Teaching students how to check the validity of their proofs

Question

The question Teaching students to find and correct their own errors and its answers address mainly calculation problems of the types typically found in secondary school and the lower levels of undergraduate mathematics. What tips do you have for students to check the validity of their proofs, apart from showing them to a more experienced mathematician, and what approaches can we use to help them spot mistakes and errors, and then correct them?

Answer 1

One tip is to have students think about whether their proof really uses all the hypotheses of the theorem. You can even put this in the question, e.g. "Part (b): where would your response to part (a) fail if $f_n$ only converged pointwise?"

Answer 2

Set an exercise with questions of the following form:

a) Read the proof below and identify the flaws in the proof.

...

b) Write a correct proof

Include as many questions in this style as you think appropriate so you can include several common types of errors, for example: not considering all cases. Depending on a fact that is false (the proof that there is no largest prime by assuming $p!+1$ is prime for example), incorrect use of converse (eg the use of if there is no change of sign, then there is no intersection), sign errors, incorrect generalisation instead of formal induction, use of a numerically calculated value instead of algebraically derived value ... you can probably think of plenty more. Edexcel has resources for proofs at A-level standard that you can use.

Explain that the purpose of the exercise is to practice the skill of checking proofs and understand the expectations for proofs.

However, you should know that "proof" is an area that students find especially difficult. Having marked iA-levels, the "proof" question was done correctly by about 1% of candidates, so don't expect this to be an easy process for students. This pons asinorum may be a bridge too far.

Answer 3

There's two aspects of your question. How students check proofs and how to teach students how to check proofs. Your question title is literally asking about the later (i.e. pedagogy), but from the text, you are more concerned with the former.

And I agree with this priority. It's actually a more strong unknown. (For once.) Just making a clarifying comment. And not to be pedantic or fussy at you. But it's important to think about what the real question is to move forward, in issue analysis.

Quite often there is this confusion here about the method versus how to teach the method. For an analogy, consider EVF (early vertical forearm) versus how to teach EVF, in freestyle swimming. It actually takes more than just understanding exactly what EVF is to change one's stroke and start doing it.

On topic, my recommendation is to think about it from first principles, yourself. And make some checklist for the student to use. Note (to my critics) it's not about some mechanical checklist that will uncover every issue. Even the act of using an IMPERFECT checklist, forces one to look at the problem again and allows the subconscious a chance to figure out the one little tail of yarn coming out of the ball...that will unwind it all. For similar reasons, I used to make my technicians do a trip point and calibration check with malfunctioning electronic equipment. It's not JUST that it might uncover something (even when they though the TP&CC was not germane...which often was a wrong assumption), BUT that it would slow them down a bit and allow the subconsious to engage. A similar thing is the advice to rewrite the problem statement, when starting to answer...even totally clueless, it allows you to put something down and gives the subconsious a chance to engage.

I recommend to make your own list and adjust it over time (prioritize, etc.) Here is mine.

1. Have we covered all the cases?

2. Domain and range issues?

3. Divided by zero kvetches.

4. Does each connection have a valid rationale (just mechanically looking at what you did!)

P.s. I would not 100% expect that checking is the key issue. In exam (or graded homework) situations, students may be trying to get partial credit and be well aware that their proof is flawed. but a partial is better than a zero. All that said, even in this case, the checking may help their subconscious to figure out how to fix their proof (that they know is flawed).

Answer 4

One thing that you should teach them is to "play the opponent's side". Normally the student (and even a professional mathematician writing an article) is in the mode of trying to convince the teacher and/or everybody else that his/her approach, computations, and logical argument make sense. In this mode mistakes are very hard to notice unless they are truly outrageous. In order to really catch the bugs, one should switch to the opposite mode of harsh criticism and reluctant acceptance of only the parts that are absolutely clear and foolproof.

The psychological trick is to imagine that the text was written by your worst enemy, whom you want to destroy at all costs, so if you cannot get at him directly, you should at least try your best to convince everyone that this particular work of his is total rubbish making no sense whatsoever. Then you start seeing awkward passages, unjustified leaps, and other things like that. Once you have marked them all with bright red pen and as negative comments as you can produce, you can switch to the proponent's side again and see how and if you can refute that criticism by changing and clarifying the corresponding places.

Usually it takes me 2-3 rounds of switching sides to make the text acceptable for showing to other people in the case of a medium length professional article. For short proofs like the ones we usually require students to produce 1 or 2 may be enough, but one is necessary under any circumstances. I'm in the profession for 30+ years and am considered a reasonably good writer and proofreader by many, but you will be amazed by what nonsense finds its way into my first drafts (if I ever show them to you, that is; the point is that I won't. :lol:).

Answer 5

On request from the asker, I will give my answer based on my decades of experience in teaching mathematics.

Frankly, anyone who is unable to check the validity of his/her own proof (excluding careless mistakes) simply does NOT understand what a genuine proof really is. Modern mathematics is based on FOL (first-order logic), and one must at least know a complete deductive system for FOL to be able to truly understand what a rigorous proof is. Informal proofs are nothing more than informal proofs. Once you know how to use a deductive system, it's trivial to check your own proof; just check that you are following the rules! I have personally taught using this foundational system.

