comment by shminux ·
2014-12-28T19:04:20.514Z · LW(p) · GW(p)
Only tangentially related, sorry.
I am incredibly slow at computation (such that on every math test I have ever taken, it took me at least 3 times as long as the second slowest person in the class to finish).
I have tutored a fair amount, and my experience is that when someone says "I am incredibly slow at computation", the real issue is their lack of fluency. After you simplified your first thousand of trig expressions, when you look at the expression 1001, you already see most of the answer, all you have to do is write out the steps.
My model of a learner is that a person is a sort of Markov chain, where you try to go from the NO-SKILL state through the LEARNING state into the HAVE-SKILL state, also known as fluency or mastery.
Learning goes like this:
NO SKILL ----- learning rate ----> LEARNING- ---- internalization rate ----> HAVE SKILL
Forgetting goes like this:
HAVE SKILL ---- slow forgetting rate ---> NO SKILL <---- fast forgetting rate ---- LEARNING
People are extremely different in their learning and forgetting rates, which are also individually subject-dependent. Some learn a new skill quickly, others take awhile. Some retain 90% of the skill from one session to the next, others barely 1%. The internalization rate is less variable. As long as you manage to mostly keep in the LEARNING state for some period of time, you eventually get to the HAVE SKILL state.
The slow forgetting rate is well,,, slow for almost everyone, so it takes a long time to forget a well-mastered activity. You can probably do long division still (maybe after 5-10 min ramp-up), even if you haven't done any in a decade and it was a real pain to learn the first time.
The apparent learning and forgetting rates also depend on already having the skills similar to the one you are learning, like when building the jig-saw puzzle.
Anyway, my point is that you have likely misdiagnosed yourself. The symptom "incredibly slow at computation" could be a manifestation of one or more of the following:
- being in the LEARNING state instead of the HAVE SKILL state
- having relatively low learning rate and/or fast forgetting rate for the usual amount/frequency of repetition
- trying to learn the skill in isolation, which slows down learning significantly
If you figure out which of these apply to you, odds are you will no longer consider yourself "slow at computation", but, say, "requiring more frequent repetitions than average to master a new computational skill".
This model is, of course, rather simplified, as everyone appears to have their limits which they hit eventually for an advanced enough skill, but hopefully helpful enough for the initial diagnosis.
comment by Vladimir_Nesov ·
2014-12-28T20:26:17.925Z · LW(p) · GW(p)
I think it's also useful to clarify the relevant meaning of "fluency" in a technical topic, so that we can talk about fluency in smaller topics and work the ratchet of getting more and more stuff towards "have skill", without juggling too much at once in the "learning" state and forgetting things before they are fixed.
The relevant sense of fluency is not about speed or quality of results or diffculty of the problems that can be solved, even though these things come with fluency, but about skill at answering most simple questions and performing recurrent tasks that's mostly offloaded to System 1, that's intuitive and doesn't require too much attention to keep going. Solving simple problems has to become easy. Even if you can solve hard problems perfectly and quickly using a method novel to you that was just explained, that's not yet fluency, because you'd be leaning on attention and working memory. If you commit those same capabilities as System 1 skills, they would do most of the work for you while leaving enough attention to plan the process, and won't be forgotten in a year, allowing accumulation of vast expertise. Compare this practice with learning a topic just well enough to become capable of solving some problems (cramming for an exam is an important example of this failure mode).
Things one can become fluent at can be as small as seeing the structure and motivation for a solution to a particular (kind of) exercise or to a standard lemma. As skills add up, knowledge of how particular words translate becomes ability to speak a new language. This never happens if you keep relying on a combination of working memory and a dictionary.
comment by shminux ·
2014-12-28T20:59:20.836Z · LW(p) · GW(p)
Right, I agree. Being able to answer simple questions almost instantly, without even having to think through it consciously, is a good indication of the System 1 at work, which is the goal (or maybe the definition) of learning a skill. Some programs and tests are deliberately structured in this way. The Kumon program is one.