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What's the most harmful heuristic (towards proper mathematics education), you've seen taught/accidentally taught/were taught? When did handwaving inhibit proper learning?

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In view of many of the answers to this question, it might help to have in the statement a definition of heuristic as it is applied to mathematics. –  Pete L. Clark Apr 26 '10 at 3:56
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In fact, the harmful entity in most answers is not a heuristic at all! –  Victor Protsak May 22 '10 at 15:07

30 Answers 30

Not the most harmful, but a fun example (credit due to Tony Varilly):

"You can't add apples and oranges."

False. You can in the free abelian group generated by an apple and an orange. As Patrick Barrow says, "A failure of imagination is not an insight into necessity."

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This almost belongs in the mathematical jokes question. ;) –  Grétar Amazeen Oct 25 '09 at 2:13
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Two apples plus three oranges equals five pieces of fruit. What's the problem? –  Gerry Myerson Aug 19 '10 at 5:50
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Indeed. Take the free abelian group A generated by the set of all types of fruit and consider the natural homomorphism onto the free abelain group generated by {Fruit} induced by sending each generator of A to the single generator of <Fruit> ... –  Steven Gubkin Oct 2 '10 at 23:59
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So according to Steven's remark, there is of course a universal way to add apples and oranges. (If this observation is not in Mathematics Made Difficult, then it ought to be.) –  Todd Trimble Apr 17 '13 at 19:59

A tensor is a multidimensional array of numbers that transforms in the following way under a change of coordinates...

I saw that for years, and I never understood it until I saw the real definition of a tensor.

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What's "the real definition of a tensor"? Element of a tensor product? A section of a tensor bundle? –  Victor Protsak May 22 '10 at 15:09
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I second this wholeheartedly!!!!!! @Victor Protsak: For me, "the real definition of a tensor" is something like the following. "Let M be a smooth manifold; let FM be the space of smooth functions on M, let VM be the space of smooth vector fields on M, and let V*M be the space of smooth covector fields on M. A (k,l) tensor is a multilinear map from (VM)^k x (V*M)^l to FM." There might be more abstract and versatile definitions, but this one seems to work pretty well in the context of general relativity, which is where the definition Darsh Ranjan quoted tends to show up (in my experience). –  Vectornaut May 22 '10 at 22:01
    
OK, a proposed intuitive point of view: say you have an ordered pair of vectors. If you multiply one of them by a nonzero scalar and the other by the reciprocal of that scalar, you've got a different ordered pair of vectors. But in both cases you have the same tensor. All the other algebraic requirements that are supposed to be satisfied are just there to make sure the algebra works out neatly the way it should. But this one is where the basic intuition is. –  Michael Hardy Jun 16 '10 at 15:08
    
For that matter, teaching linear algebra and doing echelon forms and so on does not strike me as very enlightening. When I saw a matrix for the first time, it was already in the context of linear maps and for given bases of source and target spaces. –  Thierry Zell Aug 18 '10 at 21:54
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I second, third and fourth that, Darsh. That particular definition of tensor set back my understanding of differential geometry by at least a year. –  Cosmonut Oct 3 '10 at 3:39

This isn't really a heuristic, but I hate "functions are formulas." It takes a lot of students a really long time to think of a function as anything other than an algebraic expression, even though natural algorithmic examples are everywhere. For example, some students won't think of f(n) = {1 if n is even, -1 if n is odd} as a function until you write it as f(n) = (-1)^n.

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I think this actually comes up even as late as upper division linear algebra, when they start to talk about general linear transforms, or, something even harder for students, the space of linear transforms from V to W. I worked with a student for quite a long time on this –  Michael Hoffman Oct 24 '09 at 21:27
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I think that by the second year of high school normally smart students should perfectly get the point with this. –  Qfwfq Apr 25 '10 at 19:06
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<p> I'm a high school student and I can safely say that most of my peers just don't get what a function is. The only ones who do seem to have learned from programing. Then again, all the really mathematically talented students in my very small school also program... </p> <p>Functions seem to get slipped in somewhere along the line without a proper introduction, and then it is assumed that students know it from there on in.</p> –  Christopher Olah Apr 25 '10 at 22:21
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Actually I still have a lot of trouble going back the other way, to "functions are polynomial formulas, not maps" in algebraic geometry and/or combinatorics. –  Elizabeth S. Q. Goodman Jan 27 '12 at 8:31

Along the same lines as Qiaochu's and Zach's responses, the commonly taught heuristics pertaining to functions, differentiability and integration are a pet hate of mine.

