Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (100 page)

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 university 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
).

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”.

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

Both of these problems are solved by carrying out exactly the same operation — namely, subtracting 27 from 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.

Let’s now compare the following four word problems:

1. If we break a stack of 200 photos into piles of height 50, how many piles do we get?

2. If we break a stack of 200 photos into 50 piles, how many photos are in each pile?

3. If we break a stack of 200 photos into 4 piles, how many photos are in each pile?

4. If we break a stack of 200 photos into piles of height 4, how many piles do we get?

The first two are solved by the division “200/50”, and the last two by “200/4”. That is quite obvious. But are some of these problems easier or harder than others? Perhaps it seems as if we are asking if the division “200/50” is easier (or harder) than the division “200/4”. If so, the first two problems would be easier (or harder) than the last two. But things are trickier than that, as a bit of mental simulation will show.

Let’s try to envision the first situation:
If we break a stack of 200 photos into piles of height 50, how many piles do we get
? We first imagine a tall stack of 200 photos; now we want to break it into smaller stacks of height 50. In order to find the answer by simulation (as opposed to doing it by formal division), we take the number 50 and add it to itself until we get 200. 50 plus 50 makes 100, and then another 50 makes 150, one last 50 to make 200.
Four
50’s altogether — that’s our answer.

Now let’s try to imagine the second situation:
If we break a stack of 200 photos into 50 piles, how many photos are in each pile
? Formally speaking, this problem also involves the division “200/50”. We had a stack of 200 photos and we divided it up into 50 smaller ones. We need to add up
some
number 50 times, but we don’t know
which
number. In fact, not only do we have to do 50 additions to find the right answer, but we may have to do it a bunch of different times, guessing about what to add to itself! 2 + 2+ 2…? After much toil, we get 100, and see that “2” was wrong. 3 + 3 + 3…? Again a lot of toil to wind up with the wrong answer. 4 + 4 + 4…? Well, this time, if we add right, we’ll get 200. But solving this problem is not a piece of cake.

So although the first two problems are both solvable by the same division (200/50), from the point of view of seeing what’s going on in one’s mind’s eye, the first is much easier than the second.

Now let’s look at the third one:
If we break a stack of 200 photos into 4 piles, how many photos are in each pile
? Here we break our tall stack into four shorter stacks. How many photos in each short stack? This is very similar to the second situation, where we want
to do a repeated addition in order to reach 200, but this time we only need to add the mystery number to itself
four
times, instead of 50 times. On the other hand, we have to guess at the mystery number’s identity. But if we’re clever, we may hit on “50” without too much trouble. For instance, if we recall from everyday life that 50 + 50 = 100, then we can quickly figure out that 50 + 50 + 50 +50 = 200, and we’re done.

Finally, the fourth problem:
If we break a stack of 200 photos into piles of height 4, how many piles do we get
? We know we’re dealing with repeated addition of the number 4, but the question is:
how many additions
? We know that 4 + 4 + 4 + … + 4 + 4 + 4 = 200, but the mystery is how many copies of “4” there are in this sum. This is tough, because we’ll need to keep track of two things in our head at once — firstly, how many copies of “4” we have added up so far, and secondly, what the running tab is. We thus see that mentally simulating the third situation is far easier than mentally simulating this one.