Webcomics and Teaching

Webcomics

So, I read Irregular Webcomic!, a webcomic run by DMM, pretty regularly. And often, there are totally geeky annotations, including (to point out only a few)...

here, where the joke involves vector calculus;

here, talking about the impossibility of StarWars using Thermodynamics;

here, explaining three Laws of Thermodynamics;

here, explaining how awesome Maxwell's equations are;

here, using the Shroedinger's Cat idea;

here, making Quantum Mechanics jokes;

here, making Fourier Transform jokes;

and now, here, using History and Math!

I hearts this comic so much!

Also awesome in the Math department is xkcd, by Randall Munroe! I've even used this comic in an animation I'm working on.



Teaching

So, being Saturday again, I went to the Maths school to TA; and, again, it aggravated me. But we had a meeting today, and our principal mentioned some of the duties of the Teacher and TA, which I was wanting to beat my teacher over the head about.

Some things mentioned...

Our school's philosophy/signature: Since we have such a high student-to-teacher-or-TA ratio, we have a "human element"; there is more human interaction: student-teacher/TA and student-student.

Roles of the Teacher and Techniques:

• don't lecture
  
  the kids are 13 and younger, they can't handle lecture-style yet!
  

  create lesson plan
  

  have authority
  

  avoid dead silence; talk! Fill the silence with teachery sounds!
  

  don't talk into the board
  

  do not read lesson plan
  

  don't work out of a textbook
  

  never lose face
  

  never let the student "win"
  

  lose once, lose all

Roles of the TA and Techniques:

• don't supersede the teacher
  
  if the teacher's wrong, don't point it out, no matter how frustrating
  

  make sure the students pay attention
  

  encourage participation
  

  supplement the teacher
  

  interact with the students; learn their names

Tips for Creating Lessons:

• don't copy down examples or problems
  
• not just formulas
  
• walk the students through it all

My birthday party

My birthday parties have been nick-named "Awkwardfest", but this year was pretty okay. In fact, we had a focus on awkwardness, and I even donned my "Awkward Fairy" costume (it's skimpy black spandex with a hole shows off what would be cleavage on a more endowed woman, and a black tie worn over the hole). Awkward! Especially when one of the guests flips the tie up!

But it was good. Kevin and Mark came.


Other Stuff

There's a guy at work who makes tobacco smell handsome. Or, maybe, just the way it smells on him is handsome. I don't know.


--Charisa

Arrrrmy

Fitness test went well yesterday. I just want to make a list of reminders for next year:

• bring a lock
  
• bring change of clothes
  
• bring an extra bra
  
• there are showers we can use
  
• wear blue
  
• bring a headband (or stop sweating so much)
  
• don't drink booze the night before
  
• sleep well the night before
  
• limit caffeine intake that morning
  
• sign in at the Armoury before heading to the Base

I passed, barely. First was the run / beep test. I needed a minimum of level four; made it up to four and a half, but that had me so beat. After a cool-down, during which I scared the staff because my pulse hadn't come down (my rest HR was 88 that morning!), but it turned out fine. ... I'm pretty sure I could have done better if my heart rate wasn't so high (ie: if I weren't still asleep but running on adrenaline and caffeine).

Next was a grip test, and I passed that, too. After that was a push-ups test, which was the biggest threat to my passing. I managed to meet the minimum, but only because the coach/examiner let me get away with push-ups that weren't quite up to standard (ie: I need to go lower). Finally, there was the sit-ups test, which I passed quite easily (no surprise).

My Results:

Resting Pulse (bpm)	088
Resting Pressure	Systolic: 112	Dyastolic: 062
20 MSR
	Last Stage Completed	4.5
	Minimum:	4
	Predicted VO2:	34.2
Muscular Strength
Right Hand	36	34
Left Hand	36	34
	Total:	72
	MPFS:	50
Muscular Endurance
Push-Ups	9	MPFS: 9
Sit-Ups	30	MPFS: 15

Statistics Canada

When I got home, I found that Statistics Canada had called and left a message, so I called them back. Turned out, they were having entrance exams again that night, so I went.

Unfortunately, I didn't realize I was still expected to go to work at the Reserves that night! And I had no idea the procedure for calling in to let them know I wouldn't come! So, of course, I got a (short, thankfully) lecture on that today--but I'm getting ahead of myself.

I wrote the StatsCan entrance exam and they called later to inform me I'd passed (wasn't worried for a moment). I have a job interview on Thursday at 13:00.


I got my army gear today!!!

That was all kinds of exciting! I'm worried, though; because there was no sign-in sheet for me... It's not so much that I care about getting paid, but I don't want to muddle up their paperwork/system/etc. ...I'm a little worried...

I got quite a scare when the phone rang at 08:05, and it turned out to be someone from the Armoury. She said she'd gotten the appointment wrong, and that it was actually at 09:00 instead of 10:00. I told her my bus was leaving in three minutes, and I'd arrive at the Armoury at 09:15. She said that wouldn't be sufficient; but she'd call the Clothing Store to double-check. I was so worried that I'd miss the bus; but she called back immediately and my appointment was actually 10:00.

We went out to the Base (the difference between a Base and Armoury is mostly this: A base is an area and many buildings, where an Armoury is one building for a few units) where the Clothing Store was.

I rode in a truck (of some kind) with my "boss", MCpl E--. She spoke very sternly to me, and explained, "When you join the army, you're a Private, and right now, you are the lowest of the low--yeah, you're actually the lowest one in the Armour right now. Everyone who is higher than you--that's everyone--you have to treat with respect. I'm not your friend, I'm not your buddy, I'm your boss. [The recruiter] isn't your friend either. She's friendly, because she recruited you; she's got to reel you in, you know what I'm saying?"

