Surfaces and Essences

  For each of my 4 children, I buy one blue pen, one green pen, and one red pen. How many pens do I walk out with?

  Seeing things purely formally would involve instantaneously filtering out the store, the narrator, the pens, the colors, the children, and the money, and jumping directly to the bare-bones idea that “These problems all involve multiplication of 4 and 3”, and then simply carrying out that operation. Casting the problems this way should involve no asymmetry between the two factors, because if these problems are seen on a purely abstract level, they are all just multiplication problems. And indeed, given the above problems, few us would wonder, “Is this a ‘4 x 3’ situation or is it a ‘3 x 4’ situation?” We would instead simply tend to think, “Multiply 3 and 4 together.” Or at least this is one’s first introspective impression when, as an adult, one tackles these problems. But do we really perceive, in our mind’s eye, nothing but a “Multiply 3 and 4” situation in all three cases? Do we really see and do exactly the same thing when we solve these three problems? Let’s take a closer look at them.

  In the first problem, where every item costs 4 dollars, I buy one pen for $4, then another, and then another. Thus in the end, I spend 4 + 4 + 4 = 3 x 4 = 12 dollars.

  In the second problem, where I’m buying pens for my children, I buy 3 pens for each child. Thus I buy 3 pens for the first child, 3 for the second, 3 for the third, and 3 for the fourth child. Altogether, then, I buy 3 + 3 + 3 + 3 = 4 x 3 = 12 pens.

  In the third situation, there are two possible perspectives. From the first perspective, we focus on what each child will get — that is, we see things much as in the second problem. Each child gets 3 pens, and that makes 4 cases of “getting 3”, which means 3 + 3 + 3 + 3 = 4 x 3 = 12 pens.

  From the second perspective, I focus on colors rather than on children. I’m buying 4 blue pens (one for each child), 4 green pens, and 4 red pens. Therefore, since there are 3 colors concerned, I am buying 4 + 4 + 4 = 3 x 4 = 12 pens.

  If a person solving one of these word problems immediately perceived the abstract idea that it involved multiplication, then the way in which the problem happened to be concretely embedded in the world should have no effect on the order of the factors. By contrast, if these word problems are subliminally perceived through the filter of the naïve analogy repeated addition, as opposed to being perceived formally as multiplications, then the naïve analogy should guide the way the problems are solved, and the solutions that people find should be influenced by the preceding considerations.

  To test this hypothesis, we asked older elementary-school students and also university students to solve these problems without using multiplication. It turned out that nearly everyone, children and adults alike, used repeated addition, which is no surprise, and moreover that the additions that were chosen depended crucially on how the problems were stated, which may be more of a surprise.

  Thus roughly 90% of the subjects used the addition “4 + 4 + 4” for the first problem, and roughly the same percentage solved the second problem using the addition “3 + 3 + 3 + 3”. For the third problem, involving colored pens, roughly 50% of the subjects went for “3 + 3 + 3 + 3” (thus seeing it in terms of children), roughly 40% of them went for “4 + 4 + 4” (thus seeing it in terms of colors), and the remaining 10% didn’t group anything together explicitly; that is, the repeated addition they saw was “1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1”. Of course those subjects still made unconscious groupings — either “(1+1+1) + (1+1+1) + (1+1+1) + (1+1+1)” (each group representing a child) or else “(1+1+1+1) + (1+1+1+1) + (1+1+1+1)” (each group representing a color).

  This finding clearly supports our prediction, which is that given a word problem, people will try to solve it not just by perceiving its formal structure, but also by doing their best to find, based on the way the problem is worded, an analogy to repeated addition. In summary, to solve these word problems, people tend to mentally simulate the situations described in them.

  If, on the other hand, the situation had been spontaneously perceived “formally”, meaning that the abstract concept multiplication had been instantly evoked in subjects’ minds, then (since using multiplication had been explicitly banned) the repeated addition “4 + 4 + 4” would have been suggested by all the subjects for all the problems, because that sum involves the least computation, and there would have been no good reason to opt for the repeated addition “3 + 3 + 3 + 3”, since it uses more summands.

