but not to human recipients — say, the sorting of cookies into bags, or the arrangement of chairs into rows. It can also involve non-countable substances, such as flour, water, sugar, or land, which get split up into several equal-sized portions. Here there is no sharing in the marked or narrow sense of the term — that is, a counting-out of items, similar to dealing cards out to players in a card game — but there is still sharing in a more general or unmarked sense of the term, in which a whole is divided, through some process of measurement, into smaller chunks. But in any case, none of the responses given by students to the first challenge, whether they involved the marked or the unmarked sense of the concept of sharing, was a division whose result was larger than the initial quantity. And this is no surprise, because the nature of sharing is that it makes something smaller. Sharing involves breaking an entity into smaller parts, with each recipient necessarily receiving less than the whole that was there to start with. A part cannot be larger than the whole from which it came.

  By contrast, in word problems that successfully meet the second challenge, a different naïve analogy operates behind the scenes — that of measuring something. (In mathematics education, such problems are said to involve “quotative division”.) Division problems of this type can always be cast in the form, “How many times does b fit into a?” This is a measuring situation, in the sense that b is being treated as a measuring-rod with which a’s size is being measured. If the size of b is between 0 and 1, then there will be more b’s in a than the size of a, which means that the result is bigger than the initial size. For example, the calculation 5/0.25 can be phrased: “How many times does 1/4 go into 5?” The answer, 20, is of course larger than 5. What all this shows is that if a division problem is of the sharing sort, then its answer can’t be larger than the starting value, but if it is of the measuring sort, then its answer can be larger.

  It turns out that from a historical and scholarly point of view, measuring is a more fundamental way of looking at division than sharing. The definition of division given by Bezout in his 1821 treatise is quite explicit: “To divide one number by another means, in general, to find out how many times the first number contains the second.” Indeed, the etymology of the terms involved in division reflects the view of division as a measuring process. As readers will recall from elementary school, the result of a division is called its quotient. (As Bezout explained it: “The number to be divided is the dividend; the number by which one is dividing is the divisor; and the number that tells how many times the dividend contains the divisor is the quotient.”) The English word “quotient” stems from the Latin word “quotiens”, which is a variant of “quoties”, meaning “how many”, and which derives from “quot”, a word that refers to the counting of objects. In sum, today’s terminology echoes the conception of division as measurement, since “quotient” means “how many times”.

  Bezout is aware that seeing division as measurement is not the only possible point of view, but he wants his readers to act as if it were: “One’s goal in doing a division is not always to find out how many times one number is contained in another number; however, one should always carry out the operation as if this were indeed one’s goal.” This shows that the view of division as being primarily a kind of sharing did not come from mathematicians, for they tend to favor the view of division as measurement or counting. To the contrary, the origins of the naïve analogy of division as sharing lie outside of mathematics. As we mentioned earlier, dictionaries tend to define “to divide” in its everyday sense along the following lines: “to separate into parts; split up; sever; to separate into groups; classify; (Math) to separate into equal parts by a divisor” (this taken again from the 1988 edition of Webster’s New World Dictionary).

  Is Division Mentally Inseparable from Sharing?

  The experimental results we’ve just described show that for most people, division is understood through the naïve analogy of sharing; after all, most people find the first challenge very simple and invent word problems that involve sharing, while the second challenge, which is easily handled if one simply uses the analogy of measuring, is much harder for most people. Although children spend years learning about division in school and are thus presumed to have mastered this basic operation by the end of middle school (and adults are assumed to know division yet better), it turns out that people of all ages have trouble thinking of division other than through the naïve analogy that equates it with sharing.

  Most people use the term “division” not to describe a concept that they learned in school, but to describe a category of situations that was part of their lives before they started school — sharing. When sharing comes up in a mathematical context, they have learned from school to use the term “division” instead. In other words, most people think that “division” is just a technical term to denote the concept of sharing, especially when a calculation is called for, and that’s all there is to it. When one is in math class, sharing has a fancier name, just as in certain arenas of life people use various special terms to designate familiar concepts, even though such terms don’t lend any particular insight. Thus one learns that when one is at the opera, it’s better to say “aria” than “song”, and likewise, when one has truck with wine connoisseurs, one soon gets used to hearing about the “bouquet” rather than the “smell” of the wine; one also gets used to the fact that one’s doctor will tend to speak of “apnea” rather than of “having trouble breathing”, or of “hypertension” instead of “high blood pressure”.

  To summarize, although it is tempting to think that schooling teaches people the full-blown concept of division, thus allowing them to throw away the naïve analogy of sharing like a no-longer-needed crutch, the truth is that the crutch remains the central way of understanding division — it merely disguises itself by donning the more impressive-sounding mathematical label of “division”.

  Mental Simulation in the Driver’s Seat

  To solve either of the following two word problems is a challenge as easy as they come:

  Paul had 27 marbles. Then during recess, he won some, and now he has 31. How many marbles did he win?

  Paul lost 27 of his 31 marbles during recess. How many does he have left now?

  Both of these problems are solved by carrying out exactly the same operation — namely, subtracting 27 f
rom 31. At first glance, they thus seem identical in terms of what is going on mentally when we solve them, but let’s set aside the formal operation by which we solved them; instead, let’s try to visualize these situations in our mind’s eye — that is, we’ll try to mentally simulate each of them. What happens?

  In the first case, it’s easy to imitate what happened by counting on one’s fingers or in one’s head. Paul’s marble count moved up from 27 to 28 (“1”), then to 29 (“2”), then to 30 (“3”), and finally it reached 31 (“4”). The solution takes four simple steps.

  The second case, however, is very different. This time, starting from 31, one has to move downwards 27 steps: first to 30 (“1”), then to 29 (“2”), then to 28 (“3”), then to 27 (“4”), then to 26 (“5”), … , and after a long time one will finally hit 4 (“27”).

  We thus see that these two word problems, although they’re both solved by the same formal operation (31 – 27), are not imagined or mentally simulated in the same fashion at all. One process involves just four easy counting steps, while the other takes 27 steps, which, to make matters worse, involve counting backwards.

  This contrast should recall a similar one from earlier in the chapter — namely, that of the teen-aged street vendors in Brazil. As we saw then, the product of 50 and 3 can be mentally simulated either as 50 + 50 + 50 or as 3 + 3 + 3 + …… + 3 + 3 + 3, depending on how the problem was stated; here, likewise, the subtraction “31 – 27” can correspond to two very different mental simulations, one very short and one very long.