"Similarly Complicated", and other mathematical niceties, Part 2

But the first thing that made me want to write today happened much earlier, in one of my favorite classes – Algebra. I could talk about this subject all day. But I'm going to talk about one kind of proof in particular, of which we just happened to have an example today, and also about one of the funny comments made by my slightly crazy professor.

Math is all about proofs once you get done with the stuff they make you do in high school. Proofs are about, well, proving that something is true or not true, proving that a certain object has certain properties, proving that certain sets of objects all have something in common, or proving that various definitions or statements when combined lead to a much stronger statement. One fun type of theorem (that leads to a fun type of proof) has the following syntax:

Theorem (Name): Let X be a thing. Then the following statements are equivalent:

(i) X has property A.

(ii) X has property B.

(iii) X has property C.

(iv) X has property D.

“The statements are equivalent”– What does this mean? This means if you successfully prove something like this and you come across a beast (in the mathematical wilderness) with the quality of B, then you automatically know that it also certainly has the qualities A, C, and D. That’s the whole damn point. If I know that when bears (X) have blue fur (A), they also automatically are vegan (B), then if I see a blue bear I don’t have to ask it if it’s vegan or not. I already know, since A implies B.)

And how do you prove such a thing as this? Well, you start with whichever of the options A-D you are most comfortable with, say property A. Assume X has property A. Then you have to prove that if X has property A, it must also have property B. From there, show that if X has property B, then it must also have property D, and so on. At the end, you need to lead back to the property you started with — complete the circle. It doesn’t matter what order you go in as long as you can get from any one statement (A-D) to any other statement by the time you’ve finished.

If this is confusing, don’t worry. The whole point of this type of proof is this: Let’s say you come across a bear in the wild (yes, we’re back to the bear). You really want to know what blood type it has, but you are (understandably) hesitant about taking a blood sample. But because Blood Type = O Negative happened to be property D in the theorem you just proved, you know that since the bear has blue fur, it must have blood type O Negative. You saved yourself a lot of work (and maybe your life).

This style of theorem and proof has always fascinated me. Even though the real bulk of the proof might just be to find an easier way to show property D (like the example above), you still get to explore the other properties along the way, and maybe derive some neat consequences.

(If you are interested in these mathy things, please write to me - I can give examples that don’t involve bears, or more examples involving bears - or just talk about all these things. :) )

So, this is the type of proof we were tackling in Algebra at 8:00 this morning. Our professor rolled up his sleeves and wrote the whole theorem statement on the board, and then wrote those great letters: “Proof:” and we waited for what was to come next. “A implies B” he wrote. He turned to us. “This is a great proof,” he said. “Ready?” Collective, halfhearted nods from the crowd. Then he wrote “Obvious.”  on the board. And that was it.

After another one of the implications (say, C implies D) in the proof was also just “obvious” (or, another favorite word of mathematicians, “trivial”– oh, how this word sounds to math students. Can you see the self esteem shriveling into nothing?), we had to have a quick discussion about it. Here’s what he told us.

First of all, these statements “are only ‘obvious’ if you know which definitions or other theorems to apply and in which order”. But once you do, the answer is clear. Second of all, it’s not “supposed to be obvious to you today when you hear it in the lecture”. He continued: “The difficulties are hidden in the notation.” Quite clever and quite true, not only of mathematical things. 

 And finally, the real stuff:

“What I do here in the lecture is 10% of the work, of the material.” 

I think this is what a lot of people don’t realize about math courses and that is a reason that many don’t do well in them. Just the notes from the lecture, just passively writing them down isn’t enough. Even just mechanically doing the homework isn’t enough. There is much, much more work — hidden in the details. You have to work with it enough to understand it from the inside, not just from the outside.

And that is exactly why I need to stop writing this right now and go read over my notes from this morning. Oy vey!

"Similarly Complicated", and other mathematical niceties, Part 1

Today's a Monday, which for me means four lectures in a row, almost without break (there is generally a fifteen minute break between classes, but with going to the restroom, maybe going to another building or buying something to eat, this vanishes quickly). Each lecture is two hours long (roughly), so it's quite a long block of note-taking from 8 a.m. to 4 p.m. But today, each of my math professors have just been so...mathematician-y that I felt the need to use one of my fifteen minute breaks to write this.

