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How to recruit PhD students when industry pays so much more?
I'll be starting as a tenure-track professor in a few months where I'll be doing research on software developer tools. I have seen that it is extremely difficult to recruit good PhD students in this topic area (even at elite schools), since most of them are heavily recruited by industry. Understandably, it is hard to say no to 120k+ USD salary straight out of undergrad.
How should I go about attracting talented grad students in a topic area that has to compete with such high-paying companies? I do have funding for these students but it isn't anything like industry.

Please read my similar question here: https://academia.stackexchange.com/questions/103171/fear-of-missing-out-how-to-maintain-a-good-attitude-towards-research-in-academi

Are you sure you are competing against industry and not other schools or research groups? I ask because there seem to be plenty of grad students pursuing PhD programs in top 50 schools (esp in CS).

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I love the amount of indignation attached to this simple, true observation and related, relevant question. I wonder at the ratio below of fields to which this challenge applies vs fields to which it does not... also, I am curious, @Austin Henley, why didn't you take the 120k?

@Industrademic Freedom! I wanted to work on what I wanted and to explore projects that interested me. I knew that the industry jobs would still be there if I ever changed my mind. I like to think of academia as leading a startup where I have funding for ~5 years, and I don't have to be concerned about short term returns on investment.

We agree on this, and it may be the core of a useful answer.  I considered not going to grad school, and again considered quitting with my masters degree. Freedom to pursue my own interests kept me in both times.  Getting the grad student is only one step: you must keep them. Freedom, and the love of it, seem like the most likely argument that spans both decisions...

You can maybe improve your situation by actively creating research topics that are relevant to the industry or collaborations with companies. Those do not need to be software companies, just to show that your field can be applied in real life.

The person who wants to go into industry is a different sort of person than one who wants a PhD. That isn't a universal, as some people want to do one to enable or enhance the other, but it is pretty generally true. You are likely a pretty good example yourself, so look at how you differ from those among your peers who chose as you did or otherwise.
Therefore, think of the possible pool of candidates as bifurcated and try to appeal to those characteristics that are more likely to appeal to those in the desired half (well, less than half, probably).
Future professors are likely more interested in ideas and personal growth and less interested in money. Both may want to do something to improve the world, of course, but in different ways. The potential PhD is more interested in deeper issues and longer term results than the general public. Future professors, especially, are obsessed with ideas and their development.
Another positive aspect of the professorate and the desire for a PhD is the ability, in the large, to control your own schedule. You work all the time, of course, but you get to decide when to work and (mostly) what to work on.
The people you probably are not going to attract are those with heavy life responsibilities already, and who really need that paycheck. On a more positive note, they are also more likely to want to do something now, rather than to work toward more distant goals.
If you actually have access to some of the people you want to attract, then introduce them to interesting parts of your research and those of colleagues. Take them to an advanced class that you teach in some esoteric subject - maybe even one for which you are the premier source in the world. That is very cool. If you have research labs and advise advanced students, take them to a lab and ask them to contribute - ideas if nothing else. Introduce them to your advisees and assistants. Show them how wonderful it is to work with other smart people on important problems.
When you speak to groups, emphasize the excitement of developing new ideas and how a new PhD is, at that moment, the world's foremost authority on some (perhaps small) thing.
Finally, convince them that they will never be bored and that if they work with you, you will never waste their time. Then live up to that promise yourself, of course.

This is a nice answer (upvoted). I do have one objection- in the third paragraph, you seem to be equating PhD candidates with future professors. This is a false equivalence, and it may be prevalent, but it is still wrong. One doesn't do a PhD to become a professor, neither is one entitled to go down that path because of a PhD. Many PhD holders will go on to do research but not become professors.

@user153812 That may be country-dependent, but I would say the career path PhD - Professor is the rule in Europe or, if you want, that a PhD is useless outside academia.

@user153812, you are correct, of course. I focused on that since my guess is that the OP is also focused there. See my second sentence, also. Lots of google-folk hold doctorates, of course. Ditto other big companies with serious research departments, IBM, Oracle, etc etc

I think some might also need a little help or a little nudge.  Perhaps asking undergrads if a graduate degree is something they desire and if so, become an unofficial advisor on the things one should do to get into grad school.  If their desire is sincere they should only need a little guidance to help navigate the waters and will be less tempted by industry.

I kind of disagree with the sentiment of "Well just give up on the pay, try to focus on the other parts." Your incentives are probably selecting for the worst candidates (those who couldn't get the $120k+ salaries). I'm skeptical that there exist that many people who are so attached to the perks of acedemia, that they would take what, a $50,000/year hit? Possibly more?

@Miguel- Interesting. What about jobs at national labs, R&D at corporations etc?

@Miguel No, it’s not “the rule” by any objective measure. Most PhDs don’t become professors, by a large margin (nor is a PhD useless outside of academia, even if you just count it as a qualification rather than the — rather unique — experience gained).

@KonradRudolph I would like to see those objective measures. In Spain it is certainly a common, if not objective, belief.

@user153812 The size of national research center is not comparable to Universities, but, sure, I include those into the general term academia. R&D at private corporations is exceptional in Spain, to say the least.

@Miguel I seriously doubt that: the numbers simply don’t add up. Anyway, in the UK 56% of PhDs leave academia. Of the remaining 44%, only a small fraction completes the tenure track. I couldn’t find any more detailed numbers but a back of the envelope calculation suggests that less than 10% of PhD graduates end up as professors.

@Miguel How many people graduate with a CS PhD in a typical year? How many new tenure track faculty members are hired in a typical year? In the US, the former number is often fives times the later number. Those numbers need to be the same for all PhD students to become professors.

@Stella Biderman The very concept of tenure track is radically different in Spain from the UK-US perspective. The point is that an abismal majority of PhD students in Spain aim to stay at academia. So, in the context of the OP, any encouragement to pursue a PhD because it will be useful outside academia, would be misleading in Spain (and I guess that in most of continental Europe, with varying degrees of certainty).

@Miguel “Most PhDs want to go into academia” is a very different statement from the one you started with. It might be the case that in the US most PhDs want to go into academia, but in practice less than 10% actually do.

@Stella Biderman It is, almost literally, by tollendo tollens :)

@Miguel Modus tollendo tollens says that P->Q and NOT Q together imply NOT P. It has nothing to do with wether or not “most PhDs want to go into academia” and “most PhDs do go into academia” are the same statement. Those two statements are very obviously different, and no amount of Latin obfuscation will change that. In any event, you don’t have any implication that you’ve demonstrated.

To never be bored is difficult to promise. ;) Also PhD studies usually contain some mandatory pretty boring stuff. But you can promise that he will be learning such cool things he has only dreamed about during BSc & MSc. Stuff which you almost never get time to learn in any private company.

@Miguel Whether something is a “common belief” is unfortunately very different from whether the path PhD→Professor is “the rule”. (Consider this article, though it's focused on the humanities and if you have better stats for other fields I'd like to know.)

@StellaBiderman not all countries have same tenure track system.

@ShreevatsaR Again, a UK-US based study. My point is that the OP should be tagged for context.

@Miguel Do you think that the statistics are different in Europe?

@ShreevatsaR I am completely sure. To begin with, the structure of tenure is radically different.

While there are people who would only go into industry and never Academia, and persons who for which the reverse is true - most capable researchers would not pursue PhDs if they were not paid any salary/stipend to support themselves while doing so, and a great number of them would prefer academia over industry if the pay wasn't so low and the stable employment prospects so slim. So, -1.

"The person who wants to go into industry is a different sort of person than one who wants a PhD". I know you are generalizing, but I don't accept this axiom. Instead, you can generalize people as falling along a spectrum, where only the extrema could be easily classified as Camp PhD or Camp Industry. For the majority of people, the decision is difficult and marred by uncertainty. If you model people as a spectrum, it is easy to see how lucrative industry salaries can push people over the edge.

@ScottLawson, true enough, but my answer remains the same. Pitch your appeal to those most aligned already just to avoid wasting time and effort. The question wasn't about "why this happens", it was about attracting candidates. So, yes, high salaries are attractive, but not everyone is primarily motivated by money. In fact, one reason industry salaries are so high is that the academic life (and research in general) has its own attractions and so money is used to woo the best graduates into industry.

