View original postThe second example seems better to me. A big issue in teaching math, one you reference below, is that different people do conceptualize the operations and relations differently, so illustrations that convey them to some people are often useless to others.
Yes but there are common methods, if I show someone 4x4 and they say 20 I know what the mental error probably is and can test it pretty quickly. Once I know the error I can pick from a wide array of tools to help fix that. People who use something totally different are rare enough you won't often have to encounter them, there's usually only a handful of methods people use.
View original postThat is a very interesting phenomenon in itself, and one that has always fascinated me: That most people look at a randomly distributed group of up to five items and think "number," but look at anything else and think "lots." Most people can make it to six if presented that many objects arranged like the vertices of a hexagon, but must count the sides/corners of anything larger to know their number.
I think overall it is more representative of a thresh hold between conscious and subconscious thought. There's still subconscious thought and memorization going on, because people tend to remember 4x4 =16 rather than 4+4+4+4=16 and explains why on a 4x4 grid of dots removing one or two dots would significantly slow down their answer. They can 'count' the stars on a 8x8=64 grid a lot faster than the US flag of two overlapping grids, 6x5 and 5x4 = 30+20=50 setup. Of course, I see a flag as two rectangle grids overlapping, to be counted separately then added, other might view it as 4 double rows of 11 stars plus half a remainder, 4x11+6=50. The first method will likely be the better one, except in cases where one has a very high ratio of width to length, in which case that double row method is probably faster.
View original postView original postI can attest to that since I was well into college before I ever learned 'long division' because the small gifted class I was in had the teacher opt to teach it when I'd go in for speech class (can't pronounce R's) because she showed me a couple of bigger division problems and I solved them on raw rapid multiplication skill and she assumed I knew how already, I was our star math student after all and the class was composed of kids who tended to have already learned a lot of stuff outside of school. Later in homeschooling it simply never came up and it wasn't till we were going over synthetic division that my math instructor realized I didn't know what the hell long division was. I was essentially doing it backwards and not as efficiently but retraining me to do it 'properly' was non-advantageous at that point, and so even though I know how to now I still use the old method. I learned formal geometry after learning trig and calc, and it alters my way of viewing geometric problems rather significantly, with advantages and disadvantages.
View original postInteresting; makes me wonder what I was missing learning remedial handwriting while my classmates were, well, LEARNING.
' /> I have mixed feelings about formal geometry; exposure to formal logic was helpful, but it is definitely far more intuitive to me on a numerical basis.
I wasn't even aware geometry included a logic section. I'm also in the bad handwriting club, but never had remedial, just bad grades. 
View original postView original postThis may factor into why I do not like the math/lang analogy because it rings false to me, I already know I start with the same representation and end with the same answer but follow a very different course to get there than most, and I keep my nose out for that in others and sometimes finding it, I tend to attribute some of my skill at instruction to that expectation and sensitivity to non-standard approaches by others. But in language this is totally different, I don't think two people read the same sentence, achieve an identical interpretation, yet process it differently. That just isn't how it works, they might get different interpretations of what was said but they don't start and end in the same place if they take different routes... I assume anyway
View original postI am not so sure; different people tend to focus more, or first, on different parts of sentences, take a different perspective on each word/phrase/clauses significance, yet often arrive at the same destination. The real breakdown in the analogy is in your second point: Different people derive significantly different meanings from the same sentence far more often than from the same equation.
View original postView original postIn a place where there was no formal education system for most and a very non-standard one for those who did get educated it would, IMO, be very probable for someone to casually invent positional notation in their own informal way and use it strictly for counting eggs by the dozen, teach it to their proximate colleagues an successors, and they all use it but never for anything else. Techno-speak is definitely not a modern invention.
View original postLogical, though standardization promotes communication; a discoverys profundity matters little if the discoverer cannot "show their work" and transmit it to others.
Absolutely on the latter, the best way to master a subject is to teach it, but I only agree in part on standardization, I think it better to teach all major standards, more flexibility, less 'oops' mental barriers to trip over.
View original postView original postView original postRegardless, it LOOKS like they could have used multiple dice to generate larger numbers, and perhaps did, but the other dice would have used a completely different symbol set, additively. Another point of interest here is that the ancients generally mapped the digits 1-9 and their multiples of 10 and 100. That means there were 27 possible digits (forcing Koiné to modify three letters for use as numbers,) so a d20 could not contain all of them. Further, even if it held, say, the digits 1-10 and 10-100, it would have to omit any digits >10 that were not multiples of 10. To function as our d20 it would have to be enscribed with the digits 1-10, then the 10 digit followed by the digits 1-9 (or the reverse; by the Commutative Property of Addition it does not matter,) then the 20 digit.
