to be more creative, try becoming a more boring person?

first of all, apologies to my readers for the dearth of posts in the last few weeks months and especially the lack of posts with any personal depth in them. i won't go into detail about it today, but this change in my life from a "gun-for-hire" postdoc to a "lifer" prof has bent my mind awry and i'm still learning how to cope with the job. sometimes it just feels .. crippling.

more precisely, it's not the actual job that's hard, but the stress and overthinking of this faraway goal called tenure. the more i think about it, the more it feels like i'm getting my ph.d. again.

all of that said, i'm going to go the lazy route again: i'll pass someone else's well-written piece to you (instead of writing my own).

---

when i was younger and a newer hand at research, i'd make a startling insight in my work. immediately aftewards, i'd lament why it took me so long to figure it out .. especially when the outcome appeared very simple.

i've become less critical of myself over the years, but the question still remains:

what if there were ways to become better at solving problems?
is it more than just a pipe dream to improve oneself?

below are excerpts from an essay i found, through one social media engine or another. what struck me about it was how i've unconsciously kept some of these habits and gotten more insights, in the last few years.

"A 2004 study published in Naturε examined the role of sleep in the process of generating insight. They found that sleep, regardless of time of day, doubled the number of subjects who came up with the insight solution to a task. (Presented graphically above.) This effect was only evident in those who had struggled with the problem, so it was the unique combination of struggling followed by sleep and not sleep alone that boosted insight.
..
"There’s a good reason for this: mind-wandering fosters creativity. A 2012 study (results pictured below) found that any sort of mind-wandering will do, but the kind elicited during a low-effort task was more effective than even that of doing nothing .. This, too, is congruent with my experience. How much insight has been produced while taking a shower or mowing the lawn? Paul Dιrac, the Nobel Prize winning physicist, would take long hikes in the wood. I’d bet money that this was prime mind-wandering time."

~ from "The Science of Problem Solving" @rs.io

of course, this could all just be a manifestation of confirmation bias (or if you will, the fallacy of the consequent). what i do recall, however, are periods of correlation: i was highly uncreative during those times when i was sleeping very little and had no time to exercise.

lastly, the tl;dr at the end of the essay is suggestively helpful. i'd recommend newcomers to research at least to try a few of the habits listed, if only to see what works (or not) for you.

sometimes negative signs matter ..?

after browsing this article, suddenly the difference between mεasures, sιgned mεasures, and vectοr mεasures come to mind.

The answer to this question takes us to the heart of quantum mechanιcs, to the part that popular explanations usually mangle. Quantum mechanιcs wasn't the first theory to introduce randomness and prοbabilities into physics. Ironically, the real novelty of quantum mechanιcs was that it replaced prοbabilities — which are defined as nonnegative real numbers — by less intuitive quantities called amplitudes, which can be positive, negative, or even complex. To find the prοbability of some event happening (say, an atom decaying, or a photon hitting a screen), quantum mechanιcs says that you need to add the amplitudes for all the possible ways that it could happen, and then take the squared absolute value of the result. If an event has positive and negative amplitudes, they can cancel each other out, so the event never happens at all.

The key point is that the behavior of amplitudes seems to force prοbabilities to play a different role in quantum mechanιcs than they do in other physical theories. As long as a theory only involves prοbabilities, we can imagine that the prοbabilities merely reflect our ignorance, and that a “God’s-eye view” of the precise coοrdinates of every subatomιc particle would restore determinism. But quantum mechanιcs’ amplitudes only turn into prοbabilities on being measured — and the specific way the transformation happens depends on which measurement an observer chooses to perform. That is, nature “cooks prοbabilities to order” for us in response to the measurement choice. That being so, how can we regard the prοbabilities as reflecting ignorance of a preexisting truth?

~ via "Quantum Randomness" @ AmericanScientist

ARR, MOAR!.. on writing.

the following excerpt by s.a.pιnker isn't terribly representative of his original article about writing, but i liked it anyway and thought to share it:

For example, everyone knows that scientists overuse the passive voice. It's one of the signatures of academese: "the experiment was performed" instead of "I performed the experiment." But if you follow the guideline, "Change every passive sentence into an active sentence," you don't improve the prose, because there's no way the passive construction could have survived in the English language for millennia if it hadn't served some purpose.

The problem with any given construction, like the passive voice, isn't that people use it, but that they use it too much or in the wrong circumstances. Active and passive sentences express the same underlying content (who did what to whom) while varying the topic, focus, and linear order of the participants, all of which have cognitive ramifications. The passive is a better construction than the active when the affected entity (the thing that has moved or changed) is the topic of the preceding discourse, and should therefore come early in the sentence to connect with what came before; when the affected entity is shorter or grammatically simpler than the agent of the action, so expressing it early relieves the reader's memory load; and when the agent is irrelevant to the story, and is best omitted altogether (which the passive, but not the active, allows you to do). To give good advice on how to write, you have to understand what the passive can accomplish, and therefore you should not blue-pencil every passive sentence into an active one (as one of my copyeditors once did).

