Mathematics and Perfectionism?

Question

Background

I am an undergraduate and I have just finished my first calculus class (Calc I) this summer. While this class has gone very well for me by any objective standard, I find myself drifting towards a pathological obsession with perfection. This manifests itself primarily on tests where I feel compelled to get a perfect score to even consider that particular test as a success. Getting something like a 96 has begun to feel as if I'd received a 40. While I find calculus to come very naturally to me conceptually, I am somewhat prone to making minor computational errors (i.e. I'm not a human computer) and thus, I find it difficult to attain the level of perfection that I expect of myself. My test average is about a 96 which my rational self can recognize as "good" but my emotional self interprets as a crushing blow, especially when it was such a simple error that kept me from a perfect score.

Now, despite all of this, I still very much love mathematics in the sense that it is the only thing I've ever experienced academically that brings me intrinsic joy. I spend a good deal (at least 3-4 hrs./day) of my time learning mathematics on my own just because I want to know it. As such, I'd consider myself fairly advanced and mathematically mature for where I am in my formal academic progression. Rationally, I realize that real mathematics is generally invariant under computational errors but I still feel as if any error somehow compromises my credibility. It has come to the point where I've finally swallowed my pride and seen a therapist who encouraged me to reach out to some mathematicians or students of mathematics in general who have struggled with similar perfectionistic tendencies. I am mortified at the thought of laying all of this on one of my flesh and blood professors so I am hoping that this post might serve as a proxy.

Note: I realize posts like this are generally frowned upon but I really have no alternative. To keep this somewhat within the bounds of the guidelines, I am NOT interested in a debate about various schools of psychology or really any psychology for that matter. I'm more looking for personal anecdotal strategies for overcoming perfectionism, relevant historical anecdotes (I feel like mathematics self-selects for perfectionism), etc. If this gets closed/deleted then I understand but hopefully there is a place for this question somewhere on this site.

Answer 1

In the bigger scheme of things (beyond grades assigned in undergraduate courses), mathematics does not self-select for perfectionism, nor does it award perfectionism to any realistic extent. Yes, you need to be able to calculate accurately. However, there's a lot more to mathematics than calculation.

To be an accomplished mathematical professional, you have to learn that mathematics is a rather messy topic. Of most immediate importance to you, you have to learn how to accept that you will make mistakes and how to learn from those mistakes. In fact, I would dare to say that it is not possible to prove an important mathematical theorem without constantly making mistakes and learning from them along the way. I've seen graduate students who refuse to learn this lesson; they did not succeed. Of less immediate importance but more long term importance, to be a successful mathematical professional you have to learn to think creatively about mathematics, not just how to calculate accurately; but I won't go further into that here.

If one wants more immediate awards for one's intrinsic perfectionist streak, there might be other branches of inquiry to follow. Certainly computer science awards perfectionism more directly, e.g. a computer program that does not compile must be immediately rewritten. But even programmers make logical mistakes that are subtler than just compilation errors. For example, programmers have to learn to think cleverly in order to hunt down logical errors and correct them, making the art of debugging very important.

Alternatively, try the cold turkey method. Try diving into an advanced mathematics course where rote calculation is de-emphasized, theoretical understanding is emphasized, and creative thinking becomes important.

Answer 2

A great mathematician once said to me : making mistakes is not a big deal. What's important is to make the right mistakes.

You can interpret this in whatever way you want, but I think the point is this : getting a 96% because you computed something fast in Calculus and disregarded a + or - sign is not relevant. You will still get A+ and there is a reason why it is stupid to say that the student that got 100% is "4% better than you" because he is not. At this point the exam wasn't sufficiently hard to distinguish who is better than the other, and anyway there is no point in distinguishing the both of you since you clearly both mastered what the professor evaluated you on.

What you, from a personal point of view, should want to achieve, is to make mistakes at home while studying and correct them in that moment, so that when it begins to matter not to make these mistakes, as in :

• subsequent courses where this theory is taken for granted
• qualifying exams for scholarships (from an intellectual point of view it is irrelevant, but you want to eat food)
• perhaps notes you take for yourself to prepare a talk
• your own research, since you don't want to be lead astray by a mistake you made

then these mistakes should happen as rarely as possible. On the other hand, you should never strive to get to a point where you never make mistakes anymore ; this would be a sign that you stopped learning anything new. Learning requires making mistakes to a certain extent, since it requires you to change your way of thinking and usually requires a certain adjustment period where mistakes may (and should) happen.