An individual's intrinsic desire to do a PhD is not infinite. When there is a pay discrepancy between industry and academia, you are going to lose people who would have otherwise been stellar PhD students. The larger the pay discrepancy, the more people you will lose. If the discrepancy is small, then sure, you can claim a "person who wants to go into industry is a different sort of person than one who wants a PhD" but I posit that your generalization breaks down when the pay discrepancy is sufficiently high. In your answer, you seem to gloss over the importance of this.

@Miguel BTW I never followed up after asking my question, and if you have any statistics, I'd like to know, but from the references I could find (collected here) it appears that in Spain, of many thousand science PhDs surveyed, about 37% were working in universities. Just thought you may like to know.

Remember that a PhD is intended to be a training position, not a job. Unfortunately, too many supervisors see PhD students as workers rather than as trainees in education. To get excellent PhD students, you need to convince undergrads that they personally will benefit from the training you will provide (rather than just that they will have the 'opportunity' to do lots of work for you). You cannot compete on money, and it is unlikely you can compete on non-monetary benefits either (what university offers free food, free transport, free gym, high-quality office space...?). But perhaps you can compete on the training you can offer.
In your context, this either means:

Find bright students who are not computer scientists and offer to train them to be computer scientists. For example, take on a talented biologist, sociologist or physicist who is motivated to learn your field.
Design a cutting edge project that will push the limits of the field. Such a project will put the student in a position to apply for competitive faculty positions, a higher level entry to a company or to start their own company perhaps.

Of course, this all depends on the amount of effort you are willing to put in yourself. If you just want an assistant or someone who wont get in the way whilst you work on your own things, then you will need to accept a lower standard of student.

To add to this... My personal recommendation would be to grab engineers and mathematicians since they already have a lot of the relevant skills

On the contrary, in most cases - being a PhD candidate is very much a job, in which you carry out work, there are goals (although they're flexible), there's a salary (even if it's not called that), you're required to show up and contribute to group or PI efforts et cetera. True, there's more of a training aspect to it than many positions outside academia, but it does not lose its character as employment. This issue is the focus of intense collective-action and labor-law battles in several countries in recent decades; see this story for example.

@einpoklum is right, but I know no position in industry where you can get even near the amount of time to specialize your knowledge as you can when a PhD student.

@mathreadler: That's true; but I would compare the junior, pre-PhD researcher to his/her senior colleague, the post-PhD researcher. They are similar to each other much like any junior or apprentice craftsman and his/her senior.

In what aspects are they similar? All crafts publish papers? All crafts apply for grants?

I am in the process of getting postdoc job in bioengineering field. Industry jobs are definitely an option here, so I have to consider why exactly I want to work in academia. I can share few reasons, hopefully it will help you understand the process from PhD student's perspective.
Why PhDs in engineering are willing to work for 5ish years at about 50% of industry wage rate?

Lucrative project that will get them a faculty position. You can't get a "University Professor" title and your own lab in industry (vanity+independence)
Good project that will build diverse skillset. In the industry job you will be pigeonholed into limited skillset. In academia you can learn hardware, software, wetware (molecular bio) all on the same project. Some people like that flexibility of learning
Interesting project. My postdoc project will be either unique and advanced method (2x-10x current popular method) or another unique bio/translational project. Even if I don't make it in academia and quit, I will be able to say "I worked on this craaazy project" at parties. In software industry you are either prohibited from talking about work or work on some silly boring stuff like new billing system.
You can potentially negotiate more independence. In academia work is abundant and good people are rare, hence good student will be able to ask for more independent project.
Big name university that will potentially land you higher-paying industry job. Imagine working as software postdoc at Stanford versus NoName Uni. In Palo Alto the density of people is incredible, hence more potential to land very good software job (Google), especially if you as as PI know right people.
Some medium-skilled people would probably appreciate laziness of academic job and lack of competition. I am not sure if that is good or bad for you.

#3 is possible in the industry too. A bright fresh out of undergrad doesn't have to work on a billing system if they don't want to, they could work on a self-driving car instead.

@DmitryGrigoryev I see your point. I don't have enough knowledge of industry, but can imagine that "working on self-driving car" will mean to some extent writing billing system for the self-driving car company. My point is more stem from my experience with academia, compared to industry jobs that I've seen

@DmitryGrigoryev as someone who works on self-driving cars, i can assure you it is not as flashy as it sounds. 99% of it is good old software engineering and compatibility wrangling, like any other decently-sized industry project.

@taylorswift And as someone who stayed in a research lab for some time I can assure you that it's not as flashy either. Unless you have students or assistants to do the ground work for you, you'll spend a lot of time preparing experiments, collecting data and writing reviews of existing work. Have you ever tried to repair your own car? Test instrumentation task is just as tedious and dirty, plus the parts you have to install don't quite fit.

A true investigator is moved by knowledge and curiosity.
This is what you ought to focus on. Unfortunately there are too many PhD candidates facing research as a job, under the wings of professors who see themselves as some kind of entrepreneurs running a start-up venture.
Consider yourself blessed as in your field you are more likely to attract those who are interested in learning and investigation, the true PhD-seekers. Now, if you're truly invested in your field of research surely you can attract other hungry brains to your cause. Never try to buy a student, for you will sell yourself in the process.
UPDATE:
A note to all commenters emphasising on the fact that a PhD student needs money to survive -- this question is not about not paying salaries, nor does it imply that the OP is offering a meagre salary. I assume his students are perfectly able to pay for their bills and grab a beer.
It directly discusses the quality of students seeking for the highest possible salaries.

are you a professor, PhD student, PhD grad in academia, or PhD grad in industry?

@aaaaaa PhD grad in academia, yet perhaps since tomorrow a professor.

Congrats! I still dont think you can sell science to people finishing college, it is a bit too late

@aaaaaa Perhaps. I bought it then, and did I pay for it...

"PhD candidates facing research as a job, under the wings of professors..." or maybe trying to pay the bills. dunno. as an idea.

there are also people who are "true investigators" AND need money. Please don't disregard that not everyone in academia is a single 21 years old. Also, everything is a factor when choosing path in life. Following this logic, working for jerk face Nobel laureate should be considered privileged not abuse

@aaaaaa I understand OP's question is about those seeking the highest salary, and not about an absence of it. If that Nobel prizer can offer enough enlightenment ( status being irrelevant), many obsessed minds might think putting up with some jerkiness is a cursory price

I'm very moved by knowledge and curiosity. I am also moved by the need to pay the bills and put some money aside or support my family. I am also moved by the state of barely making ends meet in fixed-term contracts on soft-money funding.

@einpoklum I suggest you seek a better institution, but I assume the OP here pays his students enough. OP seems worried about "not paying the top possible salary = not getting top students" which is surely not true. I hereby state why I believe this is so.

@Scientist: 1. The point of this question is that, typically, research institutions are not the "better institutions" you suggest. 2. At the "better institution", they are probably not doing the research I'm interested in doing.

@einpoklum sorry if misinterpreted, but I never suggested any specific institution is the best. I recommended you changed to a better place because your last phrase sounded like you're not currently being paid enough (e.g. barely making ends meet). Most PhDs I've met were paid more than just enough although not the highest salaries in the category.

@Scientist: I finished my Ph.D. in 2012... but I was very involved in graduate union activities during that time and for a while later, so I'm still fired up about it :-)

Other answers are somehow good, but I think they might miss a practical approach. I try to elaborate a bit.
In most cases, your question has a simple answer: hiring PhD students from overseas, specially third-world countries.
Why? Because most of them have visa limitations and they can't think of a job for the moment. Their options are limited and they have to be good to survive. This gives the professors a vantage point to easily take advantage of.
Please don't get me wrong. I am not saying that this a common practice and the only reason for hiring foreign PhD student and I hope nobody gets offended. But as a resident of a third-world country who has seen that many of his highly talented friends are so eager to get an overseas PhD, even from a low-rank university, I think I have some valid points here.

Actually plenty of students even from nice-to-live countries wouldn't mind coming to US to for a PhD. The key is to make the PhD offer visible to such students.

@DmitryGrigoryev yes, and note that the term overseas in my answer has a broader meaning than the US only.

To extend the previous answers, which have already given good points:
Grant your students full Open Source rights, offer similar license rights for the data they collect and produce. That is what kept me a while in science. If they have a well distributed software product in the end, this is good for the CV and gives more job opportunities as well.
A collaborative project with international partners in attractive countries where your students can stay for a few month will be an attractive plus as well.
Promote the freedom and life experience in science. That is the only advance over well paid industry jobs. If you trust your students you can also offer more freedom in form of regular home office opportunities.

