## Teaching proofs in the era of Google

Dear members,

Way back in the stone age when I was an undergraduate (the mid 90's), the internet was a germinal thing and that consisted of not much more than e-mail, ftp and the unix "talk" command (as far as I can remember). HTML and web-pages were still germinal. Google wouldn't have had anything to search, had it existed. Nowadays Google is an incredibly convenient way of finding almost anything -- not just solutions to mathematics problems, but even friends you lost track of 20+ years ago.

My question concerns how Google (and to a lesser extent other technological advances) has changed the landscape for you. Specifically, when you're teaching proofs. More details on what I'm getting at:

A "rite of passage" homework problem in the 2nd year multi-variable calc/analysis course at the University Alberta was the Cantor-Schroeder-Bernstein theorem. In the 3rd year there was the Kuratowski closure/14-set theorem. It's not very useful to ask students to prove such theorems on homework assignments nowadays, since the "pull" of Google is too strong. They easily find proofs of these theorems even if they're not deliberately searching for them. The reason I value these "named" traditional problems is primarily that they are fairly significant problems where a student, after they've completed the problem, can look back and know they've proven (on their own) some kind structural theorem - they know they're not just proving meaningless little lemmas, as the theorems have historical significance. As these kinds of accomplishments accumulate, students observe they've learned to some extent how an area develops and what it takes in terms of contributions of new ideas, dogged deduction, and so on.

I'm curious to what extent you've adapted to this new dynamic. I have certainly noticed students being able to look-up not just named theorems but also relatively simple, arbitrary problems. After all, even if you create a problem that you think is novel, it's rather unlikely that this is the case - sometimes students find your problem on a 3-year-old homework assignment on a course webpage half-way around the planet, even if it's new to you.

As Jim Conant mentioned in the comments, this is a relatively new thing. When I was an undergraduate, going to the library meant a 30-minute walk each way, then the decision process of trying to figure out what textbook to look in, frequently a long search that led me to learning something interesting that I hadn't planned on, and frequently not finding what I set out to find. But type in part of your problem into Google and it brings you to the exact line of all the textbooks in which it appears. It brings up all the home-pages where the problem appears and frequently solutions keys, if not Wikipedia pages on the problem -- I've deleted more than one Wikipedia page devoted to solutions to particular homework problems.

Of course there are direct ways to adapt: asking relatively obscure questions. And there's "denying the problem" - the idea that good students won't (deliberately or accidentally) look up solutions. IMO this underestimates how easy it is to find solutions nowadays. And it underestimates how diligent students have to be in order to succeed in mathematics.

Any insights welcome.

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The difference is that it is much much easier to find answers using Google than it is to pore over lots and lots of library books. Using Google, an answer is often a few mouse clicks and keystrokes away, whereas finding an answer in a well-stocked library requires diligence and good luck. After all, it's not so easy to figure out which book to look in for an answer, and it may be the case that the relevant book has already been checked out by someone else. I speak from personal experience. – Jim Conant Oct 9 2010 at 11:26
Well, one way to do it is to not name the theorems. Break the theorem up into a bunch of small lemmas if necessary, recast all of them as questions without naming them. Only tell the students they have proven X theorem after you hand back the homework. (Yeah, this is only a stop gap that will presumably break if "semantic web" really arrives---if it arrives at all.) – Willie Wong Oct 9 2010 at 12:25
"I've deleted more than one Wikipedia page devoted to solutions to particular homework problems." - interesting... I presume there was at least some resistance to this from people other than the page's author? – J. M. Oct 9 2010 at 23:04
If the questions are set in such a way that looking for solutions takes a considerable amount of time, then the students will spend some time in an attempt to solve them, before they start looking for solutions. If after several hours of not succeeding, they decide to look up the solution, that's fine by me. Besides, I see myself more as a service provider than as their parent. I offer them assistance in honing their problem solving skills. If they decline, that's fine by me. – Alex Bartel Oct 11 2010 at 1:42
R.L. Moore demanded from his students a pledge never to open a textbook. Presenting a textbook proof as your own was out of the question, as it would be way too slick to be believable, and there are anecdotes of him soundly rebuking students who thought they could get away with it (stopping them at the first sentence, basically, and so denying any payoff from the transgression). You could be the R.L. Moore of the internet age. – Thierry Zell Dec 2 2010 at 3:51
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How would you teach anything in an age when the "arcana" or guild secrets had been made public? Well, you would teach. And you would not ask questions that had answers that could be called "answers" on the basis of some look-up.

