## How do I know if I’m good enough to go into math?

*Hi Cathy,*

*I met you this past summer, you may not remember me. I have a question.*

*I know a lot of people who know much more math than I do and who figure out solutions to problems more quickly than me. Whenever I come up with a solution to a problem that I’m really proud of and that I worked really hard on, they talk about how they’ve seen that problem before and all the stuff they know about it. How do I know if I’m good enough to go into math?*

*Thanks,*

*High School Kid*

Dear High School Kid,

Great question, and I’m glad I can answer it, because I had almost the same experience when I was in high school and I didn’t have anyone to ask. And if you don’t mind, I’m going to answer it to anyone who reads my blog, just in case there are other young people wondering this, and especially girls, but of course not only girls.

Here’s the thing. There’s always someone faster than you. And it feels bad, especially when you feel slow, and especially when that person *cares* about being fast, because all of a sudden, in your confusion about all sort of things, speed seems important. But it’s not a race. Mathematics is patient and doesn’t mind. Think of it, your slowness, or lack of quickness, as a *style* thing but not as a shortcoming.

Why style? Over the years I’ve found that slow mathematicians have a different thing to offer than fast mathematicians, although there are exceptions (Bjorn Poonen comes to mind, who is fast but thinks things through like a slow mathematician. Love that guy). I totally didn’t define this but I think it’s true, and other mathematicians, weigh in please.

One thing that’s incredibly annoying about this concept of “fastness” when it comes to solving math problems is that, as a high school kid, you’re surrounded by math competitions, which all kind of suck. They make it seem like, to be “good” at math, you have to be fast. That’s really just not true once you grow up and start doing grownup math.

In reality, mostly of being good at math is really about *how much you want to spend your time doing math*. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine. Plus, thinking about things overnight always helps me. So sleeping about math counts as time spent doing math.

[As an aside, I have figured things out so often in my sleep that it's become my preferred way of working on problems. I often wonder if there's a "math part" of my brain which I don't have normal access to but which furiously works on questions during the night. That is, if I've spent the requisite time during the day trying to figure it out. In any case, when it works, I wake up the next morning just simply knowing the proof and it actually seems obvious. It's just like magic.]

So here’s my advice to you, high school kid. Ignore your surroundings, ignore the math competitions, and especially ignore the annoying kids who care about doing fast math. They will slowly recede as you go to college and as high school algebra gives way to college algebra and then Galois Theory. As the math gets awesomer, the speed gets slower.

And in terms of your identity, let yourself fancy yourself a mathematician, or an astronaut, or an engineer, or whatever, because you don’t have to know exactly what it’ll be yet. But promise me you’ll take some math major courses, some real ones like Galois Theory (take Galois Theory!) and for goodness sakes don’t close off any options because of some false definition of “good at math” or because some dude (or possibly dudette) needs to care about knowing everything quickly. Believe me, as you know more you will realize more and more how little you know.

One last thing. Math is not a competitive sport. It’s one of the only existing truly crowd-sourced projects of society, and that makes it highly collaborative and community-oriented, even if the awards and prizes and media narratives about “precocious geniuses” would have you believing the opposite. And once again, it’s been around a long time and is patient to be added to by you when you have the love and time and will to do so.

Love,

Cathy

Yes, yes, and yes. Right on, mathbabe!

Nice reply. One of the most thoughtful math-brains I ever met was not a fast learner or fast problem solver. But they had the patience and passion to go way, way further than many who were quick at doing exams – but didn’t really care about the topic in the same way.

Educational assessment rewards the fast over the deep – particularly in secondary and college levels. And I think this may indeed be selecting for the wrong thing. For one thing it means that people with a deep and slow style become excluded from the system, and are not in a position to give feed back their insights.

p.s. fun to find another person who solves problems when asleep – ! my best time is when I kind of almost wake up at 4am – but then dive back…;-)

Thank you so much for addressing this important question. I couldn’t agree more with your perspective. The culture of math competitions can paint a narrow view of a broad and creative field. I am close to several talented mathematicians who did very well in math competitions and now struggle with the internal expectations is set up for them. Great post.