  To summarize, we have shown that even in a bare-bones mathematical situation, people are very seldom able to ignore all of its superficial, concrete aspects and to home in on just its abstract formal structure. For better or for worse, people are influenced by how situations are concretely described, by their familiarity with similar situations, and by the naïve analogies that these situations evoke naturally.

  Sometimes Situations Do Our Thinking for Us

  Here is a problem to solve, which contains no hidden trick:

  Lawrence buys an art kit for $7, and also a binder. He pays $15 altogether. John buys a binder and also a T-square. He pays $3 less than Lawrence did. How much does the T-square cost?

  Of course you’ve already gotten the correct answer, but that’s not the main point here. We would instead ask you to think carefully about this problem and try to find the most streamlined, efficient way to solve it, showing exactly how the shortest, simplest solution would work, step by step.

  Usually, people suggest a solution that involves three calculations, as follows:

  Price of the binder: 15 – 7 = 8 dollars.

  Price paid by John: 15 – 3 = 12 dollars.

  Price of the T-square: 12 – 8 = 4 dollars.

  This a perfectly correct way to solve it. But now, how about tackling the following problem, again looking for the shortest route to the solution:

  Laurel took ballet lessons for 7 years and stopped at age 15. Joan started at the same age as Laurel but stopped 3 years earlier. How long did Joan take ballet?

  Once again we are interested in the most economical way of solving this problem, and in seeing what exactly are the steps that must be taken. We thus ask you to indulge us once again in trying to find and spell out the minimal pathway to the solution.

  One idea is to carry out essentially the same steps as in the preceding problem:

  Age at which Laurel (and hence also Joan) started ballet: 15 – 7 = 8 years.

  Age at which Joan quit: 15 – 3 = 12 years.

  Total time that Joan took ballet: 12 – 8 = 4 years.

  Although this pathway to the solution is totally correct, another pathway might have come to your mind. If Laurel and Joan started taking lessons at the same age and Joan stopped 3 years earlier than Laurel did, then Joan took ballet 3 years less than Laurel did, so she took lessons for 7 – 3 = 4 years (we were explicitly told that Laurel took lessons for 7 years). This pathway involves just one arithmetical operation!

  The existence of this alternate route to the solution of the second problem suggests to most people that the two word problems differ fundamentally from each other, since the ballet-lesson one can be solved in just one step, whereas no similar shortcut exists for the school-supply problem. Is this really true, though?

  What would it mean to use the subtraction “7 – 3” in the context of the first problem? To be sure, it gives the right answer — 4 dollars — but does it mean anything? Most people who are given this problem tend to think it doesn’t, or that if it is a meaningful thing to do, then it would take a long time to figure out why, and it’s not worth it. And yet, the situation can be described as follows: “Both Lawrence and John bought a binder plus some other item”; this view leads one to a solution using just one single operation. One of them paid 3 dollars less than the other one paid. Therefore, the difference between what the two boys shelled out is due totally to the other item. Hence the price of John’s other item (the T-square) must be 3 dollars less than the price of Lawrence’s other item (the art kit): 7 – 3 = 4.


  What’s remarkable, when one compares the solutions of these two problems, is that fewer than 5% of elementary-school children and fewer than 5% of adults (in fact, of highly educated adults — namely, university students and schoolteachers) find the direct one-subtraction solution to the school-supply problem, whereas roughly 50% of the children and also 50% of the adults spontaneously find the one-subtraction solution to the ballet-lesson problem. And so, although all that’s required here is to carry out very trivial subtraction operations in extremely concrete situations, nonetheless the angle of attack that yields a one-step solution is almost never found in the first context while it is very often found in the second context.

  What Does It All Mean?