I certainly left school thinking of functions as formulas involving combinations of elementary functions and having a very poor understanding of the relevance and correct relationship between integration and differentiation, the worst manifestation of which, now that I'm a bit older, seems to have been that

Differentiation is a nice, computable operation and tells you about functions; integration is hard and tells you about areas under curves.

Areas under curves never seemed interesting. As an analyst, my personal feelings towards them are now almost entirely reversed and I think of integration as my friend and differentiation as the enemy.

Differentiation uses up regularity; integration smooths.

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I went through the same reversal as you recently. Slightly different but related reason. My explanation is <a href="mathoverflow.net/questions/11540/…;. –  Dan Piponi Mar 1 '10 at 5:03
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That's because on formulas differentiation is nice and integration is hard, but on computable functions differentiation is hard and integration is nice. In theory, we have a denotational semantics between formulas that functions that should transport these notions back-and-forth, but we really really don't. There are tons and tons of papers in computer algebra which basically boil down to this massive gulf between abstract analysis (the study of functions given by properties) and concrete analysis (study of functions given by formulas). –  Jacques Carette Mar 13 '10 at 3:50
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I'm upvoting this partially because I agree, but mostly because you used the term "pet hate" as opposed to "pet peeve". –  Jamie Weigandt Apr 26 '10 at 0:51
    
@Jacques: that's really well-phrased! I had an "a-ha" moment reading your comment. –  Neel Krishnaswami Apr 26 '10 at 9:40
    
I had a similar reversal in a different area. In numerical analysis, numerical integration is (relatively) well understood, stable, and generally nice, while numerical differentiation is kind of a mess. But until graduate school I would have said the opposite. –  Andrew T. Barker Apr 18 '13 at 10:02

"Stacks are schemes with groups attached to points."

I don't know how much damage this has caused, but I never understood how it was actually helpful to anybody. Not only is it hand-wavy (which is okay for a heuristic), but it's hand-wavy in a way that can't really be corrected (because it's false). My feeling is that people who adopt this heuristic are trapped. If they use the heuristic to come up with a result, it's very hard to sharpen the reasoning to turn it into a proof. You have to just start from scratch and not use the heuristic.

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How do I up-vote answers multiple times?!?! –  Kevin H. Lin Oct 24 '09 at 23:02
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By leaving a comment explaining that the answer is so great others just have to upvote it. You convinced me, by the way, to give my last daily vote :). –  Ilya Nikokoshev Oct 24 '09 at 23:50
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Anton: ok, the heuristics of "groups attached to points" is very incomplete, but... so how do you (heuristically) imagine a stack, you really think of it as a forest of objects and arrows over the category of schemes?? [*/G] ? Orbifolds? Orbifold curves? Gerbes? –  Qfwfq Apr 25 '10 at 19:14
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This seems to be a carbon copy of a very useful, in my opinion, heuristic "orbifolds are manifolds with groups attached to points". –  Victor Protsak May 22 '10 at 15:15
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This seems to me to be one of those heuristics which is very useful as a first approximation, but very misleading if one starts to think of it as the whole story. –  Peter LeFanu Lumsdaine Sep 27 '10 at 18:21

Two-column proofs

Usually the only proofs that students see upon graduating from high-school are the geometry "two-column" proofs, and trying to convince them that the essence of mathematical proof lies not in the form but in the logical deductive argument takes a lot of convincing.

