I am currently self-studying, so I learn using a textbook which I work through in a linear fashion. As I am introduced to new concepts I ask myself questions and I attempt to answer these questions and if all else fails I ask Maths SE. Only after I've understood a certain topic do I then formulate all that i've learnt in a Q&A format: i.e. formulas, definitions, theorems, etc. so that it's ready to be inputted in Anki (a spaced repetition system) that is used to create flashcards and schedule them optimally so that you don't forget them.

Anki is very helpful in ensuring that I retain most of what I've learned. My concern is that I'm essentially storing all that i've learnt in memory as bits of discrete knowledge that I can recall on demand. I feel like this removes the "connections" that are made between concepts; the knowledge does not seem "networked", just disparate bits of knowledge i can recall. I want to cultivate a strong "network" of knowledge.

So my questions are:

  1. How can I utilise a spaced repetition system to help me learn Mathematics most effectively?
  2. What are the best ways of making strong connections between mathematical knowledge you've learned?
  • $\begingroup$ @SueVanHattum Thank you. $\endgroup$ – seeker Nov 8 '14 at 15:36
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    $\begingroup$ Your second question is quite broad: It concerns a large proportion of all of mathematics education research... $\endgroup$ – Benjamin Dickman Nov 8 '14 at 18:47
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    $\begingroup$ @BenjaminDickman Can you recommend me some literature about it, if possible $\endgroup$ – seeker Nov 8 '14 at 18:48
  • $\begingroup$ I think you need to somehow turn this into a more focused question: I am voting to close as too broad (which is the reason your previous question on very similar material was closed: matheducators.stackexchange.com/q/4200/262). $\endgroup$ – Benjamin Dickman Nov 8 '14 at 19:06
  • $\begingroup$ @BenjaminDickman, I think you're misunderstanding my question. All that I'm asking for are methods that will allow me to make better connections between things I've learned. $\endgroup$ – seeker Nov 9 '14 at 17:53

Making connections between concepts can be done with something called (simply enough) a Concept Map. (In fact, this is a key part of some current research that I'm involved in, and we plan to present at both the AMTE 2015 and NCTM 2015 conferences in Orlando and Boston, respectively.)

Concept mapping was developed by Joseph D. Novak in the 1970s. His book Learning How to Learn, coauthored with D. Bob Gowin, talks of how you can create these concept maps yourself. Additionally, he talks of creating a knowledge Vee, which may also be another helpful visual to make connections between ideas. (Though, I must confess, I don't know much about the knowledge Vee apart from its existence.)

Note that Concept Maps are not Mathematics specific, and can be applied to any number of fields, but Novak himself holds degrees in Science and Mathematics.

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  • $\begingroup$ Agreed! In fact, concept mapping is one of the key ideas behind the online Expii initiative (expii.com). $\endgroup$ – Victor Wang Nov 9 '14 at 4:59

I've been tempted to try to use spaced repetition in the past, as there is certainly some kind of linguistic component to learning mathematics. This always meets the paradox that many mathematicians work nonverbally, as pointed out in Hadamard's study of the mental habits of mathematicians. Mathematics is accumulated less by mnemotechnics than by a slow and difficult struggle to use all of one's mental capacity to produce reasonably faithful mental models, as observed many places by William Thurston, e.g. as expressed in this quote:

"Mathematics is amazingly compressible: you may struggle a long time, step by step, to work through some process or idea from several approaches. But once you really understand it and have the mental perspective to see it as a whole, there is often a tremendous mental compression. You can file it away, recall it quickly and completely when you need it, and use it as just one step in some other mental process. The insight that goes with this compression is one of the real joys of mathematics"

What worries me about trying spaced repetition/mnemotechnics is that these try to get the compression/memory without the long struggle to form the mental structure. Such a process is precisely the opposite of trying to store things cleanly in order to remember them quickly and accurately. Thurston's quote suggests that memory of a topic is won by engaging very painstakingly for a long time with an extraordinary mess.

