A Mathematician's Lament

by Paul Lockhart

  Are things really any different today?

  SIMPLICIO: I don’t think that’s very fair. Surely teaching methods have improved since then.

  SALVIATI: You mean training methods. Teaching is a messy human relationship; it does not require a method. Or rather I should say, if you need a method you’re probably not a very good teacher. If you don’t have enough of a feeling for your subject to be able to talk about it in your own voice, in a natural and spontaneous way, how well could you understand it? And speaking of being stuck in the nineteenth century, isn’t it shocking how the curriculum itself is stuck in the seventeenth? To think of all the amazing discoveries and profound revolutions in mathematical thought that have occurred in the last three centuries! There is no more mention of these than if they had never happened.

  SIMPLICIO: But aren’t you asking an awful lot from our math teachers? You expect them to provide individual attention to dozens of students, guiding them on their own paths toward discovery and enlightenment, and to be up on recent mathematical history as well?

  SALVIATI: Do you expect your art teacher to be able to give you individualized, knowledgeable advice about your painting? Do you expect her to know anything about the last three hundred years of art history? But seriously, I don’t expect anything of the kind, I only wish it were so.

  SIMPLICIO: So you blame the math teachers?

  SALVIATI: No, I blame the culture that produces them. The poor devils are trying their best, and are only doing what they’ve been trained to do. I’m sure most of them love their students and hate what they are being forced to put them through. They know in their hearts that it is meaningless and degrading. They can sense that they have been made cogs in a great soul-crushing machine, but they lack the perspective needed to understand it, or to fight against it. They only know they have to get the students “ready for next year.”

  SIMPLICIO: Do you really think that most students are capable of operating on such a high level as to create their own mathematics?

  SALVIATI: If we honestly believe that creative reasoning is too “high” for our students, and that they can’t handle it, why do we allow them to write history papers or essays about Shakespeare? The problem is not that the students can’t handle it, it’s that none of the teachers can. They’ve never proved anything themselves, so how could they possibly advise a student? In any case, there would obviously be a range of student interest and ability, as there is in any subject, but at least students would like or dislike mathematics for what it really is, and not for this perverse mockery of it.

  SIMPLICIO: But surely we want all of our students to learn a basic set of facts and skills. That’s what a curriculum is for, and that’s why it is so uniform—there are certain timeless, cold, hard facts we need our students to know: one plus one is two, and the angles of a triangle add up to 180 degrees. These are not opinions, or mushy artistic feelings.

  SALVIATI: On the contrary. Mathematical structures, useful or not, are invented and developed within a problem context and derive their meaning from that context. Sometimes we want one plus one to equal zero (as in so-called ‘mod 2’ arithmetic) and on the surface of a sphere the angles of a triangle add up to more than 180 degrees. There are no facts per se; everything is relative and relational. It is the story that matters, not just the ending.

  SIMPLICIO: I’m getting tired of all your mystical mumbo-jumbo! Basic arithmetic, all right? Do you or do you not agree that students should learn it?

  SALVIATI: That depends on what you mean by “it.” If you mean having an appreciation for the problems of counting and arranging, the advantages of grouping and naming, the distinction between a representation and the thing itself, and some idea of the historical development of number systems, then yes, I do think our students should be exposed to such things. If you mean the rote memorization of arithmetic facts without any underlying conceptual framework, then no. If you mean exploring the not-at-all obvious fact that five groups of seven is the same as seven groups of five, then yes. If you mean making a rule that 5 × 7 = 7 × 5, then no. Doing mathematics should always mean discovering patterns and crafting beautiful and meaningful explanations.

  SIMPLICIO: What about geometry? Don’t students prove things there? Isn’t high school geometry a perfect example of what you want math classes to be?

  High School Geometry: Instrument of the Devil

  THERE IS NOTHING QUITE SO VEXING TO THE AUTHOR of a scathing indictment as having the primary target of his venom offered up in his support. And never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as high school geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous.

  Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way.

  The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”

  All metaphor aside, geometry class is by far the most mentally and emotionally destructive component of the entire K-12 mathematics curriculum. Other math courses may hide the beautiful bird, or put it in a cage, but in geometry class it is openly and cruelly tortured. (Apparently I am incapable of putting all metaphor aside.)

  What is happening is the systematic undermining of the student’s intuition. A proof, that is, a mathematical argument, is a work of fiction, a poem. Its goal is to satisfy. A beautiful proof should explain, and it should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light—it should refresh the spirit and illuminate the mind. And it should be charming.

  There is nothing charming about what passes for proof in geometry class. Students are presented a rigid and dogmatic format in which their so-called “proofs” are to be conducted—a format as unnecessary and inappropriate as insisting that children who wish to plant a garden refer to their flowers by genus and species.

