Yesterday, I came across an interesting post by Allison at Infinigons. She reminds us that we must remain fearless in the face of difficult problems, and attempt to do something, rather than give problems a cursory once-over, then run to Google. It is worth reading her post for the details.
The example she gives is a fun problem, which I would be happy to discuss with any of my three or four readers, should they wish to comment below. The answer that I got was \(\frac{4\sqrt{2}}{3}-\frac{5}{3}\approx 0.219\), if you want to check your (or my) work (I’m pretty confident of that solution).
That being said, I think that the emphasis of the linked post should really be placed upon the willingness to think about and work with hard problems, in mathematics and beyond, without instantly looking for an authority to answer the problem.
I have noticed (though I don’t have any statistics to back this up) that my students are often unwilling to spend time working through problems. All of their homework is done online, as per the department’s wishes, and the website offers many resources that make it easier for students to get through the problems without struggling. Then they come to class and do poorly on quizzes, and often turn in problems that have not even been attempted! They will literally sit through a 15 minute quiz with three problems, and write nothing more than three frowny-faces and an apology for not knowing how to complete the problem. I find this behaviour to be very frustrating, and I think it reflects an unwillingness face hard problems head on.
Of course, I am guilty of the same crime myself. I have begun my thesis research, and have been working through a book on dimension theory (hence the last major post I wrote). At the end of chapter two, there was a particularly difficult problem that I was working on. I struggled with it for a couple of days, but felt like I wasn’t making much progress. So I spent two hours trying to hunt down a solution on Google.
Fortunately for me, there was no solution, and after a couple of meetings with my advisor (who also struggled with the problem), I think that I finally managed to come up with a reasonably solid proof. The feeling of accomplishment is profound, and I am now quite happy that I could not find a proof online.
It what may be a truly sad example of the same behaviour, I picked up a copy of a game called Shadow of the Colossus a few years ago (if you play video games, and have a PS2, this is a must-play game). I struggled mightily with the final boss, and after two or three attempts, gave in and looked over one of the walkthroughs on GameFAQs. It turns out the that solution was mind-bogglingly simple, and I felt kind of dumb for not figuring it out myself. Moreover, defeating that final boss never gave me the sense of accomplishment that it might have otherwise. I still feel a tinge of shame whenever I play that game (which I do quite a lot, because it is a very good game). I wish that I had struggled with it for more time, and figured it out on my own.
I think that there is value in working through difficult problems without appealing to authority. Ultimately, we expect our students to be the leaders and decision makers (even if only by voting for elected officials and ballot measures) of our society, and making the right decisions requires that they are capable of thinking through hard problems without giving up and blindly following pundits and demagogs. We all need to be able to think and decide for ourselves.
So let it be resolved: my students need to work on hard problems, and I need to be a better role model. For the remainder of my tenure as a graduate teaching assistant, I will do everything in my power to provide my students with problems that require them to think and struggle, and I will provide only as much feedback as is required to get them to think through the problems on their own. Moreover, in my own research I will cease to use the internet, the library, and the back of the book as sources of answers. When the problems get difficult, I will buckle down and work them out on my own. I will do everything that I require of my students.