Definitely a good idea, so +1, but remember universities sometimes have their own restrictions on open-sourcing/free-licensing research findings and try to commercialize and patent things.

@einpoklum This pressure is a bit known to me. If the university prohibits free licensing, the responsible chair leader could only fight with limits to change those politics. The topic creator seems to be US citizen and at least the bigger university names are known to be also suporitve of free licensing. MIT might be the biggest name in the field of informatics world wide. And MIT shares. If private unis dont allow, they simply have to pay more. Or live with the fact that the named problem of the topic creator will even further increase. Not enough phds.

This question is a bit like asking "How can I get a brand new car for $5,000 when they all sell for $20,000?" You can't, unless you get lucky with some special situation with unique circumstances. The market has decided that the sort of qualifications you want are worth far more than the sort of money you're willing to pay. There's nothing you can realistically do to change that.
This simple premise implies that you must do one of the following:

Pay more
Let the students work very part time (thus paying more per hour)
Settle for less qualified students

Obviously one is not an option for PhD students because you couldn't offer a meaningfully higher stipend. Second might actually be feasible if you can arrange for some sort of industry partnership where the bulk of the student's work is actual work for a company (perhaps one that develops the tools you research) and the PhD related duties become a secondary priority. But it's probably hard to get away with this in practice.
Third option is the more sensible one. Either you have to reduce your research goals, or take in bright but unqualified students and train them. Of course, at some point after being trained, the student will realize that they now are qualified and could quit anytime for a much better job. So you have to be very nice to your students and make sure their frustration does not exceed their intellectual satisfaction.
I did say the first option is not realistic, because PhD stipends don't really vary that much and are often outside your control, and even if you could somehow arrange to pay even double the usual salary to your students, it would probably trigger riots on campus when other students hear about it. But you do have more flexibility with postdoc. If you really need to hire the sort of person that could get six figure salaries in the industry, the obvious answer is to hire such a person as a postdoc, and compensate competitively.
Also, I think the "$120k+ straight out of undergrad" is a myth. Computer science pays well, but not that well. Undergrads will easily work for half that, even the median salary (for experienced workers) is far below it. The only grain of truth in it might be the Bay Area, but cost of living is also very high there.

As i wrote in my answer, there are some unique qualities in academic job that can help with salary difference

120k out of undergrad is not a myth.  Top CS students with in-demand skillsets from top schools presently command that, and more.

Yes, but very few people are top CS students. 120k is an exceptional case and while of course not "mythical" should not be paraded around as even close to standard.

The standard offer for fresh-minted bright (but not exceptional) CS undergrads for SWE positions ranges, today, between 100 and 120 in Seattle, WA.

More standard than academics like to admit. ;)  Spend some time on Glassdoor. 50th percentile of salary for software engineering jobs, readily obtainable, I'd argue, for someone in the top 20% of a top 50 CS program in Philly is 132k. In Houston is 192k, and of course in SFO is 224k. Even in Dayton, OH it's 96k. It's shocking.

This is completely and demonstrably wrong. Many people do go for a PhD (despite getting offers at companies), so it is definitely possible. Many of my students have received offers from top companies (such as Google) and decided to go for a PhD anyway, so it's also simply untrue that you can only get the less-qualified people.

-1 for the implicit assumption that all value can be measured in money ...

There's a large and growing pool of potential postgrad students who have all the characteristics described in Buffy's answer but who are "unqualified" academically: self-taught industry professionals for whom getting an undergrad degree would be an absurd waste of time and money.

@xLeitix then OP's answer is to make sure OP's program is as good or (preferably) better than xLeitix's program.  It is easier to conceptualize than to implement.  If OP can make OP's program better than xLeitix's program than all those student who would have otherwise enrolled in xLeitix's program will enroll in OP's program.

@Industrademic: It's true that Glassdoor makes salary data available, but the data they have is very different from the numbers that you implied were sourced there.

@BenVoigt Let me help you with that: 1-put Philadelphia in the location box, 2-  top dropdown to 'salaries', 4-  'software engineer' in the search box, 5- press 'search' button, 6-press the 'salary range' button to see the histogram of salaries for your search, 7-drag it halfway. Value: still 132k.

@Industrademic: Got this: https://www.glassdoor.com/Salaries/philadelphia-software-engineer-salary-SRCH_IL.0,12_IM676_KO13,30.htm   Mean in Philadelphia is $91k, median is significantly lower than that.  For software engineers with experience, not straight out of school.  In Houston: mean is $90k, median is lower.  https://www.glassdoor.com/Salaries/houston-software-engineer-salary-SRCH_IL.0,7_IM394_KO8,25.htm  Again, that includes those with experience and is not reflective of entry-level positions.  Don't see anything supporting your numbers.

@BenVoigt, it depends on how you slice the data.  If you saw nothing, you did not follow my instructions. That's fine.

@Industrademic: Your "drag it halfway" step is very unclear.  It's not even clear which page on Glassdoor you are using, there are many similar search forms (hence the reason I provided actual links to data).  But if you mean to put the cursor halfway between the two extremes, that is not what 50th percentile means.  And I still believe you are mixing in data for those with 30 and 40 years experience and trying to claim that new graduates earn the same.

My long term goal is to lecture for software engineering, but since it's a practical skill, I am gaining practical experience in the industry (having been a QA, engineer, team lead, contractor), so that I can teach based on real world experience, rather than just having had read the texts before my students.
As a team lead, one of the things I learnt was what motivates people, including myself.
People have their own definition of success, whether they know it or not, and you'll never motivate someone whose motivation is cash to put in the enormous amount of effort required for a PhD where the reward is sating curiosity and building some new knowledge for the future.
@jamesqf's comment:

One option is to recruit people who have already spent some time in industry, have enough money, and are now interested in more challenging work than they might find in industry.

is pretty on the money. The best way to attracting talented grad students in a topic area isn't to compete with high-paying companies with pay, but with more interesting, rewarding work.

Disclaimer, this is my first answer the site and my experience of academia is a BA in Philosophy and Computing for 11 years ago, but hopefully my massively rambling subjective answer is of some use!


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Under what circumstances would a lecturer 'bump up' a student to pass?
When would someone marking papers (lecturer, etc.) bump a student from just below pass up to the pass line? Do teachers consider things like attendance?
Regardless of the consideration (attendance, participation, etc.) for giving a student a better grade than they deserve, what is the justification?

I usually set my lines for passing after I see the final exams.

One of the things I like about teaching is that while we draw fine lines at numbers such as 65.0 and 89.5, teachers also get the final word on putting grades into the system, and the ability to look at a student's performance holistically to include non-tangible paper marks such as attendance (in class and office hours), intrinsic motivation, and future goals justifies "bumping" grades for particular students.
I don't do it often, but if a student is on the line and showed me high motivation but didn't quite get to the higher grade, I will consider bumping up the grade (but I won't do it because of grade-grubbing). Sometimes, it comes down to when I'm entering the final grade into the system, and I think, "You know, that 89.3% for Alice really should be an A-." I don't lose sleep over it, and it really is relatively rare.

This is exactly how I handle the issue.

source: Stack Exchange

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/3.0/

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Do sample solutions increase student 'productivity'?
I'm not 100% sure whether this belongs here, but since I am a PhD student (teaching) in the TCS/Algorithms department, I'd like to know what fellow, maybe more experienced, teachers think.
The question at hand is whether we should offer sample solutions to all our exercises for, e.g., a Data Structures & Algorithms lecture.
I am convinced that this would be a highly beneficial service for our students whereas my advisor is against it. Here are the pros and cons that we came up with:
Pros:

Students have access to high-quality answers when they don't understand something.
Our expectations on verboseness, conciseness, depth of proofs, etc. can be communicated more clearly.
A student can individually study using the solutions and is not forced to attend the tutorials if this is not his preferred style of learning
We (the professor/tutor) have a clearer idea of what solutions to expect since we have to work the problems ourselves.
The tutor (me) is more free in the design of the tutorial. Without sample solutions, the tutorial basically boils down to writing the sample solutions on the blackboard. Otherwise I can't be sure that everyone has at least seen the correct way how to solve it. Little interaction is involved.