This, I find a little strange--not her personally, but the... "unfriendliness" aspect. I can only make friends with those in my rank--three other Untrained Privates--apparently.

Aside: I'm reminded of my first day, when there was mostly lounging around and no real work for me. I went with the rest of the Unit to the Junior Mess, and someone called one of the girls I was with "a bitch" in a sort of joking manner (why are soldiers so profane?); and I said back, in "a sort of joking manner", "Y'know, if she were a guy, you wouldn't have called her a bitch; you wouldn't even call her a bastard or anything else, because there's no male equivalent of bitch in that sense; and you only call her that because of the availability of the word. Like, I couldn't call you a bitch. 'You are a bitch!' See, it doesn't work!"

Of course, everyone laughed and some people even looked strangely at me, as though wondering where that all came from, or how dare I, or something like; but I'm wondering now if I was actually out of line. It was pretty casual, though... But I'll try to be more mindful of ranks from now on.

Of course I knew this in the back of my mind, but it's much more vivid when getting the talk in that tone of voice like you've done something wrong--which, I had, since I didn't inform anyone that I wouldn't show up to work yesterday. She said that I and the three other Untrained Privates would be getting "shit work" to do--and that, since she's in charge of us four, she'll make sure we get "shit work", because she has an Education degree, civi-side, and she knows what's she's doing, and this is how it works. ... Sobering.

Getting my gear took about an hour and 45 minutes, and it was all kinds of exciting! MCpl. E-- spoke in a completely different tone to the Supply Clerk we met up with at the Clothing Store, and I got a taste of this inter-rank thingummy; though the Clerk was friendly and professional with me (I wonder how much of that was in her job, too).

I filled a large bin with gear (the SC said to try to keep track of what she gave me, because I'd have to sign for everything, and if for some reason, she forgot to give me something, and I'd signed for it, the onus was on me); tried on different sizes of uniforms and equipment (OMG I am so excited about the boots, you have no idea!); and when that was all done, I was given three garbage bags to carry it all.

Naturally, there were some items that would've been stupid to put into the garbage bag--namely, this large bag (hockey bag size) into which we'd already stuffed a lot of gear; the box containing my boots; the sleeping bag; etc.

I hope I--well, not "impressed", but--showed MCpl E-- that I wasn't inefficient or stupid or unfit; and I hope-hope that she saw I'd be a good soldier. While putting my gear into the bags, I was thinking what should go where; how heavy it would be; where the bag it should go to minimize protrusions and rips... I may have goofed it in the end, though; because at the bottom of the bin (which was about four and a half feet tall) were some smaller items that I could not reach with my arm; so I first tried standing/leaning on one leg, and when I saw that wouldn't work, I gave a sort of goofy and embarrassed smile to anyone who might be looking. And then I tilted the bin and got my stuff in the smart way.


I'm sort of that way, now. I smile a lot, I'm...friendly, not an intimidating soldier--ARRR! I smile a lot. When someone is speaking--about anything, almost!--I smile to acknowledge my understanding. I think I should try to nod instead; or blink decidedly; or, I dunno, something. Maybe smiling is unprofessional?


And then, I waited for Mark and Evan to pick me and my gear up. After all the packing, I had: three full garbage bags, a large green carry-bag, a small/medium green carry-bag, my backpack and my jacket. I definitely would not have been allowed to take these on a bus!


And now, I'm tired and my hands are cold and I'm still sore from the fitness test yesterday. I should buy some bananas and find a work-out buddy.

In the meanwhile, I'm going to have Mom's beef soup and play Grim Fandango.
--Charissa!

Checklist and Dream

I want to make a checklist of things never to do as a teacher. The first item on it will be: "End a sentence with 'Right?'" I wonder how obvious is my inspiration.

Never:

• End a sentence with, "Right?"
  
• Ignore advice or suggestions.
  
• Think I know everything.
  
• Think I don't need help doing anything ever.
  
• Touch a whiteboard with my skin.
  
• Lie.
  
• Be half-assed.
  
• Not have a lesson-plan.
  
• Use someone else's lesson plan without going over it beforehand.
  
• Ask if material is either too easy or too hard.
  
• Talk loudly when frustrated.
  

Avoid:

• Talking about teaching when students are nearby and still.
  
• Silencing students--especially when it's not affecting the entire class.
  
• Treating everything as though within my power.
  
• Treating everything as though out of my power.
  
• Talking just outside the classroom door as though it were private.
  

Endeavour to:

• Be available.
  
• Be responsive; quick to respond.
  
• Be helpful.
  
• Be enthused (no problem there!)
  
• Use many methods.
  

I had a dream this morning

I had a dream this morning, about someone from "long-ago". I've blogged about him all over the place, always trying not to use his real name, so I guess I'll do that again. Let's call him Lexus, because he's luxurious in a sort of way.

Backstory:

When highschool started for us, Lexus came to our school for Grade Nine Math although he was in grade eight; and for the rest of my highschool career, it was that way. In grade nine, though, everyone called him smelly and a grade-eight-er and a show-off; and it was mostly true.

From what I've gathered: he played chess competitively until about two years ago; was a lifeguard; was very athletic, being on volleyball and curling teams; played baseball in the summer; had terrible communication skills; liked to be well-read; had an amazing imagination / ability of imaging; and, of course, was fantastic at Math.

Around grade eleven, he joined the school choir, and this is probably when I first noticed him.

