Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (99 page)

What happens if one changes the numbers in this problem? If one merely replaces “0.26” by “5” and asks the question again (“If one gallon of petrol costs 2.47 pounds, what is the price of 5 gallons?”), then 100% of the middle-schoolers solve it correctly. This discrepancy is due to the fact that the first problem doesn’t meet the naïve analogy’s image of
adding a number repeatedly
, since the idea of adding a number to itself 0.26 times makes no sense. On the other hand, using the naïve analogy of repeated addition works just fine in the modified problem (2.47 + 2.47 + 2.47 + 2.47 + 2.47). Discrepancies between participants’ performances on the two problems reflect the fact that the naïve analogy is of no help in the first one, yet is appropriate in the second one.

It’s enlightening to compare the preceding findings with some experiments carried out in Brazil. The participants were teen-aged boys who had dropped out of school and were making a living as street vendors. The following simple problem was given to a group of them:

The same problem was also given to a different group, except that the two numbers were interchanged, as follows:

To all readers of this book it will surely be trivially obvious that each of the two boys will have to fork over 150 cruzeiros, even if one of them winds up with far fewer chocolates (at least in number) than the other one. When we read the two problems, they appear equally easy; the scenario is the same, they involve the same numbers (50 and 3), and they both involve the same arithmetical operation: multiplication. But were they equally easy for the two groups of street vendors? Not in the least.

The first problem was handled pretty well by most: 75% got it right. The second problem, by contrast, was not solved by any of the street vendors. The reason behind this discrepancy is relatively simple; it comes down to reliance on the naïve analogy of repeated addition. To solve the first problem, all one needs to do is add 50 + 50 + 50, to get 150 cruzeiros. This is just two additions, and it involves only very simple facts: first, that 50 + 50 = 100, and second, that 100 + 50 = 150. The variant problem,
however, is another ball of wax entirely. To compute the answer, one has to carry out a very long process of iterated addition — namely, 3 + 3 + 3 + …… + 3 + 3 + 3, involving fifty 3’s. As one can easily imagine, this is not a challenge that an elementary-school dropout would be very likely to be able to handle.

This might seem the moment to shower praise on our educational system, thanks to which we educated adults are all instantly able to solve a problem that to school dropouts seems impossibly hard. We all know in a flash that 50 x 3 equals 3 x 50, end of story. Given this contrast with the 0% success rate of the school dropouts, one might be tempted to conclude that schooling very effectively gets across the true nature of multiplication. However, things are not that simple.

There is no disputing the fact that schooling teaches us that the two numbers in a multiplication can be interchanged — we all know that multiplication is commutative, that
a
x
b
=
b
x
a
— and we all carry out such switches without a moment’s thought. However, carrying out multiplications using one’s knowledge of commutativity doesn’t mean that one’s understanding of multiplication, as an adult, goes far beyond that of elementary-school children. Indeed, a quick informal survey reveals that almost no one, aside from serious math enthusiasts, knows
why it is the case
that, for instance, 5 x 3 equals 3 x 5. Middle-school students, high-school students, even university students are generally unable to say
why
the two numbers in a product are switchable. How, then, do they justify to themselves the idea that five threes equals three fives, or symbolically, the fact that 3 + 3 + 3 + 3 + 3 = 5 + 5 + 5 ?

Most people, if asked this question, will answer readily that one can check this out in any specific case (“Just go get a calculator and try it out for whatever pair of numbers that you wish!”). Some people will state it more as an axiom: “In multiplication, you have the right to switch the two factors”; others will baldly assert, almost as if some kind of magic were involved, “That’s just how it is” or “Well, it’s known to be a fact.” In short, for most well-educated adults, since multiplication is conceived of as repeated addition, its commutativity appears simply as a kind of miraculous coincidence, lacking any clear explanation or reason.

The above-cited treatise on arithmetic by Étienne Bezout provides a somewhat wordy and obscure justication for the commutativity of multiplication. If one’s vision of multiplication is rooted in the naïve analogy of repeated addition, then asking why the order of the factors makes no difference in multiplication amounts to asking why two different repeated additions give the same answer, and there is no obvious symmetry between the two operations involved. Bezout tries to resolve this dilemma, but his words are not terribly clear:

Most adults consider the fact that
a x b
always equals
b x a
to be a very useful but unexplained coincidence, which simply is empirically true. They “understand” the commutativity of multiplication in much the same way as they “understand” why bicycles don’t topple over and why airplanes can stay airborne: simply because they’ve seen such things for most of their lives and have long since forgotten that these phenomena are mysteries that crave explanation. And so, although education certainly drills into students the rote fact that multiplication is commutative, it fails to instill a deep
understanding
of multiplication’s nature; instead, it leaves them dependent on their initial naïve analogy.

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.