  It’s no accident that the very same one-operation method can be used to solve both problems, as both involve situations that could be handled by using the following formal rule, which has the feel of a theorem one might find in a set-theory textbook:

  If two sets overlap, then the difference between their sizes equals the difference in the sizes of their non-overlapping parts.

  If we apply this rule to the school-supplies problem, it tells us that the difference between what Lawrence paid and what John paid must be equal to the difference between the non-overlapping parts of their purchases — that is, the difference between Lawrence’s art kit and John’s T-square. If we apply the rule to the ballet lessons, it tells us that (since Laurel and Joan had equally long periods with no lessons), the difference between their ballet-quitting ages is equal to the difference between their ballet-lesson periods. If this formal rule were learned and fully absorbed by everyone, then we would expect that both problems would be always solved by the single-subtraction method. As we have seen, though, nothing of the sort happens. How come?

  Simply because the formal rule is not part of most people’s mental repertoire. Even people who discover the one-operation method for the second problem are unlikely to be aware of any such rule. Rather, they just allow the problem itself to direct their thoughts. If they come up with a one-step solution, it’s because that is what they are naturally led to. Each situation is defined, in a person’s mind, by the categories it effortlessly evokes, and that perception, rather than the application of any formal rule, is what guides the person’s thinking.

  The ballet-lesson problem is thus perceived as follows. If two events start at the same moment, and one of them lasts N time-units less than the other, then it will end N time-units before the other one does. This is so patently obvious to us all that the sentence tends to sound like a mere tautology, a vacuous triviality. Let us nonetheless restate it slightly differently: if two events start simultaneously, then the difference between their durations equals the interval between their cutoff moments. People’s perception of time is so deeply anchored and they so intuitively understand this basic principle — taking less time means ending earlier — that they often recall the statement of the ballet-lesson problem in a distorted fashion.

  To be specific, if students who have read the statement “Joan started ballet lessons at the same time as Laurel but took them for three years less” are asked to write it down by memory, they often do not reproduce it correctly, writing instead: “Joan started ballet lessons at the same time as Laurel but quit three years earlier.” Their deduction is so deeply fused with their perception of the situation that they don’t see it as such; they are unaware of having transformed the sentence in committing it to memory.

  Indeed, the transformation of the problem from the initial phrasing (“Joan took three years less”) to the final phrasing (“Joan finished three years earlier”) converts a difference between two lengths of time into a difference between two temporal stopping points. In the isomorphic school-supply problem, this would be comparable to someone converting the three-dollar price difference between Lawrence’s total outlay and John’s total outlay into the difference between just the art kit and the T-square. However, in our experiments we have never run across any subject who read the phrase “John’s total outlay was three dollars less than Lawrence’s” and subsequently wrote it down by memory as “the T-square costs three dollars less than the art kit”.

  What we see here is that prior knowledge about relations involving time (“three years less” can be converted into “three years earlier” or vice versa) allows people to solve the temporal word problem in one single step. In this sense, one could quite reasonably claim that it’s the situation that’s “doing the thinking” for the subjects. Finding the one-step solution is not a consequence of having mastered some general set-theoretic rule such as we quoted at this section’s outset.

  As is probably quite clear, students who solve the ballet-lessons problem do not rely on its abstract, formal structure (as expressed in the above-stated “theorem”); they have no need at all to evoke the abstractions of set theory in order to solve it. When they carry out their reasoning, they don’t perceive the time intervals involved as sets (if they did, each set would contain infinitely many infinitesimal moments!), nor do they perceive the common age at which Laurel and Joan started their lessons as the overlap of two sets (their “intersection”, in set-theoretic language), nor do they see the lengths of time that the two girls took lessons as the non-overlapping parts of sets, nor do they see the girls’ ages when they stopped their lessons as the sizes of two sets. In fact, it would take careful intellectual work to recast this problem in set-theoretical language, because set theory is not the framework in which humans naturally perceive it, and that’s why efficient solving of the problem by a person should not be taken as showing that the formal rule (the “theorem”) was correctly applied. Rather, the act of perceiving the ballet-lesson situation in terms of familiar time-categories does the bulk of the work for the student, and there is no need whatsoever to code the situation into an arcane, abstract technical formalism such as set theory.