I mean, during the annual Jazz festival, I had taken a photo of him sleeping on the bus; and I had already noticed that he played trumpet in a very precise manner and as though there were something very funny--a sarcastic "inside-joke"--to his precision. But I had never paid much attention to him until he joined the choir.

Being one of the few male voices, his seat was near the back, and it happened that he stood almost directly behind me. His voice had the same sarcastic timbre as his trumpet-playing, and Chris once had to point out to Lexus that it isn't by forcing that you sing lower and better tones!

So, I heard his voice more. And I also saw him more (though I'd seen him in Grade Ten Pre-Calculus), and through this, came to be infatuated.

Then, in my graduating year, there was a slight problem in my schedule. Because of the courses I wanted, I was forced to take Pre-Calc before Calculus. Of course, my previous grades and having a father who teaches Math (badly) once a week quickly convinced Admin that this would not be a problem. It so happened that Lexus had the same happen to him.

We took Calculus together, then; and somehow, the three of us with this scheduling conflict sat mostly alone on the same side of the class (it was a small class, about fifteen).

Obviously, he began to mean something to me; I noticed and fell in love with details of him--the way his hair made his eyes sharper; the curve of his nose; the way his facial hair started simply as fuzz on his chin.

I also started a graphic journal, because I wanted to capture details. Fridays were our treat-days, where we all took turns bringing treats. One day, we had chocolate pudding--or maybe Lexus just brought his own, I forget--and he spilled some on his bright yellow shirt. I heard a slight commotion, and turned to see; he and lifted his shirt to his mouth to clean it off; and, lest I be obvious, I had to turn back, when I saw his abs.

I eventually wrote a journal entry, which turned out to be a poem:

Math and Symbols

I'll be sitting in class, and you'll
be behind me, where I can't
mentally undress you.

Not
that it stops me from trying, but
it would be so much more satisfying,
I think,
if I could stare at your body. Or
at your face while you
puzzle over a new problem.

I love that expression on your face:

It's not quite "relaxed",
it's not quite "intense",
but it's almost
pouty,
as though all the
muscles in your face just went dead
and limp like
your entire being
is focused on the problem
and you can't spare the energy to look
awake.

You make me want to shout and sing and
write bad poetry,
but I don't feel words can actually
articulate all that I want
to express.

I want
to write it out in large and esoteric
math and symbols:

"YOU ARE THE LIGHT OF MY MIND, YOU INSPIRE
AND CHALLENGE ME, YOU KEEP ME BURNING, AND
ALSO, YOU ARE GOOD-LOOKING."

But I wonder
what you would think of that,
and, if I gave you that
paper with my heart translated
into math and symbols,
would you find me odd,
and then not
be the same person you
were
in my head?

Eventually, I realized someone was going to notice the way I couldn't even look his direction with a straight face.

I wrote him a letter to a similar effect as the journal entry, mostly pointing out that I found him attractive (mentioning a few to show I was serious); that I just had to tell him because I didn't want rumours; that if he had a significant other, I didn't mean to interfere; that if I was being too bold, he should let me know, because, "since when are you afraid of girls?"

This was pretty disastrous, but not at first.

It was entirely my fault, because he didn't send any sort of reply, and I ruined the whole thing a few days later by first giving him a "secret" note telling him I'd phone tonight, and then phoning.

When I'd been put through to him, he said slowly that he was going to tell me that tonight wouldn't be a good night to call. It was then that I learned he had a girlfriend; and my lack of response betrayed my expectations. I think I managed to say, "Oh," and, after a pause, "Have fun with that". I forget what happened immediately after, but we eventually hung up and nothing was much changed, save that I couldn't look his direction for embarrassment instead of infatuation.

I'm not sure what happened after I graduated. In University, I heard short second-hand stories about how he had a girlfriend, and such-and-such happened to them, so now such-and-such took place. I visited the highschool once--a class he took, too. He made a point of announcing something about his girlfriend--but, at this time, I was dating James anyway.

I've heard from someone he used to play chess against, that perhaps he's now in the University of Waterloo, and wouldn't doubt it, with his brain.

The Dream

The dream was confusing, and I remember only to parts. The first part, there was Lexus, James, some others, and me; and we were at some sort of social gathering; and, for some reason, I had to keep it unknown to James that I had a thing for Lexus; and Lexus was making that very difficult, because he flaunted all the things I'd adored. Somehow, the topic turned to hair and sideburns--not quite sideburns, but the hair that grows over top your ears--and I'd called Lexus a hypocrite because he had that, too; and I'd grabbed him by that hair and I think I kissed him.

The other part was after the party, when Lexus and I were alone for some reason. And I remember thinking to myself, "You're not real, and if you kiss me, it won't feel real--I won't be able to feel your lips, your heat, your movement--and I must really stop imagining you." And hen he kissed me, and it felt entirely real--texture, heat, motion and all. And then he asked me if I wanted him to stop being real, and I don't remember what happened next.

Eventually I awoke, very confused because my mind, body, heart and spirit were all pulling in opposing directions so that even inaction fell into one of them.

I'm not sure what to make of the dream. This is the second dream I've had where someone I either love or am in love with has asked me whether it would be okay to be romantic. If anything, this should be a symbol or warning of my own indecision; but it doesn't help that I know not where to go.


I'm not going to let it worry me too much yet. More urgent is:

Tuesday 09:00 - Fitness test

Wednesday 10:00 - get Army gear!

Friday 18:00 - my "birthday party"!

Adieu!
--Charissa

Gnomes!

There exists love!

I'm not sure why, but I felt inspired to do this again. "There exists love" was my gnome last year, and it suddenly felt appropriate all over again! As I waited in a bus shack tonight, I wrote that in the dust on the panes. And then I wrote "there exists truth", "there exists beauty", "there does not exist silence in bus".


