What's an efficient way to learn simple formulas by heart without understanding?

Question

Is there an unorthodox way to memorize formulas for daily use, similar to the assimilation of multiplication tables in primary school? I included 'without understanding' because I do understand the formulas: given pen and paper and time to think (about the physics involved and/or dimensions) I can figure out the formula. I need to be able to use them instantly, without thinking.

Context

I am a Physics PhD student who has spent about 7 years in the field. I thought that I would by now automatically have assimilated formulas that I use daily, but this hasn't happened. If anything, the situation progressively worsens because I vaguely remember so many formulas that I mix up similar ones. I can never come up with a formula when asked on the spot, which is embarrassing. On my desk I have stacks of formula sheets that I have to use continuously, including highschool-level formulas, which is very time-consuming.

Examples include

• λ = c / f;

• k = Ω² m;

trigonometric identities; hamiltonians; derivatives; etc..

A long time ago when I studied Latin I managed to learn meaningless sequences by heart by repeating them hundreds of times (orally an in writing), rhyming, making songs etc. This is also the only approach I have found thus far for physics formulas, but the problem is that I need them every time in another form (k = Ω² m, Ω = (k/m)^1/2, Ω² = k/m, m = k/Ω, and so on). This means that I would have to separately learn by heart all different permutations of a formula, which is too much.

Answer 1

Note: this answer was migrated from physics.SE, so I've had to re-render the formulas as images. The answer is also specific to learning physics, though it might work for other fields too.

This looks like a classic XY problem. I don't know of any practicing physicists that use a special memorization scheme 'without understanding', and I think such a thing would be counterproductive. Here are a few tips that might help instead.

1. Use conceptual handles

Here are formulas for the speed of sound in a gas, the speed of waves on a string, and the frequencies of oscillation of a mass on a spring and a physical pendulum.

The intuition for all of them is the same: the numerator is some measure of a restoring force, whether that's how hard the gas pushes back, how hard the rope pulls, the strength of the spring, or the torque of gravity. The denominator is always some measure of inertia, proportional to the mass of the system. Thanks to this intuition I don't have to remember anything, except that a square root is involved.

As another example, consider all the annoying conversions between wave quantities,

along with many others. To remember the conversions between (ω,k) and (T,λ), I just use

which is the fundamental definition of the wavenumber four-vector. This fixes the factors of 2π, since that's the change in phase of one cycle. There's no place all these little tricks are written down; everybody has their own and it's best if you make them yourself as you go.

2. Don't worry about judgment

I was just watching a summer school lecture where a renowned theoretical physicist took a good 30 seconds to flip a fraction. This is completely typical and not embarrassing at all. Some people function better rearranging symbols in their heads and some function better using chalk or paper. Personally I can't do anything in my head; I have to use paper or write it out in the air, but I have never felt judged for doing this. If your colleagues are being judgmental, they are being rude and you shouldn’t let them get you down.

When I see somebody able to remember or rederive something much faster than me, I often ask them what their conceptual handles are. Unless the person is exceptionally rude, they’re typically happy to explain. (This is especially true in physics, where mere memorization is uncool.)

3. Chunk concepts, separately from tools

If you're having trouble writing a grant proposal, the solution is not to memorize the exact sequence of muscle activations needed to write every individual letter. Similarly, if physics feels too 'big', the very worst thing you can do is to make it even bigger, by unpacking every equation into eight separate equations. Learning is instead done by chunking things together.

For example, take the derivation of the wave equation, which is a full two pages of math written out. You don't want to store every line in your head. Instead, you just want to store a general intuition that "curvature means a restoring force because strings under tension straighten out", which gives you the ∂²y/∂x² term. To get from that to the final result you need to know Newton's second law (giving the ∂²y/∂t² term), the small angle approximation, and the binomial approximation. But none of these are specific to the wave equation -- they're just general tools.

As another example, I was lost when first exposed to tensor notation. It looked like there was an enormous amount of stuff to memorize! But it faded away once I sat down and wrote out all the allowed manipulations. It turns out there aren't that many, ten common ones at most. All tensor calculations up to graduate level are just using the same ten steps over and over again, so in a technical sense it's actually easier than high school algebra, which has many more allowed manipulations. Once you have this understanding, many derivations get shorter; they get chunked into "use the standard steps, plus this one trick in the middle".

Then all you have to do is remember the trick, ideally with a conceptual handle.

4. Construct your own understanding

As I emphasized above, the best way to get this kind of understanding is to construct it yourself! There's no shame in revisiting a subject that's "basic" and rebuilding it yourself from the ground up; I've done this with calculus and mechanics several times whenever I felt I was getting rusty. Make a formula sheet, or if you think visually, try drawing a mind map or a dependency diagram. Challenge yourself to rederive key equations without a reference. If you do this a lot, you'll naturally construct the necessary conceptual handles and chunks, and get better at recognizing which tool to use.

