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Comment by andai | Hacker News Reader

> Son & Simon 2012: Furthermore, even after acknowledging the benefits of spacing, changing teaching practices proved to be enormously difficult. Delaney et al (2010) wrote: “Anecdotally, high school teachers and college professors seem to teach in a linear fashion without repetition and give three or four noncumulative exams.” (p. 130). Focusing on the math domain, where one might expect a very easy-to-review-and-to-space strategy, Rohrer (2009) points out that mathematics textbooks usually present topics in a non-spaced, non-mixed fashion. Even much earlier, Vash (1989) had written: “Education policy setters know perfectly well that [spaced practice] works better [than massed practice]. They don’t care. It isn’t tidy. It doesn’t let teachers teach a unit and dust off their hands quickly with a nice sense of ‘Well, that’s done.’” (p. 1547).

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It's strange they would use math as an example or where spaced reptition would work.

It's specifically one of the least applicable domains. Things that are mainly conceptual, that build on previous concepts, don't benefit from spaced repetition at all.

With math, the hard part is the conceptual understanding. But once you understand a concept, you understand it. You don't really need to review it again.

This is why the part of the article that lists what spaced repetition works well on is all factual/associative -- vocabulary, trivia, medical terminology, etc.

Spaced repetition is for simple associative facts -- "x is y" -- not conceptual things.

For whatever reason, once we've understood a concept, it seems to be part of our thinking forever. But of course the hard part is understanding it. Some people struggle for days, weeks, or even years in school to truly understand things like derivatives, or pointers, or trigonometry. Spaced repetition does nothing for that.

I'd like to respectfully disagree.

A big part of math is building up your toolkit: when I see x, I should do y. Feynman famously made this a big part of how he approached quantum mechanics, saying that it was his toolkit that allowed him to solve problems others saw as impossible.

While the initial understanding can't be done with spaced repetition, remembering what's in your math toolkit and how to use it can very much be aided by spaced repetition. I've personally used this to great effect teaching people math through my spaced repetition app.

Can you give an example?

I'm not saying math practice isn't useful -- after all, that's what homework problems are for, to figure out which concepts apply and use them. Repetition itself is useful.

But in my experience, this doesn't benefit from being spaced over time, and it's more about achieving full understanding of the concept.

You seem to be describing pattern recognition, but pattern recognition is conceptual -- you need time/experience to build that up, but it can be done in a day and doesn't need to be revisited days/weeks later in order to not be forgotten.

What kind of math content do you think benefits from repetition that is necessarily spaced over time?

In my experience, which is graduate level mathematics, spaced repetition is essential. Once you get beyond basic mathematics and algebra, recalling and understanding concepts and definitions is essential. This is true at the calculus level where you need to remember things like the chain rule, integration by parts, various theorems, etc. But it is even more true at the higher levels of mathematics where there are many more theorems, lemmas, and definitions to remember. I find that at that level, just having a huge depth of recall for definitions alone is incredibly useful. If you remember enough definitions you can start to piece things together. If you can recall theorems, lemmas, and some key proofs you can achieve quite a bit.

Personally, I only understood logarithms after memorizing the log laws using spaced repetition. I had been taught logarithms before descriptively, but only once the equations were in my head could my brain make sense of how they work.

Good example. I know people frown on memorization, but personally I would say the following are useful to know deeply:

- Log laws

- Exponent laws

- Derivative rules

- Probability laws

- Combinations/Permutations

- Matrix algebra

- more?

When you're reading through math derivations, or trying to work one out yourself, you absolutely need to have some basic literacy in these rules.

> But once you understand a concept, you understand it. You don't really need to review it again.

I disagree - if this were true I would still understand all of calculus and linear algebra, which I took years ago.

I think part of what you say is true - some concepts that are understood stick with you more than the sort of fact memorization you might find in a history class. But there are still many math skills that would benefit from spaced repetition. What are line integrals used for and how do I compute one? What does the determinant of a matrix represent and how can I find it? What is Stokes' Theorem?

It's one thing to understand a high level concept (e.g. an integral can be used to model continuous accrual of some quantity) versus details (e.g. how do I integrate common trig functions?).

I frequently used spaced repition in my maths/physics courses to great success. The types of things I would memorize:

- Key theorems

- Proofs

- Any kind of visualizations (e.g. statistical distributions)

- Properties

- Trig identities

A lot of time in math or math heavy fields, there are a lot of tiny details and nuances. Using an SRS properly really solidifies those concepts in your head (See my comment below on manufacturing 'aha' moments). It also helped because a lot of those key concepts were immediately available in my mind and I was 100% I had them correct. Memorization reduces a lot of mental friciton.

In math, there can be a lot to memorize! There is a lot of notation, and associated definitions and facts. Also formulas, and not to mention the values certain important constants.

Mathmatics textbooks work well as a sort of reference volume. If you want to repeat earlier sections, you should, and you can, but it doesn't mean the text book needs to repeat the content. The lesson plans can use the book non-linearly.

Depending on the specific math course, it can be the case that later topics contain earlier topics anyway, so a final with only problems from the end will also cover techniques from the beginning and middle. Certainly, that's not always the case.

Yup. It's also really difficult for students. I use it in my tutoring practice, and I always have to convince my students that it doesn't mean they're backsliding.

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