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If each reversible step was clearly marked as such, I don't see a problem with starting at the goal and then working towards a trivially true statement. Usually the problem is that the steps are written down without explaining how they are supposed to relate to each other, so you are left to guess based on the order they are written down.

Across two math departments that I have taken courses with, this was handled by having an official document that explained how to write proofs ("Read this if you want to pass the homework requirement."), then mercilessly marking down any missing details (e.g. whether two statements are equivalent).

This was probably quite a shock for many who hadn't needed this kind of rigor before, but they did learn pretty quickly to state their proof attempts according to the guidelines. (There were still mistakes, of course, but at least they were less ambiguous.)

A proof is more than just a collection of true statements: it is a means of communication. And in a classroom it is also a demonstration of knowledge. Even if every step is technically correct, the presentation may not be sufficient for full marks. Depending on the class, we may or may not have high expectations for a proof. Also I don't believe the students ever once made this setup but remarked that the steps were reversible, but I would give that full points with a little note that it could be better organized.

I've never encountered a formal "do these steps to make a proof" document. I'd have to see it, but it could be a good idea.

Yes, proofs by contradiction appeal to new mathematics students because one can just wander around until a contradiction pops up and then you're done! Of course, these meandering proofs are harder to understand, and it's easier to make mistakes when there are lots of unnecessary derivations along the way.

I've always found proofs in algebra and topology to be less intuitive and consequently often had to resort to trying to find a proof by contradiction and then unraveling it for its essential elements to make a better, clearer proof. Proofs in combinatorics (often by contradiction) don't even seem to have an better, clear proof buried inside; they are just essentially inscrutable.

> So how was I supposed to grade that?

Short digression: It's been interesting to read the criticisms of Uber on HN. Every single one has been accompanied by harsh criticisms of the taxi medallion system. Now I'm quite certain that a tiny proportion of HN have any working knowledge of that system (where working knowledge would be something like your reading, teaching, and genre literacy in proofs), but the general uninformed opinion seems to be that Uber's disruption of that industry is welcome and has no significant drawbacks to speak of.

With that in mind, what do you think is going to happen when someone in Silicon Valley disrupts higher education accreditation and/or secondary degrees? Don't get me wrong-- I think there is a lot of value in university education and university life. But if people who have never even heard of taxi medallions are quick to write off the costs of it being replaced by an unregulated contractor, they are going to be even more willing to say goodbye to a system whose instructors (who themselves are mostly independent contractors) cannot or will not accurately assess the correctness of a student's work?

I don't mean to single you out here-- I've certainly done the similar. The point is everyone on HN has their own stories of experiencing this problem-- where the intellectual environment advertised by the university was violated. Coupled with the absurd debt students today carry (at least in the U.S.), that is more than enough ammunition for a disruptive company to put an enormous dent in the undergraduate degree. My fear is that will strip the money needed for grad programs-- where the real learning and research happens-- and the modern uni will fall like a house of cards.

Have you ever graded? If you have you would know that there is no such thing as "the correctness" in anything that isn't close to perfect. The OP had to decide to give a number somehow to something that was in small parts right, but completely misguided. And then they wrote a long comment explaining the problem.

That's far more useful to the student then just numbers. Maybe the student could actually read the comment and learn from it. Any violation of the intellectual promise in OP is from the student who doesn't put in the effort to learn.

> ...whose instructors... cannot or will not accurately assess the correctness of a student's work?

Have you graded proofs? GP's approach is entirely sensible.

Proofs are written for humans. Writing a proof upside down is wrong, even if the derivation is correct.

The underlying cause of that incorrectness could be a simple matter of poor communication, or the student might have a more fundamental misconception about how implication works (and/or quantifiers). Sometimes the instructor knows enough about the student to determine which is the case. Sometimes it's impossible without talking to the student. Hence the long note and (perhaps implicit) invitation to office hours.

I can't find anything in GP's post that indicates he "cannot or will not accurately assess the correctness of a student's work", aside from the obvious impossibility of projecting a multi-objective assessment of a fundamentally human task onto an integer in [0,100].

