I was in a library and by opening at random a book about university tests I crashed against this page

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This is meaningless, a finite set of numbers can't define a succession and this is a multiple choice test that doesn't ask to justify the choice: the student can answer peacefully randomly. For example in the question 3 the book say that only E is correct, but we can answer A, if we consider for example the sequence

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or B, if we consider for example the sequence

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or any other answer. The authors of the test implicitly consider the simplest law the only correct: is this way of proposing the thing acceptable? Am I too pedantic? I only contest the fact that this kind of question is not compatible with the closed form. It would be correct to reject a student who provocatively check all the answers? Please note this is not a question about mathematics (things are simple about that), it concerns the right, or maybe the duty, of a student to respond provocatively to a question he deems wrong. Would it be inappropriate?


Of course strictly speaking this is not a university book, this is a book thought to admission tests. Anyway the target are men and women who are preparing to enter the university, decidedly not kids. I add that if "quickly" is the point, I could think to:

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This is simple, quick, correct, but it would be marked as wrong. I didn't know the Stack Exchange room about teaching mathematics, is it possible migrate the question there?

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    Yes, you are too pedantic. – Kathy Nov 14 '19 at 21:02
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    Clearly the exam question is not the best, as multiple answers could be justifiable (though, I suspect the intended meaning is clear). What exactly are you asking? Whether you should mark off for students who feel duty-bound to protest the bad question by selecting multiple answers? – cag51 Nov 14 '19 at 21:43
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    That looks as a practice book for some undergraduate admission tests in engineering. If it is, it would be better to specify it in the question, because it's otherwise misleading. – Massimo Ortolano Nov 15 '19 at 2:11
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    You mention both the possibility of rejecting a student who marks all answers as well as a student responding provocatively. These are two completely separate issues. What a marker should do when a student has already marked all answers doesn't tell you what a marker would actually do and that answer doesn't tell you what a student should do when faced with such a question. It largely seems like you already know what the "correct" answer is and this is a rant more than something you want an actionable answer to. – NotThatGuy Nov 15 '19 at 12:10
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    I don't read Italian so I have no idea about the detailed content of the page, but engineers should be looking for simple solutions to problems not over-complex ones, and in that sense the test is a "good" one (but far from optimal, of course). Someone who ticks all the answers as correct may be better suited to a career in law or politics than engineering! – alephzero Nov 15 '19 at 12:10

I actually think this is a question about mathematics, or more precisely about mathematics education (and in the limited scope in which it is presented it would be a better fit on the Math Educators stack exchange), but there is a larger academic issue which is perhaps what you are “really” asking about.

The point is this: as educators we try very hard to instill in our students the values of precision, logical reasoning, and attention to detail. When they are sloppy in their answers to our carefully phrased exam/HW answers, we mark off points and wag our fingers at them. We go to great lengths to provide them with examples of well-written, thoughtful, precise, polished answers (and other content) that pertain to the topics being discussed, in the hopes that they will pay attention and seek to emulate that style of discourse.

Given all that, and especially in mathematics where that precision is an essential value that goes to the core of the entire discipline, we cannot afford to ask sloppy (and formally incorrect) questions ourselves. It is a terrible example to give students and undermines the very goal we are trying to achieve. A teacher who is sloppy cannot expect to produce students who are any less sloppy than the teacher.

Coming back to your textbook example, it may depend on the age group this textbook is aimed at, but for a university-level audience the problem you cite is indeed unacceptably sloppy. You are not being too pedantic; a student who answers that all answers are correct will have answered the question correctly (to the extent that the question is sufficiently well-posed that it can be said to have a correct answer) and deserves to get full points for the question and to not have anyone be describing their behavior with the word “provocatively”. Such a student will have shown themselves to be more thoughtful (unless they arrived at their answer by pure luck) and to have a better grasp of mathematical precision than the authors of this textbook.

