Make Friends With Your Enemy: Embrace Mistakes!

As Girls’ Angle’s sixth year gets underway, I’ve got one message for our members:

Mistakes, especially public ones, can be embarrassing. People might even laugh at you. That’s just human nature.  But to improve at math, you must try things, and when you try things, you will make mistakes. So when you make a mistake, why not just laugh along with everyone else, learn what you can from the mistake, and move on?

Every mathematician, no matter how successful they seem to be, has erred many times. Even the mathematician who finally put the Q. E. D. on Fermat’s last theorem, Andrew Wiles, made a mistake, and quite publicly. If Prof. Wiles had been too afraid to err, Fermat’s last theorem may still be conjecture today.

If you want to learn something, be unafraid of making mistakes. Don’t be so afraid of embarrassing yourself that you won’t dare to sing a tune, won’t dare to practice an overhead smash, won’t dare to paint a portrait, etc.  To learn how to paint good portraits, you must try, and when you try, you are bound to make a lot of funny looking faces.

In fact, sometimes it is impossible to avoid error. For example, suppose you need to calculate the area of a circle of radius 10 centimeters. You can use the area formula $\pi r^2$, where $r$ is the radius. So you substitute 10 centimeters for $r$. But what value do you use for $\pi$? If you really need to know the answer as a decimal, you’re doomed to make an error! You could use the approximation 3.14, but we know that 3.14 isn’t really equal to $\pi$. It’s too small. In engineering applications where one must know the area of a circle as a decimal number, it’s really quite impossible to get the exact answer. People have no choice but to live with error. So, instead, they have to design the machine so that it will still work despite the inevitable errors.

Even more, there are situations where it can help to make errors on purpose! Scientists and mathematicians do this when they simplify a complex situation in order to gain at least some understanding of it. For example, the angle $\theta(t)$ of a pendulum bob of mass $m$ that swings at the end of a rod of length $L$ under the influence of gravity $g$, as a function of time $t$, satisfies an equation of the form:

$\frac{d^2 \theta(t)}{dt^2} = -\frac{g}{L} \sin \theta(t)$.

Just in setting up this equation, there are already many simplifications. Friction’s ignored. The gravitational field, which varies with altitude, has been turned into a constant. The rod is made from an imaginary, rigid, massless material. Errors, errors, errors!

But if these simplifications aren’t made, we might confine ourselves to a limited understanding. In fact, if one wishes to get an explicit formula that approximates the motion of a pendulum, another deliberate error is often made: $\sin \theta(t)$ is replaced with the approximation $\theta(t)$. Even though $\theta(t) \neq \sin \theta(t)$, it is within 1% of accuracy when $\vert \theta(t) \vert$ is less than 0.1 radian. But practical purposes aside, that error buys us understanding.

So, the next time you are really stuck trying to solve a complex math problem, see if making a deliberate error helps. Perhaps eliminate an assumption or constraint and solve the simplified problem instead. Or add a simplifying assumption that enables you to take some step forward. Even though you know that you won’t get the answer to the original question, the answer you do get could help you to deduce how the desired answer should look.

In summary, don’t be afraid of making mistakes. Instead, make them freely and without shame. They are inevitable, so why not use them to your advantage?