One such hilarious (to me) moment today already made its appearance in the title of this post. In Analysis II (very abstract calculus, roughly explained) this morning, we worked our way at one point through a very tedious example. Lots of indices — let me say what I mean by indices. Indices (plural of index) are the subscripts we use to know which entry of a particular vector we are talking about. 

For example, the vector x:

The first entry in that vector is 5, the second 4, and the third -1. In math speak, we would write x₁=5, x₂=4, etc. This vector has three entries in it, so we say it has length 3 . Alright? Bear with me now, it’s going to get complicated. First, imagine you have a vector of length n. What’s n? Could be anything. Got that? Kinda? Good. Now imagine that you don’t have just one vector of length n but a whole series (or list) of them. So, now we might want to talk about the entry at index i  of the kth vector in that sequence…You get the idea. Subscripts of subscripts of indices. 

Most people in class were either pulling on bits of their hair as they tried to keep track of all the tiny k's and n's and x's, or they were just staring into space after having given up. (Let me note -these kinds of details are not, in my opinion, what makes math difficult. They are not abstract concepts that you need to wrap your head around. Rather, they are the tiny details (that are indeed very important) that we use to be incredibly precise about what it is we are calculating and though they are not in themselves difficult, these tedious indices and things like them certainly can, and do, make math problems discouraging). Finally, we finished the example in the lecture and the professor smiled at us and said we were about to do a different example now that was "ähnlich kompliziert", which means 'similarly complicated'. Also complicated, and in exactly the same way. Sometimes phrases just catch my eye (or ear) and I think they are worth noting in their perfect suitableness. 

Thoughts from the first week back in school.

Classes started this week. I’m finally officially matriculated and I can tell it’s going to be a full semester. I was going to try and write down here some of the classes that I’m taking - but in translation and explanation it seems to sound like I’m only taking things I’ve had before, but that’s not quite true. Some it’s more advanced versions of some things, review of some things, and since I’m switching academic systems there’s also just a bit here and there that I need to repeat just to satisfy the official requirements. Some courses I am only going to take and turn in the homework for but I don’t need to take the exam, since I already have a grade for it from Mills. And so on and so forth.

But the point is, I’m taking (in varying degrees of depth - i.e. lecture, homework, exams) four math courses and one computer science course. The very first class this week was one that I think will be quite difficult, but also probably one of my favorites. An Algebra course. Now, I’ve had some Algebra at both Mills and in Budapest, but I can tell from the first week that the pace in this course is quite different. We’re going to cover what I already know (or knew when I was in those other courses!) within the first few weeks, and then it will be all new after that. That lecture is at 8 a.m., twice a week. When I sat down in the first lecture on Monday, I had flashbacks of studying here three years ago.

I remember the enormous theater-style lecture halls, the multiple blackboards that slide up by twisting various knobs near the chalk tray at the bottom. I remember the professors actually carrying around huge sponges that they wet in the sink near the blackboards and use them to completely wipe the board clean, then use the enormous squeegee to get rid of the excess water. And I remember the pecking order of the rows in the hall. 

There are at least twenty-five rows of seats, two aisles, in the steeply sloped hall. About 80 students in a lecture. No one sits in the first row. Are you crazy? You’d be like five feet from the lecturer. That makes no sense. Plus, when the blackboards are slid up to the top of their tracks so that folks in the back could see, you’d have to crane your neck to read. The second row? I hate to be so harsh, but I’m only reporting what I see (and this is often the case with American students as well). The second row is full of guys who happen to be paradoxically long-haired and balding at the same time, all while being in their early twenties. Some of them are using fountain pens and get the ink all over their hands, they have the homework done before it’s been assigned, etc. Oh, and there are also sometimes girls who wander into the lecture hall late, looking like they forgot they signed up for a math class, and then when all the other seats are taken, they end up sitting next to these characters. The third row is for people who want to pay attention but aren’t quite up to the level of nerdity necessitated by those second-row seats. After the third row, all bets are off. There, devoted students are mixed with those who are just fulfilling some distribution requirement to those who are taking this course for the third time after having failed it twice.*

As for me - what’s your guess?

