Do names given to math concepts have a role in common mistakes by students?

Perhaps this question overlaps with similar ones, ... but I want to focus on a particular possible cause of confusion. I notice that students are often confused by the concepts of "infinite" and "unbounded". So, when asked if the set of invertible matrices is compact, they reply "no, because there are an infinite number of matrices with non zero determinant, therefore the set is unbounded". Actually this happens in Italian, where the corresponding words ("infinito" and "illimitato") are almost synonyms in everyday language. Does this happen in English too, or other languages?. I wonder: what if we chose another name for the two concepts? Would they make this mistake anyway? One way to check this would be to compare with what happens in other languages, where perhaps the words chosen do not create the confusion. Do you have other examples of this situation? Can you suggest different math concepts which in one language are named with synonyms, but not in another? Do you know if this problem has been studied anywhere?

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Your actual question seems to be more specific and linguistically oriented than your title question. (I'm sure everyone would agree that the answer to the title question is a resounding yes.) Would you consider editing the title to better match the question? – Pete L. Clark Jun 18 '10 at 12:00
Infinite and unbounded probably don't have much of a distinction for nonmathematicians in English either, but they make sense once one gets used to them. Most of my fellow students (even those not intending to become mathematicians) find the terminology mostly logical, although Spiro's answer may be an exception...nevertheless, people make mistakes with definitions without regard to other definitions anyway, and it's part of learning the definition in the first place, so I think that for the most part students should just be encouraged to think longer. – Jason Polak Jun 18 '10 at 13:25
@Pete L. Clark. I don't know how to change the question. The reference to languages is just a suggestion on how one could find out whether it is the names which cause the confusion. Simply because other languages may have chosen better names. The answers that are coming in are in theme. Thank you all... – Diego Matessi Jun 18 '10 at 14:09
Suppose, in your language, the word for "multiplication" resembles a nasty word. Perhaps that would be distracting for students. In English, in some 18th century writing, we find a polynomial is called "sexual" if it has degree six. We don't say that any more! – Gerald Edgar Feb 27 '12 at 19:08

"Open" and "closed". Every reasonable human being on the planet, who has not studied topology, will assume that something can either be open or closed, but not both. This often causes students to make statements like "Set A is open, therefore it is not closed, thus ..."

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Perhaps open sets should be called exposed sets, and closed sets should be called clothed sets (you can see the interior through a dotted line, but the solid line prevents you from looking inside). This would be consistent with some pictures in $R^2$ and $R^3$ students see early on. And then, later, you can talk about sets being clothed and exposed. If a student says that a set cannot be exposed and clothed, you just have to mention Lady Gaga. – Marty Jun 18 '10 at 15:18
At least concluding a set is not closed because it's open makes sense for a proper, non-empty subset of a connected space. Worse is the reasoning "Since $A$ is not open, it must be closed." – Chris Phan Jun 18 '10 at 16:03
Can't resist quoting a comment from another thread: Munkres is fond of saying: "Sets are not doors!". Also, there are, of course, spaces with clopen sets. – Victor Protsak Jun 19 '10 at 2:22
It's a bit clunky, perhaps, but I always thought 'enclosed' for closed and 'edgeless' for open captured the intuition from open/closed intervals without introducing problems that occur by the natural linguistic analogy to doors. 'Enclosed' because you can't get out (with a sequence), 'edgeless' because, well, there's no edges. – Ketil Tveiten Jun 19 '10 at 12:12
But it can be ajar! – timur Jan 20 '11 at 2:54

My favourite is this: some books use "completely reducible" for semi-simple and "irreducible" for simple. As a result, every irreducible module is completely reducible.

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"Semisimple" is not without its perils: for example, 1-dimensional abelian Lie algebra is simple, but it's not semisimple! (There are good reasons for making this choice: I just want to point out that making sure that naive parsing works isn't always the goal.) – Victor Protsak Jun 19 '10 at 2:20

If you count "or" as a mathematical concept, the the fact that it is fundamentally inclusive in mathematics but often exclusive in most other uses of English can lead students to mistakes.

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This reminds me about the French version of "or" and a related puzzle which I learned from Don. He can't find an explanation of why one of the extended letters on French keyboards does not come as the whole word in which it is only used. And this is the word very often used in maths! (As this is a puzzle I am not supposed to provide the answer, but I know that it is not very obvious: ù.) – Wadim Zudilin Jun 18 '10 at 12:13
in fact translates to where, which is nevertheless common in mathematics. (Ou, without the accent, is French for or.) – Niel de Beaudrap Jun 18 '10 at 13:04
"Or" is not the same as "exclusive or" in English if there are three or more disjuncts. "A or B or C" in English commonly means "exactly one of A, B, C" whereas "A xor B xor C" is true if and only if an odd number of disjuncts holds. – Timothy Chow Jun 18 '10 at 14:04
And it gets even more xorsting as the number of disjuncts increases! – Q.Q.J. Jun 18 '10 at 16:00
You just need a little xorcise... – Victor Protsak Jun 19 '10 at 9:08

I'm surprised nobody has mentioned "one-to-one function" for injection and "one-to-one correspondence" for bijection.