Cons:

It costs time and/or money.
Students may stop being engaged in the exercises since they know they can always look at the sample solutions.
Students may stop coming to the tutorials.
We can't reuse exercises from past years since students might have access to (and use) past sample solutions.
If we do it once, the students might expect we do it for every lecture.

To be clear, in both cases the students are expected to solve the exercise sheets on their own, and they will be graded. I'm merely interested in what to offer after this has happened.
I think that's all. Optimally, I would like to find some kind of empirical study that proves that sample solutions increase the "productivity" of students. Data always wins. However, so far I couldn't find anything like this.
To discuss the points mentioned above, my general opinion on these matters is that if we can offer more services using little work, we should always do it. If someone really misuses it (as stated in the cons), he or she will notice that this is the wrong approach the latest in the exams. My advisor, however, wants to minimize the time spent on lectures and have me rather do the research relevant to my PhD. Since I have to do the solutions anyway, the overhead for providing a sample solution is maybe 2-3 hours/week.
What do you think?

Tutorials? Tutors? We don't have them where I teach.  This sounds like a U.K.-specific question.

We have teaching assistants, who run recitations. But in the U.S., you generally can't give the solutions out before the recitation because the exercises count towards the grade. This (and the terminology) is why I thought you were in the U.K. Giving out solutions after the exercises are due is extremely common.

With regards to some of your cons, once you get into college, my observation is that if students stop caring (jsut look up the solutions) they're either going to quickly learn how bad of an idea that is by bombing a test, or will otherwise get weeded out. Most students who don't make an honest attempt at their work don't graduate, and I don't feel bad giving them the decision to pass or fail.

One more (in my experience) often quoted disadvantage is that students won't necessarily read the solutions, or not to the degree they should. Having the correct answer to file in their folder, they forget to sufficiently file it in their brain. If they watch/listen to you writing out the solution, they are likely to process it at least on some level.

Some teachers will distribute solutions only on paper (not electronically). This allows them to use the same questions year after year. Their mode of thinking might be along the lines of, "The solutions are always out there somewhere. If a student is going to go through the trouble of tracking down an old student and using their answers, they're beyond my help."

This is a topic that educational theorists are regularly waging war over.  The internet has only worsened the situation, as it often doesn't even matter which choice you make: students will just pop onto a stackexchange or another forum to get the answer, or use google, etc.  That's what I think you actually need to discuss and worry about.  There's a limit these days to how much you can stop exercise problems making their way onto the internet and getting full solutions.  An educator's task is now set at getting students to use those responsibly and properly.

I have three types of exercises in my algorithms classes: homework problems, exam problems, and discussion problems.  I'm not entirely sure which you're asking about.
I release detailed solutions and grading rubrics for all homework and exam problems, in part for the advantages you list, in part to speed up grading, and in part to better calibrate my own expectations for the students.  (If it takes too long for me to write up the solution, the problem is probably too hard for them.)  I take them all down again at the end of each semester.  I don't actually mind if students have access to my old solutions—as long as they write in their own words and cite their sources—because homeworks are only a small part of the course grade.  (Students who are stupid enough to submit my old solutions verbatim, typos and all, are not quite publicly fed to the wolves.)
On the other hand, I deliberately do not release solutions for discussion problems (which we discuss in, you guessed it, discussion sections) because the solutions are not the point.  The point is to practice finding the solution.  I know students are adults, but it takes a lot more discipline to practice hunting when someone just regularly hands you the meat.  Also, some discussion problems reappear later on my exams.
But this is really an individual choice.  I know plenty of algorithms instructors who don't give students solutions, and others who hand out solutions on paper but don't distribute them on the web, and others who distribute them on the web but behind a firewall, and others who beg people like me to please for the love of god stop giving away homework solutions because coming up with good algorithms homework problems is really really HARD.

Update:  Starting in 2017, I now regularly release solutions for my discussion problems, typically a few days after each discussion meeting.  (Just like homework solutions, I take these down at the end of every semester.)  Perhaps as a result, these discussion problems are now effectively fixed from one semester to the next -- in a typical semester I replace 5%-10% of them -- and discussion problems almost never appear on exams.  (I should also clarify that discussion problems do not contribute to the final grade.)
I also include an extra solved problem in each homework, with a complete grading rubric.  Again, these solved problems rarely change (as opposed to the problems the students need to solve, which change every semester).
In both cases, the idea is to provide concrete examples of the structure, precision/formality, and level of detail expected from their own work.    Realistically, once the discussion sections are over, unsolved discussion problems are not as valuable as the solutions; students are busy!  And I have lots of other unsolved exercises in my lecture notes for students who want unsullied practice.
Writing all those lab solutions (just over 100 pages of text) was a lot of work, but now it's done.  The net effect of releasing all these solutions seems to be positive—more clearly for teaching evaluations, but also for student performance.

In addition to other good points made, let me say that (in mathematics, at all levels) I myself make many "model solutions" and put them on-line.
Obviously the availability of model solutions has positive potential... The issues are the genuine downsides.
One reason for my decision to take this approach was that, especially in upper-division and graduate-level mathematics, enthusiastic students acting in good faith often put either flawed or misguided solutions on-line, and other students look at those, ... thus "learning" low-quality versions.
Another reason is according to an over-simplified reasoning: important examples should not be left to students to mess up, and unimportant examples should not be used to waste students' time. I realize this is over-simplified and has implicit hypotheses, but after 40 years of watching people diligently spend time on exercises _without_thinking_critically_ about any sort of larger picture, I am ever more fond of this pseudo-principle.
Yes, a fundamental objection is that on-line solutions allows laziness/cheating/whatever. And, yes, as JeffE noted, in some venues it's hard come up with good "training exercises". Thus, I can certainly envision scenarios in which a cyclic putting-them-up, taking-them-down could be justified. However, dedicated lazy/cheating people can maintain copies ... And so on. Thus, in effect, it is impossible to prevent laziness/cheating in the face of even modestly motivated lazies/cheats. Thus, I reason that elaborate strategies aimed at foiling laziness/cheating, at the expense of making people acting in good faith have to jump through hoops, etc., are bad.
In mathematics at least, I'd claim that many traditional contexts for "exercises" are somewhat missing the point, anyway, so that moving away from the weekly problem sets wouldn't be so bad! That is, to make a large number of "exercises" feasibly do-able by nearly everyone in every class, and in a short period of time, the issues must be contrived, not natural. Students understand this, even if only subliminally, and many of the "successful" ones have managed to squelch their critical faculties ("why are we doing this?") to be more economical in their approach to these fairly-random exercises.
Or, at the opposite end, there are the occasional much-admired slim texts where 2/3 or more of the things one needs to know relegated to exercises! Crazy! In this case, the student's disadvantage is even worse in some ways, because the issues are more real, and there're even fewer "model solutions available", and they may come away with deeply flawed or misguided pseudo-understandings.
At least in modern mathematics, I think that the inarguable "engage with the material" is too often denatured, to something like "try to prove all the theorems yourself". Supposedly, the side information of knowing assertions of true theorems is enough of an advantage. But this is a strange presumption... proof mechanisms, concepts created to enable proof mechanisms that are humanly comprehensible, are as significant as the bald assertions themselves, I think.
So, to advance collective human understanding, putting "models" on-line is good. Yes, there are downsides, and hazards, but this is just the new reality.

"Training exercises"?

I learnt English (as a required subject) in a non-English-speaking country for nearly 11 years. We were asked to solve millions of artificial multiple-choice exercises instead of to learn how to use English in real contexts. (Perhaps mainly because those English teachers don't have practical English skills either...) With hindsight, this kind of teaching/learning is extremely inefficient. Maybe the same idea applies to mathematics, which also has a language component.

@ZhouFang, indeed, despite math-culture tradition to pretend that mathematics is independent of ambient natural languages, while one could write denatured formal-logic mathematics, this would be very inefficient... and unreadable, too. Especially, such writing would stupidly fail to take advantage of all the information compressible into natural language effects: suggestion, insinuation, contraposition, etc.

As a CS major who had to take 2 theory classes, I really appreciated the prof who handed took class time to explain and solve each question after the assignment (he didn't hand out the answers so he could reuse them).  The prof would usually hand out 3 or 4 questions Monday and solve them Friday (after you turned in your answer).
He would basically act like he was doing the assignment, and was very good about not skipping steps.  He'd also explain why he wanted to see the answer in the format, which help several non-theory students (like me) learn how to appropriately write proofs.
At least in America, CS Theory is generally a difficult subject because the rest of CS education is very applied and students are generally unprepared for it.  Especially in this course, going the extra mile for students is noticed and appropriated, and will provide a counter-point to the theory prof. who calls students idiots and gets in yelling matches whenever a student ask a question (which also happened to me).

