How to Remember Formulas (by Learning them Better)

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In this post, you’ll learn how to remember formulas for any problem-solving subjects in two steps:

  1. Encode formulas; and
  2. Retrieving them (with Anki and with actual problem-solving)

The problem with formulas is that they seem like “memorization stuff” — but usually they’re not.

So what do most students do when learning formulas? Well, they resort to cheat sheets and try to remember everything!

And this is a HUGE mistake for one big reason:

Gaining knowledge is easy, but applying knowledge requires coherence. Without coherence (that can only happen when you learn formulas) then you’ll end up knowing formulas but not being able to apply them.

Since were talking about coherence, there’s no better place to learn than from a model reading comprehension — The simple view of reading.

Encoding formulas: The simple view of reading as analogy

The simple view of reading asserts that the reading process can be broken down into two equally important parts:

  1. Decoding. It’s the ability to interpret words from a set of symbols.
  2. Linguistic comprehension. The ability to interpret a set of words to extract meaning.

In mathematical subjects, instead of words we have symbols/notations, and instead of a set of words we have an entire equation.

So we can say that: comprehending formulas means decoding each notation, and then interpreting a set of notations to extract meaning.

Obviously, the first step is knowing what each symbol stands for. But the more crucial part is the meaning — how “rich” or “integrated” the formula is in our memory.

And there are three tools we can use to make any formula “integrated” into memory:

  1. System: elements, relationships, how they work
  2. Derivations
  3. Application

Let’s take these lenses to understand a bit of how the world works, shall we?

Example #1: Demand for domestic goods [System]

In Macroeconomics, there’s an equation for the demand for domestic goods, as follows:

Holy crap is that a mouthful. So let’s break it down. First, let’s decode what each symbol means.

  • Z = demand for domestic goods → well, that’s literally it. It tells you how big the market is for a domestic good.
  • C = consumption
  • I = investors
  • G = government spending
  • IM = imports
  • ϵ = foreign exchange rate
  • X = exports

As it turns out, they’re already the parts of the “system.”

So now we can start to look at the relationships and how they affect the whole.

  • Relationship: Most of them add to the demand, except the imports.
  • Whole: It makes sense because when people consume your goods, invest in those goods, and the government spends on them, it means that there is a demand for the goods. Exporting also means that there is demand from other countries, because why the hell do exports, right?

Okay, that’s understood. Upon closer inspection, the imports variable is divided by the foreign exchange rate. That makes sense, because we’re dealing with units that are in a single currency.

  • Relationship: Imports divided by foreign exchange rate
  • Whole: Because we’re dealing with a single currency right now

And while we now know why the other variables add to that, we’re still unsatisfied with the “minus”: why do imports subtract from the demand for domestic goods?

This requires some prior knowledge, but the answer is that imported goods will compete against domestic goods, and since the reason you’re importing in the first place is because it’s cheaper to do so, then it means that the demand for domestic goods will decrease.

  • Relationship: Imports subtract from demand
  • Whole: Because of price/competition tradeoff between domestic goods

Ah, it’s actually simple! Now, recalling this formula will be easy, because it’s now enriched with a ton of concepts.

Do we need to make flashcards for the individual insights we’ve gained from this process? You could, but it would be unnecessary because you’re going to do practice problems, anyway, and thus you will practice recalling them even if you didn’t make flashcards.

Let’s take another example.

Example #2: Work Equations [Derivation]

Like you saw above, “formulas” are simply the compression of many concepts and their relationships. They are not “formulas that you need to remember”, but rather statements that you need to understand. Once you understand how a formula works — or better yet, how it is derived — then you will remember it more easily.

Of course, some formulas will require Calculus or even higher mathematical background to derive, but that’s not really something to worry about if you’re learning in University.

Let’s take an example from Physics.