View original postI'd almost have to see it written out and annotated to grasp it. Remember that a lot of card players casually think in a parallel of base 13 superbase 4 but never view it that way and never apply it to anything but cards, even though they cheerfully make card analogies to life. If our clocks consisted of 4 periods, morning, afternoon, evening, and night, divided into 13 segments (27 minutes) and 52 'minutes' of 32 seconds subdivided into 52 'seconds' of .6 normal seconds you could be almost assured that card games and time would have all sort of common analogies and comparisons. "I'll meet you at club king for the film, I might be a suit late though" referring to a period of about half an hour and saying he might be abut 6 or seven minutes late. Or alternatively expressions like 'high noon' could work their way into cards. Any sort of competitive game or religious ritual are going to encourage those involved to rapidly assimilate the concept even if it has no outside parallel or logic and I think predispose them to try to graft that onto the outside world wherever there is any perceived overlap. Witness that 2d10 or d% is used to get a well known concept but a d20, with no daily use equivalent, generates them as 'natural 20!' or 'fuck, rolled a 1!' or even snake-eyes or boxcars. I don't think a game or religious divination would lead to adaptation for math or practical use but I could easily see existing math or common concept being brought into a game the way a d% is.
View original postThough I feel there's something confused, rambling, and very much a massive digression to everything I wrote here
Do not feel obliged to reply point for point
View original postIt is something of a sickness with me.
I will say THIS card player does NOT think in base 13 superbase 4, even when playing. Perhaps I SHOULD, but my memory is not in good enough shape to count every card; usually I just count honors so I know what is high in each suit, and distribution so I do not lead anything CERTAIN to be trumped (or worse, give the bad guys a rough-slough.) Sometimes that gets me in trouble once all the honors are gone and I cannot remember if an 8 or 7 or whatever is good or a higher non-honor spot is still out there. One such occasion proved especially embarrassing because I had lost count of the distribution as well, which left me wondering if my heart 8 (or 7, forget which) was high when it was not just the high heart, but the LAST heart.
View original postFrom what I can tell, most people tend to think in terms of "un/somewhat/very likely," and do not go further absent the incentive you reference. In AD&D a natural 20 is a crit success and 1 is a crit fail (or vice versa,) while in GURPS a 3 or 4 is crit success and a 17 or 18 is a crit fail*. Most people will look at that and think "makes sense; criticals are supposed to be rare, or at least uncommon," some might even opine that the ability to produce either with two rolls rather than one makes them more common in GURPS. However, the chance of rolling 20 (or 1) on a d20 is a fairly respectable 5%, while the chance of rolling 17 or 18 on 3d6 is <2%—even though there are 4 times as many ways to do it! People who are not veteran gamers (or mathematicians) seldom realize that.
View original post*GURPS further complicates things because a natural 17 is only a "normal" failure for skills >15, and any natural roll 10+ below an unmodified skill is a crit success. Both incentivize buying skills past 15, which would otherwise be almost pointless since there are only 11/216 ways to roll >15 on 3d6.
Ah, gurps... the only RPG think mentioned in the conversation thus far
Well your odds of rolling a 17 or 18 on 3d6 are 4 in 216, just under 2% as you say, but so many systems let you have 'drop the lowest' or 'reroll one' and those massively change those outlier spreads. Classic D&D, need a 17 CHR for the paladin, you had a 4 in 216 chance of getting it, or 6 times that if you got to assign your rolls to a stat. With 'drop the lowest' there are 1296 possible combinations instead of 216, and there 21 combinations for 18 or 3 each and 54 more for 17,4 each. Your odds of rolling 17+ jump to just under 6% instead of just under 4%. Alternatively 3 and 4, by dropping the lowest, actually only have 1 and 4 combinations, since you drop the lowest. Your average value shifts from 10.5 to ~12.5 and you don't mirror around that value, your odds on 3d6 of getting a 9 equal those of 12, drop the lowest and you don't just move the middle up but make it assymetric. Your odds of getting a 7 or less, previously as probable as 14+, now drop to less than 6% itself. Getting the four lowest number is less probable than getting the highest two numbers. So many games, especially for stat building, use that method and even a lot of dice intensive games offer 'drop the lowest' as a special ability without most players realizing how breaking it is. It's like reroll save abilities, versus those that let you add +5 to a die roll, it's more than passing difficult to figure out which is better especially since it would depend on the circumstance.
" "Well I'M not; Cleo, put your top back on! 
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