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I hadn't realized that in teaching (maybe because I teach subjects where those terms rarely arise), but now that you mention it that did confuse me when I was a student... – Michael Lugo Feb 29 '12 at 0:13
I didn't even realize this was a thing! For me, the word "correspondence" just means "binary relation between two sets." – goblin Mar 5 at 8:02

My favourite example is "complex analysis" (as well as "complex" and "imaginary" numbers). Students, mostly in advance, feel it too complex. There should be probably a better name but it's too late to change...

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I agree that "complex" has unfortunate overtones, but it does allow one to say things like "Let's complexify this to simplify it.". On second thoughts, perhaps that's a good reason to find a new term for it! – Loop Space Jun 18 '10 at 11:46
And how did we ever allow ourselves to say "complex simple Lie algebras"!? – John Stillwell Jun 18 '10 at 12:03
Niel: Gauss complained about the real/imaginary terminology. A quote: "That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question." – KConrad Jun 18 '10 at 19:49
Wadim, I am sure you've heard about the professor who wrote a very complicated equation on the board and with his characteristic modesty, called it $\mathit{ordinary}$ differential equation (cited by V.I. Arnold). – Victor Protsak Jun 19 '10 at 2:16
Excellent point, except for the "too late to change" part. If it helps that much, we should probably just do it. There's only a finite amount of historical work to rephrase, but there's a potentially unbounded number of future students to benefit. – Noah Jun 19 '10 at 22:47

It's not just students who get confused by terminology. I was recently puzzled for quite a while until I realised that finite von Neumann algebras can be infinite-dimensional.

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And separable von Neumann algebras need not be separable. – Richard Borcherds Jul 19 '10 at 4:08
And a hyperfinite von Neumann algebra can be properly infinite. – M Mueger Nov 16 '15 at 23:27

Not only is the word "complex" a problem, the word "simple" is too. I mean, how "simple" are the sporadic simple groups? I have to tell my students that "simple" does not mean "not complicated", but rather "cannot be simplified further."

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Is a simple polyhedron one that (a) has genus zero, (b) is triangulated, i.e., simplicial, or (c) has all vertices of degree 3? One can find all three in the literature. – Joseph O'Rourke Jun 18 '10 at 13:13
Worse still, what could a "complex simple Lie algebra" possibly be? – Allen Knutson Jun 18 '10 at 14:12
Or simplicial complex? – Victor Protsak Jun 19 '10 at 2:13
Or my favorite from Lang's "Algebra": "a ring $R$ is said to be simple if it is semi-simple and ... ." – Chris Leary Feb 29 '12 at 3:47

The article "Surprises from mathematics education research: student (mis)use of mathematical definitions" by Edwards and Ward addresses some of your concerns in the context of U.S. undergraduates.

From the introduction:

...[T]asks involving the definitions of "limit" and "continuity," for example, were problematic for some of the students. Ward's intuitive reaction was that those words were "loaded" with connotations from their nonmathematical use and from their less than completely rigorous use in elementary calculus. He said, "I'll bet students have less difficulty or, at least, different difficulties with definitions in abstract algebra. The words there, like 'group' and 'coset,' are not so loaded."

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He was surprised to see his algebra students having difficulties very similar to those of Edwards's analysis students. (So he lost his bet.) In particular, he was surprised to see difficulties arising from the students' understanding of the very nature of mathematical definitions, not just from the content of the definitions.

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Here's a link to the pdf from Ward's website, in case you don't have JSTOR: wou.edu/~wardm/FromMonthlyMay2004.pdf – Jonas Meyer Jul 19 '10 at 0:47
That is an interesting article, but its relevance to the present question seems to be confined to the distinction the authors draw between word definitions that are "extracted" (from use) and "stipulated" (by arbitrary rules). If anything, their "surprising" conclusion, based on limited data, is that even in the absence of an obvious connotation, students presented with a new, entirely abstract definition will try to relate it to mental images of things they have seen and used before rather than proceed literally from the definition. Duh! (They passed the Turing test.) – Victor Protsak Jul 19 '10 at 6:39
There are many questions in the present question, but among them is to what extent do the words' ordinary meanings affect students' misunderstandings. The authors of this article claim that the effect didn't show up in their study; "coset" has no ordinary meaning as opposed to "limit", and yet similar problems were observed among undergraduates regarding misuse of both. At least, that is what I took out of it, and why I thought it was relevant. – Jonas Meyer Jul 19 '10 at 17:30
The idea that 'group' (or 'ring' or 'field' or 'sheaf') doesn't have everyday meaning is a little strange …. – L Spice Feb 17 at 21:14

I don't know of any research on this question specific to learning about mathematics. But the question opens up a big can of academic worms, outside of mathematics.

