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My impression to students majoring in mathematics is, whenever we teach them a theorem, a proof should be given in the class, or at least as a reading assignment. However, how about students not majoring in mathematics? One extreme is, proving everything, treating them as students majoring in mathematics. The other extreme is, not teaching any proof at all, only introducing the conclusions and their applications. Of course one can always teach some proofs while omitting others. Now the question is, what proofs should we teach and what proofs should we omit? What are the factors to consider?

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    $\begingroup$ It's not a binary choice. We can give plausibility arguments or sketches of proofs, or leave out steps and say "a lengthy but straightforward calculation shows that..." In first-semester calculus, I'm not aware of any popular textbook that even attempts to give rigorous proofs of theorems that require the non-elementary theory of the real numbers. $\endgroup$ – Ben Crowell Jan 3 at 23:33
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    $\begingroup$ There may be students who first try a major in engineering, but then switch to a major in mathematics. We may want to avoid the situation where such students have to repeat their math courses, by taking the courses with proofs. Similarly, I have heard stories told by mathematicians saying: I thought I would study physics, but when my math course included proofs, I realized that was what I really wanted. $\endgroup$ – Gerald Edgar Jan 4 at 1:39
  • $\begingroup$ I think the math curriculum already handles this fine wrt calculus. you do a normal calculus course. Then later real analysis. $\endgroup$ – guest Jan 4 at 1:52
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    $\begingroup$ There can also be the case that it is easier to learn the proof stuff when having some familiarity with the topic itself (see many mathies struggle doing theoretical diffyQs without an intro course first.) $\endgroup$ – guest Jan 4 at 1:54
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    $\begingroup$ Middle school students are not majoring in anything yet. Nevertheless, proofs are the standard part of geometry curriculum. Not in the U.S. though. If anything, proofs teach how to build the whole puzzle out of smaller parts, how these parts relate to each other, how some parts follow from another, how to look for the parts that you need to build other parts... It is good just for building a coherent argument in business or politics, not necessarily in "doing math". $\endgroup$ – Rusty Core Jan 4 at 22:40
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I would like to offer a different way of thinking about the question. The idea that you have to do the proofs for maths students tends to come from people who have a fixed idea of what it means to 'do maths' and what a maths degree is.

Instead, I believe the place to start is asking 'what skills do we want the students to learn?' (followed by 'why do we want them to learn these things?'). I wrote down a list (not to hand) of about 20 different possible reasons for including a proof in a lecture. The value of each will vary according to the makeup of the class (whether maths students or otherwise). You might, for example, want to include a proof to give students the idea of how mathematicians prove things (and how this differs from experimental science), or because seeing the proof makes it easier to remember/use correctly the result etc.

Another part of the question is 'why is including this proof more valuable than the alternative uses of the time?' Maths courses tend to be heavy on content, without time for students to really understand what they are being told. Everything you put in comes at the cost of something you take out.

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  • $\begingroup$ Great answer. Moral plus one. $\endgroup$ – guest Jan 4 at 18:17
  • $\begingroup$ @guest But you're not prepared to give me an actual +1? $\endgroup$ – Jessica B Jan 4 at 22:11
  • $\begingroup$ I can't. I'm a guest commenter. Second class citizen. Attaboys and attagurls only. $\endgroup$ – guest Jan 5 at 1:34
  • $\begingroup$ @guest: Entirely up to you of course, but why not register? You're a regular contributer here after all. $\endgroup$ – J W Jan 5 at 13:17
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    $\begingroup$ I think this is a good answer, especially insofar as it does not get lost in, yet recognizes, the traditional mythology of "proof or not", not to mention the problem of quality of proof, and/or issues of persuasiveness of a purported proof... and, then, persuasiveness to whom... E.g., effective explanation for the audience at hand is surely always optimal... as opposed to wishing for a different audience, or, worse, berating one's audience for not being otherwise. :) $\endgroup$ – paul garrett Jan 5 at 23:33
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It seems to me pointless and perhaps even irresponsible to teach results/algorithms/methods without explaining them, but between rigorous proof and adequate expanation there is a lot of terrain. Explanation and proof are somehow two extremes of the same thing, the difference being one of level of detail and formalization.