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text:
How can I counter a student response saying "Why are we bothered to reinvent the wheel when proving mathematical identities?"
I am teaching an undergraduate mathematics course to software engineering students. I often ask my students to prove some mathematical identities as their homework. The identities could be, for example: trigonometry identities, Laplace transform identities, Menelaus', Stewart's, Ceva's theorems, etc.
One special student often just copies other works obtained from the Internet without doing any effort to prove them by himself. When I asked him, he replied "Why are we bothered to reinvent the wheel?". I have not replied yet and I will do later.
Could you give some advice for countering his argument?

This question is probably better at [matheducators.se], actually.

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The question surely has sense for the Math Educators site, but there is also mathematical content and possibly more general content for academe... I upvoted, and do not think it should be closed.

I'm now wondering what kind of a course is being taught that includes such a diverse range of topics, from elementary geometry through relatively advanced calculus.

@KevinArlin: I also teach (among others) Gauss-Bonnet theorem and how to derive quadratic formula $x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ to them.

@user69652 So the course has no specific remit within mathematics? It sounds like quite a smorgasbord.

@KevinArlin: Yes. But it is not too important for my question.

Someone said:*

When you reinvent the wheel, you end up learning a great deal about why wheels are round.

And that is really the point. When you’re at university to get a degree in wheels, you should fully expect not just to be told that wheels exist but to be asked to think deeply about them so that you develop an understanding, at the deepest level, of why they work the way they do. That is literally the whole point of going to college.
*Source: something I read recently somewhere on the Stack Exchange network. The quote is from memory, I can’t find the exact source now unfortunately.

Maybe this statement? https://blog.codinghorror.com/dont-reinvent-the-wheel-unless-you-plan-on-learning-more-about-wheels/

@user111388 not quite, but thanks for the link.

That is a really good quote for exactly this situation.

@TobiasKildetoft yes, and actually I vaguely recall bookmarking the thread with the thought that this is a quote I’ll want to use at some point, but I’m not finding it at the moment.

@DanRomik Perhaps this?

@GoodDeeds good one! But no, that’s not it either I’m afraid :-|

I would add to this quote 'and you learn how to explain why wheels are round'.. Formulating logical proofs in mathematics requires the same high-level mindset that formulating a logical argument for a debate or project pitch does, so it can be a very effective way of practicing the logical thinking that's required for effective communication.

In mathematics, the point of understanding how a theorem is derived is NOT to understand a particular theorem. It is to understand what constitutes the CONSTRUCTION of such an argument in an unambiguous and COMPLETE framework. To know the steps to complete a particular proof is meaningless in the mathematical postgraduate/peer reviewed ecosystem. The whole aim is the understand 1) what you need to prove 2) how to prove it thoroughly to demonstrate you achieved the goal you set about.

I think this is an exceptional answer for students who want to continue in Mathematics. I however hated doing proofs in my discreet math course required for my software engineering degree, I also still to this day do not understand why I had to do them and they have never helped me my professional career. Similar to @Graham's answer below, a tie in to why this is taught to software engineers would have benefitted me a huge amount, since I don't doubt the benefit exists, I just still don't understand it.

@jdf I think that’s an important and valid perspective, but at the same time I’d hazard a guess that doing proofs helped to teach you to think more logically and ultimately made you a better software engineer, in the same way that practicing musical scales and studying “boring” music theory helps to make a music student into a better musician. And this despite the fact that many students hate those activities and don’t understand why they can’t just spend their time practicing playing the songs/pieces they like. But your point about the need for good motivation (“tie in”) is very true.

When I first read the quote I actually thought you meant that reinventing the wheel is a bad thing. "learning a great deal about why wheels are round" sounds like "learning a great deal about why gloves are in the shape of hands". They're round because round things can roll! I guess I'm the only one who doesn't think this quote does a good job of addressing the issue.

ie, you can learn to do math, or you can learn to understand math. Which you are after, depends on what your interests and goals are.

@Behacad: Should Dan Romik add that you ought to also end up learning why we use spokes in wheels instead of having solid wheels? And why spokes and not a grid mesh? And why tires on wheels? And for the advanced classes: Why we use a differential? Why use low gear for low speeds and high gear for high speeds? ...

Re-inventing the wheel in education is about the process of re-invention, and not about the resulting wheel.

The student seems to have the misconception that mathematics is about "facts". Early education stresses elementary facts a lot, so this is pretty natural.
But mathematics is about understanding relationships, not memorizing facts. If you don't know why something in mathematics is true, you really don't understand it. The proofs in mathematics are more important than the theorem statements, actually, as the latter just capture the essence in a simpler statement.
Until he understands that it will be difficult for him to advance as a mathematician, assuming that is a goal. If it isn't a goal, then he has little incentive to change.

This is the appropriate answer if the students are actually Math students, but it applies less if they study a different subject, in which they eventually become mere end-users of Math results. For example, the central limit theorem has manifold implications in all sciences and in everyday life. To understand it, it's not important at all to understand its proof.

@lighthousekeeper I think their program is irrelevant here. They are in a math class, presumably run by the math department rather than run by the department of their program (in which case I would wonder why unnecessary things are being taught). I'm an EE and we take mechanical dynamics because motors spin. We were literally told that's the reason why. We take it in the course taught by the ME department but we don't go in and ask why friction and springs are taught when EEs don't need it, because it's a class taught by the ME department.

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I was that student. And the reason is mainly not because I was a bad student, or a bad engineer (a 2:1 degree and 25 years in industry should answer those points). It was because my lecturers were bad teachers. And yes, you too may be a bad teacher right now, without realising it. The difference is that you've seen this student's question as a challenge to improve your teaching, and that's something you should be proud of.  What matters for all of us is how we learn to do better.
You're teaching engineers, who are without exception practical people. (Or the good ones anyway. I assume you want your course to produce good engineers.) Remember that they aren't at university to become mathematicians, they're there to become engineers. If you want them to learn a subject which seems of no real value to them, you need to convey the value in doing it. It seems you haven't managed to get this across.
To divert slightly, think of The Karate Kid. How does Mr Miyagi teach Daniel? He gives him apparently pointless exercises to do. Daniel puts up with this for a while, but eventually tells Miyagi to shove it. Miyagi then demonstrates that there was some value to the exercises after all. The thing is though, this isn't the only way to learn, nor actually the way which works for most people. Go into a regular dojo, and you get taught how to block and punch by, well, blocking and punching.
You've got the Miyagi method though. You've given them seemingly pointless exercises to get practise in basic techniques. Now your Daniel has called you on it though. So what's the application for that technique? If it's widely used, what's even one application for it?
For concrete personal examples, we spent ages at university learning about phase lead/lag networks, PID controllers, and so on. What we didn't do was get an actual motor and position sensor, and actually control something. The result for me was that it didn't really stick, and I've had to relearn it all at work when I've had projects which needed it. Practicality and relevance are the keys for engineering.
You mention Laplace. Have you had them do any DSP? It doesn't get more hands-on with Laplace than filters. And that step response? That's what happens when your wheel hits a bump, or someone pops the microphone. Do you get a (Butterworth) ringing or a (Bessel) smooth step? And which do you accept, if the depth of filtering is also important?
There are lots of ways you can go which makes your subject relevant. All you need to do is find them.  This may require you to talk to the engineering staff to find how they use these techniques (per @KevinArlin's comments), to give you examples you can hang your lectures on. If it means students they'll be teaching or supervising later have a better grasp of these basic techniques which they'll need, I'd expect them to jump at the chance.