I'm not involved in such things these days, but when I was, I wrote my own questions for students. I did not expect to take questions down off the shelf from anywhere, and for that reason my questions perhaps had a few rough edges. But then I was in an institution that actually thought teaching quite demanding.

It is an answer, though it probably betrays a lack of sympathy: if you don't want students simply to look up the answer, don't simply look up the question.

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Sound advice,if stated without much mercy. – Andrew L Oct 9 2010 at 7:08
But especially with beginning mathematics, there's an element of muscle building for which you need a large number of relatively simple problems. This is true in algorithms as well: it actually helps to work out 5-6 instances of dynamic programming to get a feel for it. So while it's good to create your own problems, this doesn't work for the muscle memory section of the homework. – Suresh Venkat Oct 9 2010 at 7:44
Hi Charles, even problems you might consider novel are frequently easy to look up. As Google is cumulative, every problem you create by-and-large becomes indexed by Google before you teach the same course again. So if this is a strategy of any sort, it is one that makes your life more difficult the longer you live by it. – Ryan Budney Oct 9 2010 at 17:29
All we're talking about is making it easier to think through the problem than search for it, though. I don't think it's so hard to put a false moustache on an old chestnut. – Charles Matthews Oct 9 2010 at 18:18
Charles, I agree with you. Using and reusing questions from or based too closely upon textbook questions is "looking up the question". It would be nicer to always construct new questions; however, at the beginning level classes, or even geometry for high-school level students), there are always students who need to be taught the early steps. There are only a finite number of variations possible on the ways to teach the early steps. Thus I cannot denigrate those who use questions for which solutions are available. Students have to want to learn. Teachers can't make them want to learn. – sleepless in beantown Oct 9 2010 at 18:57

There is a technique for teaching the meaning and understanding of important mathematical theorems that is highly dependent on computer technology that I have found particularly effective, and it is unrelated to Google. Namely, some theorems are of a constructive nature; they say that you can reconstruct certain kinds of mathematical objects up to isomorphism algorithmically from associated critical data ("invariants"). These are often quite abstract sounding and students often have a difficult time really understanding them in a more than formal way. But sometimes it is possible to get the students to actually program these algorithms (using one of the three Ms---Matlab, Maple or Mathematica) and when this is the case I have found (and the students agree!) that the process of actually developing the algorithm as a program gives them a deeper understanding of the theorems in question. This may sound rather abstract, so let me illustrate it by an example. The classical course on basic differential geometry, often called "Curves and Surfaces" has as its heart three core so-called "Fundamental Theorems", the fundamental theorems of (i) plane curves, (ii) space curves, and (iii) surfaces. These three theorems say respectively that you can reconstruct these three types of objects uniquely, up to rigid motions from a knowledge respectively of their (i) curvature (as a function of arclength) (ii) their curvature and torsion and (iii) their first and second fundamental forms. I have taught this course three times using the above technique, once in Taiwan, once at Brandeis, and once at UC Irvine, and as I suggested above, I felt the results were far superior to the "old way". I have made available ALL the material I used when I taught the course at Brandeis in 2003 (course prospectus, lecture notes, exercises, programming projects, how to get started programming, etc.) and you can find all this material here:

http://vmm.math.uci.edu/Math32/

You are welcome to use it either for learning on your own or for teaching a similar course.

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I want to repeat the following from the question above: "the process of actually developing the algorithm as a program gives [a] deeper understanding of the theorems in question". Also, mathoverflow.net/questions/26950/… – Steve Huntsman Oct 9 2010 at 15:19
I've been a TA for a multi-variable calculus course that was quite computation-intensive -- getting the students to use things like Kantorovich's theorem to prove they had approximations to solutions of equations. This IMO is a great strategy but it's also not very universal. – Ryan Budney Oct 9 2010 at 17:33
I like using the big ["O" for octave](en.wikipedia.org/wiki/Octave_software) and the large ["Scilab"](en.wikipedia.org/wiki/Scilab) as they are free open-source software packages capable of matching the 3 M's in ability. Sage en.wikipedia.org/wiki/… can beat them in many aspects. – sleepless in beantown Oct 10 2010 at 1:15
I know this is quite an old post but ... ?! Three M's !? What about Sage? You can program in Python which is easier to learn and better suited for math and moreover it is free so any of the students can put it on their computers! – Matt Dec 5 2011 at 6:18
@Matt You are completely correct, Matt; the fact is that at the time I wrote those remarks I had never heard of Sage, and only became aware of it a few months ago. – Dick Palais Jan 22 2012 at 18:14
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A student who has looked something up on google and copied it out has probably learned something from doing this, so why worry? In fact, may be better for them than copying the answer from a friend.