Also--Chinese people are hilarious--there's a Chinese saying that goes something like this:

"I can't poo, the ground is too hard."

Translation: People can think up the strangest excuses to avoid work. Also: Chinese people are hilarious!

And if you are Chinese, you should try this at least once with your White friends:

Hover around an intensely shallow conversation, and when the chatter allows, step in wisely and speak thus, slowly and with a heavy Chinese accent, and maybe punctuating with deliberate gesticulation:

"Ancient Chinese wisdom: Man with one chopstick goes hungry."

And then leave.

Ironic Phrases I've collected over the years

- I hear he's a real gossip!

- None of us like to generalize.

- I hate intolerance!

- You're not very suggestible, are you?

- You're making assumptions!

- Everyone's a pessimist!

- It's good to have standards!

- I bet you're prejudiced!

- You need to be more open-minded!

Reading/sleeping time!
--Charissa

Numerals and Teachery Things

I took out the book The Abacus: a pocket computer from the library (because my brother had gotten me an abacus for my 18th birthday, or Christmas, I forget). In the chapter "The Ancients", it discusses ancient number systems, most of which I'd already heard about in "Mathematical Sorcery".


There was an explanation of Babylonian numerals and Egyptian numerals, as well as others; but what I didn't know was the meaning behind the Egyptian numerals, and this book explained some.

After some quick Googling, I've found three slightly different interpretations of the symbols:

1. http://members.tripod.com/kangwei1a14/egyptian.htm
2. http://www.recoveredscience.com/const102egynumerals1.htm
3. this book.

Picture	Explanation 1	Explanation 2	Explanation 3
	A rod.	Upright.	"just a vertical stroke, not much different from [our 'one']."
	Cattle hobble.	Vault.	Heel bone.
	Coiled Rope.	Rope coil.	Scroll.
	Lotus flower.	Lotus.	Lotus flower, "a plant the Egyptians loved."
	Finger.	Finger.	(None)
	Tadpole.	Tadpole.	(None)
	God with raised arms.	Heh-god.	"a man holding up his arms in amazement. To the Egyptians, one million was a tremendous sum."

Awkward sidenote

My good friend Mark told me about this assignment he had (a long time ago), to convert Babylonian numerals into Greek or Roman or something. When converted, the numbers spelled out the words "clitoris", "penis" and "airplane", which confused the heck out of him!

Aren't numerals fun?



I watched the movie Proof last night.

It sucked. It was almost good, and maybe if it hadn't been so "Hollywood", it would have been better; but it truly, truly sucked. Also, movies in which getting it on is part of the actual plot tend to suck.

Plot summary:

• Girl thinks she's crazy, because her dad was a crazy mathematician, now deceased.
  
• Boy is going through the deceased dad's work.
  
• Boy and girl have sex.
  
• Girl has a proof she wrote, but nobody believes she wrote it, not even the boy.
  
• Drama!
  
• Turns out, she wrote it; and maybe she's not crazy after all!
  
• Reconciliation with boy!

Did I mention the movie sucked?



Teaching

As usual, Saturday was another day of TA-ing for the Maths school. We've started on Trigonometric Identities, and I don't think the teacher adequately taught what an identity is; nor what a proof is. He later asked if this was too easy or too hard, and nobody responded.

Sometimes, I want to just jump in front of the class and tell him to shut up, and start teaching. But, the principal must have hired him as a teacher (instead of me) for a reason... At least, I hope there was a reason other than "he's older, so he'll be the teacher."

I think we should walk the kids through what an identity is; walk them through a few proofs. I'm marking their papers, and I can tell that a few just have no idea what's going on. Someone wrote "1 / 0.5" as a final answer, which is wrong on so many levels!


The format I'd like my students' work to follow

Question number. Question

Work Work Work

/ Side computations / Side computations

Answer

That would be just amazing and so much easier to mark. Maybe I'll have to make photocopies from now on, just so it's easier to explain this to them. Yeah, I'll do that--I'm going to do that RIGHT NOW!

--Charissa!

Cars vs. Goats

There was an episode of NUMB3RS (113 - Man Hunt) that included a brief explanation of the Monty Hall Problem.

Monty Hall's "Let's Make A Deal" is gameshow where you choose a door to win either a goat or a car--the objective being to win the car.

Wikipedia has a pretty thorough article on the Monty Hall Problem, but I want to make my own little problem and proof using PMI, because I wuvvy wuv wuvs PMI so much, yes I do!


Cars vs. Goats (AKA: Monty Hall Problem)

Suppose there is a gameshow in which the player chooses a door. Behind any door is either a goat, or a car; and the objective is to pick the door with a car. After a door is chosen (but not opened), the host opens a door, behind which is a goat. Theplayer must then make a choice: stay with the chosen door, or switch to another door.

More specifically, let's deal with the case where there are c doors and only 1 car. That means, there are c - 1 goats, and the player can choose to switch to one of c - 2 remaining doors, with the objective of finding the car.

Scenario One: c = 3
In this case, there are three doors, and suppose the car is behind door three. There are three initial choices the player can make: door one, door two or door three. Then, the player can choose to either switch or stay.

Here are the results:

initial:	1	2	3
switch:	Car!	Car!	Goat
stay:	Goat	Goat	Car!

If we look at the switch row, we see that 2/3 result in cars.
If we look at the stay row, we see that 1/3 results in a car.

Thus, it is better to switch when there are three doors and one car.

