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In an Advanced Calculus course, students were asked to prove $$|a_1 +a_2+...+a_n| \le |a_1|+|a_2|+ \ldots +|a_n|$$ for $n$ real numbers $a_1,a_2,\ldots,a_n$

I am teaching assistant for this course, and one of my students replied like this:

$|a_1+a_2|\le |a_1|+|a_2|$ by triangular inequality.

Then, $|a_1+(a_2+\ldots+a_n)| \le |a_1|+|a_2+\ldots+a_n|$

$|a_1|+|a_2+(a_3+\ldots+a_n)| \le |a_1|+|a_2|+|a_3+\ldots+a_n|$

Repeating this,

$|a_1 +a_2+\ldots+a_n| \le |a_1|+|a_2|+\ldots+|a_n|$

I gave him a bad grade (5 from 10), so he complained. It is my turn to reply.

  1. Is the student's proof correct?
  2. Is the point deduction justified? The professor and I expected the students to use mathematical induction for this proof.
  3. How can I most clearly communicate the grading rationale to the student in our upcoming conversation? How can I defend the grading policy against student objections?

I posted the question on math.stackexchange too: https://math.stackexchange.com/questions/4083896/grade-on-proving-a-1-a-2-a-n-le-a-1a-2-a-n?noredirect=1#4083901

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    $\begingroup$ Whether mathematical induction should be used to prove the result at this specific place in this specific course depends on expectations for proofs in the course, and thus is not something we can competently answer. And if mathematical induction should have been used, the amount taken off should be proportionate with the perceived importance of using mathematical induction at this time in this course --- some would argue that the use of mathematical induction is obvious and thus not the main idea, but maybe this part of the course is still training students to use mathematical induction. $\endgroup$ Mar 31, 2021 at 11:15
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    $\begingroup$ I think that this is a completely valid proof and I would have given it full marks. As to whether it is inductive, I'd argue that the style is more recursive than inductive but that this is more about fitting the proof into a template rather than any of the actual content. $\endgroup$
    – Adam
    Mar 31, 2021 at 13:36
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    $\begingroup$ This proof is correct and deserves full credit unless the students have been taught (and are expected to demonstrate here that they have learned) that "repeating" (and synonyms like "continuing") are abbreviations for rigorous inductive proofs. $\endgroup$ Mar 31, 2021 at 17:16
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    $\begingroup$ Agree with above comments. Additionally, I will say that the arbitrariness of the level of rigor demanded of a student can feel malicious, and definitely turn students away from thinking of mathematics as a "reasonable" field. $\endgroup$ Mar 31, 2021 at 17:55
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    $\begingroup$ Did the professor give you a rubric for grading? $\endgroup$
    – Sue VanHattum
    Mar 31, 2021 at 21:51

3 Answers 3

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In my opinion whether this proof is correct or not depends on the actual wording of the question. This recursive argument shows that the statement is true for 100 numbers (since it is true for 2, 3, 4, so we eventually reach 100) It also shows that the statement is true for 1000 numbers (we eventually reach 1000). For any fixed natural number this is a correct argument. However, if the students were asked to show that the statement is true for all positive integer $n$, then an extra step is needed. This is where quoting the principle of mathematical induction comes in. It is important for students to understand that the principle of mathematical induction is not just a shortcut to shorten a proof. It is a necessary axiom to move from "true for any fixed $n$" to "true for all $n$".

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    $\begingroup$ Can you give an example of a statement that is at the level of an advanced calculus student that is true for any fixed n but not for all n? This explanation is citing a very subtle distinction. $\endgroup$
    – Steve
    Apr 2, 2021 at 17:26
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    $\begingroup$ @Steve The distinction is subtle but real, at least to people interested in foundations. The student solution follows the idea of presenting an algorithm to verify the validity of a statement for any given finite set. This settles nicely with finitism. But if we acknowledge the existence of an infinite set N as in ZF and want to prove the statement hold for the infinitely many cases in N, then we have to bring out the axiom of infinity. Some answers here clarify: math.stackexchange.com/questions/2850064/… $\endgroup$
    – user52817
    Apr 2, 2021 at 21:58
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    $\begingroup$ @user52817 It's not that /I/ don't believe you, but part of the purpose of this question was to solicit a plausible explanation or defense that the student can at least comprehend even if they disagree. I don't think "the quantifier is in the metatheory, which doesn't always translate to a quantifier in the formal theory, and that is important not for any reason I can articulate to you but because it is important to some remote field to which you have not yet been exposed and likely never will" meets that threshold. $\endgroup$
    – Steve
    Apr 2, 2021 at 22:34
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    $\begingroup$ @Steve I think we are in agreement here. I am suggesting that we as educators delude ourselves when we think students in calculus are developmentally ready for this foundational issue. Requiring students to formally invoke and chant the axiom of infinity or principle of induction is unnecessary, unless we are specifically trying to teach the foundational issue. $\endgroup$
    – user52817
    Apr 2, 2021 at 22:56
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    $\begingroup$ @steve It is the statement itself that makes the difference. Here is an example. "For 100 real numbers, the magnitude of the sum is at most the sum of the magnitudes." This is one statement and it has a long proof (when we replace the "repeating this" of the original post with the appropriate steps. The proof is similar (but longer) when we change 100 to 1000, or any number. These are infinitely many statements with similar (but different) proofs. Some extra tool is needed for proving these at once "For any positive integer $n$, the magnitude of the sum is at most the sum of the magnitudes." $\endgroup$ Apr 4, 2021 at 23:02
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The proof is hard to follow because there is no equality sign or words between the second and the third line.