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I'm a mathematics graduate currently working in computer science and data analysis.  In my experience it is difficult (if not impossible) to memorize every single aspect of mathematics, especially for identities which could be combined with one another in various complicated ways to produce infinite results.  Instead I find it useful to start with the most basic parts for which I have a deep understanding and put them together - in essence "reinventing the wheel" - until I have my desired result.
Years after taking a certain course, I have found myself working on projects that need certain identities and facts (I've needed a surprising amount of geometry and trigonometry for processing image data).  There is no way I would remember them all, but I can still remember the basic parts and the process needed to put everything together.  Sure I could look it up (as long as I remembered the name or enough details about the identity, which isn't trivial), but that is not always an option...
Even more important, there are times when I am working with very well known identities or processes, but I need to put two things together, add something new, or make a change to get a slightly different result.  If I only ever used what could be easily looked up, I would be out of luck in this new situation.  But hopefully after "reinventing the wheel" many times over, I will have developed the problem solving skills necessary to tackle the problem.
Do that enough times, and complex things like "the wheel" become the basic parts for which you have a deep understanding, and you can use it to build greater and grander things, a "car" for instance.

This post illustrates the distinction between getting a degree and getting an education.  Unfortunately too much emphasis these days is on former, and not enough is on getting the latter.

You can tell the student the following:
The goal of homework is not to prove new things that the instructor doesn't know, but rather to give the students knowledge and experience in using the tools given to them. This is called "learning" - probably your course/institute has some goal the students "learn" something. You could refer him to that.
It's the same when a teacher teaches: The teacher does not reinwent a wheel but rather tells/shows known facts. Yet schools, universities are important.
This is also not related to mathematics or university. Most likely, the students had to learn a foreign language (or their own language) at school. Then they had to apply it (by speaking or by doing grammar exercises). Here they also do not invent new  things (as one could say a novel does), but deepen their knowledge.

Counter by

Having the student consider why a musician or athlete practices routinely, and
offering an unsolved problem to prove as an alternative.

Together these points should prove compelling, assuming the student's objections were truly sincere.

This is what I would do, which is a good argument for it not fitting the role of "what a professional educator should do".

@wizzwizz4: You might make a better professional educator than you think.  ;-)

Other than the "roles" thing, there's a certain kind of person that would not quite get that the unsolved problem wasn't an actual task, and would not admit that they couldn't solve it, and would either burn out or drop out trying to solve it.

@wizzwizz4: I suggested offering, not assigning.  Elaborate that challenging problems, some never before solved even, await but will first require mastery of the basics.

I know. It's probably not on you if you get that sort of student, though.

Possible downside to (2) is that the student may submit garbage and insist that it's a proof that you don't understand, or something (crank-style). Related, I had a possibly learning-disabled college student who discovered the Collatz conjecture mentioned in our textbook, and kept submitting nonsense and asking if it was a proof, no matter how much I recommended that he stop.

Feynman cut to the heart of this issue in The Feynman Tips on Physics:

I have a few moments left, so I’d like to make a little speech about
the relation of the mathematics to the physics—which, in fact, was
well illustrated by this little example. It will not do to memorize
the formulas, and to say to yourself, “I know all the formulas; all I
gotta do is figure out how to put ’em in the problem!”
Now, you may
succeed with this for a while, and the more you work on memorizing the
formulas, the longer you’ll go on with this method—but it doesn’t work
in the end.
You might say, “I’m not gonna believe him, because I’ve
always been successful: that’s the way I’ve always done it; I’m always
gonna do it that way.”
You are not always going to do it that way:
you’re going to flunk— not this year, not next year, but eventually,
when you get your job, or something—you’re going to lose along the
line somewhere, because physics is an enormously extended thing: there
are millions of formulas! It’s impossible to remember all the
formulas—it’s impossible!
And the great thing that you’re ignoring,
the powerful machine that you’re not using, is this: suppose Figure
1-19 is a map of all the physics formulas, all the relations in
physics. (It should have more than two dimensions, but let’s suppose
it’s like that.)
Now, suppose that something happened to your mind,
that somehow all the material in some region was erased, and there was
a little spot of missing goo in there. The relations of nature are so
nice that it is possible, by logic, to “triangulate” from what is
known to what’s in the hole. (See Fig. 1-20.)

And you can re-create the things that you’ve forgotten perpetually —if
you don’t forget too much, and if you know enough. In other words,
there comes a time—which you haven’t quite got to, yet—where you’ll
know so many things that as you forget them, you can reconstruct them
from the pieces that you can still remember. It is therefore of
first-rate importance that you know how to “triangulate”—that is, to
know how to figure something out from what you already know. It is
absolutely necessary. You might say, “Ah, I don’t care; I’m a good
memorizer! I know how to really memorize! In fact, I took a course in
memory!”
That still doesn’t work! Because the real utility of
physicists—both to discover new laws of nature, and to develop new
things in industry, and so on—is not to talk about what’s already
known, but to do something new— and so they triangulate out from the
known things: they make a “triangulation” that no one has ever made
before. (See Fig. 1-21.)
In order to learn how to do that, you’ve got
to forget the memorizing of formulas, and to try to learn to
understand the interrelationships of nature. That’s very much more
difficult at the beginning, but it’s the only successful way.

It is a very good response. However, it is not an answer that students will want to hear! :)

For the last student who asked me this, I pulled out Gradshteyn and Ryzhik (my copy from 1992).  My first line (in joke voice) was "Alright, get to memorizing."
Then I flipped to page xxiii and pointed at the text leading up to "We then kept only the simplest formula."  Then jumped to page xxiv to the line "Thus, before looking up an integral in the tables, the user should simplify as much as possible the arguments ... in the integrand."
The student responded that there exists software containing the entire book.  I replied with my experience : "There are only a few pieces of software whose marketing departments make this claim.  To date, none of them have been correct, as I have found fairly simple integrals that each version of each piece of software I have used fails to integrate."
If you can't bring your work in to a form that can be found in references, you are crippled compared to someone who can.  This may involve algebra, trigonometric identities, Fourier, Laplace or other integral transforms, the Cauchy integral formula, et c.  If you don't understand those tools and how they work, you will not understand when you should or should not be using them.  This knowledge comes only from practice.

Good tables are worth their weight in gold.

My answer would go along the line:

It's a bad idea to blindly believe everything you read. We all know that there's
a lot of nonsense out there in the internet. So you better be
able to judge yourself, and this exercise is part of teaching you how
to evaluate whether some mathematical claim is true or not.

Maybe it would raise motivation to have the students decide whether some claim is true or false, and to present sound reasoning (proof) for their answer.

So what if the student says he checks it before copying it? (Many students do.)

Doing math is different from 'knowing math'. With knowing math I mean memorizing formulas and concepts. We all probably have experienced this in an exam that contained math: you think you know some formula but when you're in the exam you notice you don't actually know how to implement it in practice. Proving identities is one of the best ways of getting some hands on practive with the material. Even though it might not be the most direct way to practice with it. One benefit you get by proving those identities is that you better know when you are allowed to apply these identities because now you know exactly what conditions are required to make the proof work.
A second argument is that proving an identity is an effective way to remember said identity. Personally proving a formula about 3 times over an extended time is enough to memorize it for me.
I'm still a student so I don't know how important it is to be able to make proofs in your career and this will probably vary a lot depending on your department. (it is very important in the maths/physics department though)

Let me formulate a contra-answer to the one from Graham:
I am also a mathematician, and I have worked as a software engineer for several firms. One of them had created their own development environment, which generated a whole bunch of binaries and one textfile. All those automatically generated things together were the actual product.
After six years, I was assigned as a functional tester, and one of the things I did, was parsing, reading, grepping, ... the automatically generated textfile, and my colleagues were completely astonished of the amount of bugs I found while doing that.
But most of all, I was astonished by the fact that litteraly no-one (we were several hundreds of colleagues) had ever thaught of reading that file as a human being: people were so focused on the fact that that file was auto-generated that they did not even think of doing that. My insight went as a shock through the company!
Therefore I'd like to encourage you continuing what you are doing: try to increase the insight of your students. They may resent you for it, they might even claim hating you for it. But later (I was about 35 years old), there might come a moment where that insight comes in very handy and for the rest of their lives, they'll love your for it!

+1 I think this gets to very important aspect of these "When will I use this?" questions. Well, I have no idea what work you'll do, what job you'll get, or even what occupations will exist in 5-50 years. All I know is that gfor almost all of us there will be one day where the need for math knowledge will be an emergency and make or break you (if not actual life-or-death consequences).

