31
$\begingroup$

I am interested in finding examples of poorly written proofs that exemplify the types of mistakes made by undergraduate students in their first year or two of writing proofs. I am interested both in examples of poor writing that introduce logical errors into the proof as well as those that just make the proof hard to read.

I am aware that there is already a question on this site asking for examples of bad proofs. I find the answers there kind of unsatisfactory (one answer simply says to look at student assignments and the other just consists of a broken link), but perhaps it's because there aren't many great examples of poorly written proofs on the internet. Also, that question is only focused on proofs that are wrong or "could be simplified" whereas I am also interested in more general problems with the way proofs are written.

In case there are no good examples online of poorly written proofs, I would like to try to generate some myself. Ideally, for each general type of mistake that undergraduates are prone to make when writing proofs, I want to write a bad proof that isolates and exemplifies that mistake. So I am interested in compiling a "taxonomy" of ways that proof writing by students can go wrong.

Here's my current taxonomy:

  1. Backwards proofs: Proofs that start from the desired conclusion and work backwards until they arrive at the premises. Not only can this be confusing to read, it is easy to introduce errors when some step of the proof is not reversible.
  2. The "obvious" worst case: It is popular to warn students against "proof by example" but I have found that students are usually already aware that an example is not a proof... unless of course they think that the example they chose is obviously the worst case scenario without bothering to justify it.
  3. The tangle: The proof that keeps going and going, introducing a complicated series of manipulations, definitions, hand-waving, etc. until somehow arriving at the right conclusion. Often some step in the middle is wrong (or possibly nonsensical) but it's hard to find because it's buried under a pile of more-or-less-correct but irrelevant details. A special favorite of students taking exams.
  4. The mysterious stranger: Some object is used without ever being introduced or defined. Often there is a reasonable way to fill in the definition of the object but the proof becomes a guessing game of what the student meant the object to be. Sometimes this is caused by a student trying to use a variable outside of its proper scope. In that case it can also lead to the student giving the same name to two distinct objects.
  5. Inappropriate types: The student claims that some ideal is an element of the ring. Or tries to divide by a set of real numbers. Etc. The point is that they are trying to do something with an object which is of the wrong type to do it with. Or sometimes they are trying to perform a typecast that cannot automatically be done in an unambiguous way like if they try to apply a function of type $\mathbb{R} \to \mathbb{R}$ to a complex number. Often in these cases the student does have in mind a way to do the typecast (although sometimes they have just misunderstood a definition or don't even really grok that mathematical objects have types) but problems occur when there are multiple mutually incompatible ways to do it.
  6. The incantation: You've just learned a big, important sounding theorem in class so you should probably use it on the homework, right? You don't quite understand it or know when it applies, but it seems like you should be using it so you just go ahead and invoke it anyways. It's pretty common for students to invoke big theorems without checking that the hypotheses of the theorem hold and sometimes in situations where it doesn't even make sense to invoke the theorem.
  7. Implication confusion: The student is supposed to prove $X \implies Y$ but instead proves $\lnot X \implies \lnot Y$ or $Y \implies X$ or tries (and typically fails) to prove $X\land Y$. Or the student is told to assume that $X \implies Y$ and then concludes that $Y \implies X$. Most common when the statements of $X$ and $Y$ are a bit complicated or involve implications as well.

Which leads me to my question.

Question: What other common pitfalls of student proof writing should be added to this taxonomy? What are some poorly written proofs that exemplify one of these pitfalls (either one that I have already listed or a new one)?

Feel free to argue with my current list. Also, if anyone wants to add an answer to the old question on "bad proofs" I'd be very happy.


EDIT: After discussing this with one of my friends, I've added two more entries to the taxonomy.