Scenario One: c = 4
Let's suppose the car is behind door number four. The only real difference now is that there are two possible switch choices, but that's easy to take into consideration.

Results:

initial:	1	2	3	4
switch1:	Car!	Car!	Car!	Goat
switch2:	Goat	Goat	Goat	Goat
stay:	Goat	Goat	Goat	Car!

Looking at both switch rows, we see that 3/8 result in cars.
Looking at the stay row, we see that 1/4 results in a car.
Since 3/8 > 1/4, switching is still a better choice.

The remaining cases
Let's skip over the details of the next few cases. Let S be our probability of success in finding the car when choosing to switch, and c be the number of doors; still having only one car.

Summary:

c	3	4	5	6	7
S	2/3	3/8	4/15	5/24	6/35

Noticeable patterns:

• S_c = (c - 1) / [c (c - 2)]
  To simplify things let's separate the Numerator and the Denominator,
  
  let N_c = c - 1
  
  let D_c = c (c - 2)
• N_(c+1) = N_c + 1
  
• D_(c+1) = D_c + 2c + 1

Proof of pattern of S:

1. Prove that N_(c+1) = (c + 1) - 1

First, look at the best case of N_c = c - 1, ie: when c = 3

N₍₃₎ = (3) - 1

N₍₃₎ = 2

This is true!

Now prove all following cases are true; ie: prove N_(c+1) = (c+1) - 1

N(c+1) = Nc - 1
N(c+1) = (c - 1) + 1	Substitute.
N(c+1) = c	Simplify. Hoorays!

2. Prove that D_(c+1) = (c+1)(c+1 - 2)

First, look at the best case of D_c = c (c - 2), ie: when c = 3

D₍₃₎ = 3 (3 - 2)

D₍₃₎ = 3 (1)

D₍₃₎ = 3

This is true!

Now prove all following cases are true; ie: D_(c+1) = (c+1) (c+1 - 2)

D(c+1) = Dc + 2c - 1
D(c+1) = [c (c - 2)] + 2c -1	Substitute.
D(c+1) = c2 -2c + 2c - 1	Expand.
D(c+1) = c2 - 1	Simplify.
D(c+1) = (c + 1) (c - 1)	Difference of Squares. Hoorays!

Therefore,

S_c = (c - 1) / [c (c - 2)]

This doesn't really mean anything relevant yet, though. We've just proven that we always know our success rate, and not that switching is better than staying. Let's look at the staying data now.

Let T be the success rate of staying, and we'll compare with S.

c	3	4	5	6	7
T	1/3	1/4	1/5	1/6	1/7
S	2/3	3/8	4/15	5/24	6/35

Pretty dismal, eh? Let's prove that S > T. But first, we must prove that T always follows a pattern...

Patterns noticed in T:

T_c = 1 / c

T_(c+1) = (T_c) * c / (c + 1)

Prove that T_(c+1) = 1 / (c + 1)

Best case: c = 3

T₃ = 1 / 3

This is true!

All following:

T(c+1) = (Tc) * c / (c + 1)
T(c+1) = [1/c] * c / (c+1)	Substitute.
T(c+1) = c / [c (c + 1)]	Multiply through.
T(c+1) = 1 / (c + 1)	Simplify. Hoorays!

Prove that it is always better to switch:

Here's the final piece! We must prove that S_c > T_cProof by Contradiction.
Go!

Sc	<	Tc	Assume.
(c - 1) / [c (c - 2)]	<	1 / c	Substitute.
c (c - 1) / [c (c - 2)]	<	c / c	Multiply by c.
(c - 1) / (c - 2)	<	1	Since c > 2, this is most definitely false! In fact, (c-1)/(c-2) cannot even be equal to 1.  It is always greater than 1.   Since all our steps after the initial assumption of Sc <>c, that assumption must have been incorrect. Therefore, Sc > Tc  for all integers c > 2. Hoorays!

Thus, it is always better to switch.


I'll write something more journal-like later.
--Charissa

Dreams

Around spring this year (2007), I noticed I wasn't feeling the same. I felt normal, but, somehow, less something, and consistently so. So I went to the doctor, and was prescribed thyroid hormone.

One of the strangest "side effects" happened to me: lucid dreaming.

I'd wake up in the morning, not knowing what to think anymore. When my dream-experiences contradict my waking paradigm--which they often do--then I'm left totally confused, because my dream-experiences feel so valid!

Most notably, I've had ponderous remarkable dreams about James.


Backstory

James and I met in Honours Calculus last September, and we got along quite easily. We started dating in December, and it was totally awesome geek-love, complete with writing geeky love notes to eachother. This was actually during my coming-out as "curly", for the first time, to close friends. James didn't mind, though, because he had a close friend, Kay, who identified as a lesbian.

Spring rolled around, and the school year was ending. I had noticed that James was beginning to withdraw, and perhaps actively hold something back from me; but his silences had always confused me, and I wondered if it was just nothing. On the last day of school before exams, we talked, and he told me he had developed feelings for another girl. We decided to take a break; especially with exams coming up and the school year ending; and we'd talk more about this after some time apart.

About a week later, we decided we'd be friends from now on.

We kept in touch a little, mostly through e-mail, though we met on occasion over the summer. He and Kay started going out. It was a little amusing for me to wonder--because she had thought I was attractive, and James and I were dating, and I'm bisexual--about all sorts of possibilities.

And over the summer, James was planning on moving into the city with Kay.


The First Dream

The first dream I had was terribly confusing; mostly because I'm not sure if it was me leading the dream, or otherwise.

I'm not sure by what circumstance, but James and I were talking in the dream. I think we were doing something romantic. Suddenly, I stopped, and I asked something about Kay.