I agree with your student.
I have a PhD in physics and used math as a toolbox. A wonderful, useful, shiny toolbox. The kind of toolbox where you know that in order to find the roots of 3x^2-9x+2=0 you would find delta etc. I do not care how it was found, because this is a tool (I am not interested how a screwdriver is built either).
This is really, honestly not to diminish the value of your field. Someone had to invent these tools in the first place.
My son is going through 2nd degree polynomials as we speak. I had to suffer through the whole proof part and he knows that at the end of the day, it is three formulas he will blindly use.
One thing you could consider doing - and this is a very useful exercise: to set up a problem where blindly using the formula will make them fail, because the formula has an introduction part ("If x belongs to ..., then (formula)"). This will probably not make them study the proof (who knows) but will teach them a valuable lessons on limitations.
And someday, as they will be engineers, they may remember that story with limitations when building the bridge you will walk on. This happened to me (not with a bridge, fortunately) and helped me to always remember that models have contraints.
What I am trying to say is that the proof may not be the most valuable thing to teach them.

This is one of these cases where I would appreciate very much a comment on downvotes to understand the thinking of the ones that do not agree.

Because you need it to pass the exam.
Call my cynical, but that's the ultimate reason. Not wanting to reinvent the wheel is often very reasonable, e.g. if you were asked to prove Fermat's Last Theorem, you would be foolhardy to attempt to prove it yourself instead of searching up Andrew Wiles' proof.
The great benefit of attempting to prove the identities yourself is that you become familiar with the thought process, the basic equations, and so on. For example with trigonometry, one might see an expression involving sin^3(x) and think of converting that one of sin(3x). Without this insight, one might never make progress. Working on similar proofs before helps provide this signpost when it matters during the exam.
If the student is capable of learning the proofs enough that they can prove similar identities, under time pressure & in an exam setting, then there's nothing to worry about. Whether they get to this stage by working through someone else's proof, or by figuring it out themselves, isn't important.

I wouldn't call the first sentence cynical, but I would say that it showcases a lack of sensibility for questions such as: What is the purpose of education? How is it supposed to make the world better?

I would not tell the student sentence like "if you were asked to prove Fermat's Last Theorem". Nobody ever will tell you "Prove Fermat's Theorem" in real life and stusents know this.
Examples like this are mathematically sound but make the instructor sound like they are in an ivory tower. This might reinforce the idea that the education is pointless.

I wouldn’t call you cynical, but I would call this the most uninspiring answer imaginable to the student’s quesrtion. I hope if/when you are either a teacher or a parent and your student or child asks you a “why?” question, you make a little bit more of an effort to think of something that’s not only factually correct in a “gotcha”, technical sense, but actually helps your student or child develop their inner creativity and sense of curiosity.

@DanRomik I'm of opinion inspiration is overrated - it's what makes people do things that aren't in their best interests and come to regret it later.

@Allure hmm, ok. Well, best of luck with that philosophy, I’m sure there are contexts when that makes some sense. But given what you’re saying here, perhaps I’ll call you cynical after all...

The last name is Wiles, not Wile.

I was that student
NOTE - Since I posted this there has been a similar answer by @Graham  -  However Graham's answer is so much better explained than mine that I have upvoted it and recommend it over mine.
Not as an undergraduate but before that in secondary education.
I resented doing chemistry and physics experiments because they were printed in books that we had to work through. As far as I was concerned, they weren't experiments because experiments are discovering something new, not repeating what had already been done. Instead I would do my own experiments at the back of the class.
I refused to learn a proof for Pythagoras' theorem that we were supposed to one year. I ended up having to prove it almost from scratch in an exam.
I rarely did maths homework because I had "understood" it when it was explained in the class and didn't feel the necessity to demonstrate the fact.
I did pass the exams in these subjects - barely.
By the time came that I was expected to apply for university I was so fed up with education that I wasn't interested. I ended up going to music college instead. It was only years later that I decided to pick up my studies again. I was working full-time and simultaneously did a part-time degree. By this time I was more mature. I worked hard, followed the rules and got a first and went on to do a Masters.
I really don't think anything could have changed my mind at the time. I was lazy and a rebel.
I suspect that this student is similar and has got by so far by surviving on natural ability without rigour. It may be that he will be more suited to researching one specific difficult problem and will make the effort at that time to put in all the necessary work. Think Andrew Wiles.
Suggestion
If like I was, this student is a lazy rebel (and somewhat immature), it will be difficult to motivate him to take the standard route. The best I can suggest is to find something that really motivates him and show how following the course will aid him in achieving that.

Nice answer, upvoted. But I don’t think Andrew Wiles fits your lazy rebel stereotype, nor that a student who thinks proving identities is a “bother” is likely to become anything like a future Wiles (although he may succeed in other ways). So that particular analogy isn’t very apt.

@Dan Romik - Neither of us know what he will become from being a budding Wiles to sleeping on street corners! My point about Wiles was that he, from an early age, had a narrow focus that he saw as important - so much so (I believe) that every bit of maths he learned was to that end. He had more motivation than just a teacher saying "Do this because it's what everyone does." He did it because it all led towards his life's work. That's why I'm suggesting finding a focus for this student so that learning isn't just a scattershot approach but a means to an end. It may or may not work.

Maybe your student is seeing only the traditional aspect of the proof: a mean of verification, validation, conviction. If this is the case and the student believe the identities are true, there is no reason to bother to prove them. Then, you could try to explain and work with the other functions of the proof: explanation, systematisation, discovery, communication and intellectual challenge. See this paper by Michael de Villiers. Here are two quotes:

Who has not yet experienced frustration when confronted by students asking "why do we have to prove this?"

The question is, however, "what functions does proof have within mathematics itself which can potentially be utilized in the mathematics classroom to make proof a more meaningful activity?" The purpose of this section is to describe some important functions of proof, and briefly discuss some implications for the teaching of proof.

Since they are software engineering students, ask them if it is still relevant to understand different sorting algorithms. After all, there are libraries.
The answer, of course, is that one should be able to understand how one's tools work, at least when one has reason to look. I recall a bug where a programmer was unaware that quicksort was not stable.

I don't understand your point. The students might say "yes, understanding sorting algorithms is important to us but not mathematical identities".

@user111388, they might say "why should I understand this? I just call .sort() and it would be quite unprofessional to reimplement the wheel. Existing libraries are better than anything custom for the project."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


text:
How can I survive a postdoc's probation period?
On Tuesday, I will be starting a postdoc at a highly prestigious UK institution. I have already been a postdoc for a year at a lower tiered institute, and I believe the probation was not much of a concern mainly due to that very reason. Additionally, a very nice and friendly PI and a casual (yet productive) environment, did not make the probation period much of a concern to me.
However, the current position that I will be starting is at my dream project / institution, and the environment seems more tense and fast paced (and add to that a bit of imposter syndrome). For this reason, the probation period has become an irrational cause for stress and anxiety. I suppose what I would like to know is what to avoid at all costs to ensure that this period passes without any issues.
Note: The probation period is six months and postdoc contract is three years.

Does this answer your question? How to effectively deal with Imposter Syndrome and feelings of inadequacy: "I've somehow convinced everyone that I'm actually good at this"

You shouldn't worry about "surviving the probation period". Your goal should be to do a good postdoc - overall, not only during or primarily during the probation period. After all, what is it worth to make it through the probation period, if you're not doing well in the rest of the postdoc - would you be any better off in that case?
Overall, I thus wouldn't worry too much about it - at least not more than striving to do a good postdoc altogether. It might even be that the probation period is a legal requirement for any kind of contract (not just postdocs) in the UK.
(One more comment: Given that you don't state at all how long the probation is, it is hard to give any concrete advice: The relevance of a  one-month probation is quite different from a 6-month or 1-year probation.)

Thank you for your useful advice. That is very true, however, I have come up with this crippling anxiety for somewhat of an irrational reason. Also you have a good point regarding the probation period; I can confirm that it is indeed 6 months (the post has been edited to reflect this).

Yes I noticed the mistake as soon as I inadvertently edited your comment. I have included the period of contract as well.

I wouldn't be too concerned with a 6-month probation - just make sure that after you start, you get engaged soon in the topics you should work on during the postdoc (again, might be very field- and supervisor-dependent how specific that is), show interest and independent thought, be engaged, contribute to scientific discussions in the group ... all the things which you (hopefully) anyway do as a postdoc. But it really depends on the subject.

Thank you very much for your input and the swift responses! May not seem like much, but very helpful.

Just be yourself. Pretending to be someone else for 6 months won't work, and then you don't want to work for 3 years with someone who wants to have (and after the probation believes they got) someone else than you are.