If anything, GP provides an excellent example of why auto-grading for upper division courses -- especially courses that are trying to teach students how to write for a human audience -- will probably require fundamental scientific advances in AI in addition to innovation on top of existing tech stacks. My money's on self-driving cars happening first.

> With that in mind, what do you think is going to happen when someone in Silicon Valley disrupts higher education accreditation and/or secondary degrees?

This already happened! University of Phoenix has been around for a while.

> The point is everyone on HN has their own stories of experiencing this problem-- where the intellectual environment advertised by the university was violated

If you want to know how people really feel, look at where they send their kids rather than they say/say they will do in online debates. Again, University of Phoenix et al have been around for a while.

> My fear is that will strip the money needed for grad programs-- where the real learning and research happens-- and the modern uni will fall like a house of cards.

Masters programs are cash cows.

Top Ph.D. programs usually aren't appreciably funded through undergraduate programs. And when they are, it's almost always backed by Ph.D. student labor. Which is crazy cheap, and which startups are going to have a very hard time competing with.

> Short digression: It's been interesting to read the criticisms of Uber on HN.

The same is true of higher education :-)

> Have you graded proofs? GP's approach is entirely sensible.

GP wrote characterized the mistake as a "significant misunderstanding" of the material. GP use of the word "just" strongly implied that the assessment inside the long, probably unread note was not reflect in the grade. If that is indeed the case then the student's "significant misunderstanding" of this material wasn't reflected in the calculation of their final grade, which is the textbook definition of grade inflation.

Again-- this kind of grade inflation happens often. (If the class size is greater than 75 and fulfills one of the uni's writing requirements, I'd argue it's almost a requirement that it happens.) It's a truth about an impossible situation with only bad choices, not an ad hominem.

> This already happened! University of Phoenix has been around for a while.

By disruption, I mean Uber/AirBnB style disruption. Disruption resulting in customers who leverage the technology saying things like, "Dude, it's changed the way I go out in Atlanta." I seriously doubt the existence of University of Phoenix has ever made an employer say anything close to that. (E.g., "it's changed the way we hire employees.")

I mean an accreditation service that basically tells an employer: look, do you really want to know if that employee is worth hiring?

I doubt this even requires any testing whatsoever-- just require incoming Freshman to get a Chromebook. But the end of their studies there could be a new Google "Fasttrack" app. You could make it almost like a digital scratch ticket-- giving Google the relevant permissions and you can see whether you really graduated with honors.

Edit: added example

> If that is indeed the case then the student's "significant misunderstanding" of this material wasn't reflected in the calculation of their final grade, which is the textbook definition of grade inflation.

I don't know; it depends. I typically write my rubrics so that there are "style" points. In a really easy proof, style might be up to half the available points. In a very difficult proof perhaps only 1/10 or less.

But even that doesn't totally solve the problem. Some mistakes are on the border between style and substance. Grading and assessment aren't a perfect science.

> I seriously doubt the existence of University of Phoenix has ever made an employer say anything close to that.

Call me old-fashioned, but IMO the actual education is still the hard part of the Education Industry.

My comments have all assumed the goal is student understanding. IMO the obsession over grades (by students and by external observers), testing, and accreditation is completely mis-guided. Focus on exceptional student outcomes and the rest is easy.

I remember a physics academic I knew giving all his colleagues a short question and lots of sample incorrect answers and asking them how they would grade them out a certain mark. There was absolutely no agreement whatsoever. On the other hand I imagine that they all agreed that a correct answer would merit full marks.

Does it matter? In the UK, your final degree generally has one of only four classes and I expect most people end up roughly the right class. There are obviously borderlines but a first class student isn't going to get a lower second or vice versa because of subtleties in the marking scheme.

It's also an issue for me with too much weighting on coursework - work during the course should be for things like getting feedback on how to write proofs properly (if you are studying mathematics). By the time, you come to sit actual exams, you really shouldn't have any excuse for not knowing how to.

In terms of disruption to accreditation, I think one of the main things that a degree of a certain standard signals is whether a person works hard in an environment less structured that school and can learn independently. That is much harder to measure than knowledge or raw IQ.