I also do not agree with teaching natural language proofs first. This does not mean that one cannot use English words and phrasing in proofs. One can definitely do that. Technically the most important ingredients in a deductive system that is pedagogically beneficial are simply:

• Fitch-style contexts: There are only two, the ⇒-subcontext and the ∀-subcontext. Not only is nothing else needed, but also the ability to produce a proof outline with the correct logical structure (which is entirely captured by these contexts) directly corresponds to the genuine understanding of rigorous proof. The usage of logical symbols does not matter, but any student who is incapable of producing the correct logical structure cannot have a proper understanding of proof validity. For example, proof by contradiction is totally obvious and undeniable once you understand Fitch-style contexts (and accept LEM for the relevant sentences), and proof by induction can only be properly understood (why it is sound and yet why it is an extra assumption) when one has a full grasp of logical structure.

• Multiple sorts: A common mistake by many people is to teach just plain (one-sorted) FOL to students. Although many-sorted FOL can be reduced to one-sorted FOL, it is actually the superior form of FOL for logical reasoning. Think about it; "every human is an animal" directly translates to "∀x∈Humans ( x∈Animals )" where Humans and Animals are sorts, and it is actually a detour to express it in one-sorted FOL as "∀x ( Humans(x) ⇒ Animals(x) ). This is obvious in the syntactic symmetry between statements and their negations; the opposite of the above example is "∃x∈Humans ( ¬x∈Animals )". Worse still, insisting on unrestricted quantifiers will make it nasty or impractical for actual mathematics (e.g. the general recursion theorem is "∀A,S∈set ∀c∈A→S ∀f∈A×(ℕ×S)→S ∃g∈A×ℕ→S ∀x∈A ( g(x,0) = c(x) ∧ ∀k∈ℕ ( g(x,k+1) = f(x,(k,g(x,k))) ) )", which is clean like this in my system but would be humanly horrible to parse if forced to use one-sorted FOL).

• Definitorial expansion: This is supported by actual proof assistants in some form or another, but is unfortunately not often taught to students, and so they simply do not understand the difference between definitorial expansion and ∃-elimination, which are the 2 ways of defining things in mathematics. Some people who might think that we can do away with definitorial expansion in a sufficiently strong set theory, but this is a horrible mistake in pedagogy. Not only would one have to go all the way to class functions in order to get the powerset operation from the powerset axiom, or cartesian product, or Ordinals, or countless other useful things that are not granted as inbuilt primitives, it is extremely misleading to invoke set theory when it is totally unnecessary. For example consider defining oddness over PA, which is a simple but instructive example demonstrating the need for precise teaching of definitorial expansion (i.e. what exactly are the rules governing how to define new predicate/function-symbols).

Again, let me emphasize that although I use symbols in my teaching, this is not necessary. I observe that in the long run the familiarity with logical symbols does make students competent in handling statements of arbitrarily high logical complexity, which is quite unattainable with natural language. However, if one does not have the time (a full semester) to properly learn FOL, then the next best approach is to use natural language phrases together with the above ingredients. For example the following is a valid rigorous proof that follows the above principles.

Define that k is odd iff there is some x∈ℤ such that k = x·2+1, for each k∈ℤ.  [definitorial expansion]
Given any k,m∈ℤ:  [∀-subcontext]
    If k is odd and m is odd:  [⇒-subcontext]
        Let x∈ℤ such that k = x·2+1.  [∃-elim]
        Let y∈ℤ such that m = y·2+1.  [∃-elim]
        k·m = (x·2+1)·(y·2+1)
         = (x·2+1)·(y·2)+(x·2+1)·1
         = ((x·2+1)·y)·2+(x·2+1)
         = ((x·2+1)·y)·2+x·2)+1
         = ((x·2+1)·y+x)·2+1.
        And (x·2+1)·y+x∈ℤ.
        Thus k·m is odd.
For any k,m∈ℤ, if k is odd and m is odd then k·m is odd.

Invariably all students who learn to use a deductive system permanently become capable of perfect logical reasoning (i.e. no more flawed thinking except careless mistakes), including the ability to check their own proofs with confidence, whereas most students who are unable to use a deductive system are also incapable of checking validity of their own proofs. After one becomes good at using a deductive system, one no longer needs to use it explicitly.

There are exceptions; 1% of students who never learnt any deduction system manage to acquire an unconscious grasp of sufficiently many logical reasoning rules covering all those needed for FOL, which hence makes them capable of constructing correct proofs and checking purported proofs. The only thing they would lack is the understanding that their logical reasoning is in fact sufficient for all mathematics. Unfortunately, many professors fall into this "exception" category but are unaware of it, so they incorrectly believe that students would be able to grasp what they unconsciously grasp.

What tips do you have for students to check the validity of their proofs, apart from showing them to a more experienced mathematician, and what approaches can we use to help them spot mistakes and errors, and then correct them?

So my answer is essentially that you need to change your teaching and then they automatically would not need any tips at all to be able to check proof validity.

Note that this does not invalidate the usefulness of heuristics in rapid guess-evaluation of proof validity. But the crucial point here is that the primary and only criterion for logical reasoning is that one follows the deductive rules. In addition, students need to understand that those rules suffice for all mathematical reasoning. (Most do not.) Heuristics are beneficial to them only after that. One useful heuristic is to check every condition of a theorem to see how it has been used (if at all) in a proof and to figure out whether it is actually required or not; if there is a counter-example without it then it must be used at least once in the proof. The deductive rules eliminate conceptual mistakes, but do not eliminate careless mistakes, so heuristics are useful to catch the latter. I just want to emphasize that heuristics must never be taught as a part of logical reasoning.

For additional related resources, take a look at this post about logical puzzles and games and this post about concrete interesting mathematics. You may also be interested in some other stuff linked from my profile page.