The 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.

That 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.

Interesting; 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 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.

Logical, though standardization promotes communication; a discoverys profundity matters little if the discoverer cannot "show their work" and transmit it to others.

It 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.

From 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.

Anyway, to see it written out and annotated, try the below link.

*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.