Sorry if the reply above was unclear. I once scored a 0.

Having been one of those ‘fast’ people all my life (and to make it worse, caring about being fast for most of my life…) I wholeheartedly agree with mathbabe. Fast or slow makes a different only in relatively artificial situations like competitions or tests; in real life it just isn’t important. Of the total time from ‘start’ to ‘finish’ of a real world challenge or task, the time it takes to ‘solve the core problem’ is always just a very small part: a much bigger part is checking, explaining, convincing, educating, specifying the implementation, doing the implementation, checking the implementation….if you enjoy what you do, math or anything else, and you care about doing it well, how fast you are won’t matter for your happiness or even for your success. (And btw, the sleep learning thing is well documented: you can ‘prime’ your brain by casually reviewing material you’re trying to internalise before you go to sleep, and your brain will continue to work on it whilst you sleep. Totally awesome and very, very useful if you’re a student!)

Wholeheartedly agree.

The best measure of your mathiness is how you face a problem you’ve never seen before, and have no tools to reason about. From what it sounds like, you have been exhibiting more mathiness than your “quick” peers.

The truth is that speed will come in time with exposure and hard work, even if it doesn’t feel like speed. You will learn to ask the right questions (which you still won’t have answers to) and try the right examples, and you will re-learn the same subject many times over as you hone your understanding (can’t say how many times I’ve learned calculus).

Expect your whole mathematical life to be one of not knowing things, and the more advanced you get the more you find you don’t understand. It’s good that you get comfortable with this feeling early on. And be proud that you continue to love math in spite of it.

Hello,

I’m a fifty six year old man. I am going into a field related to my years of construction, Geomatics/geographic data collection. I only used basic math like fractions and such. I am going back to school getting my generals done and on to a BS in Geomatics. My question to you is, how do I know if my mind is up to the task of gaining proficiency in Trig and Calculus to go along with my desire to change careers. As far as I know I have no brain damage to prevent me from going down this path.

Paul

One of your best post ever. I do hope that you helped him (or her) and many others. Most of my high school life I was myself as Jan-Peter, on the fast side. Later I realized this was silly, and also that there are a lot of ways or styles of approaching a problem in math. This last bit represents a huge thing for me, because sometimes I sar people solving problems with approach that I found it hard to grap, and later on, I found another away to solve the same problem, and that I found it much easier. (I, for instance, love combinatory reasoning, and learned about it studying Feller’s book on probability).

One of the hardest things in math is to start when you don’t know the answer, indeed when you don’t know what technique will get you the answer. This is very different from math problems, where people can be smart and confident because they have the answer drilled into them (multiplication worksheets) or they know the right formulae. At some point in everyone’s career, they hit the point where they can no longer know instantly how to solve a problem, and they have to slow down and learn the essential skill of proceeding without knowing the answer. Some people hit this in high school, some in college, some in graduate school, and the longer they’ve been able to put off confronting it, the greater the jolt when they do. The faster students can panic when suddenly they don’t know the answers, and the slower students, who learned years earlier that the heart of mathematics is trying to figure out what methods to use, pass them by, working steadily. Sometimes the fast students learn the lesson and catch up again; sometimes they fall by the wayside. Mathematics, after all, is not like politics: it cares about right answers, not fast ones.

took me a while to understand what you want to say here, but I think I got it now :-)

I was always one of the slower ones

Putting aside the point about speed, which has obviously touched a nerve . . .

How do you know if you are good enough to go into math (or some other field)? Well, what does it mean to be good enough? Two possibilities stick out for me:

(1) You can make a living doing it

(2) You can contribute to the field

For (1), math has an easy answer in most countries right now: if you are enthusiastic and put in a reasonable amount of time, there are many things you could get paid to do that are related. “Going into math” for this student probably keeps open many more options than it closes. If, instead of math, the question were about basketball (or most other sports, many areas of art), I suspect the answer would be very different.