James does this thing with his posture sometimes, like closing up; and I've come to interpret it as a sign he's sad, distressed, scared. He did that, in the dream, and I asked him what was going on. He explained something about Kay, though I don't remember what, and I got the sense he was worried, but also hurt or offended or outraged and something like he wanted to do something about it but couldn't, or didn't know what.

A few months later, I visited James at his new address in the city. After some general chatting, I asked where Kay was; and he explained that her parents were not permitting her to either move out of her house, or into this one. And he did the same thing he did in my dream--the closing up--and I got the same readings from him.


The Second Dream

After hearing about Kay, I was even more rattled about the dream, because it had started off with a very close and probably romantic feeling between me and James. I don't know what to make of this, but the second dream was almost entirely romantic.

In this dream, James and I were swimming, not just in a beach, but something that seemed farther, larger and deeper. It reminded me of Venice: salty ocean water.

I don't even remember all the details, and there are some I don't want to mention; but we kissed, and there was a time he asked if this was okay--because of this, that and the other thing--and I said yes, so we kissed again.

Well, I guess it wasn't okay, because I woke myself up immediately; and I was frightened and bewildered.


Charissa probably thinks too much

I don't know what to make of these dreams.

Probably the most likely explanation is that, if these dreams mean something true about James, they mean that James still has affections for me (as I do for him), but we simply cannot do anything about them because of the choices we've made and the lives and responsibilities we have now.

Or, I'm simply projecting what I want, and just happened to be right about Kay being in some sort of trouble that affected James in a very predictable way.

I wonder how much of the first dream was meaningful, though, and if the relevance of one dream is an indication of the relevance of all other dreams--ie, if one dream came true, maybe all my dreams can come true.


With my birthday party coming up, I'm wondering whether or not to invite James. I mean, I want to, because I still want to be friends with him, but he still affects me too deeply, maybe--too deeply to know if it's real.

I'm going to try meditating tonight.
--Charissa

Dressup!

I was at work, wondering what I'd like to dress up as for Halloween. Even though I'm too old for trick-or-treat-ing, I still like to dress up for fun.

Last year, I went to University as a drunken wench--bottle and everything--which was neat, because we got our assignments back that day, and I got 100%.

I couldn't decide on a costume.

So I pondered...	...and pondered.

Aha!

I decided I'd dress up as my dad!

Actually, that's not entirely accurate. I decided I'd dress up as one of those old-days Chinese martial artists -type guys; but when I grew out my goatee, I looked disturbingly like my father.

How appropriate for Halloween!

--Charissa!

Charissa thinks too much

Kevin came over tonight.

Kevin went to school with my good friend Mark, and I first met Kevin at Mark's sixteenth birthday party. We didn't see eachother again until this year; so a lot has changed since then. Mark and Kevin came over to watch a movie tonight.

Right now, Kevin's majoring in English, and has been interested in film for a while--so he's a Theatre Kid! He's also very much a Virgo, and a fun person to be around.

Do I like him?

Yeah, definitely; I mean, he's a great guy and all.

Do I like him?

That's a bit more complicated. I like spending time with him, and I love that he knows how to move/mime/act; he's graceful and articulate that way. And, sure, I think he's physically attractive. The truth is that, yes, I like him; but for one trait. He's such a Virgo. I mean that he's not very passionate; he's a perfectionist for the sake of being a perfectionist, not because he is compelled.

On a side note, I guess this also says something about me. I like to be able to affect people--especially romantic partners. I'm not sure if this means I have some kind of power complex where I need to be able to control people; but in any case, I like to know I cause change or reactions. (This actually caused me some problems in my last romantic relationship, because my sweetheart wasn't a very expressive person, and so I had difficulty interpreting the lack of responses.)

Anyway.

Coupled with the fact that I have never been prepared for romantic relationships, is the knowledge that Kevin is so completely not appropriate for me.

I wish I had the ability to let myself simply enjoy whatever -ship we have or will have; but I'm so worried I'll do something stupid--again. I wish I could run my fingers through his incredibly wiry hair and tell him how handsome he is. I wish I could tell him how much I admire his ability to express with motion. I wish I could kiss him. And then, I wonder why I want all these things.

The words "boyfriend" or "girlfriend" never meant anything real to me; mostly because every time I was introduced to someone's -friend, they spent the rest of the time doing private things together in our company, so maybe I've associated those words with those private actions--kissing, playing with hair, affectionate petting. So, somewhere in my mind is this picture of what a romance is supposed to look like. I think the whole problem is just about labeling; but I can't simply not-label anything. That's the opposite of learning.


My best friend 
 
This is how Eric and I became best friends.

We met eachtoher in the winter of 2005 when our high school's choir needed more males. Later, our Music groups took a trip to Chicago on a bus. Eric and I were bus-buddies, which basically meant that we sat together on the bus for the 17 hours there, the 17 hours back, the umpteen hours traveling from one destination to another, and spent umpteen hours together at the actual destinations. The only times we weren't together was when we slept in different rooms (boys with boys, girls with girls; though this arrangement presumes a bit) or when one band went here while another went there.

So, on that trip, we became pretty comfortable with eachother.

However, it was the Summer of Eric that made the difference.

Over the spring and summer, we spent time together; mostly biking. When the summer of 2006 rolled around, and I'd graduated, we still managed to find time. We spent many summer nights just sitting, standing or laying on his front lawn and talking; sometimes as last as midnight (which was late by his standards).