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Do students even see the two-column proofs any more? From some things I've read I've gotten the impression that those have been pushed aside in favor of just not proving anything at all. –  Michael Lugo Oct 28 '09 at 23:11
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They certainly do. –  Akhil Mathew Oct 29 '09 at 0:06
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If students are taught that two-column proofs are the only kind there is, then I agree that they could be harmful. However, I think the framework of two-column proofs can be extremely helpful in teaching students to think through the underlying structure of a proof before trying to write it out in paragraph form, because it helps them avoid vague hand-waving arguments. When I teach undergrads how to do proofs, I have them write two-column proofs first, and then explain that "This is what the proof looks like naked. But to take it out in public, you need to put clothes on it." –  Jack Lee Aug 18 '10 at 17:30
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...what is a two-column proof? –  Piero D'Ancona Aug 18 '10 at 19:57
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A two-column proof is a proof arranged as a series of numbered statements, with the statements in the left-hand column and corresponding justifications in the right-hand column. This used to be the way proofs were universally taught in US high-school geometry courses. They're still taught this way, but somewhat less universally, I think. –  Jack Lee Aug 18 '10 at 20:39

Linear algebra purely as row manipulations. I've written about this here:

Students stuck in a rut of thinking of matrices as a clever way to arrange numbers will get lost and confused; I know this because I was one of those students. I had to “de-program” what I was taught in high school before I could grasp what was going on.

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Agreed. It's really hard to internalize what all those intermediate steps in a row reduction actually mean. –  Qiaochu Yuan Oct 24 '09 at 21:11
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I had no idea why matrices would exist until beginning the linear algebra class I'm currently in. They seemed perverse and non-sensical. They really don't belong in high school math, frankly. I didn't even remember how to multiply them until I refreshed myself recently. –  DoubleJay Oct 25 '09 at 16:47
    
By the time I got to linear algebra last year, I had already totally forgotten how to multiply matrices. Luckily, for proofs, the definition of matrix multiplication is a better way to prove something than drawing out (with ...'s) a big nxn matrix. –  Harry Gindi Apr 25 '10 at 15:43
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I didn't come across matrices until university, but I wholeheartedly agree that linear algebra should not begin with matrices and their operations. I didn't get a proper view of linear algebra (especially the determinant, which was basically taught by giving the definition and making the students calculate the determinant of a general four-by-four matrix by hand) until I read Sheldon Axler's "linear algebra done right". There the pedagogical idea was to begin with linear mappings and noting as a side note how they can be presented with these funny squares of numbers etc... –  Rami Luisto Apr 17 '13 at 20:13
    
This. Matrices in high school were one of the things that pushed me towards going for Physics instead of Math in college. Only when I had my linear algebra course did I get what matrices were about. –  finitud Apr 25 at 15:36

"Truth is binary. If a theorem has been proven once, there is no need in a second proof."

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The "FOIL" (first+outside+inside+last) mnemonic for multiplying two binomials is terrible. It suppresses what is really going on (three applications of the distributive property) in favor of an algorithm. In other words, it is teaching a human being to behave like a computer.

The legacy of FOIL is clear when you ask your students to multiply three binomials, or two trinomials. Students usually either have no idea what to do, attempt it but get lost in the algebra, or succeed but complain about the arduousness of the task.

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I can't stand FOIL! It seems to indicate to students that order matters here. I don't see what FOIL adds, but it certainly detracts from the idea of just multiplying all the pairs and adding. Instead of teaching the idea (which they'll never forget), they now have something memorized (easy to forget). And I once had a student erase their correct work because they accidentally did FLOI or something and rewrite the same thing in a different order. –  Matt Apr 25 '10 at 18:43
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As much as I dislike teaching mathematics "algorithmically", there is a reason why FOIL is taught as such: by forcing the user to adopted an algorithm, you can minimize mistakes. Doing things "in order" is a good habit, which should be encouraged. It is unfortunate the trend where "educators" take good practices, and distil from it something all but recognizable... –  Willie Wong Apr 25 '10 at 21:18
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As a high school teacher, I usually encountered students after their first exposure to FOIL, so I made a point to revisit the process and introduce "Super-FOILing" (which, of course, was just applying the distributive property to two polynomials of any length). Yes, yes: I hammered proper terminology and all the conceptual stuff, too, but starting off with "Ah, so you can FOIL ... but can you *Super*FOIL?" really made the ears perk right up! In a way, prior exposure to FOIL was helpful to me, providing an accessible object lesson that math is always "bigger" than any of us are ever taught. –  Blue Apr 25 '10 at 21:23
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"teaching a human being to behave like a computer" -- or like a dog. Show students an expression like $(x-1)(x-3)$, and many will have a Pavlovian response "FOIL!" even if it doesn't do them any good. –  Todd Trimble Aug 25 '12 at 20:43
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Todd: Ask students on an exam to solve an equation such as $(x-1)(x-2)(x-3)(x-4)=0$. I've done this a couple of times. A very common attempt of solution was to expand things out (often making mistakes along the way), contemplate the new, messy equation, and declare, "It can't be factored!". Sad... –  Pedro Teixeira Apr 17 '13 at 21:35

