**Career advice by Terence Tao**

**Advice is what we ask for when we already know the answer but wish we did not. (Erica Jong).**

Here is my collection of various pieces of advice on academic career issues in mathematics, roughly arranged by the stage of career at which the advice is most pertinent (though of course some of the advice pertains to multiple stages).

*Disclaimer*: The advice here is very generic in nature; I don’t pretend to have any sort of “silver bullet” that will solve all career issues. You will of course need to evaluate many factors, contexts, and needs specific to your own situation, as well as employing a healthy dose of common sense, before making any important career decisions. I would in particular recommenddiscussing such decisions with your advisor if you have one, as he or she will be familiar with your situation and will likely be able to provide pertinent advice. Also, it should be clear that most of this advice is targeted towards academic careers in mathematics; of course, there are many other career options available besides this, but I have no particularly informed advice to offer for such alternatives.

**Talk to your advisor**

**It is the province of knowledge to speak and it is the privilege of wisdom to listen — Oliver Wendell Holmes — The Poet at the Breakfast Table.**

Your advisor is one of the best sources of guidance you have; not only in directly assisting you with your research topic, but in directing you (both explicitly and implicitly) to relevant researchers, conferences, publications, open problems, folklore, or other pieces of good mathematics. Your advisor also knows your situation well and can give career advice which is tailored to your specific strengths and weaknesses (unlike the generic advice in these pages).

If things get to the point that you are actively avoiding your advisor (or vice versa), that is a very bad sign. In particular, you should be aware of your advisor’s schedule, and conversely your advisor should be aware of when you will be available in the department, and what you are currently working on.

For similar reasons, you should give your advisor some advance warning if you want to take a long period of time away from your studies.

If your advisor is unavailable, you should regularly discuss mathematical issues with at least one other mathematician instead, preferably an experienced one. [Also, it is not uncommon for a student to have both a formal advisor, who handles all the official paperwork, and an informal advisor, with which you discuss research and career issues.]

Of course, you should not rely *purely* on your advisor; you also need to take the initiative when it comes to your mathematical career.

Primary school level

- Advice on gifted education
**If you can give your son or daughter only one gift let it be the gift of enthusiasm. — Bruce Barton.**

Education is a complex, multifaceted, and painstaking process, and being gifted does not make this less so. I would caution against any single “silver bullet” to educating a gifted child, whether it be a special school, private tutoring, home schooling, grade acceleration, or anything else; these are all options with advantages and disadvantages, and need to be weighed against the various requirements and preferences (both academic and non-academic) of the child, the parents, and the school. Since this varies so much from child to child, I cannot give any *specific* advice on a given child’s situation. [In particular, due to many existing time commitments and high volume of requests, I am unable to personally respond to any queries regarding gifted education.]

I can give a few *general* pieces of advice, though. Firstly, one should not focus overly much on a specific artificial benchmark, such as obtaining degree X fromprestigious institution Y in only Z years, or on scoring A on test B at age C. In the long term, these feats will not be the most important or decisive moments in the child’s career; also, any short-term advantage one might gain in working excessively towards such benchmarks may be outweighed by the time and energy that such a goal takes away from other aspects of a child’s social, emotional, academic, physical, or intellectual development. Of course, one should still work hard, and participate in competitions if one wishes; but competitions and academic achievements should not be viewed as ends in themselves, but rather a way to develop one’s talents, experience, knowledge, and enjoyment of the subject.

Secondly, I feel that it is important to enjoy one’s work; this is what sustains and drives a person throughout the duration of his or her career, and holds burnout at bay. It would be a tragedy if a well-meaning parent, by pushing too hard (or too little) for the development of their child’s gifts in a subject, ended up accidentally extinguishing the child’s love for that subject. The pace of the child’s education should be driven more by the eagerness of the child than the eagerness of the parent.

Thirdly, one should praise one’s children for their efforts and achievements (which they can control), and not for their innate talents (which they cannot). This article by Po Bronson describes this point excellently. See also the Scientific American article “The secret to raising smart kids” for a similar viewpoint.

Finally, one should be flexible in one’s goals. A child may be initially gifted in field X, but decides that field Y is more enjoyable or is a better fit. This may be a better choice, even if Y is “less prestigious” than X; sometimes it is better to work in a less well known field that one feels competent and comfortable in, than in a “hot” but competitive field that one feels unsuitable for. (See alsoRicardo’s law of comparative advantage.)

My own education is discussed in the following articles. While I am very happy with the way things turned out for me, I would again caution that each child’s situation, strengths, and weaknesses are different, and that my experience might not necessarily be the ideal template to follow for others.

- “Terence Tao”, Ken Clements, Educational Studies in Mathematics, August 1984, Vol. 15, No. 3, 213-238
- “Parental involvement in Gifted Education”, Billy Tao, Educational Studies in Mathematics, August 1986, Vol. 17, No. 3, 313-321
- “Radical Acceleration in Australia: Terence Tao”,Miraca Gross, G/C/T, July/August 1986
- “Insights from SMPY’s greatest former child prodigies: Drs. Terence (“Terry”) Tao and Lenhard (“Lenny”) Ng reflect on their talent development”, Michelle Muratori, Julian Stanley, Lenhard Ng, Jack Ng, Miraca Gross, Terence Tao, Billy Tao, Gifted Child Quarterly, Fall 2006, Vol. 50, No. 4, 307-324

For professional advice on gifted education, I can recommend the Center for Talented Youth. See also my page on career advice.

