When should you do math in your head vs writing it out on paper?

There is an asymmetric tradeoff between 1) blowing your working memory capacity and leaving yourself unable to make progress, versus 2) wasting a couple extra seconds writing down a bit more work than you need to. When in doubt, write it out.

When learning math, when should you solve a problem in your head vs writing it out on paper?

I would say that the most technically correct general rule is this:

Do a problem (or sub-component of that problem) in your head when

1. you're able to do it in your head reliably, with very little effort, and
2. it feels like you're only holding one thing in your head -- like, you're working with a solid, cohesive "chunk" of information as opposed to having to "juggle" multiple components of that information to keep it in your brain.

Here’s the science behind that.

It’s well established in cognitive science, specifically under the umbrella of “cognitive load theory,” that your working memory has a limited capacity to hold new information – and when you push your working memory close to that limit, you become more likely to make mistakes and less likely to complete the training task, which impedes your learning.

Reducing cognitive load is the goal – not just “a” goal, but in fact “the” goal, the whole point – of structured education. The more instructional scaffolding is provided, the lower the student’s cognitive load, and the less cognitive “friction” there is to slow the student’s acquisition of the knowledge covered in the curriculum.

It’s the same way with solving problems on paper: the goal is to lower your cognitive load. By writing down intermediate steps on paper, you can temporarily remove information from your working memory to make room for new information, and then quickly load up the original information by looking back at the paper when needed.

(In a sense, the paper functions as an artificial long-term memory bank where you can store new information that you don’t already have encoded in your brain’s long-term memory. It takes a lot of time and effort for your brain to encode information to its own long term memory, but writing information on paper enables you to sidestep these biological limits.)

The Asymmetric Tradeoff

To be clear, there is a hidden tradeoff. If you overdo the scaffolding, or you write down more than you need to on paper, then it’s going to inflate the amount of time that it takes you to complete the curriculum or solve the problem, respectively. If your cognitive load is already low, there is no benefit to lowering it further – all that does is create more mechanical work for you that burns your time.

However, the tradeoff is asymmetric:

• If you undershoot the scaffolding or don't write down enough work on paper, and you blow your working memory capacity, then you hit a brick wall. You're simply unable to complete the task. Your learning progress grinds to a halt, full-stop. Even if you just "come close" to full capacity, your error rate skyrockets, impeding your learning.
• On the other hand, if you overshoot the scaffolding or write down more than you needed to on paper, then sure, it will technically be suboptimal, but typically not by much. You wrote down an extra line or two on paper than you really needed to? Big whoop, it took you an extra couple seconds to solve the problem. You could have saved a couple seconds by not writing those steps down, but that would also have put you dangerously close to holding too much in your head and making a mistake. Just like in your bank account, having a little buffer is not a bad thing.

Because the tradeoff is so asymmetric, it’s best to err on the side of caution, writing down potentially a bit more than you need to. When in doubt, write it out.

Building Good Habits

In addition to guarding yourself against being on the wrong side of the asymmetric tradeoff, another reason why you should err on the side of caution (i.e., writing down too much as opposed to too little) is that you need to build good habits for the future.

As you climb up the levels of mathematics, the level of technical sophistication increases, and consequently, so does the level of cognitive effort. Even if you are able to do problems entirely in your head at lower levels of math, you will not be able to do so indefinitely into the future.

You will eventually get to a point where you are unable to do problems in your head without blowing your working memory capacity, and at that point, the only way to continue making progress will be to use paper and pencil as a tool to reduce your cognitive load.

However, the longer you go without using paper and pencil, the more solidified that habit will be, and the harder it will be to get yourself to change it.

So, even if it’s not strictly necessary, it’s a good idea to get in the habit of writing at least some work out.

If you don’t, then you may cling to the habit of doing all the work in your head for too long, well past the point when you really need to start writing work down on paper – which will gradually eat away at your performance and progress, eventually bringing you face to face with a “day of reckoning” where your entire mathematical future is on the line.

I can’t tell you how many times I’ve seen a bright student breeze through basic math refusing to write down any work, only to struggle in intermediate or advanced math simply because they stubbornly continue refusing to write down their work.