In linguistics, the Whorf hypothesis (sometimes called the Sapir-Whorf hypothesis) can be summarized as the notion that different peoples have different languages (syntax, lexicon, etc..), and these differences influence how they think about things. For example, different languages have different tenses available for use -- does this affect how speakers perceive time?

So, I'd say to start by looking up the Whorf hypothesis -- maybe it's been considered by some applied linguists studying education.

The other linguistic can of worms is the use of metaphor in mathematical language. Some words we use are directly visual, like "smooth" and "compact", some are strange (to me) analogies like "sheaf" and "flabby", and others are part of larger metaphorical systems like "consider a variety over a finite field" (the use of the positional word "over", to express dependence like a building resting on its foundation).

If you want to read up on these aspects of mathematical language, I'd recommend books by the Berkeley linguist Lakoff -- the classic "Metaphors we live by", and the application to mathematics in "Where mathematics comes from". Not that I agree with everything in the latter book, but it's an interesting read. I don't think you can address your question seriously without reviewing the linguistics literature.

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I do not think that the Whorf hypothesis is taken very seriously these days. – Chris Godsil Jun 18 '10 at 22:22
I don't think it's taken seriously by those who follow Steven Pinker's criticisms -- certainly it's been taken down on a number of occasions. But I think it still rears its head in some contexts. The question definitely suggests Whorf to me... – Marty Jun 18 '10 at 23:26
I agree with Marty, this Whorf hypothesis seems related. Also, it seems like one of the take-away messages I got from reading 1984, though I never knew it had a name. Thanks for writing this answer – David White Feb 27 '12 at 23:02
Actually these days it seems linguists (patterning after Noam Chomsky) think that "language affects cognition in /no/ way". So yes, a can of worms. – Elizabeth S. Q. Goodman Feb 29 '12 at 0:06
I don't think this question has anything to do with what Whorf was suggesting. (And for that matter, I'm under the impression that what people refer to as the "Whorf hypothesis" is arguably rather stronger than Whorf himself would have argued for.) In any case, what's at issue here is simple confusion of terminology. That happens in any language, in any field, and has nothing to do with the idea (quite likely wrong, or at least vastly overstated) that the inherent structure of a language has some determining effect on how its speakers view the world at large. – Carl Offner Feb 29 '12 at 2:02

This is very elementary, but I find it surprisingly common: many students talk about "infinite" numbers. You know, like

$$0.3333333\ldots$$

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The problem here is not terminology, but that while such elementary concepts such as this are taught, teachers stay as far away as possible from talking about infinity. There is no opportunity to discuss the difference between an infinitely long representation of a number, and a number which is 'semantically' infinite. Not that I advocate teaching infinite cardinals in grade-school, but if we did (or taught some other notion of infinity with some amount of seriousness), an immediate remedy would be to point out that 0.333333... is bounded between 0.3 and 0.4. – Niel de Beaudrap Jun 19 '10 at 9:35
Right. I suspect that teachers themselves are far from comfortable with infinity, and not just grade school teachers, either. – Victor Protsak Jun 19 '10 at 10:29
I am far from comfortable with infinity as well... Fortunately understanding infinity is not a necessary condition to work in mathematics. – darij grinberg Jul 18 '10 at 21:11
I think this is a symptom of the teaching of most students wherein Calculus is the pinnacle of their education in mathematics. If someone takes a calculus class, doesn't get an A on the sequences and series chapter and never thinks about sequences and series again, what can we expect them to pass on to others besides "funny things happen at infinity" ? – stankewicz Jan 20 '11 at 14:59

Oh, and “convex”. As far as convex sets go, the mathematical usage accords well with everyday language. Not so with convex (and concave) functions. Educationists (?) have tried to remedy this by using terms “concave up” and “concave down” in calculus textbooks, a usage that I detest. (I have a hard time remembering which is which of those two.)

Edited based on comments: It seems that in Russian, "convex" can refer a surface curved outward, where in English usually "convex" refers to a solid whose surface curves outward. Perhaps that's why in complex geometry, one might consider a "convex domain" or its boundary, a "convex hypersurface".