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It's a little like losing one of your favourite toys, there is an emotional element to it. I very much like the feeling that it is "me against the problem". Sure students will have learned something but they'll have lost the satisfaction of knowing they came up with an idea -- their idea was to type their problem into Google. – Ryan Budney Oct 9 2010 at 17:44
Richard, I agree with you. Particularly if the student has actively tried to solve the problem first, and run into a problem or an insurmountable level of frustration after whatever amount of effort was reasonable to them. Even more so if reading and copying the solution really does teach them the "trick", or how that difficult step could have been conceived of if they had tried for a little longer or tried when they weren't so tired.. – sleepless in beantown Oct 9 2010 at 18:48
Kind of assumes the answer was correct. Don't try this for the contractibility of the sphere in Hilbert space. The malicious solution would be to post fallacious answers outnumbering the real ones, making Google the largest multiple choice test ever. – Charles Matthews Oct 9 2010 at 18:54
@sleepless, if they find unsurmountable levels of frustration, then either they are not taking the correct course or the problems are badly chosen. Having the students google for solutions in either case is quite orthogonal to any kind of sensible approach to the problem! – Mariano Suárez-Alvarez Oct 9 2010 at 22:24
@Charles Matthews, the correctness of what search engines find for you has always been in question. Try to look up a fact which you are certain is true; there will often be many hits for sites that will say the exact opposite, even if the fact in question is easily verifiable by going back to the original source. Students and researchers who take a search engine answer page as a valid and true result are making a very unreasonable assumption. – sleepless in beantown Oct 9 2010 at 22:26
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I think this is a very valid question. My response is very simple and it's best phrased by another question: How is the situation currently with Google different from previous eras when students have access to a well-stocked library?

At major universities, where just about every major text on any subfield is present in the library, we've all seen mathematics students with piles of texts and monographs looking for proofs. By the time they get to that level, they're supposed to understand that building mathematical muscle is done by banging your head against problems and proofs for weeks until some insight is gleaned. If they really are serious about being mathematics majors-and someday mathematicans-they'll willingly refrain from such things until all else fails. And if they look at other sources-it will be as an absolute last resort and most of them won't look at the full proof, just the beginning to get a "hint" of how to get started. Surprisingly, most of the time,this is more then sufficient.

Otherwise, they fall into the category of people I called in my blog "constudents" -- a hybrid term from conmen and students -- students whose primary concern is grade excellence, regardless of whether or not it's earned. Such students hand in completely copied answers; claiming complete authorship without remorse and fully expecting perfect grades. Typical constudents are premeds, prelaw students, pre-engineering-any discipline with the promise of substantial financial and/or sociopoltical reward and puts absolute emphasis on grade level rather then actual talent. Sadly, they are NOT one and the same, regardless of what inexperienced people outside of academia think. But I seriously digress, I apologize.

My point is that Google should be irrelevant to students who truly wish to become mathematicians since they'll realize such shortcuts will only hinder their development as mathematicans. But if you're really worried about such things, here's a possible answer: In my graduate courses, we were given very difficult, lengthy problem sets and encouraged to work in groups. This worked very well and in my experiences, we students never looked up answers.

Well, we'd usually look up whether or not the answers we came up with ourselves were correct or not.

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Maybe a premed saw it. – Richard Borcherds Oct 9 2010 at 17:16
Yes, stay healthy, and don't even think of litigation. – Charles Matthews Oct 9 2010 at 18:22
And I don't have health insurance,so since I don't live in a civilized country with garanteed health care,it doesn't really matter how I live,does it? – Andrew L Oct 10 2010 at 2:03
I think that one should have some sympathy for these people. I do not believe they are all really so aware that their habits actually hurt them (besides being extremely irritating to an educated observer). I think they often feel a lack of confidence that their thinking about math really will amount to anything. It's easy to have this attitude about math when you're a mathematician, but then how many of us were really so sincere when we endured courses in other, less appealing subjects? And at any rate, isn't it better to try to help rather than to foster a disdain? – Phil Isett Dec 5 2011 at 3:54
The trouble is that you indirectly make life difficult for the serious students if you make life too easy for the constudents. The serious student may know the right way to do things, and may want to do things that way, but if doing so will mean failing the course because all the constudents get much better grades with much less effort, then the serious student is put in a quandary. You can't become a mathematician if you fail out of school. An analogy may help: Professional athletes who want to stay "clean" will have trouble staying competitive if there is rampant steroid use. – Timothy Chow Dec 5 2011 at 15:55
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In most Italian universities, grades are not based on homework but on a final (written and/or oral) text. There are different levels of what you are allowed to use during the written text (textbooks, notes, a limited amount of notes, or plain nothing), but definitely anything that can connect to Google is not authorized.