(for more articles of this kind, visit edge.org.)

are the dollar signs worth it?

so i spent most of the day LaTeX'ing, which went well enough.

the best thing about writing up ready results in LaTeX is that one feels smart, that maybe all those days of banging one's head against these damned technical lemmas [1] were worth it after all.

..
.. on the other hand, there's something else I should write up, but i can't convince myself to do so:

whenever i think about the proof, I don't feel smart. instead, it just seems .. trivial.

[1] technically, the plural of lemma is "lemmata."

my younger self used to be so smart ..

yesterday i printed out some research notes i wrote in february ..

.. all 24 pages of them, just to prove one lemma. [1]

ye gods; the ideas still make intuitive sense, but the proofs are painfully technical .. much more so than i recall. [2]

they're not even a complete set of notes! for the lemma to have a complete proof, i'll have to read another set of 5 pages of notes for a sub-lemma, as well as a short 3-page section of an old preprint of mine ..

no wonder why i couldn't convince myself to work hard in april. after that and then that trip to spain in march, i had little to no energy left.

anyway, it's time to write up the result(s) ..

.. because what currently exists are the notes, not a preprint. there's a lot of exposition and simplification before i have anything remotely readable!

[1] at this point maybe i should promote it to "theorem" status. (before i thought it was .. well, obvious, and the proof would be short.)

[2] one series of estimates involved 4-5 indices. i was comparing intervals at different dyadic scales, as well as the same scale, which explains two indices .. but the intervals aren't nested in one another, which requires two more indices. lastly, these estimates were "fibrewise," so an additional fifth index kept track of which fibre was which.

ARR, MOAR!.. or: "active" = passive.

one of these days i'll start posting regularly again .. and about research too.

until that day comes, however, readers of this blog will have to suffer through the occasional repost about maths education, like this one:

"Lectures Aren't Just Boring, They're Ineffective, Too, Study Finds" by a. bajak @science

The meta-analysis, published online today in the Proceedings of the National Academy of Sciences, concluded that teaching approaches that turned students into active participants rather than passive listeners reduced failure rates and boosted scores on exams by almost one-half a standard deviation.

i don't doubt that active learning causes students to retain information and to improve their understanding of the concepts of the course.

my contention is that the only novelty of this teaching style is that it institutionalises what students should already be doing:

studying.

you see, it's been suggested that today's college students simply don't hit the books as often as previous generations. when i was an undergrad, i was told that 1 credit hour translates to 2 hours of self-study outside of the classroom, in order to keep up with the class.

(by this rule, a full-time student with four 3-credit courses should be studying 24 hours per week; combined with class times, this should be close to the hours for a full-time job .. which is exactly how seriously college should be treated by college students.)

to me, what active learning amounts to is ..

(a) moving the act of self-study from the student's prerogative to inside of the classroom,

(b) eliminating the temptation of distraction (from tv, internet, dorm buddies) by forcing the student to focus on clear goals with immediate rewards/punishments.

in other words, active learning caters to the modern, distracted, undisciplined mind. they are bringing studying back into college by rebranding it as a new, structured feature of a course.

to their credit, proponents of this method are being efficient. I have no doubt that this does encourage (read: force) students to study, inside and outside of the classroom ..

.. but it does so, at the cost of maintaining the status-quo of short attention spans and immediate gratification, while further reducing personal responsibility.

An open letter to panicked calculus students, after their final exam.

[to put this in context, i had a more vicious version of this letter in mind .. but after some contemplation, it seemed wiser to be diplomatic.]

---

Dear Student,

This is to confirm receipt of your email which was sent shortly after the final exam of our course. I realise that Finals Week is a stressful time for you, as well as every other student at our University.

Being that our final exam ended very recently, the final exams for our course are not fully graded. In fact, they are not even halfway done. Fair grading, especially that which rewards students for their understanding of the concepts treated in this course, necessarily warrants care, and therefore, adequate time.

That said, barring a rather poor semester-long effort on your part (in which case your grade will be already close to failure) I have no idea yet what your final grade is. Based on the scores preceding this final exam, it looks likely that grades for this course will be put on a curve, and it will not be clear how the distribution will look until the final exams are fully graded.

That said, it is futile for you to ask about your final grade. Either be patient or stop asking because I, your instructor and grader, simply have no answer for you yet. In fact, your (and your fellow students's) sending me these sudden emails is causing a great annoyance to me; it even affects my focus on grading exams, which further delays any outcome you would like to know.

Keep in mind that if you are writing me now, then very likely other students are doing the same. All of you point to extenuating circumstances in your situation, many of which do not pertain to University guidelines for exceptions.

By the very definition, exceptions are rare. Please think twice before you consider your situation a truly rare one.

---
Regarding your inquiry into "extra credit" there is none available. If there were such a possibility, then it would have been properly announced on the course syllabus, as distributed on the first day of classes.

To offer you an additional opportunity now would be unfair to other students and therefore unethical of me; being that it is Finals Week and other students will likely have arranged their schedules to best prepare for their other exams or term papers, to offer everyone a chance for extra credit would be unreasonable.

