Surfaces and Essences: Analogy as the Fuel and Fire of Thinking (101 page)

To recapitulate, it turns out problems #1 and #3 are fairly easy to simulate in one’s mind’s eye, while problems #2 and #4 are challenging. If we go back to our earlier distinction between division as
sharing
and division as
measuring
, we see that two of these problems are easy to simulate mentally: there is an
easy sharing problem
(#3, where 200 photos are shared among 4 piles) and there is an
easy measuring problem
(#1, in which a 200-photo stack is measured using big stacks of height 50). There are also two problems that are very hard to simulate mentally: there is a
difficult sharing problem
(#2, where 200 photos are shared among 50 piles), and there is a
difficult measuring problem
(#4, where a 200-photo stack is measured using small stacks of height 4).

Let’s rephrase this in another way. One problem involving the division “200/4” is
easy
to simulate mentally (#3: 200 photos shared among 4 piles), while another problem that involves exactly the same division is
hard
to simulate mentally (#4: “Measure a 200-photo stack using piles of size 4”). Likewise, one problem involving the division “200/50” is
easy
to simulate mentally (#1: “Measure a 200-photo stack using piles of size 50”), while another problem involving exactly the same division is
hard
to simulate mentally (#2: 200 photos shared among 50 piles).

Here is a short summary of what can be said about these four word problems, all of which are very similar in form (each involves a division whereby a pile of photos is broken into smaller piles), yet differ greatly in how one conceives of them (some involve sharing, some involve measuring) and also in terms of their difficulty for a solver who is mentally simulating them (some are easy, some are hard):

1. If we break a stack of 200 photos into piles of height 50, how many piles do we get? [200/50; easy measuring problem]

2. If we break a stack of 200 photos into 50 piles, how many photos are in each pile? [200/50; difficult sharing problem]

3. If we break a stack of 200 photos into 4 piles, how many photos are in each pile? [200/4; easy sharing problem]

4. If we break a stack of 200 photos into piles of height 4, how many piles do we get? [200/4; difficult measuring problem]

Let’s presume that when subjects solve such a problem, they do so by making a mental simulation rather than by instantly carrying out an arithmetical calculation. If that’s the case, then the first and third problems should both be
easy
, while the second and fourth should be
difficult
; moreover, knowing which arithmetical calculation is involved (200/4 or 200/50) does not tell us how hard the problem is.

This way of looking at word problems stands in marked contrast to the traditional view, which takes the formal arithmetical operation needed to solve a problem as a gauge of the problem’s difficulty. In place of this, the new perspective highlights the spontaneous way in which the situation is framed — that is, the analogy that allows one to solve the problem in a very direct way, namely by counting in one’s head. This point of view shows why different word problems, even if they are equally down-to-earth, and even if they all involve exactly the same
formal
arithmetical operation, can nonetheless have extremely different levels of difficulty. The surprising findings that we described earlier concerning Brazilian teen-aged street vendors tackling multiplication problems now seem to apply more generally, both to other kinds of word problems and to other groups of people. The key variable is seen to be the simplicity of the mental simulation that will yield the correct solution.

Rémi Brissiaud, a developmental psychologist, has done pioneering research into these ideas in the context of learning to do arithmetic, and he has written very innovative and efficient new mathematics textbooks inspired by his discoveries. In collaboration with him, we have studied how seven-year-olds who are just beginning to learn the basic arithmetical operations tackle different kinds of word problems. Our results show a clear distinction between problems that tend to be solved by mental simulation and ones that tend to make children resort to formal arithmetical operations. In cases where the mental simulation is not unwieldy, it is always preferred. This principle is illustrated by the following examples:

The first problem, if solved by simulation and not multiplication, requires adding 4 to itself 10 times. In such a simulation, children might imagine Paul at the grocery store, taking boxes one by one off the shelf and placing them in his shopping cart. Thus it would go as follows: “Box #1 (4 cookies); #2 (8); #3 (12), … , #9 (36), #10 (40)”. This process requires one to count and to keep a running tab at the same time, and for children just learning how to add, repeatedly adding up all these 4’s is far from easy. Mental simulation is hard here. By contrast, the second problem is solved by mental simulation quite handily. It takes just four additions, and what’s more, they’re all easy; in fact, each sum along the way echoes the counting number just preceding it, as follows: “Box #1 (10 cookies); #2 (20); #3 (30); #4 (40)”.

Our experiments have shown that children are much better at solving word problems in which mental simulation comes easily than they are at solving problems in which it does not. Not only does this hold
before
the relevant formal arithmetical
operation has been taught to them (not too big a surprise!), but it also holds
after
it has been taught (this, by contrast, is quite surprising). We observed that even two years after the relevant arithmetical operation has been taught, if mental simulation provides a short solution path, the problem is solved much more easily than via formal calculation.

Some of our findings fly in the face of received ideas about the relative difficulty of arithmetical operations. For instance, subtraction is usually considered to be an easier operation than division, and is thus taught in schools already roughly at age 6, whereas all mention of division is put off for another two years or so. And yet our experiments have shown that children who have supposedly mastered subtraction but have heard nary a word about division manage to solve certain division problems (those that can be done via mental simulation) better than they can solve subtraction problems in which mental simulation is inefficient. Here is an example of what we are talking about:

This division problem is much more easily solved by children at the above-described stage than the following subtraction problem:

This is not simply due to the fact that 40 breaks easily into 10 + 10 + 10 + 10, because the problem “Jill has 40 cookies and wants to make little packets of 4 each. How many packets will she make?” turns out to be far harder than the one given above, involving 40 cookies given out to 4 children, although both have the same answer (namely, 10).

