in the trenches .. or so it feels.

i should probably leave the office soon, if only because ..

there are students *everywhere* in the department.

i hear many young voices from the few office doors that remain open.

the math help room is full of people.

young men and women have taken over the couches in the middle, watching a tutor intently as she's explaining demοrgan's laws on a nearby whiteboard.

needless to say, 'tis the season of midterms .. and as time goes on, fewer office doors will be open. probabilistically, it's only a matter of time before someone stops by my door and asks me a maths question ..

that said, i should probably get out of here before i have to say "NO" a lot to many slightly desperate faces.

in which it is not easy to be an adult.

this morning, after finalising my calculus lecture for the day, i stopped by a local cafe and started working on a proof of a statement that i had jotted down as a claim, months ago.

it was frustratingly fun. i almost had the proof .. when i realised that, soon enough, i would have to reach campus and give my lectures.

then i started speculating:

what if i .. just didn't show up?
what's the worst that could happen?

.. but then i sighed and realised that, deep down inside, i respect responsible people that can be counted to show up when they need to show up, and that i want to think of myself as such a person. that said, there is only one way to get what you want ..

.. and that is to put in the effort, to work hard, in order to attain that goal.

i've told my students that before, and that achieving their goals is conceptually simple but emotionally nontrivial. those things said, that means that i have to do the same ..

life would be so much easier if one could pick and choose one's convictions, at moment and whim. it is far harder to be consistent.

it isn't easy. it takes a commitment and somehow we educators do it .. but like anything worthwhile, it shouldn't be easy .. and for me, it isn't.

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on a mostly unrelated note, i really enjoy visits to that cafe. the coffee is incredible.

ARR, MoAR! On computers and proofs.

so today i learned what a "discrepancy" is:

"Adding up the numbers in a sub-sequeηce gives a figure called the discrepaηcy, which acts as a measure of the structure of the sub-sequeηce .."

~ from " Wikipedia-size maths proof too big for humans to check" @newscientist

as for how this came up ..

Erdös thought that for any infinite sequeηce, it would always be possible to find a finite sub-sequeηce summing to a number larger than any you choose - but couldn't prove it.

It is relatively easy to show by hand that any way you arrange 12 +'s and -'s always has a sub-sequeηce whose sum exceeds 1. That means that anything longer – including any infinite sequeηce – must also have a discrepaηcy of 1 or more. But extending this method to showing that higher discrepaηcies must always exist is tough as the number of possible sub-sequeηces to test quickly balloons.

Now Konev and Lisitsa have used a computer to move things on. They have shown that an infinite sequeηce will always have a discrepaηcy larger than 2. In this case the cut-off was a sequeηce of length 1161, rather than 12. Establishing this took a computer nearly 6 hours and generated a 13-gigabyte file detailing its working.

grade school, of a kind.

often when meeting new people, i just tell them that i teach maths.

it's a more expedient answer during those times when you don't feel like explaining your research to someone who doesn't seem particularly adept at basic algebra, simply wants to be friendly, subsequently asks about your work, and only expected a simple answer (but as to why they expected such a thing is beyond me). [1]

that kind of answer gets misconstrued, if you're not specific. occasionally i'm asked if i teach junior or (senior) high school.

so the next time someone asks me what grade i teach, i think i'll say:

"oh, 13th through 16th, and the occasional 17th grader." (-:

[1] just now i re-read that sentence and realised how long it is. did anyone get it on the first attempt? (likely the grammar is incorrect; if anything, the style is poor.)

Seasonal Affective .. Re-order.

odd. every february i get new ideas to struggle with.. or what i like to think as creating form out of chaos.

i wonder if it's a seasonal matter, if the isolation of winter stirs deeper thinking and contemplation.

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when i think about it, time seemingly and magically slows down when snow is falling: i perceive it so, at least.

rain doesn't fall the same way, nor do baseballs and rocks. i'm too slow to realise that the structure of snowflakes allows an exception; falling leaves, too.

the alternative would be that the laws of gravity have been suddenly shut off or gone on holiday, allowing solidwater to float briefly on air.

it makes me believe that improbable things can happen, urges me to try and create impossible things, if only to prove a point (via contradiction).