On the second day of that particular lecture (Wednesday) I slid into an empty seat in the third row (did you guess right?) and hear some of the Row 2-ers talking in front of me about the homework (which at that point had only been posted online for about twelve hours). They were already done, discussing the subtleties of the last two proofs. I, meanwhile, had printed out the sheet, jotted down a few notes for each problem and had a solid plan for at least two, but hadn’t written any up in their entirety. So, I felt a little sheepish, worrying that I wasn't at the same level as the rest of the class. All of a sudden, from the row behind me, I heard the low voices of two guys, barely awake at this 8 a.m. class:

‘You here those two up there? Are they talking about the homework?”

—“Yeah, I think so. They’re already done.”

“What? Jeez… I haven’t even printed it yet.”

—“I know, me neither. God, they’re motivated.”

“I know. Horrible, right?”

I smiled to myself. I think I’m sitting in the right row.

——

After managing the first four days of the week (where all my classes are scheduled, though some homework is due on Friday) and finishing two of the first four homework assignments, I was feeling good, but exhausted. Thursday evening I went to babysit - not the two girls I’ve talked about before, but a young boy (5) and his younger sister (3.5). Both of them already are learning to speak both Russian and German because of their parents, and their knowledge of English is very limited (understandably!) . The boy has learn the words to some children’s songs, such as Head-Shoulders-Knees-and-Toes, The Itsy Bitsy Spider, etc. Neither one of them understands all the words to these songs, but they sing along anyway - the boy knowing fairly well how to pronounce everything but his sister just throws caution to the winds. When a kids version of ‘Yellow Submarine’ came on, the sister was singing loud enough for me to understand ‘sumbasine’ ‘tubmarine’ and even ‘subtartine’ during the course of the song.

The boy is a real mathematical mind - you can tell already. The way he builds with blocks, they fascinated way he stares at the rubics cube in their playroom, and - the one thing really on his level at the moment - when he plays cards. Right now, the card games we can manage together are Uno and War. I do believe we played about 30 games of Uno on Thursday. It started with him saying just the names of the numbers or the colors when he put his cards down, but then after playing for an hour and hearing me talk about the game, during the last hand when he kept getting a bunch of +4 cards from me (he can take it :) ) he finally exasperatedly held up his tiny hands full of the cards and said “Too many cards!!!”. Considering he didn’t know the word ‘card’ at the start of the game, I thought this was pretty successful. And after the week of classes and bureaucratic errands at the university, Uno was about as complicated a game as I could handle anyway, so we made a great team.

*If I haven’t mentioned it before, failing a class is something completely different here in Germany. Your grade in a course is determined entirely by your grade on the final exam. You are required to score a particular amount of points on the homework to be allowed to take the exam, but other than that, your homework doesn’t count for anything. Also, to make things more interesting, those who write the homework problems are traditionally TAs, the same ones who run the weekly workshops where that homework is due. But the person who writes the exam? The professor. Do you know how the professor likes to write problems before then? No. Also, are people paying exorbitant amounts of money to take these classes? No. So do they kick and scream and fuss (or do parents kick and scream and fuss) when students get grades below an A? No. You don’t need to study math to combine the above facts and come up with the result that failure is not an uncommon outcome in a course. Because of that, though, the attitude around failure is also quite different - you’re not expected to be perfect. Very few people (but some) get grades equivalent to 'A's in math. However, I’m coming into this system with my US-grade mindset, and thus I find the system frankly terrifying. 

A weekend of culture.