This would solve the problem completely, wouldn't it?

In fact, there is no (mandatory) homework at all: students may skip all the lectures, study on their own, and arrive prepared at the final test. When I first learned how heavily universities abroad rely on homework problems (and how many TA hours are devoted to grade them), I was shocked. To my eyes, this method looked like high school, baby-sitting students through the coursework.

Granted, with the Italian system, some students get lost around the way, like in "drink and play World of Warcraft the whole day and then fail miserably". But those who don't, they learn how to organize themselves and work autonomously towards a goal.

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This is my answer too and I am surprised it was not posted earlier. – Timothy Chow Dec 5 2011 at 15:59
This reminds me of a story I heard while an undergraduate. A professor told me we would have an oral final exam, and that such and such would be expected to be known. He then recited a tale of a student he had, who while standing before him during the final oral, was asked to state the Fundamental Theorem of Algebra. The student scoffed and said "I can look that up in a book anytime I need", to which the professor replied, "unfortunately for you, you need it right now". – Steve D Mar 21 2012 at 18:14
The primary thing I don't like about the Italian system is that exams have strict time constraints, so they generally can't give the students enough time to be very contemplative. A homework assignment you can keep on coming back to, over a week or two, trying new ideas. Exams and Google searches in a sense both make it easier to be less introspective. – Ryan Budney Dec 6 at 9:07

The era of searching on the internet has increased the ease of looking up answers and solutions to textbook questions, and so perhaps it has increased the percentage of students who go to the trouble of looking for a shortcut rather than working out the problem on their own. But I do not think that it is a new problem.

There have always been resources to turn to which had solutions to problems:

• copies of the teacher's editions (for the elementary algebra, calculus, and physics classes)

• workbooks which provide problems and detailed solutions

• archives maintained by groups (fraternities, sororities, societies such as student mathematics groups or physics clubs or biomedical engineering clubs)

• solution manuals are often published to go along with a text, either by the same publisher or a competitor

• geometry has always (in the last few centuries) been taught in a programmatic step by step fashion, with proofs building up on proofs. There are multiple centuries (millennia) of history and texts to look at and study from. Sometimes simply by looking ahead in the book, it's possible to get a clear answer on what the structure and details of a proof ought to include.

In all of these cases, the students are either (1) cheating themselves out of exercising their brains and coming up with a solution on their own, or (2) helping themselves past a hurdle which they could not overcome on their own and which they've decided to bypass by taking someone else's answer. Only the student can know if they've spent hours or days working at it and found it too frustrating, or not worth the effort of waiting on it / sleeping on it / approaching it again on another day.

So, yes, the internet and search engines have increased the fraction of students who might use a shortcut instead of doing the work themselves. However, the good students (the ones who would want to go on to solve problems on their own initiative, or try to solve the same ones again in different ways, perhaps even become mathematicians and scientists) will probably not be the ones who would takes those shortcuts and bypasses around the obstacles. Hmmm, I realize this is sounding like I'm saying not to care about the ones who would cheat. The problem with teaching, and caring about teaching, is that a teacher would like students who want to learn and find joy in knowledge and problem solving. However, teachers do not get to choose their students, as they did back in the days of Hippocrates when doctors literally could choose or refuse to teach particular students. Teachers get the students who sign up for their classes. Teachers cannot improve the motivation or attitude of their students. Students have to be responsible for their own education at some point. We can provide lessons or be the water; the horse has to actually drink.

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 I don't think anyone here can improve on that answer.Well said,Sleepless. – Andrew L Oct 10 2010 at 2:05

One advantage of the current era is that several proofs of the same result can be found. Reading those proofs and comparing them is very instructive, both when only details differ and when different approaches are used. Just stating the correct query in Google or MathSciNet is instructive.

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Representing someone else’s work as one’s own is plagiarism.  Students can be expelled for it, and tenured professors can be fired for it.