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landscapes trans-form into their mollified versions, where sharp corners of peaks and cusps are gone. in contrast, jagged edges appear from broken sheets of ice, and once clean lines along rooves are interrupted by icicles.

what was rough is smooth; what was void is now full of matter, with sharp corners. this is a different world, an inverted world.

a cynical rant on .. you guessed it: teaching!

i hate to say this .. but the more i think about it, the more it makes sense that good teachers slowly become negligent, if not bad, teachers.

at the moment i'm grading exams. for one problem, i'm quickly realising that most of my students weren't paying attention to me when i was patiently .. doing my best .. to explain how to deal with this one particular concept.

that's not the only exam problem where this phenomenon has come to pass.

if the students aren't paying attention to you, then what's the point of putting in the effort to teach them carefully?

coupled with the constant excuses of "i had to miss class because .." or "but my high school teacher told me .." it's enough to make you wonder how students ever learn mathematics at all, and if you were some sort of social mutation to whom maths was somehow natural ..

(to their credit, students rarely (if ever) complain further,
once they realise you are being fair with them .. well, in my experience, anyway.)

[sighs]

teaching well is as equally frustrating as doing good research .. the problem is that the former task has no real guarantee of working. it really depends on the students in your class, particularly their disposition. i've heard pundits made casual demands of educators, where

..if you're a good teacher, then you always get through to your students and they can succeed.

i believe that as much as if you work hard enough, then you can become a multi-millionaire. it is surely possible, but the odds are bad and most of the time, circumstances don't favor that outcome.

epilogue (as of 15 feb): on these student papers i seem to be writing "irrelevant" just as often as i write "false" or "incomplete" ..

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on a related note, i wish i had some mind to talk about my research .. but this new professorial life makes it all but impossible to get anything done ..

on alter-egos..

all I want to do lately is have enough time to write up this one result. in fact this morning I woke up extra early to $\LaTeX$ a lemma and most of its proof, before having to start my day job of being. . well.. a "professor."

odd..

I didn't mean for it to come out that way, but there seems to me a sharp divide between the 'me' that teaches university maths and the me that does the research.. or tries to, anyway.

it's a little like how Batman has to play at being Bruce Wayne, if only to make sure that he can be whom he thinks he should be, for at least some of the time.

"either you die a hero, or live long enough to become the villain" ..

i think i give good lectures.

there's enough evidence that points to this, in the form of student evaluations and comments over the years and from different universities .. and even from a few different countries.

apparently i have the ability to be very clear, which is fine.

what i've been struggling with is:

i can give the best lectures that i can .. that i care to give.

when i do, then i feel expert and in my full powers. many of my students would look up to me and i receive their esteem. simply put, it makes me feel good.

it will probably impress the students, put me in a good light, and continue to give me good evaluations. it will be good for my career if i keep doing this, and only make stronger my case for tenure.

the question is whether that really matters.

if my students don't get anything out of the clearest, most intuitive lectures that anyone can give, then really: what's the point?

some of my colleagues may argue that if the student is committed, then s/he will put in the effort to get what they can from our lectures and the course in general.

i would rather say that if the student is committed, aware, and well-trained..

awareness, i believe, is not only a personality trait; with time, it can be learned. the awareness that is relevant to a student in my course is being aware of what problems are hard, why they are hard, and what steps are needed to overcome these difficulties. the point of the clarity in my lectures is so that my students can cope with the more difficult aspects of the course, by means of a few basic but useful principles.

study habits are precisely their namesake: they are habits, and they can be learned too. some students never learn these habits to do well at the university level [1], which means that it is up to me and other university instructors to promote these habits. in particular, it means convincing students to change their ways, which often means that they should do things that they are not comfortable doing or simply don't want to do, like ..

.. reading the textbook ..

.. asking questions, answering questions ..

.. "showing enough work to demonstrate understanding of the problem, the solution, and related concepts to them" [2] ..

that said, commitment is a two-way street. who in their right mind would commit to responsibilities that offer no reward, tangible or otherwise?

so it's not clear to me:

when i'm teaching, should the lesson be so clear that the students don't realise the underlying difficulty, and fail to pay it adequate attention? if the students don't struggle with the material on their own, then will they really learn it as well as i'd like?

i have played a hero before, been given a stage to strut;
should i, for the greater good, play a villain instead?

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[1] habits form due to need and from reinforcement. i've seen plenty of students with the 'wrong' study habits, if only because they've been rewarded by their previous teachers for skills that i would not reward. high school mathematics in the united states can be taught rather formulaically, and some students have been taught to do nothing more than operate a sophisticated calculator. the point is to replace old habits with new ones .. which can be an even more daunting task for the educator!

[2] i always put this policy on the cover page for my exams. sometimes i even pass out printouts of this cover page, a week or two before the exam, and explain what it means.

a former colleague of mine once referred to a syllabus as a list of threats and promises; s/he's not that far off ..!

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on an unrelated note: since my (unexpected) fall hiatus from blogging, it's become less easy to return to the habit and have things to share with you readers.