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The "working backwards" one was what I always noticed students doing. It's super frustrating as a teacher since they're simultaneously so close to being right (

by tgb 6y ago

The "working backwards" one was what I always noticed students doing. It's super frustrating as a teacher since they're simultaneously so close to being right (when they manage to avoid an irreversible step) and yet also have completely misunderstood a basic idea in what it means to even attempt to prove something.

faceplanted 6y ago
Do you factor that into the teaching now?
- tgb 6y ago
  I no longer teach. I'm also skeptical of how useful this page is except as therapy for teachers - students have to make mistakes to learn from them. Though it could be a good reference to point to after the fact instead of explaining the problem from scratch.
voxl 6y ago
"Working backwards" is a completely valid method of proof, used all the time in automated reasoning. If a student successfully reasons backwards then there is no issue, it is a proof.
- tgb 6y ago
  It's a completely valid work to do to figure out how to write a proof, but is not a proof itself. The section in the article states all my thoughts on that matter more clearly than I can. https://math.vanderbilt.edu/schectex/commerrs/#Backward https://math.vanderbilt.edu/schectex/commerrs/#Backward
  - voxl 6y ago
    As a PhD student myself, I would die on this hill, and you would lose. Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.
    - aesthesia 6y ago
      Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.
      - voxl 6y ago
        Sure, that's fine, but to say that it is an invalid method of proof is wrong. If I want to prove A, and I prove A <=> B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!
        grayclhn 6y ago
        It is a kind of backwards reasoning that students empirically screw up all the time. Edit: so much so, that I would definitely recommend students rewrite these proofs on assignments as an exerecise to make sure it's correct. By the time they reach a math PhD they probably don't need to do that anyomre. :)
    - tgb 6y ago
      Say a student is completing a problem set and is asked to prove P. If the student writes "P therefore Q therefore True since Q was proven in class. QED" then the student has committed a significant error in reasoning. If the student instead just writes "P. Q. True. QED" then they are writing the above in a more terse manner and have equally committed an error in their reasoning. If they first write that but then amend in some "iff"s between each step I'll give them full credit (assuming iffs are valid) but be slightly worried that they just learned that you need to put an "iff" in those spots to keep the teacher happy but don't understand why. If a strategy is not guaranteed to give proof, then you need to verify afterwards that the putative proof is in fact a proof. Just as if you get a potential solution to an equation (via solving a more general equation, perhaps), you have not "solved" the original equation until you actually check that solution, even if your putative solution is the true one. If a student does not do the "check if steps are reversible" part, then they have not written a proof even if every step is reversible! That's what's lacking form their proof.
  - zozbot234 6y ago
    "P implies thesis (proof) hence it suffices to prove P: (proof of P)" is a completely valid proof step, and is backwards chaining.
- 6gvONxR4sf7o 6y ago
  Nobody’s saying it’s totally invalid. I think you’re reading that into what they’re saying and you’re talking past each other. The complaint is that students work backwards without understanding why working backwards is different than working forwards. So if they happen to do it correctly (e.g. by using reversible steps), then it’s only success via accident. Done correctly, and on purpose, with the different care that isn’t required working the other way, yeah it’s legit, but that’s not what’s being complained about.
  - watwut 6y ago
    When you are doing equality and such all steps are reversible. Working backwards is exactly the same as working forward. You develop reasoning where you see it easier until to figure out to connect sides.
    - Caylio 6y ago
      Having equality signs at each step does not mean you can reverse the steps. You need bidirectional implication for that.

by tgb 6y ago

faceplanted 6y ago

Do you factor that into the teaching now?

tgb 6y ago
I no longer teach. I'm also skeptical of how useful this page is except as therapy for teachers - students have to make mistakes to learn from them. Though it could be a good reference to point to after the fact instead of explaining the problem from scratch.

voxl 6y ago

"Working backwards" is a completely valid method of proof, used all the time in automated reasoning. If a student successfully reasons backwards then there is no issue, it is a proof.

tgb 6y ago
It's a completely valid work to do to figure out how to write a proof, but is not a proof itself. The section in the article states all my thoughts on that matter more clearly than I can. https://math.vanderbilt.edu/schectex/commerrs/#Backward https://math.vanderbilt.edu/schectex/commerrs/#Backward
- voxl 6y ago
  As a PhD student myself, I would die on this hill, and you would lose. Moreover, the first sentence: "This is an unreliable method of proof" implies that it can be a proof if done correctly.
  - aesthesia 6y ago
    Well, it's a proof as long as you stipulate somewhere that all the logical connectives are "if and only if." This doesn't mean that a student who writes down a long series of equations beginning with some identity to be proven and ending with some known fact has proven the identity.
    - voxl 6y ago
      Sure, that's fine, but to say that it is an invalid method of proof is wrong. If I want to prove A, and I prove A <=> B for some proven statement B, then I have proven A! There is no question! Feels like crazy pills to think otherwise. This is a kind of backwards reasoning!
      - grayclhn 6y ago
        It is a kind of backwards reasoning that students empirically screw up all the time. Edit: so much so, that I would definitely recommend students rewrite these proofs on assignments as an exerecise to make sure it's correct. By the time they reach a math PhD they probably don't need to do that anyomre. :)
  - tgb 6y ago
    Say a student is completing a problem set and is asked to prove P. If the student writes "P therefore Q therefore True since Q was proven in class. QED" then the student has committed a significant error in reasoning. If the student instead just writes "P. Q. True. QED" then they are writing the above in a more terse manner and have equally committed an error in their reasoning. If they first write that but then amend in some "iff"s between each step I'll give them full credit (assuming iffs are valid) but be slightly worried that they just learned that you need to put an "iff" in those spots to keep the teacher happy but don't understand why. If a strategy is not guaranteed to give proof, then you need to verify afterwards that the putative proof is in fact a proof. Just as if you get a potential solution to an equation (via solving a more general equation, perhaps), you have not "solved" the original equation until you actually check that solution, even if your putative solution is the true one. If a student does not do the "check if steps are reversible" part, then they have not written a proof even if every step is reversible! That's what's lacking form their proof.
- zozbot234 6y ago
  "P implies thesis (proof) hence it suffices to prove P: (proof of P)" is a completely valid proof step, and is backwards chaining.
6gvONxR4sf7o 6y ago
Nobody’s saying it’s totally invalid. I think you’re reading that into what they’re saying and you’re talking past each other. The complaint is that students work backwards without understanding why working backwards is different than working forwards. So if they happen to do it correctly (e.g. by using reversible steps), then it’s only success via accident. Done correctly, and on purpose, with the different care that isn’t required working the other way, yeah it’s legit, but that’s not what’s being complained about.
- watwut 6y ago
  When you are doing equality and such all steps are reversible. Working backwards is exactly the same as working forward. You develop reasoning where you see it easier until to figure out to connect sides.
  - Caylio 6y ago
    Having equality signs at each step does not mean you can reverse the steps. You need bidirectional implication for that.