This issue is addressed eloquently and emphatically in Section I of the Ethical Guidelines of the American Mathematical Society.

It is true that it’s easier to commit plagiarism now than before computers existed, just as it’s easier to rob a bank now than before automobiles existed.

Assuming that you’re permitting your students to look up solutions, and that they are providing proper citations of sources as necessary, there might be considerable pedagogical benefit to this exercise.  For instance, they might discover that certain useful ideas and techniques occur over and over in solutions to some sort of problem.

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Although it might be easier to commit plagiarism now, it is also easier to detect it. At least, that's why my very scant experience with this problem tells me. – José Figueroa-O'Farrill Dec 29 2010 at 20:47
"[T]enured professors can be fired for it [plagiarism]." This is technically true, but it seems that in practice it almost never happens. When I was an undergrad, it came out that a tenured prof had stolen a graduate student's work in the most egregious way. The result was that he was "censured" (I don't know what this means exactly; I can't help but picture the Klingon ritual wherein they cross their arms and turn their backs on you) and was, for a time, not permitted to have any graduate students [!!]. Does anyone know a single instance of a tenured math prof being fired for plagiarism? – Pete L. Clark Dec 5 2011 at 5:42

Actually, I think it is rewarding to look some things up, even if you use Google and Co.. For a student it is also a psychological effect to find solutions on the Internet. Namely, if you have some problem in symplectic geometry courses, these problems are normally rather specific, so you will have trouble finding solutions on the web easily. However, when using Google books, you can easily find parts of books which may contain relevant information for your proof or even give a good starting point for a proof. If someone simply uses the Internet to copy a proof then he/she will have problems to really understand mathematics. But, admittedly, there are proofs that I haven't understood or, this more often the case, wouldn't be able to reproduce offhand. In this case, the internet proves to be very useful, especially if you also have students that try to autodidactically learn some further topics in mathematics. Personally, I would design problem sheets as follows: -Make 3 to 4 easy problems which simply consist of getting familiar with the definitions and the rules in the respective field. -Then make one problem that is rather technical and requires the student to make some longer steps in proving the statement. These calculations shouldn't be too complicated as otherwise the student will probably lose patience and simply look the solution up. -Then design 2-3 problems that are more far-leading and require using calculation rules, and a bit creativity. Still, they shouldn't be too complicated.

Why am i always telling you about the complexity of the problems? Well, at least in Germany, we have only 30% of the students obtaining a degree in mathematics. I don't know how things are handles in the U.S. or elsewhere, but most of our students are frustrated becuase they simply don't find a starting point for the exercises. And this shouldn't happen.

I think, most of the students have the will to solve problems on their own. Especially mathematics and physics are subjects you study because you're passionate about them. Biological research has shown indeed that motivated apes are more thankful and curious then demotivated apes. I think this applies to students as well (and to us as well). The typical student involved in a biologically complicates process. The post-adolesence. The years 15-30 are the years where you are requires to pass a lot of tests. And for doing so you have to be motivated and self-confident. At least in Germany, for a lot of students, this is a problem (But I don't think it's much different elsewhere, at least I hope so, because otherwise, we're doing something wrong here).

Most cheating can be avoided if questions are motivating and asked in sch a manner that the first ones are easy to answer, and the following ones increase slightly in diffculty.

This is my opinion. I talk to a lot of students in physics which suffer from depressions because they believe they won't get anything. This belief is apparenbtly that deep that they simply copy homework or use Google. Mostly, when these students continue their studies they are very succesful later on. But a lot of them simply stops studying math and physics. A lot of potential is wasted here.

All I've written may sound a bit offtopic, but I think that there is the real problem. If you motivate students to think on their own, they will rejoice in proofs and all that. This was also my (personal) experience.

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As many people pointed out here, copying proof of well-known results is not a new problem. Just that whatever it took days to do it in the pre-Internet age would only take minutes today.

My suggestion is to improve the way you grade the student's homeworks. When you give them the problems, you tell them to give you step by step detail proofs. The solutions they can find on Internet usually are not detailed enough. You can tell from the sheets they submitted whether they absorbed the answers and re-wrote them in full details or simply straight copying.

Many years ago (stone age), I took a graduate math class. I found some homework problems in the Schaum's series book, I copied the answers. I got a C grade back. I went to the professor to ask what was wrong. He showed me his Schaum's book. Next time, I took time to re-write the answer to the extent that he knew I did spend efforts. I got A.

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