- High School Education
- Advice on mathematics competitions
- Sports serve society by providing vivid examples of excellence. — George Will.
- I greatly enjoyed my experiences with high school mathematics competitions (all the way back in the 1980s!). Like any other school sporting event, there is a certain level of excitement in participating with peers with similar interests and talents in a competitive activity. At the olympiad levels, there is also the opportunity to travel nationally and internationally, which is an experience I strongly recommend for all high-school students.
- Mathematics competitions also demonstrate thatmathematics is not just about grades and exams. But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does.(While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)
- Also, the “classical” type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different from the “modern” mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see that the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modern algebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)
- In summary: enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful.
- For advice on how tosolve mathematical problems, you can try my book on the subject.
- Some collected quotes on mathematics competitions can befound here.

Which universities should one apply to?

**A college degree is not a sign that a one is a finished product but an indication that a person is prepared for life. (Edward Malloy).**

Going to college is a major event in one’s education, but the choice of exactly which college to go to is not as critical as it is sometimes portrayed to be; usually, there will be *several* good choices that suit your specific strengths and weaknesses, and it is not absolutely necessary to secure the “best” choice for your undergraduate or graduate education. I would recommend a flexibleattitude towards this decision; by focusing too much on one institution, you might overlook others which may in fact be a better fit for you.

It is common to focus on the general prestige of the institution, but actually it is the *specific* strengths of an institution which should play a more important role in your decisions. Examples of specific strengths include particular research strengths, teaching programs or initiatives, campus resources, academic culture, location, flexibility, affordability, availability of financial assistance, and so forth. On the other hand, given that your interests or situation may change somewhat as you learn more about your chosen field, one should not be too narrowly focused in one’s selection criteria; for instance, if you wish to go to an institution purely because of a single faculty member there, then you might run the risk that that faculty member moves, or is no longer accepting students.

I do however strongly urge that you study at different places; it’s good to move a little bit out of your own “comfort zone” and broaden your education. It’s also good to talk to your advisor about these matters.

My final advice is to have no regrets once one has made one’s choice; get the most out of the place one has chosen, and don’t spend too much time worrying about whether the grass would have been greener elsewhere. In particular, I would not recommend trying to “have the best of both worlds” by somehow trying to study simultaneously at your two top choices; this is very complicated to execute and usually does not work out very well.

I myself earned my undergraduate degree at Flinders University in my hometown of Adelaide, Australia – a small and not widely known institution, but one which was very friendly, close to home, and whose maths department was willing to accommodate my unusual educational experience. My graduate degree was at (the somewhat better known) Princeton University, which turned out to be a good fit for me, as I ended up with an excellent advisor and a challenging, self-driven environment which shook up my complacency about my own mathematical knowledge. My first postdoctoral position was at UCLA, which I liked so much that I have stayed here ever since, even though some of the faculty that I originally came to UCLA to work with have since left. Of course, there are many other good schools, which each have their own strengths and weaknesses. (For example, if the activity of big-city life is important to you, then Princeton does not fare terribly well in this regard.)

Undergraduate level:

- How can one become better atsolving mathematical problems? Note that there is more to maths than grades and exams and methods; there is also more to maths than rigour and proofs. It is also important to value partial progress, as a crucial stepping stone to a complete solution of a problem.
- Don’t base career decisions on glamour or fame. But you shouldstudy at different places.
- Does one have to be a genius to succeed at maths?
- Solving mathematical problems:

*Chaque vérité que je trouvois étant une règle qui me servoit après à en trouver d’autres [Each truth that I discovered became a rule which then served to discover other truths].* (René Descartes, “Discours de la Méthode“)Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one. Later in your research career, you will find that problems are mainly solved by knowledge (of your own field and of other fields), experience, patience andhard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills. I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies. There are of course several other problem-solving books, such as Polya’s classic “How to solve it“, which I myself learnt from while competing at the Mathematics Olympiads.

Solving homework problems is an essential component of *really* learning a mathematical subject – it shows that you can “walk the walk” and not just “talk the talk”, and in particular identifies any specific weaknesses you have with the material. It’s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.

I find that “playing” with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See “ask yourself dumb questions“.

It’s also best to keep in mind that obtaining a solution is only the short-term goal of solving a mathematical problem. The long-term goal is to increase your understanding of a subject. A good rule of thumb is that if you cannot adequately explain the solution of a problem to a classmate, then you haven’t really understood the solution yourself, and you may need to think about the problem more (for instance, by covering up the solution and trying it again). For related reasons, one should value partial progress on a problem as being a stepping stone to a complete solution (and also as an important way to deepen one’s understanding of the subject).

See also Eric Schechter’s “Common errors in undergraduate mathematics“. I also have a post on problem solving strategies in real analysis.

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Prof Terence Tao blogs at https://terrytao.wordpress.com

If you like the above, do send a word of appreciation to him…

More later,

Nalin Pithwa