  Let’s look at some specific instances of this insanity. We’ll begin with the example of two crossed lines:

  Now the first thing that usually happens is the unnecessary muddying of the waters with excessive notation. Apparently, one cannot simply speak of two crossed lines; one must give elaborate names to them. And not simple names like ‘line 1’ and ‘line 2,’ or even ‘a’ and ‘b.’ We must (according to high school geometry) select random and irrelevant points on these lines, and then refer to the lines using the special “line notation.”

  You see, now we get to call them and . And god forbid you should omit the little bars on top—‘AB’ refers to the length of the line (at least I think that’s how it works). Never mind how pointlessly complicated it is, this is the way one must learn to do it. Now comes the actual statement, usually referred to by some absurd name like:

  PROPOSITION 2.1.1.

  Let and intersect at P.

  Then ∠APC ≅ ∠BPD.

  In other words, the angles on both sides are the same. Well, duh! The configuration of two crossed lines is symmetrical for crissake. And as if this weren’t bad enough, this patently obvious statement about lines and angles must then be “proved.”

  Proof:

  STATEMENT REASON

  1, m∠APC + m∠APD 180 m∠BPD + m∠APD = 180 1. Angle Addition Postulate

  2. m∠APC + m∠APD = m∠BPD + m∠APD 2. Substitution Property

  3. m∠APD = m∠APD 3. Reflexive Property of Equality


  4. m∠APC = m∠BPD 4. Subtraction Property of Equality

  5. ∠APC ≅ ∠BPD 5. Angle Measurement Postulate

  Instead of a witty and enjoyable argument written by an actual human being, and conducted in one of the world’s many natural languages, we get this sullen, soulless, bureaucratic form-letter of a proof. And what a mountain being made of a molehill! Do we really want to suggest that a straightforward observation like this requires such an extensive preamble? Be honest: did you actually even read it? Of course not. Who would want to?

  The effect of such a production being made over something so simple is to make people doubt their own intuition. Calling into question the obvious, by insisting that it be “rigorously proved” (as if the above even constitutes a legitimate formal proof), is to say to a student, “Your feelings and ideas are suspect. You need to think and speak our way.”

  Now there is a place for formal proof in mathematics, no question. But that place is not a student’s first introduction to mathematical argument. At least let people get familiar with some mathematical objects, and learn what to expect from them, before you start formalizing everything. Rigorous formal proof only becomes important when there is a crisis—when you discover that your imaginary objects behave in a counterintuitive way; when there is a paradox of some kind. But such excessive preventative hygiene is completely unnecessary here—nobody’s gotten sick yet! Of course if a logical crisis should arise at some point, then obviously it should be investigated, and the argument made more clear, but that process can be carried out intuitively and informally as well. In fact it is the soul of mathematics to carry out such a dialogue with one’s own proof.

  So not only are most kids utterly confused by this pedantry—nothing is more mystifying than a proof of the obvious—but even those few whose intuition remains intact must then retranslate their excellent, beautiful ideas back into this absurd hieroglyphic framework in order for their teacher to call it “correct.” The teacher then flatters himself that he is somehow sharpening his students’ minds.

  As a more serious example, let’s take the case of a triangle inside a semicircle:

  Now the beautiful truth about this pattern is that no matter where on the circle you place the tip of the triangle, it always forms a nice right angle. (I have no objection to a term like “right angle” if it is relevant to the problem and makes it easier to discuss. It’s not terminology itself that I object to, it’s pointless, unnecessary terminology. In any case, I would be happy to use “corner” or even “pigpen” if a student preferred.)

  Here is a case where our intuition is somewhat in doubt. It’s not at all clear that this should be true; it even seems unlikely—shouldn’t the angle change if I move the tip? What we have here is a fantastic math problem! Is it true? If so, why is it true? What a great project! What a terrific opportunity to exercise one’s ingenuity and imagination! Of course no such opportunity is given to the students, whose curiosity and interest is immediately deflated by:

  THEOREM 9.5.

  Let Δ ABC be inscribed in a semicircle

  with diameter .

  Then ∠ABC is a right angle.