Hope that helps,

Answer 3

Let me add to Lee Mosher's remarks about math and CS: First, I've done both, and made plenty of mistakes in both places. But in CS, because there are ways of testing things relatively easily, the mistakes are often easier to find. A consequence of this is that you end up developing habits that keep you from making mistakes in the first place. Here's a (trivial) example: when I'm typing up a paper using LaTeX, I often get to a point where I need an "align" environment, intending to write something like this:

\begin{align}
y &= 2x + 3x \\
&= 5x. 
\end{align}

One way to produce this is to type

\begin{align}
y &= 2x + 3x

and then continue on. An alternative is to type

\begin{align}
\end{align}

and then fill in the stuff in the middle. The latter approach prevents a very common mistake, namely, something like

\begin{align}
y &= 2x + 3x \\
&= 5x. 
$$

where old habits of ending a displayed equation with "$$" end up introducing an error.

In much the same way, as you do mathematical computations (e.g., you multiply a couple of polynomials and then simplify), you can do the simplest of checks: plug in $x = 2$ to both the starting and ending steps of your algebra. If the answers differ, you made a mistake. If they don't, odds are that you did OK. (Plugging in $x = 0$ is easier...but it only checks the constant terms.)

Developing habits like this can be really useful. Pretty soon you don't even notice that you have them...you just spend a lot less time fixing things.

Another good habit: when you apply a formula, make sure you know what every symbol in it means. Here is a nice example of a failure to do this. It's certainly easy to make mistakes like this, but if you, while writing or working, always have a tiny part of your mind asking "Wait a minute...does that actually mean what you think?", you can save yourself a lot of grief.

I want to briefly advocate the opposite point of view: the "let's go wild and try stuff and forget about whether it really makes sense or not" kind of exploratory mathematics. For this, you turn off that little "editor in your head" and just do stuff that looks interesting. Sometimes a beer helps free up your mind for this kind of thing ... otherwise you're constantly in "acid mode," picking away at every thought lest it be wrong in some way. Here's an example: when you have a square matrix, $A$, you can compute a polynomial in a single variable $x$ called the "characteristic polynomial" of the matrix. The roots (real or complex) of this polynomial are called the eigenvalues of the matrix, and turn out to tell you a great deal about the matrix. But suppose you took the polynomial, something like $$ p(x) = x^2 + 2x + 3 $$ and say "what happens if I plug in $A$ for $x$?" Well, you get $$ A^2 + 2A + 3 $$ which doesn't make sense, because $A^2$ and $A$ are matrices, but $3$ is a number. But if you rewrite $p$ as $$ p(x) = x^2 + 2x^1 + 3x^0 $$ and plug in $A$, you get $$ A^2 + 2A + 3A^0 $$ and now you could say "Let's just say that $A^0$ means the identity matrix", so you get $$ A^2 + 2A + 3I $$

And the amazing thing is that if you do this, for any matrix and its characteristic polynomial, the result will always be the zero matrix. That's one version of something called the Cayley-Hamilton Theorem, and it's the kind of thing you discover by messing around outside the boundaries of "what's allowed". Proving it requires getting back to more formal mathematics, but discovery can arise from not being a perfectionist about everything.

Answer 4

I'm a perfectionist, but I unintentionally make lots of careless mistakes all the time. Like you, I feel a sharp twinge of disappointment or embarrassment, depending on the situation, but it goes away almost immediately. The reason is that I consider intention and not outcome to be the deciding factor, and so as long as I had put in what I deem sufficient effort to avoid careless mistakes (such as checking my work), whether I actually make mistakes will be irrelevant to my judgement of myself. This is why I can justify and succeed in suppressing any feeling of chagrin quickly if there was no reasonable way I could have prevented some mistake.

Of course, being a perfectionist, I would still attempt to fix any mistake I am made aware of in as perfect manner as possible. This can be a second avenue for you to channel your thoughts and energy into; instead of harping on your failure to yourself you could focus on rectifying the fault. This includes trying to figure out ways to prevent subsequent recurrence of the same situation that encourages mistakes (such as tiredness, distractions, style of working, ...) as well as actually fixing the consequences of your fault (wrong computation? do it right this time! claimed a false statement to someone? make a corrected statement!).