"Generalization for the sake of generalization is a waste of time"

I think that generalization for the sake of generalization can be rather fruitful.

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Whoever first said that had in mind one or two specific examples of empty or shallow generalizations, and generalized based on those examples, purely for the sake of generalization. –  Tracy Hall Aug 18 '10 at 22:18

"Categories can be specified by objects alone." It's easy to get this impression, because people who are familiar with the categories in question already know the morphism structure, and don't bother to specify it. There is a related heuristic concerning the composition law, but it doesn't seem to burn people as often.

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Similar abuses of language include naming a model category by its fibrant objects ("the model category of quasicategories") or a 2-category by its 1-morphisms ("the 2-category of spans"). –  Reid Barton Oct 24 '09 at 22:03
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yet nobody is brave enough to name categories from the name of arrows, like if we said "category of continuous mapping" for Top, etc. –  Pietro Majer Jun 25 '10 at 10:55
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@Pietro With the exception of Ehresmann and his school. :-) –  Robert K Mar 13 '11 at 15:24
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I'd like to hear a convincing example where this has really been a problem. Usually there's a default notion of morphism (think of the category of sets, for instance), and in my experience, when anyone departs from the default, they make a point of it (e.g., the category or bicategory of sets and relations -- see, I didn't specify the 2-cells just now!). I hope Thierry can remember the details of his tale. –  Todd Trimble Aug 25 '12 at 19:40
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Ironically, I just had an example the other day (linear codes) where it wasn't completely clear to me what the correct notion of isomorphism should be!! So this is me answering my former (August 25 2012) self. –  Todd Trimble Apr 18 '13 at 15:24

One extremely harmful heuristic I held until fairly recently: identifying math with algebraic manipulation. When asked to prove an identity or an inequality I would often dive straight into algebraic manipulation of the relations that I knew, wasting many many hours of my time. I have found that it is much more useful to try and test statements against examples I already know, and to try and rephrase identities and inequalities in terms of a statement in natural language that I have some intuition for.

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"Vectors are directed line segments." When worded this way, this utterance is only acceptable if the student is satisfied with getting on his or her bicycle at the end of class and never returning to mathematics again.

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Well...in principle, you could define a vector of, say, R^2 to be an equivalence class of "directed line segments". –  Qfwfq May 11 '10 at 12:26
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That's verbatim how I learned the definition of vector. But the "equivalence class" part of it changes everything (and did not go over too well with many of the other students; it was junior-high after all...) –  Thierry Zell Aug 18 '10 at 22:06

That there is something weird and unsavory about field extensions that are not separable and that serious contemplation of such things should be put off to the indefinite future.

(In fact, much of the richness and "pathology" of geometry in characteristic p is easily understood once one has a firm grasp of how field extensions behave.)

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Moreover, the heuristic that there is something weird about the "theory of the automorphism groups" of inseparable extensions. Rather, the automorphisms that do exist are perfectly fine; it's just that inseparable extensions are more rigid, so there are fewer of them. –  Jay Pottharst Apr 27 '10 at 2:36
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@Jay True in one sense, false in another. I remember in grad school several of us got interested in computing the group scheme of autmorphisms of an inseparable extension. It's length is more than the degree, although all of that length is nilpotent, so you don't see it in the actual automorphisms. –  David Speyer Apr 11 '11 at 12:16

Similar to Tom's answer,

a vector is a mathematical quantity with both a magnitude and a direction.