On Friday this week, I got to attend a ballet (modern-esque ballet) of one of my favorite children's books - Momo, a German novel by Michael Ende. While sitting here this afternoon, I have tried several times to come up with an adequate summary of Momo and I keep disappointing myself, so I won't really try. I'll just say that it's about a girl named Momo and a mysterious and dangerous group of beings called the Grey Gentlemen, who come into the world and try to convince all the people to "save time" by investing it in the Time Bank, but as most of us know, when you try to 'save time' you don't have a big pile of time waiting for you at the end of the day, or week, or month, or year. Most of the people in the world start to be driven mad by this constant rush, trying to be productive all the time and save as much time as they can - and Momo has to try and stop the Grey Gentlemen. I haven't been to a dance performance in years. After all my dancing in my youth, they always really affect me deeply, and this one was no exception. Plus, I think the experience of begin at a live performance of any kind can be so fascinating, from seeing the transformation of the performers in character (acting, dancing, singing, etc.) to them just being people when they take their bows, and when the house lights come up and you realize you are surrounded by strangers but you still just shared something with them by watching collectively. And dance. We should just dance more. I should dance more. Period.

  Click here if you want to see the preview of the ballet (try to guess which ones are the Grey Gentlemen :) ), and if you have a young child who you really like and who could use a new book for whatever reason, I really recommend Momo. (I have not read the English version, but I think it would be good, too. )

Then, on Saturday, I visited the Frankfurt Buchmesse, or Frankfurt Book Fair, which is the world's largest book trade fair (according to google). After being there during the day, being overwhelmed but enjoying ourselves, Claudia and I watched the news the next day and heard about the attendance for that very day we were there (somewhere in the 10s of thousands)--- no wonder it had felt so crowded!

There was everything at this fair. Some kind of comic book convention so that we saw at least three different Pikachus walking around in the train station on the way there, stands for self-publishers or those who hope to do so, stands with all kinds of subject-specific books from various publishers (for math, astrophysics, cooking, computer science), and rows and rows of stationary (!!!! If you know me at all, you'll know that this was one area I would not miss) and book binding companies, new education technology companies showing off their developments -- oh, and stands with ice cream and pretzels every now and then. :)

But during the day, we saw two rather spectacular things. One was a live cooking demonstration from an Indonesian chef, a charismatic woman who cooked three dishes and a dessert (I know for a fact that two were delicious, didn't get to try the other two)  the space of an hour and had drawn an enormous crowd of people from the enormous exhibition hall (there were eight such halls, with multiple levels. So many books!) and we were riveted for the hour! After that, we went and saw a 'Translation Slam' -- something I've never heard of before but Claudia was interested to see. It was three professional translators sitting on armchairs in front of the small crowd who had gathered and a screen behind them that showed various quotes (in English). Then, on the spot (alone and in collaboration with the others) the translators had to translate those lines into German. And they weren't easy things like, "Wow, the sky is blue today".

No, we are talking about elements from music with tricky wordplay:

"But in the end, the love you take is equal to the love you make."- The Beatles 

Positively crazy and fantastic sentences like:

"The sun shone, having no alternative, on nothing new." - Samuel Beckett

"You used to be much more...muchier. You've lost your muchness." - Lewis Carroll, Alice in Wonderland

And, to our positive delight, the best chunk of the Monty Python 'Dead Parrot' Sketch. You know it, but I'll put it here anyway:

" 'E's not pinin'! 'E's passed on! This parrot is no more! He has ceased to be! 'E's expired and gone to meet 'is maker! 'E's a stiff! Bereft of life, 'e rests in peace! If you hadn't nailed 'im to the perch 'e'd be pushing up the daisies! 'Is metabolic processes are now 'istory! 'E's off the twig! 'E's kicked the bucket, 'e's shuffled off 'is mortal coil, run down the curtain and joined the bleedin' choir invisible!! THIS IS AN EX-PARROT!!"

And to wrap it all up, there were a series of Shakespeare quotes that the professionals had to translate but in a particular style drawn from a hat - including the beloved 'that which we call a rose' quote in the style of today's young people's German slang, and 'If music be the food of love, play on' in the style of a children's book. Quite a fantastic and impressive presentation!

Autumn

Abby, at home in Meadville. I miss that one quite a lot.

The leaves are turning.

I think they all fell on the ground at the same time. Or, at least the first wave has. The trees aren't bare yet, but all of a sudden the ones that have come down cover my shoes while I wait at the bus stop.

