i'm still without firm conclusions,

only guesses and impressions.

i'd like to think that my students have begun to see proof as a means of discovering truth in mathematics, much like how scientists think in terms of experiments in order to make new discoveries.

*we know the world is out there,*

that it obeys certain laws and patterns,

but how do we find out what they are?

that it obeys certain laws and patterns,

but how do we find out what they are?

i should emphasise:

*begun*. many students tried very hard, but they were not often successful [0].

however, i think it very important that they tried. many of them improved substantially.

when i was a graduate student and more cavalier, i used to say:if you think of maths as a language, then it takes a lot of time to become fluent at it. if you think of a maths ph.d. as "mastery" (which i discourage you from doing) then it takes quite a few years .. on top of having completed an undergraduate degree for its preparation.

"you know what half a proof is? half of nothing."

my older self would now disagree. maths is not something one learns instantaneously, nor is much else in life. even a video game take days of obsession to figure out and win.

that younger self did always underestimate the value of progress .. to the point where he (the old me) was stunned, after some years, that he earned a ph.d.

so i thought of my students' proofs much as i did compositions in a first course in spanish: there are always a few made-up words, errors in grammar, a lack of style from a lack of experience .. and occasionally you read a sentence that sounds .. real, that you could hear someone actually saying ..

[0]

*even induction proofs were tricky to some.*

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