Useful for distinguishing between speed and velocity but little else. The above is a typical definition from a physics textbook I had on the shelf; here in British Columbia, vectors are introduced in high school physics but not high school math. By the time students get to linear algebra in first- or second-year university, it can be hard to convince them that a real number (much less a polynomial) can be a vector. Usually, you have to resort to "a real number does too have a direction: positive or negative" and even then they don't believe you because

a scalar is a mathematical quantity with a magnitude and no direction

and so if real numbers are vectors, how can they be scalars?

Don't even ask about function spaces.

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My mother had an old "Advanced Calculus" book lying around when I was in high school. It mentioned this old chestnut and commented that it is a poor definition because some things are vectors but have neither magnitude nor direction (like scalars) and some things have both but are not vectors (like trains). –  Ryan Reich Nov 19 '11 at 7:06
    
+1: it's just wrong for so many reasons. For one thing, it sounds sort of like a reduction of math to physics or something. For another, you need something like an inner product to make sense of it. But worst of all, it's totally ass-backwards when it comes to abstract mathematics, because "vector" has no independent meaning. Rather, a "vector" just means an element of some given vector space, which is a set equipped with ... so it's the concept of vector space which is primary, not vector! Paul Halmos had a similar rant in his automathography. –  Todd Trimble Aug 25 '12 at 19:54
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If you are trying to say that $\mathbb R$ is a real vector space, do people really object that $-3$ and $+3$ only have magnitudes, and not directions? I prefer an actual definition over a misleading characterization, but I don't think this one leads to big problems. –  Douglas Zare Apr 17 '13 at 20:47

The opposite of Qiaochu's dictum is just as misleading - "formulas are functions". There are a lot of non-denoting expressions! It's just that mathematicians don't tend to write non-denoting terms very often. Of course, there's a good reason for that - you can't prove anything interesting about non-denoting terms (or rather, way too much). But then students never get the intuition that there are expressions which are 'junk', nor tools to prove that something is 'junk'.

My favourite 'junk' expression is $$1/\frac{1}{\left( x - x \right) } $$

Lest you think this is not very important, try to "teach" first-year calculus to a computer, and you'll see how these non-denoting terms are most troublesome.

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"A continuous function is one you can draw without raising the pencil"

This has terrible disadvantages when generalizing functions defined on a real interval to non connected sets, non compact sets and in general topological spaces.

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oh and I heard of a student claiming that "x+1" is not continuous because you need to raise the pencil at least twice whn you write it. –  Pietro Majer May 22 '10 at 16:57
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@Pietro: Se non e vero, e ben trovatto! –  Victor Protsak May 23 '10 at 7:06
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Victor: compliments, very good knowledge of Italian -and Italians –  Pietro Majer May 23 '10 at 22:35
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Pietro, that's just too funny (albeit in a sad way). For that matter, $x$ is discontinuous, unless you're in the habit of making your $x$'s look like $\alpha$'s. –  Todd Trimble Aug 25 '12 at 19:58

Not sure if this qualifies exactly, but I can never remember which theorems of group theory apply to finite groups, and which ones apply to groups in general. Anytime I remember a result, I have this sinking feeling that it appears in a textbook preceded by "for the remainder of this section, let G be a finite group." I'm not sure how well-founded this fear is (other than the theorems that obviously don't make sense for infinite groups, like the Sylow theorems).

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By the way, the Sylow theorems make sense (and are true, I think) for infinite groups if you make a few modifications. A p-Sylow subgroup is a maximal subgroup which is a p-group. The first theorem (existence) is obvious by Zorn's lemma. The second (that all p-Sylows are conjugate) is interesting. The third is interesting if the index of a p-Sylow is finite or if the number of p-Sylows is finite. –  Anton Geraschenko Oct 24 '09 at 21:47
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There are also profinite Sylow theorems, yielding the existence of a maximal pro-p subgroup. The proofs are relatively straightforward extensions of the finite proofs. –  S. Carnahan Oct 24 '09 at 21:54
    
This got me in a lot of trouble in my first-year graduate algebra class. I also had a habit of forgetting that infinite groups even exist, which is the same sort of thing. –  Michael Lugo Oct 24 '09 at 22:06
    