Transforming a statement using a series of equavilence transforms into an always true statement isn't a valid proof? After all, you've shown that the original statement is the same as a true statement. Isn't that a valid proof as well? It is often easier to go in that direction, since you have a clear starting point. In the other direction, you'd have to guess from where to start.

1. Many, many theorems are not full equivalences. Students who write proofs bottom-up are, ime, MUCH more likely to mistake implications for equivalences when using theorems/lemmas that are not equivalence conditions.

2. Writing proofs upside-down is bad form. Sometimes it doesn't matter -- e.g., for very short proofs or for proofs that are just a lot of symbol pushing. But for proofs that benefit from intuition, writing the proof in the correct order (from assumptions to conclusion) is infinitely easier to follow.

3. Even when all the steps are honest equivalences, it's a matter of convention and familiarity. Yes, community norms are sometimes arbitrary and silly. And there are all sorts of exceptions to those norms. But I don't walk around my neighborhood naked or in a speedo, and I don't write relatively short direct derivations upside down ;-)

4. This is not unique to mathematics. Legal texts, philosophical texts, and even business plans tend to follow the same structure when presenting arguments.

> In the other direction, you'd have to guess from where to start.

But a proof is NOT a brain dump of your process! It's an exposition of your final result.

It's fine to write it that way as long as you justify that the steps are equivalent and thereby showing you know what you are doing. If you don't write something like 'by equivalence' then the assumption should be that you don't know what you are doing. Or, when there comes a time where there is actually a proof or step of a proof where there is no equivalence, mark it wrong. Don't patronize your students but do write better problems, perhaps.

>It's fine to write it that way as long as you justify that the steps are equivalent and thereby showing you know what you are doing.

Have you heard of cargo culting? Some students will just learn that writing "by equivalence" sometimes gets them most of the points. Correct proofs don't necessarily indicate understanding. That's not patronizing, it's just true.

But that's really beside the point.

The primary problem with these proofs is not that they are logically wrong. Re-read #2 and #3 above.

These proofs are usually difficult to read and understand, in part but not exclusively because they break with traditional style. And not just mathematical style (#4).

Don't be the guy who complains that he doesn't need to follow style guides or write maintainable code because "my code doesn't have bugs".

I don't want a brain dump of the student's thought process. I want a clear exposition.

Sometimes upside down proofs provide a clear exposition; in #3 I mention these exceptions! But more often, the proof is upside down because the student is narrating their problem solving process instead of communicating a valid argument.

Unreadable code is technical debt.

Unreadable documentation will lose you customers.

Unreadable proofs are wrong.

Developers who write easy-to-understand code and clear documentation are strictly superior, all else equal, to developers who write in an idiosyncratic and difficult-to-follow style.

Math is no different, and one goal of proof-based courses is to teach students how to communicate technical content clearly.

> Don't patronize your students but do write better problems, perhaps.

At this level, the problem is usually just a bog standard statement of a theorem in established mathematical prose.

Some number of students new to proof writing will write upside down proofs regardless of what you ask them to prove.

The wording of the question does not prompt this behavior.

A student writing mathematical proofs is not a developer writing client code.

It's a metaphor.

One goal of proof-based courses is to teach students how to communicate technical content clearly.

> So how was I supposed to grade that?

Tell them they're required to include equivalence markers (⇔) and implication markers (⇒) in their proofs. The arrows make it obvious when the proof is correct and when it's wrong.

I've noticed this (backward steps) with some friends. I think it's because people don't think logically with math notation--they just play with symbols until they get the sentence they want.

To get around this, I just get people to argue aloud. People wear their logic hats again when they have to talk to a human and convince them of something.

Alternately, it's a very useful technique to identify the penultimate step; it's often a useful rephrasing of the problem, and gives something easier to prove. And then just repeat...

(It's probably usually a hangover from poorly taught rote proof exercises in a lower grade, though.)

This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.

Writing the proof in the wrong direction is useful in some kind of problems, for example to prove by induction that 1^3+2^3+3^3+4^3+...+N^3 = N^2 * (N+1)^2 / 4.

My recommendation to my students is to write to "proof" in the wrong direction in a scratch paper, the rewrite it carefully in the correct order in the final paper checking that all the implication go in the right direction, and then destroy the scratch paper.