During this time, we talked about anything: God, ourselves, family, relationships, people, bands, music, our futures... I think Eric was the first person I "came out to" (I remember saying, "What would change if I started calling myself 'bisexual'?"). Or maybe Mark was first, I forget.

In any case, we developed an ease of understanding eachother; of communicating and reading.

I used to make this gesture whenever I found something unbelievable: I'd put my hands just behind my head, with my fingers touching my thumbs, and then bring them forward suddenly and open my palms. The first time I texted "?!" to Eric, he replied with a lol, saying that he could see exactly what that expression looked like.

One night, Eric and I were talking, and one of us wanted to ask a question, and the other said, "Yeah," right away, and we both understood exactly what the other wanted to convey. And then Eric sort-of paused and said suddenly, "I think this is what marriage must be like."

We talked once about "going out", and we both have our own reasons--whether or not they coincide is of little consequence--but my strongest reason is that, to have any other relationship with him, now, after our best-friend-ship, would be less. It just--it wouldn't.

Eric currently has a girlfriend, and I was extremely happy when I found out (he'd never had a girlfriend before, and felt he was missing out; maybe because I had a sweetheart at the time and prattled on and on, and on, and on). Eric explained to me that, right off the bat, they had talked about what their relationship would be like; and they decided that it didn't matter what "girlfriend" or "boyfriend" meant; they were going to do as they liked, regardless of others' expectations.



This is what I would love to have with Kevin--no expectation. If we're friends, we're friends; if we laugh, we laugh; and if we kiss, we kiss.

What disturbed me about my affections for him was that he is so completely not what I'd prefer in a lover. (Ideally, I'd want a brooding math genius; but that's for an entirely different blog entry.)

So, I think I need to learn that it's okay to be however I am; it's okay for him to be however he is. ... I hope I learn that quickly and well!



Other things I've thought about tonight:

- This was the first time Kevin was in the house. I showed him around, and he commented, twice, "This is a big house." So after he left, I walked around and stood where he had stood and tried to see as he might have seen. (Both he and Mark come from much smaller houses than this.)

- I was talking with Mum after they'd left; and suddenly said, "Man, I don't think I could have kids. I mean, I don't understand the kids one year younger than me; and we--you're thirty years older than me, and we can barely communicate! How could I ever hope to communicate with people thirty years younger than me if I can't understand the kids one year younger?"


And that's all for tonight. As the title of this entry so plainly states, Charissa thinks too much.

--Charissa

Boobfest and Private

"What are you doing?"

"Curve-sketching."

"Are those breasts?"

"Your point?"
I've been working on the "Boobfest" series off-and-on; and I'm not yet accustomed to uploading my artwork to this blag yet. So I'll just take a moment here to mention that I've finished Boobfest III.

It's pictured to your right, and if you have a dA account, you may also view it here. It's oil pastel on yellow construction paper.

I'm pretty proud of how it turned out!


Why "Boobfest"?

Well, one night, I made this sketch (pictured to your left; and again, if you have a dA, you can also view it here).

I think I did it mostly for the sake of drawing the naked female form, and then coloured it mostly for practice with oil pastels--the point being that this was an entirely gratuitous and un-planned picture. It's also of terribly low calibre.

A while later, I decided I wanted to do another study of the nude female figure (which isn't unusual for me) but I wanted something of better quality.

One of my artist friends had recently shown me a piece she did in oil pastels on construction paper, and I was eager to try that out for myself. I put more planning into this one (pose, composition, colour) and this was the result, pictured on your right.

Now, I had no idea what to call this, but I was reminded of the "boobfest" sketch from earlier, and could not be inspired to think of a new name, so I called it "Boobfest II". Again, if you have a dA, you may also view it here.

In short: laziness is the reason I'm calling this the "Boobfest Series". I've actually got number IV sketched, but it isn't a nude piece; so I may re-name the series when I've finished that one. I guess I also like the way it's silly.

Whee!



Last night, I got enrolled into the Canadian Forces as a Signal Operator.

I am now a Private (Recruit)! Arr! I mean, YES SIR!

I almost forgot that part of joining the Militia means taking orders and that whole "superiors" thing. I'm not sure how I managed to almost forget that. I wonder, though, if I'm more enthusiastic than I should be? If I'm forgetting that I'll be an inferior, then maybe I've got the wrong ideas. Maybe.


Why I'm in the Military

• I need more structure--especially since I'm not attending school regularly right now.
  
  
• I need to be learning things--especially since I'm not in school right now.
  
  
• I want and need more discipline.
  
  In this specific case, I'd consider "discipline" to be "being able to do well the things I don't necessarily want to do well".
  
  

  I need to find out who I am.
  

  I have a basic idea; but, I feel that if I'm not challenged, I won't know more fully.
  
  

  I want more personal strength.
  

  My best friend has told me that I'm "hardy", which means: if he were to stick me in a large pit of destruction, I would survive; last a longer time than most people; something like that. And, I feel that one of my "callings" is to be a hardy person; as hardy as I can be.
  
  

  I need to get out of this house.
  

  I live with my parents, but it's too sheltered, and I'm not independent of them when it comes to soft things. I still ask my mom what pants I should wear; if my father says something carelessly, I might be greatly upset by it; if I'm going to be home late, I worry that my folks are worrying.

  
  

  I feel that physical distance would be the best cure for this.

  
  

  I want more training (which would be the combination of education and practice).
  

Those are the important ones--to me, at least. Less important would be:

• I could use the work-out.
  
• Communications involves Math (and we all know I wuvs Math!).
  
• I get to play with impressive and nifty machinery!
  
• Army buddies! Also: lesbians!
  
• Doing important things both locally (Canada) and internationally!