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    Actually, the OP does not seem to be totally honest: for what I can tell, that's not a textbook, but a practice book for some undergraduate admission tests in engineering. – Massimo Ortolano Nov 15 '19 at 2:05
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    @MassimoOrtolano yes, I realized that how acceptable this way of presenting things is may depend on the target audience. It also depends on specific issues of wording that I can’t speak to since I don’t speak Italian. In any case I hope my answer addresses the larger issue I was extrapolating to from OP’s situation. Also, even for a practice book like you are describing, I think it wouldn’t be good practice to frame the question as being about finding the “correct” next term in the sequence (but I don’t know if that’s what’s done here). Even high school students deserve more respect than that. – Dan Romik Nov 15 '19 at 3:08
  • I guess there is also the philosophical (but also practical) issue of whether one views mathematics as a particular methdological approach to finding out what is true, or as a standard for Xtreme certification of particular kinds of "truth". I myself am a bit fond of the latter, but in my later years far more interested in the former. – paul garrett Nov 15 '19 at 4:40
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    If this is indeed a practice book for admissions, there is little point scolding the authors of the book - if the admission questions are indeed that silly, the book needs to prepare students for that, otherwise it's not doing its job very well. – xLeitix Nov 15 '19 at 11:34
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    @xLeitix perhaps you have a point that the book isn’t a proper textbook and its authors aren’t really educators and should not be held to the same standard as educators. Fair enough. Nevertheless, I can imagine a way to explain to the readers how to “correctly” answer the question (meaning to get full points) while still managing to put the point across that the question is sloppily stated and has no “correct” answer, so I don’t fully agree the authors are beyond reproach. (And regardless, even if they are, I can still be unhappy that students are exposed to such poorly thought out content.) – Dan Romik Nov 15 '19 at 14:00

Although not reliably explicitly made a part of the question, it is widely understood that the "correct" continuation of a sequence is the (allegedly) "simplest". There is still a problem with the notion of "simplicity", since it is relative to context. (The notion of "Kolmogorov-Solomonoff-Chaitin complexity" is a rigorous approach, but certainly not something that kids should be expected to know.)

Nevertheless, in the implicit context of school mathematics, for example arithmetic sequences are very simple, so if the initial segment fits into an arithmetic sequence, just continue it. Also geometric sequences.

Beyond that, it quickly becomes murky, because I don't think students are taught "polynomial interpolation" (also known as "Lagrange interpolation"), although I may be mistaken.

So for a student to check "all" as a sort of "protest vote" is not really justifiable, since a student who knows that any finite sequence can be interpolated arbitrarily will surely also know that that's not really the point... :)

EDIT: and, no, _of_course_ there's scant excuse for an awkward version of the question being posed, where there is no unique very-simple extrapolation. Yes, of course, in principle this could happen, and then we might imagine that a student is justified in a protest-response. In practice, in my experience, such questions most often really do have a plausibly-best response, while, yes, in principle, being ambiguous. The fact that, in principle, there's a problem, does not, in practice, create a problem.

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    What about cases where there are multiple reasonable interpretations, and the student chooses one which isn't the standard or expected one, but nonetheless makes sense? I find "that's not really the point" a very weak argument imho, and it does not make for a reasonable excuse for the question being set in an imprecise and unspecific manner. – YiFan Nov 15 '19 at 9:34
  • Of course, I can understand your point for the questions specifically listed out in the OP, which are clearly arithmetic/geometric progressions. Even so, I don't think it is appropriate to mark a student down for putting down an answer which is technically correct, even if one might be tempted to write it off as pedantic. (Things get worse if the sequences aren't so obvious even if relatively simple, for example with sequences a(n) given by a quadratic in n but only the first few terms given.) – YiFan Nov 15 '19 at 9:38
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    What @YiFan said. I've absolutely seen examples of this where there were multiple very plausible choices and which one comes to mind is just a matter of how your brain works. The protest here is fully warranted because, short of something like it to fight the entire practice of giving this type of question, students are going to be unjustly punished for it silently. – R.. GitHub STOP HELPING ICE Nov 15 '19 at 13:09
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    While a student at this level is unlikely to be able to prove it, I find it plausible that some students know that for any n points in the coordinate plane, unless vertically aligned, that a polynomial can be found to pass through them all. (1, 3), (2, 9), (3, 81), (4, k), for any value of k. It might even be mentioned as a "teaser" in a relatively low level class. – Buffy Nov 15 '19 at 14:02
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    The fact that "simplest answer" is sometimes ambiguous does not mean that it's always ambiguous. I'm pretty sure that most of the folk who are hair-splitting here could in fact figure out what answer the examiners intended, if there was something of value at stake. Yes, there are poorly-written multiple-choice questions where it's genuinely unclear what the "correct" answer should be. No, these are not among them. – Geoffrey Brent Nov 15 '19 at 22:18