@ML: right. I don't think textbooks can be fairly construed to be confusing about which results apply only to finite groups. BUT most undergraduate algebra textbooks I have seen certainly give the impression that finite groups are more important, more natural, and more studied than infinite groups, when many if not most mathematicians would say that the reverse is true. –  Pete L. Clark Mar 1 '10 at 0:08
    
I was tempted to try adding something like this to the false beliefs question. At a higher level, the same becomes true with the properties "finitely generated" or "residually finite". –  Jonathan Kiehlmann May 9 '11 at 8:35

Almost any heruistic can be "most harmful" if used by a teacher in a situation when the audience does not know why it makes sense, and without an explanation. This is especially dangerous in the frequent case that the heruistic does not actually seem reasonable to a person seeing it for the first time, since it makes sense only in some ways but not others. It might require months of experience for an uninitiated person to understand how and why it applies.

For example, the heuristic of schemes as manifolds is such -- every algebraic geometer understands it, but it actually is harmful to a person who is seeing schemes for a first time (such a person would vary likely interpret this heruistic as saying that affine schemes are trivial to understand). Same applies to "integration is the inverse of differentiation", and some of the other answers to this question.

Of course, these heuristics are also the most useful ones, once you (and any audience you might have) actually understand them. The whole point of learning math is to gain more such heuristics, and to makes the ones you have more precise. For this reason, it seems to me that the use of such heruistics on an unprepared audience is the most common problem in the lectures by the very best mathematicians.

A related problem is the an abundance of statements that are not strictly true, but "correct in spirit". Again, this may be very useful in research or when talking to a person of appropriate sophistication, but it is very bad for students if such statements are used carelessly and without explanation.

P.S. This whole answer is generalization for the sake of generalization. Was it a waste of time, I wonder?

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Also not really a heuristic, but "differentiation is easy," as encoded in the following two sub-heuristics:

  • Differentiation is just repeated application of the product and chain rules, and
  • Most functions are differentiable most of the time.

Edit: Someone doesn't seem to like this answer, so I'll expand. Students who leave calculus with this impression enter analysis with a disadvantage: differentiation is not a property that "most" functions have in any reasonable sense, not even continuous ones, and to compute the derivative of a function that isn't given as a sum of compositions of "elementary" functions requires an entirely different mindset than the one that values the product and chain rule.

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I think your argument is more effective against a slogan like, "all interesting functions are differentiable". In my (limited) experience, differentiation tends to be algorithmic in practice, although it can be unstable in numerical applications. This is in contrast to integrals, which exist much more often and tolerate numerical error well, but are generally very difficult to compute. –  S. Carnahan Oct 24 '09 at 22:38
    
Somewhat related is the assertion that "differentiation is more fundamental", since it is "easier" and usually taught first. Not only is this misguided for the reasons you and Scott cite, but following Roger Penrose we can also turn the argument upside down in the complex plane by using Cauchy's theorem to define the derivative of a function by means of a contour integral. I've always hoped there was some alien civilization in another spacetime where derivatives were actually introduced this way. –  jvkersch Mar 1 '10 at 12:39

Writing a proof as a chain of expressions connected by equals signs whether they are appropriate or not.

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That's not really a heuristic, that's a misunderstanding of the equals sign. –  Anna Varvak Oct 28 '09 at 15:46

"Differentiation and integration are inverse operations."

To many calculus students, this is their conception of the fundamental theorem. There's truth to this heuristic, of course, but one needs to be constantly informed by a much deeper understanding of integration (and differentiation) in order to properly wield this correspondence in most situations beyond those encountered in a first course in calculus.