I'm still negotiating the transition from one academic system to another but I think the end is in sight. One more exam and if/when I pass that, I can start math again. I feel like it's been forever. My friends encourage me that I'll jump right back in and it will be okay, but the nervousness I have is the kind that doesn't go away until you actually prove it false, as nice as encouragement can be.

I got to go hang out with the girls I mentioned in the last post again this week. This time, to learn more English (specifically to practice some writing and reading) we had a very fun game. Each person wrote down the name of an animal in English (this took a considerable amount of time, especially since the youngest one doesn't even know the full alphabet yet, even with the German names for the letters, let alone the English ones). I coached the four year old, who frequently had the right idea for how a letter was shaped but didn't trust herself to get it right, so she would "write" it in the air with her hand first and run that by me before committing it to paper. After making all these flashcards, we would hold one up at a time (not looking at which one it was) and whoever was "it" had to read the word on the card and pretend to be that animal so that we could guess which one it was.

Pretending to be animals seemed to work as a good motivator for some language learning. The 7-year-old, however, is too clever and has too broad of a vocabulary for her own good - or at least for mine. She made me have to act like a snail. If anyone knows how to do that convincingly, please do tell me.

Somewhere Between Monday and Tuesday

One of life's modern pleasures, quite different from the almost immediate gratification of the internet and other entertainment forms we currently have, is waiting for the delivery of furniture to your home. I am supposed to be getting a desk in the mail -- and according to the tracking information, it should arrive "between Monday and Tuesday". Well, it's Wednesday. Someone has to be home to sign for it when it arrives, so here I am.

And as I opened the window to let some of the morning air in, even though it feels like it will be quite a warm day outside, there was that definite crispness in the air, a scent that -- well, can something smell 'cold'? I feel like that makes no sense, but it smells like cold and autumn. I suppose I'm back in a place with real seasons. I can't wait!

And yesterday, I had quite the interesting experience. I've been looking for some part-time work here in Mainz and though I have been looking primarily for math tutoring (one of my favorite things) I've also found quite a few families who would like to have English-speaking babysitters for their children. I haven't babysat (if you don't count living with the youngsters this summer or with my ((second? Once removed?)) cousins the summer before) in years, but I thought I'd give it a try. So, last night, I met two adorable youngsters - one 4, one 7 - daughters of two nice German parents in a suburb of Mainz. But here's the interesting part. I was asked not to let the children know that I spoke any German. The parents told them I would only understand English (which they have already started to learn, to various extents) -- and let me tell you, it was an interesting experience.

First of all, I watched the older girl do something pretty neat for a person new to a language. She really believed I spoke absolutely no German and knew that her sister couldn't help her if she couldn't come up with a word in English, so she had to explain what she meant when she didn't know exactly what something was called. For example, at one point, her sister was looking around desperately for (what turned out to be) her stuffed rabbit, but when the youngest one was running around and saying the words for exactly that in German, I asked the older one what she was looking for and she stood the, puzzled, and then finally said "Her going to bed teddy bear rabbit", which is a fairly fantastic description. I have a lot of respect for this particular kind of speaking, as it's something that I also have to deal with.  I was having a discussion about this recently with C. The thing is, as a native English speaker, I can be incredibly, incredibly lazy. Well, I could have been so lazy as to never have learned a second language at all, but even though I have gone to the lengths to learn German, I still get to be lazy in ways people with a different first language can never be -- i.e. when I come across a word in the German sentence I want to say that I don't know how to say, I can just casually say the English word instead and most of the time, I will be understood.

I do this less than I used to (because my German has gotten better) but it still happens. In my own defense, I don't do it necessarily out of laziness -- more because I don't want to interrupt the flow of the conversation by saying 'huh' or something equally eloquent. Anyways. It's a hard thing to do and a very interesting thing to do from a learning standpoint - to be able to describe something even if you don't know how to say it exactly, and this seven-year-old was pretty damn good at it.

And the other hard part for me yesterday was when the youngest had trouble falling asleep and wanted to know when her mother would be home, and I had to speak to her in English even though I know she couldn't understand everything I said. She got a lot of it, though, and eventually, did fall asleep. But it was very hard to not just switch to German, which I am sure would have been more comforting to her.

Well, still no desk. I guess I'll keep waiting.