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Qiaochu, this is a classic situation where a picture is worth a thousand words. When I teach it, positive derivative $\iff \cup$ and negative derivative $\iff\cap.$ Any language-based metaphor, no matter how clever, is bound to fail. – Victor Protsak Jun 19 '10 at 4:07
Sometimes I just throw up my hands and say "has positive second derivative"; the time it takes to say the few extra words is repaid in the larger number of students who understand. – Michael Lugo Jun 19 '10 at 13:46
Ah, right. It's concave up/down, not convex up/down. I'll edit the answer accordingly. Now if we insist on language-based metaphors, wouldn't “smiling” and “frowning” work better? (At least in 1 dimension.) – Harald Hanche-Olsen Jun 19 '10 at 14:05
Curiously, it is convex up/down in Russian. And which is which is clear, due to the ordinary (non-mathematical) meaning of the word. – Sergei Ivanov Jun 19 '10 at 18:59
No, "convex up" is concave (has negative second derivative). The word "convex" (adjective) in Russian means "bent outwards", and "convexity" (noun) typically means a fragment of a surface resulting from such bending. For example, a low quality table surface may have "convexities" (small hill-shaped defects), and they are clearly convex up (i.e., a layer of the surface is bent up from its normal place). – Sergei Ivanov Jun 19 '10 at 20:50

I nominate “trivial”. It can be rather confusing that something can be trivial, but not trivially so.

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"Trivial" as an adjective to a result is indeed subjective. But "trivial" as math concept is not and is what you expect: trivial group, trivial subgroup, etc. – lhf Jun 18 '10 at 13:12
@LHF: First of all, in your examples of trivial being used as a math concept, it is also being used as an adjective (and indeed, this is always the case). Grammar aside, I think I disagree with your premise: the adjective "trivial" applied mathematically is context dependent and open to interpretation. For instance is a "trivial torsor" a torsor under a trivial group? what does it mean to say that a group or a space has "trivial cohomology"? Also we had a MO question about Bombieri's intended meaning of "trivial solution" to a certain Diophantine equation. – Pete L. Clark Jun 18 '10 at 17:00
...and is the 'trivial map' a constant map, or the identity? – Ketil Tveiten Feb 28 '12 at 8:30

Sequence vs series? Particularly if the two notions are introduced one right after the other in a calculus course, students are doomed to mix them up.

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But kids are doomed to mix left and right (which are in most cases introduced to them more or less consecutively) and at that point the two words probably do not mean anything to them. Maybe the problem is that when one is introduced to too many things (i.e. two) essentially at once confusion is to be expected? – Mariano Suárez-Alvarez Jul 19 '10 at 2:30
It's a difficult distinction if it isn't stressed: a sequence is a mathematical object. A series is a type of expression. Equality and identity coincide in the former, not in the latter. – AndrewLMarshall Dec 21 '10 at 5:40

In italian, the words bound and limit sound the same, "limite". This often causes confusion, like in the limit points and the boundary points of a set or a bounded function and its limit.

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An example is French module monogène for what is "cyclic module" in English. I can't prove that the possibility that a module monogène for a given ring may not be a groupe cyclique would be a stumbling block for a student; but it shows the phenomenon (avoid overloading). (I learned this when Serre pulled up an anglophone lecturer speaking in French on one occasion.)

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In french, a "monic polynomial" is a "polynôme unitaire", and a "unit vector" is a "vecteur unitaire". What happens to the students when they consider a dot product on a space of polynoms?

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Similar situation in German (with 'normiert' instead of 'unitaire'). – user9072 Feb 29 '12 at 0:52

In french, a "unitary endomorphism" is an "endomorphisme orthogonal"... and an "orthogonal projection" is a "projection orthogonale" (the 'e' is for female, it's pronounced the same). You can hardly imagine how bad I feel when I have to tell my students that orthogonal projections aren't unitary...

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French has thousands of homonymic pairs of words: most French people seem to understand pretty well the difference between aimait, aimais, aimé, aimée, aimés, aimées, say. Why would university students have any trouble dealing with a few words that have two meanings?! – Mariano Suárez-Alvarez Feb 28 '12 at 23:34
Oh, another one where the same happens in German. – user9072 Feb 29 '12 at 0:54
@Mariano S-A: the words you cite are not all pronounced the same for one, and second, they're just variants of the same word depending on grammar, so there's definitely no trouble with them. – Julien Puydt Feb 29 '12 at 9:55

I think it takes sometime for starters to realize the tensor product between representations and tensor product between modules, despite the similarity on the surface that we can treat $g$-representations as $g$-modules. Usually a confusion appears when a concept which makes perfect sense in one area was re-defined or used in a more subtle way in the other area, which might be counter-intuitive in some sense. A concrete example comes up in my mind is Segal's paper on representations of compact Lie groups, where a lot of definitions are rather ad-hoc in modern literature but makes perfect sense when one read his paper careful enough.

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Parallel to most people is a precise and useful term, describing for instance railway tracks, even when curved. But when we learn "Euclid's parallel postulate", it merely means "never meeting even if extended indefinitely". This causes difficulties for students introduced to non-Euclidean geometries.

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