In particular, I have reiterated the same consistent policy to other students in our course, earlier in the semester, and in light of this, they have chosen to withdraw from this course. So to offer you extra credit would be a disservice to those students who made the most reasonable choice with the options made available to them at the time.

You are asking something unfair and unethical. I will not grant this request.

---
I realise that you may be writing me not on your own rational consideration, but in the heat of panick or perhaps on the suggestion of your parents, your friends, or even your academic advisor. In that sense, it is possible that you initially did not want to send me your message in the first place.

That said, please understand my position in this scenario. The grade breakdown in the syllabus has been clear from the start. Barring exceptions given in the University guidelines, every student is graded in the same way. In particular, that means I grade every student based on his/her written work, and that alone.

So by writing this panicked email to me, you have an answer that you could have deduced on your own. Moreover, the time I spent writing this reply could have been more efficiently spent on actual grading of exams. This means that you have delayed the answer to your question; worse yet, it means that you have delayed a definitive answer to those other students who have the same question; in particular, you have just inconvenienced your fellow classmates.

There has been no benefit to you by writing to me now. This has been a waste of your time and mine.

Sincerely,
Your instructor.

I'd rather students be honest, but if they're going to be efficient about it...

the more i think about it, the less sense it makes:

if you were a mediocre calculus student, then why wouldn't you invest in a complete solutions manual for the textbook of your course?

even if you copied all the answers, the percentage of exam problems similar to textbook problems is rather high .. enough, say, for a C grade or probably a B.

after all, solutions manuals are perfectly legal products to buy. the only setback I can think of is the additional cost.

[scratches head]

sometimes i just don't understand students.

down, for now.

more than eight years ago in a cold, snowy december I sat alone in a cafe, trying to make sense of what had happened.

a little later, a fellow grad student stopped by, looked at me, and remembering, asked me how it went.

"well," i said, "i didn't fail, but .." [1]

i doubt i finished the sentence. we went back and forth in polarised roles, he attempting to be cheerful, me still down.

i wouldn't feel "up" for a while.

it would take a while of forgetting, earning small separate achievements, before i could look back at it without pain and malaise.

-----
that scene from my life came to mind because right now, i have a similar mood .. only less severe.

today was my last day of class .. until August, anyway.

i don't feel like celebrating, though much of me feels so glad ..

i don't feel mournful, but it feels like I lost something .. maybe a chance to have made a great beginning. instead at best i managed a mediocre but trouble-prone one.

all I feel are regrets .. that i could have done better or more, that I put to much effort in pointless things, that i never really understood or appreciated what was really going on, under my own nose and in my own classroom ..

.. that i just had no real sense of "how it all works."

i feel wrong about many things that don't matter, and i don't know how I feel about the things that do.

all this past academic year had taught me is doubt and stress and futility.

-----
on the other hand, those were the same lessons i learned after my first year in graduate school. knowing that i'd be there a while, i worked at it and eventually it all made more sense .. not quite everything, but it got better.

part of me suspects that there is no purpose in life .. in that there is no one canonical purpose. to quote the existentialists, everyone must choose for himself.

i have not determined my purpose yet, but i believe there is one that suits me, in this academic life. it will take time .. but trusting on the pseudo-repetitiveness of life, it gives me a certain illusion of faith.

so i'm down right now. it won't be forever.

[1] it was right after my comprehensive exam, which was an oral, 1 1/2 hour friendly firing squad of questions. i distinctly remember feeling awful that i couldn't prove certain facts on the fly .. and advantageously yet unfairly, the committee never asked for them.

another brick in the wall.

the more I teach liberal arts students, the more convinced I am that my own history is .. slightly out of the ordinary.

simply put.. mathematicians, let alone academic scholars, are a significant minority of the population .. even when one restricts to the college-attending sub-population of the western world only. we aren't ordinary in the sense that we are probabilistically rare. [1]

i guess that's just one way of saying (read: justifying) that i'm weird and i shouldn't be surprised that my students and i don't understand one another.

-----
today i taught two lectures and it felt like i was instructing two confused yet brick-made walls.

it's a very isolating, unnerving experience. maybe i take too much pride in delivering clear, motivating lectures .. where "clear" means clear to me, but apparently not to anyone else.

i'm not being fair, though. it's not equal footing between them and i, because i'm the one setting the agenda and I know what's coming next; to them, though, i'm speaking a foreign language and during class, it's all they can do to copy what's on the blackboard.

isn't it fair, though, that after most of the semester, that they keep up with me?

put otherwise, if i'm going to interact with a brick wall, why should i leave my office to do so? why should I talk about things i know well when i can explore this i don't know but that i want to?

what's the damned point?!?

[1] the more i think about it, the more irrationally obsessive of a student i must have been. i remember studying with friends for exams by quizzing each other, asking each other how to prove this or that statement on the spot. i remember checking out maths books from the library, reading what seemed fascinating at the time. i remember being incredibly upset that, after two years and then graduating, i couldn't solve one particular open problem that my mentor suggested to me. (as far as I know, though, even now nobody's solved it either.)