Never do anything that bores you. My experience in science is that someone is always telling to do something that leaves you flat. Bad idea. I’m not good enough to do something I dislike. In fact, I find it hard enough to do something that I like. (James Watson)
A high school student who is preparing for college recently asked me a few questions by email. He wants to study computer science. I thought that his email was great so I will reply publicly. Let us call him John.
John says… I intend to major in Computer Science (though I’ve also considered majoring and/or minoring in neuroscience or math, of some sort). So, I’ll obviously be taking math courses. Now, for most of my life, I’ve pretty much hated math. I had a poor early educational experience in general and came to despise math, even though I typically got good grades.
I don’t think we should be surprised by John’s feelings about mathematics. We spend a great deal of time teaching stupid things like how to divide 46712 by 13 using a long division, with a pen and some paper. Or adding fractions like 3/5 and 7/9. That would be like getting students to take 5 years of Microsoft Office classes and then be surprised that they hate “computer science”.
How do you approach learning new math concepts? And, in particular, how do you understand formulas? (…) I feel like I am probably missing the bigger picture somehow, and failing to gain a true understanding of math.
I work from examples that I care about. In fact, I avoid learning new mathematical concepts for their own sake. It is a matter of motivation. Having a practical problem I care about activates my intelligence. Boredom makes me dumb.
Have you learned “new math” since obtaining your Ph.D.? When you go to work every day, do you gain new insights and learn things about math, like, say, a programmer would?
I do not use a lot of mathematics as a computer scientist beyond the basics: set theory, Boolean logic, elementary probability theory, some descriptive statistics, functions and relations, linear algebra. You can go a very long way with this small amount of mathematics.
In my experience, mathematics suffers from a diminishing return effect. The basics are very, very useful… but more advanced mathematics often turns into solutions seeking problems. It is best to learn the most advanced mathematics as needed.
John says… A lot of math involves calculations, but I’ve begun to realize that calculating isn’t really what math is about. It seems math is a far more abstract topic and calculating is just an “unfortunate” chore that must be done before having the real fun. Problem is, up to now I’ve always thought that the calculations were the math, and that was all there was to it. How do you abstract away from the rote number crunching and gain an intuitive grasp of math concepts? Must one be good at quick mental calculations and basic math to truly understand and enjoy math?
Fast and accurate number crunching might be useful to impress teachers. But computers are about a billion times better than the best human being could ever be.
John says… How do you learn new math? When you do, have you always found it necessary to work through lengthy sets of problems involving the new concept? If so, how do you learn math outside of textbooks and college courses when problem sets are not present? Have you always just learned from college courses? Do you find math as presented in textbooks dry? Is there, perhaps, another, better way to learn math?
I have gotten rid of my books several years ago. I am a university professor, but if you come to my office, I have no textbook. For some classes I teach, I expect the students to get a textbook, but that’s mostly to satisfy students’ expectations.
Solving problems is definitively the best way to learn mathematics. For most topics, there are thousands of great problems online. There are also great YouTube videos and so forth.
It is not that textbooks are bad… but learning mathematics from a handful of textbooks is like trying to learn about love by reading a handful of romance novels. You need to get out there, find your own references.
Obviously, you have a Ph.D. in math. If you could do things over again, would you get the Ph.D., or not? Would you major in another field?
Degrees, especially advanced degrees, are overrated. You sometimes need degrees to get through the door. I bet that it is a lot easier to get an interview with Google if you have a computer science degree. If you want to get a research job, the Ph.D. makes things easier. But, the degree itself is a small thing.
Think of it as being clean. It is hard to get a job if you are dirty. But if you clean yourself up, you may still never get the job you want. The fact is that is hard to get good jobs, even with a computer-science degree.
Whether you have a college degree or not, only about a third of all workers are enthusiastic about their work. A degree gives you access, potentially, to higher paying jobs… but it won’t make your career.
So don’t go just do a degree. Do something cool and fun on the side. Build a mobile app. Build a VR mini-game. Design a website. Build a new AI. Find something you like doing and get good at it. It comes down to the same idea: work on examples you care about.
Would I still get a Ph.D.? That’s a bad question. A better question: Do I advise students to get PhDs? No. The rule is that you should only get a Ph.D. if you cannot imagine living without one. Would I discourage my own boys from getting a Ph.D.? Yes. Do I encourage them to think about college? No. I repeatedly, tenaciously, discourage my kids from focusing on diplomas and degrees. I keep telling them not to wait for the teachers. Do cool stuff now, on your own.
A degree can be a useful tool, but that’s all it is. If that’s all you have, you have no career. You are at a high risk of ending up in a dead-end job if you get a job at all. There are more boring jobs out there than you know.
Do you have any suggestions or advice for a soon-to-be college student? I’m a pretty solid introvert, and mostly keep to myself – so I’m not looking for the “Don’t party, focus on your studies” advice :). I’m more looking for intellectual sort of advice. Perhaps suggestions on how to get more out of college than I otherwise might?
- Courses, professors, and grades matter less than you might think. Make sure you get the required grade. Make sure you take the right course but don’t worry more than needed about it. In the real world, employers don’t care whether you aced your math. courses.
- The people you interact with (and that’s not necessarily the professors) matter a great deal. If you are studying a topic, try to meet people either on campus on online who are interested in this topic.
- Don’t focus on the degree and the courses. Build strong side interests. Acquire a demonstrable expertise. Do something that people might care about and then tell the world. Remember that it is how Linus Torvalds started. It is how the web started. Who cares whether Linus got a good grade in math? Just do stuff, and do not wait passively for your teachers to tell you what to do. Again, by doing projects you care about, you are going to learn a lot more than you could otherwise. And learn to tell the world about what you do. Communicating your ideas is just as important as doing the cool work. Fanciful mathematics should be far down the list of priorities unless that is your thing.
What if you are not interested in anything, and you prefer to just go to your classes and maximize time spent in front of the TV? If you don’t know what to do, and you prefer being told what to do, that’s fine. Just don’t complain later when people pick your work for you.