$\endgroup$
9
  • 2
    $\begingroup$ Two examples I can think of that might be slightly outside your scope (because even strong mathematicians can do this from time to time) are not considering all possible cases when dividing the proof into separate cases, and in overlooking sometimes problematic exceptional situations (what if the variable is zero, what if the set is the empty set, etc.). $\endgroup$ Feb 17 '20 at 19:33
  • 2
    $\begingroup$ @amWhy The fallacies you link to are rhetorical fallacies, which is explicitly not what I'm asking about. I suppose it's possible that an undergraduate student could include an instance of the ad hominem fallacy in a proof they write, but I have yet to see it. $\endgroup$ Feb 17 '20 at 22:01
  • 1
    $\begingroup$ There is far more overlap than you acknowledge (circular reasoning, argument by authority, etc.) $\endgroup$
    – amWhy
    Feb 17 '20 at 22:03
  • 1
    $\begingroup$ You do need to understand that at the root of many students' logical fallacies are rhetorically/cognitively/psychologically based. Certainly not all. But we can see this comes into play often in combinatorics and probability and stats. Also, wrt the material conditional $p\to q$ which is true except in the case that p is true and q is false, students often expect $p$ to be relevant to $q$, or for $q$ to be dependent on $p$. So some have a hard time accepting the truth of, e.g., "If pigs can fly, then purple unicorns can swim", and even, "If pigs can fly, then $2+2 = 4$." $\endgroup$
    – amWhy
    Feb 17 '20 at 22:35
  • 3
    $\begingroup$ I love this. I'm a TA for vector calc and I most often see backwards proofs and "the tangle". Beautiful example for explaining backwards proofs is one I saw last year: The question asks you to prove that if $r(t)$ is a parametric curve and $r(t)$ is always orthogonal to $r'(t)$, then $r(t)$ traces out a circle. Many students started with $r(t) = \langle R \cos(t), R \sin(t) \rangle$ and then proved $r'(t) \cdot r(t) = 0$ to conclude that $r(t)$ and $r'(t)$ are orthogonal. $\endgroup$ Feb 25 at 16:39
14
$\begingroup$

Perhaps related to "The Tangle" is what I call "Wishful Thinking". This most often happens when the student has a correct algebraic expression/equality and knows the correct final expression/equality, but makes great (or incorrect) leaps to get from one to the other, e.g. in proofs by induction.

There is also the "Using What You Are Proving" (maybe similar to your "Backwards Proof," but more subtle). Example: in proving two elements $a$ and $b$ commute, they use $(ab)^2 = a^2b^2$. Or in an onto proof where they assume a general element of the range has the form of an image.

$\endgroup$
4
  • $\begingroup$ Those are both great! I think "Wishful Thinking" is different from "The Tangle" since in the tangle, the errors are usually not caused by trying to get to any particular goal; it's more like lots of unnecessary complexity is introduced which often leads to errors, which often in turn leads to a false proof. And I also think "Using What You Are Proving" is different enough from "Backwards Proof" to constitute its own entry. $\endgroup$ Feb 17 '20 at 18:50
  • $\begingroup$ Also, it occurs to me that "The 'Obvious' Worst Case," "The Mysterious Stranger" and "The Incantation" are all at least sometimes caused by cases of "Wishful Thinking." $\endgroup$ Feb 17 '20 at 18:52
  • $\begingroup$ I suppose "using what you are proving" can be better described as "circular reasoning"? $\endgroup$
    – GoodDeeds
    Feb 17 '20 at 18:57
  • 2
    $\begingroup$ @GoodDeeds It's true that "Using What You Are Proving" is an instance of "Circular Reasoning," but what is interesting to me about it is the circumstances that cause students to make that mistake. Most students can recognize bald instances of circular reasoning and know it is wrong. But they still end up engaging in circular reasoning anyway and I am interested in how that occurs. $\endgroup$ Feb 17 '20 at 19:32
4
$\begingroup$

The Q&A so far has gone into a lot of relatively sophisticated stuff. The students I teach are mostly engineering majors at a community college, so their mistakes are a lot simpler. They have generally been exposed to proof writing in high school geometry. However, my experience is that this does not translate well into a more general skill. Many of them learned "two-column proofs," which are pretty artificial and constrained.

Often they will "prove" something by giving an example, e.g., they'll prove that the dot product is commutative by making up a couple of vectors and doing the computations. This also comes up in physics labs, where they claim that their lab results prove Newton's second law. They don't want to believe that a counterexample can disprove a general statement, but one piece of evidence can't prove it.

They write something that they claim is a proof, but they never make use of one of the assumptions, and this assumption is necessary.

