Advice on Advice

She generally gave herself very good advice (though she very seldom followed it)

                                        —Alice’s Adventures in Wonderland, by Lewis Carroll

I've given advice elsewhere. Here, I'll give some advice about advice.

My most important advice on advice is to value advice from those who have experience and expertise more than advice from those who don't.

When I arrived at the University of California at Irvine as a professor, I was told that the math department had a program where math majors could request a mentor to advise them on applying to grad school, and faculty could volunteer to mentor students. 

Over the years, students and colleagues had sought my advice and seemed to value it. So I volunteered to be a mentor, and was assigned a senior.

The senior wanted to get a PhD in mathematics, but insisted on only applying to Masters degree programs. She thought that a Masters degree was a prerequisite for applying to a PhD program.

I advised her to apply to PhD programs, for several reasons. Such programs would provide funding, while Masters programs were more likely to make students pay. At my previous university, Ohio State, the bar was higher for students who already had Masters degrees than for those who didn't, so a Masters degree could be a disadvantage when applying for a PhD program. If she got a Masters degree and then decided to go elsewhere for the PhD, she could still do so. Plus, graduate admissions committees that felt she wasn't ready for the PhD program could decide to admit her to the Masters program.

The senior told me I was wrong. Why? Because some first year grad students told her that one doesn't get into a PhD program without having a Masters degree. She believed that they knew more than I did about getting into grad school, even though I had served on graduate admissions committees and the students hadn't.

Other professors gave the senior the same advice that I had. I hoped that by getting the same advice from other faculty, she'd learn to trust me and find my mentoring more useful. But the senior only applied to Masters programs, and didn't come to see me again.

Sometimes the right people to ask for advice are the people who know you well. (Knowing you well also counts as experience or expertise.) I occasionally get emails from students I don't know, in various parts of the world, asking me for advice specific to their situations. While I can sometimes give them very general advice, I emphasize that for specific advice, much more valuable is the advice they should get from professors who know them well and the people from whom they've taken courses. 

My advice to advisors is to make clear the limits of your relevant expertise and experience, and state your best guess as to how much confidence the advisee should or shouldn't have in your advice. If appropriate, suggest who else might be able to give more reliable advice.

I'll end by emphasizing the limits of my experience and expertise to advise on advice. Don't just listen to me. Be open to advice from anyone, paying special attention to those with expertise and experience, and to those who know you or your situation well.

An All-Star Cast

The first mathematics talk I ever gave was a job talk at Brown University on February 1, 1984, and my second talk was in the Harvard Number Theory seminar exactly a week later. It had an all-star cast, including Harvard Professors Barry Mazur and John Tate and Yale Professor Serge Lang.

Right before the talk, I noticed that some of my friends, who were sitting in the front row, were giggling. It was a bit disconcerting when they refused to tell me what they were giggling about; they said they'd tell me afterwards.

I began my talk, and started to write on the blackboard. 

Serge Lang screamed, "Stop! Stop! I can't stand it any more!" 

I stopped. I had barely said anything. Could I have already said something wrong? What could I have said that would upset Serge Lang so terribly?

"The chalk is screeching on the blackboard!" he explained. 

Tate told me to break the chalk in half so it wouldn't screech. I pointed out that all the chalk was tiny---too small to break in two.

We all waited while Barry Mazur ran out of the room, and ran up and down the fifth floor hallway, looking for an open office with chalk. 

Eventually, Barry came back with the report "No chalk". What was I to do? 

Tate handed me a chalk holder into which I could insert the tiny chalk and use it in a way that wouldn't screech. 

Serge calmed down. (He probably went to sleep; I didn't hear from him again.) My talk proceeded uneventfully. But I learned how many Harvard professors it takes to calm down Serge Lang.

What were my friends in the front row giggling about? They were trying to decide whether to tell me that the newly-famous mathematician Gerd Faltings was in my audience. I hadn't yet met or seen him, so I didn't know he was there. That's just as well. There were enough stars to worry about.