That said, the idea of the proof is correct.

If the goal is to teach rigorous mathematics, then this is a good opportunity for seeing how to go from the ideas to the rigour.

If you have to give points (which in general does not lead to learning, but rather the opposite), then you should not do that on ad hoc basis, but rather write a list of criteria which are mutually independent and how many points they are worth. Then you can check this solution and others against those criteria.

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Evaluating "Correctness"

There are many things that I dislike about the argument given, but these things are (perhaps) largely related to (1) the amount of time a student might have been given to produce that argument, (2) they style of the argument, and (3) the general level of mathematical thinking which is on display (with respect to rigor).

Time

In a timed setting, such as an exam or quiz, where time is a resource which must be used efficiently, I think it is wise to be more forgiving than on assignments where time is not a major factor (e.g. homework assignments).

Is this argument perfect? No.

But are those imperfections evidence of a student not understanding the assignment, or are they a symptom of the student not having enough time to more fully flesh-out their ideas? If the latter, I'd let it go.

Style

As Tommi points out in their answer, the argument is disjointed, in the sense that it isn't clear how one line relates to the next. The student just presents two unconnected inequalities. I would prefer to see something like

\begin{align} \bigl\lvert a_1 + a_2 + a_3 + \dotsb + a_n \bigr\rvert &= \bigl\lvert a_1 + (a_2 + a_3 + \dotsb + a_n) \bigr\rvert \\ &\le \bigl\lvert a_1 \bigr\rvert + \bigl\lvert a_2 + a_3 + \dotsb + a_n \bigr\rvert \\ &= \bigl\lvert a_1 \bigr\rvert + \bigl\lvert a_2 + (a_3 + \dotsb + a_n) \bigr\rvert \\ &\le \bigl\lvert a_1 \bigr\rvert + \bigl\lvert a_2 \bigr\rvert + \bigl\lvert a_3 + \dotsb + a_n \bigr\rvert. \end{align}

I might even go so far as to argue that, without the connecting "verb" (the equality between the second and third lines), the proof is not actually correct. It is, at the very least, more difficult to follow.

I am also a stickler for grammar and punctuation, and am annoyed by the unnecessary paragraph break after the first sentence and the lack of punctuation in the displayed equations.

To be clear, these are quibbles about the presentation, not the underlying correctness of the argument. The mathematics appears to be correct, even if it isn't communicated very clearly.

Rigor

Terrence Tao has written a really good piece on stages of mathematical rigor.

  • In a context where students are expected to be pre-rigorous, the proof is fine (e.g. this is exactly the kind of hand-wavy proof I would give to a precalculus class, where the formal idea of mathematical induction has not even been taught).

  • In a context where students are expected to be rigorous, or are being taught to write rigorous mathematical arguments for the first time, the proof leaves quite a lot to be desired. I would penalize this presentation fairly heavily, I think, because one of the goals of such a class is to get students to write more complete and rigorous arguments.

  • In a context where students are expected to engage in post-rigorous thinking, this is fine. If you point out to the student that the argument is incomplete, they should be able to reply "Oh, sure—but it's a simple induction argument, so I left it out."

So, depending on the objectives for the class, the work may be perfectly correct, way off base, or perfectly correct. It really depends on what you expect the students to know.

Assessing the Student's Work

Grading should always come down to the learning objectives for a particular assignment. What is the goal of asking students to prove this fact? What are they meant to learn by writing up the proof?

Though it is not given in the question, this looks the kind of assignment that is given in a "Proofs 101" kind of class (i.e. a kind of bridge course from lower-division to upper-division mathematics). In such a class, the primary goal is to get students to think and write rigorous, formally correct mathematics. In that context, a lot of the problems things which would otherwise be brushed aside as "style" are actually the thing that we are trying to teach.

No one should be surprised that the triangle inequality can be extended to any finite sum—it seems "obvious", and no one is going to ask you to prove it unless they are really worried about whether or not you know how to completely justify every step. So the fact that the induction argument is missing here is actually a big problem, since that induction argument seems to be the entire point.

As such, penalizing students for not outlining that argument seems appropriate to me.

That being said, I don't give students written assignments without also giving them the opportunity to revise and resubmit their work for full credit. So, in one of my classes, this would probably be returned to a student as "satisfactory, but please revise for a better score" (on a scale of "needs improvement", "satisfactory", "excellent").

In a points out of ten scenario, I think that 5/10 seems a little harsh (that's an F on the American scale). I'd probably ding 'em half a point (maybe even a whole point) for the annoying lack of punctuation and linking between lines, and another couple of points for the lack of induction. 7.5/10.

Communicating to Students

The most effective way to communicate grades to students is via a rubric. The rubric should outline the goals of the assignment, and how it will be graded. The rubric doesn't need to be published ahead of time (though you should give some indication ahead of time regarding what is expected), but it definitely needs to be shared after you have finished grading.

This helps to ensure (a) that students actually understand their grade, and (b) that you are grading fairly and consistently. Both of these things are important.

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