I think that's all from me tonight. I'm trying to get into a better habit of blogging; but I also want to get into a better sleeping routine. Right now it's more "go to sleep whenever-the-hell I feel" and "wake up before 13:00", which isn't very disciplined.

So, I'm off for tonight. I've spent just over 24 hours as a Private, and I've gotta say, it doesn't feel much different yet. I guess it's like turning eighteen, and you're suddenly considered an adult; but nothing feels different yet. So I'll grow into a Private the way I'm (hopefully) growing into an adult.

Speaking of which: I've been scheduled to get my army gear on October 24th, which is just five days before my birthday! So it'll be like a great-big birthday present! Hopefully!

--Charissa

"One Million Beans!"

This is an old question I was given by a Psychology student. I just changed it into a Math question, because I'm a Math student!

This is a very simple-to-solve problem, and requires no advanced Math. If you can do one of those "Billy has two apples" -type problems, then you will be able to do this one!

One Million Beans!

There are two jars: Jar A and Jar B.

To begin, Jar A contains P red beans, and Jar B contains P green beans.

From Jar A, Q red beans are removed and put into Jar B. Then, Jar B is shaken, and Q mixed beans are removed from Jar B and put into Jar A. There is no way to control or know how many of which colour beans were moved.

Now, there are n green beans in Jar A, and m red beans in Jar B.

1. Find | n - m | for all Natural P, n, m and Natural Q ≤ P.

2. Prove this is always the case.

This is one of my most favourite questions, mostly because it looks terribly harder than it actually is! The trick is to not over-complicate it, and to not rely on intuition. Rather, keep track of what you know, and you'll find the answer very easily.

I'll post the solution to this in a few weeks; because I'm submitting a more specific version of this problem to a workplace newsletter. Hehe!

--Charissa

When I grow up

"Mum, do you know what I want to be when I grow up? I want to be someone who can talk to anyone. I want to be able to understand what anyone says, and I want for anyone to be able to understand what I say. Because, some part of me realizes that until something I experience is real to someone else, it's incomplete. Even I am 'someone else' every passing month, week, day; and I forget; and when I forget something, maybe it didn't happen. But if someone else knows it, or if I can make a perfect record, something more than simply experience has been accomplished."

At least, that's what I wish I'd said.

Last night, I took a bus home from work. I kinda had this feeling that I should leave work on time tonight--sometimes, I stay logged in for an extra few minutes, especially on Sundays, because I could get paid overtime if I wanted. But, last night, I just felt like I'd benefit more from something if I left work on time.

On the bus home, an elderly Oriental man took a seat in front of me.

Maybe I should explain...


Why Buses are Important to me

I meet a lot of people through the buses.

I'm not quite sure why it happens that way, but I just do. Once I jumped over a bank of snow to get to the bus stop, and a young man laughed, so we started talking. Turns out, he's a friend of the principal of the Maths school where I work.

People like to talk to me for some reason. Mostly, these are older folks, and it's eye-opening to hear what they think of the world. I never knew there were so many pessimists!

Several times, I've talked to folks, and they've told me stories of how there are such injustices in the world, and they're the only ones fighting back! There was a lady whose mother was robbed while they were downtown, so the lady clobbered the thief so badly that the next time he saw them, he crossed the street to avoid them. There was a man who entered a restaurant and saw a young lady leave her bag on the table and a ruffian pick up the bag and make as though to leave with it. The man picked the thief up by the collar and made such a commotion that the thief could never show his face at the restaurant again. There was a woman who saw a lady in a wheelchair was wanting to board the bus. There was a young boy sitting on a fold-away bus seat, and the woman told the boy, rather severely, that he'd have to move.

These struck me as painfully loveless solutions to painfully loveless problems. And it was on the bus that I decided that if I want to change this, I'd have to be the opposite. If someone steals from me, maybe that means I'll catch him and give him money instead of a beating.

...Anyway.


Talking to People

On the bus home, an elderly Oriental man took a seat in front of me. Near the end of the bus ride, he began coughing violently; so I tapped him on the shoulder and passed him a pack of cough drops. He thanked me over his shoulder; and we talked a little.

His English was poor, but he spoke Cantonese. Actually, his English was better than my Cantonese (which actually might not be saying much). I never asked his name--mostly because he never asked mine, and it really isn't important--but I'll call him Jim for the purposes of this blog entry.

Jim's from Vietnam. He told me he has a younger sister living in the States, and other siblings back in Vietnam. He works for a company that makes chairs; and on Sundays, he goes to a casino by himself. He has two sons; and he's a grandfather at 62 years.

And I wouldn't say I have amazing communication skills, but I'm even a little impressed we managed to exchange that much information. At one point, we switched to Cantonese, and I had to explain in my undoubtedly Canadian-accented Cantonese, "Sik tang siu-siu, sik gong siu dee," which means I can listen/understand a little, but I can speak less. I live at home with my parents and am the youngest of my siblings. My "dai-goa" lives in Germany, and my "yee-goa" lives on a different street.

It felt unreal to talk to Jim. We were talking across a fairly steep language barrier, two countries, two generations and 43 years. I could see bits of his world, but not understand their depth; and I'm sure it was the same for him.

So, when I got home that night, I told my mum how awesome it was. I'm not sure she quite understood it all, but that's okay, because I can keep explaining things to her and she'll keep trying to understand; and she'll try to explain things to me, and I'll think I understand them (though I probably don't), and it's okay, because she's still my mum and I'm still her daughter. I find that so incredibly comforting.

And when I grow up, I want to be someone who can communicate with anyone, no matter the kind of separation between us.

--Charissa