Our experimental findings show that whenever it’s possible, children opt for using analogies to real-world situations rather than making formal arithmetical calculations. If a word problem can be conceived of in such a way that formal calculations can be bypassed, then simulation is the pathway that children tend to follow. The formal technique will be wheeled out only when there is no alternative — that is, in situations where mental simulation would be inefficient, either because it would require too many steps (adding up ten 4’s) or because it would require the use of arithmetical facts about which the child is still a bit shaky (
e.g
., “4 + 16 = 20”).

Does
sharing
’s dominance over
measurement
as a naïve analogy for division mean that the former is a simpler concept than the latter? Is
sharing
such a natural and familiar idea that it automatically and irrepressibly jumps to mind as a ready-made analogue for division? And is
measurement
such a rare and unfamiliar idea that it is unlikely to be used as an analogue for division? Is this why
sharing
enjoys the lion’s share of mental imagery for division?

The answer is no. The predominance of sharing as the naïve analogy for division is not due to intrinsic simplicity, but merely to an accident of language. Division is unconsciously associated with sharing in the minds of most speakers of English because the English word “division” has both a mathematical meaning and an everyday meaning, and connotations of the everyday meaning inevitably spill over into the technical meaning; as a consequence, the naïve analogy of
division as sharing
overwhelms that of
division as measuring
.

Suppose there were an arcane mathematical notion called “surgery” (indeed, it exists). If you were told that surgery sometimes involves smoothly tying things together and other times involves tearing things asunder, it seems likely that, thanks to your prior familiarity with medical surgery, the naïve analogy that you would unconsciously exploit in trying to make sense of the notion would tilt more towards
tying together
than
tearing apart.
A similar story can be told about division and sharing. Suppose that hundreds of years ago, the English word assigned to the mathematical concept of
division
had not been “division” but “measurement”. Had that been the case, then children today, on first hearing about the arithmetical notion called “measurement”, would tend to create a very different primary naïve analogy for it. And the idea of sometimes getting a larger answer than what they started with (that is,
a/b
being sometimes greater than a) wouldn’t strike them as strange or confusing in the least.

In sum, the predominance of the naïve analogy “division is sharing” doesn’t imply that envisioning a measurement (“How many B’s will fit inside A?”) is cognitively more demanding than envisioning an act of sharing (“If I cut A into B parts, how big will each part be?”). Indeed, quite to the contrary, we’ve seen that a measurement problem such as “How many 50’s are there in 200?” is much easier than a sharing problem that involves exactly the same numerical values: “Dole out 200 candies to 50 kids!”

What we learn from this example and similar ones is that the prevalence of
sharing
as the naïve analogy for division is not because sharing is easier to imagine than measuring; it is because there is an unconscious bleed-through of the everyday meaning of “to divide” into the technical term “to divide”, and this semantic contamination gives a big head start to
sharing
as the source of the naïve analogy, even if there are many cases where
measurement
would be a more apt analogy. In other words, it’s easy to solve “How many 50’s are there in 150?” in one’s head (“50 and 50 and then 50 again — that makes 150, so the answer is 3”), but to realize that one has just solved a
division
problem is not easy at all, because the usual feeling of carrying out a division is pervaded by the everyday sense of that word, which has no connection with the idea of measuring anything.

Does what we learn in school profoundly affect how we see situations? Does school teach us to think “formally” about situations? By this, we mean acquiring the ability to zoom straight to the abstract core of a situation, not deflected by its concrete details. We all do this in some aspects of daily life — that is, we routinely ignore many aspects
of certain situations, though we are fully aware of them intellectually. Thus, we know but we forget that our closet door was once part of a tree, that Adolf Hitler was once a baby, that the meat on our table was not long ago inside an animal grazing in a field, and so forth. Even if we accept the
truth
of these facts, we systematically ignore them, so that it’s fair to say that we simply don’t see an ex-animal in the steak, nor an ex-tree in the door, nor an ex-baby in photos of the Führer. This is not stupidity but intelligence.

In the same way, we pay no attention to all sorts of properties of the objects that surround us. Who would think of using a painting hanging on the wall as a tray on which to carry the dirty dishes into the kitchen, or as a bulletin board onto which we could post a bunch of family photos, or as a throw rug that might decorate our floor? And yet in theory, and in case of extreme need, any of these uses might come to mind, perhaps even seeming eminently reasonable. Only when one is extremely angry or frightened does it ever occur to one that a candle, a plate, a vase, a glass, a statuette, a chair, and a mirror are all potential weapons, and this “forgetfulness” is as it should be.

Categorization involves taking a certain point of view, and once one has chosen a category for something in one’s environment, that act tends to suppress the perception of all sorts of properties that are irrelevant to the chosen category. Who ever wonders if the hamburger in their bun came from a male or a female cow? And who would ever care if it’s a left or a right shoe when one is so starved that one is desperate to eat it? In short, we are constantly abstracting and thus constantly ignoring thousands of potentially observable facets of things and situations. Once again, this is not stupidity but intelligence. Does school teach us to use this kind of “intelligent forgetting” or “intelligent ignoring” more systematically, especially in mathematics?

When most people are given a mathematical word problem — even a very simple one — they have great trouble ignoring some of its irrelevant aspects. Instead of treating such a problem in a
formal
manner, they tend to be influenced by some of its salient concrete features. Even if someone eventually discovers the abstract mathematical structure in a word problem, that recognition never fully overrides the person’s more spontaneous initial view of the situation; various concrete aspects get blurred in with more abstract ones. Our ability to perceive mathematical situations formally — that is, in such a way that our thinking is not contaminated by some of their irrelevant, surface-level aspects — is very limited.