Answer 2

I really like @knzhou’s answer, which I think does a great job of addressing the issue in a more or less canonical way. To add to that, part of your false premise (what @knzhou refers to as your XY problem) is that you are trying to compartmentalize “memory” and “understanding” as two separate and independent things. That’s not how the brain works. In practice, remembering things and understanding things are two aspects of the same set of mental phenomena. When you properly understand the derivation of a formula, it’s not just that you are able to reconstruct the derivation and the formula on demand given time and pen and paper, but it also becomes much easier to remember the formula without the derivation. This is because of the conceptual handles (to use @knzhou’s very apt terminology) that you develop through the process of acquiring that good understanding.

In fact, I think whether you can easily memorize a formula or not can in many cases be used as a good signal to test whether you have acquired the necessary understanding of the derivation/reasoning behind it. If you still find the formula unintuitive and difficult to remember, or generally feel uncomfortable with what the formula says, it’s possible that you haven’t internalized the logic behind it (maybe because you are focusing too much on memorizing the derivation itself rather than really thinking and understanding why each step is natural and how the different parts fit together). The mental exercises that @knzhou suggests can be a good way to bring yourself up to that level, which is a higher level of understanding than just reading a proof or derivation and verifying that it is correct.

In addition to that, @knzhou’s advice to simply care less about your ability to come up with formulas on the fly when other people are watching also sounds very sensible.

Answer 3

For me (a mathematician), the process is based on repeating the formulas often enough. At some point, I begin to guess what the right answer might be, then I will do the (mental or paper-based) calculation to verify. Some repetitions more, and I will not need to verify any more.

For some formulae, there are specific tricks, which usually relate to testing with very simple special cases and checking that the result makes sense. For example, either sin(θ) = x or sin(θ) = y, but which was it? I do remember that sin(0) = 0, and remember where the zero angle points in the unit circle, which allows me to deduce which alternative is correct. (True solution left as an exercise.)

In physics, I need to remember whether I should divide or multiply by some quantity. In addition to checking the units, I might let the quantity go to zero or infinity and consider what should happen then. v = s/t or v = t/s; if distance stays constant but time goes to infinity, then the velocity must be small, which allows me to pick the right alternative. (I actually check this by looking at the units, but the example should illustrate the technique.)

Answer 4

The method I have always found most effective is to make a word or initialism out of the letters in the formula, filling in vowels/consonants as needed.

Examples:

1. To remember the Laplace domain formula for the first-order lag element PT1 in control engineering

   

   note that the denominator spells out "1st"!

2. To remember the formula for time invariance

   

   I treated the uppercase delta as an "A" and remembered "futa" (Japanese word for "hermaphrodite") which of course only makes sense if you're familiar with the word.

3. Similarly, I use "Uli" (German first name) and "ICU" (Intensive Care Unit) to remember this pair of formulas in electronics, while imagining my friend in the ICU:

   

   

I also use a similar idea to remember lists (use their first letters, for example PAGE to remember that rational agents have perceptions, actions goals and an environment) and definitions (find a mnemonic in its name, for example, in bioinformatics, BLAST extends a match in both directions while FAST calculates oFFsets).

Answer 5

As far as I know there is no established, classic memorization system for formulas. Most people in memory arts have focused on memorization of numbers or text (or cards).

However, there have been a few sketches of how it can be done in the memory arts community. This thread discusses various suggestions, including defining images for the symbols (integral sign is a violin), peg systems for lists, or using the method of loci to wander around the formula. Similar ideas can be found here.

This method presupposes that you know other basic memory arts. That is of course the problem, it takes a fair bit of training before you will be great at them. I did my PhD on memory, have worked on human enhancement for years, and often have need to recall various formulas... yet I do not use memory arts for them. A mixture of laziness and alternative cost have kept me from it; the tradeoff may be different for others.

Generally when I do memorize formulas I try to get familiar with them - get to know each factor, term and symbol and why they are there. Once you have a web of links you can reconstruct the formula fairly easily unless it is something arbitrary and structureless like

• 4.652 x⁴ + 0.4587843 x³ - 0.434343 x + 17.

But if you need to rattle off such formulas again and again you might want to reconsider what you are doing.

Finally, making and having a cheat sheet of often used formulas is nothing to be ashamed of. It is useful. I recently created a standard file to include with physical constants and formulas in my code, and it made my current project (a book writing one) noticeably easier and less frustrating. There is nothing wrong with outsourcing your mind to the Rubber Book if you can look things up fast enough.

Answer 6

For the permutations of the formulas, you are better off getting more familiar with algebra and making the transformations that you need on the spot.