Actually, it would be wrong to reject such an answer, but the student had better have evidence to back it up. This is the sort of thing that a teacher might be misled by if they haven't actually developed the question themself. Or worse if they are a "tester" rather than a "teacher". Worst, if they are teaching "by the book", keeping one day ahead of the students as is done too often.

I fear for the student in a situation like this. The "expected" answer is likely looking at only simple progressions and not sophisticated ones.

I once encountered a similar situation, though, thankfully, not on a test.

But if a student had an enlightened instructor they could answer outside the bounds similarly to what you point out here. But an uncommented "checking" of all the boxes will probably not be seen as provocative, but as unprepared.

Gauss was famously hated by his teachers for his brilliance. Or so the story goes.

Let me add, for people who aren't mathematicians here (taken from a comment elsewhere):

While a student at this level is unlikely to be able to prove it, I find it plausible that some students know that for any n points in the coordinate plane, unless vertically aligned, that a polynomial can be found to pass through them all. You don't even need to get to the point of "a unique polynomial of minimum degree".

So, consider (1, 3), (2, 9), (3, 81), (4, k), for any value of k. There is a polynomial, which is a simple sort of function, passing through all of those points. It might even be mentioned as a "teaser" in a relatively low level class. It shows how relatively simple functions can have great power and applicability.

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    Indeed, a good teacher should be able to come up with better questions. But it is usually a mistake to make a model more complex than necessary. See for example: en.wikipedia.org/wiki/Overfitting and en.wikipedia.org/wiki/Occam%27s_razor – louic Nov 14 '19 at 21:28
  • @louic Overfitting is the very reason why all of these answers are potentially correct, though; it’s possible to construct a polynomial sequence that exactly matches any finite set of points, as long as they all have different values on the x-axis. – nick012000 Nov 15 '19 at 1:34
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    @nick012000 Sometimes the gap between "possible" and "advisable" is very wide indeed ;-) – Geoffrey Brent Nov 15 '19 at 3:57
  • Just curious: if you had received this as an answer to an exam question, would you have marked it as correct? gocomics.com/calvinandhobbes/1995/01/09 – Allure Nov 15 '19 at 5:15
  • @Allure If the student can explain how this translates to an accurate description of Newton's First Law, they should have it marked as a snarky but correct answer. Of course that requires more interaction between the student and the teacher than is often allowed. – user253751 Nov 15 '19 at 11:30

The authors of the test implicitly consider the simplest law the only correct

Google-assisted translation from the text at the top of the page, emphasis added:

"...these are questions that intend to assess the candidate's ability to quickly discover the criterion by which numbers and letters are arranged within a given succession. Also in this case, no mathematical or linguistic concepts are required other than the ability to perform elementary arithmetic operations."

IMHO that text makes it pretty explicit that they are looking for quickly-discoverable answers which don't require the use of non-elementary operations. Of the two alternate answers you suggest, both require significantly more time than the obvious/expected answer, and one requires non-elementary operations (square root).

(Also: you mention that you opened the book directly to this page, so I guess none of us here know whether there's any further guidance earlier in the book on how to approach such questions.)

This is meaningless, a finite set of numbers can't define a succession

It's not so much meaningless as slightly non-rigorous. Yes, there are infinitely many different sequences with the same beginning, which cover all of the answer options given. But even mathematicians don't need to be 100% rigorous 100% of the time; it's commonly accepted practice to use "..." to mean "continuing this sequence according to the most obvious/simple rule", in cases where there is one rule that is much more obvious and simple than the alternatives.