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Generalizing differentiation and integration lead us to see that they differ as left- of right- sided inverses. One side generalizes to Lebesgue differentiation theorem, on the other side generalizes to bounded variation and absolute continuity. –  Colin Tan Apr 25 '10 at 14:41
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I disagree with this: I think it is a fantastic heuristic, indeed the single most important heuristic of first year calculus. To argue against it is mostly to say "I don't like heuristics", it seems to me. –  Pete L. Clark Aug 2 '12 at 8:21
    
Well, I didn't really have first year calculus in mind when I wrote this answer. Sure, it's a great heuristic at that level, but it's not so great later on. I guess the lesson here is that you can't really talk about a heuristic without talking about the context as well. My answer was less about the heuristic being bad, and more about it being bad to cling onto a heuristic as you transition into territory where it ceases to be so fantastically useful. –  Zach Conn Nov 18 '12 at 5:21

I wish to point the attention on Pete Clark's very relevant initial comment. The term heuristic is often taken as synonymous to non-rigorous method, only based on intuition or experience. I personally dislike this acceptance of the word in mathematics, and I suspect it is not even historically correct (now I'm curious to check the use of it in the classic authors). The etymology of the adjective, from the verb εὑρίσκω (to find, discover) means "aimed to find". As I see it, it is exactly the method we follow when looking for a solution of a problem: using all implications of being a solution in order to identify a candidate solution. Of course, the heuristic is only half the job, and it is only rigorous if followed by part 2: checking the solution. But there's a very smart idea in it. For instance: solving an equation, transform it, but do not check the equivalence of each single step, just follow a chain of implications. So, what is harmful is not the heurstic method, but leaving out the (often less creative) part 2. That said, here's my example: let F be a smooth function bounded below (or a functional) with only one critical point. Then one would argue:

Any minimum point of F(x)=0 satisfies F'(x)=0, whose only solution is x0. Hence, x0 is the minimizer.

False!, if one does not check that F(x0)≤F(x) for all x ("direct method in Calculus of Variations") or if one has not proved the existence of a minimizer (indirect method). Many students make this mistake... but not only them!

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In elementary school, there are false principles which take a lot of effort to overcome:

  • Math problems have one answer.
  • There is one right method.

These may be ok (though the second is debatable) when you are working on $1+2$, but not when you are supposed to isolate a variable, to graph a function, to recognize how you can apply the chain rule, to solve a complicated word problem, or to prove something. Many students don't think math is a place to experiment or to apply creativity. They are afraid to take incorrect steps even when it is no longer convenient or possible to say what the right first step is.

There is an interesting app called Dragonbox. It is very popular in Norway. When children think of algebra as a puzzle or game, they feel free to experiment, and they quickly learn to do things like isolate variables which usually give algebra students trouble. See also Terry Tao's blog posts on gamifying algebra. Students can learn to solve the problems, but have difficulty because these incorrect principles get in the way.

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"you'll need a computer for that".

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I don't think Zeilberger would disagree with that "heuristic"/advice! –  Quadrescence Oct 3 '10 at 2:15

Any attempt to draw a fat Cantor set is a bad heuristic in my opinion. I saw such a diagram as an undergrad and believed for a while that there were intervals contained in the fat Cantor set. I don't think it's possible to express in a picture that a fat Cantor has positive Lebesgue measure and has empty interior.

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From Keith Devlin's article

http://www.maa.org/devlin/devlin_06_08.html

"Multiplication is repeated addition."

This is true when multiplying natural numbers, but is a special case of a scaling operation in the reals. We know it is also a rotation in the complexes, but that should probably be left out at the beginning, although it might interesting to think about how one would include them at the beginning.

Devlin also mentions "exponentiation is repeated multiplication."

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It's an incomplete heuristic, one that does work only for very special cases. But does this mean it is a bad heuristic? The only case where I can imagine getting bitten by it is when defining a linear map, forgetting the $f\left(\lambda x\right)=\lambda f\left(x\right)$ condition. On the other hand, here is a much more malign heuristic: Lie brackets are commutators. Very dangerous when you consider the tensor algebra of a Lie algebra. –  darij grinberg Apr 10 '11 at 21:20
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On the other hand, "the exponential map is an infinitely repeated infinitesimal multiplication" is a very good heuristic to have, particularly in Lie groups... –  Terry Tao Dec 13 '11 at 19:20
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But this rule has such a nice direct application: it shows that all rings (with unit) admit a map from $\mathbb Z$. –  Elizabeth S. Q. Goodman Jan 27 '12 at 8:48

The excluded middle ( A Law or an Heuristic) .

On a more general level given any closed question: Is it A or B ? , the heuristic says it is one or the other disregarding the option : the question is wrong or stupid or irrelevant or incomplete.