For this question there is nothing special about calculus. This question is applicable to nearly everything one teaches in mathematics, at every level. (Generally speaking the least useful research talks are those that (try to) give all the details of the proofs ...)

The worst case scenario is the way my daughter was taught to take square roots in middle school. The students were shown a step by step algorithm for obtaining square roots and learned to perform it. No explanation of why it worked was given. The students learn something, albeit very little, by learning to correctly perform the operations and obtain the correct answer, but I am not sure there is not done more harm than good - the same students who can correctly calculate $\sqrt{34}$ to two decimal places cannot say - beforehand - whether $\sqrt{34}$ is bigger or smaller than $6$. Also the students learn this way that mathematics is a formal game with opaque rules, the objective of which is to get the correct answer. The essence of such a results only approach is authoritarian and anti-educational. (After dissecting the algorithm in question, it automates writing the putative square root in its decimal expansion and solving iteratively for the unknown digits - something that certainly could be explained to middle school students, and that could probably profitable serve as a way to improve understanding of decimal representations of numbers. A separate question is whether the finding of square roots should be taught in middle school, but this is traditional and universal in Spain, and I think also many other European countries.)

In calculus the situation is the same. One does not want to simply assert that a function with a positive derivative is nondecreasing. A student who has simply memorized this learns nothing. At a practical level such a student is likely to run into problems understanding critical points. At a deeper level, the student learns to treat mathematics as a purely deductive exercise of purely formal steps. However, the same statement can be explained reasonably well without a formal proof involving the mean value theorem if one has developed intuition relating the derivative to the slope of the tangent line, and it can be explained in such terms to a physicist's level of rigor. For many students, particularly those in engineering, such an approach may be more productive than dwelling on the formal details. On the other hand, as anyone who has tried to do this knows, this is more difficult to do than it sounds - precisely, giving a careful, clear intuitive explanation of the significance of the sign of the derivative without entering into formalities is a difficult task. In fact, I am not completely convinced by any attempt I have seen or made, and so, when I teach this to engineering students, I give both such intuitive explanations and something close to a formal proof. With proper motivation the formal proof is largely accessible and introduces tools useful in other parts of the course. This of course requires having developed previously the language of continuity and diferentiability.

With students from social sciences, computing, or biological sciences the situation is more difficult (generally speaking such students have less interest in mathematics than do engineering students). I think the solution is not to omit explanations so much as to omit content. Better to explain well a few things than to cover a lot of ground without explaining anything. Very little of what is taught will be instrumentally useful for more than a handful of students, and so the goal is to teach a student the skills and background necessary to learn later, on her/his own, whatever aspects of the material might be needed later.

In any case, the classroom is not really the place to give all the details of most arguments. The goal in class is to communicate the main ideas and the motivations. The student who cares about the details is expected to learn them on her/his own, using a book, in problem sessions, or in office hours, etc. (In high school the pace is different and more is done in class, but the situation is not fundamentally different.) A good explanation is a "proof" if it contains the ideas/motivations, and a good "proof" really amounts to making concrete the steps necessary to formalize the ideas and motivations - in how much detail to develop each of those steps is really the hard decision for the teacher - and the answer will always depend on the context of who the students are - but identifying them and pointing them out seems essential if anything is to be really gained from the exercise.

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  • $\begingroup$ Excellent answer. Moral plus one. $\endgroup$ – guest Jan 4 at 18:16
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For the sake of making the question more definite, let's assume we are looking at calculus students. Let's assume we are looking at a generalist class. One with advanced strong high school students or decent college students. Not calculus for business students). But a standard course that meets the needs of physics and engineering students. But also likely has future math majors in it (many will not even be declared by then). But also chemists, oceanographers, and maybe even business students (for schools that don't do a weakened version). But not Caltech or MIT. Something close to the AP curriculum. Standard course of Thomas, Stweart, Granville, etc.

