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I've asked a previous question about math habits that would give a more organized and less cluttered mind when doing mathematics. I asked it because I feel like I'm not making the most out of my time when doing math problems (there are some really good answers there for those who share my problem). This question is a more specific follow up on that earlier question, specifically about the problem of "trivial mistakes"

Whenever I'm doing derivations on some new problem I'm not experienced with, those derivations always contain certain elementary steps that I've been doing for years. But because I've been doing it for years, it feels like they are so easy, that I'm basically doing them on "auto-pilot".

The problem.

As a result, I'm making the most silly of mistakes sometimes: clumsy mistakes like:

$$\begin{array}{rcll} \text{WRONG: } & e^{f(x)}e^{g(x)}=e^{f(x)g(x)} \\ \text{WRONG: } & f(x)+g(x)=0 \implies f(x)=g(x)\end{array}$$

And earlier today I was solving the general linear first-order ODE: $$\frac{\partial x}{\partial t}+f(t)x(t)+g(t)=0$$

I tried to simplify it by using the chain rule in reverse: $$\frac{\partial (e^{F(t)}x(t))}{\partial t}=f(t)e^{F(t)}x(t)+e^{F(t)}\frac{\partial x}{\partial t}$$ However, in order to substitute this chain rule in the ODE, we have to do something with the $e^{F(t)}$.

In some bizarre mental quirk, I applied both approaches at the same time, by both dividing the chain rule by $e^{F(t)}$, and multiplying the ODE with it when doing the substitution, so that I got to: $$\text{WRONG: } e^{-F(t)}\frac{\partial (e^{F(t)}x(t))}{\partial t}+e^{F(t)}g(t)=0$$

How do we learn to stop making such mistakes?

I don't think the solution is to just "pay more attention", because while that would in principle be good, it is not a very "actionable" principle.

Of course we should always double-check answers, but this is time-inefficient and certainly not fool-proof.

Instead, I'm wondering if there are some specific (mental or practical) habits that people can develop (or that you've already developed) in order to stop making these clumsy unnecessary mistakes?

Michael Hardy
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user56834
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    well... in my opinion an organized mind it is not the best to do math because in the end you need a lot of creativity. Then my conclusion is that mistakes are unavoidable, just see the errata of many (elementary) textbooks of maths. –  Apr 20 '17 at 08:27
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    Clumsy mistakes are the fruit of divided attention, in my opinion. I've never repeated the same mistake, it's always some new unique fact about math I come up with :D – AlvinL Apr 20 '17 at 08:29
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    Things happen. I would rather try to find efficient ways to double-check, I don't think there is a way to never do mistakes. – Dirk Apr 20 '17 at 08:36
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    Some mistakes always happen. Sometimes thinking conceptually can help - e.g., a fundamental, defining property of $f(x) = e^x$ is that it is a function that takes addition to multiplication: $f(x+y) = f(x) f(y)$. Not $f(xy) = f(x)f(y)$. – Jair Taylor Apr 23 '17 at 00:14
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    But you're like me, you will always make small mistakes regardless. You just have to slow down and learn to "debug" your mistakes by noticing that something is wrong and then going through computations line by line until you find the error. Also, computers can help. – Jair Taylor Apr 23 '17 at 00:16
  • Drawing diagrams of different test functions $f(x)$ and $g(x)$ might help... – Adam Rubinson Aug 16 '22 at 12:03

4 Answers4

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I find its quite helpful to pause every once and a while and think "that makes sense, right?" then dive back into the problem. I have caught many mistakes this way, and it more than makes up for the lost time in asking the question. (Its also a useful habit in a larger scale, asking questions like "does attacking the problem this way really make sense? before pouring hours into the approach).

Aperson123
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I think that this has something to do with skipping intermediate steps of calculations. Instead of jumping straight to conclusion from memory (which can be rather foggy), perform the intermediate steps, even mentally if you do not bother to write them down.

Algebrica
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    Corollary: Once you've practiced skipping steps enough, your memory is better and you tend to make mistakes less. Or, solidify a rule in your memory explicitly, e.g. $1-x = y \iff 1-y = x$. – Zaz Feb 19 '22 at 22:50
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A small useful thing I've found is to think of mathematical notation as both a book-keeping device, and a narrative notation/language for "math itself". The book-keeping is about doing algebra or arithmetic algorithms correctly... while the narrative aspect is about telling what you did.

Of course, there is not a clear demarcation...

By this point in my life, I try to avoid brutal computations, to minimize the book-keeping aspect, which I trust less-and-less. Rather, I aim for narrative of coherent concept...

paul garrett
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I was always impatient as a kid and wanted to do as much in my head as possible. Doing a lot in your head is not necessarily bad, but it is harder to improve your accuracy than to improve your speed.

As an adult, I went back and did Khan Academy from the ground up (started by counting bunnies). I focused on accuracy rather than speed, but my speed naturally improved over time.

The tenet that I've found the most useful is, regardless of whether you're solving mentally or on paper, is to solve a problem in discrete steps, just like a pilot going through a checklist. Each step should be something you are certain about.

For example, I used to reason "$1-x=y$, plus, minus, $\implies 1-y=x$" (i.e. I was vaguely remembering to add $x$ to both sides and subtract $y$ from both sides). This kind of sloppy thought process is what causes mistakes. Instead, just prove to yourself and then memorize that $1-x = y \iff 1-y = x$. If don't 100% recognize a step as an identity, prove it.

Zaz
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