In most textbooks, the equation for Work is W=Fdcos(θ). Now, you can leave it at that and simply remember the formulas and its hundred variations. But when you know how it’s really derived, you’ll find that this equation (and a LOT of its variations) come from only two things:

  1. The cosine equation, cos(θ)=adj/hyp
  2. The work equation, W=F×d

Why is the cosine equation present here?

Ah, because by definition, F is the component of force in the direction of motion. If it’s not in the same direction, then the force didn’t do any work. So if you look at the equation above, it makes sense.

This example might be extremely simple (simplistic, even) but the same thing could be applied to anything else especially in Physics subjects:

Prior knowledge (even newly learned ones — like the definition of work) can be used to derive equations for problems. And the act of deriving formulas encodes it better into your memory.

It’s like knowing how a story unfolds, and instead of knowing that “Eren becomes a villain in Attack on Titan,” you end up knowing how everything arrived to that scenario. And that often eliminates the need to “memorize” it as much.

(For AOT fans who haven’t finished the anime: Is Eren really the villain? Well, find out…)

Example #3: The Ellipse Equation [Application]

If a formula doesn’t make sense and the variables don’t seem to logically explain the entire system, then it probably means that it’s a derived formula already. But while learning the derivations do help, sometimes that would be unnecessary.

For example, formulas in geometry usually have long proofs and/or derivations, but an engineering student does not usually need to remember these proofs. (For someone who majors in Math, however, doing the same thing might lead to a failing grade.)

So instead of memorizing them as fact, we apply them so we can remember them better.

For instance, the ellipse equation…

…can be derived from the distance formula, which is derived simply from Pythagorean theorem.

But instead of trying to memorize it by brute force, we answer problems with ellipses so we know how to use the equation even with different unknowns.

But learning isn’t enough — you have to maintain knowledge, too

You’re probably familiar with this: people will tell you that the key to retention is simply understanding. Memorization alone isn’t enough.

But surely you can do both. And in fact, that’s what I recommend you do, because our access to old memories fade in two ways:

  1. Time
  2. Learning new things

And look, especially if you’re in Engineering (like me back then), then you’ll most likely have subjects that take multiple semesters before they’ll be used again.

Take for example, Physics:

After learning classical mechanics, if you have Physics 3, then you’ll learn solely about electricity, magnetism, electrostatics, and electromagnetics — you won’t be using those free-body diagrams that much, not to mention the formulas you use for collision, momentum, buoyancy, heat, etc.

So unless you do something to maintain that knowledge, you’ll forget it over time. And the HUGE problem is that problem-solving subjects are cumulative. They build on top of each other.

Thus, we need to retrieve, because retrieval is the way we strengthen our memories. That makes sense, because the things we need to recall are often the things that are important. In a way, retrieving is like “telling the brain that this is important so you don’t forget it.”

And like I said in my other article, Learn Math on Your Own, you can do that by using Anki.

Retrieving formulas: Making formula flashcards in Anki

Now comes the “candy” part — the card formulation. If you don’t know how to use Anki, you can start here.

In the study system course and in the effective flashcards guide, I argued that effective flashcards share three characteristics:

  1. Effective flashcards are encoded (i.e. holistic/integrated into prior knowledge)
  2. Effective flashcards are atomic (my definition: quick enough to answer)
  3. Effective flashcards are future-proof

The good news? You have already done the work to make it encoded.

And the other good news is that formula cards don’t need that much thought to create. So long as you have encoded them and applied them to a problem, you’re all good!

Heck, the entire mathematics curriculum in Engineering only took me 277 cards:

That’s because I’ve used my cards to aid my purposeful practice. (More on this later)

So what types of flashcards should you create? Let me share with you 2 out of the 3 types I laid out in the better solving course:

  1. The “formula” card (which is not necessarily just a formula)
  2. The “situational” card

Card type #1. The formula card

If you’re thinking that “I’m not sure if I’m doing the best way to create cards for memorizing formulas” then remember what I just said earlier:

Formulas don’t need much thought to, well, formulate.