For (2), I don’t see how anyone can set a reasonable test for a high school student that will help predict this. Frankly, my guess is that most people in most fields don’t really contribute much and just push the field along in pedestrian ways. However, most great advances seem to come from some combination of this background effort, individual effort, social interaction within the field and other factors (luck?) and are not really predictable. Being enthusiastic, keeping at it, and enjoying the process along the way over an extended period of time seem the only reasonable ways to increase your chances.

The concept that “most people in most fields don’t really contribute much and just push the field along in pedestrian ways,” is entirely true, but not in a negative way. It is only in a culture (as exists in the US) that glorifies the superstars but minimizes the contribution of the rest of the team, that students think that they shouldn’t go into a field in which they can’t hope to win a Nobel prize (or Fields medal, etc).

I can’t speak to math as a field (I’m in medicine), but if only the superstars did research, we wouldn’t be advancing much in science. It is the mundane discoveries that provide the platform upon which someone (with brilliance, collaborators and colleagues, a professional network, and luck and timing) makes that final jump that results in a “great advance” and recognition on the larger stage. .

“Whenever I come up with a solution to a problem that I’m really proud of and that I worked really hard on, they talk about how they’ve seen that problem before and all the stuff they know about it. ”

Of course, everything seems easy if you already know the solution. The real learning process (and the true joy of math) comes from working really hard, and finally discovering the solution on our own.

Honestly it is very difficult for you to know, because the math environment has changed a lot in the last few decades, and therefore high school math may not be a reliable indication of your ability to do advance math in the future. Let me be more specific :

A lot of what constitutes high school math has been thoroughly automated by mathematical software such as Mathematica or Sage or Matlab. These tools, and indeed a deep understanding of computer science, shifts the burden of work towards different areas.

To give you a warrior analogy, high school trains you to be a samurai, expert in wielding the sword, whereas in the real battlefield, one asks from you to be a sniper. It doesn’t mean that there are not common qualities between the samurai and a sniper – actually there are many (situational awareness, courage, hand eye coordination) – but it is not exactly the same sport. In uni and the lab, you will of course take “sniper courses”, but it is a good idea to get acquainted with them quickly so :

a) don’t hesitate to read “popular math” books written by luminaries of the subject. If you click while reading then, it is a good sign

b) become good at coding.

An additional advantage of b) is that a coder with strong math background has the potential to be an extremely productive coder (when I say extremely, it means that you are able to do the work that is usually done by a team of 10 to 50 people), and that can be a lucrative endeavor if academic math turns out not to be your thing

Love this post. Will show it to my daughter.

Every word she says is true!

I am a mathematician but I work as an engineer.

We do not put problems on the board.

Except our own problems on our own boards.

We get weeks or months to work problems that have no Right answer.

They have to be broken into smaller and smaller problems whose solutions help solve the bigger problem. No one much cares where the solution to a single step came from or how you got it.

The special skill I bring to engineering along with my background in math ( which impresses the hell out of engineers!) is the ability to see which approach is going to reveal the most informative results in the end. Anyone can learn to solve a familiar problem, the trick is getting real life broken down and organized into a solvable problem.

If you love math, do math.

Be familiar with many areas of math as you go forward. Then if no specific field of math calls you in ways you can’t say no to, don’t do research. There are many many other areas of work that need your skills too.

Late in life I started attending drawing classes. There, I learned to forget about the outcome, and to enjoy the process (whereupon the outcome began to more-or-less take care of itself). Artists draw all the time, not because they want to pile up a stack of drawings, but because the process of drawing is its own reward.

Therefore, I think this part of your post is very important: “being good at math is really about how much you want to spend your time doing math. And I guess it’s true that if you’re slower you have to want to spend more time doing math, but if you love doing math then that’s totally fine.”