In this case, I would suggest that 95%+ of competent high-school maths students could figure out what answer the examiners are looking for.

Insisting on absolute rigor in all circumstances is just a recipe for unreadable tedium, even for mathematicians - let alone undergraduate engineers.

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    You make a reasonable case to support your conclusion, but it’s worth pointing out that polynomials can be evaluated using just elementary operations. So putting the emphasis on that wording is a bit of a non sequitur. – Dan Romik Nov 15 '19 at 4:01
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    @DanRomik I didn't say that the polynomial answer required elementary operations, and my answer doesn't assume that. The "quickly discover" is the part that excludes that one. – Geoffrey Brent Nov 15 '19 at 4:03
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    Well, why did you emphasize the part about elementary operations? I don’t see how it’s relevant to the discussion. – Dan Romik Nov 15 '19 at 4:08
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    @DanRomik Because it's relevant to the other answer OP proposed. Square-root is not generally considered an elementary operation. – Geoffrey Brent Nov 15 '19 at 4:46

If a student is knowledgeable enough to recognize that all the options are correct, (s)he should also be knowledgeable enough to recognize which is the answer that's desired.

As the exam-setter, you can never exclude the possibility that there's something you aren't aware of which could make a multiple-choice question have multiple answers. For example in the 3rd series, even if you write "what is the next number in this power series", how certain can you be that in some obscure field which you've never heard of, "power series" has a different meaning? To illustrate, try this question, which is similar:

Q: Which of the following is a metal?

A) Bromine B) Oxygen C) Carbon D) Calcium

Technically, all four of these elements are metals - by the astronomer's definition of "metals" as all elements heavier than hydrogen & helium. Of course, by the much more common chemist's definition of "metals", only calcium is a metal. If you had never heard of the astronomer's definition, you would never have dreamed of this objection. I'd actually venture that most astronomers would also never think of this objection and will happily mark (D) as the answer. Indeed, I first heard of this question as a gameshow question where the contestant got it wrong, then discovered via Google that there's actually a definition of metals where his answer was correct, and filed a lawsuit alleging he should be given the prize. Similarly, I will give anyone who tells me "none of the above, metal is a type of music" zero marks (sorry).

You can't rule out every possible objection, but you can say that anyone who's familiar with these obscure "other interpretations" of the question really should be able to recognize what you're trying to ask, and therefore get the right answer. In the absolute worst case, they should object during the exam. The student should know that marking one of the other answers and then objecting after the exam is asking for trouble, and that marking all of them is just as bad (I'm sure the instructions say to only mark one answer).

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    Sorry, but I don't buy the argument. You are just guessing what is in the mind of the instructor. "desired" answer in mathematics is repugnant. – Buffy Nov 15 '19 at 0:50
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    I agree with @Buffy. Saying that an answer is “technically” correct but is not what’s “desired” may be acceptable for a physicist or in other areas (although I don’t think it should be), but it’s not acceptable in a university math class. See my answer for additional thoughts. – Dan Romik Nov 15 '19 at 1:42
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    @Buffy It's not really "guessing" in this case, because the accompanying text gives some information about how these questions are supposed to be answered (quickly, without use of non-elementary operations). – Geoffrey Brent Nov 15 '19 at 3:44
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    The example is a bad one, imho. If the question comes up in a chemistry class, it is obvious what definition of metal is assumed. It's like taking a course on group theory and objecting that "group" can also be interpreted in the colloquial sense as merely a collection of objects. That's not really relevant to the discussion taking place here, which is about whether or not a mathematically correct answer can be marked wrong because it is not the desired one. – YiFan Nov 15 '19 at 9:47
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    Another (far more elementary) question to illustrate your point: "We want to build a cannon, what material is required?" A steel, B wood, C plastic, D glass. You can build a cannon out of all 4 materials that sort of maybe works, but there is only one that should be used. – Zizy Archer Nov 15 '19 at 14:07

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