Note that my advice applies to all areas of life and not just mathematics. That said, some people may find it harder to adopt this attitude than others. Do not be afraid to recognize that, and also know that different people may find different ways of de-stressing to be effective. Yes, extreme perfectionism leads to an unhealthy level of stress, so we should both regulate our perfectionism and also have an outlet to release perfectionism-driven stress. Also do not be concerned about how others think of you when you seek help. After all, nearly everyone has to rely on others for various things, and social interaction is one of them.

After seeing Jack D'Aurizio's comment, I would like to add that you should also learn to see mathematics (like many other fields) as a cooperative endeavour. It is not only easier to work together than alone, it is easier to help each other spot and fix mistakes.

Answer 5

Well because you only have exams a few times a semester I can see where the perfectionism might come in. A couple of computational errors and you already are down to an A- (or worse!) depending on the grader. Also if the courses are not challenging you might feel like you are obligated to get a perfect score because of your aptitude. Also because math is such a precise subject, if you let up and lose interest you can hastily start failing. If you start becoming a perfectionist in your studying, in your in-class discusssions, etc. etc. then I think you may have a problem (as this sub-game creates a lot of stress and drains your energy, hurting your general cognitive abilities). Note Norbert Wiener was hit by his father when he was young and he still made mistakes as an adult.... So I wouldn't worry about it. The issue is when you make a mistake but cannot identify it; there you have an issue with understanding logic (which is just the English language!) or concepts.

Answer 6

Even if you can obtain a "100%" grade, don't be complacent about it: a stricter examiner might have given you only 98%!

You can still be a perfectionist; but what you are presently trying to perfect is the wrong thing. You are trying to perfect the output, whereas what you should be trying to perfect is something like the ratio of output to input (according to some reasonable valuation measure). For example, consider the process of refining a metal. Ideally you want 100% purity. You start with crude metal that is 90% pure. At reasonable cost, you can refine it to 99% purity. At a rather higher cost, you can get 99.9%. Next, 99.99% is achievable at considerable cost. At each stage, the cost escalates, until the entire world's economy is involved in producing (and perhaps never quite reaching) an atomically pure sample. The value of the produced material goes up, but eventually is overtaken by the cost.

Accuracy is important, but balance your effort on improving accuracy with effort on other important aspects of scholarship.

Answer 7

Overall I agree with Jack D'Aurizio's comment: the best way to snap out of this is to realize and internalize the fact that mathematics is not about "doing well," it's really about collaboration and discovery. It's about talking to people, some of whom you might consider to be absolute geniuses, asking questions, getting stuck and then asking more questions.

In the real world (I am counting research as part of the real world here), you aren't on a 90 minute time-limit. If you make a mistake, you will eventually notice it. You can go away and come back to review your work another day; and, best of all, you can get somebody else to help you.

There are two important points here, which must be viewed in-tandem:

1. Everybody else knows this. That is, if you apply somewhere else and people see your transcript, they too will understand that losing four marks means very little with regards to how well you understand the subject.

2. You should aim for your feeling about your marks to correlate well with how others might feel about your marks. Ultimately, your mark on a calculus exam means very little to you. You know exactly how well you understand calculus without an examiner having to "tell you." The purpose of the exam mark is to convey your level of understanding to others, so the opinions that really matter are the opinions of others (and ultimately others will agree with your "rational self").

   a. Caveat: On the other hand, if someone is reading this actually did badly on an exam and is sure they understand the subject well (say, they did well all year on coursework and homework), then it is worth noting that an exam mark is not the be-all-and-end-all of the situation. Ultimately you are likely you know and understand much more about a subject than you can get across on a whole exam script, nevermind just the mark on the front.

The important word here is internalize. From what you describe, it sounds like it will take some work convincing your "emotional self" of these things. My best advice here is to take a break. Stop thinking about maths in-detail for a few weeks. You don't have to stop thinking about mathematics entirely, just don't do much (or anything) mathematically taxing. In the meantime, you could try, for example, reading about famous mathematicians and the history of mathematics, practicing some other interest (perhaps chess or another board game, if you're that way inclined), trying to read a novel or two - that kind of thing. When you come back to mathematics, you'll have a fresh pair of eyes.

I hope this helps. And, by the way, congratulations on your exam success.