I would say do the proofs you can (don't do ones that use concepts kids don't have...leave that for a later theoretical course). Some, more intricate proofs may be put in an appendix. I would treat the proofs as motivation, NOT objective. The proofs give the kids a feeling that what they are doing is true. And in some cases, they may help solidify knowledge and make people remember something. But they are NOT the objective. Doing problems is the objective. (Like if we were in a war, shooting bad guys would be the objective, not explaining how the rifle works.) I wouldn't even test on proofs much. It's more a form of motivation.

If you want an actual put your finger on it answer my advice is to go with Thomas level of proofs. Not "real analysis in sheep's clothing" Spivak. But a little more rigor than classical Granville or current Stewart. Based on what I have gleaned about your class, Zuriel, between the lines, I think that would work well.

But again...the objective is to teach, drill, and test problem solving. So don't overthink this proofs issue.

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I think there's also an issue of how you present the proofs. As an engineering student—knowing that I didn't need to remember proofs, just results—I often sat in class happily following each step of a derivation, and I could see that because all the steps were right the result must be true, but most of the time I didn't have a clear idea of what the proof outline was—the proof or derivation was almost a magical sequence of steps leading to the result.

But I would definitely have benefitted from understanding the proofs.It would have made the results easier to remember and understand, and knowing how the proofs worked would have given me a means of checking for myself whether I was remembering a particular formula or equation correctly.

So I think: where possible, focus on giving a clear understanding of the main concepts and arguments of the proofs—but for people who will be using results rather than proving them, avoid getting bogged down in too much detail of the actual manipulations.

Though of course there might be students whose experience is the opposite: ones who see the outline straight away and have trouble following details so need to see every step written down. For me, being given all the details was actually an obstacle to understanding.

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    $\begingroup$ Yeah. It's a balance. Ideally you would give the outline and the details. I was definitely one of the engineering students who felt like "show me the damned algebra". Don't trust it unless I see it. Maybe I am more of a math type than I realize. Or at least a scientist. Engineers seem to be very willing to push the "I believe" button. $\endgroup$ – guest Jan 4 at 18:20
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Some lovely answers are here already. I would like to look at an example below the level of calculus. In beginning and intermediate algebra (at a community college), I love working through the proof of the quadratic formula with my students.

Here's why:

  • They are highly likely to think of it as a complete black box.
  • It is the nastiest looking formula they are likely to ever use if they are not going on in math. (Well, standard deviation looks worse, but statistics is sadly full of black boxes for them.)
  • They have seen it before.
  • It is one of the reasons we work on completing the square, which has few uses for those not going on, but is a powerful tool worth learning, if put in a good context.

I use James Tanton's method (using an area model) which avoids fractions. (And fractions raise the anxiety level for many students.) We refer to the area model frequently as we go. His youtube videos on this are here, here, and here. [Note: His method does not always work as described, you may need to multiply by 4 a second time. Watch the videos and see if you can find examples of this glitch. I am trying to figure out which types of problems cause it, and do not have an adequate description yet.]

I think proving this can help them see how math works. Our process allows them to see this proof / derivation as a sensible explanation, and as something they might be able to do themselves, if they cared enough. As we are working on it, I talk about us building this thing ourselves. I do it one day, and the next day, I have them tell me each step. My hope is that this gives them a better understanding of why we prove things.