Don't worry about the probation period. This is mostly a formality.
Probation periods at UK universities exist mostly for policy reason (because the University imposes them on all employment contracts). They exist to have the legal option of terminating a contract, in truly problematic cases. They are not a tool (at least not for postdoc hires) to evaluate the quality of a hire.
In practice terminating a postdoc contract creates a lot of hassle for your supervisor. Not least because, they have go through the hiring process for a replacement, which will probably delay whatever project you are supposed to work on by a year. Moreover, if your position funded from a grant, there may not be enough money left in the grant to offer a full three year contract to your replacement, making it harder to find a high quality replacement.
So, unless you do something that makes your supervisor actively want to get rid of you at all costs, you will probably make it through probation.

The phrase that stood out for me in your question was "... an irrational cause for stress and anxiety". It seems that you are pretty much on top of your professional game, but that the mental stress is the main problem in its own right.
Have you considered talking to a coach/therapist? Professional help for anxiety is not reserved for people diagnosed or burned out. A few sessions might give you better insights than you can hope to get here.
Most importantly perhaps, enjoy and have fun. You have got your dream position at a prestigious institution. It seems that the imposter syndrome gets more intense the higher the prestige, but try to find comfort in the fact that out of all other potential candidates, the chair has your name on it. And in academia that is almost always for a reason. If you bring that confidence in your own ability with you into your work you'll most likely have a great time.

"It seems that the imposter syndrome gets more intense the higher the prestige": That's the point, isn't it?

It's your dream institution and your dream project, but this is just a patina over the work and the working environment you will be living in the next 3 years. And this patina is unfortunately not that relevant, in the long term.
Why?
If you are interested in the academic career, the postdoc is much less relevant with respect to what you do as research than the "management" aspects (i.e. applying for funds, delivering the bureaucracy on time, supervising younger guns, etc...).
My advice is to think of the probation period as a mutual evaluation period: they are evaluating you, but you are also evaluating your new workplace.
Focus on the first "long-term" deadline (1st year project report? 1st year paper? whatever), work towards that goal, at the same time be yourself, be open, be curious, try to get in touch with the research of your peers in the department .... and then after 3 months make your decision.
I think you will need 3-4 months to scratch the patina of your new working environment. If, after three months you are enjoying the project, the working environment, your work-life balance, the place and the project, it is unlikely you will be performing so badly that you do not pass the probation period. It is very easy to assess if a person is enjoying the work, it is very hard to assess quality of a PostDoc work after 6 months, and no one has the time to do that, so the evaluation will be on your motivation to keep on working on the project for 3 years [1], rather than on some metrics.
If you are not enjoying it, just focus on yourself and work hard towards getting your own funds for your own project, make your own plan B. If you have no academic interests, use the remaining months to sharpen your CV for your exit to the private world.
[1] pay attention not to corner yourself in a hole. They hired you for 3 years to work on a certain project, however your CV in 3 years down the road will benefit if you showed some independence in getting funds, supervising students, etcetc ... keep your possibilities open.

I agree with most of the answer, but very strongly disagree with the first sentence "If you are interested in the academic career, the postdoc is much less relevant with respect to what you do as research than the "management" aspects". Maybe that's field dependent - but when, if not during a postdoc, do you want to produce lots of research results, rather than wasting your time with bureaucracy? So, to the OP: This is the time to excel in research. Once you are a professor, you will be caught in bureaucracy, supervision, funding, etc.

@user151413 to put it simply: I absolutely agree that being a postdoc is the best time to do research, and it will be most likely the last thing you will do in the academia, if you do not prepare your professor career. Sad but (nowadays) true.


text:
Practical Tips: Mathematical research and discoveries

How to behave when you have the feeling of working on something innovative? What to do if there is a chance (even the 1%) that your
  work is leading you to something original?

For example what if I don't know mathematicians I trust to ask? Is a good idea to talk about your results (even if are not real results) to someone?

Question:
What do you do if you have this feeling?
I'm looking for a list (or links to related SE questions/guides/books or others)
  of practical tips to use while studying a certain subject you
  have the feeling of having found an innovative approach that can
  provide a new solution.

I am interested too in books about the mathematicians' way to research. (I think is related)

I already asked this question on Math.SE but was closed two times. Here the link to MathematicsSE

What do you mean by trust? Is this question related/duplicate?

@DanielE.Shub yes this qeustion is a bit related thanks for the link . Anyways that was only an example, my qeustion was more general.

There is an art to balancing "diverging" and "converging."  By diverging, I mean using your intuition to figure out what "should" be true - what you think you can prove, if you try.  Converging means actually proving it.  You don't want to get too far out there into the unknown without proving milestone lemmas to anchor you down, or you may find that much of your research is predicated on a false assumption.  On the other hand, forcing yourself to prove every little thing while suppressing your imagination simply won't lead to interesting research.

@AlexanderGruber yea, that is one of my problems too, if I try to formalize all I my ideas, the work becomes long and more i continue and more i lose the meaning of what I'm doing, but I'f travel with fantasy I continue to find a great number of ideas and connections and I come in a huge landscape of conjectures... Do you know some reading about this? there is maybe some point that I must remember while working in order to find a balance between  "diverging" and "converging."?

@MphLee Actually, the terminology "diverging and converging" comes from my previous career in design.  I don't know a good place to read about this, but it is a relatively common topic of conversation in art school.  The rules on when to diverge and converge aren't set in stone and are a little different for most people; however, my rule of thumb is not to get more than about "two steps ahead" of what I've already proven before I try to converge.  (After all, sometimes going back and proving the conjectures helps you think of new ones, too.)

0. To give original contribution to the understanding of mathematical objects is what mathematician do, and talk about all the time. There is a priori no need to fear for ideas being stolen; if you want to play secure, you can publish a preprint to ensure priority.
Given the wording of your question, I assume that you are an amateur mathematician, and that you feel that you may have found something about an important, well-known question. Sorry if this is not the case.
As most mathematicians, I receive demand for advice of this kind from time to time, so here is my usual answer.

Be prepared to have made a mistake,
be prepared to have found something known for a long time,
be prepared to have found something that will not attract interest.

This may sound very negative, but these are the worries that are much more likely to be relevant than seeing your contribution stolen. I have seen recently an amateur publishing on viXra after seeking advice from me, that was afraid of having her ideas stolen. It turned out that her contribution was a few hundred years behind current knowledge.
Researchers in mathematics only succeed in advancing knowledge because they spend much time learning their specialty and keeping up with what is being proven, and we do sometimes reinvent the wheel (I got scooped by 130 years once, realizing that Camille Jordan already solved a cute problem I was interested in), or make mistake, or do things that do not interest our colleagues. It is tremendously difficult to avoid this pitfalls when you don't have access to the literature, don't have colleagues to speak with about your research, don't have a regular seminar to listen to, don't have had a PhD advisor to guide you through your first problems.
So, for a positive piece of advice:
4. learn the field you are interested in (e.g. read books, from the point you are in your knowledge to the field you are interested in). Be prepared for this step to take much time.

1,2,3 are all very true. And sometimes, re-inventing the wheel can result in a highly cited paper. E.g. "Preferential Attachment" (http://en.wikipedia.org/wiki/Preferential_attachment) was re-invented several times (1925,1955,1976 etc) but still managed to get 16,787 citations as a Nature Paper in 1999.

@Benoît Kloeckner In order to be more concrete, I talk about me: 1st I don't know if my question is know, but I know that I never saw something similar (and I searched keywords), I feel to have found a way to look in this direction like a new theory, but more I work on this idea and more I undesrtand that I must study totally (at the first look) different fields because are all linked. So In thit moment I'm seriously asking myself if is good to ask someone or not (because I do not have many theorems):

if I don't ask probably I'll stay forever on a fake problem building useless and wrongs constructions (or already known with other names) and anyways very slowly, If I do and there is a good idea, an expert mathematician can do 100 times faster what I was going do and "cut me out" forever.

But I must remember that my question was more general too, was more about pratical sides of the whole process, from the beginning to the pubblication. But thanks for the answer Is very usefull.

@MphLee: if you feel that someone else could easily appropriate your ideas, there is even less probability that they avoid 1.2.3. : innovative mathematics are often (though not always) hard to understand for everyone, possibly (but not necessarily) except for its author.

Another point: you are reluctant to discuss your ideas, but it is often more difficult to get people interested in them than likely that they steal them from you.