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.

---

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).

---

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" ..

---

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?

---

[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 ..!

---

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.

my teaching routine, lately.

---begin:teachingmode---

$CONFUSIONPARAMETER=LOW

#include badjokes.lib

define class spring;
type spring.level = nonmajor;
type spring.time = 13012014;
define notes lesson;
type lesson.example = introduction;

while (spring.time != 01052014) {
lesson.example = workout(lesson.example-1);
lesson.outline = summary(badjokes.day(spring.time),lesson.example);
class.print(lesson);
spring.time++;
}

---end:teachingmode---

so, if anyone asks ..

like all other working mathematicians out there, i am often asked ..

.."so what do you do all day?"

lately i've felt like answering:

"i've been trying to build impossible geometric objects, in order to show that certain mass distributions [1] cannot possibly exist."

as you may guess, my recent obsession has to do with finishing a proof by contradiction .. which i suspect is a means of inference that most people are uncomfortable with.

if my experience in teaching basis analysis has any weight here, i think that even mathematics majors at university have trouble with this method of proof.
..
..
.. thinking about it, that kind of answer isn't terribly helpful .. that is, to me. more often than not, conversations of this kind only on go downhill after that question; either (a) i say something too complicated and the other person, not understanding, feels dumb, or (b) i make it sound too easy and the other person wonders why i bother working on that kind of problem.

more likely, the other person would probably be wondering what would drive a person to think all day about things that might not exist ..?

often i just can't win with this kind of thing.



[1] this is my colloquial expression for what's known as a measure; i think i borrowed (read: plagiarised) the term from falcοner's book, in fact. before settling on this, i tried to use the term "prοbaβility distributιon" but this usually misled my audience that my work is related to statistics of some kind ..

so it begins .. again, yet not again.

odd:
after posting a few days ago, i suddenly feel like posting again.

---

as i mentioned last time, yesterday was my first day of teaching for the semester. i can only say that i felt very boring.

come on: everyone knows what vector addition is!
do i really have to go over the first section of the first chapter?

surely there is a more efficient way .. maybe i should have built a worksheet, have everyone work it out in a few minutes, go over the answers, and then go on to something more interesting ..

.. thinking about it, maybe i should have done exactly that! [1]

---

apart from that, there's little else to say. today was the first department meeting for faculty and i learned exactly how out of the loop i am about departmental and administrative affairs.

..
it's a strange thing, being a tenure-track faculty member. it's like being suddenly thrown into the real world and realising that you have to grow up.

[1] that said, if you're reading this and will actually try this approach in your own first day of class, then let me know how it works ..!

what i didn't.

i don't know what happened.

tomorrow is my first day of teaching for the spring semester;
today i spent 8 hours in the office, i was busy all day ..

.. and yet i still haven't written out my lecture notes!

oh well: my classes meet in the afternoon, so i guess i'll write them tomorrow morning.

---

it's been a while since i've last posted in this blog. i thought i'd spend the winter break making sense of my life and all that's happened, this past fall ..

.. what with this new position at a new university and all ..

.. but things still don't make sense. most days of the week i'm making it up as i go along, just trying .. trying my best to get it all done and stay sane at the same time.

there never seems enough time to do it all: teaching, research, faculty meetings and advising and so on. more precisely, there is never enough time in the sizes and shapes that i want them [1].

if i could identify a change in my life, then i'd say that time now comes in fractured form.

---

so i shouldn't talk about what i did during winter break [2]. it would be more appropriate to say what i didn't do.

i didn't go to the office,
i didn't answer any student emails that i didn't have to answer.

i didn't make sense of my life,
i didn't travel out of town,
i didn't make any new goals.

for the most part, i didn't want to do anything.

i wanted to, i tried to write up notes for a research idea. there ended up being a flaw in the argument and so i thought, off-&-on about the problem ..

.. but not so deeply as to make it too much like work;
i think i got somewhere with it.

this past week i realised that, starting tomorrow, i will have to start doing things for a while: commitments, duties, promises ..

it's starting again. whether it makes sense or not, this new job and life, there are things to do, again.

[1] if you spend enough time staring at weekly schedules, such as the default format for gοogle calendar, then time stops feeling 1-dimensional and linear. instead, it becomes more and more like a very weird tetris game in 2-D, fitting commitments into rapidly dwindling empty spaces.

[2] today was also the first day of spring classes, so quite a few colleagues asked me that question anyway.

ARR!.. imagine computers as research collaborators?

sometimes i wonder if i should be a mathematician at all .. lately i've been reminiscing about the dreams my younger self had: among them was to be a successful novelist, perhaps in science fiction.

so when i read news articles like this one ..

"Some might argue that computers will never be able to match human ingenuity but it is difficult these days to argue they can't at least mimic many of our skills.

Take the eDavid painting robot. The computer-controlled arm - adapted from a welding machine - chooses from five brushes and 24 colours to create impressive artworks on canvas.

It works by snapping a photo of its subject matter and then making the necessary calculations to turn the image into a drawing or painting in a wide variety of styles.

Its creators admit that it has no awareness of what it is doing. But it is able to make decisions about things like shading and brushstrokes as it goes, tweaking its moves based on how the picture is evolving, rather than just creating a pre-determined image."

~ from "The quest to turn computers into creative artists" @bbc_tech

.. then i immediately begin to imagine the possibilities:

if the strength of a computer lies in being very efficient with a finite, fixed set of tools, then imagine if we could get computers to prove simple lemmata for us, just by giving them suitable hypotheses and a fixed set of axioms and existing lemmata ..

yes, it is hard enough to build a robust proof-checking program .. and some researchers have spent years of their lives focusing on a single, specific verification .. but i'm not talking about a universal engine:

i liken it to writing programs that play go or chess well. it's not that the computer can think on its own, but rather that it can traverse through the decision tree of possible games very efficiently. in fact, a competitive program mightn't even run through all the possibilities, but simply the games that grand masters have played before.

so imagine coding in all the basic, rigorous proofs from mathematics (e.g. the proof of the triangle inequality in Euclιdean space) and adding shortcuts into the space of all proofs. i wonder what lemmas a computer could tell me ..?

being a mathematician, sometimes i have mathematical daydreams.

a shot in the dark (but no updates yet: stay tuned)

(yes, it's been a while .. and no, i don't have time yet to write about what's happened since the last two posts ..)


[sighs]

i'm almost convinced that there is no good way to teach a first course in proofs to undergraduates. sometimes i even wonder if it's something that can be "taught" .. in the sense that if the student really wants to learn and to understand, then (s)he has to commit to a minimum amount of time for self-study and development.

it's like teaching someone how to be paranoid: as a skill, it only develops with time, experience, and stimuli ..