  Proof:

  STATEMENTREASON

  1. Draw radius OB. Then OB = OC A 1. Given

  2. m∠OBC = m∠BCA m∠OBA = m∠BAC 2. Isosceles Triangle Theorem

  3. m∠ABC = m∠OBA + m∠OBC 3. Angle Sum Postulate

  4. m∠ABC + m∠BCA + m∠BAC = 180 4. The sum of the angles of a triangle is 180

  5. m∠ABC + m∠OBC + m∠OBA = 180 5. Substitution (line 2)

  6. 2 m∠ABC = 180 6. Substitution (line 3)

  7. m∠ABC 90 7. Division Property of Equality

  8. ∠ABC is a right angle 8. Definition of Right Angle

  Could anything be more unattractive and inelegant? Could any argument be more obfuscatory and unreadable? This isn’t mathematics! A proof should be an epiphany from the gods, not a coded message from the Pentagon. This is what comes from a misplaced sense of logical rigor: ugliness. The spirit of the argument has been buried under a heap of confusing formalism.

  No mathematician works this way. No mathematician has ever worked this way. This is a complete and utter misunderstanding of the mathematical enterprise. Mathematics is not about erecting barriers between ourselves and our intuition, and making simple things complicated. Mathematics is about removing obstacles to our intuition, and keeping simple things simple.

  Compare this unappetizing mess of a proof with the following argument devised by one of my seventh-graders:Take the triangle and rotate it around so it makes a four-sided box inside the circle. Since the triangle got turned completely around, the sides of the box must be parallel, so it makes a parallelogram. But it can’t be a slanted box because both of its diagonals are diameters of the circle, so they’re equal, which means it must be an actual rectangle. That’s why the corner is always a right angle.

  Isn’t that just delightful? And the point isn’t whether this argument is any better than the other one as an idea, the point is that the idea comes across. (As a matter of fact, the idea of the first proof is quite pretty, albeit seen as through a glass, darkly.)

  More important, the idea was the student’s own. The class had a nice problem to work on, conjectures were made, proofs were attempted, and this is what one student came up with. Of course it took several days, and was the end result of a long sequence of failures.

  To be fair, I did paraphrase the proof considerably. The original was quite a bit more convoluted, and contained a lot of unnecessary verbiage (as well as spelling and grammatical errors). But I think I got the feeling of it across. And these defects were all to the good; they gave me something to do as a teacher. I was able to point out several stylistic and logical problems, and the student was then able to improve the argument. For instance, I wasn’t completely happy with the bit about both diagonals being diameters—I didn’t think that was entirely obvious—but that only meant there was more to think about and more understanding to be gained from the situation. And in fact the student was able to fill in this gap quite nicely:Since the triangle got rotated halfway around the circle, the tip must end up exactly opposite from where it started. That’s why the diagonal of the box is a diameter.

  So a great project and a beautiful piece of mathematics. I’m not sure who was more proud, the student or myself. This is exactly the kind of experience I want my students to have.

  The problem with the standard geometry curriculum is that the private, personal experience of being a struggling artist has virtually been eliminated. The art of proof has been replaced by a rigid step-by-step pattern of uninspired formal deductions. The textbook presents a set of definitions, theorems, and proofs, the teacher copies them onto the blackboard, and the students copy them into their notebooks. They are then asked to mimic them in the exercises. Those that catch on to the pattern quickly are the “good” students.

  The result is that the student becomes a passive participant in the creative act. Students are making statements to fit a preexisting proof-pattern, not because they mean them. They are being trained to ape arguments, not to intend them. So not only do they have no idea what their teacher is saying, they have no idea what they themselves are saying.

  Even the traditional way in which definitions are presented is a lie. In an effort to create an illusion of clarity before embarking on the typical cascade of propositions and theorems, a set of definitions is provided so that statements and their proofs can be made as succinct as possible. On the surface this seems fairly innocuous; why not make some abbreviations so that things can be said more economically? The problem is that definitions matter. They come from aesthetic decisions about what distinctions you as an artist consider important. And they are problem generated . To make a definition is to highlight and call attention to a feature or structural property. Historically this comes out of working on a problem, not as a prelude to it.

  The point is you don’t start with definitions, you
start with problems. Nobody ever had an idea of a number being “irrational” until Pythagoras attempted to measure the diagonal of a square and discovered that it could not be represented as a fraction. Definitions make sense when a point is reached in your argument which makes the distinction necessary. To make definitions without motivation is more likely to cause confusion.

  This is yet another example of the way that students are shielded and excluded from the mathematical process. Students need to be able to make their own definitions as the need arises—to frame the debate themselves. I don’t want students saying, “the definition, the theorem, the proof,” I want them saying, “my definition, my theorem, my proof.”

  All of these complaints aside, the real problem with this kind of presentation is that it is boring. Efficiency and economy simply do not make good pedagogy. I have a hard time believing that Euclid would approve of this; I know Archimedes wouldn’t.

  SIMPLICIO: Now hold on a minute. I don’t know about you, but I actually enjoyed my high school geometry class. I liked the structure, and I enjoyed working within the rigid proof format.