most recent 30 from http://mathoverflow.net 2013-05-25T05:45:48Z http://mathoverflow.net/feeds/question/28612 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students Diego Matessi 2010-06-18T11:01:45Z 2012-02-29T00:21:40Z

<p>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? </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28614#28614 Wadim Zudilin 2010-06-18T11:26:58Z 2010-06-18T11:26:58Z

<p>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...</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28616#28616 Charles Matthews 2010-06-18T11:34:24Z 2010-06-18T11:34:24Z

<p>An example is French <em>module monogène</em> for what is "cyclic module" in English. I can't prove that the possibility that a <em>module monogène</em> for a given ring may not be a <em>groupe cyclique</em> 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.) </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28619#28619 Q.Q.J. 2010-06-18T11:52:30Z 2010-06-18T11:52:30Z

<p>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. </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28621#28621 John Stillwell 2010-06-18T12:11:02Z 2010-06-18T12:11:02Z

<p>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."</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28624#28624 Harald Hanche-Olsen 2010-06-18T12:48:23Z 2010-06-18T12:48:23Z

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

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28625#28625 Harald Hanche-Olsen 2010-06-18T12:53:47Z 2012-02-29T00:21:40Z

<p>Oh, and “convex”. As far as convex <em>sets</em> go, the mathematical usage accords well with everyday language. Not so with convex (and concave) <em>functions</em>. 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 <em>those</em> two.)</p> <p>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".</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28628#28628 Spiro Karigiannis 2010-06-18T13:10:33Z 2010-06-18T13:10:33Z

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

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28630#28630 Rune 2010-06-18T13:27:03Z 2010-06-18T13:27:03Z

<p>"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 ..."</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28637#28637 Marty 2010-06-18T15:46:03Z 2010-06-18T15:46:03Z

<p>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.</p> <p>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?</p> <p>So, I'd say to start by looking up the Whorf hypothesis -- maybe it's been considered by some applied linguists studying education.</p> <p>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 <em>over</em> a finite field" (the use of the positional word "over", to express dependence like a building resting on its foundation).</p> <p>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. </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/28715#28715 Victor Protsak 2010-06-19T02:17:46Z 2010-06-19T02:17:46Z

<p>This is very elementary, but I find it surprisingly common: many students talk about "infinite" numbers. You know, like </p> <p>$$0.3333333\ldots$$</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/29752#29752 Alfonso Gracia-Saz 2010-06-28T07:38:57Z 2010-06-28T07:38:57Z

<p>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.</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/32394#32394 Jonas Meyer 2010-07-18T20:18:36Z 2010-07-18T20:18:36Z

<p>The article "<a href="http://www.jstor.org/stable/4145268" rel="nofollow">Surprises from mathematics education research: student (mis)use of mathematical definitions</a>" by Edwards and Ward addresses some of your concerns in the context of U.S. undergraduates.</p> <p>From the introduction:</p> <blockquote> <p>...[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."</p> <p>...</p> <p>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.</p> </blockquote>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/32435#32435 Peter Shor 2010-07-19T01:41:17Z 2012-02-28T00:13:01Z

<p>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. </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89685#89685 alberto.bosia 2012-02-27T17:53:51Z 2012-02-27T17:53:51Z

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

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89717#89717 Changwei Zhou 2012-02-28T01:52:11Z 2012-02-28T01:52:11Z

<p>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. </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89755#89755 maproom 2012-02-28T11:19:52Z 2012-02-28T11:19:52Z

<p><strong>Parallel</strong> 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. </p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89788#89788 JeffE 2012-02-28T19:26:53Z 2012-02-28T19:26:53Z

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

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89796#89796 Julien Puydt 2012-02-28T20:56:39Z 2012-02-28T20:56:39Z

<p>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...</p>

http://mathoverflow.net/questions/28612/do-names-given-to-math-concepts-have-a-role-in-common-mistakes-by-students/89797#89797 Julien Puydt 2012-02-28T20:59:19Z 2012-02-28T20:59:19Z

<p>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?</p>