Surfaces and Essences

by Douglas Hofstadter

  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.

  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:

  Paul has 10 boxes containing 4 cookies apiece. How many cookies does he have in all?

  Paul has 4 boxes containing 10 cookies apiece. How many cookies does he have in all?

  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:

  Jill passes out 40 cookies to her 4 children. How many cookies does each child get?

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

  Paul has 31 marbles. He gives 27 to his friend Peter. How many does he have left?

  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”).

  The Influence of Language on Naïve Analogies

  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.

  What Schooling Leaves Untouched in Our Minds

  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 d
esperate 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.

  We will illustrate this using one of the easiest possible cases for an adult — multiplication, which, as we stated earlier, is one of the first operations taught in school, and which is a concept that few people feel they have anything more to learn about. To make it even simpler, we’ll look only at cases of multiplication involving positive integers. We compare three situations:

  I go into a store where every item costs 4 dollars, and I buy 3 pens. How much money do I spend?

  For each of my 4 children, I buy 3 pens. How many pens do I walk out with?