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  • $\begingroup$ Going a little deeper into the videos: I'm worried that "multiply by 4 here" will feel like a rule rather than a thing they have control over. How do you motivate the "times 4 trick"? It seems like "I want a 16 here so I will add 9 to both sides" is a good way to think about fixing the constant, but "I want to multiply by 4 to fix the x term" seems like a big leap. $\endgroup$ – Chris Cunningham Jan 10 at 0:14
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    $\begingroup$ Odd numbers in the b term are one cause of fractions. You would multiply by 2. But then the a term isn't a perfect square. It seems sensible to me. But I see your concern. $\endgroup$ – Sue VanHattum Jan 10 at 16:47
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I like this comment:

@DanFox: "between rigorous proof and adequate explanation there is a lot of terrain"

(1) I would try to sprinkle in a few very clear proofs of non-obvious theorems. So I would avoid the intermediate value theorem, because it seems so obvious. But proving that the geometric mean is at most the arithmetic mean, $$ \sqrt[n]{ a_1 a_2 \ldots a_n} \le \frac{a_1 + a_2 + \cdots + a_n} {n} \;' $$ for positive reals $a_i$, is certainly not obvious. And usually requires induction.

(2) Another example is described in this response to an MESE question, Good, simple examples of induction?. The line of logic is clear once explained, but certainly not obvious de novo.


      3-coloring


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That’s not a yes or no question. The proper way to teach mathematics is to give all (or all main) derivations, sometimes even different ways of getting those derivations and formulas. The students must understand why the formulas are the way they are. They must have a feel of the logic behind. They should be able to prove and derive formulas with confidence. At the same time some books may be either excessively rigorous or not rigorous enough. Some books may contain too many proofs while others may cut down on proofs way too much. A good balance is needed. Much more serious problem is the lack of sound knowledge of of precalculus. Teaching calculus theorems and their proofs or theorems from linear algebra is hardly acceptable if a student can’t derive the $\sin(a+b)$ formula or prove the law of cosines. Some can’t even derive the formula for the quadratic equation – totally unacceptable but it is common! So, basic stuff and most glaring gaps should be addressed right away, otherwise we are going to erect a building without any foundation that will tumble down in a light breeze.

This is a serious problem in teaching math. It can only be solved by filling out all the fundamental gaps and by filling them properly. It basically means that students should optimally take College Algebra (precalc) and should study it properly, i.e. with understanding all the basic proofs and derivations. Students should also be confident when they reason and make derivations. They should have a feel and a solid grasp of what they are doing. They should be able to debunk false proofs or false steps if those are intentionally thrown at them for confusion. It’s difficult when standard curriculum cuts down on proofs, derivations and explanations. Yet, just following algorithms or doing Schaums problems is not enough in itself. It’s not the right approach to understand math. Just plugging numbers in or following algorithms without real understanding won’t cut it. The balance between the amount of rigor and derivations should boil down to the course books chosen. I never use books that severely lack derivations and explanations. On the other hand a text like Calculus by Apostol might be overkill for students in engineering. It depends on students and the college. We have to choose middle ground. I gave extensive recommendations on books but that post is not available. Good books will have the right amount of rigor and derivations for engineering students.

Still, I'm not going to leave this answer in the dark and I'm going to list a few books. For engineering students a good "dip stick" of how many proofs, derivations and how much rigor is a good balance will be the books on Calculus by Larson or Calculus by Marsden; Linear algebra by Strang or Linear Algebra by (this one is probably overkill) Kolman; Algebra and Trigonometry by Larson or the same title by Stewart (this course is usu. called College Algebra and is a foundation without which no calculus or linear algebra can be studied because without it everything is just rote-learning in the worst sense of the word). That's the frame of reference in terms of derivations and proofs. It' s the right balance for engineering students. What is important here is not to omit derivations, proofs and explanations in these books. Said books already contain derivations/proofs in a very digestible form and in the minimal amount. Just doing template problems is a huge malpractice. Derivations and explanations are also part and parcel of studying math just like doing standard problems. Derivations should never be skipped or turned a blind eye to even if they are in the book's addendum! Your students are better off by skipping some chapters rather than explanations and derivations/proofs.

All the necessary proofs/derivations are in respected books I gave. Okay, you can cut down a bit more on them but I'd rather not. It's already the bare minimum. I suggest cutting down on chapters and material, not on explanations/derivations/proofs.

Important note. All students say they know precalc. but they never do! All the know is a bit of selective rote-learning. And then they plunge into calculus, linear algebra, etc.

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