  The diagrams on the facing page show three ways of conceiving these isomorphic word problems. The diagram at the top shows how the school-supplies problem is imagined by nearly everyone to whom it is given. The typical assumption is that in order to figure out the price of the T-square, you have to subtract the price of the binder from John’s total outlay (which itself is found through a subtraction), and the binder’s price is found by subtracting the price of the art kit from Lawrence’s total outlay — thus three subtractions in toto seem necessary. In this diagram, one doesn’t see that the difference between John’s and Lawrence’s total outlays is identical to the difference between the prices of the T-square and the art kit. That key idea, a prerequisite to solving the problem in a single step, is missing, and so three arithmetical operations seem to be needed.

  The middle diagram shows the way that many people very naturally envision the problem of the ballet lessons. They begin with the idea that Joan and Laurel took up ballet at the same age, and so the difference between the lengths of time they took ballet has to equal the difference between their ages when they quit. This idea is screamingly obvious in the diagram, and that explains why the one-step solution is often found for the ballet-lessons problem. (Let’s not fail to note the frame blend on which this diagram is tacitly based, and which Gilles Fauconnier and Mark Turner would delightedly point out — namely, the fact that we are imagining aligning the two girls’ lives on a single horizontal time axis. Aligning their lives means placing their births at the same spot on the time axis, and as a result of this maneuver, their first lessons will also coincide.)

  Each of the two upper diagrams (the top one, involving three operations, and the middle one, involving just one) was tailored to fit one specific problem, and it is not apparent that they have much in common with each other. Looking at the upper two diagrams alone, one might think that the problems they represent are of very different sorts; in fact, this is what most people claim who try to solve them both. But the bottom diagram not only shows how the two problems can be seen in a single unified manner; it also reveals how a one-step method works j
ust as naturally for the school-supply problem as for the ballet-lesson problem. While the “theorem” enunciated above was abstruse and difficult to grasp, this visual encoding of the two problems in one single picture reveals their isomorphism (i.e., the fact of having the same underlying structure), as well as how they both can be solved with just one arithmetical operation.

  These three diagrams show how the way a person visualizes a word problem can either bring out or hide a pathway to the solution. The bottom diagram could be said to be more abstract than the upper two in that it unifies them in a single diagram, but on the other hand, the image of two boxes standing on the same pedestal is very concrete, and as soon as these problems are cast in a form that involves a shared “pedestal” (the binder, in the first problem, and the starting age, in the second one), the abstruse idea expressed in the “theorem” suddenly becomes crystal-clear, as it has been fleshed out in a concrete manner, using simple, everyday images, such as boxes resting on a shared pedestal. Teaching a young student to see the “pedestal” in these two word problems imparts an elegant insight that is unavailable to most untrained adults.

  A key challenge for educators is thus to take into account the way people manage to adroitly sidestep the formal encoding of situations by exploiting the way their familiar categories, built up over years of interactions with the world, work. Although most teachers are quite aware that the way in which a problem is “dressed” can profoundly affect its difficulty, educators have not yet figured out how to make the art of dressing mathematics problems into a powerful teaching tool. To achieve this would be a great advance, but of course doing so constitutes a great challenge as well.

  A Naïve Analogy that has Ill Served Psychology

  Do you find it hard to see naïve categorizations such as multiplication is repeated addition and division is sharing as constituting analogies? Despite the multiple arguments we’ve mustered and the many situations we’ve dissected with a fine-tooth comb, perhaps a little voice inside you keeps insisting, “I’m sorry, but categorization and analogy-making are just not the same thing. Taking two notions that initially seem very far apart and then building a mental bridge between them because one sees that they have certain abstract qualities in common is a profoundly distinct type of act from seeing something and merely recognizing that it belongs to a familiar category, such as table.”