The principle of excluded middle disregards intuitionist logic. And has been harmful in not providing direct (constructive) proofs which are often more clear - yet can be harder to find.

Intuitionism is is also rather natural : being against anti-communists does not means you are a communist.

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Perhaps more proponents of intuitionism should have readily available examples for the glaring question: what are some natural settings where classical logic is faulty next to an (intuitionistic) alternative. Compelling answers to this question are much scarcer than suggestions to consider intuitionism. –  AndrewLMarshall Aug 10 '11 at 23:51
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A topology is an example of a Heyting algebra, not a Boolean algebra. How's that? –  Todd Trimble Aug 25 '12 at 20:02

"Mathematical knowledge is contained and communicated primarily by documents."

I'm not sure if this is a heuristic, but in terms of beliefs that inhibit learning, this is definitely the one that hurt my mathematical development the most.

I would say the correct statement is "Mathematical knowledge is contained primarily in the minds of mathematicians and communicated primarily by informal oral communication."

This problematic belief grew out of the way that I (and pretty much everyone else) was taught mathematics at the undergraduate and beginning graduate level. In this setting texts are a central authority and a complete, well-written resource for the knowledge needed to solve any mathematical problem encountered.

In the world of mathematical research, this is no longer the case. I finally figured this out by reading Thurston's essay "On proof and progress in mathematics", which I would strongly recommend for any beginning mathematician.

Maybe it is possible to do research mathematics using papers as a primary resource, but I believe this is highly inefficient. I spent several years trying to learn the noncommutative standard model by reading the available papers on the subject and made no real progress. Looking back, I don't think I ever had a chance of succeeding with this approach.

I would guess that to be successful in mathematics, it is absolutely vital to become regularly involved in conversations with working mathematicians, as awkward and intimidating as that might be.

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This has nothing to do with heuristics, whatsoever! –  Mariano Suárez-Alvarez Apr 17 '13 at 20:52
    
Sorry but I disagree. There is so much buried knowledge in the unread works of the past that I wouldn't be surprised if it surpasses the knowledge of currently living mathematicians. Take into account that many authors forget their own papers after a couple of decades... –  darij grinberg Apr 17 '13 at 22:56
    
I disagree with the disagree-ers. Sure, this answer is a little more "meta" than the question likely intended, but not overwhelmingly so: if we take "a heuristic in math" to mean "a rule of thumb for how to prove things in math", then this answer is arguably on target, even though it is more methodological and less domain-specific. Besides, I think it's an important message to have out there, a realization that every mathematician will have to come to in order to be successful. Even though it's not the sort of issue you'll find discussed in papers :). –  Tim Campion Nov 14 '13 at 21:52
    
Especially a lot of the intuition is communicated orally and informally. And it is impossible to do mathematics without having an intuition about what you're doing. –  Turion Sep 10 at 15:13

"A set is a collection of elements".

Firstly, this does not distinguish sets and classes. Next in ZFC, sets are characterized more by their relation to each other by $\epsilon$, rather than that they contain anything.

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I think this an excellent heuristic. If someone asked you what a set was, what else would you tell them? –  Pete L. Clark Apr 25 '10 at 20:23
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-1, as the ZFC axiom of extensionality says precisely that two sets that have the same elements are indistinguishable. –  Qfwfq May 11 '10 at 12:23
    
I agree with Colin. In strictly ZF terms, a class is a collection of elements. A set is an element of a collection. –  Tom Ellis May 31 '10 at 13:42
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@Clark: I think "collection" does indeed give the wrong idea. I tend to explain sets as a concept that links items together by properties they have, even if the property is arbitrary. Instead of thinking "$x\in A$" as "$x$ is in $A$", I try to explain "$x$ has a property $A$ and everything has this property or doesn't." Sometimes the property is meaningful: "$x$ is even", or arbitrary: "I'm tagging $x$ with $A$." So the prop. is "$x$ is tagged by $A$." It helps people understand things like dense/uncountable sets better. Also, it departs from the arbitrary "sets only contain unique items". –  Quadrescence Oct 3 '10 at 2:33

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