In my experience students in elementary classes like first-semester calculus need to get some experience writing really simple, easy proofs so that they can get past these initial stages. Unfortunately, I don't think most instructors assign any such work in these courses. I would imagine that this is a big reason for the meltdowns I hear about when math majors start their upper-division coursework.

Here's an example of what I would like them to be able to do. Suppose they know enough about the dot product to know how to compute it from cartesian components. Then if I ask them to prove that the dot product is commutative, ideally they would be able to write something like this: "The dot product can be expressed in terms of the products of the two vectors' components. Since the components are real numbers, and multiplication of real numbers is commutative, it follows that the dot product is commutative."

But even this extremely simple-looking argument contains some pretty sophisticated mental activity. Students are trying to relate everything to their previous education, and their previous education contains a couple of possible models of this. One model is concrete computation involving numbers, so they produce an example involving numbers and call that a proof. Another model is two-column proofs in high school geometry, but that model doesn't carry over here, because it doesn't help them to focus on the essentials, ignore inessential elements, and write a sentence meant for a human to read.

$\endgroup$
5
  • $\begingroup$ Thanks for adding your perspective. At my current university, lower division math students are not really expected to be able to read or write proofs so the issue of having trouble producing correct proofs doesn't come up directly (though it comes up indirectly when they employ faulty reasoning in the course of solving a problem). I agree with you that it's a shame that there is often no opportunity for students to really understand how to write a rigorous proof in a environment which is slow-paced enough that they can just focus on mastering that skill. $\endgroup$ May 14 at 3:46
  • $\begingroup$ Perhaps it would even be beneficial for lower-division, computation-heavy courses if students had already been through some course on logical reasoning and proof writing. I feel that in such classes (i.e. computation-focused ones) the instructors are often math professors who attempt to use mathematical language to teach the students and often give at least sketches of proofs. But since the students have not been formally taught how to understand mathematical language or how to read proofs, they have trouble following this kind of instruction. $\endgroup$ May 14 at 3:48
  • $\begingroup$ At my current university, lower division math students are not really expected to be able to read or write proofs Here in the US, students are generally expected to write proofs in high school geometry. I've edited my answer to discuss this a little more. $\endgroup$
    – user507
    May 14 at 15:25
  • $\begingroup$ Perhaps it would even be beneficial for lower-division, computation-heavy courses if students had already been through some course on logical reasoning and proof writing. I don't think this is realistic. Nearly all students in these courses are either engineering majors or life science majors. The engineering majors in particular have a very heavy load of required courses. You can't just add another semester-length course to their requirements. $\endgroup$
    – user507
    May 14 at 15:26
  • 1
    $\begingroup$ I agree with you that it's a shame that there is often no opportunity for students to really understand how to write a rigorous proof in a environment which is slow-paced enough that they can just focus on mastering that skill. I actually don't think I agree with you here. There is no need for them to be "rigorous." They just need to get experience writing correct arguments that are understandable to another human. I don't think it's a good idea to focus on just the skill of proof-writing. Writing simple, easy proofs should just be a normal part of every class. $\endgroup$
    – user507
    May 14 at 15:30
2
$\begingroup$

My friend, who is currently teaching an abstract algebra course, recently provided me with an example of an incorrect proof that I think showcases a few types of faulty reasoning that are common among students. I'm sharing it here with his permission.

Proposition: Suppose $R$ is a ring and $I$ is a prime ideal in $R$. For all ideals $J$ and $K$ in $R$, if $JK \subseteq I$ then either $J \subseteq I$ or $K \subseteq I$.

Proof. Suppose $JK \subseteq I$. Then $\sum_{n, m} a_nb_m \in I$. Since $I$ is an ideal, it is closed under addition so each $a_nb_m$ is in $I$. Since $I$ is prime, either $a_n$ or $b_m$ is in $I$. Therefore either $J \subseteq I$ or $K \subseteq I$.

There are a few obvious problems with this proof.