This sort of thing is even reported in interviews with more algebraic thinkers like Alain Connes:

" I always proceed in the following way : whatever the complexity of the problem, instead of trying it first on a piece of paper, with a pencil, I just go out for a walk, and try to have all the ingredients present in my mind, in order to start manipulating them mentally. Only after this exercise, am I able to see clearly, think about the various steps and begin to get a mental picture. This is a painful process which consists in gathering in your mind, in your memory, all the elements of the problem, in order to begin manipulating them. It is an exercise that I recommend - well, of course, different people function differently- if one wants to be able not to depend upon paper and pencil. Because with paper and pencil, you get tempted to start writing immediately, and if you haven’t thought long enough before, you will get nowhere. You will get discouraged before having had enough time to create in the linguistic part of the brain specific mental pictures that you can then manipulate, as usual, by zipping them, transforming them into something smaller, and then moving them around."

This compression is again won by struggle. I'd be wary of "learning" mathematics via ANY sort of mnemotechnics. It's probably best to struggle with things and work for their compression without looking for a shortcut! Of course, this is just my opinion.

Perhaps there is one honest mnemotechnic mathematicians use: the simplest nontrivial example of a given phenomenon. Such an example is analogous to a model of a finite geometry, in that any such example encodes the whole theory and the example can be generated from a relatively small seed that is more compatible with our mental modules, perhaps computationally or visually. One gets the impression that such an example is more concrete since its interrelations are less disembodied, or could be generated from the simpler rules used to define the example.

Edit: The following may be helpful regarding the use of spaced repetition. It might be helpful to internalize "atomic facts" with which to work (think arithmetic facts) but is not likely to be helpful for working with these facts in creative ways, (think fancy mental arithmetic a la Art Benjamin). This said, the following excellent quote of Paul Halmos about the mathematician's work points to a potentially useful application of spaced repetition:

For the professional pure mathematician, mathematics is the logical dovetailing of a carefully selected sparse set of assumptions with their surprising conclusions via a conceptually elegant proof.

From this standpoint, it may be useful to employ spaced repetition to quickly internalize the "sparse set of assumptions" in order to begin thinking about them. This is interesting in light of Alain Connes's quote above, as well.

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    $\begingroup$ It's not an either-or. If you ask SRS fans, they'll tell you that SRS is at its best when it's used in conjunction with good mental models that come from that "slow and difficult struggle." The intent of SRS, as expressed in rule #1 of Wozniak's Rules (supermemo.com/en/articles/20rules) is to help you retain knowledge that you have already mastered. $\endgroup$ – SigmaX Jan 2 '18 at 17:12
  • $\begingroup$ The property of compression attributed to mathematics in this answer sounds a lot like what the psychologists call "chunking", which is thought to occur in every domain of learning, not just mathematics $\endgroup$ – Calculemus Jul 23 '19 at 19:17
  • $\begingroup$ @Calculemius: indeed this is true, but the distinction comes in the ability to "move" the compressed objects around mentally. These amount to pseudo calculations that have been described as like "muscular contractions" in Hadamard. One might describe these as stable mental structures that can be manipulated like objects. This is distinct from strictly linguistic or conceptual chunking and seem special to mathematical practice. $\endgroup$ – Jon Bannon Jul 23 '19 at 19:43

For the past eight years or so I have used spaced repetition to maintain proficiency in a number of areas. primarily it is language study, but I also use it to maintain a grasp on many scientific concepts, although I have not yet begun to focus entirely on math.

I think that learning mathematics would be similar in principle to learning a language. When I am learning a new language, I have to focus on two areas: grammar and vocabulary.I have found that creating multiple flashcards that test my knowledge of a particular grammar point is very helpful at making sure that your knowledge forms a cohesive network, one flash card by itself does not seem to be sufficient for something as broadly "connectable" as grammar is.

Also, when learning about a more difficult subject, I need to create flashcards that connect each of the pieces together. Imagine that your concept is a large dinosaur skeleton, and flashcards or the cables and glue that hold the statue together. For every connection, you need at least one flash card. While this might seem excessive, over the next 30 years reviewing that single flash card barely takes minutes of your cumulative time, so add many flashcards and delete the ones you find to not be useful as you review them.

Using spaced repetition seems to be the most efficient way to maintain synaptical connections in the brain that humans have at their disposal right now. You are definitely on the right track, you just need to learn to capture the connections of a more nuanced discipline in flashcard form. I highly recommend you read this article that is basically the 20 Commandments of flash card creation: http://www.supermemo.com/articles/20rules.htm

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