  1. The reason why you’re memorizing them is because you’re going to apply them to solve a problem — a higher cognitive process that improves retention of the formula
  2. The reason why they’re in your cards is because you’ve understood how to use them already

Think of formulas simply as shortcuts. They’re handy dandy equations that allow you to plug and play numbers.

But the secret to remembering them comes with comprehending how each element in the formula is logically connected to the others, or at least, being able to use them to solve a problem.

For this reason, Math is probably the easiest subject to Anki-fy. Physics is another story, because it has more concepts embedded than procedural knowledge.

For example, because I already solve problems that involve Half-Wave Ripple Voltage at filter output, I can afford to make cards like these:

Obviously, you could get away with using image occlusion on textbook screenshots for this. They don’t need much thought as long as you understand them using the learning methods I shared with you in the Encoding part.

And like I mentioned…

Formula cards aren’t just limited to mathematical formulas. Again, think of them as shortcuts. So other cards that can fall into formula cards are:

  1. Theorems you’ve already learned (Mean Value Theorem, for example)
  2. Identities, after you’ve encoded them
  3. Common Graphs (but hey, this should be understood also)

The next one I call a situational card.

Card type #2. The situational card

Sometimes, application of knowledge could be situational, therefore if there isn’t any coherent understanding about when you should use a formula or when to apply a certain technique, you can just create a card for a specific situation.

In my deck, it’s usually in the format of “What formula to use when [situation]?” but you can format it however you like.

If I’m studying Physics, for example, and I’m confused about when I should use negative or positive signs, I could make a situational card that allows me to recall the right convention:

  • When should I use a negative sign when drawing vectors in free-body diagrams?
    • Left and down direction

As you can see…

These are flashcards that aid problem solving. If you don’t structure your cards in a way that aids your problem solving skills, then you’re just going to end up knowing a lot of formulas but not having the mathematical intuition to use them in problems.

You still need to practice — the right way!

As I’ve been emphasizing…

You can certainly maintain your knowledge of formulas by using Anki, but do know that problem-solving itself is equally important. After all, that’s what you’re maintaining your knowledge for!

If you don’t learn your formulas the way I showed you in the encoding part, and if you don’t apply your formulas in problem-solving, you’ll NEVER be able to use them when you encounter a new problem you’ve never seen before.

Just like how you won’t learn self-defense by studying martial arts book and memorizing the steps.

Just like how you will never be good at business just by reading business books and doing nothing.

But like any other skill learning, there’s the right and wrong way to practice.

In Better Solving with Anki, I tell my students to do purposeful practice instead of grinding an endless amount of problems.

This requires a “solved problems” book, and the goal is to train your mathematical intuition so you’ll be able to solve any variation your professor can think of. You can learn more about it by clicking here.

And again, if you just stop at understanding the formula but don’t give yourself the opportunity to retrieve them (either via practice or via retrieval with Anki) then you’re going to forget them eventually especially if you’re learning a lot of things unrelated to them.

To be honest, 3 years after my board exams, I’ve forgotten most of the formulas I’ve studied in Engineering. But that’s okay. If you’re the same way, know that it’s not a memory defect. It’s just that the memory requires you follow the attention-encoding-retrieval process to be able to maintain and build new knowledge.

In summary…

Overall, in order to commit formulas into memory we just need to learn how a formula works — which in this lesson, you have learned in three ways:

  1. By looking at it as a system
  2. By learning the derivations
  3. By applying the formula to a problem

Only then do we make flashcards for that formula, so they can remain accessible in memory and ultimately, useful for problem-solving.

To quickly impart some food for thought…

Math is the type of subject you don’t learn in Anki — you simply use Anki to help your Math, because problem-solving skills require procedural knowledge 🙂

NOTE: This article is a curated excerpt from Better Solving with Anki: Get Amazing Grades on Problem-Solving Subjects (Without the Endless Grinding).