“Math is patient”. Marvelous!

Hans said, “Of course, everything seems easy if you already know the solution.”

When I was a High School Kid, some schools selected kids who appeared (to someone) to have an aptitude, and coached them to excel on speed tests, for the honour and profit of themselves and their schools. Other kids, who either were not selected or who attended other schools, sometimes read and asked teachers and practiced on their own, sometimes had opportunities to excel on speed tests too. Too often we told to consider ourselves to be smarter, better kids. Too often, having been coached or having been told the right answers, ie not really having derived satisfaction from solving problems, we had to fall back on telling others how much better we were.

To sharpen the point, Kids who do it slowly, but do it themselves, might be better Mathematicians. And if they avoid taking discouragement to heart, they are more likely to be happy.

dm00’s point about drawing… or playing violin… is important. If you love it, do it for you.

However, artists and musicians (excepting a miniscule few) are not valued very highly by our culture. They often end up waiting tables or selling tractors to buy roof and food. Mathematicians, with papers attesting, have more options. Also, you can translate math skill into other more marketable skills. But you need to think about how to earn a living, if your skills are insufficient, or not fashionable, or you live in the wrong neighbourhood, or have the wrong skin-colour or chromosomes. You do not need to decide now, but you do need to keep it in mind.

Finally, if you live in USA, some other backward countries, you need to think about whether you can justify signing on for a crippling debt load before going to a university.

The point about debt can be translated into two other interpretations of “are you good enough:”

(1) is it a good investment to spend a given amount of money (potentially borrowed) to pursue your study of mathematics?

Staying just with the monetary costs, I’d be curious to hear other people’s thoughts on the different choices of how to invest in a math education. For example, what do you think of paying full expenses to study at a top tier university? What about a solid state school? Others? Working outside a formal institution?

I guess Cathy will say Berkeley is best (smileys)?

(2) Should you spend your time doing math instead of other things? Should you choose to not pursue other opportunities in favor of studying math?

Money is one element of the opportunity cost and time is the other clear component. I doubt any of us would tell HSK to spend all of her discretionary time on math and never do anything else.

Of course, that begs a retelling of the classic joke:

Q: Why does a mathematician (M) have a lover (L) and a spouse (S)?

A: So that M can tell L s/he is with S and tell S s/he is with L, but actually sneak off to do some math!

Note, we have to assume that L and S exist and are two different people, to exclude the degenerate case.

Being slow is sometimes an advantage. You may see things by going slowly and not knowing much about a field that others miss. The famous German physicist, Wolfgang Pauli, once remarked when he was asked why he wasn’t doing much recently, that he knew too much! Another important talent in Mathematics is asking good questions. This comes from experience.

In summary, people have different talents, learn to use yours.

I must add that unless you know for sure someone really found the answer before you or found the right answer, don’t believe them. I know for a fact that many people pretend they are faster or smarter than their classmates when they are not! I know this because I am the professor grading the work these kids are handing in. Let me add that two kids submitting a quiz and both earning a perfect score, the faster student often doesn’t justify their work as neatly or show all the steps. That faster students will probably do really well in finance. But the students who do super well in mathematics, are the careful cautious ones who provide detailed proofs for all their work.

This is kind of a late response, but I totally agree with your sentiments here. It’s not about how fast you can do it, but how much time you’re willing to spend on it, as even Einstein was quoted saying “It’s not that I’m so smart, it’s just that I stay with problems longer.” Also, I’d like to say that anything you enjoy doing is worth doing, so if you enjoy doing math, keep at it.

As agreed, I too am a late-comer… but honestly, you (and this response) are one of the sole reasons I am going to study to be a mathematician. I am not terribly slow, but I like to “feel” my way through mathematics… and was not sure if because of that reason I should major in it, since the program is at a top institute and very prestigious… therefore also extremely difficult… but I think that I can do it.

Honestly, this is a beautiful post and you helped me in my life greatly. Thanks again.