About Homework Solutions

Some Perspective

I am not a big fan of preparing homework solutions. I have written a "problem book" --- the world-famous Cauchy-Schwarz Master Class, and from that experience I know that writing the solution to a problem is no easier than writing any other part of a book. In some ways, writing a good solution may even be harder than writing the linear text, just as poetry is typically harder to write than prose.

So, writing good solutions is hard, but that is not the only reason not to give out solutions. Other reasons are perhaps more important:

• In a good graduate class almost all of the students should get almost all of the problems almost perfectly right. This means that solutions are not needed.
• If a person simply made a slip, then the mark on the graded homework will set one right --- again no need for a solution.
• If a person did not know how to solve a homework problem because material from the lecture or the text was not understood, then a solution does not help ---only studying the text or the notes can help.
• Finally, reading a solution is an entirely different kind of action than finding a solution. Great skill at reading solutions has limited usefulness, while great skill at solving problems is one of the most powerful things you can learn.

And More --- on HW Questions via E-mail

In general I am happy to comment on a HW question via e-mail (if there is not a deluge!), but there are two versions of requests were I can't be of much help.

• Version 1. "Can you help me find my mistake?" Such a exercise forces me to go down a trail of (often messy) errors. All I can really say here is "if you write this out clearly you will find your own mistake."
• Sometimes I say, "Here is a hint." Rarely will I have the time or patience to sort out someone else's hard argument when I already know an easy one. All of the arguments in this course are easy (mmm... if you know them).
• Version 2. "Is this right ... followed by uncompiled latex." You do have some chance of getting a useful answer from me if you send me a short, carefully written pdf, but I don't like to compile any latex in my head that is more than six keystrokes long... e.g. x \leq y is the limit.

In office hours I can sometimes get talked into a "find the mistake" exercise, but you'll usually see me growl and grimace. I do know that sometimes a person can get "hooked" and really wants to know if a certain idea can work. On the other hand, it is really best not to get "hooked." If an idea gets too confusing or too messy, be lazy --- look for an easier argument. That's what the "Pro's" do.

In my experience, if you think your solution may be wrong, it is probably even worse than you think. It is good to have doubts about solutions. By resolving those doubts for yourself, you greatly deepen your understanding. This is the much of mathematical discovery proceeds.

Also, if you are tempted to cite some theorem that we have not developed in class, then even if you happen to be "right" you haven't really landed on the place you really should want to land. What we do here is pretty pretty basic --- but the goal is mastery of the basic material we do cover.

Examples of comments that I love? "On Problem 15, don't we have to assume that X has a finite variance?" Quite often, you are right and I left out a hypothesis. Other times, I can explain, "No, the charm of this problem is that we can overcome the objection that there is no finite variance."

It is also fine to say, "I have worked on this for two hours, are you sure it is right? Can you give me a hint?" if this appeal does not come too often, I am usually happy to oblige.

Back to the 530 Homepage