How to evaluate a proof that seems to be using induction informally?

Question

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

Answer 1

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 $n$".

Answer 2

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.

Answer 3

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.