  • First, it is not clear what $a_n$ and $b_m$ are. Presumably the $a_n$'s are some finite set of elements in $J$ and the $b_m$'s are some finite set of elements in $K$, but this should be stated explicitly. This is an instance of The Mysterious Stranger.
  • Next, saying that "since $I$ is closed under addition, each $a_nb_m$ is in $I$" is not correct. It is mixing up "closed under addition" with its converse, so this is an instance of Implication Confusion.
  • The conclusion of the proof is more or less a non-sequitor. All that the proof has really established is that either $J$ or $K$ contains some element which is in $I$, which is not enough to show that either $J$ or $K$ is entirely contained in $I$. I would say this is an example of Wishful Thinking.
  • Finally, there is an element of The Tangle to the whole proof. It is not really clear what the strategy of the entire proof is, why the sum $\sum_{n, m} a_nb_m$ is being introduced or what the goal of each step is. It's likely that such a proof is mostly the result of pattern-matching the appearance of proofs given in class in the hopes that this will lead to an approximation of a correct proof (and thus partial credit).
$\endgroup$
3
  • 3
    $\begingroup$ I'm accustomed to tangles being much longer than this. $\endgroup$ May 13 at 2:30
  • $\begingroup$ @AndreasBlass Yeah, I agree. That's why I just said "there is an element of the tangle." But it would be nice to have better examples. $\endgroup$ May 13 at 2:34
  • $\begingroup$ I think this useful and highlights what I came to say: there should be a category of "The Totally Incoherent" which is likely a synthetic gumbo of all the various errors. $\endgroup$ May 15 at 4:28
2
$\begingroup$

Three categories of error I often encounter are "Incorrect negation", "trouble with quantifiers" and "Confusing general parameters with specific objects". Everybody knows that students have trouble dealing with negating quantifiers in the context of, for example, an $\epsilon-\delta$ proof in an Analysis course. We also see these issues arise in Linear Algebra, where student work like the following is rampant:

Let $\vec{v}$ be any nonzero vector; prove that $\{\vec{v}, \vec{0}\}$ is linearly dependent.

Solution. Let $a\vec{v} + b\vec{0} = \vec{0}$. If $a = b = 0$ then this is a trivial relation; therefore the vectors are linearly dependent.

What's happening in this muddle is something like this: Students know that "linearly independent" means "there is no nontrivial linear relation on the vectors"; they try to negate it, but get tangled up in the triple negative ("it is not true that there is no nontrivial..."), and end up thinking that "linearly dependent" means "there is a trivial linear relation on the vectors". The use of the letters $a$ and $b$, which seem like general parameters in the first sentence, but are subsequently revealed in the very next sentence to have been very specific constants all along, helps to obscure the obvious error.

In fact on our most recent final exam we included, as a true/false question,the statement

"If $S = \{v_1 \, \dots, v_m \}$ is a set of vectors, and $a_1v_1 + \cdots a_nv_n = 0$ is a trivial relation, then $S$ is linearly independent."

A distressingly large fraction of our students marked this as true. They seem to have trouble understanding that "There are no nontrivial relations..." means something different from "There is a trivial relation..."

$\endgroup$
3
  • $\begingroup$ Thanks for the nice answer and for the example. I think that "incorrect negation" and "trouble with quantifiers" feel related to "implication confusion." In all three cases, the problem is that students don't understand (or at least don't have an operational understanding of) how to manipulate formulas of propositional or predicate logic. $\endgroup$ May 14 at 3:30
  • $\begingroup$ I like "confusing general parameters with specific objects." It feels like it is similar in nature to "inappropriate types" and "the mysterious stranger." In all three cases, the error seems to arise from misunderstanding how mathematicians talk about mathematical objects. I think what sometimes causes it is that students are trying to imitate the language that they saw instructors using in class without completely understanding the underlying logic (e.g. they see instructors sometimes talk about "setting a and b equal to 0" and so they think it is something that is valid in any situation). $\endgroup$ May 14 at 3:32
  • 2
    $\begingroup$ For the true/false question that you mentioned, I would guess that part of the reason students got it wrong is that they didn't parse it correctly. Some of them may have read it as "If $S = \{v_1, \ldots, v_m\}$ is a set of vectors and $a_1v_1 + \ldots + a_nv_n = 0$ is a trivial relation for any $a_1, \ldots, a_n$ for which the relation holds then $S$ is linearly independent." This is adding a quantifier that is not in the original statement, but mathematicians have fairly particular conventions about when there are implicit quantifiers which I don't think all students are aware of. $\endgroup$ May 14 at 3:37

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.