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These are primarily for algebra mistakes. Teaching a proof based course, the most frustrating error I saw was people doing a fine proof but in reverse (this is

by tgb 9y ago

These are primarily for algebra mistakes. Teaching a proof based course, the most frustrating error I saw was people doing a fine proof but in reverse (this is related to the article's point about irreversible steps). If I wanted them to prove that the square of any number was nonnegative, they'd start by assuming that x^2 >= 0 and then driving equations from that until they got something obviously true like 0>=0. This was the proof written "upside down" if every step were reversible. They should have just taken the last step and started from there until they derived their first step. So how was I supposed to grade that? It showed a significant misunderstanding but also often literally had every step necessary for a correct proof. Usually I just wrote overly long notes about how they should write proofs in the future that I'm sure the students never read. The other fun one was unnecessary proof by contradiction. They'd start by assuming not P. Then they'd prove P directly, not using that assumption. But they, that contradicts the assumption of not P. Hence not P is false, so then they conclude P. Nothing was wrong there, they just had a couple extraneous sentences that overcomplicated things.

ucarion 9y ago
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.
- sdenton4 9y ago
  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.)
  - arielb1 9y ago
    This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.
  - gus_massa 9y ago
    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.
yorwba 9y ago
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.)
- tgb 9y ago
  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.
jancsika 9y ago
> 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.
- wbl 9y ago
  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.
- throwawayjava 9y ago
  > ...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 :-)
todd8 9y ago
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.
bennofs 9y ago
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.
- throwawayjava 9y ago
  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.
  - pishpash 9y ago
    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.
    - throwawayjava 9y ago
thaumasiotes 9y ago
> 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.

by tgb 9y ago

ucarion 9y ago

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.

sdenton4 9y ago
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.)
- arielb1 9y ago
  This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.
- gus_massa 9y ago
  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.

yorwba 9y ago

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.)

tgb 9y ago
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.

jancsika 9y ago

> 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.

wbl 9y ago
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.
throwawayjava 9y ago
> ...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 :-)

todd8 9y ago

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.

bennofs 9y ago

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.

throwawayjava 9y ago
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.
- pishpash 9y ago
  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.
  - throwawayjava 9y ago

thaumasiotes 9y ago

> 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.